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The Separation Of Multicomponent Mixtures In Thermally-Coupled Distillation Systems

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The Separation Of Multicomponent Mixtures In Thermally-Coupled Distillation Systems

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Content THE SEPARATION OF MULTICOMPONENT MIXTURES THERMALLY-COUPLED DISTILLATION SYSTEMS by Walter John Stupin A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Chemical Engineering) June 1970 70- 25,068 STUPIN, Walter John, 1939- THE SEPARATION OF MULTICOMPONENT MIXTURES IN THERMALLY-COUPLED DISTILLATION SYSTEMS. University of Southern Califor-ia, Ph.D., 1970 Engineering, chemical University Microfilms, A X E R O X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED UNIVERSITY O F SO U TH ER N CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CA LIFORN IA 9 0 0 0 7 This dissertation, written by Wa l t er John S t up in......... under the direction of h i s . . . . Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R O F P H I L O S O P H Y Dun Date... June .1970 DISSERTATION C O M M T T E Chairman TO BARBARA ii ACKNOWLEDGMENTS 1 wish to express my gratitude to all those people who have helped in both large and small ways in the com pletion of this dissertation: especially. Dr. F. J. Lockhart, Who has shown such a great interest in the sub ject while directing the research work and who has contri buted considerably to my understanding of engineering problems and their solutions; Dr. C. J. Rebert, Who has spent many hours discussing and offering helpful sugges tions on various aspects of the work; and also, the other members of the Dissertation Committee, Dr. R. B. McGhee and Dr. G. V. Chilinger. I would also like to acknowledge the help from the National Science Foundation Traineeship Program which provided me with financial support during the period, 1965-1967. In addition, I acknowledge the assistance and computer time provided during the study by the Computer Sciences Laboratory of the University of Southern California. I wish to express a special thanks to Mrs. Ruth Toyama, Who has been extremely helpful in many ways, and ill especially in completing the final typed draft in record time. Finally, I am particularly grateful to my wife, Barbara, who was very helpful in the preparation of the rough draft of the thesis, and my children, Steven, Gregory, and Nancy, for their patience and understanding. iv TABLE OP CONTENTS Page ACKNOWLEDGMENTS ..................................... iii LIST OF TABLES ...................................... vii LIST OF FIGURES ..................................... ix NOMENCLATURE ........................................ xiii CHAPTER I. INTRODUCTION ................................ 1 II. CONVENTIONAL DISTILLATION .................. 6 A. The Distillation Column ............... 6 B. The Separation of Multicomponent Mix tures in Systems of Conventional Columns ................................. 19 III. REVERSIBLE, BRUGMA AND THERMALLY-COUPLED DISTILLATION PROCESSES ..................... 26 A. Reversible Distillation ................ 26 B. The Brugma Process ..................... 31 C. Thermally-Coupled Distillation Systems . 31 D. Comparisons of Various Schemes and Literature Information on Thermally Coupled Distillation ................... 37 IV. ANALYSIS OF THERMALLY-COUPLED DISTILLATION SYSTEMS ..................................... 41 A. Equilibrium Stage Model Design variables 43 Page B. Alternate Simplified Model .............. 48 C. The Design and Rating of Thermally Coupled Systems ......................... 50 V. DESIGN METHOD ............................... 52 A. Material Balance ........................ 52 B. Minimum Number of Stages ................ 55 C. Minimum Reflux .......................... 62 D. Intermediate Reflux ..................... 77 VI. CASE STUDY OF THERMALLY COUPLED DISTILLATION. 95 VII. CONCLUSIONS ................................. 117 REFERENCES ........................................... 118 APPENDICES ........................................... 122 A. Independent Variables in Distillation Systems Separating Multicomponent Mixtures .. 123 B. Reversible Distillation ..................... 150 C. Rating Method for Thermally-Coupled Distillation Systems ........................ 167 D. Minimum Reflux Conditions in Thermally- Coupled Distillation Systems ................ 229 E. Stripping Factor Method for Column 1 of the Thermally-Coupled System ................ 258 vi LIST OF TABLES Table Page I Sample Comparison of Thermally Coupled and Conventional Dxstillation ................... 38 II Summary of Example Calculations at Infinite Reflux ........................... 63 III Summary of Minimum Reflux Example Calcula tions, Case 1 ................................ 74 IV Summary of Minimum Reflux Example Calcula tions, Case 2 ................................ 75 V Summary of Particular Solution to Example Minimum Reflux Calculation, Case 1 ......... 78 VI Vapor and Liquid Flows: Example at 1.3 Times Minimum Reflux ........................ 87 VII Preliminary Material Balance ................ 88 VIII Summary of Example Equilibrium Stage Design . 94 IX Comparison of Conventional and Thermally Coupled Systems at Minimum Reflux .......... 97 X System Design for Rating Calculations 104 A—I The Determination of the Number of Independent Variables in the General Distillation System . 141 B-I Determination of Component Distributions to the Products for a Reversible Distillation Column....... 157 B-II Material Balance for Example of Figure B-3 .. 161 C-I Typical Main Program for Thermally Coupled Distillation ................................. 200 vii 201 208 210 211 214 216 217 219 220 221 222 SUBROUTINE STAGWZ SUBROUTINE INPUT SUBROUTINE INTLIZ SUBROUTINE OUTPUT SUBROUTINE MOLPRA FUNCTION FTHETA . SUBROUTINE BETTA FUNCTION BUBPT .. INPUT FORMAT ___ Sample Input .... Sample Output ... viii LIST OF FIGURES Figure Page 1 Flow diagrams for the separation of ternary mixtures by conventional and thermally- coupled distillation schemes ............... 2 2 Equilibrium stage model of a conventional distillation column (with a total condenser). 8 3 A system which approximates the behavior of the equilibrium stage ....................... 10 4 Stages as a function of reflux for a typical distillation ................................. 16 5 Cost as a function of reflux ratio for a typical distillation. From Robinson and Gilliland (21) .............................. 17 6 Sequences of conventional columns for separating a single feed into four products . 21 7 Reversible distillation process for the separation of a ternary mixture.............. 30 8 The Brugma Process ........................... 32 9 Thermally coupled schemes for the separation of quaternary mixture ....................... 36 10 Equilibrium stage model of the three product thermally-coupled distillation system ...... 42 11 Uncoupled model of the thermally-coupled distillation system ......................... 49 12 Example calculations for column 2 at a system reflux of 1.3 times the minimum ..... 90 ix Figure Page 13 Sample problem: stage requirements of the thermally coupled and conventional schemes as a function of vapor rate ................. 99 14 Sample problem: stages in the thermally coupled system as a function of reflux ..... 101 15 Sample problem: overhead product composition as a function of internal distribution of vapor and liquid ............................ 105 16 Sample problem: intermediate product compo sition as a function of internal distribution of vapor and liquid ......................... 106 17 Sample problem: bottoms product composition as a function of internal distribution of vapor and liquid ............................ 107 18 Sample problem: the range of internal flow distributions for design at a reflux ratio of 1.3 times the minimum .................... 108 19 Sample problem: the effect of column 1 stages on system performance ....................... 110 20 Sample problem: the effect of column 1 feed location on system performance .............. Ill 21 Sample problem: the eftect of feed composi tion on system performance .................. 113 22 Sample problem: the effect of reflux ratio on system performance ....................... 116 A—1 Sectionalized representation of a thermally- coupled distillation system ................. 131 A-2 An equilibrium stage ......................... 133 A-3 A column section............................. 136 x Figure Page A-4 Reboilers and condensers ..................... 140 A-5 Conventional distillation columns ............ 146 B-l Distillation column model .................... 154 B-2 Reversible distillation column ............... 160 B-3 Vapor and liquid rates for example reversible distillation .................................. 162 B-4 Liquid compositions in example reversible distillation .................................. 163 B-5 The separation of a ternary mixture by a reversible distillation process .............. 164 C-l The equilibrium stage model for the rating of the three product thermally-coupled distilla tion system................................... 173 C-2 General column unit ........................... 175 C-3 Stagewise calculations ........................ 178 C-4 General layout of computer program ........... 194 C-5 General layout of SUBROUTINE STAGWZ .......... 196 C-6 Typical convergence of the computer calculations .................................. 227 D-l Schematic diagram of the thermally-coupled distillation system .......................... 237 D-2 Stage calculations by Underwood's equations .. 241 E-l Model for the design of the thermally- coupled distillation system .................. 263 E-2 Model of an absorbing or stripping column .... 266 xi Figure Page E-3 Lower part of the thermally coupled system ... 271 E-4 Upper part ot the thermally-coupled distil lation system 275 xii NOMENCLATURE components of a hypothetical multicomponent mixture. effective absorption factor for component i in the rectifying section of column k. effective absorption factor for component i in the stripping section of column k. absorption factor for component i on stage j of column k, Lj,k/(K1#j <k). molal rate of component i in the distillate or overhead product from column k, v^ 1 ^ - 1i,0,k* distillate or net overhead rate from column Vl,k “ L0,k* absorption factor function for component i in the rectifying section of column k. absorption factor function for component i in the stripping section of column k. stripping factor function for component i in the rectifying section of column k. stripping factor function for component i in the stripping section of column k. molal feed rate of component i to column k. molal feed rate to column k. number of components in mixture. equilibrium ratio for component i on stage j of column k. y*, j j >k- xiii - molal liquid rate of component 1 from stage j of column k. - molal liquid rate in the rectifying section of column k under constant molal overflow condit ions. - molal liquid rate in the stripping section of column k under constant molal overflow conditions. Lj - molal liquid rate from stage j of column k. Nk - number of stages in column k. N' - number of stages in a column section. q^ - thermal condition parameter for the feed to column k, ( Qj - heat duty of stage j. Rfc - reflux ratio in column k, Lq s^ - molal rate of component i in the intermedi ate product of the thermally coupled system. S - molal rate of the intermediate product in the thermally coupled system. ^ “ effective stripping factor for component i in the rectifying section of column k. k - effective stripping factor for component i in the stripping section of column k. SA . ^ - stripping factor for component i on stage j of column k, Ki>Jk vJk/LJ k . vA j k - molal rate of conponent i in the vapor from stage j of column k. xiv - molal vapor rate in the rectifying section of column k under constant molal overflow con ditions. - molal vapor rate in the stripping section of column k under constant molal overflow con dit lone. Vj ^ - molal vapor rate from stage j of column k. w^ ^ - molal rate of component i in the bottoms from column k, l1>N,k - vi#N+1>k. - bottoms or net downward flow from the strip ping section of column k, Lu<k _ vN+l,k* xi,j,k " mole fraction of component i in the liquid ‘ on stage j of column k, 1^ j k/Lj k* ^ - mole fraction of component i in the vapor from stage j of column k, Vi,j,k/vj,k* z^<k - mole fraction in the feed to column k, £ i./V - volatility of component i relative to that of the reference component, the relative volatility. - relative volatility of component i with respect to component ii. - change in liquid rate across the feed stage of column k, - L^. AN - number of stages separating two points in a column. 0. - common root to Underwood's equation for ^ * column k. xv NOTE - When the subscript, k, denoting the column is unnecessary for identification, it is left off. SPECIAL SUBSCRIPTS C - refers to a condenser. D - refers to the distillate or overhead product. f - refers to the feed stage. F - refers to the feed. H - refers to a heavy component, one of low volatility. L - refers to a light or volatile component. m - refers to the minimum reflux case. N - refers to the bottom stage of a section. N+l - refers to the vapor entering the bottom stage of a section. R - refers to a reboiler s - indicates the product is of the system and not an intermediate stream. S - refers to the intermediate product of the thermally coupled system. T - refers to the total number of stages in the system. W - refers to the bottoms product. 0 - refers to liquid entering the top stage of a section. xv i CHAPTER I INTRODUCTION Distillation, being a costly part of nearly all chemical processing plants, is a prime target for optimi zation studies. Typically, the operating variables, flows, pressures and the like in a given system are varied to give the desired objective. However, greater potential exists, in many cases, in alternative processing schemes. For the separation of a multicomponent mixture into several prod- cuts, the thermally coupled distillation schemes are po tentially very good alternatives to conventional schemes (2,3,4,18,20). The thermally coupled scheme for the sep aration of a ternary mixture is compared to the convention al system of columns in Figure 1. A distillation system contains a thermal coupling when a heat flux is utilized for more than one fractiona tion and the heat transfer between the fractionators occurs by a direct contact of vapor and liquid. Such a coupling is common in side strippers on petroleum fractionators. Thermally coupled distillation systems use such thermal couplings to carry out close fractionations with the 1 2 A CONVENTIONAL SCHEME B CONVENTIONAL SCHEME (ALTERNATIVE SEPARATION SEQUENCE) CV QUID FEED LIQUID ST - COOLING FLUIO - HEATING FLUID C THERMALLY COUPLED SCHEME Fig. 1 Flow diagrams fox the separation of ternary mixtures by conventional and thermally- coupled distillation schemes. 3 objective of savings in heating and cooling costs. Thermally-coupled distillation systems have been around for a long time. In 1937, such a process was pat ented by Brugma (2,3). However, the process has not been used much (16), probably due to its relatively unknown status and difficulties in its design and analysis. Re cently, new interest has been shown by several authors (4,18,20). The improved efficiency of utilization of heat for the thermally coupled system over conventional systems can be explained through considerations of the reversible mul ticomponent distillation schemes proposed by Grunburg (8). In fact, the thermally-coupled distillation system can be simply derived from the corresponding reversible multicom ponent separation scheme. The separation of a multicomponent mixture is con ventionally accomplished in a series of columns, numbering one less than the number of products, each having a con denser and a reboiler. In contrast, the thermally coupled system requires only one condenser and one reboiler and a number of column sections which may or may not be in the same column. 4 Literature information on thermally-coupled distil lation is limited to descriptions of various possible ap plications and comparisons with conventional schemes. There are no design methods or quantitative information which are helpful in understanding the details of these processes or are useful in determining their flexibility. The objectives of this research were to fill in the above mentioned voids. Since this is quite a large under taking, maximum simplicity was required in order to allow for completion of the study. For this reason all work was restricted to calculations assuming constant relative vola tility and constant molal overflow. Some of the methods developed apply only to systems where these assumptions are valid, but other parts of the method can be easily extended to the more general case. Even though these assumptions are only approximations for most real problems, they are useful in obtaining initial estimates for the process design. The study of thermally coupled distillation columns presented here is founded upon the concepts used for con ventional distillation. Because of this, a brief review of these concepts as typically applied is presented 5 first. Then, a review of published information on thermal ly coupled distillation is presented. This is followed by a discussion of the analysis of the general aspects of the design and rating of these systems. Then, the proposed design method is presented. This is followed by a case study of the separation of a hypothetical ternary mix ture. Finally, a brief summary of the conclusions reached during the study is presented. CHAPTER II CONVENTIONAL DISTILLATION The conventional distillation column and the conven tional multicomponent separation schemes are reviewed in this chapter. A. The Distillation Column General Description The conventional distillation column separates a single feed stream into two different products and con sists of a column, a reboiler and a condenser. The frac tionation is obtained by the contacting of vapor and liquid on trays or over some packing material in the column. In most cases, the distillate or overhead product is with drawn from the system as part of the liquid condensed in the condenser. But, in some cases, a vapor phase overhead product is taken from the condenser. The bottoms product is usually withdrawn as a hot liquid from the bottom of the column or from the reboiler. The feed to the fractionator is introduced in the middle of the column. For convenience, the column is divided into three 6 7 sections; rectifying, stripping and feed. In the rectify ing section, above the feed entry, the net flow is in the direction of the vapor flow. In the stripping section, be low the feed entry, the net flow is in the direction of the liquid flow. Equilibrium Stage Model Distillation columns are designed and studied using a model based on a hypothetical contacting unit, the equilibrium stage (26). In this stage several streams are brought into intimate contact and heat is transferred to or from the mixture, producing two phases, vapor and liquid, in equilibrium. The stages are interconnected in the same way the actual trays, reboiler and the condenser are inter connected, as illustrated in Figure 2. Partial reboilers and partial condensers are considered equivalent to equi librium stages. Total reboilers and total condensers are considered heat exchangers producing total vaporization and total condensation, respectively, and are not equilib rium stages. The equilibrium stage model is related to real distillation by the use of strictly empirical quanti ties such as "overall tray efficiencies." There is very 8 CONDENSER 51 ° c RECTIFYING SECTION °f-1 FEED F V Lf f Vi S*»2 _l r 3 J V2 | Vi 1 Vi - Qr ------ 1 FEED STAGE REBOILER DISTILLATE D STRIPPING SECTION BOTTOMS Fig. 2 Equilibrium stage model of a conventional distillation column (with a total condenser). 9 little similarity between equilibrium stages and real trays. A device which could be made to closely approximate an equilibrium stage is illustrated in Figure 3. In this device there is a mixing section in which the feed streams are mixed, and the heat, if any, is transferred. Leaving this mixer, the streams are in thermal and chemical equi librium. This mixture is then separated and then, the phases leave the stage as separate streams. Variables and Problem Formulation The design or analysis of a system is accomplished by specifying values to certain variables and then deter mining numerically all others by the equations forming the mathematical model of the system, for the equilibrium stage representation of distillation, the following list identifies the pertinent quantities. 1. Liquid compositions 2. Vapor compositions 3. Temperatures 4. Pressures 5. Vapor rates 6. Liquid rates 10 FEED HEAT PHASE STREAMS EXCHANGER MIXER SEPARATOR LIGHT PHASE AVY PHASE Fig. 3 A system Which approximates the behavior of the equilibrium stage. 11 7. Heat fluxes 8. Numbers of stages An Important consideration is, "How many of these variables can be assigned values independently?" This important question is studied in detail in Appendix A and the results of the study are utilized in the discussion below. In order to simplify the discussion, certain re strictions are placed on the equilibrium stage mode? as discussed in this dissertation. These restrictions apply to the typical analysis of most industrial applications of distillation but may be relaxed for specific unusual cases. The restrictions are: 1. The pressures on all stages of the equilibrium stage model have been predetermined. 2. The heat duties of all stages except reboilers and condensers have been predetermined (heat leaks are usually considered zero). 3. Partial reboilers are used with the bottoms product withdrawn as a bubble point liquid. With these restrictions there are 1 + 7 independent vari ables describing the equilibrium stage model of a conven tional column (see Appendix A). Of these, 1 + 2 are 12 usually specified by the feed to the column as followst Quantities Number of Independent Variables Feed temperature 1 Feed pressure 1 Feed composition 1-1 Feed rate 1 Total 1+2 This leaves 5 Independent variables which are chosen ac cording to the type of problem formulation. There are two distinctively different approaches applied to solving the equilibrium stage model. In one, the design oriented approach, the objective is to deter mine the model which results in the specified performance of a distillation system. In the other, the rating ap proach, the objective is to see how a given model performs. This rating approach is analogous to building a complete distillation unit and then operating it to see how it works on a given feed (25). The remaining independent variables for the design problem are assigned as follows: 13 Quantlty Number of Independent Variables Reflux rate or ratio 1 Distillate condition 1 Optimum feed location 1 Product specification 2 5 The reflux rate or ratio is arbitrarily chosen within cer tain limits or as based on past experience. For a total condenser the distillate condition is specified in terms of the degree of subcooling in the condenser. For partial con densers the distillate condition is specified by the per cent of overhead product that is vapor or the temperature of the overhead product. The feed location is specified to be an optimum, defined as the location which allows the specified separation with a minimum number of stages at the given reflux rate or ratio . Generally, the product speci fications are converted into the distributions of 2 key components Which define the separation. These may be de fined in terms of percentages in the products or in percent recoveries to the products or otherwise. The solution of this design problem has been adequately presented in the literature (for instance, references 13, 21, 25 and 35). 14 The remaining independent variables for the rating problem are assigned as follows: Quantity Number of Independent Variables Reflux rate or ratio 1 Distillate rate or bottoms rate 1 Distillate condition 1 Number of stages 1 Feed locatIon 1 Total 5 Generally, the rating approach takes less judgment In get ting to a valid solution and Is readily handled on a com puter. For this set of specifications, there Is always a single solution. In contrast, the solution to the design problem may not be unique or may not exist. Several methods for solution of the rating problem have been presented in the literature (for instance, references 11 and 25). Design Reflux Ratio In the design of a column for separating a given mixture, all independent variables except one, the reflux IS ratio, describe the constraints put on the design in order to meet the processing specifications. Then the column design for the separation of this mixture depends only on the chosen reflux ratio. The number of stages required is plotted against chosen reflux ratio in Figure 4, for the typical design case. The relationship is described by the hyperbolic shaped curve with asymptotes at infinite stages and at an infinite reflux ratio. The limiting case with infinite stages occurs with minimum reflux. The limiting case with minimum stages occurs at infinite reflux and is referred to as the total reflux case. All designs lie be tween these two extremes. Empirical relationships relating the vapor and liquid rates and physical properties to column capacities are used to determine the column sizes (see references 21,25,35) for the various reflux ratios. This complete design, in turn, can be used to estimate the cost of the column. The results of such an economic study, by Robinson and Gilliland (21) are shown in Figure 5. This figure illustrates the following generalities which apply to most distillations. NUMBER O F STAGES 16 TO 00 z m 0 REFLUX RATIO, R - L^O TO 00 Fig, 4 stages as a function of reflux for a typical distillation. COSTS, DOLLARS PER UNIT TIME 17 SEPARATION OF METHANOL AND WATER 0.8 TOTAL COST 0.6 HEATING AND COOLING COSTS O . h COST Of DISTILLATION COLUMN 0.2 COST OF REBOILER AND CONDENSER 3^0 2.0 Fig. 5 Cost as a function of raflux ratio for a typical distillation. From Robinson and Gilliland (21). 18 1. The optimum reflux ratio is about 1.15 times the minimum reflux ratio. 2. Heating and cooling costs amount to about 80 percent of the total. 3. For reflux ratios of 1.05 to 1.5 times the minimum, the cost curve is relatively flat. For distillations carried out under unusual operating conditions, requiring columns of special construction or materials, the optimum reflux ratio and the fraction of the total costs attributable to heating and cooling may be different than stated above, but the trends in cost with reflux ratio would be similar. Whereas the optimum reflux ratio is found at about 1.15 times the minimum reflux ratio, most columns are de signed at about 1.3 to 2.0 times the minimum. This design at higher reflux ratio is justified by the following facts. 1. The vapor-liquid equilibrium information used in the equilibrium stage modeling contains considerable uncertainty. 2. The minimum reflux ratio and the design near minimum reflux are quite sensitive to the 19 relative volatilities of the components present and therefore, designs close to minimum reflux are quite uncertain. 3. Designs closer to minimum reflux contain more contacting elements and therefore, have more holdup of liquid and vapor. This makes the system less responsive to control and less flexible. 4. The equilibrium stage model is not an exact representation of distillation. B. The Separation of Multicomponent Mixtures in Systems of Conventional Columns Generally, the separation of a particular feed material into several products is accomplished in a series of distillation columns, each having a condenser and re boiler, and producing two products from one feed stream. For this type of system, the number of columns needed is equal to one less than the number of products being pro duced. For systems producing many different products many different column arrangements are possible. For instance, 20 if three products are to be obtained from the system, there are two possible arrangements, as shown in Figure 1 (Chap ter I). In one arrangement, the lightest fraction is taken out as the overhead product of the first column and the bottoms product contains both heavier fractions. Then, the second column serves to separate these heavier fractions. In the other arrangement, the heaviest fraction is taken out as the bottoms product of the first column and the two lighter components are separated in the second column. For systems producing more three products, the number of variations becomes quite large, For instance, in a system producing four products, there are five pos sible variations as illustrated in Figure 6; in a system producing five products, there are 14 variationss and in a system producing six products, there are 42 variations (9). Typical Design Practice The typical design practice, in the design of dis tillation systems, is to, more or less arbitrarily, pick one of the possible configurations and then to assign the separations of two key components in each column of the system. Each column is designed independently of the 21 A B C ABC AB ABCD. BCD CO A. ABCD B, a ac ABC A B ABCD BCD C ABCD BC D AB A ABCD CO c . Fig. 6 Sequences of conventional columns for separating a single feed Into four products. 22 others by the general methods described In the literature. The configuration In which the lightest fraction Is taken overhead In the first column and the next heavier fraction Is taken overhead In the second column, and so on, Is the one most frequently utilized. This configuration Is commonly referred to as the linear or normal sequence. In those Instances where the condensation of the overhead vapors of the columns requires pressures above atmospheric the first column Is operated at the highest pressure (due to the low boiling point of the lightest product), and each succeeding column Is operated at lower pressures. Since the separation of a mixture by distillation Is usu ally easier at lower pressures, It Is argued that the best system Is the one which decreases the pressure by the maximum amount when proceeding from column to column. Generally, the normal or direct sequence allows for the pressure to drop at a maximum rate through the system. Under conditions where vacuum operation is necessary or atmospheric pressure is sufficiently high for easy con densation of the vapors, the above arguments for the nor mal sequence do not apply. 23 The Optimum Column Saqutnce The rather arbitrary assignment of a distillation sequence for separating a multicomponent mixture, in many cases, leads to designs that are not optimum. Whereas several recent studies have been made into techniques for optimizing a chemical plant for a fixed sequence of units, studies pursuing techniques for arriving at the optimum sequence of units in a process seem to arouse less inter* est. But several authors have studied the problem of find ing the optimum sequence (9,14,22) of distillation columns. Lockhart (14) has carried out a detailed economic analysis of the sequences in the de-ethanizing de-propa- nizing, de-isobutanizing and de-butanizing of natural gaso line and found that in some instances, the direct or normal sequence was not optimum. One of his conclusions was that the "rule of thumb," to reduce the pressure level in the distillation system as soon as possible, seems to be true only for large pressure increments at pressure above 200 psia. Furthermore, at the lower pressures, the relative amounts of products was more of a factor in deciding Which sequence to use. Harbert (9) and Rod and Marek (22) have developed 24 generalized methods for determining the optimum sequence. For this, a simple criteria for the optimum sequence is necessary. For the typical process, in which a product or pro ducts of given amounts and purities are to be produced, the system giving the maximum profit has the lowest total cost. As stated by Herbert (9), there are those costs that are fixed and are the same in all distillation systems separating the same mixture, and those that vary from sys tem to system. Of those that vary from system to system, the cost of the heat utilized and rejected is of overwhelm ing Importance. He went so far as to state, "Indeed, it can almost be said that no other variable exists." Based on this observation, systems can be compared on the amount of heat utilized. If we assume that the latent heat of vaporization is constant and neglect sensible heat effects, we can use the amount of vapor produced in the system as a measure of the heating duty and the amount of vapor condensed as a measure of cooling duty. On this basis, the most eco nomical system requires the lowest vapor boilup. Obviously, this is a great oversimplification of 25 some complex economics applied to a complex industrial pro cess. The above stated generalizations, therefore, can be applied only When substantial differences in vapor require ments are found. The approach used in this dissertation is to design the several possible configurations of a distillation sys tem with each column operating at a reflux ratio of con stant factor times the minimum reflux ratio for that col umn. The sequence which uses the least heat or vaporizes the least liquid is chosen as the optimum. CHAPTER III REVERSIBLE, BRUGMA AND THERMALLY-COUPLED DISTILLATION PROCESSES A. Reversible Distillation A completely reversible process for the separation of binary mixtures by distillation was first described by Hausen (10) in 1932, and at a later date by Benedict (1). A reversible multicomponent distillation system was pre sented in 1956 by Grunberg (8), and later by others (17, 19,23). The fundamental requirement of a reversible process is that conditions at all points of the system have to be displaced from the equilibrium conditions by infinitesimal amounts and the process can be reversed by changing condi tions by infinitesimals to the "other side" of the equilib' rium point. It follows that a requirement of a reversible distillation is that the vapor and liquid compositions change by infinitesimals from stage to stage. A detailed discussion based on this requirement is presented in Appendix B of this dissertation. The analysis shows that for the separation of a 26 27 given mixture into two products by a reversible distilla tion process, the following requirements must be met. 1. The internal reflux ratio, l/V, varies continu ously from stage to stage. 2. Heat is transferred to each stripping stage. 3. Heat is transferred away from each rectifying stage. 4. An infinite number of stages is required for a finite change in composition anywhere in the column. 5. The lightest and heaviest conponents present in the feed are the key or determining compo nents. This contrasts with the conventional distillation in which all heat is transferred to the system in the re boiler. All heat is transferred from the system in the condenser, the separation is made in a finite number of stages, and any two components in the mixutre can be chosen as the key components. A comparison of a conventional distillation column at minimum reflux and a reversible distillation column shows that the internal reflux ratio at the feed stage is 28 the same in both cases for the same separation of the key components (required to be the components of extreme vola tility) . Heat balances around the stripping sections of these columns show the same total heat consumption in both cases. And heat balances around the rectifying sections show the same total heat rejection to the surroundings in both cases. However, the reversible distillation process degrades the heat through a minimum temperature difference and only a small fraction of it is degraded from the high temperature at the bottoms to the low temperatures at the top. In contrast, the conventional distillation system degrades all heat through the maximum tenperature differ ence. Generally, the cost of heating in industrial plants is more a function of the quantity of heat to be trans ferred and is only slightly affected by the tenperature level for the modest temperatures required in distillation. Also, the cost removal of heat is typically only slightly affected by the temperature level. Under these circum stances, the more reversible process offers no advantages but requires more equipment, such as more stages and inter mediate heat exchangers. However, under special 29 circumstances, the concept of intermediate heat exchange can be attractive. This is especially true in plants oper ating at subambient temperatures, where heat removal be comes the controlling cost and is strongly affected by the temperature level. The extension of the concept of reversible distil lation to the separation of multicomponent mixtures is accomplished by the following logic. Consider the separa tion of a ternary mixture of A, B and C into three pro ducts. Component A is the most volatile and component C is the least. Since the reversible separation in each column is made between components extreme in volatility, the initial separation is between components A and C. Then, the overhead product from the first column containing components A and B is separated in the second column and the bottoms, containing B and C is separated in the third column. In order for the reversible separations to be made in second and third columns, a complete separation between A and c has to be made in the first column. The reversible scheme for the separation of this ternary mixture as pre sented by Grunberg and others (8,17,19,23) is shown in Figure 7. 30 A COLUMN 2 AB LIQUID COLUMN FEED ABC COLUMN 3 8C LIQUID NOTE 1 ALL COLUMN SECTIONS HAVE INFINITE STASES 2 ♦ DENOTES NEAT AOOITION 3 DENOTES HEAT REMOVAL Pig. 7 Reversible distillation process for the separation of a ternary mixture. 31 B. The Brugma Process The Brugma process uses approximately the sequence of separation of the reversible process as shown in Figure 8. The feed is first split into two fractions which are then sent into the second column. The second column then fractionates these two intermediate streams into either three or four products. These three or four products could then be further separated in a third column to give a maximum of eight products. The second and third columns are made up of several thermally coupled units, and the system is therefore a type of thermally coupled system. C. The Thermally-Coupled Distillation System In this dissertation, the particular scheme shown in Figure 1 is referred to as "the thermally coupled dis tillation system." The configuration of this process follows directly from the reversible multicomponent dis tillation scheme. The thermally coupled distillation sys tem holds the same relation to the reversible multicompo nent distillation process as does the conventional binary separation process relative to the corresponding revers ible binary distillation. Also, the thermally coupled 32 A SEPARATION OF A TERNARY MIXTURE B SEPARATION OF A QUATERNARY MIXTURE Fig. 8 The Brugma Process 33 distillation system can be considered a form of the Brugma process with additional thermal couplings. In this process the following points are of im portance. 1. All heat is added to the system at one point. Therefore, only one reboiler is required. 2. All heat is removed from the system at one point, and therefore, only one condenser is required. 3. As in the Brugma process, vapor from one section is used to reboil another and liquid from one section is used to provide reflux to another. 4. The final products are produced from sections where the distillation is essentially binary. 5. The thermally coupled system generally requires less total vapor than other systems. Inasmuch as the thermally coupled distillation sys tem can be considered a derivative of the reversible multi- component scheme, several of the qualities associated with the reversible scheme are retained in the thermally coupled scheme. For instance, the transfer of heat be 34 tween the columns occurs by a direct heat transfer mecha nism, thus eliminating the need for a temperature gradient across some Impervious solid material. A more Important source of thermodynamic Improve ment In the thermally coupled system over the conventional distillation occurs at the feed stage. In the conventional column, the separation Is generally made between adjacent components In the mixture. Under these conditions, the composition at the feed stage Is quite different than the composition of the feed. Then, the Introduction of the feed Is quite Irreversible. In contrast, In the thermal ly coupled system, the separation In any column Is general ly between the components, In the feed to that column, extreme In volatility and under these circumstances, the feed stage composition Is close to the composition of the feed to that column. This more reversible mixing at the feed stages improves the thermodynamic efficiency of the process. Then, the improvements in the distillation system introduced by the thermally coupled system are 1. The use of heat in more than one fractionation by direct contact heat transfer. 35 2. Reduced irreversibilities in mixing at the feed stage. The actual hardware for thermally coupled distilla tion systems could be constructed in various possible con figurations. The equipment may consist of three individ ual columns with vapor and liquid lines interconnecting them, or two columns as shown in Figure 1, or a single column divided by a vertical partition (20). The separation of mixtures into more than three products can be easily envisioned. However, as was the case for systems of conventional columns, as the number of products increases above three, the number of possible schemes grows rapidly. Thermally coupled schemes for the separation of a quaternary mixture are presented in Fig ure 9. The scheme presented in Figure 9a corresponds to the same separation sequence as the Brugma process for the separation of the four component mixtures. The separation in column 1 of Figure 9a is a separation between adjacent components B, C. An easier separation is between compo nents A and D, and the scheme of Figure 9b makes this easier separation first in column 1, then further separa tions are made with key components in each section being AB reco a.b.c.d HCAT A HCAT ■ J ut ab ABC FCCP A.B.C.D BC BCD CO B Pig. 9 Thermally couplad schamas for the separation of quaternary mixture. 37 those extreme in volatility. This corresponds to the separation sequence required in the reversible process. The scheme of Figure 9b will, in general, require less heat than that of Figure 9a. However, the difference is not great and the increased complexity of the system would tend to make it unattractive (18). D. Comparisons of Various Schemes and Literature Information on Thermally Coupled Distillation Cahn and Di Miceli (4) present an analysis of the separation of a mixture of pentanes and hexanes at 40 psig. Their results are summarized in Table I. The number of actual plates required by the thermally coupled and the Brugma processes are comparable to the number required in the conventional scheme. But the heat and cooling water requirements for the inproved schemes are considerably less. The reduced number of reboilers and condensers and the reduction in utilities are considerable economic factors favoring the thermally coupled system. At this point, it is important to note that Cahn and Di Miceli (4) appear not to have knowledge of the work of Brugma and have claimed the invention of the process referred to 38 TABLE I Sample Comparison of Thermally Coupled and Conventional Distillation From Cahn and Di Miceli (4) All columns at 40 psig Material balance all cases Component iC5 nCR * 5 nC6 Total feed iC5 cut nCg cut iCg cut nC6 i b./s.d. b./s.d. b./s.d. b./s.d. b./s 458 439 18 1 970 41 870 59 607 2 517 88 775 13 762 2810 480 890 590 850 Actual Plates Column 1 Column 2 Column 3 Total Comparison Conven tional System Brugma Process Thermally Coupled (Fig. 7c) (Fig. 9b) (Fig. 11a) 42 68 69 179 42 155 197 42 137 179 Reflux Requirements, b./s.d. Column 1 4520 Column 2 3600 Column 3 4250 Total 12,370 Onsite Investment $600,000 Utility Requirements Steam lb/hr at 25 psig 25,600 Power 100 Cooling water, GPM 1,500 4520 4250 8770 4520* 9750 9750 $480,000 20,300 100 1,230 * Also included in the reflux to column 2. 39 here as the Brugma process. Their calculations are labeled as for the Brugma process, even though they have claimed it. For the particular case, their calculations shew a savings of about 20 percent for the thermally coupled sys tem over the conventional scheme. They did not complete the analysis for the Brugma process. Petlyuk, Platonov and Slavinskii (20) have calcu lated minimum reflux for both the conventional and the improved distillation processes for complete separation of three component mixtures. The composition of the most volatile was taken equal to the composition of the least volatile and the relative volatilities were taken as 1.2, 1.1 and 1.0. The improved distillation system showed from 40 to 50% less vapor requirements than the conventional system for the range of composition of the intermediate component of 10 to 90%. Their calculations were based on constant relative volatility, constant molal overflow, and a saturated liquid feed. Petlyuk, Platonov and Avet'yan (18) have compared the thermally coupled schemes to conventional schemes for the separation of several commercially important mixtures. In all cases, substantial savings for the thermally 40 coupled systems are realized. In general, these savings amount to about 30 percent. CHAPTER IV ANALYSIS OF THERMALLY-COUPLED DISTILLATION SYSTEMS The equilibrium stage analysis of this chapter is limited to the thermally coupled system for producing three products from a single feed as shown in Figure 1. This particular system was chosen since it is the simplest and probably the most useful in terms of practical applicabi lity. To simplify the presentation, the components in the feed to the system have been classified as light compo nents, mid components, or heavy components, depending on their volatilities. The light components are the most vol atile and concentrate in the overhead product. The mid components cure intermediate in volatility and concentrate in the intermediate product. And the heavy components are the least volatile and concentrate in the bottoms product. The equilibrium stage model of the three product thermally coupled distillation system has been sectional- ized as shown in Figure 10. 41 42 FEED COLUMN VAPOR I □ LIQUID RECTIFYING FEED VAPOR RECTIFYING O n ] FEED OVERHEAD PROOUCT ► COLUMN 2 STRIPPING LIQUID INTERMEDIATE PROOUCT STRIPPING □ RECTIFYING LIQUID 1 I feed T I | VAPOR FLOW | LIQUID FLOW RE80ILER IS THE BOTTOM STAGE OF COLUMN 3 S' STRIPPING LIQUID V COLUMN 3 BOTTOM PROOUCT Pig. 10 Equilibrium stag* model of the three product thermally-coupled distillation system. 43 A. Equilibrium Stage Design Variables The analysis, given in Appendix A, shows that for the thermally coupled system producing three products from a single feed, there are 1+14 independent variables under the following restrictions, which are similar to those stated for the conventional column. 1. Pressures above all stages in the system are known or have been predetermined. 2. Heat leaks or heat duties on all stages except the reboiler and condenser are known or have been predetermined (heat leaks are usually as sumed zero). 3. The intermediate product is withdrawn as a bubble point liquid. 4. The system is equipped with a partial reboiler with the bottoms taken off as a bubble point liquid. Of these independent variables, I + 2 are used to complete ly define the feed to the column. This leaves 12 independ ent variables for the specification of the details of the process or the separation to be obtained. If, in addition 44 we fix the distillate condition as a bubble point liquid, there axe 11 independent variables. The design oriented approach is to specify the de sired separation and then to determine the process which gives this separation. As was the case with a conventional column, this would involve specifying a separation and optimizing the remaining independent variables in terms of minimizing the cost to gain this separation. It follows that increasing the number of independent variables to be optimized, by decreasing the number of specifications, results in more latitude and a lower cost optimum design. The complexity of solution, however, also increases due to the greater number of variables which have to be considered in the optimization. Considering one independent variable per product stream as the minimum for the specification of the separa tion, the design of the three product process is then de termined in terms of an 8 dimensional optimization. If we specify a reflux ratio, which is some multiple of the mini mum reflux, such as R ■ 1.2 we fix another variable which then leaves a 7 dimensional optimization of the system. 45 The subsequent analysis of the limiting cases of minimum stages and minimum recycle strongly suggests that for ease of solution, it is desirable to specify one quan tity each for the lowest and the highest boiling products and two specifications for each of the intermediate pro ducts. On this basis, which corresponds to the 7 dimen sional optimization of the three product system, the design specifications for the purities of the products would be fixed by four independent quantities. Then, the design problem would be defined by the assignment of the following variables. Number of Quantity or Quantities Independent Variables Feed variables (temperature, pressure coiqposition, and rate) 1 + 2 Product purity specifications 4 Reflux or boilup rate 1 Overhead product condition 1 Optimum liquid distribution 1 Optimum vapor distribution 1 4 r Optimum distribution of stages 4 Total 1+14 ^Determined by the optimum feed locations in the three columns and the optimum relative numbers of stages in column 1 and the total of the rectifying stages of column 3 and the stripping stages of column 2. 46 The specification of the optimum quantities for six of the independent variables in the table above would be quite difficult to implement. This contrasts to the problem of a conventional column in which, at a given reflux ratio, a single dimensional optimization to optimize the feed stage is carried out (i.e., vary feed location to minimize total number of stages). In summary of the above discussion on the optimum design of thermally coupled distillation systems, the fol lowing facts are pertinent. 1. The objectives of the distillation have to be converted into 3 or 4 purity values, or equiva lent . 2. A criterion for optimization has to be developed. 3. A multi-variable optimization has to be carried out to exactly fulfill the design specifica tions. However, experience with conventional distillation systems indicates that in the region of the optimum, the total cost is not sensitive to small changes in design variables 47 and, in fact, the design at conditions other than the opti mum based on typical economics is intentionally done to get more stable design. Then, with "Engineering Judg ment," designs based on reasonable quantities rather than optimum quantities for the distributions of vapor, liquid and stages between the various column sections will be adequate. In the rating analysis of the thermally coupled distillation system, the stages, rates, etc. are specified and the distributions of components to the products are calculated. In this case, the independent variables cure assigned as follows. Number of Quantity or Quantities Independent Variables Feed variables (temperature, pressure composition, and rate) 1 + 2 Overhead product condition and rate 2 Reflux or boilup rate 1 Bottoms product rate 1 Distribution of liquid between column 2 stripping section and column 1 rectifying section 1 Distribution of vapor between column 3 rectifying section and column 1 stripping section 1 Number of stages in each column section 6 Total 1+14 48 B. Alternate Simplified Model At this point, it is convenient to consider a sim plified model of the thermally coupled distillation sys tem. Referring to Figure 10, we see that the thermally coupled system uses a vapor sidestream or product to pro vide vapor for another column. Similarly, liquid reflux for one column is provided from a product of smother. The primary purpose in using a vapor sidestream from one section as a stripping vapor in another is to eliminate the need for one heat exchanger and an externally supplied quantity of heat. A similar purpose can be stated for the utilization of a liquid sidestream from one section as a reflux for smother. Since the various recycle streams in the thermally coupled system act as modes of heat transfer between columns, the substitution of ficti tious heat exchangers for these recycles is a convenient approximation to make. Based on this argument, the pro cess flow diagram shown in Figure 11 is an approximate re presentation of the process shown in Figure 10. In the simplified process, the columns have been uncoupled as far as the recycle of material between the columns. The COLUMN 2 INTERMEDIATE PRODUCT FEED COLUMN t COLUMN 3 HYPOTHETICAL INTERMEDIATE NEAT EXCHANGERS Pig. 11 Uncoupled model of the thermally-coupled distillation system. 50 thermal coupling is represented as the transfer of heat in heat exchangers and the material flows sequentially through the units to the products. The three columns shown in Figure 11 are solved by any of several available methods for conventional columns. In column 2, the heat transferred to the feed stage can be accounted for by adding it to the feed, making it a very superheated feed. In column 3, the heat transferred from the feed stage can be accounted for by taking it from the feed, making it a very subcooled feed. C. The Design and Rating of Thermally Coupled Systems The design or rating analysis of the thermally coupled system can be undertaken by either model, Figure 10 or Figure 11. The model depicted in Figure 10 more closely approximates the actual system, but the model depicted in Figure 11 is simpler to evaluate, since the units of this model are conventional distillation columns. The design of systems according to the model of Figure 10 is covered in Chapter V. The design, according to the model of Figure 11 has one shortcoming. The pro 51 duct compositions would be specified, but since the compo sitions at the overhead and bottoms of column 1 are un known, a method would be needed to insure that assumed values for these quantities are reasonable. The rating of thermally coupled systems by the model of Figure 10 using an extension of the Thiele-Geddes (28) type calculation as presented by Holland (11), is presented in Appendix C. This was included in the appendix since the scheme is computer oriented and its details would not be of much interest to the process engineer. The rating using the model of Figure 11 presents no problem since the system can be modeled using existing methods by solving for the products from each column sequentially. CHAPTER V DESIGN METHOD The design method presented in this chapter is based on the fact that like a conventional distillation column, the thermally-coupled distillation system has limits of operation, a minimum reflux and a minimum number of stages. Constant molal overflow and constant relative volatility are assumed at various points in the development. Other assumptions are made when necessary. A. Material Balance The first step in the design of a processing system is to make a preliminary material balance. For the thermally-coupled distillation system given in Figure 10, there are 4 streams to be considered in the overall material balance. Each of these is described by I independent quantities giving the composition and flow rate. Then, there are a total of 41 quantities describing the overall material balance and I material balance equa tions. This leaves the overall material balance described by 31 independent variables. Of these I are defined by 52 53 defined by the feed and either 3 or 4 key component quanti ties define the product purities. With 4 quantities de fining the key component distributions, 21-4 independent quantities remain undefined and with 3 defining the key component distributions, 21-3 independent quantities remain undefined. In order to complete the preliminary material bal ance, values have to be assumed for these undefined quanti ties. The assumed values should be as close to the results of the equilibrium stage calculations as possible. The following set of facts is useful in approximating these undefined quantities. 1. Components lighter than the light key appear only in the distillate. Their mole fractions in the intermediate product or bottoms product are very small. 2. components lighter than the mid key component or components appear only in the overhead pro duct or the intermediate product. Their mole fractions in the bottoms product are very small. 3. Components heavier than the mid key component 54 appear only in the Intermediate product and bottoms product. Their concentrations in the overhead product are very small. 4. Components heavier than the heavy key appear only in the bottoms product. Their concentra tions in the overhead or intermediate product are very small. Example of Material Balance The following ternary mixture la to be separated into three liquid products each of 90 mole percent purity. System Feed - Example 1 (a bubble point liquid) Component Mole Fraction in Feed Relative Volatility A 0.3333 9.0 B 0.3334 3.0 C 0.3333 1.0 This separation of a ternary mixture is defined by 3 purity quantities. Therefore, according to the analysis 21-3, or 2(3)—3 ■ 3, undefined quantities need to be assumed before the material balance is completed. According to the general properties of the system, the mole fraction of component A in the bottoms and the mole fraction of component C in the overhead product are very small, and are assumed to be zero. One more quantity has to be assumed to bring the total to 55 3. This last quantity la althar tha sola fraction of component C or component A in the intermediate product. Assuming these mole fractions to be equal completes the set and allows the material balance to be completed. Basis: 1 mole of feed 1. Material balance for component A 0.90(D) + 0.05(S) + 0(W) - 0.3333 (X-l) 2. Material balance for component B 0.10(D) + 0.90(S) + 0.10(W) - 0.3334 (X-2) 3. Material balance for component C 0(D) + 0.05(S) + 0.90(W) - 0.3333 (X-3) 4. Solving simultaneously D - 0.3541 moles/mole of feed S - 0.2918 moles/mole of feed W - 0.3541 moles/mole of feed B. Minimum Number of Stages In general, as the reflux rate increases, the number of stages required for a given separation decreas es. The minimum number is approached as the reflux ratio approaches infinity. This limiting case is commonly 56 referred to as total reflux, since it can be simulated by operating a column with no product rate. Since the product rate is zero, reflux rate relative to the amount of product produced is infinite. At this limiting case, any product withdrawn or feed added is insignificant compared to the liquid or vapor traffic in the column. Because of this, the thermally coupled system can be modeled at this limiting case using the equations and calculation procedures developed for conventional distilla tion columns. The number of stages, in column 2 of Figure 10, are determined by applying the total reflux equations and the compositions of the overhead and intermediate products. The stages, required in column 3, are similarly determined from the compositions of the intermediate and bottoms pro ducts. Since the feed to columns 2 and 3 at this limiting case are insignificant compared to the liquid and vapor traffic in these columns, the positions of the feed entries have no effect on the separation there. Furthermore, the degree of separation of the system feed into these inter mediate feeds has no effect on the separation accomplished. 57 Therefore, for a system with minimum stages, column 1 degenerates to a stream divider and contains no stages. Since it contains no stages, no rectifying liquid or strip ping vapor is used in it. In terms of applying the equilibrium stage model of the thermally coupled system for determining the minimum stages, the independent variables are assigned values, as follows. Quantities Independent variables Number of Remarks Temperature, Pres sure & Composition Rs -0° Nx - 0 Insignificant since Nj-0 ■ 0, since Nj-O vx - 0, since Nj-0 Insignificant since Rs" oo Feed 1 + 2 System reflux Column 1 stages Column 1 feed location Column 1 reflux liquid Column 1 vapor Columns 2 and 3, feed locations 2 Distillate condition 1 Product compositions ____4 Total 1+14 For this limiting case, a number of these quantities be come insignificant and have no effect on the location. For instance, in addition to those stated above, the feed temperature, the feed pressure and the distillate condi tion need not be assigned values in order to solve the 58 mathematical equations describing this special case. As stated earlier, only 3 independent variables may be used to characterize the separation under some cir cumstances. Then, only I + 13 of the independent variables are fixed and the minimum number of stages has to be deter mined by an optimization of the one remaining degree of freedom. The analysis of total reflux or infinite recycle in the conventional column is presented in general references such as Robinson and Gilliland (21). For systems in which the assumption of constant relative volatility applies, the relationship between the number of stages separating two points in the system and the compositions at these points is given by the Fenske-Underwood equation (7,29). AN - (1) Log * i#11 This equation can also be used for cases where the relative volatility is not constant by use of some appro priate average relative volatility. For a discussion of the appropriate value, see Robinson and Gilliland (21). The specific application of this equation to the 59 thermally coupled distillation system at infinite recycle is as follows. The number of stages required in column 2 of the system presented in Figure 10 is determined by the overhead product and the intermediate product compositions. And the number of stages required in column 3 is determined by the compositions of the intermediate product and the bottoms product. Both equations (2) and (3) can be written for 1 - 1 independent combinations of components i and ii. These equations combined with the I equations defining the material balance for each component are solved for the minimum stages and the complete compositions of the pro ducts. N2 = Log (2) Log Cti,ii (3) Log Oti,ii 60 Example Calculation of Minimum Stages The process specifications used earlier in this chapter define three fixed quantities, the mole fractions in three products. But according to the discussion on minimum stages, there are four independ ent variables which may be used to define the products at the infinite reflux limit. Therefore, this problem becomes one of optimization with one degree of freedom, and is most conveniently solved by specify ing various values of mole fraction for either component A or component C in the product of intermediate volatility. The number of stages is then calculated, and the minimum number of stages is found by trial and error. In this case, the mole fraction of component A in the inter mediate product is taken as the independent variable. A mole fraction of zero is obviously a poor choice for an assumed value since it would require an infinite number of stages in column 2. A mole fraction of 0.10 would also be a poor choice since it would require that the mole fraction of component C in the intermediate product be equal to zero and lead to an infinite number of stages in column 3. 1. Assume x^ g ■ 0.02. (X-4) 2. Then the compositions of all product streams, by material balance are, 61 Mole Fractions In Product Component Overhead Intermediate Bottoms A 0.90 0.02 * B 0.10 0.90 0.10 C * 0.08 0.90 (*a very small number) 3. The stages In column 2, at finite reflux, by equation (2) ffBj f0.90U 0.90j N _ Log *«BlD 1 XA*S „ Log iQ.02/lQ.10* (x_5) 2 L°g 04AB Log 3 N2 - 5.45 4. The stages in column 3, at infinite reflux, by equation (3) [ 10.90110.90| N3 - Log XC*S *B W . Log »0.08* *0.10* (x.6) Log «tCD Log 3 N3 - 4.20 5. The mole fraction of component C in the distillate was assumed to be very small. Checking this with a rearranged form of equation (2) *b,d|£I o.io ( £ 22) *XB'S *0.90» «c.d — i f * -----rtf'-2-2 - 10-5 <x-7) «*l 3 and it is small. 62 Similarly, the amount of component A In the bottoms can be calculated. Calculations are carried out for other values of x ^ g end from these the optional is chosen. The calculations are summarized in Table II. The minimum stages occur at x ^ g ■ 0.05, and the system requires a minimum of 9.26 stages. Assumed values of the concentration of component A in the intermediate product were varied by a factor of A. However, the num ber of stages required varied only by about 5% over this range. This illustrates that if a reasonable value for this concentration were assumed (one which most likely would have been right on the optimum) a reasonable approximation of the minimum number of stages would have been obtained. C. The Minimum Reflux In general, the reflux requirement of a distilla tion system can be reduced by increasing the number of stages in the system. However, there is a lower limit be low which the separation cannot be made. An important case of this minimum reflux occurs when infinite numbers of stages are used in all sections of the system. Under these conditions, the minimum reflux can be calculated by the 63 TABLE II Summary of Example Calculations at Infinite Reflux _________________Case________________ Quant ity xA#s(assumed) XA,D XB,D ,D XB ,S ^ ,S XA,W XB,W *C ,W 1 0.02 0.90 0.10 2.22x10“5 0.90 0.08 2.22xl0“4 0.10 0.90 5.45 4.20 2 0.05 0.90 0.10 3.41xl0”4 0.90 0.05 3.41xl0-4 0.10 0.90 4.63 4.63 3 0.08 0.90 0.10 2.22xl0“4 0.90 0.02 2.22xl0-5 0.10 0.90 4.20 5.45 9.65 9.26 9.65 64 equations derived by Underwood (31-34) for minimum reflux in a conventional column. The derivation of these equa tions and their application to thermally coupled distilla tion is given in Appendix O. The Underwood equations cure based on constant molal overflow and constant relative volatility. These equa tions, arranged in a form suitable for the analysis of thermally-coupled systems, are tions. Underwood (33,34) has shown for a conventional column at minimum reflux that certain of these roots are 1. For the feed to the system (4) 2. For a rectifying section (5) 3. For a stripping section (6) The quantities 9 | v represent roots to these equa- 65 identical for the rectifying section, the stripping sec tion and the feed. These roots lie between the volatili ties of the components that appear both in the overhead product and bottoms product at minimum reflux. If two components distribute between the overhead and bottoms, there is one root common to equations (4), (5) and (6). If there are three components that distribute, then there are two common roots, etc. The analysis of Appendix D shows that in the ther mally-coupled system as presented in Figure 10, the Under wood equations apply at minimum reflux. And furthermore, the analysis shows that if the overhead product from one column is used to feed a second column and reflux for the first column is withdrawn from the feed stage of the sec ond column, the roots to equation (5) written for the first column also are common to equations (4), (5) and (6) for the second column, as long as they have values between the relative volatilities of the components distributing to the overhead and bottoms of the second column. Simi larly, if the bottoms product from the first column feeds a third column and the vapor for the first column is with drawn from the feed stage of the second column, the roots 66 for the components that distribute in the third column are common to equation (5) for the first column and equations (4), (5) and (6) for the third column. These facts allow the minimum reflux to be determined easily. The minimum reflux condition in the thermally- coupled distillation system is calculated by the following steps. 1. The common roots, 0 , which lie between the relative volatilities of the components in the feed are calculated from equation (4). 2. The minimum boilup and reflux to the system are then calculated from equations (5) and (6). 3. Since the reflux to the system is related to the boilup by a material balance and enthalpy balance, the values calculated can be compared to see which actually corresponds to a minimum reflux. This minimum reflux is then determined by the greater of the two cases. These equations, as described in more detail in the appendix can be applied to any two product single feed column in the system using the net flows to and from the column or to the overall system. However, these equations 67 apply only in those cases in which the values of common roots, 9, lie between the relative volatilities of com ponents which appear in both the bottoms and overhead from the particular column. If in a particular column only one component appears in both the distillate and the bottoms, there are no common roots and the Underwood equa tions do not apply. Consider the system shown in Figure 10 with a ter nary feed of components A, B and C, where A is most vola tile and C is least volatile. This feed is to be sepa rated into 3 fractions; an A fraction, a B fraction and a C fraction. At minimum reflux there will be an infinite number of stages in all sections of the system. To calcu late the minimum reflux we determine two roots of equa tion (4). These roots will be 9^ Which lies between otA and Otg and 92 which lies between and olq. Under most circumstances in which a separation is made between com ponents adjacent in volatility in a column, only these two key components will distribute to distillate and bottoms in that particular column. Therefore, we assume that 9^ applies in column 2 and 92 applies in column 3. Then, equations (5) and (6) are used to calculate the minimum 68 vapor rates in the rectifying section of column 2 and the stripping section of column 3. If we assume constant molal overflow, the vapor in the stripping section in col umn 3 is equal to the vapor in the rectifying section of column 2 less the vapor introduced with the feed. On this basis, we can compare to see which of the two calculated vapor rates corresponds to a true minimum. If, for in stance, the vapor rate in the stripping section of column 3 is limiting, the actual vapor at minimum reflux in the rectifying section of column 2 will be greater than that calculated. This indicates that the common root, did not apply to column 2. This is so because under the stated conditions, only components B and C distribute in column 1, and component A does not appear in the bottoms of column 1. As was the case with the infinite reflux calcula tion, it is informative to consider the number of inde pendent variables which define the minimum reflux. The thermally-coupled distillation system at a given pressure and with specified heat losses from each stage is fixed by I + 14 independent variables, as discussed in Chapter IV. For the minimum reflux case, assuming that 4 inde pendent variables are used to define the purities of the 69 products, the independent variables are assigned values by the following quantities. Number of Quantity or Quantities Independent Variables Feed quantities (temperature, pressure and composition) 1-4-2 Distillate condition 1 Numbers of stages in all sections (Nk = oo ) 6 Product specifications 4__ Total 1+13 According to this analysis, the thermally-coupled distil lation system is not completely defined at minimum reflux. However, since the independent variables specified are sufficient to define the minimum reflux rate by the Under wood equations, this degree of freedom does not affect the actual value of the limiting reflux requirement for condi tions of constant molal overflow and constant relative volatility. This added freedom allows the rather arbitrary assignment of internal flow distribution among the various columns of the system in this case. A particular solution may be found by assuming an additional specification or specifications to bring the 70 total to 5. This is simplified if the additional specifi cation is taken to be the distribution of a particular component in column 1. Then, with material balances and equations (4), (5) and (6), the complete solution to the particular case can be found. Example of Minimum Reflux Calculation The minimum reflux for the example problem presented earlier in the chapter is found by determining the values of the various quantities describing this limit as follows. 1. Since the feed is a bubble point liquid, q ■ 1. 2. The common roots defining the minimum reflux are found by equation (4). For the ternary mixture equation (4) is + «?.*.£,« . ! _ - 0 a B - 9 - 0 Substituting in the values, _ 0 —— 9 9(0.3333) _ l. 3(0.3334) ^ 0.3333 . , , 9 - 0 3 - 0 1 - 0 (X-9) and solving 0X - 4.6641 02 - 1.3359 71 Assume the common root, 9^, applies to column 2 In the system and that at minimum reflux only components A and B appear In the over head product (then X£|D - 0). Under these conditions, the minimum vapor in the rectifying section of column 2 is given by equation (5). I— ] ■ (x-10) *Ds’m °^A~ ®1 “ ®1 Substituting in the values 1 ^ 1 ■ 1.6876 (X-ll) »Ds'm 9 ~ 4-6641 3 - 4.6641 In a similar manner, assume the lower root, 62, applies In column 3 and that only components B and C appear In the bottoms product at minimum reflux. Under these conditions, the minimum vapor in the stripping section of column 3 is given by equation (6). l a + (x_12) I W 8 lm C t B “ 9 2 <XC - 9 £ J Substituting in the values IXlI - - [ - 3(0.10). + 11Q,.9)____ 1 - 2.4991 (X-13) » V m [ 3 - 1.3359 1 - 1.3359 J For conditions of constant molal overflow and taking off the intermediate product as a liquid, the rectifying vapor In column 2 Is related to the stripping vapor of volumn 3 by V, - V, + <1 - q )F (X-14) Since q_ ■ 1, this reduces to 8 72 V2 - V3 (X-15) Assuming conditions set by both equations (X-ll) and (X-13) are re quired at the minimum reflux, the ratio of overhead product to bottoms product is _ 2^499 m 1.481 (x-16) Wg 1.688 v ' It is apparent from the feed to the column and the product specifications, however, that the overhead product rate is approximate ly equal to the bottoms product rate. Therefore, both (X-ll) and (X-13) cannot apply. Since these facts indicate that the minimum vapor calculated by equation (X-13) will be larger than that calculated by (X-ll), the value calculated by (X-13) must be the minimum. The fact that equation (X-ll) does not apply indicates that the 9^ cannot apply to column 2. This occurs if component A does not distribute between the overhead and bottoms of column 1 and therefore, at minimum reflux XA,N,1 " °* As stated in the example calculations on the material balance, for the given problem, the specifications were insufficient to complete ly define the material balance. For the assumed value of 0.05 mole fraction for component A in the Intermediate product, the minimum reflux is by equation (X-13). V3 - 2.499 (0.3541) ■ 0.8849 moles/mole of feed. From this, the liquid and vapor flows at other points in the system 73 can be calculated. Furthermore, other values of mole fraction can be assumed for component A in the Intermediate product. For Instance, If x^ g ■ 0, the material balance calculations give Wg “ 0.3379 and V3 ■ 0.8446 moles/mole of feed. The material balance shows that as the mole fraction of component A in the intermediate product Is reduced, the bottoms product rate decreases and the minimum reflux will also decrease. The minimum occurs when the mole fraction of component A in the intermediate product is zero. This condition of minimum reflux will be referred to as case 1. If the original specifications required the mole fraction of component A in the intermediate product to be 0.03, then the former value of column 3 stripping vapor would be the minimum. This condition will be referred to as case 2. These minimum reflux calculations are summarlr ad in Table III for case 1 and Table IV for case 2. Because the minimum reflux case as solved still retains a degree of freedom, the vapor and liquid rates in column 1, in the stripping section of column 2, and in the rectifying section of column 3 are not defined. In both cases, none of component A can be allowed to reach the bottoms of column 1. Then, w^ ^ ■ 0 at minimum reflux. The additional degree of freedom can be taken care of by assuming an additional specification on column 1; for instance, let dj, ^ ■ 0.01 moles/mole of feed. Then the minimum reflux equation for the rectifying section of column 1 is 74 TABLE III Summary of Minimum Reflux Example Calculations, Case 1 Material Balance Basis: 1 mole of feed _______________ Products___________ ponent Feed Overhead Intermediate Bottom moles moles MF moles MF moles MF A 0.3333 C.3333 .90 0 0 0 0 B 0.3334 0.0370 .10 0.2625 .90 0.0338 .10 C 0.3333 0 0 0.0292 .10 0.3041 .90 1.000 0.3703 0.2918 0.3379 Vapor and Liquid Rates, Moles/Mole of Feed, at Minimum Recycle Column 1 Rectifying section Stripping section Column 2 Rectifying section Stripping section Column 3 Rectifying section Stripping section Vapor 0.8446 0.8446 Liquid 0.4742 1.1826 75 TABLE IV Summary of Minimum Reflux Example Calculations, Case 2 Material Balance Basis: 1 mole of feed Products_______ Component Feed Overhead Intermediate Bottom moles moles MF moles MF moles MF A 0.3333 0.3187 0.90 0.0146 0.05 0 0 B 0.3334 0.0354 0.10 0.2626 0.90 0.0354 0.10 0.3333 0.0146 0.05 0.3187 0.90 0.3541 0.2918 0.3541 Vapor and Liquid Rates, Moles/Mole of Feed Column 1 Rectifying section Stripping section Column 2 Rectifying section Stripping section Column 3 Rectifying section Stripping section Vapor 0.8849 0.8849 Liquid 0.5308 1.2390 76 + + °-cdc.i _ (x.17) “» - 8 «» - 8 *c - 8 Because component A does not distribute in column 1 at minimum reflux, only the value 62 applies in this equation. Substituting in the values 9(°-3» 3) ♦ . — + .» » • ” > . (X-18) 9 - 1.3359 3 - 1.3359 1 - 1.3359 Simplifying -1.8028 dR , + (V.) - 0.3616 (X-19) 0,1 x m The common root for column 2 can be calculated from the determined minimum vapor rate and the overhead product composition. By equation (5) — ■ + - *3-d. Bi2- - (V2)m (X-20) *A - *1,2 - ®|f2 Substituting the appropriate values for case 1 1(0/3333; + 3(0,0320) . 0 .8A46 (x.2i) 9 - ® 1,2 3 - 0|>2 Solving el,2 " 5,618 or 2,698 Components A and B distribute in column 2 and the appropriate root, 0j'2> has a value between their relative volatilities. 77 Therefore, the larger value applies. Since the net overhead of column 1 is the feed to column 2, the common root for column 2 must also satisfy equation (X-17). Substituting in the values 9(0.3333) + _ 3dB,l + 1(0.01) _ (y > (x_22) 9 - 5.618 3 - 5.618 1 - 5.618 1 m Simplifying 1.1459 d. . + (V.) - 0.8848 (X-23) Dpi in Solving (X-19) and (X-23) simultaneously dB,l " O -1774 (V.) - 0.6816 i n With this information, this particular solution to the minimum reflux problem can be completed. The results are shown in Table V. D. Intermediate Reflux The limiting cases of infinite reflux, and minimum reflux provide a basis for arriving at the design of the thermally-coupled distillation system. The procedure relies heavily on our experience with the design of con ventional single feed two product systems. The first step in determining a system design is to 78 TABLE V Summary of Particular Solution to Example Minimum Reflux Calculation, Case 1 Additional Specification - Set the net rate of component C at 0.01 moles/mole of feed in the overhead from column 1 Rates in Moles/Mole of Feed Column 1 Material Balance Component di,l =vi,l,l- 1i,0,l wi,l= 1i,Nf,l ” vi,N+l,: A 0.3333 0 B 0.1774 0.1560 C 0.0100 0.3233 0.5207 0.4793 Vapor and Liquid Rates Vapor Liquid Column 1 Rectifying section 0.6816 0.1609 Stripping section 0.6816 1.1609 Column 2 Rectifying section 0.8446 0.4742 Stripping section 0.1630 0.3133 Column 3 Rectifying section 0.1630 0.0216 Stripping section 0.8446 1.1826 79 choose some operating reflux rate such as a reflux r?tio of 1.5 tiroes the reflux ratio at minimum reflux. Then, the splits in liquid between the rectifying section of column 1 and the stripping section of column 2 are chosen so that a reasonable loading occurs in both sections of column 1, the rectifying section of column 3 and the strip ping section of column 2. Similarly, the split of vapor between the rectifying section of column 3 and the strip ping section of column 1 is chosen so that a relatively good balance in vapor loading occurs. This rather arbi trary division of vapor and liquid flows is justified since there is no unique solution to the design problem. The choice may be aided by considering the design relative to complete solutions of the minimum reflux case. The key to the design procedure is setting up a material balance around column 1. The primary function of this column is to split the system feed into a fraction containing the light and mid components and a fraction con taining the mid and heavy components. A reasonable mate rial balance around column 1 is obtained by assuming all the light materials appear in the overhead vapor and none in the bottoms liquid, and assuming all of the heavy 80 materials appear in the bottoms liquid and none in the overhead vapor. Then, the amounts of mid components in the net feed to column 2 are determined by difference; that is, the net feed rate less the net rate of feed of the light components to this column. Then the material balance is completed to give the feed to column 3. If there is more than one mid component, then they are distributed so that the lighter ones appear predominantly in the feed to column 2 and the heavier ones appear in the feed to column 3. Columns 2 and 3 are designed as conventional col umns, based on the simplified model given in Figure 11, using either an approximate design procedure such as the Gilliland correlation (2), or the Erbar-Maddox correla tion (6), or by stage to stage calculations (21,25,35). If the system is separating a ternary mixture into three relatively pure products, then columns 2 and 3 are carrying out essentially binary separations and sure easily designed using the McCabe-Thiele graphical method, or possibly an analytical solution. Since the function of column 1 is to remove the light components from the feed to column 3, a reasonable estimate of how much lights to allow to enter column 3 is 81 necessary. Similarly, the amounts of heavies in the over head product from this column are needed. This degree of separation needed in column 1 is quite a function of recycle to the system, since at one limit, infinite reflux, no separation is needed at all and at the other limit, minimum reflux, a very good or complete separation is needed. Estimates which have been found to be reasonable and adequate for design are based on the following assumptions or propositions. 1. The mole fraction of the light key component in the bottoms liquid of column 1 at infinite reflux is equal to the mole fraction in the intermediate product. 2. The mole fraction of the light key component in the bottoms liquid in column 1 is zero at minimum reflux. 3. The mole fraction of the heavy key component in the overhead vapor of column 1 is equal to the mole fraction in the vapor in equilibrium with the intermediate product at infinite reflux. 4. The mole fraction of the heavy key component in the overhead vapor of column 1 is equal to zero 82 at minimum reflux. 5. The mole fractions at intermediate reflux ratios can be found by linear interpolation with the ratio of minimum reflux to operating reflux, (RgJn/Rs* as the independent variable. Based on this set of assumptions, the concentration of light key component in the bottoms liquid in column 1 is ^■N,i°raV R8)m) <7) Ks And the mole fraction of the heavy component in the over head vapor from column 1 xs V l . l - | R a - ^ ) V S V S < 8 > Since the separation in column 1 is being made be tween the two components of the extreme volatilities, the degree of separation is not high. For such sloppy separa tions, generally all components distribute between the overhead and bottoms products at minimum reflux conditions. This corresponds to the class I separations as described by Shiras, Hanson and Gibson (24), in which the composi tions on the feed stage at minimum reflux are equal to the 83 compositions of the corresponding phases in the feed. Therefore, a reasonable assumption is that under typical operating conditions not too far removed from minimum re flux, the compositions on the feed stage of column 1 close ly correspond to the compositions in the feed. The numbers of stages in the rectifying and strip ping sections of column 1 are determined from the composi tions of the feed stage and the estimated concentrations of a heavy in the overhead vapor and of a light in the bottoms liquid using the following modified forms of the Kremser equations (12,27). The derivation is presented in Appen dix E. N'+l Es . Sl.. k, - s. l,* (9) i'k eN +1 _ i,k 1 a *N1+1 » *1.* - *1.* (10) i»k .ll'+l . Ai,k " 1 In the stripping section o± column 1, for a light com ponent XL|N|1 „ I1. ' (12) *«.*,! ! . Vi S.l va ” l3/Kt 84 The equilibrium ratio, is evaluated at the £eed stage temperature of column 3 and KH is evaluated at the condi tions of the feed stage of column 2. These equations are solved by trial and error by assuming the numbers of stages, calculating the E* ^ and E® ^ values, and then the values of mole fraction of the heavy component in the over head vapor and mole fraction of the light component in the bottoms liquid. If these calculated values are less than or equal to those estimated by equations (7) and (8), the numbers of stages are satisfactory. The average stripping and absorption factors can be estimated for each section by the Edmister method (5) SL k *’ V Si,l(Si,N' + + °*25 “°*5 (13> Ai>k ="VAi,N’(Ai,l + 1) + °*25 _0*5 (14) where the subscripts 1 and N* represent the top and bottom stages of the section, respectively. Example of Calculations at Intermediate Reflux The stage requirements for the example presented earlier in the chapter are to be determined at a reflux ratio of 1.3 times the 85 minimum. Inasmuch as the material balance changes considerably with re flux ratio for the problem, as first stated, It cannot be determined easily at this intermediate condition. However, If we consider the material balance as fixed by the total reflux calculations, then the minimum reflu': should be based on the same product compositions; that is, a mole fraction of 0.05 for component A in the intermediate pro duct. This minimum reflux corresponds to the case 2 calculations. Therefore, the design calculations presented here are based on the following specifications. a) All products of 90 mole percent purity. b) Five mole percent component A in the intermediate product. 1. Material Balance. The preliminary material balance for the system is given in Table IV. 2. Internal Flows. The operating reflux rate is L2 - 1.3*(L2)m - 1.3 (0.5308) - 0.693 and the vapor rate is V2 - L2 + Dg - 0.693 + 0.354 - 1.047 The recycles to column 1 are chosen so that a reasonable balance exists between the vapor and liquid streams in column 1, the rectifying section of column 3, and the stripping section of column 2. A rectifying liquid in column 1 of 0.221 moles/mole of feed 86 and a stripping vapor in column 1 of 0.635 moles/mole of feed satisfy this requirement. The liquid and vapor rates in other sections of the system are calculated by material balance. These are summarized in Table VI. These flows should be chosen so that the net flow of material is always toward a product and the above set satisfies this requirement, That is, the net flow is down ward in the lower sections of column 2 and is upward in the tipper section of column 3. However, if flows of 0.221 and 0.4 moles per mole of feed were chosen for the liquid and vapor, respectively, in the rectifying section of column 1, the flow in the stripping section of column 2 would be upward and it might be very difficult if not impossible to make the separation. 3. Column 1^ Material Balance. The net overhead of column 1 is D1 “ V1 ” L1 “ 0.635 - 0.221 ■ 0.414 moles/mole of feed Assuming none of component C gets into this net overhead and none of component A gets into the net bottoms, the material balance around this column is completed as shown in Table VII. 4. Column 2 and Column 3 Material Balances. Combining the results of the column 1 material balance with the material balance for the system as given in Table IV, the material balances for columns 2 and 3 are completed. These are presented in Table VII. 5. The Stages in Column 2 and 3. The number of stages in these columns are estimated as if they were conventional columns. However, there are differences around the intermediate product 87 TABLE VI Vapor and Liquid Flows: Example at 1.3 Times Minimum Reflux Rates Moles/Mole of Feed Vapor Liquid Column 1 Rectifying section 0.635 0.221 Stripping section 0.635 1.221 Column 2 Rectifying section 1.047 0.693 Stripping section 0.412 0.472 Column 3 Rectifying section 0.412 0.180 Stripping section 1.047 1.401 88 TABLE VII Preliminary Material Balance Column 1 Feed Net Overhead Net Bottoms Component fi#s di,l xi,D wi#l xi ,w A 0.333 0.333 0.804 0 0 B 0.334 0.081 0.196 0.253 0.432 C 0.333 0 0 0.333 0.568 1.000 0.414 0.586 Column 2 Feed Overhead Product Net Bottoms Component fi,2 di,s xi,D wi,2 xi,w A 0.333 0.319 0.90 0.014 0.233 B 0.081 0.035 0.10 0.046 0.767 C 0 0 0 0 0 0.414 0.354 0.060 Column 3 Feed Net Overhead Bottoms Produc Component f i» 3 di,3 xi#D wi,s xi,w A 0 0 0 0 0 B 0.253 0.218 0.94 0.035 0.10 C 0.333 0.014 0.06 0.319 0.90 0.586 0.232 0.354 89 withdrawal point. In general, the composition at the bottoms of column 2 will not correspond to the composition of the net bottoms and the composition of the net overhead of column 3 will not correspond to the vapor at the top of this column. The stagewlse calculations are illustrated in Figure 12 by the McCabe-Thiele construction for column 2. The construction Indicates that 6 stages are required in column 2 with stage nunber 2 as the feed stage. A similar construction for column 3 shows that 9 stages are required with the 5th stage from the top as the feed stage. 6. Column 1_ Stage Requirements. The numbers of stages in column 1 are estimated by the stripping factor method described by equations (7) through (14) in the following steps, a) The mole fraction of light component in the liquid from the bottom of column 1 is by equation (7): - 1 x . l ' A ~ — i (0.05) - 0.0115 (X-24) A ’N’1 V ( R s>m A,S 1,3 The mole fraction of the heavy in the overhead vapor is estimated by equation (8): C>1>1 Rg/(Rs)m C-S ‘ 25> By a bubble point on the Intermediate product, ■ 0.308. Substituting the values, the design concentration is MOLE FRACTION COMPONENT A I N THE VAPOR Y 90 1.0 RECTIFYING SECTION OPERATING LINE EQUILIBRIUM LINE 1 4 LINE1 «/(1-*> 0.8 MOLE FRACTIW IN OVERHEAD PROOUCT MOLE FRACTION IN NET FEED 0.6 STAGES Y-X STRIPPING SECTION OPERATING LINE O.k 0.2 COMPOSITION OF NET BOTTOMS 0.6 1.0 0.2 MOLE FRACTION A IN THE LIQUID X , BOTTOM LIQUID COMPOSITION Pig. 12 Example calculations for column 2 at a system raflux of 1.3 timas the minimum. 91 i i ■ *'? i 1 (0.308)(0.05) - 0.0036 C)l|l X • J b) The compositions of the feed stage streams are estimated as equal to the compositions of the corresponding phases In the feed. Since the feed Is a bubble point liquid, the composition of the vapor has to be calculated. BUBBLE POINT CALCULATION xi,F * «i *ixi,F yi,F “ ®(ixi,F^ E **iXi,F Ki,F “ A 0.333 3.0 0.692 2.08 B 0.334 1.0 0.231 0.693 C 0.333 0.333 0.077 0.231 4.33 1.000 c) The average stripping and absorption factors are determined by equations (13) and (14). For the rectifying section, the equilibrium ratio at the top is assumed equal to the equili brium ratio at the feed stage of column 2. From Figure 12 KC,f,2 " KA,f,2/ClA " y ^ <XA®lA) “ °*8/(0-58*9) - 0.15 (X-26) The equilibrium ratio at the bottom of the section Is assumed equal to that of the column 1 feed stage. Then absorption and stripping factors are calculated for the top and the bottoms of the section. At the top 92 Ac i i - V (KC , l , l V “ 0.221/(0.15-0.635) - 2.26 (X-27) (X-28) At the bottoms Ac.f-l.l " Ll^^C,f-lplvl^ ’ 0.221/(0.231-0.635) - 1.50 (X-29) SC,f-l,l “ 1/AC,f-l,l • °-666 The average absorption and stripping factors are calculated as Sc>1 - ¥0.443(0.666 + 1) + 0.25 -0.5 - 0.494 d) Assume 5 stages are required in the rectifying section. e) The composition in the overhead vapor for 5 stages is estimated by equations (9), (10), and (12). By equation (9) -0.5 (X-30) Substituting the values -0.5 * 1.77 and -0.5 (X-31) Substituting the values By rearranging equation (10) - Ar i - 1 1.77 - 1 , (1 - E* ) - . . 0.0258 (X-33) C>1 - 1 1.77® - 1 Substituting the values into equation (12) y ■ — ----(X-34) , . H «g.i L, - KaV Av2 Then _ (0.0258)(0.077)___________ yZ,l,l i 0.221(0.491) 0.693 - 0.15(0.412) - 0.024 e) Since the calculated yA . . is less than the design value, 5 Ay X y X stages are sufficient. If these calculations are carried out, assuming 4 stages in the rectifying section of column 1, i 1 i® calculated to be 0.0436 and this is insufficient. f) In a similar set of calculations, 8 stages are estimated to be required in the stripping section of this column. g) The total stages required in column 1 is then the nunber rectifying plus the number of stripping stages plus 1 for the feed stage, giving a total of 5 + 8 + 1 * 14. Summary. The design is summarised in Table VIII. 94 TABLE VIII Summary of Example Equilibrium Stage Design Column 1 2 3 Rectifying stages 5 1 4 Stripping stages 8 4 4 Total* 14 6 9 System total = 29 stages *Total = Rectifying + Stripping + 1 (for the feed stage) CHAPTER VI CASE STUDY OF THERMALLY COUPLED DISTILLATION The separation of multicomponent mixtures with the system of thermally coupled distillation columns given in Figure 1 has been studied using the design and rating tech niques proposed in this dissertation on a hypothetical ternary mixture. The results are compared to a similar study on a system of the conventional type. The calculations were completed for cases of minimum stages, minimum reflux and both rigorous and approximate solutions to the model for intermediate reflux ratios. The calculations were carried out as examples of the analysis and were not meant to be considered an extensive testing of the method. The sample problem, used in the previous chapter as the example, is as follows: A three component mixture of equal amounts of com ponents A, B and C at its bubble point, is separated into three streams each of 90% purity. The relative volatili ties of the components are 9, 3 and 1, respectively. The mole fractions of components A and C should be roughly 95 96 equal In the intermediate product. All three products are withdrawn as bubble point liquids. The details of the preliminary material balance and other quantities are given in Chapter V. 1. Minimum Stages The minimum number of stages in both the thermally coupled and the conventional systems are the same. Accord ing to the Fenske-Underwood equation, 9.26 stages are re quired at infinite reflux in all of the various systems. 2. Minimum Reflux The minimum reflux conditions for the thermally coupled and the various possible conventional schemes are compared in Table IX. These are based on a system in which the intermediate product contains 0.05 mole fraction A. The calculated values for the thermally coupled systems were determined by the extended Underwood equations as de scribed in the previous chapter. The minimum reflux for the conventional schemes were calculated by applying the Underwood equations as originally proposed to each column in the system. The results show that the thermally coupled system requires considerably less vapor boilup than the TABLE IX Comparison of Conventional and Thermally Coupled Systems at Minimum Reflux Percent increase Minimum Vapor Over Thermally Conventional System Boilup Required Coupled System a. Scheme of Figure la 1.124 27.0 b. Scheme of Figure lb with a total condenser on column 1 1.430 61.5 c. Scheme of Figure lb with a partial condenser of column 1 1.120 25.6 Thermally Coupled System 0.985 98 conventional schemes. The conventional scheme presented in Figure la and that from Figure lb with a partial con denser on the first column require about the same vapor boilup, but require considerably less vapor boilup than the system of Figure lb with a total condenser on the first column. The differences between the various systems range from 26 to 62%. 3. Intermediate Reflux The stage requirements for both the thermally coupled system and the conventional system of Figure la cure presented in Figure 13 as a function of vapor boilup relative to the vapor boilup of the thermally coupled system at minimum reflux. Figure 13 was developed using the approximate design method of Chapter V for the thermally coupled system and the Lewis-Matheson (13) calculation procedure for the con ventional system. In the computations for the thermally coupled system, columns 2 and 3 were designed as binary separations using the analytical solutions by Underwood (30,31). In the computations for the conventional system the total vapor boilup was divided between the columns to NUMBER O F STAGES 99 MINIMUM REFLUX FOR CONVENTIONAL SYSTEM THERMALLY COUPLED SYSTEM THERMALLY COUPLED SYSTEM WITH REFLUX OF 1.3 TIMES THE MINIMUM REFLUX CONVENTIONAL SYSTEM CONVENTIONAL SYSTEM WITH REFLUX OF 1.3 TIMES MINIMUM REFLUX IN EACH COLUMN 10 MINIMUM VAPOR FOR THERMALLY COUPLED SYSTEM 1.0 VAPOR BOILUP RELATIVE TO THE VAPOR BOILUP OF THE THERMALLY COUPLED SYSTEM AT MINIMUM REFLUX Fig. 13 Sample problem: stage requirements of the thermally coupled and conventional schemes as a function of vapor rate. 100 operate both at the same reflux ratio relative to each minimum. It is apparent from Figure 13 that the thermally coupled system can be used to produce the same separation as the conventional system with considerably less vapor boilup and with slightly greater numbers of stages. Since the cost of producing and condensing this vapor is the major portion of the distillation costs, we would expect the thermally coupled system to be more economical. Fur thermore, the thermally coupled system is no more sensitive to design vapor rate than the conventional system and should give satisfactory performance as long as properly designed, built and controlled. The general trend for stages versus reflux ratio for the thermally coupled system is very similar to the trend in a conventional column. There is minimum reflux at which infinite numbers of stages are required, and there is minimum number of stages at a condition of infinite reflux. These facts are further illustrated in Figure 14 for the particular example of this study. Also shown on this figure sure the distribution of these stages to the three columns of the system. The stages required in 101 DESIGN METHOO RATING METHOD TOTAL STAGES 20 COLUMN 1 STAGES 10 COLUMN fSTAGES 2.0 3.0 1.0 ^N. Fig. 14 sample problems stages in the thermally coupled system as a function of reflux. 102 column 1 are most affected by reflux ratio. At reflux ratios of less than 1.15 times the minimum, the curves are very steep, and subsequently, the small differences in equilibrium data which would be reflected by the minimum reflux ratio, will give substantial differences in the num ber of stages required. However, the curve is not exces sively steep at higher reflux ratios and the system could be designed at reflux ratios of 1.2 times the minimum or greater with confidence. The solid points on Figure 14 indicate rating calculations at which the system meets or betters the required purities of the products. These cal culations show that the design method tends to be conser vative for this particular example. Since part of the design method was not developed until after the rating calculations were completed, the design used in the rating calculations is not identical to that resulting by the design method, although the total numbers of stages are quite similar. The rating calcula tions do show that the system is quite flexible. And, changing the recycles to one section to increase the separation there, reduces the recycles to another section and the separation in that section becomes poorer. 103 However, the net result Is a relatively small change In the product compositions and can be considered a combina tion of compensating effects. Similarly, a few stages in one section could be transferred into another section and the overall change would probably be quite small because of these compensating effects. The fact that the total stages agree quite well indicates the design method is reasonable at least for this particular problem. These rating computations were obtained by numeri cal solutions of equations describing the equilibrium stage model of Figure 10, using the technique described in Appendix C. Many of the solutions were obtained with various versions of the rating method during its develop ment. The design used in these computations is summarized in Table X, and the results of these calculations are pre sented in Figures 15, 16, and 17. These figures illus trate how the fractionation can be varied by changing the amounts of vapor and liquid recycles to column 1 from columns 3 and 2. In Figure 18 the range of acceptable amounts of vapor and liquid recycles to column 1 which maintain the required purities of the products are shown. Over much of the range of calculations, the 104 TABLE X System Design for Rating Calculations Feed: Bubble point feed qs - 1.0 Component Feed rate, moles A 0.3333 B 0.3334 0.3333 1.0 Relative volatility 9.0 3.0 Design Variables 1.0 Reflux Ratio at Rs/<Rs>n Rates. = 1.3 Moles/Mole of Feed Column Section vapor liquid st aces Column 1 Rectifying varies varies 7 Stripping varies varies 6 Column 2 Rect ifying 1.047 0.693 1 Stripping varies varies 4 Column 3 Rectifying varies varies 4 Stripping 1.047 1.401 6 The total number of stages In each column Is the sum of the rectifying, the stripping and 1 for the feed stage. V,. molcs/Molc o f feed 105 0.80 0.75 0.70 0.20 0.10 I,, MOLCVtoLC OF FEED Fig. 15 Sample problem: overhead product coraposition as a function of internal distribution of vapor and liquid. M0LE5/M0LE O F FEED 106 0.80 0.70 O.fe 0.09 0.10 0.19 0.20 0.29 L1# M0LE9/toLE OF FEED Pig. 16 Sample problem: intermediate product composition as a function of internal distribution pf vapor and liquid. MOLES/MOLE O F FEED 107 0.80 0.75 i> 0.50 o.*»5 0.10 0.25 L v MOLES/MOLE OF FEED Pig. 17 Sample problems bottoms product composi tion as a function of Internal distribution of vapor and liquid. MOLES/MOLE O F FEED 108 REGION OF ALL PRODUCTS 90 PERCENT OR GREATER PURITY 0.70 O.65 0.60 f t • > 0 .1 0 0.25 0.20 L|# HOLES/MOLE OF FEED Fig. 18 Saapla problaau the rang* of lntarnal flow distributions for dssign at a raflux ratio Of 1.3 tiass tha ainiaum. 109 separation made Is more than sufficient and the number of stages in the system can be reduced while maintaining pro ducts of sufficient purities to meet the requirements. Since the stage requirements of column 1 were obtained by very approximate design equations, the number probably could be reduced considerably. A number of rating calcula tions were undertaken to determine this. The results of these calculations cure shown in Figure 19 for a fixed ratio of rectifying to total stages in column 1 equal to 0.5. These calculations show that with this feed location, the number of stages in column 1 could be reduced from 14 to 9 while maintaining the required product purities. Fur ther improvement could be obtained by varying the feed location as hown in Figure 20. The minimum number of stages in column 1 with the design of the remaining part of the system as given in Table 10 is about 7. These re sults show that the design method is quite conservative for column 1. However, this excess number of stages in column 1 tends to broaden the range of operability and would add to the stability of the system operation. For instance, if under some upset in the operation am excess of component C entered column 2 through the feed stage. 96 95 9*» 93 92 91 90 09 88 J. i— i — i — i — i — i — i — i — r ^ - O.58O MOLES/MOLE OF FEED L1 - O.iSO MOLES/MOLE OF FEED -OLUKN 1 STAGES RECTIFYING TOTAL “ °*5 INTERMEDIATE PROOUCT OTHER QUANTITIES GIVEN IN TABLE 10 COMPONENT B BOTTOM PROOUCT COMPONENT C COMPONENT A OVERHEAD PROOUCT MINIMUM ACCEPTABLE PROOUCT PURITY (SET BY SPECS) J I I I I 1 I I L 8 10 12 lb NUMBER OF STAGES IN COLUMN 1 L9 Sample problem t tha effect of column 1 on systam performance. MOLE PERCENT I N THE 111 1 - 0.580 moles/mole or feed 2 L1 - 0.160 MOLES/toLC OT FEED 3 6 STAGES IN COLUMN 1 U OTHER QUANTITIES AS GIVEN IN TABLE 10 95 93 BOTTOM PROOUCT - COMPONENT C 92 INTERMEDIATE PROOUCT -COMPONENT B MINIMUM ACCEPTABLE PROOUCT 90 PURITY (SET BY SPECS) OVERHEAD PROOUCT COMPONENT A 89 88 0.8 o.U 0.6 0.7 COLUMN 1 RECTIFYING STAGES TOTAL COLUMN 1 STAGES Pig. 20 Sample problem* the effect of column 1 feed location on system performance. 112 then the intermediate product would go off specifications. This occurs since component C is of low volatility and would be driven down in column 2, and necessarily, some of this excess would be taken off in the intermediate pro duct . Rating calculations were also carried out to see what would happen if the feed to the system as originally designed was varied. Its composition was charged to 40 mole percent A, 30 mole percent B, and 30 mole percent C. In one case, the product rates were kept at their original values and the products went off specifications as was re quired by the overall material balance. For this feed and all conditions as given in Table X with = 0.221 and = 0.635, the compositions of the products were 0.836, 0.825 and 0.994 for the overhead product, the in termediate product and the bottoms product, respectively. However, if the material balance were modified to suit the new feed composition, the system performed well. The range of internal distribution of liquid and vapor for acceptable performance changed somewhat but was similar to the system operating on the original feed as shown in Figure 21. In this case, the system would be operating 033J J O 3T0M/S3T0H 113 0.70 0.65 0.60 • > 0.55 REGIONS OT ALL PRODUCTS 90 PERCENT OR GREATER PURITY 0.50 PEED COMPONENT AMT A O.333 B 0 . 3 3 * 1 c 0.333 0.H5 1 0.05 0.10 0.15 0.20 0.25 Lj, MOLES/MOLE OT PEED Fig. 21 Samplo probl«at th« effect of food composition on systsn pmrformanc*. 114 at about 1.3 tlines minimum reflux based on the new feed and satisfactory performance would be expected. Since the system is over-designed in terms of numbers of stages, it should perform satisfactorily at the design vapor and liquid for a relatively wide range of feed. However, the minimum reflux requirements increase for increasing B in the feed and there would be a limit at which the vapor boilup would have to be increased. For instance, if the feed composition changed to 30 mole percent A, 40 mole percent B, and 30 mole percent C, the design vapor rate of 1.047 moles/mole of feed would correspond to a system re flux ratio of about 1.2 times the minimum. Considering Figure 14 as a general correlation, the number of required stages would increase by about 3 with most of the increase occurring in column 1. But the rating analysis indicates that the design has about 7 stages to spare, and it would probably work over a considerable range on this feed. Inasmuch as the design is conservative, the system reflux ratios could be reduced keeping the numbers of stages constant. Rating calculations were carried out with a system reflux ratio of 1.2 times the minimum and other conditions as given in Table X. These results, 115 presented in Figure 22, show that the design is even con servative for this reflux ratio. However, the range in internal flow distributions which are acceptable in giving products of 90% purity has been reduced. Q33J J O 3TOH/S3TOW 116 0.80 1 --- 1---r STAGES AS IN TABLE 10 0.75 i> REGIONS OF ALL PROOUCTS 90 PERCENT OR GREATER PURITY o>5 1 J__________I __________L 0.09 0.10 0.19 0.20 Lv MOLES/MOLE OF FEED 0.25 Fig. 22 Sample problems the affect of reflux ratio on system performance. 117 CHAPTER VII CONCLUSIONS The main conclusions of the present study can be summarized as follows: 1. The thermally coupled system is a lower cost alternative to conventional systems. 2. The thermally coupled distillation system can be designed to be quite flexible in terms of the internal distributions of liquid and vapor flows. 3. The thermally coupled distillation system can be designed to be quite flexible in terms of the handling of various feeds. 4. The design method for thermally coupled system proposed in this dissertation is a resonable and well defined method of attaining initial designs. 5. The rating calculation procedure presented in the appendix is a useful convergence scheme for such distillation systems. REFERENCES 1. Benedict, M., "Multistage Separation Processes," Chem. Engr. Proctr. 1., No. 2, Trans. Am. Inst, of Chem. Enors., 43, 41-60 (1947). 2. Brugma, A. J., "Fractional Distillation of Liquid Mix tures, Especially Petroleum," Dutch Patent No. 48,850 (Oct. 15, 1937). 3. Brugma, A. J., "Fractional Distillation of Liquid Mixtures Such as Petroleum Oils," U. S. Patent No. 2,295,256 (Sept. 8, 1942). 4. Cahn, R. P., and A. 6. Di Miceli, "Separation of Multicomponent Mixture in Single Tower," U. S. Patent No. 3,058,893 (Oct. 16, 1962), Filed Sept. 1, 1959, Ser. No. 837,490. 5. Edmister, W. C., "Absorption and Stripping-Factor Functions for Distillation Calculation by Manual and Digital Computer Methods," A.l.Ch.E. Journal. 2., 165-171 (1957). 6. Erbar, J. H., and R. N. Maddox, "Latest Score: Reflux vs. Trays," Petroleum Refiner. 40, No. 5, 183-188 (May 1961). 7. Fenske, M. R., "Fractionation of Straight-Run Pennsyl vania Gasoline," Ind. Eng. Chem.. 24, 482-485 (1932). 8. Grunberg, J. F., "The Reversible Separation of Multi- component Mixtures," Advances in Cryogenic Engineer ing. 2, 27-38 (1960), Proceedings of the 1956 Cryo genic Engineering Conference, National Bureau of Standards, Boulder, Colorado (Sept. 5-7, 1956). 9. Harbert, W. D., "Which Tower Goes Where?" Petroleum Refiner. 36. No. 3, 169-174 (March 1957). 118 119 10. Hausen, H., "Economical Separation of Gas Mixtures by Reversible Rectification," Z. Tech. Phvslk. 13, 271-7 (1932), Cited by Petlyuk, Platonov, and Girsanov (19), Details see Chem. Abstr., .26, 3970 (1932). 11. Holland, C. D., Multicomponent Distillation, Prentice- Hall, Inc., New Jersey, 1963. 12. Kremser, A., "Theoretical Analysis of Absorption Process," Nat. Pet. News. 22, No. 21, 42 (May 21, 1930), Cited by Souders and Brown (27). 13. Ldftis, W. K., and G. L. Matheson, "Studies in Distil lation, Design of Rectifying Columns for Natural and Refinery Gasoline," Ind. Eng. Chem.. 24, 494-498 (1932). 14. Lockhart, P. J., "Multi-Column Distillation of Natural Gasoline," Petroleum Refiner, 26, No. 1, 104-108 (Aug. 1947). 15. L'vov, S. V., Some Problems in the Distillation of Binary and Multicomponent Mixtures. Publishing House of the Academy of Sciences of the U.S.S.R., 1960. 16. Nelson, W. L., Petroleum Refinery Engineering. 4th Edition, McGraw-Hill Book Co., New York, 1949, pp. 229-230. 17. Petlyuk, F. B., and V. M. Platonov, "Thermodynamical ly Reversible Multicomponent Rectification," Khlm. Prom., 40, (10), 723-725 (1964). 18. Petlyuk, F. B., V. M. Platonov and V. S. Avet'yan, "Optimal Rectification Schemes for Multicomponent Mixtures," Khlm. Prom., 42 (11), 865-868 (1966). 19. Petlyuk, F. B., V. M. Platonov and I. V. Girsanov, "Calculation of Optimum Distillation Cascades," Khim. Prom.. 40, (6), 445-453 (1964). 120 20. Petlyuk, F. B., V. M. Platonov and D. M. SlavInskii, "Thermodynamically Optimal Method for Separating Multicomponent Mixtures," International Chem. Enqr., J>, No. 3, 555-561 (July 1965), Article First Pub lished in Khim. Prom.. 41, (3), 46-51 (1965). 21. Robinson, C. S., and E. R. Gilliland, Elements of Fractional Distillation. 4th Edition, McGraw-Hill Book Co., New York, 1950. 22. Rod, V., and J. Marek, "Separation Sequences in Multicomponent Rectification," Coll. Czech. Chem. Comm.. 24, 3240-3248 (1959). 23. Scofield, H. M., "The Reversible Separation of Multi- component Mixtures," Advances in Cryogenic Engineer ing, 3., 47-57 (1960) , Proceedings of the 1957 Cryo genic Engineering Conference, National Bureau of Standards, Boulder, Colorado (Aug. 19-21, 1957). 24. Shiras, R. N., D. N. Hanson and C. H. Gibson, "Cal culation of Minimum Reflux in Distillation Columns," Ind. Eng. Chem.. 42, 871-876 (1950. 25. Smith, B. D., Design of Equilibrium Stage Processes. McGraw-Hill Book Co., New York, 1963. 26. Sorel, A., "La rectification de la'alcool," Paris, 1893, Cited by Robinson and Gilliland (21). 27. Souders, M., Jr., and G. G. Brown, "Fundamental Design of Absorbing and Stripping Columns for Complex Vapors," Ind. Eng. Chem.. 24, 519-522 (1932). 28. Thiele, E. W., and R. L. Geddes, "Computation of Distillation Apparatus for Hydrocarbon Mixtures," Ind. Eng. Chem. . 25. 289-295 (1933). 29. Underwood, A. J. V., "The Theory and Practice of Testing Stills," Trans. Inst. Chem. Enqrs. (London), 10, 112-158 (1932). 121 30. Underwood, A. J. V., "Fractional Distillation of Binary Mixtures-Numbers of Theoretical Plates and Transfer Units," J. Inst. Petroleum. 29. 147-155 (1943). 31. Underwood, A. J. V., "Fractional Distillation of Binary Mixtures-Simplified Computation of Theoretical Plates and Transfer Units," J. Inst. Petroleum, 30. 225-42 (1944). 32. Underwood, A. J. V., "Fractional Distillation of Ternary Mixtures, Part II," J. Inst. Petroleum. 32. 598-613 (1946). 33. Underwood, A. J. V., "Fractional Distillation of Multicomponent Mixtures-Calculation of Minimum Reflux Ratio," J. Inst. Petroleum. 32. 614-626 (1946). 34. Underwood, A. J. V., "Fractional Distillation of Multicomponent Mixtures," Chem. Enqr. Progress. 44. No. 8, 603-614 (1948). 35. Van Winkle, M., Distillation. McGraw-Hill Book Co., New York, 1967. APPENDICES 122 APPENDIX A INDEPENDENT VARIABLES IN DISTILLATION SYSTEMS SEPARATING MULTICOMPONENT MIXTURES Nomenclature Introduction Variables A Process Strear Distillation System The Various Units General Treatment of Distillation Application to Typical Systems Literature Cited 123 APPENDIX A Nomenclature number of feeds to a system. number of components In a given mixture. number of feeds to an equilibrium stage. stage number (numbered from the top)(a subscript). number of product streams from an equilibrium stage number of column sections In a distillation system. number of three Inlet stages (usually feed stages). molal liquid rate from stage j. number of stream dividers In a distillation system. number of stages In a column section. number of stages In column section k. number of total condensers and total rebollers. number of partial condensers and partial rebollers. heat duty on stage j. number of Independent variables. number of Independent Intensive variables. number of independent variables in restricted model molal vapor rate from stage j. number of phases present in an equilibrium mixture. 124 Introduction 125 Multicomponent distillation systems are studied and designed by means of the equilibrium stage model. In this model, many variables are related by a large number of equations. The number of equations and the number of var iables defining the system depend on the system configu ration, the number or numbers of stages and the number of components In the feed mixture. The difference between the total number of variables and the number of equations Indicates the number of quantities which may be assigned values Independently. These are the Independent vari ables. The problem of determining the number of Independent variables In steady state multicomponent distillation has been studied by several authors (1-3,5-7). The recent work of Howard (4) extended the analysis to dynamic sys tems. Murrlll (6) applied his analysis to the study of control systems. The analysis presented here Is similar to the recent work of Kwauk (5), and Smith (7). 126 Variables Variables can be categorized according to the types of quantities they describe. They will be classified as either equipment variables, operating variables, or physical property variables. Those quantities that define the equipment are the equipment variables. Examples of such variables are the number of stages, tray spacing, weir height, reboiler area, column diameter, etc. Those quantities which describe the operating con ditions are called the operating variables. Examples of these variables are the temperature, time, composition, pressure, flow rate and heat flux. Those quantities which define the basic behavior of the materials present are referred to as the physical property variables. For example, the relative volatility, the density, the viscosity, and the surface tension are such parameters. In summary, the equipment variables define the physical plant for the process being considered, the operating variables quantitatively describe what is oc curring in this plant, and the physical property variables 127 describe the rules to Which the operating plant must con form. The designer has a direct control over the equipment variables in terms of design and a similar direct control over the operating variables, although only a certain num ber of these are independent. However, the physical pro perties of the material being processed are controlled by the chemical nature of the material being processed and the operating variables. Although the designer does not have direct control over the values of the physical property variables, he can control their values by fixing the identities of the con stituents and the operating variables. For instance, if a material of a certain viscosity is desired, the designer is free to choose the chemical constituents, the composi tion, the temperature and pressure in such a manner to meet the requirement of the viscosity. In many cases, the designer may not need to know all the values of the variables describing the system. For instance, if he is interested in calculating the pres sure drop for incompressible isothermal flow in a certain straight pipe line, he may need to know the density, the 128 viscosity of the material flowing, the flow rate, the length and the diameter of the pipe. However, the names of the chemical constituents, the thermal conductivity of the material flowing, the temperature, and the composition may not be of any interest as tar as the particular prob lem is concerned. A Process Stream For a mixture of given chemical constituents at equilibrium, the number of independent intensive quanti ties is given by Gibbs phase rule: Uj =1-0+2 (A-1) Thus, for a mixture of given components of one phase, there are I + 1 independent intensive quantities. These may be assigned by the intensive variables, the tempera ture, pressure, and the I - 1 independent mole fractions. On the other hand, the system may be described by values of the physical property variables, such as thermal con ductivity, bubble point temperature at a given pressure, etc. However, in terms of the typical process stream, we are also interested in the amounts present or flowing. 129 For a stream containing 0 phases, we may determine 0 inde pendent rates or amounts of the phases of the stream. Then the number of independent variables required to completely define a process stream is U=UI +0=1+2 (A-2) For instance, a single phase stream is completely fixed by 1 + 2 independent quantities; of these I + 1 are the in tensive variables and the remaining is an extensive vari able. Furthermore, if two phases are present, then these same independent variables could be used to define the stream. For instance, the overall composition (I - 1 in dependent variables), the temperature, the pressure and the overall rate define this 2-phase stream. However, the compositions of the two phases are dependent variables and are determined by the equilibrium relationships for the system. On the other hand, the 2-phase process stream could have been defined by the temperature, the composition of one of the phases, set by I - 1 variables, and the rates of flow of the 2 phases. Then the composition of the other phase and the pressure would be determined by the equili brium relationships. 130 Distillation Systems The analysis of distillation systems is simplified by dividing the system into a number of sections, in which all stages have the same number of inlet streams. This type of division is illustrated in Figure A-l for a system of thermally coupled columns. These sections will be de signated as column sections, feed stages, stream dividers, reboilers, condensers, and one inlet stage. In Figure A-l the sections enclosed by the broken lines are the column sections which are made up of 2 inlet stages. These are interconnected through stages with 3 in let streams (feed stages) and stream dividers. Each of these units can be utilized independently and therefore, is described by a set of independent variables. Then, the distillation system, which is made up of these units, is described by all the variables describing all the individ ual units. However, the units combined into the distilla tion system are not independent, since an outlet stream of one is generally an inlet stream to another. Therefore, the sum of the independent variables for the individual units is greater than the number of independent variables for the system by the number of variables which describe 131 COLUMN SECTION FEED • 1 i — ~ T z r= S J 3-INLET STAGE COLUMN SECTION n i_L n L. - - SYMBOLS c n EQUILIBRIUM STAGE f VAPOR FLOW | LIQUID FLOW V HEAT FLOW $ STREAM DIVIDER TOTAL CONDENSER COLUMN SECTION 1 ■ i 3-INLET STAGE | I COLUMN SECTION INTERMEDIATE PROOUCT COLUMN SECTION 3-INLET STAGE COLUMN SECTION PARTIAL REBOILER BOTTOM PROOUCT rig. A-l Sactionalixad raprasantation of a tharaally-couplad distillation systan. 132 both an out let of one unit and an inlet to another unit. Therefore, the number of independent variables for the sys tem can be calculated from the number of each type of unit, the number of independent variables for each type of unit, when each is considered independently, and the number of interconnecting streams. The Various Units Single Stage The equilibrium stage is defined as a unit which operates on a feed made up of one or more streams and pro duces from this feed 2 or more streams in equilibrium, each of a different phase. The equilibrium stage can be visual ized as in Figure A-2. This conceptual version of the equilibrium stage consists of a mixer in which the enter ing streams are brought together and mixed with heat trans ferring to or from the mixture and a separator in which the mixture is separated into the separate phases. Consider the stage with j inlet streams producing k product streams. Since each stream is defined by I + 2 variables, a total of (j + k) (I + 2) variables describe the streams to and from the stage. In addition, the heat 133 I p MIXER AND HEAT EXCHANGER PHASE SEPARATOR Pig. A-2 An aquilibrium stag*. 134 flux Is defined by one extensive variable resulting In a total of (j + k)(I + 2) + 1 variables. For this system, there are I Independent material balances, one for each component, and one heat balance. In addition, for the pro duct streams to be In equilibrium, their temperatures and pressures must be equal and their compositions related by (k - 1)*I Independent relationships equating the fugacltles In each phase. The total number of equations Is Type Number of Equations Material balance I Heat balance 1 Equilibrium (k-1) 1 + 2 (k-1) Total kl + 2k - 1 Then, the number of Independent variables Is U = (j + k) (I + 2) + 1 - kl - 2k + 1 U = jl + 2j + 2 (A-3) For a stage with 3 inlet streams, j * 3, and U « 31 + 8 (A-4) For a stage with 2 inlet streams, j ■ 2, and 135 U ■ 21 + 6 (A-5) For a stage with a single Inlet stream, j * 1, and U = I + 4 (A-6) Column Sect Ion Consider a column section with n stages numbered irom the top, as shown in Figure A-3. Since every stage of this section has 2 inlet streams, the liquid from the stage above and the vapor from the stage below, each, when considered individually is completed described by 21 + 6 variables as shown by the preceding analysis. The total number of variables needed to describe a set of n independ ent stages is found by summing up 21 + 6 variables for each stage included, and one more for the number of stages in the section, since this number can be a variable. However, for the column section as shown in Figure A-3, the stages are not independent but are interconnected. Since 1 + 2 variables are required to completely describe a stream and a product stream of one stage is the same as the feed to another, 1 + 2 independent relationships equating these variables can be written. In the system being discussed, there are 2N - 2 interconnecting streams, and therefore, 136 STAGE 1 N-2 N-1 L ± O , Pig. A-3 A column section. 137 there exists a total of (2N - 2)(I + 2) Identities relating the various stage variables. If the number of variables describing the relationships between the stages are sub tracted from the total number of variables required to completely define all of the stages, the total number of Independent variables required to completely describe the system Is U = N(2I + 6) + 1 - (2N - 2)(I + 2) = 21 + 2N + 5 (A-7) Stream Dividers A stream divider divides a single phase stream Into 2 streams of the same phase, composition, temperature, and pressure. No heat Is transferred to or from the streams and no pressure change occurs In this device. For the description of the three streams involved 3(1 + 2) vari ables are required. There are I independent component material balances and there is one heat balance equation. In addition, since both products streams are of the same composition, temperature and pressure, I + 1 additional equalities can be included. The equality of the inlet and outlet pressures is another equation. Then, the number of 138 Independent variables describing a stream divider Is U = 3(1 + 2) - I - 1 - (I + 1) -1=1+3 (A-8) These variables would be assigned as follows, for the typical case; 1 + 2 define the feed to the divider and one is used to define the distribution of material to the 2 streams of the divider. Rebollers and Condensers Partial reboilers and partial condensers have one inlet stream and outlet streams which are considered to be in equilibrium. This is by definition an equilibrium stage with a single input stream. Then, a partial reboiler or a partial condenser is described by I + 4 independent variables. Total condensers are quite common in distillation systems, whereas total reboilers are rare. The device is considered a heat exchanger in which a total change in phase occurs, followed by a stream divider. For a heat ex changer, there are 2 streams, each defined by I + 2 vari ables and a heat duty. This heat exchanger is governed by I material balances and a heat balance. Therefore, a heat exchanger is specified by 139 U * 2(1 + 2) + l- I- l = I + 4 (A-9) The total number of Independent variables for a heat ex changer and an Independent divider Is equal to (1+4) + (I + 3). However, in the total condenser system the heat exchanger output Is the same as the feed to the divider. This can be accounted for by I + 2 Identities. Then, for the total condenser or total reboiler system, the number of independent variables is U = 21 + 7 - (1 + 2) =1+5 (A-10) As an example, a typical total condenser system would be defined by the feed stream, utilizing 1+2 independent variables, the heat duty, the pressure and the relative amounts of overhead product and reflux. Schematic representations of reboilers and conden sers are shown in Figure A-4. General Treatment of Distillation A distillation system is made up of interconnected individual units of the types discussed above. The deter mination of the number of independent variables is made in Table A-l by counting the numbers of independent 140 TWO PHASE INLET STREAM | EXCHANGER PARTIAL REBOILER OR CONDENSER VAPOR PROOUCT LIQUID PROOUCT INLET / | HEAT SI ONE PHASE STREAM STREAM | | EXCHANGER __ __ J>IVJDER_ __ TOTAL CONDENSER OR REBOILER — ■ PROOUCT NO 1 PROOUCT NO 2 Fig. a-4 Reboilars and condensars. TABLE A-I The Determination of the Number of Independent Variables in the General Distillation System TYPE OF WIT COLIMN SECTIONS 3 INLCT OR rtCO STAGES STREAM OIVIOERS NUMBER OF INDEPENDENT NUMBER VARIABLES EACH UNIT K 2 NK ♦ 21 ♦ 5 L M 31 ♦ 8 » ♦ 3 NUMBER Or VARIABLES COt/TR I BETTED TO SYSTEM 2 Z Nr ♦ 2IK 5K K-1,K 31L ♦ 8l IM ♦ 3M NUMER or INLET STREAMS PER EMI NUMBER or INLET STREAMS CXTRI BlfTED TO SYSTEM 2K 3L M TOTAL CONDENSERS ANO TOTAL REBOILERS PARTIAL CONDENSERS AM) PARTIAL REBOILERS ip ♦ 5P 10 ♦ *0 TOTAL NEMBER or VARIABLES - 2 E N ♦ I (2K ♦ 3L N-1.K 0) ♦ 5k+8l +3M+3p *>io TOTAL NEMBER OF INLET STREAMS » 2K ♦ 3L COMPUTATION or THE NUMBER OT INDEPENDENT VARIABLES FOR THE SYSTEM NUMBER or VARIABLES 2 t NK *l(2K4>3L'frM+P*Q) ♦5K + 8L+»M+«P+*« K-1.IE NUMBER OF IICET STREAMS WHICH ARE OUTLET STREAMS FROM OTHER UNITS IN THE SYSTEM TIMES THE MMCR OF INDEPEIBCMT VARIABLES PER STREAM NUMBER or IMXPEHOENT VARIABLES IN T* SYSTEM - f I (2K ♦ 3L ♦ M ♦ P * 0 - r ) » > K » < L * 2 M » 2 P * 20 - 2F j 2 Z NK +ir + K+ 2L*M+3P*2Q*2F K-1 .K 1 4 1 142 variables required to describe each unit of the system in dependently and then subtracting from this number, the number of equations describing the interconnections in the system. Since each process stream is described by I + 2 independent variables, then the interconnection of an out let stream of one stage to the feed to another stage or unit is equivalent to setting up I + 2 equalities. The total number of such equalities in a distillation system is I + 2 times the number of such interconnecting streams. The number of interconnecting streams for any dis tillation system can be determined by counting the number of inlet streams to all units of the system. Since all of these come from either other units of the system or from external sources, the number of interconnections between units is the number of inlet streams to these units less the number of external feeds. In Table A-l, the number of independent variables for the generalized distillation system has been determined using the above logic. In general U » 2 H Nk + FI + K+2L + M+3P + 2Q + 2F (A-ll) k«l,K 143 This relationship can be simplified by the following re strictions on the distillation system. 1. All pressures have been predetermined and are not variables. 2. Heat duties in all stages except partial con densers, partial reboilers and other one inlet stages are specified (heat leaks usually as sumed zero). These restrictions reduce the number of variables by 2 Y. Nk + 2L + M + 2P + Q . k=l ,K Then, the number of independent variables for the re stricted system is UR * FI + K + M + 2P + Q + 2F (A-12) Application to Typical Systems Analysis of a Flash Distillation In order to illustrate the above conclusions, an analysis of a single stage with one feed, the simplest of continuous distillation processes and usually referred to as a flash drum, is considered here. This system can be 144 considered directly from our analysis of a single stage or In light of the general analysis of a distillation system. Taking the second approach, the number of each type of unit In the system Is tabulated. The flash drum Is con sidered the same as a reboiler or partial condenser and is the only unit in the system; therefore, K = L * M = P “ 0, and Q = 1. From this and equation (A-ll), we conclude that, U = I + 4. Since any process stream is described by 1+2 independent variables, with complete specification of the feed to this device, there remains 2 more independ ent variables. Any two of the set of unspecified variables may be chosen as long as they are independent of each other. For example, both the vapor and liquid rates from the stage could not be chosen, since they are related by an overall material balance around the system. A possible choice could be the vapor rate, and the mole fraction of one of the components in the liquid phase. Absorbers and Strippers Absorbers and strippers are distillation systems consisting of one column section and two feed streams. Therefore, we conclude that the number of independent 145 variables can be calculated from equation (A-ll) with K * 1, and L, M, P. and Q equal to zero and with F « 2. Then U - 2N + 21 + 1 + 2(0) + 0 + 3(0) + 2(0) + 2(2) U - 2N + 21 + 5 In the usual case, the streams entering the system, the liquid Lq and the vapor VN + 1, are conqpletely specified (refer to Figure A-3). This fixes 2 (1+2) variables. Also, the heat addition rate to each stage, usually equal to zero, and the pressure above each stage are specified, fixing 2N variables. Then, a subtotal of 2N + 21 + 4 vari ables have been specified. Since the process is described by 2N + 21 + 5 independent variables, a single independent variable remains unspecified. Thus, a mole fraction is the overhead vapor or the number of stages, or any one of the many variables could be specified and the system would be completely fixed. Conventional Distillation Column A conventional distillation column is presented in Figure A-5, in terms of the general sectionalized system. The system of Figure A-5a is made up of two column 146 FEED FEED COLUMN SECTION TOTAL CONDENSER OVERHEAD PRODUCT R COLUMN SECTION n C FEED STAGE (3 INLETS) PARTIAL REBOILER BOTTOM PRODUCT A PARTIAL REBOILER ANO TOTAL CONDENSER / T E E . VAPOR OVERHEAD PARTIAL CONDENSER COLLMN ~ _ LIQUID OVERHEAD SECTION H Z D FEED STAGE COLUMN SECTION PARTIAL REBOILER L BOTTOM PROOUCT 8 PARTIAL REBOILER AND CONDENSER Fig. A-5 Conventional distillation columns. 147 sections, a 3-inlet stage, a partial reboiler, a total con denser and has one feed stream. Therefore, the number of independent variables is determined with K - 2 M - 0 L a p * Q * 1 and F - 1. Consider the common case in which the heat inputs to all stages except the reboiler and condenser are zero and the pressure above all units are specified. Then, the number of independent variables is determined from equa tion (A-12) UR - I + 7 The system of Figure A-5b is equipped with a partial condenser and the ability of taking off both vapor and liquid overhead product. For this system F - 1 K - 2 L = M - 1 P » Q 148 and Q ■ 2 Then, by equation (A-12), the number of Independent vari ables in the restricted system is Ur - I + 7 Analysis of a System of Thermally Coupled Columns for the Separation of a Single Feed into Three Products The distillation system discussed in this section is presented in Figure A-l. In this system, there are 6 column sections, 3 inlet stages, 3 stream dividers, a re boiler and condenser. For the case of a total condenser K = 6 L = 3 M = 3 P - 1 Q = 1 and F =■ 1 Then, if all heat transfer occurs at the condenser and re boiler, and the pressures above all stages and condensers are specified, by equation (A-12), Ur *1+14 149 Literature Cited 1. Brown, G. G., and M. Souders, Jr., "Separation of Petroleum Hydrocarbons by Distillation," The Science of Petroleum (A. E. Dunstan , A. W. Hash, B. T. Brooks and H. T. Tizard, Editors), .2, 1544-1579, Oxford University Press, London, 1938. 2. Gilliland, E. R., and C. E. Reed, "Degrees of Freedom in Multicomponent Absorption and Rectification Col umns," Ind. Eng. Chem., 34, 551-557 (1942). 3. Hanson, D. N., J. H. Duffin and G. F. Somerville, Computation of Multistage Separation Processes, Rein hold Publishing Co., New York, 1962, Chapter 1. 4. Howard, G. M., "Degrees of Freedom for Unsteady State Distillation Processes," Ind. Eng. Chem. Fundam.„ 6, 86-89 (1967). 5. Kwauk, M., "A System for Counting Variables in Separa tion Processes," A.I.Ch.E. Journal. 2, 240-248 (1956). 6. Murrill, p. W., "Degrees of Freedom Determine Control Needs for Distillation," Hydrocarbon Processing and Petroleum Refiner, 44, No. 6, 143-146 (June 1965). 7. Smith, B. D., Design of Equilibrium Stage Processes. McGraw-Hill Book Co., New York, 1963, Chapter 3. APPENDIX B REVERSIBLE DISTILLATION Nomenclature Introduction Mathematical Model Reversible Multicomponent SeparatIons Summary Literature Cited 150 APPENDIX B Nomenclature net upward flow in the rectifying section, the overhead product rate. net feed rate. equilibrium ratio for component i on stage j, liquid rate from stage j in a rectifying section. liquid rate from stage j in a stripping section. vapor rate from stage j in a rectifying section. vapor rate from stage j in a stripping section. net downward flow in a stripping section, the bottoms product rate. mole fraction of component i in the overhead product (either vapor or liquid). mole fraction of component i in the liquid from stage j. mole fraction of component i in the bottoms product mole fraction of component i in the vapor from stage j. mole fraction of component i in the feed. 152 Introduction Reversible processes have the maximum efficiency of conversion of heat into work. Therefore, they give useful but unattainable limits to what can be expected of real processes carrying out such conversions. The fundamental requirement of a reversible process is that conditions at all points of the system have to be displaced from the equilibrium conditions by infinitesimal amounts. Thus, the process can be reversed by changing conditions by infinitesimals to the "other side" of the equilibrium point. Therefore, a requirement of a revers ible distillation is that the vapor and liquid composi tions change by only infinitesimal amounts from stage to stage. According to this analysis, the fractionation oc curring at minimum reflux in a conventional column in the zones of constant composition is reversible. However, in other parts of the system, the fractionation is irrevers ible. Furthermore, for a completely reversible distilla tion, conditions at all points in the system are analogous to those in the constant composition zones in a conven tional column at minimum reflux. These conditions were applied to the reversible 153 distillation of binary mixtures by Hausen (3) and Benedict (1), and to the reversible multicomponent distillation by Grunberg (2), and others (4,5,6). Mathematical Model Consider the rectifying section of a distillation column as shown in Figure B-l. Then, the material balance for any stage, m in the rectifying section of a distilla tion column is Vm+1 yi,m+l " Lm xi,m * *B“1) For a reversible distillation, the liquid and vapor phases entering the stage are of the same compositions as the phases leaving the stage and are determined by the equili brium relationships yi.m * Kl.« xi,m (B-2) Combining equations (B-l) and (B-2), we get K - T Ki,m "ra xi,m W i i _ x i<d xi,m (B-3) 154 j.0 (NET FLOW) STAGE LIQUID FLOW RECTIFYING SECTION n+i ri,N+i f-z f-i f+i f+2 STRIPPING SECTION VAPOR FLOW N-3 N-2 N-1 LIQUID FLOW (NET FLOW) PI9 . B-l Distillation coluan nodal. 155 By a similar derivation, the combined material balance and equilibrium relationship for the stripping section is K _ x**w J ^ n (B.4) ^n+1 l - iSjUw xi ,n On any stage the ratio of vapor and liquid rates is then determined by equations (B-3) and (B-4), the concentration of a key component and the equilibrium ratio for that component on that stage. Equations (B-3) and (B-4) rearranged to calculate the composition of the overhead and bottoms products as a function of liquid to vapor ratio and composition are Ki,m " ^ v^m ,D ci,D T T T " lB“5) 1 - (h/v)m *l,n - <^>n w = --------------- X* n (B-6) 1<w i - iZ /7 )n i , n A requirement for this reversible process is that the compositions of streams entering and leaving a stage differ by only infinitesimals; therefore, the composition of the liquid and vapor portions of the feed have to correspond exactly to the compositions of the liquid and 156 vapor portions on the feed stage. It is also necessary that the temperatures and pressures be the same. Then, if the composition of the liquid portion of the feed is known, the ratio of liquid to vapor at the stages above and below the feed entry can be determined by equations (B-3) and (B-4), a specified concentration of heavy key component in the overhead product, and a specified composition of a light key component in the bottoms product. And by apply ing equations (B—5) and (B-6), the compositions of the overhead product and the bottoms product can be completely determined. This calculation is illustrated in Table B-I for an example for which the mole fraction of component D is zero in the overhead product and the mole fraction of component A is zero in the bottoms product. This corre sponds to the maximum separation attainable in the revers ible distillation column. Considering the results shown in Table B-I, the reversible distillation process giving the maximum separa tion only effectively separates the very heaviest compo nent from the overhead product and the very lightest from the bottoms product. The other components distribute in considerable amounts to the overhead product and the 157 TABLE B-I Determination of Component Distributions to the Products for a Reversible Distillation Column 1. Component A B C 2. Mole fraction In liquid portion of feed 0.25 0.25 0.25 0 3. Mole fraction In feed stage liquid 0.25 0.25 0.25 0 4. Relative volatity 2 1 0.5 0 5. Equilibrium ratio on feed stage 2.22 1.111 0.555 0 6. Composition of the overhead product 0.593 0.282 0.125 0 7. Composition of the bottoms product 0 0.228 0.340 0 (L/V) * 0.1111 - 0 s 0.1111 above 1 - 0 feed stage (L/V) = 2.22 - 0 » 2.22 below 1 - 0 feed stage D .25 .25 .1 .1111 .432 158 bottoms product. This property restricts the choice of light and heavy key components In the reversible process to the lightest and heaviest components of the mixture. In the rectifying section of a reversible distilla tion producing a complete separation of the heavy key from the overhead vapor, the Internal reflux ratio Is V " kH * \ ' P ” 1 - § (B"7) v m vm vm In general, In moving up the column, the equilibrium ratio decreases, and If stages are numbered from the top down dKH is positive. Differentiating (b/v)m with respect dm to m tjL - + D |V „ (B_8) dm v _ dm dm m Since both D and V are positive numbers, ~ must also be dm positive and the vapor flow increases going down the column and decreases going up the column. This requires heat removal at each rectifying stage. Similarly, it can be shown that there is heat addition at each stripping sec tion stage. The reversible distillation column described above 159 mathematically is shown In Figure B-2. The key components are those of extreme volatilities and heat is transferred to each stripping stage and away from each rectifying stage. It is necessary for the feed to be between the limits of its bubble point and dew point at the pressure of the feed stage. Furhtermore, total condensers can be used only if the overhead product is a pure material; otherwise, the overhead product is produced as a vapor, and if necessary, this overhead vapor can be condensed in an external condenser. Similarly, in the reversible pro cess, total reboilers can be used only if the bottoms pro duct is a pure material. These factors are further illustrated for the maxi mum reversible separation of a ternary mixture into two products as shown in Table B-II, Figure B-3, and Figure B-4. Reversible Multicomponent Separations The reversible distillation system for separating a multicomponent mixture into various products presented by Grunberg (2), and others (4,5,6) is presented in Figure B-5 and is based on the same concepts derived for a single 160 OVERHEAD PRODUCT INFINITE NUMBER OF STAGES FEED INFINITE NUMBER OF STAGES HEAT REMOVAL HEAT ADDITION BOTTOM PRODUCT Fig. B-2 Reversible distillation column. 161 TABLE B-II Material Balance for Example of Figure B-3 Component Relative volatility Mole fraction in feed Mole traction in overhead vapor Mole fraction in bottoms liquid Distillate rate Bottoms rate A B C 9 3 1 0.3 0.4 0.3 0.75 0.25 0 0 0.50 0.50 moles/mole of feed of feed (vapor) - 0.40 - 0.60 moles/mole Feed is a bubble point liquid. VAPOR O R LIQUID RATE, HOLES/MOLE O F FEED 162 10 MINIMUM STRIPPING LIQUID FOR CONVENTIONAL DISTILLATION 1.0 liquid MINIMUM VAPOR FOR CONVENTIONAL DISTILLATION MINIMUM RECTIFYING LIQUID FOR A CONVENTIONAL DISTILLATION 0.01 FOR DETAILS OF EXAMPLE SEE TABLE B-2 FEED BOTT( 0.001 U.O 2 .0 2.5 EQUILIBRIUM RATIO FOR COMPONENT A Fig. B-3 vapor and liquid rat«s for «xa*pl« reverBible distillation. HOLE FRACTION I N THE LIQUID 163 COMPONENT B COMPONENT C COMPONENT A 0.1 0.01 FOR DETAILS OF EXAMPLE SEE TABLE B-2 AND FIGURE B-3 FEED BOTTOM 0.001 2.0 2.5 3-0 EQUILIBRIUM RATIO FOR COMPONENT A 3-5 Fig. B-4 Liquid compositions in example reversible distillation. 164 A COLUMN 2 VAPOR AB LIQUID COLUMN FEED ABC VAPOR BC LIQUID NOTE 1 ALL COLUMN SECTIONS HAVE INFINITE STAGES 2 *•- DENOTES HEAT ADDITION 3 •♦DENOTES HEAT REMOVAL Pig. B-5 The separation of a ternary mixture by a reversible distillation process. 165 column producing two products. There are restrictions on this system as there were on the single column. In order for there to be a separation between components A and B in column 2 of the figure, a complete separation of component C from the overhead of column 1 is required. And none of component A can be allowed to enter column 3. In this scheme, the components extreme in volatility are the key components in each column. Reflux is provided at the top of column 1 by a part of the liquid flowing down the recti fying section of column 2 and the vapor at the bottom of column 1 is provided as a sldestream of column 3. $ufrmary 1. In each column, only the components extreme in vola tility are effectively separated. 2. Infinite numbers of stages separate all points differ ing in composition by finite amounts. 3. Heat is transferred to each stripping stage. 4. Heat is transferred away from each rectifying stage. 5. The vapor and liquid rates vary continuously when moving from stage to stage. 166 Literature Cited 1. Benedict, M., "Multistage Separation Processes," Chero. Encrr. Progr., JL, No. 2, Trans. Am. Inst, of Chem. Enors.. 43. 41-60 (1947). 2. Grunberg, J. F., "The Reversible Separation of Multi component Mixtures," Advances in Cryogenic Engineer ing, .2, 27-38 (1960) , Proceedings of the 1956 Cryo genic Engineering Conference, National Bureau of Standards, Boulder, Colorado (Sept. 5-7, 1956). 3. Hausen, H., "Economical Separation of Gas Mixtures by Reversible Rectification," Z. Tech. Physlk. 13, 271-7 (1932), Cited by Petlyuk, Platonov, and Girsanov (5), Details See Chem. Abstr., 26, 3970 (1932). 4. Petlyuk, F. B., and V. M. Platonov, "Thermodynamically Reversible Multicomponent Rectification," Khim. Prom.. 40 (10), 723-725 (1964). 5. Petlyuk, F. B., V. M. Platonov and I. V. Girsanov, "Calculation of Optimum Distillation Cascades," Khim. Prom., 40 (6), 445-453 (1964). 6. Scofield, H. M., "The Reversible Separation of Multi- component Mixtures,“ Advances in Cryogenic Engineer ing. 3,, 47-57 (1960) , Proceedings of the 1957 Cryogenic Engineering Conference, National Bureau of Standards, Boulder, Colorado (Aug. 19-21, 1957). APPENDIX C RATING METHOD FOR THERMALLY-COUPLED DISTILLATION SYSTEMS Nomenclature Introduct ion Model of Thermally Coupled Distillation Column Units Convergence Procedure Tray Compositions Constant Molal Overflow and Relative Volatility Computer Program Literature Cited 167 APPENDIX C Nomenclature A 'i k “ equilibrium factor for absorption of component i in the rectifying section of column k, de fined by equation (C-62). A'i k ~ equilibrium factor for absorption of component i in the stripping section of column k, de fined by equation (C-63). Ai 1 k “ absorption factor for component i on stage j of column k, k). ^ - tray factor for component i on stage j of column ' ' k, defined by equation (C-14). ^ - tray factor for component i on stage j of column k, defined by equation (C-15). d^ ^ - molal rate of component i in the distillate or overhead product from column k. D. . . - tray factor for component i on stage j of column k, defined by equation (C-18). E, . k - tray factor for component i on stage j of column k, defined by equation (C-19). f. v - molal feed rate of component i to column k. 1, K - equilibrium ratio for component i on stage j of column k, y^j j 1i i k “ molal liquid rate of component i from stage j of column k. 1/^ - molal liquid rate in the rectifying section of column k under constant molal overflow condi- t ions. 168 molal liquid rate In the stripping section of column k under constant molal overflow condi tions. molal liquid rate from stage j of column k. number of stages in column k. molal rate of the intermediate product in the thermally coupled system. molal rate of component 1 in the intermediate product of the thermally coupled system. equilibrium factor for stripping of component i in the rectifying section of column k, de fined by equation (C-64). equilibrium factor for stripping of component i in the stripping section of column k, de fined by equation (C-65). stripping factor for component i on stage j of column k, V._k/LJ k. molal rate of component i in the vapor from stage j of column k. molal vapor rate in the rectifying section of column k under constant molal overflow condi- t ions. molal vapor rate in the stripping section of column k under constant molal overflow condi- t ions. molal vapor from stage j of column k. molal rate of component i in the bottoms from column k. bottoms or net downward flow from the stripping section of column k. 170 j ^ - mole fraction of component i in the liquid on stage j of column k, 1^ j j^/Lj yi 1 k ” n,ole inaction of component i in the vapor from ' stage j of column k, vA j k^Vj k* 0C^ - relative volatility of component i. 9, 1 ^ 2, if5, - correction factors for material by © - method of Holland. NOTE - When the subscript denoting the column is unnecessary for identification, it is left off. Special Subscripts CA - refers to a quantity calculated by stage to stage calculations. CO - refers to values corrected by Holland's © method. f - refers to the feed stage. N - refers to the bottom stage of a section. N+l - refers to the vapor entering the bottom stage of a section. p - refers to the product from the feed stage. R - refers to a reference component in a mixture, one with the relative volatility equal to one. s - indicates the product is of the system and not an intermediate stream. S - refers to the intermediate product in the thermally coupled system. 0 - refers to reflux entering the top stage of a section. 171 Introduction The design methods for the thermally coupled sys tems presented In this dissertation are based on many assumptions; therefore, more rigorous solutions to the equilibrium stage model are needed to confirm the design. The rating methods mentioned In the text are Ideally suit ed for this, being straightforward and easy to program for computer solution. The calculation procedure developed by Thiele and Geddes (5) can be made to converge rapidly in typical dis tillation problems involving close boiling materials (3). However, the basic procedure as applied to multifeed col umns is unstable numerically (1). This basic instability in the numerical method may be overcome by applying the method of Hardy et al. (2) (see Holland (3), chapter 8), developed for absorbers and strippers. The method of cal culation proposed in this dissertation is an extension of this modified Thiele-Geddes calculation procedure as presented by Holland using the so-called & - method of convergence (3,4). 172 Model of Thermally Coupled Distillation The model of the coupled distillation system is presented in Figure C-l. The general approach is to as sume vapor profiles and temperatures in all stages of the system. The system is then divided into units. Each unit is solved with assumed recycles to give products that are feeds to subsequent units. Then, the subsequent units are solved to give the recycles to the first units. Once through the system, the products are calculated and then adjusted to be in material balance with the feed. Then, the compositions on all stages are calculated and the necessary adjustments in assumed vapor rates and tempera ture profiles are made. Then, the procedure is repeated until temperatures agree to within the tolerances speci fied. The basic units into which the system is divided are called column units. Each unit is divided further into rectifying and stripping sections, and a feed-pro- duct stage. In each unit, a liquid reflux is supplied at the top of the rectifying section and a vapor is supplied to the bottom of the stripping section. The rectifying and stripping sections are interconnected by a feed- FEED COLUMN r J UMN 1 A 173 & Q CONDENSER C RECTIFYING liquid, OVERHEAD PROOUCT VAPOR XX r n feed I LIQUID RECTIFYING n y COLUMN 2 n J FEED T STRIPPING VAPOR STRIPPING LIQUID INTERMEDIATE PRODUCT RECTIFYING LIQUID FEED XX S’ COLUMN 3 ( VAPOR FLOW | LIQUID FLOW REBOILER IS THE BOTTOM STAGE OF COLUMN 3 STRIPPING LIQUID BOTTOM PROOUCT Fig. C-l The equilibrium stage model for the rating of the three product thermally-coupled distillation system. 174 product stage. Feeds or Products are added or withdrawn at this stage. Column Units The general column unit is shown in more detail in Figure C-2. The computational procedure is to determine, for a given vapor rate profile and temperature profile in the stripping section, the relationship between the moles of each component in the liquid leaving each stage, and the moles in the liquid leaving the bottoms of the column. In a similar manner, the relationship between the moles of each component in the liquid leaving each stage above the feed entry and the moles in the vapor leaving the top of the column is determined. Then, by a matching at the feed product stage, the ratio of moles of each component in the overhead vapor to moles in the bottoms liquid are deter mined . The basic relationships describing the equilibrium stage process are the equilibrium,the material balance, and the enthalpy balance relationships. The solutions of distillation columns are usually obtained by first deter mining the vapor and liquid rates on all stages by a 175 UNIT K STAGE 1 RECTIFYING SECTION STRIPPING SECTION Fig. C-2 G«n«ral column unit. 176 preliminary enthalpy balance and then solving the combined set of equilibrium and material balance equations. After wards, the vapor and liquid rates are corrected by reapply ing the enthalpy balances. Several methods for making the corretions for enthalpy balances are thoroughly discussed by Holland (3), and will not be discussed here. The method of solving the material balance and equilibrium relation ships presented here does not depend on the enthalpy balances, and the overall computation scheme could be modified to include them. The equilibrium between vapor and liquid is de scribed by the equation *i,j - Ki,j xi.j (c-1) Multiplying both sides of this equation by Vj/Lj (C-2) (C-3) or vi,j * si,j 1i,j (C-4) 177 where S. . Is called the stripping factor. * * J • j Solving for the molal liquid rate of component i from stage j where is called the absorbing factor. Consider the first few stages at the top of the column section as shown in Figure C-3. The material bal ance for component i around stage j and the top of this rectifying section is To determine the ratio of the moles of any component in the vapor from any stage j to that in the vapor from the top stage, equilibrium and material balance equations are alternatively applied in a stagewise fashion from the top. (C—5) vi,j+l * k j + vi, 1 " 1i,0 (C-6) Dividing the material balance by 9 vij+i H . ) . , 1i,o ' 1"" » + 1 — (C-7) vi,l vl,l vi,l For the top stage (C-8) 178 1,0 rectifying section r l h j 1 ^ r‘»J It.N-3 1 f Vi.N-2 ii.N-2 t / i , N-1 f *i,N li.N ♦ 1 *^i*NPl STRIPPING section STAGE 1 STAGE 2 STAGE 3 STAGE j STAGE STAGE N-3 STAGE N-2 STAGE N>1 STAGE N Pig. C-3 Stagawlsa calculations. 179 Then, by material balance lj " (C-9) vi 2 ^-10 * A. , + 1 ---^ vi,i 1,1 vi,i For the second stage, by equilibrium, (c-10) Then, by material balance, equation (C-l), -*1.2 *1.1 + U+*1.2> l1 - ^ ! <C-U > By the equilibrium equation, (C-5), " Ai,3 Ai,2 Ai,l + Ai,3(1 + Ai,2} ( 1 “ (C-12) From this development it can be seen that, in general, for a rectifying section U.J vi ^ + Bi,J + Cifj ( 1 - ^ 7 ) (C-13) where the coefficients j and j cure given by the following recursion formulas. 180 and Bi.l * Ai,l (c Bi, 2 “ Ai, 2 Bi,l Bi, 3 = Ai, 3 * Bi,2 Bi*j “ Ai,j * Bi»j-1 Ci(1 = 0 (C-15) ci,2 * Ai,2 (1+ci,l> ci,3 = Ai,3 (1+ci,2^ ci,j * Ai,j <1+ci,j-l> Similarly, calculations for the stripping section are carried out by calculating up from the bottom. The material balance equation for the stripping section as illustrated in Figure C-3 is (c_16) 1i,N 1i,N 1i,N Then, the combined equilibrium and material balance 181 relationship for the stripping section as derived in a similar stepwise fashion, calculating from the bottom, is ~ „ /, vi,N+l » (C-17) !i,N iJ iJ 1 Xi,N 1 where the coefficients are given by the following recur sion formulas: (C-18) °i,N * 1 °i,N-l = si,N ‘ °i,N °i,N-2 = si,N-l * DifN-l DU " Si J-l ' Di»j-l and Ei.N ‘ 0 <C-19> Ei#N-l " si,N Ei,N ■ * " 1 Ei,N-2 " Si,N-l Ei,N-1 + 1 Ei,j * si,j-1 Ei J-l + 1 At the feed product stage the amounts of each com ponent in the vapor sidestream sure 182 Vp Vi.p - ^ Vi f (C-20) and the amounts of each component In liquid sidestream are The overall material balance for the feed stage is fi " vi,f + 1i,f + vi,p + Xi,p " 1i,f-l “ vi,f+1 (C”22 Then by combining equations (C-13) and (C-17), relating the feed stage composition to the overhead vapor composi tion and the bottom liquid composition, with equations (C-20), (C-21), and (C-22), the following equations are obtained. (C-23) £L(fi+Vi.N+lU+ 9iEl,f)) + Vj.N+1 Ej.f^J-i.O Bi.f Ai (C-24) (C-21) a ^i(fl + 1i.O(1^*?lCl.f)) + vi.N+l Dl,f + 11.0 Cl,f Ai 1 i,f a vl.l Bl.f El.f + H.N ci.f pi.f + fi ci,f Ei.f ci,f + Di,f + *?i ci,f Ei,f (C-25) where the following shorthand notation has been adopteds 183 ^i * Bi,f + ci,f (C-26) ^1 * Di,f + Ei,f (C-27) »y1 « V P Kii.* t (C-28) Ai - (C-29) The equations presented above reduce to simple forms when there Is a simple relationship between the feed at the end of the unit and the product from that end. For instance, when there is a partial reboiler at the bottom of the stripping section, this units acts as the bottom stage and acts without a vapor feed. Then, v^ N+1 = » 0, and equation (C-24) reduces to total condenser at the top of the rectifying section. For this case, the feed to the top of the section is related to the liquid product by the external reflux ratio, l^o/d^ * Lq/D and the overhead vapor is ^ = » 1^ q + d^. Under these conditions equation (C-23) becomes . ^ifl + Bl.f J-l.O Ai (C-30) Another case of particular interest is that of a 184 a . 'l*! + Dl.f n.w-1 ( c .3 1 ) A A . L0 + — Bi>f d+7i^i) The case of partial condenser Is quite similar to the case of the partial reboiler. Convergence Procedure The model of the system is solved by first assuming the temperatures and vapor rates from each stage in the system. The liquid rates from each stage are then deter mined by material balance. Then, absorption factors, Ai,j,k * Lj,k/(Ki,j.kvj,k)' and stripping factors, SiJ#k = Ki,j,kvj,k/Lj,k are calculated for each component on each stage of the system. With the absorption factors, the values of j ^ and ^ are determined by equations (C-14) and (C-15) for each stage above the feed-product stages and the feed-product stages in each of the three column units. In a similar manner, the values of D, , v 11 J / K and j ^ are determined by equations (C-18) and (C-19) for each stage below the feed-product stages and the feed- product stages in each of the column units. The general approach is to use assumed recycles to any unit with equations (C-23) and (C-24) to determine 185 the distribution of components to the overhead and bottoms of that unit. Then, these values are corrected by the 8 - method of Holland (3) to be in overall material balance around the unit and to sum up to the correct totals. These corrected overhead and bottoms flows replace the previous ly assumed values in the subsequent calculations. In column unit 1, the values for the distribution of components between the overhead vapor and the bottoms liquid are corrected by the following set of equations. In these equations, the values subscripted "CA" cure the calculated uncorrected values. 9 is a correction factor and is the same for all components in the mixture. Equa tions (C-32) and (C-33) are solved by Newton's method as described by Holland (3). The amounts of each component in (C-32) (C-33) (C-34) 1i,N,1 * fi + 1i,0,l + vi,N+1,1 “ vi,l,l (C-35) 186 the bottoms liquid cure determined by either equation (C-34) or (035). Equation (035) is preferred for com ponents appearing in only negligible amounts in the over head vapor. In column unit 2, the distributions of components to the overhead product, to the liquid recycle to column 1 from the feed stage, and to the bottoms liquid are corrected by the following set of equations. d, „ -------yl,l,l+yl,N+1.2---------- (c-36) i, s ----- 1 + * ' d< “ \ A . CA J-'° CA h o i --------V - 1 * ? ,N+ y I , ------r (C-37) 1 + -+- — CA CA = Ds (=-38) iL 1i,0,l “ *‘ 0,1 (C-39) di.s (C-40) » u i,S » CA 1i,N, 2 * vi,l,l + vi ,N+1,2 “ di,s ” ^ O , ! (C-41) Equations (C-35) through (C-39) are solved simultaneously 187 by the Newton-Raphson method as described by Holland (3) in the bottoms liquid are determined by either equation (C-40) or (C-41). Equation (C-41) is preferable for components appearing in only negligible amounts in the overhead vapor, whereas equation (C-42) is preferable for components appearing in small quantities in the bottoms liquid. Then, by material balance around the intermediate product, the liquid reflux to column 3 is determined. In column unit 3, the overhead vapor, the bottoms product and the vapor to column 1 are determined in a similar manner. The equations cure for d^, 1i # o,1' ^1 and ^2* The amounts of «ach component k.N,! + 1i,0.1 Vi,N+1,1 " H,N,1 + lj.0,1 (C-43) wH Q » Wo (C-44) 188 (C-45) CA w i.B (C-46) This set is identical to the set, (C-36) to (C-41) except for the names of the variables, and are, therefore, solved by the same routine. column units can be used to give estimated product and intermediate streams. However, the products may not be in component material balance with the system feed. The same technique, the 0 method, can be then used to determine corrected products that are in component material balance. The following set of equations cure then used. The results of the calculations on these three (C-48) (si,s)co * ¥j6 /si.sl 5 lwi,sJ CA (C-49) (C-50) 189 X *si,s*CO " Ss (C-51) CA (C-52) w i,s d i *s - s i,s (C-53) Again, these equations are identical to the sets used for columns 2 and 3 and the same routine is used to solve them. The compositions or component rates are calculated in a stagewise fashion from the feed stage of each unit. These calculations require that the overhead vapor, the bottoms liquid, and the feed stage liquid compositions cure known. The combined equilibrium and material balance equation, (C-13) can be rearranged to give Substituting (C-54) into the material balance (C-7) and rearranging, we get Tray Compositions H.O = Bi.J + C1,J ~ 1i,j/vi,l (C-54) C U 190 , »1.J »t.l ♦ C1.J i,J 1 + cl.j Since v^ j+^ is related to 1^ j+^ by equation (C-4), j+1 can be eliminated from equation (C-55) l, t . °1.J Vl.l + Cl.J U.j*l (c.56) 1 + cl.j Similarly, in the stripping section, the combined material balance and equilibrium relationship is arranged to give „ DU * EU ~ ( c _ 5 7 ) 1t.H Et,J Then, substituting (C-57) into the material balance (C-16) and rearranging, we get i, , - D -bi ■ 1a<» * Ei. a ,c-58) ,J i + s±.j "t.j These equations are applied as follows. In the rectifying section, the composition of the stage above the feed is determined by equation (C-56). Then, this compo sition is normalized to total up to the correct or assumed flow on this stage. Then, the process is repeated for the stages above. In the stripping section, the calculations 191 are carried out In a similar manner calculating downward from the feed stage. Constant Molal Overflow and Relative Volatility A computer program was written in Fortran to carry out the solution of these equations. Since the problems for which the program was developed involve distillations with constant molal overflow and constant relative vola tility, several simplifications were made. For instance, when constant relative volatility holds for all components in a mixture, the equilibrium relationships reduce to And the equilibrium ratio for the reference component, based on the bubble point, is (C-59) (C-60) For a dew point calculation (C-61) 192 By this procedure, temperature need not be considered. Then, the Iteration scheme would be based on adjusting the equilibrium ratio for the reference component on each stage of the system. In addition, since we are restricting the calcula tions to constant molal overtlow, the absorption factor on any stage In a rectifying section Is A, , = - L -i = A' — i-- (C-62) i.J «Av kRiJ 1 kr>j and In the stripping section A, , - — Mr — — » A', — (C-63) 1,J *lv KR,j 1 kR,J Similarly, the stripping factor in the rectifying section of a column is si,j --t V j * s'i 1' r <c-64> And in a stripping section ou V si,J £ - KR.j * S'i KR ‘°-65) This allows the absorption and stripping factors on each stage of the system to be calculated as the product of a 193 factor for each component which is dependent on the section of the system and the equilibrium ratio for a reference component. Computer Program The general layout of the computer program is given in Figure C-4. The program was written in the Fortran IV programming language for the Honeywell 800 com puter . The actual stagewise calculations and iteration scheme are performed in SUBROUTINE STAGWZ. During execu tion, SUBROUTINE STAGWZ calls the following subprograms: FUNCTION FTHETA, SUBROUTINE BETTA, and FUNCTION BUBPT. FUNCTION FTHETA is used to determine the correction factor Q, and solves equations (C-32) and (C-33) using Newton's method. SUBROUTINE BETTA solves the equations correcting the column 2, column 3 and overall material balances, by the Newton-Raphson technique. It is used to determine , 1^2 • V3 * ^4/ t5. and FUNCTION BUBPT calcu lates the equilibrium ratio for the reference component by the bubble point condition, equation (C-60). A flow chart outlining these calculations is presented in 194 ENTER INPUT (SUBROUTINE INPUT) OUTPUT (SUBROUTINE OUTPUT) YES YES NO NO STOP NEW PROBLEM MOLE FRACTIONS CALCULATED ^ CALCULATE MOLE.FRACTIONS (SUBROUTINE MOLFRA) STAGEWISE CALCULATIONS (SUBROUTINE STAGWZ) INITIALIZE VARIABLES (SUBROUTINE INTLIZ) Fig. C-4 General layout of computer program. 195 Figure C-5. A copy of the computer program Is presented in Tables C-I through C-IX. The input format is given in Table C-X, and a sample of input is given in Table G-XI. A sample of the output is given in Table C-XII. For most problems at conditions reasonably far from minimum reflux, using a direct iteration on the equi librium ratios of the reference component produces conver gence to about 5 significant figures in about 10-15 iterations. In some problems in which "pinches" occur in one or more of the sections, however, the direct iteration is un stable. But, by averaging the calculated equilibrium ratios with those used in the iteration, the problem is solved in many cases. However, the number of iterations may be quite high. If, for instance, 50 or 60 iterations are required, the question is raised to the validity of the criterion for convergence, and therefore, the solution. This problem of stability and convergence was quite evident in some attempts to solve problems with many stages in all sections at the minimum reflux ratio for the desired separation. However, for designs in the range of 196 SET M - 0 ITERATION NO CALCULATE EQUILIBRIUM FACTORS "*> S' « . K EQUATIONS (C-62), (C-63), (C-6U), AND (C-65) START NEW ITERATION CALCULATE TRAY FACTORS FOR RECTIFYING STAGES 8A,},K'C£.j.K equations (C-lfr) AND (C-15)_____ 4*A»K* (C-18) AUO (c-19) CALCULATE TRAY FACTORS FOR STRIPPING STAGES EQUATIONS STORE ASSUMED EQUILIBRIUM RATIOS FOR REFERENCE COMPONENT K‘R»i,K “ KR,j.K ENTER Pig. c-5 General layout of SUBROUTINE STAGWZ. 2 197 CALCULATE DISTRIBUTION OF COMPONENTS TO OVERHEAD AND BOTTOMS OF COLUMN 1 Vl.iy*i.N,1. EQUATIONS (C-23) AND (C«2l») FUNCTION FTHETA SUBROUTINE BETTA CORRECT COMPONENT RATES IN THE PRODUCTS FROM COLUMN 2,EQUATIONS (c-36). (c-37), (c-38). (c-39). {6-1*0), AND (C->*1) SOLVE EQUATIONS C-32 SOLVE EQUATIONS (C-36) (C-37), (C-38), AND (C-39), FOR AND fj CORRECT COMPONENT RATES FROM TOP AND BOTTOM OF COLUMN 1, EQUATIONS (C-32), (C-33), (C-3U), AND (C-35)__________ CALCULATE COLUMN 3 REFLUX BY MATERIAL BALANCE AROUND THE INTERMEDIATE PRODUCT WITHDRAWAL POINT CALCULATE DISTRIBUTION OF COMPONENTS TO THE PROOUCTS OF COLUMN 2, £^N / d i , s AN0 Ao.i/^t.s» equations (C-21), (c-23), (C-2>*), (C-25), AND MATERIAL BALANCE AROUND CONDENSER Fig. C-5 (continued) 198 SUBROUTINE BETTA SUBROUTINE BETTA FUNCTION BUBPT CALCULATE KRf),K EQUATION (C-60) CALCULATE NEW BU3BLE POINT CONDITIONS ON ALL STAGES SOLVE EQUATIONS (C-U8). (C-l»9), (C-50), AND (C-51). FOR V'c AND V SOLVE EQUATIONS (C-l»2) (C-I»3)t (C-UI»), AND (C-l»5). FORAND CALCULATE LIQUID RATES ON ALL STAGES EQUATIONS (C-56) AND (C-57) CORRECT COMPONENT RATES IN THE SYSTEM PRODUCTS, EQUATIONS (c-W). <c-»»9), (c-50), (C-51), (C-52), AND (C-53) CORRECT COMPONENT RATES IN THE PRODUCTS FROM COLUMN 3,EQUATIONS (C-U2), (C-U3). (C—HU), (C-U5), (C-U6), AND (C-U7) CALCULATE DISTRIBUTION OF COMPONENTS TO THE PRODUCTS OF COLUMN 3, V£, ***> V^^.i/iv^s,EQUATIONS (C-20), (C-23), (C-2U), (C-25), AND MATERIAL BALANCE AROUND REBOILER Pig. C-5 (continued) X DOES X *R.j.K EQUAL "'r.J.K (CALCULATED COMPARED \ t O ASSUMEDJv ^ YES (WITHIN TOLERANCE) NO RETURN YES SUBROUTINE OUTPUT PRINT INTERMEDIATE RESULTS NO YES NO STOP Fig. C-5 (continued) TABLE C-I Typical Main Program for Thermally Coupled Distillation COMMON/BLOCKE/XD.XS.XW.NJOB.M COMMON/BLOCK2/CL 10(6.50.3)»KREF(50.3).CVONE(6.3).CLN(6.3).NFEED(3) 1.N(3) .L(3).V (3).V8A R (3).LBAR(3)»CVN(6.3).CIO(6.3) REAL L.LBAR.KREF 1 FORMAT(2F10.0) CALL INPUT CALL INTLIZ 12 CALL STAGWZ CALL OUTPUT CALL MOLFRA 2 FORMAT(313) 3 f o r m a t (16 ) READ (2.3)NJOB CALL EOF(K) GC TO (lO.ll).K 10 STOP 11 REAO(2.2)N.NFEED GO TO 12 END 200 TABLE C-II SUBROUTINE STAGWZ SUBROUTINE STAGWZ COMMON/BLOCK 1/ALPHA(6) COMMON/BLOCK2/Cl IQ(6150»3),KREF(30»3>*CV0NE(6*3),CLN(6»3).NFEED13) 1«N(3)«L(3)«V(3)«VBAR(3)»19AR(3)»CVN(6*3)*CLO(6«3) COMMON/BLOCK3/FEE0(6J COMMON/BIOCK8/BETAI2) COMMON/BLOCK6/BONE<2) COMMON/BLOCKA/THETA C0MM0N/BL0CK5/DIST(S)«SI (6>tDTtS COMMON/BLOCKN/NCOMP COMKCN/BLOCK7/W(6)»WT CuMMON/BLOCKE/XD*XS»XW,,NJOBtM REAL I«LBAR«KREF REAL KCHK DIMENSION A (6 «6)tCONST (6) DIMENSION RONE(6)iRTWO(o)tRTHREE(6)»RF0UR(6)tRF IVE (6) » X (6) »KCr|K (5 J 1*3) «B (6 « 50 » 3) «C (6» 501 3) ♦ D (6* 50 ♦ 3) * E (6»50 » 3) « AIJ (613) * SIJ (6*3) » ?T0LFE0(6)»W0D(6)«S00(6)tCVAP (6) : C INITIALIZE KNOWN PARAMETERS MsO ro 104 Ksl »3 NAsN(K) DO 104 I sit NCOMP A!J(I,K)s L(K)/(ALPHA(!)«V(K>) SIJ(I,K)s ALPHA(I)«V3AR(K)/LBAR(K) C(I«ltK)s0,0 E 111NA«K)s0»0 104 D(I«NAtK)sl.O 600 M s M*1 DO 111 K»l*3 K0=N(K) DO 111 JsltND 111 KCHMJ* K)sKREF (J*K) TABLE C-II (continued) C CALCULATIONS f o a l l r e c t i f y i n g s e c t i o n s to 102 K* 1 » 3 n d * n f e e d <m to 100 1* 1.NCOMP 100 E (I»1,K>*AIJU,K>/KREF(1.IU to 101 J*2»ND tO 101 1*1.NCOMP B(I,J,IU*AI J(I.K)*3U.J-1.K)/KFEF(J.R) 101 C(I.J.K)sAIJ(I.K)«(C(I.J«1*K)«1*0)/KkEF(J«K) DO 99 1*1.NCOMP E(!»ND.K) s B(I.ND»K)*L3AR(K>/L(K) 99 C ( I»ND.K)=C (I.ND.K)*LBAR IK)/L(K) C CALCULATIONS FOR ALL s t r i p p i n g s e c t i o n s ND*N(K)-NFEEOIK) DO 102 Jsl.NO JP*N (K)-J DO 102 1*1,NCOMP D(l.JP,K)s(D(I«JP*l.K)«S!J(!«K)*K*EF(JP«1,K)) IF t(K.EQ.3).AND.(J.EQ.l))0 (I«JP«3)«D(I•JPt3)*LBAR(3)/WT 102 E(I»JP»K)*E(I»JP*l»K)*SIJ(I»K)#KREFiJPM*K) *1«0 C CALCULATE PRODUCT DI5TRI3JTION FOR SECTION ONE MD =NFEED(I) CO 103 1*1.NCOMP TClFEO<I)=FEEQU)*CLOU.l)>CVN(I,n tENOM* B U , N D , n * T O L P E O ( I ) > C ( I » N D i l ) * I F E E D U > » C V N U t i ) >*C(I vND«l)* 1CVNII.1) RONE(I)sD(I,NE.i)*TOLFEOU)4EII,NOtl)*(FEED(I)*CLO(I»l) )*C(I»ND,l> l*CL0(I,i) 103 RCNc(l)*RONE(I)/DENOM THETA*FTHETA(R0NE,L3AR(i).TOLFED) DO 130 1*1.NCOMP CLN(I,1) s TOLFSD<I>/U.O*RONE<I)*THETA> 130 CVONE(I,l)*THETA*RONE(I)»CLN(I,l) 202 TABLE C—IX (continued) C CALCULATE p r o d u c t d i s t r i b u t i o n r a t i o s f o r s e c t i o n t w o NA=NFEEI<2) DO 105 I * 1»NCOMP FELSTS*B(I.NA,2)*V(2)/DT*C(I»NA,2) RTWO(I) s FEDSTG*L(1)/L9AR(2) TCLFED(l)=CVONE(I.n*CVN(I,2) KTHSE£U)=FE0$TGoT0LFED(IUCVNU,2)«t<ltNA,2)*(RTW0(I)*1.0> DEN0MzDU»NA,2)*T0LFED<I>*E<I,NA,2)*CV0«E<I,l) 105 RTHREE(n=RTHREE(I)/OENON CALL BETTA(TOLFtD.LSAR (2)»DT»RTHREE,RTWO,BONE) 1*0 106 1*1* NCOMP DENOMsLBAR(2)«(1.0*8ONE(l)»RTHREE(n*BONE(2)«RTWOCI)) 106 CL0(I,3)=T0LFED(n*80NE(l)»RTHREE(I)*L(3)/DEN0ii C CALCULATE PRODUCT DISTRIBUTION RATIOS FOR SECTION THREE r.‘A«NFEED(3) DO 107 1*1,NCOMP FEDSTG*U(I*NA,3)*E(I,NA,3) RFIVE(I)*FEDST6*KREF{NA,3)*ALPHA(I)*VBAR{1)/LBAR(3) T0LFED(I)sCLN(I,l)*CL0(I,3) DENOM=B(I»NA,3)«T0LF£0(I)*C(I,NA,3)*CLN(Ifl) RFOUR(I)sFEDST6«TOLFEO(I)*C(I,NA,3)*CLO(I.3)*(RFIVE(I)*1.0) 107 RFOUR(l)*RFOURtI)/OENOM C CALCULATE THE DISTRIBUTION RATIOS IN THE SYSTEM PROOUCTS DC 108 1*1,NCOMP W0D(I)=l,0*RTHREE(I)«(1.0»RONE(I)*L(3)/LBAR(2)URTWOU) l,ENOM*RONE(l)<MRFOU8(I)+RFIVE(l)4l*0)♦RFOUR(I) WCD(1)*W00(I)/DEN0M 108 S0E(I)*RTHREE(I)«S/LBAR(2) C OVERALL MATERIAL BALANCE CALL BETTA <FEED,WT,DT,HOD.SOD,BETA) DO 109 1*1,NCOMP DIST(I)=FEED(I)/(1,0*BETA(1)*WOD(I)*BETA(2)*SOD(I)) W(I)=DIST(I)«NOD(I)«BETA(l) SI(I)*DIST(I)«S0D(|)«8ETA(2) 203 TABLE C-II (continued) CLN(I«2)*SI(I)»LBAR(2)/S 109 CL0U»3)*SI(I>«LC3)/S C CALC H A T E THE COMPONENT FLOWS IN THE LIQUID IN SECTION THREE STRIPPING SECTION NA=NFEED(3) ND*N(3) DO 113 JsNA«NO DO 113 Is11NCOMP 113CLN(ItJ«3)sW(I)»(0(ItJf3)«E(ItJ,3)> C CALCULATE THE COKPNNENT FLOWS IN SECTION 2 RECTIFYING SECTION ND=NFEED(2) PO 114 Js1«NO DO 114 Is1»NCOMP 114 CLia(I,J»2)sDIST(I)*(B(I.J»2)*V(2)^DT*C(I»J»2)> SUM = 0 SSUMsO C NORMALIZE THE COMPONENT FLOWS FROM SECTION 2 AND 3 FEED STAGES DO 115 Isl»NCOMP SUM=SUM*CLIQ<1»NAt3) 115 SSUrt=SSUM*CLIQ(I»ND,2> CO 112 I * 1f NCOMP CLIQ(I*NA«3):CLia(ItNA«3)4LBAR(3)^SUM 112 CLIO (I.ND»2)sCLIQ(I»ND»2)°LBAR(2)/5SUM C CALCULATE COMPONENT RATES TO SECTION ONE SUM = 0 DO 200 Ial,NCOMP CVN(I«l)sCLIQ(ItNA*3)«ALPHA(I)«KREF(NA«3)*VBAR(1)/LBAR(3) 200 5UM*SUM*CYN(I»1) DO 110 Isl,NCOMP CVN(I ,l)sCVN(I,l)«V3AR(l)/SUM CL0(I,1)3CLIQ(I«ND«2)»L(1)/LBAR(2) 110 TOLFED (I)*FEED(I)♦ CLO (I«1)♦CVN(111) C RECALCULATE SECTION ONE DISTRIBUTIIN RATIOS AND MATERIAL BALANCE NDs N F EEDU ) DO 116 Isl.NCOMP 204 TABLE C-II (continued) DENOM=B<I,ND.l)*TOlFED(I>*C(I,ND.l)<UFEED<I>*CVN<I.i>}*E<I,ND.l> 1«CVN(1'1) RONEn)=D(I.ND.t>*TOLFED(I)*E(I»NO,l)*<FEEDU>*CLO(I»l) >*C(I.ND,15 i«CL0(l,l) 116 RCNE(I)=ROHE(I)/DENOM THETA=FTHETA(R0NE.L3AR(1).TOLFED) DO 117 1 = 1. NCOMP C L N d ,1)=TOLFED(I)/(1.04RONE(I)*THETA) 117 CV0NE(1.1)=CIR(I«1)»R0N£(I)*THETA SUMs0 DO 118 I = 1.NCOMP 118 5UM=SUM*CVONE(I»1) C CALCULATE COMPONENTS FLOWS FROM SECTION ONt FEED STAGE SSUM=0 DO 119 1=1,NCOMP CVONE(I.1)=CV0N£(I,1)*V(1)/SUM CLIQ(I,ND,l)=(C(I.ND,l)»(D(I.ND,l)«CLN(I,i)*FEED(I)«E(I.ND,l)) 1 *B (I,ND,1>«E(I,ND.1)«CV0NE(1,1))/(C(I.ND.1)♦£(!,.ID.l)) 119 SSUM=SSUM*CLIG(I,ND»1> DO 120 1=1,NCOMP 120 C L N ( I ,ND.1)=CLIQ( I ,ND,1)*LBAR(1) /SSUM C CALCULATE COMPONENT FLOWS FROM SECTION ONE AND TWO STRIPPING STAiES DO 121 K=1,2 ND=NFEED(K)*1 NA = N (K) DO 121 JsNO.NA 5UM = 0 DO 122 1=1,NCOMP ClIQ(I,J,K) = (D(I»J,K>»CLN(I.K>*CLlQ<I»J-l,K)«E<l,J,lO )/E(I,J-l,K) 122 Slh=SuM*CLIQ(I,J,iO DO 121 1=1,NCOMP 121 CLI<3(I,J,A)=CL1Q(I,J,R)«LBAR(K)/SUM 205 TABLE C-II (continued) C CALCULATE OVERHEAD VAPOR FOR SECTION 3 AND BOTTOM VAPOR FORSECTION 2 SUMsC DO 123 1*1♦NCOMP CV0NE(I«3)aCL0(I«3)«CLN(Itl!*rf(I)fCVNU«i) IF (CVONE(I» 3) .LT.O.O)CVONE(I*3)=RFOUR(n*W(n 123 SUMsSUM*CVONE(I,3) CO 124 Is 1«NCOMP C VONE(I»3)=CV3NE(I*3)*V(3>/SUM 124 CVN(I,2)sCVONE(l,3)*VBAR(2)/V(3) C CALCULATE COMPONENT FLOWS FROM RECTIFYING STAGES OF SECTIONS ONE *N» THREE DO 125 Ksl,3,2 SUM = 0 WD*NFEED(K) DO 126 Isl«NC0MP CVAP(I)sALPHA(I)*KREF(ND.K)«V(K)/LBAR(A)«CLIQ(I»ND»K) 126 SUM=SUM*CVAP(I) TO 125 JCOUNTs2,ND JsND-JCOUNUl DO 127 Isl,NCOMP CVAP(!)sCVAP(I)*V(K)/SUM 127 CL I« C I,J,K) = <e<I,J,K)*CVONEU,K>*CUfJ»K>*CVAP(I) )/(l.0«CU« J,K)) SUM*0 DO 125 Isl,NCOMP CVAP (l)sCHQ(l, J»K>*ALPHAU>*KREF(JtlO*V(K>/L(M 125 SUM=SUM4CVAP(I) C RECALCULATE K-REFERENCE DO 126 Ks1,3 ND=N(K) DC 123 Js1•NO DC 129 Isl,NCOMP 129 X U>sCLia<l»J»Kl 128 KREF<J,K)s3UBPT(X) DO 933 Ksl,3 206 TABLE C-II (continued) MD = NU) EC 933 J=1»ND 933 KREF (J«K)as(KREF(JtK)**CHK(J«K) )/2. IF (M.iT. 5*OR«M.EQ*2O*OR»M.E! 5«3O.QR.rt.£Q.4O.OR»M.EQ.50.OR.M.E'1.6O. lCR.H.EQ.70«OR.M.EQ*3O.OR,M,EQ.90.OR«n.EQ.I00)CALL OUTPUT IF(M.GT.150)STOP C TEMPERATURE CHECK £0 131 Ksl,3 NC=N(K) DO 131 Jsl.ND TCHK = A E S ( K C H M J t K ) M R E F U » K > * l * ) 131 IF ( K H K . G T . 0.00001)30 TO 600 return END 207 208 r * 1 » — m «*> * • <o «© ~ - o U _J z o o * >■ uin • > » » <o I O a § o >• « r » u ui n to <o z ( S I OL ^ 0 o £ >- ^ a • » • » o. < CO X a. CL CL CL J3 %o o «D o X X X V m j * ■ * u> o o o o * * • 4 x Z z Ui Ui Ui Ui o C / > * » z z z Z m n » • X >- • » • > sO ♦ « • » • — X I I • - 4« • » * o « o < C 3: e •X » I I t l I t cr «£ » « / > <o o. M » < 4» « 4 x ^ » a ►- X 2: » m a. • —• :> Ui U ( / ) » O o « M « 0 ^ « ■ * CM 1 — _i • > uj U Ui •H • - 4 D < U u. or o z X X z CL < _l a £L ■ s .N <n \ ^ N \ V \ X X a o z z a UI Z f- i M ro m r- u Ui z m O CL Ui •X CM < f > UJ UI « - * >■ *c »C z » m Ui -1Ui X >- >- Ui u» o u • • U OT u> U Ui o Ui e • z <X u H- u. z z W O O O * x O O O * o fO o < W A z z Z «J «J c -J _1 -J u • - < CM CM f T l CD 00 —1CD CO CD D SD >D co u • —• * z •H n CM CM * 1 - ■ s .V. - j V v >V. \ V \ O CM CM CM CM • > • * CO CO 3 Z Z * -J Z 2» 22 z z z ►- i- i- 1 - X W < V IOi * —• CM CM -M ' W ' o o o o o o o o o -* <t «* ' * * * w- u Ui a: r *o —J •r y~ s r z X 2 7 2 2 -J Q =) a a O n Q O H H £ • - * *r <r jf 7 7 r 2 7 nr k o nor <t < -X <r *T * X < • " *■ - 4 r» r > C 7 n UJ 6 O o o o o o o -T ' . UUJ Ui UI u» t L f u» i t i ui a cr in u u U OC u u u u Ui u u. u. U. Ui a: X ac or or or or cr or z z Hftienh- r v j 210 REAl)(2f27) M JOB K l A0(2*J)L(1)tVBAR(l) TABLE C-III (continued) FD*=BUBPT(FEED) DO 103 K s l O NjsN(K) DO 103 Jsl.ND 103 RHF.F(J,R)aFDR CALL OUTPUT r e t u r n END to © VO 210 <n a U J U- z •» to » O I U a s N H J W E - f ui — ro IO ► * <o « — — c Ui — I z u o * >• ~ J n — <o a n - • » z c > <omu w W to z u « * CM cm ui <n * — >- OC • —*CO oc • ^ ac CD X o > z » » tn *■» ^ o lO CO >- in ro » •j •-« •«« » • > 1— *-* ♦ > ~ sO *• z <x w> » o. or CNJ ^ a u I — X i >- •-« >- >- CM N o « > < / ) • 1 QS o » —« .J (Ml « * • » u UJ «~i « - u 2» < £K Q. • —< -J x u Q z z IK ♦ CM W «o ♦ CO *T X X ^- 4 M w H v v. o \ ’ N. N _J V —* CO O o «_ Z Ui M w in r- Z ^4 >■ • + -J u f i C QC >• Z ^ >• w: • u- CM CM z • —• <r <X u C* U u u u c r o or I f CV _l ro n to X CO N Ui ^ O J M , o o o <* ♦ <1 w IK X w • r-« > >• z •J ^ ro -J — 1 -J sc cn « < • » .J -t I I > >- I I n t i I f IO •-4 A £ w X SO X - J M > ( Si : i x to I I ro ca » » — S V -J V » w w > * « - » » > CM 3 z z r > n 2 * * —/ -J ii r fsi I I I I —# ro rr o o » * k o o o c O I I *-» «r • o • »« M UI or X 2T m T7 J X sr fO Of or <o -4 z «> r r w r*~ ^ OJ •-« >• <x « -j 9 1 o z o o o z o o r> •u w < ■ - * > _i Cl ro 9- O —i *> V) u u ♦ u u u * > > j -1 >• Z O O u u u o o TABLE C-V SUBROUTINE OUTPUT SUBROUTINE OUTPUT CCMMON/BLOCU/ALPrlA (6) CCMMON/BL0C*2/CLIQ(6*50,3> ,KREF (50*3) *CV0NE <6*3) «CLN (6,3) * N F E E D (3) 1*N(3) ,LO)»V<3),VBAR(3)«L3AR(3)iCVN<6»3)*CL0<6»3) C3M10N/BL0CRN/NC0MP REAL L»LB4R,KREF C0MM0N/BL0CK3/FEED(S) C0MH0N/BL0CK4/THETA C0MMON/3LOCK5/DIST(S) ,SI (6)*DT*S COMMON/BLOCK6/ DETA1(2) COMMON/3LOCK8/BETA(2) COMMON/9LOCK7/W<6),WT COMMON/BLOCKE/XD.XS.XW*NJOB.M 1 FORMAT(1HI»10HNEW COLUMN*3X*16*/) 2 FORMAT <20HBTM LIQU10*SECT ION 2*6 (4X,E1**6)/) 4 F0RMAT(2X,I3,F15»7,4X*E12.6»4X*E12»6*4X»E12,6*4X»E12»6,4X,E12.6» 14X,E12.6) 8 FORMAT (/ * 22HSECTI ON I CALCULATICNS*/) 9 FORMAT</»22HSECTI0N 2 CALCULATIONS*/) 10 FORMATU3X.45HIMPROVED DISTILLATION PROCESS AT STEADY STATE ,/, 16 X * 60HFCR CONSTANT RELATIVE VOLATILITY AND CONCTANT MOLAL OVERFLOW , 2*/*13X.16HTERNARY MIXTURES ♦/) 11 FORMAT(15X»74HMETHOD FOURiFORCES OVERALL MATERIAL BALANCE— NO SUBS 1ECTION ITERATIONS ,//) 12 FORMAT(SHCOMPONENT*21X»1HA*15X*1HB*13X*1HC«15X*1H0.15X.1HE.15X. 11HF */) 13 FORMAT (9HFEED RATE*UX,6(4X*E12.6)./) 14 F0RMAT(19HRELATIVE VOLAT ILITY,2X«S <4 A,F12.6)*/) 15 FORMAT (/»38MLIQUID AND VA"OR RATES IN EACH SECTION*/*7HSECTION. 110X*lHVtl3X«lHL*llX*4HVBAR«9Xt4HL6AR*/) )6 FORMAT(I5*F15.7*F17*7.F17.7*F17.7*/) 17 FORMAT U1HTHE NUMBER OF STRIPPING STAGES IN SECTION, 13,1X,2HI3*1X, 1 14,/) 21L TABLE C-V (continued) 18 FORMAT(42HTHE NUMBER OF RECTIFY fN3 STASES IN SECT I O N ,13 * 1X liX'i*'/) 19 FORMAT (/*22HSECTI0N 3 CALCULATIONS »/> 20 FORMAT (13X*25HTRAY TO TRAY CALCULATIONS,/,5HSTAiSE»3X, U 2 H R E F , K-RATI0,3X,22HC0MP0NENT LIQUID FLOWS) 21 FORMAT(/»36HC0MP0NENT FLOW RATES IN THE PRODUCTS,/) 22 FORMAT (10HDISTILL ATE,10X,6(4X,E12,6)) 23 F0RMAT(20HINTERMEDIATE PR0DUCT»6(4X»E12»6)) 24 FORMAT(7HBOTTOMS,13X«6(4X«E12,6)) 25 FORMAT(12HC0NVER6ED IN,17) 26 F0RMAT(///20HINTERMEDIATE STREAMS,///20HOVHD VAPOR,SECTION 1,6(4X,E12»6)/) 27 F0RMATU9HREFLUX TO SECTION 1,1 X ,6 (4 X , E 12 .6)/) 28 F0RMATU8HVAP0R TO SECTION 1,2A,6(4X»E12.6)/) 29 FORMAT(20HBTM L IQUIJ,SECTION I,6<4X,E 12*6>/) 30 F0RMAT(19HREFLUX TO SECTION 3, 1 X,6<*X♦E 12.6)/) 31 FORMAT (20HOVHQ VAPOR,SECT I ON 3 * 6 <4X»E12«6>/) 32 F0RMAT(18HVAP0R TO SECTlON2,2X,6(‘» X . E U * 6 ) /) 33 FCRMAT«5HTHETA,4X,F10.5,/) 34 F0RMAT(19H3ETA(1) AND SETA (2),2F15,9 ) WRITE(3,1)NJOB WrtITE(3,10) WRITE(3 , 11) WRITE(3,12) WRITE(3,13)(FEED(I),1*1,NCOMP) WRITE(3,14)(ALPHA(I)»I = 1» NCQMP ) WRITE(3,15) WRITE (3,16)((R,V(K),L(K),VBAR(K),LBAR(R)) ,Kal,3) 00 190 K = l,3 N0*NFEED(K)-1 190 WRITE (3,ld)K,N0 DO 191 K*1,3 ND=N(X)-NFEED(K)+1 191 WRITE (3,17)K,ND 2HIS, 1 212 TABLE C-V (continued) 200 N0=N(1) WRITE (3*8) 201 WRITE (3*20) DO 300 J=1,ND 300 WRITE(3*4)(J'KREF(Jtl)• (CL IQ(I,J .1)* 1*1 *NCOMP)) WRITE(3,33)THETA N D = N (2) WRITE (3*9) WRITE (3*20) 00 301 J=1»ND 301 WRITE(3*4)(JtKREF(J»2) , (CL IQ(I»J,2)»I«1.NCOMP)) WRITE(3,34)BETA1 ND = N (3) WR I TE(3*19) WRITE(3 * 20) 00 302 JsltND 302 WRI IE(3,4)(JtKREF(J * 3 ) , (CL IQ(11J ,3)11*11NCOMP)) WRI (E (3,26) (CVONEd ,1) ,1* 1,NCOMP) WRITE (3,27) ( C L O U ,11 ,I*i,NCOMP) WRITE (3,26) (CVN(I,1),1*1,NCOMP) WRITE(3,29)(CLN(1,i),1*1,NCOMP) WRITE (3,32) (CVN(I,2),I=1,NC0MP) WRITE(3,2)(CLN(I,2),1*1,NCOMP) WRITE(3,30)(CL0(I»3),1*1,NCOMP) WRITE (3,31) ( C V O N E d , 3) ,I*1,NC0MP) WRITE(3«21) WRITE (3,22) (DISTd) ,1 = 1,NCOMP) WRITE(3»23)(SI(I),1*1,NCOMP) WRITE(3,24)(W(I),1=1,NCOMP) WRITE (3,34)BETA 187 WRITE(3.25)M r e t u r n END 213 TABLE C-VI SUBROUTINE MOLFRA 5JMK0UT1NE MOLFRA CO i 'IMON/BL OCKN/NCOMP CoMnoN/DLOCP2/Cl IQ<6*50,3),FREF(50»3)»tV0NE<6*3),CLN (6»3>,NFEED(3) 1« K J ) i l (3) »V (3) ,VBAR (3) »LDAR (3) »CVN (6*3) •CLO(6«3) RlAL L,tr-AP.« KkEF CJMMON/BLOCR7/W (6),WT Cni1IHiN/blorK5/DlST (6) f SI (6) «DT«S 1 FORMAT(////,lOX,53HCOHPONENT MOLE FRACTIONS IN THE CORRESPONDING $ 4Ti<EftMS*//) 0 7 100 Ke1« 3 NlsN (K) 51 . 111=0 SJML=0 SJMVso SSUMLbO DO 110 I si 1 * NCOMP S-JMeSUM+CVONEdtK) SiJMl=SUKl+ClO(I,K) SJMVb SUMV*CVN(I*R) 110 SSIJMLa5SUML*CLN(I»R> DO 102 I = 1 » NCOMP CVOHE(I.R)=CVONE(I*R)/SUM ClOU,F>=ClU<I,K)/SUML C i/H (I »K) sCVN (I »K) /SUMV 102 CLN ( I ,K>eCLN(I,K)/SSUML 00 100 J=lfND SJM = 0 » 0 DO 101 IsltNCOMP 101 SjM=SUM*CL!Q(I»JtK) DO 100 I=1»NC0MP 100 C L K K I « J«k) «CL IQ (I*J«K) / SUM 214 TABLE C-VI (continued) SiJMUcO S-JMbsO SIJMWsQ Do 103 1=1.NCOMP 5'imJ = $UKD*DlST(n SUMS=SUMS*S1(I) 10^ SUMW = SUMW<.W(I) DO 104 1=1.NCOMP L ) I ST (I) = I> I ST (I) /SUMD SI (1)=S1 ( D /SUM S 104 W(J)=W(I)/SUMW WRITE (3*1) call output DO 103 1 = 1,NCOMP CVN(!.tl)=CVN(!,l)oVBAR(l) CLU(I.1>=CL0U,1>*L<1) 103 CVN(I,?)=CVN(I,?)*VpAR<?) return END N> in TABLE C-VII FUNCTION FTHETA THETA-M'JLTIPLYING FACTOR SECTION 1 F J N C T ion F THETA(X C3M*0'I/UL JCKN/NCOMP 3IMtrISI0M X (6) »B ( 6) K THETmsQ.J 262 G»RIMt*0.0 Gs-A DJ 260 IaUNCOMP D£N0M = l«0*FTHETAa X U ) GaG*B (1 ) /DENOfl 260 GPR IH£aGPRIi1E^d (I )*X (I) / (Dt NOM*«2) OtL TaG/GPRIME FTHETAs FTHETA+DELT IF(DELT-0.0000000 01)26 It 261 *262 261 RETURN END (2*2)9QD0o3N09-(c4T)9 G9G &0M 1 9 = T n 3 G U • 2) 9C9fl* (24 T)fiQ9G-<2*2KiC9a* (I *1)909(1 = 91130 (2UON:J0<»(n03H0C)/( nMOiSe( 1)0332*2) S09Q=(2*2) CG9Q IM • — im ► — o o o r; Cl o a o ( / ) a O o o o z Cl *-• Cl CP z o a n 41 b > . m Ci Cl o r o D O C o o o o o I I — 4 “11 O m o I I Cl ► —4 o 6 o o a m —4 z z >1 2u D z TC 4 P4 z -I G o a 3 2 C a a : a o 2 a . rn c c x> n o r o 03 • — • — 4 o Z m x » os m 2 r o 2 ◦ o i i 2 2 o o I I 1 i i GJ z O Z r\» ro I I o ro > — •* - • « • * > *>4 ► ■ 4»-4 c . 1 f — i ro C O i . o « « « ii • — »CD O I I I I I I I I I I ii C3 o £ * - *• —4 « X JC ►- N h - a I I “)2 1 I — • « * « p W ►— c. ►- o I I 1 1 r o o G M W W n a a: r n *-* ♦ * ii (1 • * * ■ # 4 -* z f “ I I I I I I z n o ♦ • o z CO CJ z I I ro ro ro • O o a o o x ♦ -n O ♦ o -H h o o o ro n Q O O 3 O m ♦ o o o G o • m 7* a o o im 2 n m O 2 ac 2 O ro m z a: o> ac * * - — m C3 a~ -3 « * > . * - • a X a 6 o • o 4-4 z f \ > t —* * — IM * o o ro a- r> • • « M ♦ * * o o ►* IM *- W V < J % CD CP 2 V w •v o o rn m P ♦ 4 I O r o * 2 4 -n -4 —♦ ■n > n m 2 n Ol j > o n n pi 2 O o o • » * z rn m rn O 2 « ■ « » z ro • — 4-4 C J D O 2 M K4 n W» o a » ro w o IP* M « , # X. c/ 1 * 6 * D — t / > o —I -I • o o O X X » A S'*- O ^ N. x e e ^ m n — x x w 0 0 — Z 3 \ * —• * —■ O m r * o o x a» ■ m X 3 > m —t » N a o x n c « * ro « o —* o X X r » o LIZ TABLE C-VIII SUBROUTINE BETTA SUBKOUTINE BETTA<FE1U 10,W ♦DTOW,STOW,BETA) TABLE C-VIII (continued) DElT2 = G0NE*0GU3(2»l)-GTW0<*DGDB(l»l) OBCTA1*DELT1/OELTS D3ETA2=0ELT2/DELTS IF (DBETA 1-0,0000001) 7»6t*j 7 IF(UB£TA1*0.0000001)5,6,6 6 IF (DBETA2-0#0000001)8,9*5 6 IT (L)DCTA2*0»9000001>5»9»9 5 IF(BETA(l)*DBCTAl)11*lltll 10 BETA(1)=0.5*BF.TA(1) GO TO 12 11 8ETA(1)=BETA(1)+DBETA1 12 IF(BETA(2>*DBETA2)13»14.14 13 0ETA(2)sBETA(2)«O,5 GO TO 15 14 6ETA(2)=BETA(2)*DBETA2 17 GO TO 15 9 BETA(l)sBETA(l)*DBETAl BETA(2)=BETA(2)*DBETA2 20 RETURN END 218 219 TABLE C-IX FUNCTION BUBPT f u n c t i o n e u b p m x ) C0MM0.N/6L0CX1/ALPHA (6) c o m m o n / b l q c k n / n c o m ? D I ^ t N S I G N X ( 6 ) S'JM-0.0 s u m l = o . o CO 500 I * I« NCOMP 5UMaSUM>X(I) 500 SUMLsSUML^X(I)«ALPHA(!) PU3P TsSUM/SUilL RETURN END 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 220 TABLE C-X INPUT FORMAT DATA FORMAT NUMBER OF COMPONENTS IN MIXTURE* 13 RELATIVE VOLATILITIES OF THE COMPONENTS 6F10.5 COMPONENT FEED RATES 6F10.5 ASSUMED COMPOSITION OF COLUMN 1 REFLUX 6F10.5 ASSUMED COMPOSITION OF BOTTOM VAPOR TO COLUMN 1 6F10.5 ASSUMED COMPOSITION OF BOTTOM VAPOR TO COLUMN 2 6F10.5 OVERHEAD PRODUCT RATE F10.5 INTERMEDIATE PRODUCT RATE F10.5 REFLUX RATE TO COLUMN 2 F10.5 THERMAL CONDITION OF THE FEED F10.5 NUMBER OF STAGES IN EACH COLUMN 313 FEED STAGE NUMBER IN EACH COLUMN 313 COLUMN IDENTIFICATION NUMBER 16 REFLUX RATE TO COLUMN 1 F10.5 VAPOR RATE TO STRIPPING SECTION OF COLUMN 1 F10.5 *COMPUTER PROGRAM IS LIMITED TO 6 COMPONENTS TABLE C-XI Sample Input 6 6.0 5.0 4.0 3.P 0.15 0.15 0.25 0.15 . 6 0.3 0.1 0.0 .0 0.05 0.5 0.4 0.0 0.0 0.0 0.1 0.3 0.4 0.7 1.0 10 10 10 5 5 5 701 0.1 0.487304 10 10 10 2.0 1.0 0.6 0*15 0.15 0.15 0.0 0.0 0.05 0.0 0*3 0*6 $ TABLE C-XII. S a m p l e Output NEW COLUMN 701 IMPPCVED DISTILLATION process at steady state FOR CONSTANT RELATIVE VOLATILITY AND CONSTANT MCLAL OVtRFLOW TERNARY MIXTURES METHOD FOUR,FORCES OVERALL MATERIAL BALANCE--NO SUBSECTION ITERATIONS COMPONENT FEED RATE RELATIVE VOLATILITY 1150000E 00 6,000000 B • 15000CE 00 5.000000 •25000CE 00 4,000000 .150000E 00 3,000000 >150000E 00 2,000000 LIQUID AND VAPOR RATES IN FACH SECTION SECTION V L 1 .4873040 1.0000000 .1000000 ,7000000 VBAR .4873040 ,5126960 3 ^ 1 5126960 .2000000 1.0000000 THE NUMBER OF RECTIFYING STAGES IN SECTION 1 IS 4 THE NUMBER OF RECTIFYING stages in section 2 IS 4 THE NUMBER OF rectifying stages in section 3 IS 4 THE NUMBER OF STRIPPING STAGF5 in SECTION 1 IS 6 THE NUMBER OF STRIPPING STAGES IN SECTION 2 IS 6 THE NUMBER OF STRIPPING STAGES IN SECTION 3 IS 6 LOAR 1.1000000 .6000000 1.3000000 r .150000E 00 1.000000 222 TABLE C-XII (continued) SECTION 1 CALCULATIONS TRAY TO TRAY CALCULATIONS STAGE REE. K-P.ATIO COMPONENT LIQUID FLOWS I .2578353 •164427E-01 • 173738E—01 .297914E- 01 .167410E-01 .132793E-01 .637194E-02 2 .2592937 .155760E-01 .160903E-01 «272?13E- 01 .161738E-01 .1502C5E-01 .989802E-02 3 .2758029 •151013E-01 .155199E-01 •261246E* 01 .156518E-01 .153142E-01 .122882E-01 A .2706826 -148338E-01 .152209E-01 .255574E- 01 .15J07CE-01 . 152094E-01 . 136715E-01 3 .2820632 •161A26E 00 .165527E 00 .277604E 00 , 166C24E 00 .16546 1E 00 .163956E 00 6 .2830191 •156362E 00 .165467E 00 .280 ^66E 00 . 167382E 00 .163917E 00 .1641042 00 7 .28**821 .14S6P9E 00 .16446AE OC , H85*98£ 00 .170053E 00 .166815E 00 .164360E 00 6 .2872857 -127515E OC .U1336E 00 .292102E 00 .17544eE 00 .163772 7 00 •16AB25E 00 n .2921629 .120670E 00 •153806E 00 .4999636 00 • 16640CE 00 .173503E 00 • 163659E 00 10 .301*191 •95947IE-01 • 13761ie 00 ,30 5*696 OC .2C8029E 00 .1859COE 00 .166824E 00 THETA 1.00000 SECTION 2 CAICUIATICNS TRAY TO TRAY CALCULATIONS STAGF REF . H-RATIO COMPONENT LIQUID FLOWS 1 .2079A13 •213420E 00 . 19834AE 00 .52961 IE 00 .500939E-01 •623664E-02 ,9320096-04 2 •215J175 .179134E 00 .186A29E 00 ,253362E 00 .667987E-01 .119484E-01 .330000E-03 3 .2221008 .154603F 00 • 173224E 00 .2641BOE 00 .863483E-01 •20581*E-01 .106625E-02 A .2269751 . 137460E 00 •159946E CO •Z64313E 00 .10 1636E 00 .33159CE-01 .3284986-02 5 .2371529 •106533F 00 .12365AE 00 .419120E 00 .970094E-01 .433527E-01 •833200E-02 6 • 240 7220 .935993E-01 .12234fE CO .228293E 00 . 1C2651E 00 .445339E-01 .637669E-02 7 .24*7376 • 617561E-01 .117175E 00 ,2355i*E 00 •110474E 00 .466276E-01 .8452996-02 8 .2495387 .7092616-01 •110070E 00 ,439*496 00 • 120267E 00 .506401E*01 •864825E-02 9 .2538261 •609260E-01 .100865E 00 .538064E, 00 •13Z2A3E 00 .585733E-01 •932905E-02 10 .2631611 .514254E-01 .891886E-01 .426156E 00 .145206E 00 .740181E-01 . 120054E-01 BETA(1) AND BE TA(2 > ,999990317 ,999991652 K> KJ U) TABL£ C-XII (continued) SECTION 3 CALCULATIONS trat to that calculations STAGE REF. R-RATIO COMPONENT LIQUID FLOWS 1 .26J 3979 2 .2925334 3 .3016251 4 .3102556 5 .3191417 6 .3392127 7 .3653627 6 .4020685 9 .4613731 10 .5572357 •139576E-C1 .1271C6E-01 .120662E-01 • 115961E-01 •7Z6516E-01 .460660E-01 .27P035E-01 • K2447E-01 .644C47E-02 .530313E-03 .247766E-01 ..225126E-01 .212466E-01 •203396E-01 • 12 7 C f 3 E 00 .957496E-01 .665997E-01 .4160766-01 .222296E-0! .216092E-02 .682Z95E-01 .63C247E-01 .594422E-01 .566625E-01 •352533E 00 .322454E 00 .272621E 00 .207144E 00 .134664E 00 •159748E-01 .515990E-01 .510562E-01 .4S2832E-01 .471692E-01 .292467E 00 .322766E 00 .335604E OC .318329E 00 •26160CE 00 .398032E-01 .3395406-01 .386501E-C1 .40S452E-01 .409632E-01 .259318E 00 .306158E 00 •36T579E 00 •432675E 00 .469332E 00 .995426E-01 .7480906-02 . 116357E-01 .17111OE-O1 .2326786-01 • 195947E 00 .2067896 00 •230568E 00 •285979E 00 .405723E 00 .141988E 00 intermediate STREAMS OVHD VAPOR•SECT 1 ON 1 .123956E OC . 109146E 00 •149725E 00 •631024E-01 •333693E-01 .800598E-02 REFLUX TO SECTION 1 .177554E-01 .209423E-01 .3*5199E-01 •161662E-01 .722544E-C2 .13S867E-02 VAPOR TO SECTION 1 •521A76E-01 .760145E-C1 .168693E 00 •104963E 00 .6204416-01 .2344116-01 BTM LIQUIDtSECTION 1 .959471E-01 • 137611E 00 .3C5A69E 00 .2C6C29E OC .165900E 00 .166824E 00 VAPOR TO SECTION*. •604108E-01 .693655E-01 .196673E 00 • 111665E 00 .4898626-01 .539644E-02 BTM LIQUIDtSECTION 2 .514254E-01 .891886E-01 .226156E 00 . 145206E 00 •T40161E-0V .1200546-01 REFLUX TO SECTION 3 .171418C-01 .297296E-01 .76^521E-01 •464020E-01 .2467276-01 •400181E-02 OVHD VAPOR.SECTION 3 •604108E-01 .8936536-01 .196B73E 00 « 1 11665E 00 •489862E-01 .539644E-0? to to TABLE C-XII (continued) COMPONENT flow rates in the products distillate intermediate product BOTTOMS BET a < 1) AND BET A(2) CONVERSED IN 22 .113186E 00 .3«2638E-01 •530313E-03 .999997J82 .883799E-01 ,594«92E-01 .216092E-02 .99999C8CA .819210E-01 .133927E-01 .11I195E-02 .83058AE-05 .152104E 00 *968040E*01 .493454E-01 .800363E-02 .159748E-01 .398032E-01 .995428E-01 .141988E 00 IO to UI 226 conditions and stages that would be used for practical problems, this problem did not occur. Typical convergence of a calculation is illustrated in Figure C-6. EQUILIBRIUM RATIO FOR 227 Pig. C-6 Typical convergence of the computer calculations. 228 Literature Cited 1. Friday, J. R., and B. D. Smith, "An Analysis of the Equilibrium Stage Separations Problem - Formulation and Convergence," A.I.Ch.E. Journal, 10, 698-707 (1964). 2. Hardy, B. W., S. L. Sullivan, Jr., C. O. Holland and H. L. Bauni, "Figure Separations This New Way: Part 5-A Convergence Method for Absorbers," Petroleum Refiner, 40 (9), 237-248 (Sept. 1961). 3. Holland, C. D., Multicomponent Distillation. Prentice Hall, Inc., New Jersey, 1963. 4. Lyster, W. N., S. L. Sullivan, Jr., O. S. Billingsley and C. D. Holland, "Figure Distillation This New Way* Part 1-New Convergence Method Will Handle Many Cases," Petroleum Refiner. 38 (6), 221-230 (June 1959). 5. Thiele, E. W., and R. L. Geddes, "Computation of Dis tillation Apparatus for Hydrocarbon Mixtures," Ind. Ena. Chem.. 25. 289-295 (1933). APPENDIX D MINIMUM REFLUX CONDITIONS IN THERMALLY-COUPLED DISTILLATION SYSTEMS Nomenclature Introduction General Aspects of Minimum Reflux Use of Equations Based on Constant Molal Overflow and Constant Relative Volatility Underwood's Equat ions Rectifying Section with Infinite Stages Stripping Section with Infinite Stages Column with Rectifying and Stripping Sections with Infinite Stages Minimum Reflux in Thermally Coupled Systems Literature Cited 229 APPENDIX D Nomenclature constants defined by equations (D-20) and (D-21). net upward flow In a section divided by the vapor rate, for a rectifying section d^ and for a stripping section - wA ^/V^. distillate or net overhead rate from column k. molal rate of component 1 In the distillate or overhead product from column k. molal feed rate to column k. molal feed rate of component 1 to column k. number of components In a mixture. molal liquid rate of component 1 from stage j of column k. molal liquid rate In the rectifying section of column k under constant molal overflow condi- tIons. molal liquid rate In the stripping section of column k under constant molal overflow condl- tIons. ratio of liquid to vapor flow, L^/V^ or L^/V^. thermal condition parameter for the feed to column k, q^ ■ ( molal rate of the intermediate product in the thermally coupled system. 231 molal rate of component l in the Intermediate product of the thermally-coupled system. molal rate of component 1 In the vapor from stage j of column k. molal vapor rate In the rectifying section of column k under constant molal overflow condi- tIons. molal vapor rate In the stripping section of column k under constant molal overflow condl- tIons. molal rate of component 1 In the bottoms from column k. bottoms or net downward flow from the stripping section of column k. mole fraction of component 1 in the liquid on stage j of column k. mole fraction of component 1 in the vapor from stage j of column k. mole fraction in the feed to column k, f^ ^/F^- relative volatility of component i. change in liquid rate across the feed stage of column k, - L^. common root to Underwood's equations for column k. root to Underwood's equations in the rectifying section of column k. root to Underwood' equations in the stripping section of column k. 232 NOTE - When the subscript denoting the column is unnecessary for identification, it is left off. Special Subscripts f - refers to the feed stage. H - refers to the heaviest component appearing in the overhead product. I - refers to the heaviest component of the mixture. L - refers to the lightest component appearing in the bottoms product. N - refers to the bottoms stage of a section. N+l - refers to the vapor entering the bottoms stage of a section. RP - refers to the zone of constant composition in the rectifying section. s - indicates the product is of the system and not an intermediate stream. S - refers to the intermediate product in the thermally-coupled system. SP - refers to the zone of constant composition in the stripping section. 0 - refers to reflux entering the top stage of a sect ion. 233 Introduct ion In general, the reflux requirement of a distilla tion system can be reduced by Increasing the number of stages In the system. In the limit the number of stages becomes Infinite and a minimum amount of reflux Is re quired. It follows that a necessary condition for minimum reflux Is an Infinite number of equilibrium stages. This fact precludes the actual operation of a column with the minimum reflux, and therefore, minimum reflux Is purely a hypothetical limiting case. Nevertheless, since this case gives the minimum reflux, It Is an Important guide and supplies Important Information for the design of actual distillation systems. General Aspects of Minimum Reflux An Infinite number of stages Is a necessary condi tion for minimum reflux operation, but It Is not a suffi cient condition, since, if only one section has an infi nite number of stages, the reflux may be further reduced by adding stages to other sections of the system. In some cases, however, minimum reflux conditions can be found by considering an infinite number of stages in only one 234 section of the column, but this occurs only in cases of unusual volatility behavior of the components present. In all cases, adding stages either decreases the reflux requirement or in the limit makes no change. There fore, it seems reasonable, when considering the minimum reflux requirement, to include infinite numbers of stages in all sections of a distillation system. For the case of a conventional single feed two- product column with a total condenser, in which the pres sures above all stages the reboiler and the condenser are specified and heat losses from each stage are zero, there are 1 + 7 independent variables. In the design of such a distillation system, generally, the feed will be complete ly specified by I + 2 independent variables; the tempera ture, pressure, composition (I - 1 mole fractions), and the rate. An additional variable is specified by the condition of the distillate, usually considered to be a bubble point liquid. This set of specifications leaves four independent conditions to be assigned. Two of these independent variables are assigned to the distribution of the key components and the remaining two are fixed by re quiring infinite numbers of stages in both the rectifying 235 and stripping sections of the column, Under these condi tions, the system will generally be completed defined and the minimum reflux can be calculated. The one exception to this type of consideration of minimum reflux occurs if one or both of the key components is specified to be com pletely separation, a condition which requires an infinite number of stages irrespective of reflux. Under these con ditions, the minimum reflux can be determined by consider ing the minimum reflux for the case in which a finite amount of this key component appears in both products (a case which takes a finite number of stages at reflux ratios greater than the minimum), and then considering the limit ing minimum reflux requirement as this finite amount which distributes approaches zero. By means of these arguments, sufficient conditions for minimum reflux in a conventional single feed two-product column separating a given feed have been determined to be infinite numbers of stages both above and below the feed entry, and finite specified distribtuions of the two key components between the two products. For thermally coupled systems of distillation columns, the greater number of independent variables tends 236 to complicate the situation. Consider the three column system shown in Figure D-l. For this type of system, there are 1-4-14 independent variables (cases where the pressures above all stages are fixed and the heat leaks are zero). If a set of specifications analogous to that de scribed for a conventioanl single feed two product column is used as the set of design variables, the system will not be completed defined and further independent variables re main. For instance, the feed, the condition of the dis tillate, the number of stages (infinite) in each section and a mole fraction or an amount for each of the three key components (one for each product stream) specify a total of I + 12 independent variables. It follows that there sure still two degrees of freedom. These two unspecified in dependent variables may be assigned to the distribution of liquid and vapor between the three columns of the system. Then, the minimum reflux is determined as the minimum in the plane described by these two independent variables. If constant molal overflow and constant relative volatility are assumed, the minimum reflux for the system is independ ent of the distribution of liquid and vapor between the columns. For this case, the system is not uniquely RECTIFYING FEED LIQUID STRIPPING RECTIFYING LIQUID * I FEED FEED COLUMN 1 STRIPPING FEED LIQUID STRIPPING LIQUID | VAPOR FLOW THE BOTTOM STAGE OF ( LIQUID FLOW COLUMN 3 237 CONDENSER OVERHEAD * PRODUCT ’ s t **** COLUMN 2 INTERMEDIATE PRODUCT COLUMN 3 BOTTOM PROOUCT Pig. D-l Schematic diagram of the thermally- coupled distillation system. 238 defined, and there are an infinite number of possible solutions. If constant molal overflow and constant rela tive volatility are not assumed, a unique solution probably exists. However, this more general case leads to a model which would be very difficult to solve numerically. In many cases it may be advantageous to use one or both of these two remaining independent variables for further specifications on the distribution of key compo nents. For instance, the amounts of a certain component in both the distillate and intermediate product may be specified. If five independent specifications are imposed on the distributions of the key components in the products, the system becomes completely and uniquely defined for all cases, and the minimum reflux can then be calculated. How ever, care has to be taken that all the specified values are consistent with the equilibrium stage model and the relative volatilities of the components, Furthermore, all specifications have to be independent of each other. Use of Equations Based on Constant Molal Overflow and Constant Relative Volatility Generally, nearly all actual distillations involve mixtures which do not have constant relative volatilities 239 and any set of equations based on these assumptions has to be considered as only an approximate model. However, due to the complexities ot real distilla tion systems, equations based on this simpler, more appoxi- mate model are extensively used as approximate guides for the limiting cases of total reflux and minimum reflux. Since the behavior of systems with regular volatilities follows completely that of the model based on constant molal overflow and constant relative volatility, at least on a qualitative fashion, this type of approximate model is also useful in the study of the general operation of distillation systems. Underwood's Equations Underwood (1-5) has developed analytical equations, based on the assumptions of constant molal overflow and constant relative volatility, which describe the distilla tion of multicomponent mixtures without resorting to the "stage to stage" type of calculation. These equations re duce to a relatively sinple set of equations for the case of minimum reflux, and therefore, are used extensively for estimating this quantity. 240 The Underwood equations are general and apply to any column section and their application to simple single feed, two product distillations has been well presented by Dr. Underwood. However, the application to more compli cated systems is considerably more complex and requires a basic understanding of the equations. For this reason, a brief derivation of the equations is present first; then they are used in the consideration of minimum reflux in thermally-coupled distillation systems. The compositions on any two adjacent stages in a distillation column are related by (see Figure D-2): i=*i,l Where: 1. stage 0 is above stage 1; 2. m is equal to L/V, the internal reflux ratio; 3. b^ is the net flow of component i out of the top of section divided by the vapor rate. Underwood found that by defining a set of para meters, 0j , where 0j are the roots of (D-l) (D-2) i«l,I 241 * T vy^i. - C.o STAGE 1 VAPOR FLOW, V LIQUID FLOW 0*-1 n n *v>- *4 \ n = ^ xi.« STAGE Xy%+i 1 “i *VH1 Fig. D-2 Stag* calculations by Undarwood's squat ions. 242 the compositions of the various stages of a column can be related by analytical equations without referring to "stage to stage" calculations. In general, if there are I components, there are I such roots 0j to equation (D-2). Underwood combined equations (D-l) and (D-2) to get °*i xi.P = £i i=l.I <* - 0J ^ <*i - 0j m £ qCl xi#1 i-l,I 1=1*1 £ <*i (D-3) Writing equation (D-3) twice, with different values of 0, and dividing y l r t t y a . xy . Z—> Qtj - 0j = 10j\ Z-. Pi - Joj E «i xi.O V"1 *1 xi.l «i - *M <*i - (D-4) This equation can be applied successively to adjacent stages and the composition n stages below the original stage will satisfy Rearranging equation (D-5) y <^i xi,o i-iTx QCj "Vj xi ,n* 0 i=l, I «i - 0j J (D-6) For the case of I components present, there are I values of 0 and it follows that equation (D-5) or (D-6) can be written with I - 1 different independent combina tions of 0. and 0U. The set of I - 1 equations derived produces a set of I linear equations and I unknowns, x ^ n, which can be solved to determine the composition at any stage in the column section. If n is a negative integer, the composition calculated corresponds to a stage n stages above stage 0, or if n is positive, the composition corres ponds to a stage n stages below stage 0. J from equation (D-6) and the fact that i“l, I 244 Rectifying Section with Infinite Stages A rectifying section is defined as a column section in which the net flow is in the direction of vapor flow. If the vapor from the top stage is V and the liquid reflux to this stage is L, then the net flow is D = V - L (D-7) and the net flow for each component is di = vi,1 “ xi,0 (D_8) Then, it follows that the constants used in equations (D-l) and (D-2) are defined by di b± - ^ (D-9) and equation (D-2) then becomes Y -* 1 = V (D-10) ZL «i - 0j i*l, I For a rectifying section, the roots of equation (D-10) cure related to the relative volatilities of the components in the overhead vapor by ^ 0^ > &2 ^ 02 • • • • ^ ^ ^ 0 245 where the components are arranged in order of decreasing volatility and 0CH is the relative volatility of the heavi est component in the overhead vapor. Starting at the top stage and proceeding an infinite nuntoer of stages down the column, a zone of constant composition called the rec tifying pinch zone will be reached. The composition of this zone can be calculated by applying equation (D-5). Using in each of the equations, the set becomes, with a slight rearrangement, Z «i xi.RP oo V *i " - / M L. <*i xi.O «i - E <*1 xltRP l^j I ^ QCj xlto «i - 0H / Ctj - 0H 00 and since 0H < 0^ and they are both positive (0^/0j) -+0t therefore E i-l,H X- : 4 r > * 0 (D-ll) °ti ~ 0j for all 0j except 0R. This set of equations and the summation of the mole frac tions then defines the composition of the constant compo sition zone, x^ gp. 246 If there are components present in the feed material which are not present on the top stage, the composition of the feed stage will be different from that of the zone of constant composition and the feed stage will be separated from the zone of constant composition by an infinite number of stages. Starting with the feed stage, the composition on any stage above the feed stage can be described by For the case of I components present on the feed stage and H of these appearing in the distillate, the composition on the feed stage will satisfy the set of H-l equations described by (0-12) and the summation of mole fraction. This is then not sufficient to fix the feed and as n -n ♦ 0 and then (D-12) i=l,I for all 0. except 247 stage composition, since there are I-H more unknowns than equations. A stripping section is defined as a column section in which the net flow is in the direction of the liquid flow. If the liquid from the bottoms stage of the section is L and the vapor to this stage is V, the net flow out of the bottoms of the section is Then, it follows that the constants b^ used in equa tions (D-l) and (D-2) applied to the stripping section are Stripping Section with Infinite Stages W = L - V (D-13) and the net flow for each component is wi = 1i,N “ vi,N+l (D-14) (D-15) and substituting into equation (D-2) V (D-16) i-l.I 248 The symbol, will be used in the stripping sec tion for the roots of equation (D-2) or equation (D-16) to eliminate confusion with the roots for the rectifying section. These roots, , are related to the relative volatility of the components in the bottoms liquid by where the components are arranged in order of decreasing volatility and OL L is the volatility of the lightest present in the bottom liquid. Using the same type of analysis used in the dis cussion of the rectifying section, we find the composition of the stripping section zone of constant composition is given by i«L,I for all y'j except ^ L. and, furthermore, the composition on the feed stage satisfies * l+l> °<-l*\> ... >1' I > « I (D-17) (D-18) i«l,I for all ^ except As in the case of the rectifying section, the set of equa tions described by (0-17) and the summation of mole frac tions are sufficient to determine the composition of the constant composition zone; however, the set described by (D-16) is not sufficient to describe the feed stage compo sition. Column with Rectifying and Stripping Sections with Infinite Stages At minimum reflux conditions infinite numbers of stages cure required in both the rectifying and stripping sections. Under these conditions, the composition on the feed stage will be described by the set of equations made up from the sets described by equations (D-ll) and (0-12), and the fact that the summation of mole fractions is one. Consider a distillation of a feed mixture containing I components. If the components I through H appear in the overhead vapor and the components L through I appear in the bottoms, then there are H - 1 equations derived from equation (D-12) and I - L equations derived from (D-18) describing the feed stage composit ion of mole fractions. 250 This results in a total of H-l + I-L equations of the form - 0 (D-19) i=l /1 where k varies from 1 to H-l + i-L. For a non-trivial solution of the I values of F to exist (trivial solu tion xi F * 0) only I-l of the equations (D-19) can be in dependent. Therefore, H-l + I-L -(I-l) or H-L of these equations must be linearly dependent on the others. Inas much as the coefficients of (D-19) are calculated by Ot j a± ±--- for U X i (H-l) (D-20) <*i - 0k and a. v - — ---- jj— i------ for H / k ^ (H-l+I-L) (D-21) <Xi - ^(L-H-l+k) Furthermore *1 > ^ i > a 2> *2> ----------------- 0 and V'l-^ L > ^ L+l y**- L+l > • • • ^ a I-l> I It follows that for the H - L equations to be linearly dependent that 251 Wl+2 “ 0L+1 “ °2 (D-22) i^H * 0H-l " °H-L where 0^ are the roots coramon to both the rectifying and stripping sections. Underwood has shown that if a root is common to both the rectifying and stripping sections, it is also a root of where q is the ratio of the change in liquid rate across the feed stage to the feed rate to the stage. This allows the roots common to both the rectifying and stripping sections at minimum reflux to be calculated from the known feed composition and the thermal condition of the feed as described by q. Then, the minimum reflux is determined from either equation (D-10) or equation (D-16) and the desired distribution of the key components. If the condition arises in a column that only one component distributes between the overhead product and the (D-23) i-l. I 252 bottoms, no common roots exist and equation (D-23) has no meaning in this case. For this case minimum reflux condi tions are not satisfied and one independent variable re mains to be fixed. This variable could be the reflux. However, if desired, the minimum reflux conditions can be obtained for this type of separation by considering the distribution of an infinitesimal amount of a second key component to the product in which it does not occur. Under these conditions, two components distribute (even if only by an infinitesimal amount) and the Underwood equations apply. Minimum Reflux in Thermally Coupled Systems Consider the thermally-coupled distillation system described in Figure D-l. This system has been divided into three columns each having rectifying and stripping sections. At the limiting case of minimum reflux, each section of the system is considered to have an infinite number of equilibrium stages. If the net flows of the various components present and the characteristic roots are known for each section, then equation (D-2) can be used to determine the vapor 25$ rates in every section of the system. Applying equation (D-2) to rectifying section of column 1, I - » Vl (D-24) i-l, I < \ r ^-i “ *J.l and applying equation (D-2) to the stripping section of this column Z * 1 ( 1 j ' - H'1 - -V, (D-25) (*i - 'rj,! x 1*1,1 The minimum reflux condition for this column is determined by the roots common to both the rectifying and stripping sections of this column and defined by equation (D-23). Those roots of equation (D-23) which are the characteristic roots common to both the rectifying and stripping sections are determined by those components Which distribute between the overhead and bottoms of column A. In general, the variables i ^ o l' l and v^ N+1 ^ are not independently chosen and the minimum reflux condition for column 1 cannot be determined directly by these equations. Minimum reflux conditions are imposed on the entire 254 system, and the same relationships will hold for all col umns in the system. Applying equation (D-2) to the rectifying section of column 2, we find i-l, I Ofj dl.s V2 (D-26) and applying equation (D-2) to the stripping section of this column, Z ctl (ll,M,2 - V1,N+1,2> . - (D_27) * 1 - *1.2 1 = 1,1 The minimum reflux for this column is determined by the roots common to both the rectifying and stripping sections of it. These common roots are determined by equation (D-23), Y * " = i - q2 (D-28) *1 “ ®j,2 2 i-l, I The values of the composition of the feed to column 2, are g&ven by 255 (D-29) and the effective thermal condition of the feed is ( AL) 2 L2 ~ 1*2 P2 VX - Li (D-30) and substituting (D-29) and (D-30) into (D-28) * 1 lvi,l.l ~ 1i.O,l) = *-1 " *j,2 (D-31) i=l#I By comparison to equation (D-24), the common roots, 0j,2 and j . are the same. If the components which distribute to both the distillate and the intermediate products also distribute to both the overhead vapor and the bottoms liquid of col umn 1, then, at minimum reflux the common roots for column 2 will also be among the common roots for column 1. Furthermore, if the components which distribute to the intermediate product and the bottoms product also dis tribute to the overhead vapor and bottoms liquid of column 1, a similar analysis would show that the common roots for column 3 will also be among the common roots for column 1. 259 It follows, therefore, that the common roots for column 1 sure the key to the minimum reflux conditions for the system. Since the common roots for column 1 can be calculated directly from the feed to the system, the calculation of the minimum reflux for the system is relatively simple and straightforward. This is accom plished by calculating the common roots for the feed to column 1, and then applying them to columns 2 and 3 to determine the minimum vapor. 257 Literature Cited 1. Underwood, A. j. v., "Fractional Distillation of Binary Mixtures - Numbers of Theoretical Plates and Transfer Units," J. Inst. Petroleum, 29, 147-155 (1943). 2. Underwood, A. j. V., "Fractional Distillation of Binary Mixtures - Simplified computation of Theoretical Plates and Transfer Units," J. Inst. Petroleum, 30, 225-42 (1944). 3. Underwood, A. J. V., "Fractional Distillation of Ternary Mixtures - Part I," J. Inst. Petroleum. 31, 111-118 (1945). 4. Underwood, A. j. v., "Fractional Distillation of Multicomponent Mixtures - Calculation of Minimum Reflux Ratio," j. Inst. Petroleum, 32. 614-626 (1946). 5. Underwood, A. J. V., "Fractional Distillation of Multicomponent Mixtures," Chem. Engr. Progress. 44, No. 8, 603-614 (1948). « APPENDIX E STRIPPING FACTOR METHOD FOR COLUMN 1 OF THE THERMALLY COUPLED SYSTEM Nomenclature Introduction Kremser Equations Application to the Design of Column 1 Literature Cited 258 APPENDIX E Nomenclature effective absorption factor for component 1 in the rectifying section of column k. effective absorption factor for component i in the stripping section of column k. absorption factor for component i on stage J of column k, 1-j ,*/»<!, molal rate of component i in the distillate or overhead product from column k. absorption factor function for component i in the rectifying section of column k. absorption factor function for component i in the stripping section of column k. stripping factor function for component i in the rectifying section of column k. stripping factor function for component i in the stripping section of column k. molal feed rate of component i to column k. equilibrium ratio for component i on stage j of column k, Y1(J, molal liquid rate of column i from stage j of column k. molal liquid rate in the rectifying section of column k under constant molal overflow condi- t ions. 259 260 - molal liquid rate In the stripping section of column k under constant molal overflow condi- t ions. L. . - molal liquid rate from stage j of column k. J 9 * - number of stages in a column k. S - molal rate of the intermediate product in the thermally coupled system. s^ - molal rate of component 1 in the intermediate product of the thermally coupled system. k - effective stripping factor for component i in ' the rectifying section of column k. Si k - effective stripping factor for component i in ' the stripping section of column k. k - stripping factor for component i on stage j of column k, Ki<j/k Vj>k/Lj/k. v. . k “ molal rate of component i in the vapor from stage j of column k. - molal vapor rate in the rectifying section of column k under constant molal overflow condi tions . Vk - molal vapor rate in the stripping section of column k under constant molal overflow condi- t ions. V. . - molal vapor from stage j of column k J 9 K wi k ~ molal rate of component i in the bottoms from column k. xi,j,k “ mole fraction of component i in the liquid on stage j of column k, xi>J<k * li,j,k/Lj,k- yi J,k mole fraction of component 1 in the vapor from stage j of column k, yA j k * ^ j k/Vj k . 261 NOTE - When a system containing only one column is being considered, the subscript k, referring to the column is left off. Special Subscripts f - refers to the feed stage. H - refers to a heavy component, one of low volatility. L - refers to a light or volatile component. N - refers to the bottoms stage of a section. N+l - refers to the vapor entering the bottoms stage of a section. s - indicates the product is of the system and not an intermediate stream. S - refers to the intermediate product in the thermally coupled system. 0 - refers to reflux entering the top stage of a sect ion. 262 Introduction In developing an approximate design method for thermally coupled systems, it is soon recognized that the column providing the initial separation of the components extreme in volatility, column 1 of Figure E-l, is not analogous to conventional distillation. This situation is brought about by the following facts. 1. This column provides only the initial separa tion and does not produce any of the system products; the degree of separation required is difficult to determine. 2. The degree of separation in this column is a function of system reflux ratio. At infinite reflux, no separation is required in this column, whereas, at minimum reflux, a very high degree of separation is required. 3. Since the fractionation occurring in this column is between components of extreme vola tility, experience with multicomponent distil lation, which typically separates adjacent components in the feed, may be of limited usefulness. FEED COLUMN ' J UMN 1 A VAPOR i LIQUID RECTIFYING II i I peed VAPOR □ RECTIFYING ] FEED STRIPPING LIQUID A s STRIPPING RECTIFYING V. 1 n LIQUID - f I FEED t i f VAPOR FLOW i LIQUID FLOW REBOILER IS THE BOTTOM STAGE OF COLUMN 3 STRIPPING LIQUID Fig. E-l Model for the design of the coupled distillation system. 263 CONDENSER OVERHEAD ’product COLUMN 2 INTERMEDIATE PRODUCT COLUMN 3 BOTTOM PROOUCT thermally- 264 Inasmuch as the degree c£ separation in column 1 varies with system reflux ratio, the considerations of minimum reflux and minimum stages, as applied to conventional col umns with fixed separations, do not apply. For this rea son, the design of this part of the system is carried out using a stripping factor method which does not require this type of information. The method is based on the use of the familiar Kremser equations (2). For this design work, the light and heavy keys for column 1 design are components which concentrate in the distillate and bottoms of the system, respectively, and occur in the intermediate product in only small amounts. Their amounts in the liquid flowing into the stripping section and in the vapor flowing into the rectifying sec tion of column 1 from the feed stage are estimated and then, the stripping factor equations are used to calculate the amount of the heavy key component in the overhead va por, and the amount of the light component in the bottoms liquid from column 1. This computation is illustrated in Chapter V of the text. Kremser Equations To clarify the meaning of the Kremser equations their derivation is reviewed here. Consider the absorption or stripping column depict ed in Figure E-2. The material balance around any stage j is vi,j+1 + 1i,j-1 = vi,j + 1i,j (E_1) The equilibrium relationship on this stage is vi,j * si,j 1i,j = ^/Ai,j> (E“2) Using equation (E-2) to eliminate the component vapor rates in equation (E-l) and rearranging, si,j+1 1i,j+1 “ (i + si,j> iij + iifj-1 = 0 (E“3) If the stripping factor, si,j ® Ki,j ^ ^ j ' is constant through the column for component i, then Vi.j+l - (1 + St> li,j + "t.J-l * 0 <E-4> Dividing by and shifting our frame of reference by one stage I 266 Fig. E-2 column. li, 1 4.2 /•i.3 ^.j t u 4.N-2 STAGE 1 v;.2 STAGE 2 'i.3 5SKAGE 3 u STAGE 1 o I *t.N-1 STAGE *-1 Vi.N STAGE N Vi,NM Model of an absorbing or stripping 267 This is a 2nd order linear difference equation with con stant coefficients. Using the terminology of the calculus of finite differences, this equation is written with the shift operator B as (E2 - (1 + At) B + A±) l1#j - 0 (B-6) The non-trivial solution of this equation is (5) 1i,j = C1 ^1 + C2 (B-7) where 0^ and ^ 2 are the roots of the polynomial pre multiplying 1. Using the quadratic formula 1 J 0 , T V T -1'2 (E-8) then solving 0^ = » and = 1 Then, the molal component rate in the liquid from any stage j in the column is h.j " ci AiJ + c2 <*-*> The constants, C1 and C2 are determined by the boundary conditions. For this particular case 260 1. When j “ 0, 1^ j “ ^io* 2. When j - N. vt J+1 - v1<N r Substituting the first boundary condition into equation (E-9), we get li,0 * C1 + °2 <E-10) To apply the second boundary condition we have to consider what happens around the botto’ stage of the column. Solving the material balance around stage N, equation (E-l), for vA N+1# v * v + 1 - 1 (E-ll) i,N+l i,N i.N i,N-l Then, eliminating vi>N by the equilibrium relationship, equation (E-2), vi,N+l (1 + Si^ Xi,N “ 1i,N-l (E-12) Substituting equation (E-9) into equation (E-12), the following result is obtained upon rearrangement. - C1 A1H + C2 Si <B-W > Equations (E-10) and (E-13) form two equations in two un knowns and can be solved to give 269 1 AtN+1 - 1 N+l C~ - A1 lj.Q - Alvi,N+l (E-15) 2 Al»+1 - i Substituting these quantities into equation (E-9) and re arranging, we obtain " ( a V i - 11 K - 1 + I 1 ' Setting j a N and determining the molal rate for component i in the bottoms liquid. we get 1i,N " Ei vi,N+l + (1 ” Ei * 1i,0 (E-17) where . N+l . o N , a Aj - Aj Si — i * ' A ™ - X ' <E- 18) and N+l „ a N - i Ej = Si--- l i i _____ Ai 1 (E-19) SlN+1 - 1 A,"*1 - 1 These factors, E^a and E^8, can be considered the fraction of the molal flow of component i coming in with the vapor 270 at the bottom of the column that is absorbed, and the fraction of the molal flow of component i in the liquid flowing in the top of the column that is stripped out, respectively. By considering the overall material balance for the column and equation (E-16), the following is derived. '1,1 ' V 1i,0 + (1 - Eia» vi,N+l (E-20> These equations have been applied to many types of multistaged processes. For instance, see references 1 through 4. Application to the Design of Column 1 Column 1, Stripping Section Consider the section of the thermally-coupled system shown in Figure E-3. The material balance around the feed stage and the bottoms of column 3 is 1i,N,l + ^f-l.S “ vi,f,3 + vi,N+l, 1 + wi,s (E—21) Since both the vapor to the stripping section of column 1 and to the„ rectifying section of column 3 are the same composition, v^ ^ » ^ 1^V3^ vi f 3’ furthermore, for 271 |v.M J jAu-'.' ' — " Ifceo ffc..TT 4. 0.3 STRIPPING RECTIFYING y COLUMN 3 - r ~ | feed ^i»^»3w § **•^♦1.3 % »N,1 STRIPPING t.S HEAT TO REBOILER BOTTOM PRODUCT Fig. E-3 Lower part of the thermally coupled system. 272 a light component, wi<s is very small and therefore, equa tion (E-21) reduces to VL, f, 3 ^ + ^1^V3* " XL,N, 1 + 1L,f-l,3 (E—22) Then, applying equation (E-16) to the rectifying section of column 3 1i,f-l,3 " Ei?3 vi,f,3 + (1 " Ei % ) 1i,0,3 (E“23) For a light component, the fraction stripped out in column 3 rectifying section is very close to unity, and by choice of the key, 1T n . is small. Furthermore, the fraction L>t U , J absorbed is approximately equal to the absorption factor for this section. Therefore - El%> h.O.B5"0 <E-24> El% ^ Ai>3 (E-2M Under these circumstances, equation (E-23) reduces to XL,f-1,3 “ ^L,3 VL,f,3 (E-26) Substituting this into the material balance for a light component, equation (E-22), 273 1L,N, 1 + ^L, 3 VL,f,3 “ VL, f, 3 (1 + ^1^3* (E-27) Equation (E-27) can be solved for the mole fraction of the light in the vapor from column 3 feed stage 1li!Lt2 (E-28) y i f 3 “ — v3 - l3/k1/3 and, therefore, 1l.N,l vi,N+l,l " V1 yi,f,3 V3 L3 (E-29) Vl VlKi#3 The stripping factor equation applied to the stripping section of column 1 is Xi,N,1 = ®i?l vi,N+1,1 + (1 ®ifl) ~i,f,1 (E-30) Then, substituting for v^ ^ by equation (E-29) and rearranging, for a light component, (1 — Et l) I4 £ 1 1 - V1 *L.l r3 - L3/KL,3 If the liquid rate is constant in the stripping section of column 1, then 274 x, M , * ---------- ‘ ---_ k 1 (E-32) t1 ~ ^L.l) xi.f.l Li,N,l 1 _ V1 EL.l V3 “ V * L f3 Column jL Rectifying Section The material balance around the feed stage and the top of column 2, as illustrated in Figure E-4 is vi ,1,1 + vi , f+1,2 = ^i,f,2 + ^-i.O.l + di,s (E-33) The liquid reflux to column 1 and the liquid to the stripping section of column 2 sure of the same composition and 1^ q i = (Lj/L^) f 2- The amount of a heavy com ponent in the distillate of the system is very small, and therefore, dH,s " °* Then, for this heavy component 1 = Vhi1A * VH1f+1i. 2. (E-34) H 'f '2 1 + L,j/ L 2 Applying equation (E-20) to the stripping section of column 2 V1,f+1,2 = ®i,2 1i,f,2 + (1 ” ®i,2} vi,N+l,2 (E-35) By our choice, for a heavy component, vH^+1,2 is sma11 275 RECTIFYING > COLUMN 2 .2 COLUMN 4.N.2 St#S INTERMEDIATE PRODUCT Fig. E-4 Upper part of the thermally-coupled distillation system. 276 and the fraction of this component in the bottom vapor to column 2 that is absorbed is close to unity. Further- more, the fraction that is stripped from the liquid flow ing into this section is approximately equal to the strip ping factor. Therefore “ gH?2> vh ,N+1,2 = 0 <E-36> - s ~ - _ KH,2 V2 (k-31) H, 2 H, 2 " jT (E 37) Then, substituting this information in the material bal ance around column 2 feed stage and with some manipulation VH,1,1 1, n = T ' v (E-38) H '0'1 h2 + KH.2 v2 Li LX Applying equation (E-20) to the rectifying section of column 1, vi,1,1 = Ei,l 1i,0,1 + (1 “ Ei,l) Vi,f,1 (E-39) Then, eliminating 0 1 ^ substituting in equation (E-38) and rearranging (1 - a E„ ,) v H,l,l H. 1 H.f.1 1 - Ll EH?1 l2 “ kH,2 ^2 Dividing both sides by (1 ~ EH.l* yH,f.l yH, 1, 1 * T F S ! _ L1 EH,1 L2 “ KH,2 V2 277 (E-40) (E-41) Literature Cited Edmister, W. C. , "Absorption and Stripping-Factor Function for Distillation Calculation by Manual and Digital Computer Methods,“ A.I.Ch.E. Journal, 2, 165-171 (1957). Kremser, A., "Theoretical Analysis of Absorption Process," Nat. Pet. News, 22. No. 21, 42 (May 21, 1930), Cited in Reference 4. Smith, B. D., and W. K. Brinkley, "General Short Cut Equation for Equilibrium Stage Proceeses," A.I.Ch.E. Journal, J5, 446-450 (1960) . Souders, M. Jr., and G. G. Brown, "Fundamental Design of Absorbing and Stripping Columns for Complex Vapors, Ind. Eng. Chem.. 24, 519-522 (1932). wylle, C. R., Jr., Advanced Engineering Mathematics, 2nd Edition, McGraw-Hill Book Co., New York, I960, Chapter 5.

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Creator Stupin, Walter John (author)

Core Title The Separation Of Multicomponent Mixtures In Thermally-Coupled Distillation Systems

Degree Doctor of Philosophy

Degree Program Chemical Engineering

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Advisor Lockhart, Frank J. (committee chair), Chilingar, George V. (committee member), Rebert, Charles J. (committee member)

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