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A moving boundary model of concrete sewer pipe corrosion: Theory and simulation

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A moving boundary model of concrete sewer pipe corrosion: Theory and simulation

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA' 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A M O V IN G B O U N D A R Y M ODEL OF CO NC RETE SE W ER P IP E CORROSION: TH EO RY AN D SIM ULATIO N by Fereidoun Jahani A Thesis Presented to the FACULTY O F T H E G RADUATE SCHOOL U N IV ERSITY OF SO U TH ERN C A LIFO R N IA In P artial Fulfillment of the R equirem ents for the Degree M A STE R O F SCIENCE (A pplied M athem atics) May 1999 Copyright 1999 Fereidoun Jahani Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 13 95129 Copyright 19 99 by Jahani, Fereidoun All rights reserved. UMI Microform 1395129 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY O F S O U T H E R N CALIFORNIA T H E G R A D U A T E SCH O O L. U N IV E R S IT Y P A R K L O S A N G E L E S . C A L IF O R N IA 9 0 0 0 7 This thesis, written by Fereidoun Jahani under the direction of hhS. Thesis Committee, and approved by all its members, has been pre sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE 5 , 1999 D ate h, TH ESI? jCOMM I 'hair m ax Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o n ten ts List O f Figures iv List O f Tables vii N otation viii 1 Introduction 1 1.1 O verview ...................................................................................................................... :3 1.2 Biochemical T ransform ation of Sulfur C o m p o u n d s .............................................................................................................. 5 1.2.1 R eduction of S u l f a t e ................................................................................ 5 1 .2 . 2 M ineralization of S u l f a t e ......................................................................... 6 1.2.3 O xidation of H ydrogen S u lfid e .............................................................. 6 1.2.4 Chem ical P recip itatio n of S u l f i d e ....................................................... 6 2 M athem atical M odeling of C orrosion 8 2 . 1 Corrosion m o d e lin g ................................................................................................ S 2.2 T h e o r y ......................................................................................................................... 10 2.2.1 Application to P ipe C o r r o s io n .............................................................. 11 2 .2 . 2 Hydrogen Sulfide in th e Gas P h a s e .................................................... 12 2.2.2.1 Sulfide Gas B oundary Condition on the P ipe Surface 15 2.2.2.2 Sulfide Gas B oundary Condition on the M oving In terface 15 2.2.3 Hydrogen Sulfide in th e Liquid P h a s e ................................................ 16 2.2.4 Biological O xidation of Dissolved S u lf id e ......................................... 16 2.2.4.1 Dissolved Sulfide Boundary C ondition on th e Pipe Surface ...................................................................................... 17 2.2.4.2 Dissolved Sulfide Boundary C ondition on th e M oving In te rfa c e ..................................................................................... IS 2.2.5 Hydrogen and Sulfate Ions in Liquid P h a s e ...................................... IS 2.2.5.1 H ydrogen Ion B oundary Condition on the P ipe Surface 19 2.2.5.2 H ydrogen Ion B oundary C ondition on the M oving In te rfa c e ..................................................................................... 19 ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.6 Model S u m m a r y ..................................................................................... 20 3 N um erical M odel and Sim ulation M ethod 23 4 N um erical Sim ulation of Corrosion and R esults 30 5 Conclusions and Future R esearch 51 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist O f F igures 1 .1 Corrosion of reinforced concrete pipe. The region below the surface of water is mainly reductive. Protein and S0%~ are converted into sulfides. Sulfides are released into the atm osphere above the surface of the water. Hum id air produces condensation and sulfides dissolve on the surface. The biological oxidation of sulfide into sulfuric acid results in the corrosion of th e pipe..................................................................... 4 2.1 Control volume shown at t > 0. Portion of cross section of a pipe w ith a layer of corrosion product (gypsum) and undam aged concrete on the right................................................................................................................ 13 2.2 Diffusion, adsorption, and dispersion of H^S in heterogeneous porous concrete, a, /3,and 7 are solid, liquid, and air phases in the concrete respectively................................................................................................................. 14 3.1 (a) Example of a function u (E Uh- (b) The basis function, ~pj.......................26 4.1 C oncentration profiles at three porosities. W ater and air- filled porosi ties were equal and the rem aining volume was reaction p ro d u ct. From top to bottom : 10 %, 20 %, and 30% in each phase. N um bers on the plot indicate tim e, in years. Plots were obtained from d a ta at the end of each five-year interval. W here each curve stops is the m oving inter face between the corrosion product and the uncorroded concrete. See Table 4.1 for other param eter values. Plots indicate direct correlation between porosities and corrosion layer m easured as concrete depth. . 34 4.2 (a)pH calculated up to the m oving boundary and the corrosion thick ness as centim eters of concrete for different w ater and air-filled porosi ties. From top to bottom : 10, 20, .30 % in each phase. N um bers on the plot indicate tim e, in years. Plots were obtained from d a ta at the end of each five-year interval. See Table 4.1 for o th e r param e ter values. pH on the surface of concrete is fixed to 1 and rises to near 3 on the moving boundary as a result of the effect of alkalinity on neutralization of acid in water-filled pores. (b)In all three cases, corrosion rate is high initially and decreases w ith tim e...................................35 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 C oncentration profiles obtained for different values of bioconversion rate, K b - From top to b o tto m , K b = 0.0, 7.2, and 72 day-1 . N um bers on the plot indicate tim e, in years. See Table 4.1 for other p a ra m e te r values. Plots show no corrosion formed for K b = 0. M iddle plot shows extensive corrosion. B ottom plot shows very little increase in corrosion as com pared w ith th e middle plot when bioconversion rate, K b - , was increased by ten fold. The concentration profiles show steady-state behavior for large bioconversion rates...................................... 37 4.4 (a)pH calculated up to th e m oving boundary and the corrosion as centim eters of concrete for different bioconversion rates, K b - From top to bottom , K b = 0 .0 , 7.2, and 72 day-1 . N um bers on th e plot indicate tim e, in years. See Table 4.1 for other param eter values. (b)In the m iddle and b o tto m plots, corrosion rate is high initially and decreases w ith tim e.................................................................................................. 3S 4.5 C oncentration profiles obtained for different values of dissolution rate. Kc- From top to bottom , Kc — 0.S4, 0.0-S4, and 0.00S4 c m /d av . Num bers on the plot indicate tim e, in years. See Table 4.1 for o th er param eter values. In the m iddle and bottom plots, corrosion is slowed down at lower dissolution ra te s............................................................................ 40 4.6 {&)pH calculated up to th e m oving boundary and th e corrosion as centim eters of concrete for different dissolution rates, Kc- From top to bottom , K c — 0.84, 0.0S4, and 0.0084 cm /day. N um bers on the plot indicate tim e, in years. See Table 4.1 for param eter values. T he effect of acid neutralization in the m iddle plot is not as strong as the top plot, (b) Middle and b o tto m plots show lower corrosion rates th an the top plot initially. Furtherm ore, the corrosion shows linear correlation w ith tim e at lower dissolution rate ....................................................41 4.7 C oncentration profiles for different values of pH on the surface of concrete. From top to b o tto m , pH = 1 , 2 , 4. N um bers on th e plot indicate tim e, in years. See Table 4.1 for other param eter values. Top plot shows the largest corrosion am ong the three selected pH's. At pH of 4, there is only a slight corrosion observed.............................................. 43 4.8 (a)pH calculated up to th e moving boundary and the corrosion as centim eters of concrete for different values of pH on the surface of concrete. From top to b o tto m . pH = 1 , 2, and 4. N um bers on the plot indicate tim e, in years. See Table 4.1 for other p a ram eter values.(b) Corrosion rate a t pH of 1 is the highest am ong th e th ree plots. The plot of corrosion shows linear correlation w ith tim e a t pH of 2 or higher.............................................................................................................. 44 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.9 Corrosion layer depth com puted after 25 years based on several values of concrete alkalinity and fixed pH on th e surface of concrete. Cor rosion rate is inversely proportional to alkalinity and directly propor tional to the pH on the surface of concrete. Plots show th at increase in alkalinity provides protection against corrosion....................................... 46 4.10 Corrosion layer thickness com puted based on several water-filled porosi ties, ew. All other p aram eter values sam e as in Table 4.1. Larger corrosion thickness is due to higher w ater content of concrete pores. 48 4.11 C om puted corrosion rate of concrete based on several water-filled porosities, tw. All other param eter values sam e as in Table 4.1. The rate decreases and becomes asym ptotic w ith tim e. Larger corrosion rate is due to higher w ater content of th e concrete pores...............................50 5.1 Corrosion cell to be used in m easuring m odel param eters and concen trations of hydrogen ions in a concrete specim en..............................................56 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist O f Tables 4.1 Numerical data for several param eters. Some param eter values were obtained from the following references: a [1], b [2], c [3], d [4], and e [5]............................................................................... . . . . . . . . . ...................... 31 5.1 Effects of the N um erical d a ta on the corrosion rate given as weak, m oderate, and strong............................................................................................... 53 5.2 Results of Sensitivity Analysis for Controlling P aram eters.............................55 v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N o ta tio n A alkalinity of concrete as calcium carbonate equivalent A w interfacial area between air and liquid phases a moles of hydrogen ions react w ith calcium carb o n ate b w idth of stream m C average corrosion rate, same as Cavg in /v r C' concentration of a fluid in porous m edia m oles/liter Cc alkalinity of concrete moles liter moles 11 f p r c concentration Djk,V m olecular diffusivities 11 uCl cm2/d ay Da diffusivity of hydrogen sulfide in the air ph ase cm2/d ay D w diffusivity of hydrogen sulfide in the w ater phase cm2/d ay D h diffusivity of hydrogen ions in the w ater ph ase cm2/day d distance over which ib varies significantly cm EBOD effective biochem ical oxygen dem and m g/L en unit vector of dimension n hj partition of length yt - — y ^ i cm h subintervals of a larger interval [0 , 1 ] moles cm2-sec J diffusive m ass flux in the porous m edium K reaction rate constant in oxidation of sulfide A app apparent ra te constant, same as K b K b bioconversion rate constant for sulfide day -1 A c dissolution rate constant for calcium carb o n ate cm /day A c product of K c and the hydrogen ions concentration moles cm2-day A’ H modified H enry’s Law constant I\T mass transfer coefficient cm /day K n tridiagonal n by n m atrix A'i equilibrium constant m oles/liter A'2 equilibrium constant m oles/liter k corrosion factor in the range of 0.3 to 0.4 L characteristic length of the global system cm L n tridiagonal n by n m atrix M num ber of pores in a cross section M n tridiagonal n by n m atrix V I 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n am ount of a species diffusing through, a m edium P w etted perim eter p H negative logarithm of th e activ ity of hydrogen ion p partial pressure of hydrogen sulfide pg partial pressure of gas in th e atm osphere p exposed perim eter of pip e to th e stream ppm parts per million on a volum e basis Q wastewater flow R gas constant S area of a th in slab s(t) location of moving interface So location of corrosion interface initially s’ slope of the energy line T am bient tem perature Un{t) n dim ensional, tim e dependent vector function un n dimensional, tim e and space dependent vector function representing concentration in a given phase u n dim ensional, tim e and space dependent vector function representing concentration in a given phase V averaging volume V0 initial rate of sulfide rem oval Va volume of air phase contained w ithin concrete Vw volume of liquid phase contained w ithin concrete V n{t) n dim ensional, tim e d ep endent vector function vn n dimensional, tim e and space dependent vector function representing concentration in a given phase v concentration of sulfide in th e liquid phase U i velocity vector W n{t) n dimensional, tim e dependent vector function w n n dim ensional, tim e and space dependent vector function representing concentration in a given phase w concentration of hudrogen ions in the liquid phase XiA spatial coordinates Z defined function in the Z-form ula Z observed concentration o f sulfuric acid in the solution m oles lite r.tim e ft ppm , a tm atm m v /v ft3/se c a tm -lite r m ole-K length cm cm f t/ft ° K cm m ole liter-sec cm 3 cm 3 cm moles liter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G reek L etters a solid phase in the porous m edium j3 liquid phase in the porous m edium 7 air phase in the porous m edium , gypsum ea air-filled porosity in the corrosion layer ew w ater-filled porosity in the corrosion layer r]i value of concentration of speceies i at yt - Ai initial concentration of sulfide gas on the surface of pipe A 2 initial concentration of dissolved sulfide on th e surface of pipe A 3 initial concentration of hydrogen ions on th e surface of pipe <pj basis function belonging to set of fu n c tio n s ^ Gsw flux of sulfide to the pipe wall o sf flux of sulfide from the stream to the air Sym bols [•] represents the concentration of a chemical species moles liter moles liter moles liter S m 2-h r O ' m2-h r moles liter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract T he focus of this thesis is on the developm ent of a m athem atical m odel to predict the ra te of corrosion inside the surface of a concrete sewer pipe. In this m odel, the diffusion of sulfide (H 2 S ) inside the pores of th e concrete, its dissolution in th e liquid phase, its biological conversion to sulfuric acid, and the corrosion of calcium carbon a te aggregates are expressed using tran sp o rt equations and the law of averaging. T he corrosion front is modeled as a m oving boundary. The location of the interface betw een th e corrosion layer and the uncorroded concrete is not known in advance and therefore is determ ined as p art of th e solution to the model equations. T h e m odel consisted of a system of one dimensional reaction-diffusion partial differential equations coupled to an ordinary differential equation describing the m ovem ent of the corrosion front. T he equations were approxim ated by a sequence of ordinary differential equations using finite elem ent Galerkin approxim ation on a fixed spatial domain. The modified equations were solved num erically to obtain concentration profiles of H 2 S in the air and the liquid phases, the pH as a function of concrete depth, and the position of the m oving boundary (corrosion front). Sev eral param eters of the model such as sulfide concentration in the atm osphere, the porosity, bioconversion rate, dissolution rate, and the calcium carbonate content of the concrete were modified in this m odel and the corrosion rate was com puted. It was dem onstrated that the model predicted the corrosion depth with a reasonable degree of accuracy provided th at the p aram eter values in the m odel were chosen appropriately. A comparison between the sulfide and the hydrogen ion concentra tions indicated that the hydrogen ions were m ore significant in the corrosion process based on their concentrations in the liquid phase. A new equation for the corrosion rate was derived based on the results obtained. This equation was directly propor tional to th e pH of the surface and inversely proportional to the calcium carbonate content in the concrete. The constant of proportionality was the calcium carbonate dissolution rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p ter 1 In tr o d u c tio n Hydrogen sulfide (H 2 S ) has been known as a m a jo r contributor to objectionable odors, corrosion of w astew ater collection system s, an d safety hazards to employees in th e w astew ater industry. Extensive d eterio ratio n o f m any sewers has prom pted the m unicipal districts to take m ajor rem edial m easures. Furtherm ore, it is estim ated th a t the to tal cost of sewer repairs in Los Angeles C ounty alone will exceed a billion dollars, an d there have been substantial increases in pro p erty taxes to provide the revenue. Before the 197(Ts [6 , 7], additional concrete thickness, calcareous aggregates or com binations of the two were used to extend th e life of concrete pipe sewers in a sulfide environm ent. A dditional cover and alkalinity requirem ents were based on engineering judgm ent and experience. Too often, the results were an excessive am ount of protection at greatly increased costs. M ost recently, researchers [1-7] have focused on th e developm ent of surface coat ings, flushing of the concrete surface, and bacterial co m p etitio n for extending the life of the pipes already in service. However, there is very little research [6 , 7] done in the area of corrosion modeling. U ntil the publication of th e EPA M anual [6 ], there was no rational m ethod in existence to calculate th e ra te of corrosion. Pom eroy [6 ] developed a sem i-em pirical equation relating th e ra te of corrosion to the concrete alkalinity and flux of hydrogen sulfide [H^S) to th e pipe wall. The hydrogen sulfide flux to the surface was modeled to be a function of flow characteristics such as flow velocity, turbulence, the size of the pipe, and th e hydraulic gradient. Previous stud ies [7] have shown th a t the actual corrosion rate was 10 to 20 times higher than the 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. calculated corrosion ra te under uniform flow. T he difference in calculations was due to the turbulence at th e pipe junctions. In this research, a m ore elaborate m odel was developed for th e prediction o f cor rosion rate. Sulfide concentration in the atm osphere of a sewer pipe varied seasonally in the range of 0.5 to 4.0 ppm [6 ] irrespective of fluctuations in the w astew ater flow. Therefore, th e corrosion rate was assum ed to be proportional to the sulfide concen tration in th e atm osphere of the pipe. T he m athem atical m odel assum ed diffusion in two phases. Sulfide released from th e surface of w ater reached the pipe wall by the action of air currents. It then diffused into the air-filled pores, and dissolved in the w ater phase. The bacteria in the w ater phase oxidized sulfide to sulfuric acid, which dissociated into sulfate (SO*~) and hydrogen ions (H +) instantly. T he hydrogen ions reacted w ith the calcium carbonate in the concrete, causing corrosion. Sulfate reacted w ith the concrete m aterial to form precipitates. T he m ost com m on precip ita te is gypsum (CaSO^ ■ 2HoO). It is crystalline in shape and can expand. Under ap p ro p riate surface conditions, the expansion leads to fracture and the exposure of uncorroded concrete surface to additional acid attack. T he m odel was created to test the influence of several param eters on the corrosion rates. T he m odeler can select the concentration of hydrogen sulfide in the pipe atm osphere, biological conversion rate, and concrete characteristics such as porosity, internal surface area, alkalinity, and the w ater content of the pores. T he m odel may be used to study eith er heavily corroded or relatively new pipes. In the num erical solution, the pH of the pipe surface was initially assum ed to be in the acidic range ( e.g., 1 .0 ) in order for the m icroorganism s to actively participate in the oxidation of sulfide. It was also necessary to initially assum e an infinitesim al layer of corrosion product. The values of several param eter constants were given. These included: 1. A ir and w ater diffusion coefficients for hydrogen sulfide. 2. W ater diffusion coefficient for hydrogen ions. 3. A ir and water-filled porosities for concrete. 4. M ass transfer coefficient of hydrogen sulfide. 5. Specific surface area of w ater in concrete pores. 6 . H enry's Law constant for H2S in concrete pores. • ? Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7. Bioconversion rate constant for hydrogen sulfide. S. Dissolution rate constant for calcium carbonate. The o u tp u t of th e model was a series of curves showing air and w ater concentrations of hydrogen sulfide, and p H , as a function of d e p th in the concrete and the position of corrosion front as a function of tim e. T h e num erical results may be used to develop a m ore accurate life factor [6 . 7] for th e design of sewer concrete pipes. The advantage of a new life factor may be in designing of a pipe which may depend not ju st on the alkalinity of the pipe, but also other physio-chem ical properties which affect th e rate of th e sulfuric acid attack. These properties include reactivity, perm eability, porosity, and w ater content. In the following sections, the process of corrosion in the sewer pipes, characteristics and properties of sulfide, and im portant biochem ical reactions are discussed. 1.1 O v erv iew In the w astew ater, aerobic processes such as degradation of sulfur-containing am ino acids and anaerobic processes such as anaerobic sulfate reduction result in the pro duction of volatile sulfur compounds like hydrogen sulfide (H 2 S ) and M ethyl Mer- captan (C H3SH). Some anaerobic organisms axe also able to consum e hydrogen ions (Eq. (1.2)). T he decom position of the organic com pounds in the w astew ater always results in hydrogen sulfide or m ethyl m ercap tan [7, S] production. These com pounds en ter the sewer atm osphere through volatilization from the w ater sur face. A utoxidation of HoS results in sulfur form ation, which precipitates on the sewer wall. If enough m oisture is present, ubiquitous thiobacilli m etabolize sulfur and excrete sulfuric acid (see Fig. 1 .1 ). The acid reacts w ith the concrete to form gypsum . T he corrosion product has essentially no stru ctu ral strength and can be scraped away w ith a hard tool. Therefore, if corrosion is not detected at the right tim e, a sewer pipe m ay totally collapse. T here are cases in which corrosion has reached the soil and street pavem ents have collapsed due to undetected corrosion. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. biological conversion of sulfides into sulfuric acid and corrosion oxidation concrete control volume sewage degradation o f protein to sulfides biofilm sediment. mud reduction Figure 1 .1 : Corrosion of reinforced concrete pipe. The region below th e surface of w ater is mainly reductive. Protein and SOl~ are converted into sulfides. Sulfides are released into the atm osphere above the surface of the w ater. H um id air produces condensation and sulfides dissolve on th e surface. The biological oxidation of sulfide into sulfuric acid results in the corrosion of the pipe. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 B io ch em ica l T ran sform ation o f S u lfu r C om p ou n d s T here are two m ajor reactions tak in g place in the sewer system s: the reduction of sulfate (S'0 4 - ) to sulfide (.S'2 - ) in th e w astew ater and oxidation o f hydrogen sulfide (HnS ) to either sulfuric acid on th e concrete surface or su lfate in the w astew ater. B oth of these reactions are biochem ical in nature and require presence of m icroorgan ism s. T he oxidation of hydrogen sulfide to sulfate in the w astew ater can be achieved through addition of chemicals such as chlorine, hydrogen peroxide, and ozone. The purpose of this m ethod is to control the odors in the sewer system . T he prim ary bio logical transform ations of S 0 2~ are: reduction, im m obilization, an d m ineralization. T he chem ical transform ation for hydrogen sulfide is oxidation. 1.2.1 R eduction o f Sulfate Sulfate m ay be reduced by the anaerobic bacteria present in th e w astew ater [7]. In reducing sulfate, organism s use organic acids, fatty acids, an d alcohols as electron donors (Eq. 1 .1 ). Some anaerobic organism s are also able to consum e hydrogen ions and electrons (Eq. 1.2). T he carbon is used for ceil synthesis and grow th. The following reactions occur in the slim e layer on the surface o f th e pipe under the w ater: 2 CHzCHoOH + S 0 2~ — » ■ 2C H zCOOH + S 2~ ± 2 H 20 + C 0 2 ( 1 .1 ) Se- + S H + + S 0 2 4~ — ► S 2~ + 4 tf 2 O (1.2) T he principal species of bacteria responsible for the reaction in Eq. (1-1) include Desulfovibrio. Desulfomonas, and Desulfotomaculum [9, 10]. Also, bacteria can im m obilize the sulfur in SO\~ into th e ir cells and com bine w ith o th e r cellular organic com pounds. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2.2 M ineralization o f Sulfate H eterotrophic bacteria can decom pose organic com pounds containing sulfur. One such degradation is the reduction of m ethionine in the presence of w ater. T he m ethyl m ercaptan formed in the process is hydrolyzed to m ethyl alcohol an d by-product of hydrogen sulfide: C H 3S C H 2C H 2C H N H 2C O O H + H20 — > C H 3S H + iV f t + C H 3C H 2C O C O O H (1.3) C’H3S H + H20 CH3O H + H2S (1.4) 1.2.3 O xidation o f H yd rogen Sulfide T he chem ical oxidation of hydrogen sulfide is affected by a num ber of factors, some m etals can act as catalysts and oxidize the sulfide, while others such as glycerol inhibit the oxidation. M any o th er reactions are pH dependent an d pH adjustm ent m ay be necessary in order to o b tain the desired oxidation. C hlorine has been used as a chem ical treatm en t to reduce hydrogen sulfide concentrations in the w astew ater and prevent corrosion: Cl2 + H2S — y 2H + + 2ClT + S° (1.5) a c i2 + h 2s + ± h 2o — ► iq h + + s c r + s o 2- (i.e) O ther strong oxidants include hydrogen peroxide (H20 2) and ozone. These oxidants are also pH dependent. An advantage of H20 2 is th a t it decom poses into safe by-products such as oxygen and w ater. 1.2.4 C hem ical P recip itation of Sulfide T he solubility of m etal sulfides in the wastewater is very low [7]. Therefore, an o th er m ethod of sulfide rem oval is precipitation w ith the ionic m etals present in the wastewater. Am ong the m etals tested, iron produces the least soluble product 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F e S ) and therefore most effective in precipitating sulfides. Both ferric and ferrous salts have been used as additives to remove sulfide from th e wastewater. It has been shown th a t the chemical treatm en t can reduce the concentration of hydrogen sulfide in the wastewater below 0 . 2 m g /1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p ter 2 M a th em a tica l M o d e lin g o f C orrosion 2.1 C orrosion m o d e lin g Pom eroy and Parkhurst [11] developed a model to predict the generation of sulfide in th e partially-filled sewers. T he m odel was based on the studies conducted at the w aste water collection system s of Los Angeles County, California. T he rate of sulfide generation was found to be directly proportional to the effective biochem ical oxygen dem and (EBOD) of the w aste w ater, slope of the energy gradient, mean w aste w ater velocity, and inversely proportional to th e hydraulic radius. High rates of the sulfide production were predicted when the dissolved oxygen exceeded O.o m g/ 1 or insufficient nutrients were present in the w aste water. According to the process design m anual [6 ], th e transfer of sulfide from the w astew ater to the sewer atm osphere can be calculated using an exchange equation. T he flux of sulfide from the stream surface was described as directly proportional to tem p eratu re, stream velocity, and th e slope of the stream . T he flux of HoS from the stream was proportional to the relative saturation in the air. T ypical concentration of sulfide in sewers was w ithin 2 to 20 percent of the satu ratio n concentration. Thus the ra te of sulfide transfer was at least SO percent of th a t of a sulfide-free atm osphere. T he forecasting of sulfide buildup was developed based on th e em pirical equation generally known as the Z formula: z = ^ p ( p- /6) c2-1' S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where [EBOD] is effective BOD (m g/1), s'is slope of the energy line of th e stream , Q is w astew ater flow (ft3 /sec), P is w etted perim eter (ft), and 6 is surface w idth (ft). Using this form ula, it was shown th a t th e sulfide was com m only observed at Z values exceeding 10,000 (dim ensionless), b u t was rare below 5,000. T he effect of th e stream velocity in th e generation of sulfide was based on the solids tran sp o rt in th e sewer pipes. G enerally for a stream velocity less th a n 1.0 foot per second, there was severe sulfide build-up due to the loosely deposited solids at the bottom . Higher velocities were also shown to enhance the release o f sulfide to the sewer atm osphere as was m ixing or vertical drops when w astew ater was released to an adjoining pipe. T he corrosion ra te was assum ed to d ep en d on the rate of the release o f th e sulfide from th e w astew ater. A sem i-em pirical eq u atio n describing the flux of th e sulfide to the wall was developed. The flux was expressed as: 4>s w = <t>af(b/p) (2.2 ) where o sw is flux of sulfide to the pipe w all(g/m 2-hr), < b sf is flux of sulfide from the stream to the a ir(g /m 2-hr), and b/p is ratio of surface w idth of th e stream to the exposed perim eter of pipe wall above the water surface. Prior experim ents by Pom eroy [12] indicated th at th e corrosion ra te decreased with an increase in the alkalinity. Thus a m odel for the corrosion rate of concrete was proposed based on the flux equation and the am ount of th e reactive m aterial th a t was p resent in the pipe. T he sulfuric acid was assum ed to re a ct only partially and therefore a correction factor was included in the equation as following: C = Q A 9 k o sw± -{ (2.3) where C is average corrosion rate (in /y r), A is alkalinity of the concrete expressed as CaCOz equivalent, and k is correction factor. The value of k was based on the experience and engineering judgem ent . For rapid production of acid, k was chosen in the range 0.3 to 0.4. The alkalinity A was estim ated from the com position of the cem ent m ix used in the pipe. In concrete pipes m ade of granitic aggregate, the alkalinity was taken to be between 16% a n d 24%. If Limestone or dolom ite was used as an aggregate in the pipe, the alkalinity was taken to be 100%. T he eq uation was 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m ost applicable to th e uniform pipe reaches and away from the locations of the high turbulence. Iw am atsu et al. [13] addressed the sulfur cycle and its relation to the corrosion of sewer pipes. It was suggested th a t the sulfide production was directly proportional to the concentration of organic com pounds in the w aste w ater, tem perature, sludge thickness, and inversely proportional to the flow rate, oxygen level in the waste w ater, and th e fall in the pipe. Among several cases studied, the corrosion of pipe sections resulted from high levels of dissolved sulfide and anaerobic conditions. The authors analyzed the sulfide concentration and its corrosion dam age to the pipe using a predeterm ined m odel for the sulfide generation. T he A m erican Society of Civil Engineers (ASCE,[7]) published m anual no. 69 specifically for the purpose of providing inform ation about sulfide problem s and m ethods of quantifying and controlling them . The m anual included data, case stud ies, and inform ation about corrosion in the w astew ater treatm en t plants. The m an ual discussed the reliability of corrosion prediction m odels by com paring the results w ith the case studies. T here were two models used in determ ining sulfide build up in the w astew ater. T he Pom eroy and Parkhurst m odel [11] predicted as much as 5 tim es the m easured sulfide generation in the w astew ater. T he T histlethw avte m odel [14] predicted as m uch as 2 0 tim es, indicating no correlation. In the prediction of corrosion rate, Eq. (2.3) was used. T he d ata was obtained from as m any as 1 0 0 m anholes, and generally indicated an increase in corrosion rates dow nstream in the system . The observed rates were 2 to 5 tim es greater than those th a t were predicted. T he corrosion rate due to turbulence was not entirely included in the m odel equation. 2.2 T h eo ry Dispersion of a fluid in a porous m edium has been of interest for m any years am ong researchers. In particular, the flow of a noxious gas in an adsorbing and reactive colum n such as granular activated carbon (GAC) has been m odeled [15]. Previous work by B ear [16] and W hitaker [17] have described dispersion in porous m edia w ith 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here c is concentration (m oles/liter), ut - is velocity vector (cm /sec), x, is spatial coordinates (cm), and Djk is m olecular diffusivity (cm 2/sec). T he developm ent of above equation relied on m ixing an d turbulent flow. Because of th e com plex ge o m etry of the porous m edia, W hitaker [17] developed sim ilar results using th e law of averaging for describing the concentration and velocities. This law expresses th e microscopic equations which hold a t a point in space and fu rth er develops the m acroscopic equations which hold for som e volume V in space. In this approach, the control volume may contain a discontinuous space of fluid and therefore, th e micro scopic equations are averaged over th e entire region, thus providing a m acroscopic equation valid over a continuous space. 2.2.1 A pplication to P ip e Corrosion T h e concrete is assum ed to be a solid of low porosity w ith a w etted surface. The tran sp o rt effects and reaction m ust be considered sim ultaneously so th a t the condi tions inside the pores can change w ith respect to time. The model is created sim ilar to th a t of the unreacted-core shrinking m odel [18](see Fig. 1.1, 2.1) w here a fluid re acts w ith a solid to produce a fluid product plus a porous solid layer. T h e unreacted solid is impervious to the fluid because it is densely packed. On the o th e r hand, the solid product layer is porous so th a t the reactant fluid can diffuse and th e product fluid can diffuse out. T he inner surface of concrete p ipe above the water line undergoes com plex bio chem ical reactions. At first the surface pH falls from 11.0 to 7.0 as a result of carbonation. Then bacteria consum ing hydrogen sulfide gas grow on this neutral surface and produce sulfuric acid. T he sulfuric acid reacts with the surface of the concrete forming calcium sulfate (gypsum ). As the reaction proceeds, the reaction zone will move into the interior of th e concrete leaving behind a layer o f in ert m ate rials (e.g., gypsum, sand, and gravel, see Fig. 2.2 ). Note th a t the ex tern al diam eter of the pipe remains the sam e. However, the internal diam eter m ay decrease as a result of crystal growth and expansion of the corrosion layer. T he corrosion layer is considered a continuum and therefore, the transport laws can be applied. Because 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of w ater vapor condensation on th e surface of th e pipe, th e pores in th e corrosion layer adsorb w ater and become m oderately w et. T he tran sp o rt process inside the the corroded surface is m odelled as a m ulti-phase diffusion m odel containing liquid, gas. and solid phases. In the following sections, th e fundam ental equations describing the flux of H2S into the surface of concrete an d its biochem ical interaction w ith the concrete m aterial is discussed. 2 .2.2 H ydrogen Sulfide in th e G as P h ase T he m odel assum es the concentration of hydrogen sulfide in the atm osphere of the pipe is constant. T he value of the concentration is chosen based on th e reported d a ta [4], H ydrogen sulfide diffuses into th e porous o uter layers of th e concrete and dissolves in the w ater on the surface of th e concrete particles (see or-phase in Fig. 2.2). It diffuses according to F ick ’s Law, at a rate proportional to the second derivative of the concentration gradient. T he diffusion ra te is reduced in porous m edia since th e cross-section available for diffusion is reduced by th e presence of the particles, and the path length is increased by the to rtu o sity of th e pores. T h e diffusion rate constant is accordingly m ultiplied by th e porosity and a toruositv factor, both decim al fractions betw een zero and one. A com m only used value for the toruosity is 0.66. C oncentrations are expressed in moles per liter. D issolution in the w ater occurs a t a ra te which is proportional to th e difference betw een the sulfide concentration in th e w ater, and the concentration which would be seen at satu ratio n (equilibrium w ith the gaseous H2S). According to [15. 19], the rate of dissolution is proportional to the area of the gas and liquid interfaces in the averaging pore volume and inversely proportional to th e average pore volume in the gas phase. Assum ing no biochem ical conversion of sulfide in the gas phase, the tran sp o rt of sulfide gas in the concrete pores can be w ritten as - a r 1 = ^ . d - ^ ) - K T ^ ( i M H 2s i - i H 2s u ) (2.5) w here [H2S]g. is sulfide gas concentration in the air phase in parts per m illion (ppm ), Da is effective diffusion rate of hydrogen sulfide in the air phase (cm 2 /clay), Kj- is 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gypsum layer, with sand, gravel, and microorganisms Interface Undamaged Concrete x . direction of corrosion Figure 2.1: Control volume shown a t t > 0. Portion of cross section of a pipe with a layer of corrosion product (gypsum ) and undam aged concrete on the right. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2 .2 : Diffusion., adsorption, and dispersion of H2S in heterogeneous porous concrete, a , ,/3.and 7 are solid, liquid, and air phases in the concrete respectively. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e overall mass transfer coefficient (cm /d ay ), 4?“- is interface area between air •'a and liquid phase per unit volum e o f air phase (cm -1 ), Kfj is H en ry ’s Law constant (dim ensionless), and [H2 S]aq is dissolved phase concentration of sulfide (m oles/liter). 2.2.2.1 Sulfide Gas Boundary C ondition on the P ip e Surface Previous studies of hydrogen sulfide [7] have indicated th a t th e concentration of sulfide in the pipe varied in a narrow range (0.5-4.0 ppm ). T his fluctuation is further com pounded by the flow characteristics inside the pipe such as the hydraulic gradient, flow velocity, turbulence, an d th e size of the pipe. T hese factors together m ay predict erroneous results in com puting the hydrogen flux to th e surface of the pipe. In this study, the co ncentration of sulfide on the surface of the wall is used instead of the flux. Therefore, th e concentration m ay chosen arb itrarily in the specified range w ithout considering changes in the flow of w astew ater. In this model, a D irichlet boundary condition for sulfide was chosen. T he concentration of sulfide on the surface of pipe was taken to be 5 p pm and was held constant throughout the num erical com putations [fl,S1,(0,0 = (2.6) where [ff2 * 5 ’ ]ff(0, t ) is concentration of hydrogen sulfide on the surface of the pipe (m oles/liter), pis partial pressure of gas (atm ), R is gas constant (atm - Liter/moIe-°K), and T = am bient tem perature (e.g., 298°K). In above equation, th e partial pressure was taken to be o x 10- 6 atm (5 p p m ). 2.2.2.2 Sulfide Gas Boundary C ondition on the M oving Interface On the moving interface, x = s(f), the flux of hydrogen sulfide was considered negligible due to low perm eability o f concrete. Therefore, a boundary condition of the second kind, N eum ann condition, was chosen: - 0 (2.7) where [f/2 S']3 (.s(f), t) is hydrogen sulfide concentration at the interface, and s(i) is location of the moving interface (cm ). 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.3 H ydrogen Sulfide in th e Liquid P h ase T ransport of hydrogen sulfide in th e Liquid phase is m odeled w ith equations sim ilar to those describing tran sp o rt in the gaseous phase (see /?-phase in Fig. 2.2) w here a fraction of the pores is occupied by the liquid. This liquid is adsorbed onto th e surface of the pipe as a resu lt of w ater vapor in the atm osphere of pipe. T he hydrogen sulfide concentration in th e w ater phase changes as H2S dissolves from th e atm osphere, and as HoS is consum ed by biological conversion to sulfate. Gas transfer occurs as described in th e previous section, at a rate pro p o rtio n al to the difference between the existing; concentration in the liquid phase and th e satu ratio n concentration. This difference m ay occur as a result of continuous biological oxi dation of dissolved hydrogen sulfide. T he product of this oxidation, sulfate, reacts w ith concrete to form gypsum . 2.2 .4 Biological O xid ation o f D issolved Sulfide T h e corrosion of cem ent bonded m aterials is due to sulfuric acid form ed from the oxidation of hydrogen sulfide: H 2 S + 2 O 2 - ^ 2 H+ + S 0 2- (2.8) where Kg is bioconversion rate co n stan t (per day). This reaction is com plete in the presence of bacteria (e.g., T.concretivorous). However, the order o f th e oxidation reaction with respect to sulfide and the reaction rate on the surface of the pipe has not been determ ined yet. VVilmot et al. [20] studied the kinetics of sulfide oxidation in the w astewater. T h e experim ents were perform ed to d eterm in e the order of the oxidation reaction, ra te constant, and effects of te m p e ratu re on the reaction rate. According to the au th o rs, the general equation describing th e sulfide oxidation reaction has the form - M M k = K{HiS}” {0 ,y (2.9) where [H2 S ]aq is concentration o f hydrogen sulfide in the w astew ater (m o les/ liter), [O2 ] is dissolved oxygen concentration (m oles/liter), and A”is reaction ra te constant. It is convenient to study the effect of hydrogen sulfide or oxygen on th e reaction rate 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by determ ining the initial reaction rate. For exam ple, for constant [0 2], the Eq. (2.9) can be reduced into V0 = K a„ [ H 2S]?qfi (2.10) where V o is initial rate of sulfide rem oval ( = - [T^-S'k^o is initial sulfide concentration (m oles/liter), and K app is ap p aren t rate constant which was previously referred to as the bioconversion rate, K B for m = 1. A log-log plot of V o versus [H2S]aq.o using the initial rate above can determ ine the order of reaction w ith respect to sulfide. The slope of the line gives the o rder of the reaction. In recent studies by VVilmot et al. [20], the reaction order w ith respect to sulfide in wastewaters w ith p H as low as 4.10 was approxim ately first order. Here, the sam e order and th e reaction rate is assum ed for the purpose of num erical com putations. According to M ori et al. [21], the num ber of T. Thiooxidans rem ained the sam e up to the m oving interface . This suggests that K b may be chosen as constant throughout the corrosion layer. An equation similar to the transport Eq. (2.5) in the liquid phase (see ,J-phase in Fig. 2.2) can be obtained d[H2S]aq 8 d[H2S]aq . A aw . . r r r .-,1 v ~ d F ~ ~ ^ ( D , - 1 ^ ) + ^ T ^ - ( R H [ H 25]g - [ H 2^ q) — K B[H2S}aq (2.11) 2.2.4.1 Dissolved Sulfide Boundary C ondition on the Pipe Surface T he boundary condition for [H2S]aq on th e surface of pipe {x = 0) is a D irichlet condition. If the partial pressure of sulfide changes, the soluble fraction of sulfide also changes. However, the partial pressure of sulfide in the air phase was held constant in this model. Thus, [//2S]a,(0,f) = K h [H2S]g(Q, t) (2.12) Assum ing small changes in tem p eratu re and partial pressure of hydrogen sulfide, the H enry’s Law constant remains unchanged. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.4.2 Dissolved Sulfide Boundary C ondition on the M oving Interface On the moving boundary, x — s(t), the flux of hydrogen sulfide in the liquid phase was also considered negligible due to low perm eability of concrete. Therefore, a boundary condition of the second kind, a N eum ann condition, was imposed: = 0 (2.13) where Dw is effective diffusion rate of hydrogen sulfide in the liquid phase (cm 2/day) and [fif2 - 5’ ]a?(s(f). t) is the liquid hydrogen sulfide concentration on the moving bound ary. 2.2.5 H ydrogen and Sulfate Ions in Liquid P h a se Hydrogen and sulfate ions are both released as biological oxidation of sulfide occurs, and consum ed in the corrosion process. T hese ions react with th e calcium carbonate in the concrete to form gypsum 2HoO + H + + SO\~ + CaCOz ^ CaSOA ■ 2 H20 + H C O J (2.14) where Kc is dissolution rate constant for calcium carbonate (cm /d ay ). In this equa tion. the hydrogen ion is the rate controlling species. Because of the concentration gradient between the surface and the m oving boundary where calcium carbonate ex ists in abundance, the hydrogen ions diffuse through the corrosion layer towards the moving boundary. Here the corrosion layer is assum ed to consist m ainly of gypsum which does not react w ith hydrogen ions. A nother source of hydrogen ions is as sum ed to be due to the biological activities present in the gypsum layer. Therefore, in this layer one can assum e diffusion and form ation of hydrogen ions sim ultaneously: “ a r = + (2-i5) where Dh is the effective diffusion rate for hydrogen ions in the liquid phase ( cm 2/d a y ). For the solution of this equation, two boundary conditions are needed; hydrogen ions concentrations on the surface of the pipe and the m oving interface. IS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.5.1 H ydrogen Ion B oundary C ondition on the P ip e Surface Before the concrete pipe becomes exposed to the hydrogen sulfide and undergoes the corrosion process, the pH on th e surface on concrete above the w ater level is well above 7.0. After the exposure to sulfide, vigorous bacterial activity a n d the production of sulfuric acid will lower th e pH of the surface to 1 over a p erio d of tim e. Islander et al. [22] and [23] suggested th a t the acid production lim ited the activ ity of organism s and no evidence of fu rth e r sulfur oxidation was observed when the surface pH was between 0.5 to 1.0 i.e., K b = 0. Furtherm ore, th e tim e it takes the surface pH to fall to 1.0 m ay be different in various sections of a sewer. Therefore, in this model, pH of the surface was assum ed to be 1.0 and the num erical sim ulations were obtained based on a fixed surface pH. According to Eq.(2.S), one mole o f sulfate is produced for every mole of sulfide reacted. Given a surface pH of 1.0, th e m olar concentration of hydrogen ions o n the surface i.e., [/f+](:r = 0,£), will be 10- 1 m o les/liter. For the purpose of sim ulation, this concentration may be changed to stu d y th e effect of hydrogen ions and su lfate on th e corrosion rate. Ultim ately, a tim e-varying boundary condition m ay be necessary in studying the effect of surface coatings a n d o th er corrosion prevention m eth o d s. It has been shown [24] that when the surface pH falls below a critical value of 4, the rate of corrosion increases exponentially. 2.2.5.2 H ydrogen Ion Boundary C ondition on the M oving Interface T h e boundary condition on the m oving interface, x = s(t). can be o b tained using a m ass balance. A fraction of the hydrogen ions , a, reacts w ith concrete to form corrosion products. The rem aining m ole fraction of hydrogen ions, 6 . m oves along w ith the interface at a speed proportional to th e corrosion rate, ds/clt. A ssum ing th a t there are no sources of hydrogen ions generation on the interface and no diffusion at th e interface, the concentration balance for hydrogen ions takes the form: = a K c ^ + ( 2 - 1 6 ) 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here D m is the diffusion rate constant for hydrogen ions in the liquid phase (cm 2/day), a is th e num ber of moles of f f + th a t will react w ith 1 mole of th e calcium car b o n ate on the moving interface, b is th e rem aining mole fraction o f H + moving along w ith the interface, and K c is dissolution rate of calcium c arb o n ate (cm /day). P om eroy [25, 12] showed th a t the corrosion rate is directly proportional to the flux of sulfide in the sewer and inversely proportional to the alkalinity. H ere, a sim ilar rate equation has been adopted and an intuitive expression for the corrosion rate is given Tt = lJU T ' * > °- (2'l7> w here C c is alkalinity of concrete (m oles/liter). Initially, the corrosion interface may s ta rt at a prescribed location i.e., s ( 0 ) = So- 2.2.6 M odel Sum m ary T h e governing transport equations derived above form a system o f th ree partial differential equations and an ordinary differential equation prescribing th e corro sion rate. These equations together w ith the 6 boundary conditions and 4 initial conditions are sum m arized here: ~ 1 = h (D ° r ^ i r L) ~ K r i f 9 [ H o S ] a q ^ / n Q [ H z S ]a ( ? d a u / . . . — J + l\T-rr-(L\H[tl2j\g ~ [xl2X> dt dx dx V . W - K B[H2S}aq (2.19) aj l r “ (i-20) ± = A c I f i W 0 , t ) (-2.21) 0 < x < s(t ), 0 < t < T boundary conditions at x = 0 are: [H 2 5 ],(0 ,f) = A, ( 2 .2 2 ) 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [tf2S ]a9 (0 ,f) = a 2 (2.23) A 3 (2.24) where AL , A2, and A 3 are fixed concentrations and assum ed to be constant over the tim e period t. Furtherm ore, assum ing th a t all of hydrogen ions are used up in reaction w ith concrete, the boundary conditions at the moving interface x = 4 t) can be w ritten as: = o (2.25) = o (2.26) _ D " ^ r (s(!)’< ) = K c ;[Ff+](s(f),f) (2.27) 0 < t < T initial conditions are: [ff2 S ]„ (x ,0 ) = Al (2.28) [Ff2S']a,(x ,0 ) = a 2 (2.29) [tf+ ] ( x , 0 ) = A 3 (2.30) 5(0) = 50 (2.31) 0 < x < sq 0 < t b n VI constants used in the model are: Da = diffusion constant for [H2 S]g (cm 2/d ay ), Dw = diffusion constant for [FA Ala, (cm 2/d a y ), D h = diffusion constant for [Ff+] in w ater (cm 2/d ay ), A t = m ass transfer coefficient of [A2 -5 ,j3_J .a, ? (cm /d ay ), — exposed surface area of w ater (cm -1 ) of concrete (i = a or w ), Kfi = modified H enrv:s Law constant for [H2 S]aq, 2 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Kg = biocoaversion rate constant (day-1 ), Kc = calcium carbonate dissolution rate constant (cm /d ay ), .s(O) = location of interface at initial tim e inside concrete (cm ), Cc = calcium carbonate concentration in concrete (m o les/liter), = air phase sulfide concentration on pipe surface (ppm ), A 2 = liquid phase sulfide concentration on th e pipe surface (m o les/liter). A 3 = hydrogen ion concentration in the liquid phase (m o les/liter), the m odel outputs are: [HzSjgix.t), [ff2 S]a,(x , t), [H+](x, t), s(t ) Using the equation pH (x.t) = —log[H^](x,t), the pH can also be com puted. T he resulting equations can be solved using several num erical techniques. In particular, finite elem ent m ethod (FEM ) is used in this thesis to solve these equations. A description of the FEM technique and further m odification of th e m odel equations is given in the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p ter 3 N u m e r ic a l M o d el a n d S im u la tio n M e th o d In this chapter, the model Eqs.(2.1S)-(2.21) are modified using the fixed-coordinate tran sfo rm atio n of the variable x and th e finite elem ent m ethod (FEM ) is introduced for the purpose of obtaining sim ulation results. The m ethod of transform ation is as follows x € [0 ,s(f)] = [0 . 1], s{t l where th e length of corrosion layer is norm alized to 1. Also for convenience, the following change of variables are m ade: u(y,t) = [H2S ]5 ( x , f ) - A 1: v{y,t) = [H2S]aq(x,t) — A 2, w(y,t) = [tf+](x ,f) - A 3. A system of equations on the fixed dom ain, 0 < y < 1 is obtained. In the sim ulation results presented here, the concentrations A ,-, i = 1,2,3 defined in Eqs.(2.22)- (2.24) are assum ed to be constant. T he m odel equations given in Eqs.(2.IS)-(2.20) becom e du Da d 2u t ^ , i(f) du rK TA aw Tr ^ T t {y' t] = W ) ^ ) + W f~d~y{y' ) ~ [ ^ r k H )u {y - t) (3-D ^ a 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he boundary conditions given in Eqs.(2.22)-(2.27) become u(O.f) = 0, = 0 , (3.5) s(i) ay u(0,t) = 0 , - ^ ^ ( 5 ( 0 , 0 = 0 , (3.6) u>(0 , t) = 0 , Du dw = K c(w(s(t),t) + X 3), (3.7) t > 0 , T h e initial conditions given in (2.28)- (2.31) now take the form: u(y, 0) = 0, (3.S) v(y, 0) = 0, (3.9) w{y,0) = 0, (3.10) s ( 0 ) = Sq. (-3.11) 0 < y < 1 T he classical numerical m ethod for p artial differential equations is the difference method where the discrete problem is obtained by replacing derivatives w ith differ ence quotients involving the values of the unknow n at certain (finitely m any) points. T he discretization process here is based on the finite elem ent m ethod (FE M ). To 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solve the m odel equations approxim ately using finite elem ent m eth o d [26]. the fol lowing steps are necessary (i) variational form ulation of the given problem , (ii) discretization using FEM : construction of the finite dim ensional space, Vh, (iii) solution of the discrete problem , (iv) im plem entation of the m ethod on a com puter: program m ing. Mote th at in this study, a one-dim ensional finite elem ent num erical m ethod was used to solve above equations. However, the possibility of using a two- dim ensional finite elem ent in polar coordinates may be m ore advantageous as com pared w ith the finite difference or any other m ethod. Let 0 = y0 < t/i ■ ■ - < y\r < Um+i = 1, be a p artitio n of th e interval (0,1) into subintervals Ij = (yj-i, yj) of length hj = yj — y j - i , j = 1 , . . . , M + 1. T he quantity h is a m easure of how fine the partition is. If Uh is the set of linear and continuous functions u ( e.g., H 2 S concentration ) on each subinterval Ij an d u(0 ,f) = 0 (see Fig. 3.1). Then, to describe the function u E Uh on the interval [0,1], choose the values Uj{t) = u(yj. t) at the node points yj, j = 0 , . . . . M + 1 . Introduce the basis functions ipj E £//,, j — 1 , . . . , M, defined by { 1 if i = j where ^> j is a continuous piecewise linear function ( linear spline ) th at takes the value 1 at the node point yj and the value 0 at other node points (see Fig. 3.1). Thus the function u E (J\ has the representation: M u{y-. t) = Ui(t)<pi(y), y E [0,1] (3.12) {=1 where Uft) = u(yi,t), i.e., each u E Uh can be w ritten in a unique way as a linear combination of the basis functions y> {. In particular it follows th a t Uh is a linear space of dimension M with basis Let 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 I % Figure 3.1: (a) Exam ple of a function u £ 0 \. (b) The basis function, ipj. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. un{y,t) = t>o, o < y < 1 , j =i w "(y,f) - E V 7 (0 v » “(y), « > 0 , 0 < y < 1 , j= i u T (y ,f) = t > 0 , 0 < y < 1 , j = i and 0 -"(t) = [CT(t), <7,"(0.~, C C W f e f f i" , v“(o = [vrw, v2 nw,..., k t w F s k ”, w--(() = [»•"((), w ? (0 ,..., WT(()]r e ffi". T h e Galerkin equations are th en given by A /-C H 0 = + f!|h £ » £ T > (() - + { A T ^U «,jiV r l/„ W i (>0> ( 3 [ 3 ) a .vf’d ) = _ J^-A-rw + ^|iT(tj + {^=A'w }«*t’W + fi"s}:W” V“(f) - a'ba;, t > o. * L L ’ + K B M nV n(t) + K BXZ, t > 0, i ”(0 = + t>0, (3.16) (3.17) and the initial conditions /7n (0) = 0 (3.IS) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Kn(0) = 0 W'n(0 ) = 0 5n(0) = 5o (3.19) (3.20) (3.21) where M n, k'n, Ln G IRnXn and A£, A?, en € R n are given by M n = - I A q 0 0 0 0 0 h o 0 0 0 A I I 0 0 0 0 0 I A 2 - 1 0 0 0 0 0 1 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 0 K n -- n 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 1 i [Ln]ij = Jy<p?{y)ipj(y)'dy, i j = l,2 ,...,n , or o 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ln = - n 3 3t—1 ^ 0 _ I 3 t + l 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3i —1 3:4-1 6 3i —1 3: —1 6 6 A? = A2[ - , 1 ...K ± ] T. - -n 7 n ■ n • 2n.-> - A» = A3[A I ...A . f ] T. 3 •H n ’ n 1 n ' 2n* and en = [0 . 0 , ..., 0 . 1 ]r . Note th a t the m atrix M n is positive definite and diagonally dom inant. So inversion is sim ple. To obtain the results th a t will be presented below, the initial value prob lem Eqs.(3.13)- (3.21) was integrated on a SUN SPARC10 using the IMSL routine DIVPAG. This routine is an im plem entation of an Adam s-Gear stiff system solver. T he m atrix M n was inverted via banded Choleski decomposition. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p ter 4 N u m e r ic a l S im u la tio n o f C orrosion a n d R e su lts T he num erical results were obtained based on the param eters specified in Table 4.1. Som e param eter values are from theory and no experim ental d a ta have yet been published. O thers have been estim ated in laboratory experim ents. In addition, there is no d a ta available for the bioconversion rate of hydrogen sulfide, K b - The values chosen are an extrapolation based on the studies perform ed by YVilmot et al. [20] at m uch higher values of and pH in th e w astew ater. In order to test the m odel using the known values from literature, all of th e param eters listed in Table 4.1 were considered constant throughout the corrosion layer. Y V e also made the following standing assum ptions. • Corrosion product (gypsum ) form ed on the surface of pipe is not rem oved. • No other reacting or com peting species besides the sulfuric acid co n tribute to the corrosion process. • T he m ajor raw m aterial used in construction of concrete pipe is calcium car bonate, CaCO^. • The flux of sulfide species across the moving boundary is negligible and there fore zero at the boundary. • The adsorption of sulfide gas in concrete is less th an 2 % and considered negligible. • The tem p eratu re inside the pipe is 25° C throughout th e experim ent. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P a r a m e te r s V alu es tortuosity 2 0 . 6 6 (dim ensionless) air-filled porosity. Va 2 0 (%) water-filled porosity, Vw 2 0 (%) diffusion rate of H2S in air phase2, Da 8640.0 (cm 2/d ay ) diffusion rate of H2S in w ater phase2, Dw 0.S640 (cm 2/d ay ) diffusion rate of H + in w ater phase2, D u 0.8640 (cm 2/d ay ) surface area to volume ratio, air2 ^ L l x 1 0 4 (cm -1 ) surface area to volume ratio, liquid 2 l x 1 0 4 (cm -1 ) mass transfer coefficient, K t 8640.0 (cm /day) m odified Henry’s Law co n stan t6, K u 2.50 (dim ensionless) bioconversion rate2, K b 72 (day-1 ) [H2S]a{QA) = 5.0 (ppm v) [f/2‘ 5]a9(0. t) = X2 5.1 xlO - ' (m oles/liter) [H+](0A) = X3 1 0 - 1 (m oles/liter) corrosion thickness, s( 0 ) 0 .1 (cm) dissolution rate2, Kc 0.84 (cm /day) Cc alkalinity of concrete pipe 2 . 0 (m oles/liter) [H2S]a(x,Q) = A f 5.0 (ppm v) [H2S]aq{x, 0) = A 2 S .lx lO - 1 (m oles/liter) [H+](x ,0) = X3 1 0 - 1 (m oles/liter) Table 4.1: Numerical d a ta for several param eters. Some param eter values were obtained from the following references: a [1], b [2], c [3], d [4], and e [5]. • For the purpose of testing th e values given in Table 4.1 and their variations, the pipe thickness is assum ed infinite. • T he m odel closely represents the actual corrosion process. Any param eter values chosen m ust be w ithin physically reasonable range. T he purpose of this effort is to identify some of the m ore im p o rtan t param eters and their role in corrosion of concrete sewer pipes. Estim ating the param eters is only of value when there is indication th a t they strongly influence the process of corrosion. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this study, several sim ulations were perform ed based on th e values given in Ta ble 4 .i and the m odification of certain param eters thereafter. In these experim ents, the param eters were changed to new values and the model perform ance was then observed. The param eters of interest were porosity, hydrogen sulfide concentration on th e surface, bioconversion rate, K b , dissolution rate, Kc, expansion of corrosion product into the pipe, corrosion rate, and the pH of the surface. These num erical experim ents represent a sim ulation of the effects of the various param eters in Table 4.1 on the corrosion rate. T he model was run over a 25-year period. The calculations were m ade on a daily basis and the plots were m ade based on com pilation of d a ta at the end of every fifth year. T he results were plotted versus concrete depth for [ # 2 - S ’ ]a? 5 and pH. In each plot, the results are shown in 5-year intervals. N ote th a t the concentrations for [HoS}g were given in parts per million by volume (ppm ) and for [H2 S]aq were expressed in m oles per liter. Also, the corrosion depth (gypsum ) was plotted as centim eters of concrete versus the tim e in years. S im u la tio n 1: E ffects o f P o ro sity The porosity of concrete is in the range of 6 to 10% [27]. It is noted that with porosities greater th an about 1 0 % most of the porosity is in an interconnected form, and with porosities less th an about 5-10% most of the porosity is of the closed or isolated type. The reason for this behavior can be found in statistical laws which have been used quantitatively in percolation probability theory. In a sewer pipe exposed to acid attack , the pores may become interconnected an d porosity increases in the corrosion layer. An increase in porosity m ay affect th e corrosion rate by allowing penetration and retention of w ater or other liquids. In this sim ulation, the porosity for each phase was changed from 10 to 30 percent and the concentrations for each phase were com puted accordingly. The volume fractions were assum ed to be equal for this sim ulation. All o th er param eters in Table 4.1 were held constant. The plots in Fig. 4.1(a,b) indicate th a t the concentrations of sulfide in both phases decreased at a given depth as the corrosion depth increased. This is due to an increase in volume and drawdown of the concentration profiles. Com paring 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e concentrations of both species, it can be seen th a t they were approxim ately th e sam e for different porosities. F urtherm ore, in twenty-five years, the position of corrosion front changed from 7.21 cm to about 12.4 cm as the porosity was changed from 10 to 30 percent. This is an indication th at, based on porosity changes, sulfide concentration on th e surface had no significant effect on the position of th e m oving boundary. Therefore there existed som e o th er factor(s) which influenced the position of th e m oving boundary. T he plots in Fig. 4.2(a,b) indicates th a t the pH on the surface of concrete was kept at 1 and at the moving boundary was com puted to be 2.78. T he sm all differences in p H indicates th a t the diffusion of hydrogen ions is very rapid and its value is in acidic range up to the surface of concrete where the effect of neutralization by alkaline species of the concrete may be m uch larger. Mori et al. [21, 28] have m easured pH in th e range of 6-7 in the region betw een the corrosion product and the uncorroded concrete sam ples obtained from sewers. It is noticeable that th e corrosion front shows strong dependence w ith respect to pH. T he corrosion thickness produced in this sim ulation was approxim ately 0.5 c m /y r. O ne of the assum ptions m ade in these sim ulations was that the gypsum layer re m ained in place and therefore provided some protection from corrosion. T his ar gum ent is quite reasonable and m ay be noticeable in Fig. 4.2(b) where the ra te of corrosion is changing as a function of th e thickness of the gypsum layer. S im u la tio n 2: E ffects o f th e B io co n v ersio n R a te C o n sta n t T h e bioconversion rate constant, K b , is a m easure of how fast the dissolved sulfide can be converted into other chem icals by the bacterial activities. So far, there is no literatu re directly supporting the values chosen here in this sim ulation. However, in order to study the effects of changes in the bioconversion, the rate values were changed from zero to a high of 72 day - 1 [20]. Mori et al. [21, 28] showed th at bacteria cells responsible for corrosion of sewer concrete pipes penetrated th e corrosion layer to significant depths. Therefore, one can assum e th a t there is bacterial activ ity in at least some part of the water-filled pores in the gypsum layer. In this sim ulation it 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 20 C O CO o CM ^ X o 0 5 10 5? 4 CM £ 2 20 25 10 - 7 x 10 2 6 c r CO 4 CM £ 2 - 7 x 10 Q-4 CO n CM X 0 5 10 concrete depth, cm co 4 CM £ 2 concrete depth, cm Figure 4.L: C oncentration profiles at three porosities. W ater and air- filled porosities were equal and the rem aining volume was reaction product. From top to bottom : 10 %, 20 %, and 30% in each phase. Numbers on the plot indicate tim e, in years. Plots were obtained from d ata at the end of each five-year interval. W here each curve stops is the moving interface between the corrosion product and the uncorroded concrete. See Table 4.1 for other param eter values. Plots indicate direct correlation betw een porosities and corrosion layer m easured as concrete d ep th . 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 1 0 1 5 20 fS 0 0 5 10 4 0 0 5 10 4 0 0 5 10 concrete depth, cm 10 5 0 0 5 10 20 15 10 5 0 0 5 10 20 15 E ° _ 1 0 cf C /3 2 5 o C J 20 time, years Figure 4.2: (a)pH calculated up to th e m oving boundary and the corrosion thickness as centim eters of concrete for different w ater and air-filled porosities. From top to bottom : 10. 20, 30 % in each phase. N um bers on the plot indicate tim e, in years. Plots were obtained from data at th e end of each five-year interval. See Table 4.1 for other param eter values. pH on th e surface of concrete is fixed to 1 and rises to near 3 on the moving boundary as a result of th e effect of alkalinity on neutralization of acid in water-filled pores. (b)In all three cases, corrosion rate is high initially and decreases with time. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was assum ed th a t the bioconversion rate is constant throughout the corrosion layer. The rem aining param eters in this sim ulation were held constant as in Table 4.1. In Fig. 4.3(a,b), as the bioconversion rate K b was increased, more sulfide was consum ed by the bacteria. This effect was enhanced a t a K b value of 72 d ay -1 . The change of K b from 0 day - 1 to 72 day - 1 increased the corrosion d ep th from 0.1 cm initially to 12.34 cm over 25 years. This in d icated th a t an increase in the rate m ade a significant contribution to the corrosion. Furtherm ore, there were two possibilities th at could have enhanced or reduced the effect of the bioconversion rate on the corrosion rate: (a) if the sulfide concentration in th e air was significantly higher th an 5 ppm; m ost of the concentrations reported in [4] were in the range of 2 to 15 ppm or (b)the bioconversion rate exceeded the range used in this experim ent. The studies conducted by W ilm ot et al. [20] on w astew ater obtained a bioconversion rate of 72 day - 1 for pH values as low as 4. In Fig.4.3(a.b), pH as a function of corrosion layer and corrosion thickness in centim eters of concrete as a function of tim e were p lo tted . In Fig. 4.3(b), the plot of corrosion showed strong nonlinear behavior in the first five years. T h e graph indicates a decrease in the corrosion rate as a function o f tim e. S im u la tio n 3: E ffects o f th e D is so lu tio n R a te C on stan t In this sim ulation, the calcium carbonate in the concrete was assumed to have been m ostly consum ed to the left of the corrosion interface such th a t anything left in the corrosion layer did not affect the reaction w ith sulfate and hydrogen ions. Therefore, the effect of the dissolution rate, K c, was only tested based on the q u an tity of calcium carbonate present on the interface. This assum ption was made to the extent th at the concrete is a very dense m aterial and therefore dissolves at a constant rate. In this sim ulation, the value of 0.S4 cm /d ay was chosen. This value is the sam e as the experim ental results m easured by Devinny [5]. A ccording to Devinnv, for a given rate constant determ ination, this rate constant was inversely proportional to the surface area assum ed to be available in the concrete pores and was generally expected to be lower on the basis th a t concrete is porous and has a large surface 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 5 10 0 5 10 - 7 x 10 20 0 5 10 2 6 cr C O 4 CM S.2 10 20 25 0 5 10 - 7 x 10 n 3 p CO * CM £ o 0 10 5 •24 Si 2 0 5 10 concrete depth, cm concrete depth, cm Figure 4.3: C oncentration profiles obtained for different values of bioconversion rate. K b - From top to bottom . K b — 0.0, 7.2, and 72 day-1 . N um bers on the plot indicate tim e, in years. See Table 4.1 for o th er param eter values. P lots show no corrosion form ed for K b = 0. M iddle plot shows extensive corrosion. B o tto m plot shows very little increase in corrosion as com pared w ith the m iddle plot w hen bioconversion rate. K b , was increased by ten fold. T he concentration profiles show stead y -state behavior for large bioconversion rates. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 0 0 5 10 E ° 10 cf o c o o W o o o 5 10 20 15 4 2 5 0 0 5 10 E ° 10 cf o C O o w o u 20 4 2 5 0 0 5 10 E u 10 cf o C O o w o o 0 5 10 20 15 concrete depth, cm time, years F igure 4.4: (a)pH calculated up to th e m oving boundary and the corrosion as cen tim eters of concrete for different bioconversion rates, Kb- From top to b o tto m , Kb = 0.0, 7.2, and 72 day-1 . Num bers on th e plot indicate time, in years. See T able 4.1 for o th er param eter values. (b)In th e m iddle and bottom plots, corrosion rate is high initially and decreases w ith tim e. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. area. Here it is assum ed th a t calcium carbonate has larger surface area than in the Devinny's m odel. Thus the dissolution rate, K c , was lowered by a factor of ten to 0.00S4 cm /d av and the sulfide concentration was com puted. The alkalinity of concrete was chosen to be 2 . 0 m oles/liter as calcium carbonate at each step of calculation. All o th er param eters were chosen according to Table 4.1. The results indicate th a t as the dissolution rate decreased from 0.S4 cm /dav to 0.00S4 cm /dav. th e corrosion depth also decreased from 12.34 to 1.90 cm. For K c values lower th a n 0.S4 cm /day, sulfide concentration profiles were very sim ilar near the surface of concrete (Fig. 4.5(a,b)) . This im plied th a t sulfide concentration had steady-state behavior throughout the concrete depth for K c values larger than 0.084 cm /d ay and transient behavior for K c larger th an 0.84 cm /dav. These results indicated th a t the diffusion rate for sulfide was faster th an th e dissolution rate Kc at values near the m easured experim ental values. The pH profiles proved to be different from the previous sim ulations in most of the plots (Fig. 4.6(a)). At lower dissolution rates, the p H profiles were indistin guishable. This im plied th at the diffusion rate of hydrogen ions exceeded the rate of dissolution at low values. The changes in dissolution rate showed striking effects in the corrosion rate (Fig. 4.6(b)). In the case of K c = 0.0084 cm /day. the plot of corrosion depth appeared to be linear. This m ay have some im portant applications in the designing of a pipe when a constant rate of corrosion is m ore favorable th an in the case of a nonlinear rate. It is reasonable to assume th at the corrosion rate is constant and an estim ate of pipe thickness m ay then be calculated based on the results shown in Fig. 4.6(b). In this sim ulation, the results indicated th at decreasing the dissolution rate by a thousand tim es reduced the corrosion thickness S5 percent. S im u la tio n 4: E ffects o f Surface p H In this sim ulation, the effect of changes in the pH of conrete surface on the corrosion rate was tested. T he surface pH was varied in the range of 1 to 4. All other param eters in Table 4.1. were held constant. At surface pH of 4.0. there was hardly any corrosion form ed in the concrete and very little hydrogen sulfide diffused into the surface of concrete (Fig. 4.7(a,b)). As the pH of surface was lowered, more 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 C O rt C O C\J o 5 10 6 E Q. 0-4 ( g C O p cm ^ X 0 0 5 10 6 E Q. CL 4 eg C O p C M £l X o o 5 10 concrete depth, cm x 10'7 S! 2 0 5 10 - 7 x 10' 2 6 cr C O 4 C M £ 2 .20 25 -7 x 10 C O 4 C M £ 2 concrete depth, cm Figure 4.5: Concentration profiles obtained, for different values of dissolution rate. Kc- From top to bottom . K c= 0.S4, 0.084, and 0.0084 cm /day. N um bers on the plot indicate time, in years. See Table 4.1 for other param eter values. In th e m iddle and b o tto m plots, corrosion is slowed down at lower dissolution rates. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 25 0 10 0 5 5 0 10 0 5 15 20 2 0 10 0 5 10 5 0 10 15 5 20 0 E ° 10 c o CO S 5 w o o 0 0 5 10 0 5 10 15 20 concrete depth, cm time, years Figure 4.6: (a )pH calculated up to the moving boundary and the corrosion as cen tim eters of concrete for different dissolution rates, K c- From top to bottom , K c = 0.S4, 0.0S4, and 0.00S4 cm /dav. N um bers on the plot indicate tim e, in years. See Table 4.1 for p aram eter values. T he effect of acid n eu tralizatio n in the middle plot is not as strong as th e top plot, (b) Middle and b o tto m plots show lower corrosion rates than the top plot initially. Furtherm ore, the corrosion shows linear correlation w ith tim e a.t lower dissolution rate. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. hydrogen sulfide diffused into th e surface of the concrete and more corrosion product was form ed. The results show th a t sulfide concentration at the m oving boundary was near zero. An alm ost linear relation was observed for the corrosion rate at a surface pH g reater than 2 (Fig. 4.8(b)). T he corrosion thickness in this sim ulation varied be tw een 0.3 and 12.34 cm as the p H on the surface of concrete was lowered from 4 to 1 . S im u la tio n 5: E ffects o f C on crete A lk a lin ity In this sim ulation, the effect of changes in concrete alkalinity on th e corrosion rate have been tested. Alkalinity was varied in the range 0.2 to 10.0 m oles/liter at the m oving interface for a fixed pH on the surface of concrete in th e range of 1 to 3. T he plots of alkalinity versus corrosion depth are shown in Fig. 4.9. The results Indicate th a t reducing the alkalinity of concrete increased th e corrosion layer d ep th of the concrete. This observation agrees with the experim ental m ethod used by Pom eroy [12] to show th at the rate of corrosion is inversely proportional to the alkalinity of the pipe. L ater Pom eroy [6 ] showed th a t the rate of corrosion of cem entitious pipes can be predicted according to the equation: _ 0.45 kSsw C.„, = -------------------------------------------------------(4.1) w here C'avg is average rate of p enetration (in./year), k is coefficient o f efficiency for acid reaction (a num ber between 0.3 and 1.0). dsw is flux of HoS to the pipe wall in g /m 2-hr. and A is alkalinity of concrete as calcium carbonate equivalent. In the above equation, k and d sw change based on the w astew ater characteristics as well as the location and environm ental conditions. Pomeroy [6 ] suggested th a t to use this equation, one may need to carefully m onitor the conditions inside a pipe and therefore use sound engineering judgm ent in estim ating a value for k. In fact, this equation has underpredictecl the corrosion rate com pared w ith those m easured in the Sacram ento Study [7]. Because of the experim ents conducted here, it is now clear th a t the hydrogen sulfide flux alone at levels m easured in field experim ents has very little effect on the rate of corrosion. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 0 5 10 25 0 5 10 6 i ---------------------------- — e - C L C L 4 - iS C g C O O ■ CM ^ X o'-------------- ----------------- 0 5 10 concrete depth, cm x 10-7 0 5 10 - 7 x 10 2 6 cf 5T4 C V J £ 2 10 - 7 X 10 2 6 £ 2 O '-------------- ----------------- 0 5 10 concrete depth, cm Figure 4.7: C oncentration profiles for different values of pH on the surface of con crete. From top to bottom , pH = 1 , 2, 4. N um bers on the plot indicate tim e, in years. See Table 4.1 for other param eter values. Top plot shows the largest corrosion am ong the three selected pH 's. At pH of 4, there is only a slight corrosion observed. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 0 0 5 10 4 0 0 5 10 0 5 10 concrete depth, cm ° 10 C O 20 E o cf J O C O o o o 10 5 0 15 10 20 0 5 10 15 time, years Figure 4.8: (a)pH calculated up to the moving boundary and the corrosion as cen tim eters of concrete for different values of p H on the surface of concrete. From top to bo tto m , pH = 1 , 2, and 4. Num bers on th e plot indicate tim e, in years. See Table 4.1 for other param eter values.(b) Corrosion rate a t p H of 1 is the highest am ong the three plots. T he plot of corrosion shows linear correlation w ith tim e at p H of 2 or higher. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Here, it is shown th at m uch of th e corrosion rate is due to p H on the surface and th e alkalinity of the pipe. Therefore, a new equation for corrosion is proposed: ds r, [H +} T t = 1<C- c f - < 4- 2> w here ^ is corrosion rate in cru/year, K c is dissolution rate of calcium carbonate in the concrete in cm /day, [H+] is the hydrogen ion concentration in m oles/liter, and Cc is the alkalinity of concrete in m o les/liter. T his new equation relies on the fact th a t th e pH on the surface of concrete m ay reach as low as 1.0 and is proportional to the hydrogen ion concentration at th e surface. T he above studies suggest th at th e corrosion rate may be constant w hen certain physical param eters are changed or controlled. Therefore, the equation can be fu rth er sim plified into ds K r T t = c f ' «-3> w here K'c is Kc [H+\. T he dissolution rate, K c , is a factor depending on the q u an tity of calcium carbonate present in the concrete and its availability for reaction w ith acid. Therefore, as was shown by D evinny [5], its value can be determ ined experim entally. In addition, the eq uation is independent of the hydrogen sulfide flux proposed by Pomeroy earlier [6 ]. S im u la tio n 6: R e la tiv e C o n tr ib u tio n o f S y stem E q u a tio n s to th e C o rro sio n R a te In sim ulation 2. it was shown th at th e bioconversion rate and oxidation of sulfide were im portant in the production of hydrogen ions and corrosion of concrete. At the sam e tim e, the change in K b (7.2 to 72 per day) did not affect the thickness of corrosion layer significantly. In this m odel, the bioconversion term , A’ sf/fo-Sja,. appears only in Eq. (2.20), so its relative contribution to the corrosion was tested by setting it to zero. T he concentration profiles, pH , and th e corrosion thickness were identical to those obtained in sim ulation 2. This sim ulation showed a 4% decrease in the thickness of corrosion over 25 years. This finding suggests th at the relative contribution of Eqs. (2.18)- (2.19) to th e corrosion process may be small. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 0 25 - E o CO £ 2 0 in C \J 0) lo 115 C D *0 CD > > _C0 o 10 CO o o o .. surface pH = 1.0 surface pH = 2.0 - surface pH = 3.0 5 - 4 5 6 alkalinity as CaC03. M 10 Figure 4.9: Corrosion layer depth com puted after 25 years based on several values of concrete alkalinity and fixed pH on the surface o f concrete. Corrosion rate is inversely proportional to alkalinity and d i r e c t l y proportional to th e pH on the surface of concrete. Plots show th at increase in alkalinity provides protection against corrosion. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Furtherm ore, the hydrogen ion diffusion equation (see Eq. (2.20)) m ay be the most significant transport equation in the corrosion process. However, an experim ental procedure m ay be necessary to m easure K b before Eqs. (2.18)- (2.19) are dropped in the future studies. S im u la tio n 7: E ffect o f W ater C o n ten t o f C o n crete P o res on C orrosion Layer T h ick n ess The chem ical reaction between sulfate, hydrogen ions, and the concrete takes place in the liquid phase of the pores. This process m ay be slowed in the presence of other cem ent m aterials such as sand, gravel, and non-reactive species. In order for the calcium carbonate of concrete to become available for this reaction, sufficient ions of the reactive species i.e., sulfate and hydrogen ions, m ust become available. W ater is a strong solvent. Its presence, even in m inute am ounts, will prom ote corrosion. This sim ulation is to show the effects of w ater content of the pores on the cor rosion. In Table 4.1, the air-filled porosity was fixed at 1 % and th e water-filled porosity was increased accordingly in intervals of 10 beginning with 10 %. T he re sults indicated a strong relation exists between the w ater content and th e corrosion layer build-up. In Fig. 4.10, increase in the w ater content of the pores in the concrete increased the corrosion layer thickness from 4.47 cm to 7.82 cm (75 percent). This result may indicate th a t the relative hum idity inside the pipe plays a greater role than m any param eters th at have been tested so far. It is highly recom m ended to develop m ethods to control the relative hum idity in order to control the corrosion rate. S im u la tio n 8: C a lcu la tio n o f th e C orrosion R a te Pomeroy [6 , 7] developed a rate equation for corrosion in sewer pipes (Eq. 4.1). The rate was directly proportional to the hydrogen sulfide flux to the wall of the pipe and inversely proportional to the alkalinity of the pipe. The equation assum ed sulfide flux to be constant. T he assum ption was not necessarily true because the flux as was shown by Pomeroy depended on the w astew ater flow velocity, th e hydraulic 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. water-filled porosity - 1 0 % 2 0 % — 30 % ... 40% C O 20 25 10 time, years Figure 4.10: Corrosion layer thickness com puted based on several water-filled porosi ties. ew. All o ther param eter values same as in Table 4.1. Larger corrosion thickness is due to higher w ater content of concrete pores. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. gradient, and turbulence. In fact, in the study of Sacram ento sew er lines [7], the observed corrosion was as m uch as 1 0 to 2 0 times the calculated ra te . In this sim ulation, the corrosion ra te was obtained using very sim ilar model. As stated previously, the model equation for th e corrosion rate is directly proportional to hydrogen ions, and inversely proportional to the alkalinity of the pipe. T he constant of proportionality is the calcium carbonate dissolution rate. T he ra te equation varies w ith respect to the hydrogen ion concentration at the interface an d therefore is a m ore appropriate model of the corrosion process. T he results o b tain ed here show th a t th e corrosion rate is not entirely constant (Fig. 4.11). T h e rate decreases exponentially as a function of tim e. T he rate is considerably higher when th e water content is high. At 40 % water-filled porosity, the rate is ab o u t tw ice as great as the rate at 10 % in the asym ptotic region. The effective diffusivity is directly proportional to the volume fraction of water-filled pores. As w ater content of the pores increased, the diffusion rate also increased. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 water-filled porosity 1 0 % 1.6 30% 40% £ 1.2 0.8 0.6 0.4 0.2 25 time, years Figure 4.11: Com puted corrosion rate of concrete based on several water-filled porosities. t w. All other param eter values sam e as in Table 4.1. The rate decreases and becom es asym ptotic with tim e. Larger corrosion rate is due to higher w ater content of the concrete pores. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter 5 C o n clu sion s an d F u tu re R esea rch In this thesis, a new m odel for the prediction of corrosion rates in sewer pipes has been developed. T he m odel describes th e diffusion of sulfide gas inside the concrete pores, its dissolution in the liquid phase, its biological conversion to sulfuric acid, and reaction w ith calcium carbonate to form corrosion product. The corrosion rate is m odeled as a moving boundary. This boundary is not known in advance and therefore is com puted as p a rt of the solution to th e m odel equations. T he model consisted o f a set of th ree parabolic differential equations of one spatial dim ension coupled to an ordinary differential equation describing the m ovem ent of th e corrosion front. The equations were approxim ated by a sequence of ordinary differential equations using a finite elem ent G alerkin approxim ation on a fixed spatial dom ain. The effects of several param eters on the corrosion rate were tested. These included porosity, diffusivity, bioconversion rate, and calcium carbonate dissolution rate. These param eters were assum ed to be constant throughout the experim ents. These assum ptions may require fu rth er justification in the future or be modified to closely represent the corrosion process. T he o u tp u t of the m odel is a series of curves showing air and w ater concentrations of hydrogen sulfide, and pH as a function of concrete depth, and the position of th e m oving boundary (corrosion front). T he results indicated th a t hydrogen sulfide concentration in both phases de creased to almost zero at th e m oving boundary. In som e experim ents, the hydrogen sulfide concentration showed quasi-steady-state behavior. This observation may sug gest further sim plification of the m odel to adapt to the actual physical behavior of sulfide. A comparison of sulfide and hydrogen ion concentrations showed th at the hydrogen ion concentration as a function of concrete dep th was considerably higher 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th an the sulfide concentration. This finding m ay suggest th at the surface p H is the controlling factor in the corrosion rate. In this thesis, the m odel developed earlier by Pom eroy[6 ] was modified to show th a t the corrosion rate is directly proportional to the hydrogen ions concentration on the surface of concrete and th e dissolution rate and inversely proportional to th e pipe alkalinity. Based on the prelim inary results obtained in this thesis, the m odified rate law for the corrosion rate showed superior advantages com pared w ith the earlier m odel. T he new model is sim ple to use, is in dependent of such factors as sulfide concentration on the surface of th e pipe wall and the efficiency of acid production as suggested by Pomeroy. T he resulting expression combines the effect of the surface p H and the alkalinity to express corrosion rate. In this thesis, the physical param eters listed in Table 4 . 1 were changed to new values and their effects on the corrosion rate was observed. In T able 5.1, the results of these experim ents were listed as weak, m oderate, or strong. N ote th a t these param eters and their influence on the corrosion rate were tested individually. There m ay exist the likelihood of com bined effects of these param eters. Therefore, it is an objective of further studies to identify any com bination of these constants which may produce larger effects. The plots of p H in m any cases suggested th a t th e corrosion product layer was acidic up to the moving interface and the results agreed with the experim ental d ata obtained by Mori et a/.[21]. In addition, the p H profile may explain the experim ental work perform ed by Islander et al.[2'2] where th e corroding surfaces were flushed in term itten tly w ith sewage to control the corrosion inside the pipe. The flushing removed hydrogen ions and o ther dissolved chem ical species from the surface of the pipe. T he net effect was a raise in the pH of surface of pipe above 6 . However, as was indicated by Islander et al., the pH fell to 1 In 4 to 5 hours. According to the plots of p H , the decrease in pH may be interp reted as hydrogen ions diffusing outw ards and bringing the p H of the surface back down. Therefore, frequent flushing can ultim ately rem ove the ions and provide a control. Sim ulations showed th at th e diffusion of ionic species such as hydrogen ions was the key to the corrosion process. Hence, reducing the system of three diffusion equa tions to two equations m ay be appropriate in simplifying the m odel and perform ing experim ents. It was also shown th a t the effect of water-filled porosity on th e cor rosion was large . T he effect of bioconversion rate, K b and the con trib u tio n of the last term in Eq. (2.20) was not significant. T he result of sensitivity analysis for the 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P a r a m e te r s E ffe ct | tortuosity r- ---- ------- - not tested air-filled porosity, Va m oderate to strong water-filled porosity, Vw m oderate to strong diffusion rate of H2S in air phase, D a m oderate to strong diffusion rate of H2S in w ater phase, D w m oderate to strong diffusion rate of H + in w ater phase, D h m oderate to strong surface area to volume ratio, air phase not tested surface area to volume ratio, liquid phase A ^ not tested mass transfer coefficient, K t not tested modified Henry's Law constant K h not tested bioconversion rate, Kg weak to m oderate [HoS]g on pipe surface, At weak to m oderate [HoS]aq on pipe surface, A 2 weak to m oderate [ff+] on pipe surface, A 3 strong corrosion thickness, s( 0 ) not tested dissolution rate. Kc m oderate to strong j alkalinity of concrete pipe, Cc m oderate to strong Table 5.1: Effects of the Numerical d a ta on th e corrosion rate given as weak, m od erate, and strong. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. controlling param eters are shown in Table 5.2. A com parison of different param e ters and the corrosion depths indicated th at alkalinity of concrete pipe m ay play an im portant role in lowering the corrosion rate. E xperim ental studies may involve the application of electrochem ical techniques applied to instrum ented concrete samples. Two different approaches can be used: (i) characterization of concrete structures using electrochem ical im pedance spec troscopy (EIS), and (ii) sensing of diffusion profiles of corrosive species (Fig. 5.1). The results obtained from these two techniques provide inform ation about th e diffu sion processes. By fitting experim ental EIS data to the reduced m odel, properties of the system such as m ean pore size, porosity, and the diffusion ra te and th eir changes w ith tim e as various corrosive and non-corrosive species en ter and the corrosion product leave the system can be determ ined. According to the m odel findings, the simultaneous diffusion of hydrogen sulfide and hydrogen ions influences the corrosion rate. Therefore, one control m ethod would be to incorporate cathodic protection into the m odel by way of suppressing the diffusion of hydrogen ions into the concrete pores. If the m eth o d is successful, an optim um control can be achieved. A nother method would be to com bine th e ongoing efforts of Sanitation D istrict with the model in this thesis and develop a predictive control m ethod for use in the area of surface coatings. T h e C ounty S anitation D istricts of Los Angeles County is currently focusing on the developm ent of surface coatings for their So miles of large diam eter sewers requiring corrosion protection. Field studies subject concrete pipe sections to very aggressive acid solutions and m easure their corrosion rates. This approach together w ith th e sim plification of the model in this thesis m ay provide a better understanding o f corrosion process and corrosion prevention inside the concrete sewer pipes. In a solution prepared w ith a high concentration of sulfuric acid, one can assum e th a t additional sulfuric acid formed by the bacteria is insignificant com pared w ith th e solution provided by the experim entalist. Furtherm ore, subjecting a concrete surface to this solution simplifies the model into a diffusion-reaction-equilibrium process in liquid and solid - phases. Therefore, assum ing Ficlc’s diffusion law and the m oving boundary as before, the model described in Eqs. (2.19)- (2.20) can be reduced to: 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P a r a m e te r s T e s te d V a lu e s 2 5 -y r C o rro sio n d e p th , (c m ) Porosity {Va = Vw): (e.g., Va = 10% and Vw = 10% ) 1 0 % 2 0 30 7.20 10.16 12.40 Porosity of w ater-phase. Vw: 1 0 % 4.47 (N ote, Va = 1 % as an exam ple) 2 0 5.97 30 7.02 40 7.S2 Bioconversion rate constant, K b - 0 per day 0 7.2 12.33 72 12.34 Dissolution rate constant, Kc'- 0.0084 cm /d ay 1.90 0.084 7.05 0.84 12.34 pH on the surface of concrete: 1 12.34 2 3.80 4 0.30 Alkalinity of concrete pipe, Cc (surface pH = 1 ) 0 . 2 0 m oles/liter 1 . 0 0 5.00 1 0 . 0 0 26.98 10.69 3.92 2.43 Table 5.2: Results of Sensitivity Analysis for C ontrolling Param eters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sulfuric acid solution u yr 1 r r concrete specimen pH sensors EIS Figure 5.1: Corrosion cell to be used in m easuring m odel param eters and concen tratio n s of hydrogen ions in a concrete specimen. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d[H+}«, d d [ H * u . .. . " a T " = Tx ( D h ~ ^ ] ( , ' 1) = K c [^ k ((_i = s(i)) (. 2) [tf+]a5 (f, a: = 0) = g(t) (5.3) ~ Dv d % l ~ ( L x = s(t)) = /vc [tf+]t t ( 7(f.x = s(f)) (5.4) [i?+]a?(0 .x ) = Xaq (5.5) s(0) = 50 (5.6) 0 < x < s(t) (5.7) where [H+]aq is the liquid-phase hydrogen ion concentration, D h is effective diffusion constant for [H +] in th e liquid phase (cm 2/d ay ), K c is calcium carbonate dissolution rate constant (cm /d ay ), s ( 0 ) is location of interface at the initial tim e inside the concrete (cm ), Cc is calcium carbonate concentration in the concrete (m oles/liter), Xaq is the initial tim e liquid-phase hydrogen ion concentration on the surface of the pipe (m oles/liter), and g{t) defines the pH changes on the surface of the concrete as a function of tim e. T he equations above do not model biological conversion of hydrogen sulfide into sulfuric acid and the activities w ithin th e w astew ater such as: flow velocity, pipe slope, w astew ater level, sulfide concentration in the wastew ater, and the rate of sulfide flux to pipe wall. Furtherm ore, the m odel param eters such as diffusion rate and porosity have not yet been determ ined experim entally. The only pieces of inform ation available at th e present tim e are the theoretical diffusion rate constant and the dissolution rate constant based on aggressive acid tests conducted by Devinny [5]. E xperim ental procedures may be designed to determ ine: C... s{t). D h . and K c- Using the simplified m odel and the m easured param eters, the corrosion rate m ay be com puted and com pared w ith the rates m easured in field studies. In this thesis, m any of the param eter values in Table 4.1 were extrapolated based on the work of o th er researchers and handbooks. In th e future studies, using experim ents and a least square procedure, these param eters m ay be determ ined. An experim ental procedure described by Rosen, et a/[29] w here a concrete sam ple subm erged in a bath containing a solution of sulfuric acid m ay be used to obtain O l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m odel param eters. Considering the experim ental setup in Figure 5.1 , as sulfuric acid becom es neutralized by calcium carbonate of th e concrete sample, the pH of bath changes w ith tim e. Using electrochem ical im pedance spectroscopy (EIS) and sensors em bedded in the concrete sam ple at different depths, effective diffusion rate. D h , and corrosion front, s(t) m ay be determ ined. C olorim etric m ethods m ay be used to determ ine the dissolution rate, K c- Hence, th e d ata for this experim ent m ay be given by Z(t)[30] where Z(t) denotes th e observed interface location at tim e t. Given th e observations Z(f,-), i = 1 ,2, • - -, k one can find am ong some class Q of adm issible param eters, param eter q = (D h , K c ) th a t m inim izes : A?) = E !z (f,-) -s (f.-) |2 (5.S) := 1 ' where s is the location of moving interface (corrosion thickness) num erically obtained using the m odel Eq.(5.2). In order to validate the m odel, th e com puted param eters m ay be com pared with those obtained from th e experim ents. In addition to esti m ating param eters, experim ents may be perform ed to form ulate control strategies. For exam ple, introducing a coating of a known alkalinity into the model and sulfuric acid of a known concentration required to neutralize this coating, one m ay predict the tim e required for the pH on the surface of coating to drop to the critical value of 4. An optim um solution in this m ethod is to determ ine the most cost effective coating to achieve the longest corrosion protection before another application of the coating m ay becom e necessary. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R e fe r e n c e L ist [1] G .F . From ent and K.B. Bischoff. Chemical Reactor Analysis and sign. John W iley and Sons. W iley series in chem ical engineering, 2nd. edition. 1990. [2] W . S tum m and J .J . M organ. Aquatic Chemistry - An Introduction. John W iley and Sons. New York, NY, 2nd. edition, 1987. [3] K.S. Cho and T . Mori. A newly isolated fungus participates in the corrosion of concrete sewer pipes. Water Science and Technology, 31(7):263— 2T1. 1995. [4] G.S. Beck. D eposition and pipe corrosion. W ater Env. & Tech. Update, pages 46-49. 1995. [5] J.S. Devinny. Model of concrete corrosion. University o f Southern California. Dept, of C ivil/ Environmental Engineering. 1993. [6 ] Process design m anual for sulfide control in san itary sewerage system s. Oct. 1974. [7] Sulfide in Wastewater Collection and Treatment Systems. ASCE, New York. NY, asce m anual no. 69 edition, 1989. [8 ] W . Sand and E. Bock. B iodeterioration of m ineral m aterials by microorgan isms- biogenic sulfuric and nitric acid corrosion of concrete and natural stone. Ge omicrobiology Journal. 9:129-138, 1991. [9] S tan d ard m ethods for the exam ination of w ater and w astew ater. 1985. [10] R.Y . Stanier, J.L . Ingraham , M.L. W heels, an d P.R. Painter. The Microbial World. Prentice-H all, Englewood Cliffs, NJ, 5th. edition, 1986. [1 1 ] R.D . Pom eroy and J.D . P arkhurst. T he forcasting of sulfide buildup rates in sewers. Progress in Water Technology, 9:621-62S, 1977. [12] R.D . Pom eroy. Calcareous pipe for sewers. Journal Water Pollution Control Federation. 41 (8 ): 1491-1493, 1969. [13] J. Iw am atsu. Iv. Nishizaki, Y. K atano, and K. Horino. Internal corrosion in sewers caused by hydrogen sulfide. Corrosion Engineering. 37:221-228. 1988. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [14] D .K .B . Thistlethw ayte. Control o f sulfides in sewerage system s. A nn Arbor Science Publishers, 1972. [15] S. W hitaker. The transport equations for m ulti-phase system s. Chemical En gineering Science, 2S:139-147, 1973. [16] J. B ear. J. Geophys. Res., 66:1185, 1961. [17] S. W hitaker. Diffusion and dispersion in porous m edia. A IC hE Journal, 13(3):420-427. 1967. [18] C.Y. W en. N oncatalytic heterogeneous solid fluid reaction models. Industrial and Engineering Chemistry, 60(9):34-54, 1968. [19] W . Gray. A derivation of the equations for m ulti-phase transport. Chemical Engineering Science, 30:229-233, 1975. [20] P.D. W ilm ot, K. Cadee, J .J. K atinic, and B.V. Kavanagh. Kinetics of sulfide oxidation by dissolved oxygen. Journal WPCF, 60(7):1264-1270, 19S8. [21] T . M ori, T . Nonaka, Iv. Tazaki, M. Koga, Y. Hikosaka, and S. Noda. Interaction of nutrients, moisture and ph on m icrobial corrosion of concrete sewer pipes. Water Research, 26(l):29-37, 1992. [22] R.L. Islander, J.S. Devinny, and F. M ansfeld. Microbial ecology of crow n cor rosion in sewers. J. Environmental Engineering, 117(6):751-770. 1991. [23] E.S. K em pner. Acid production by thiobacillus thiooxidans. J. Bacteriol., 92(6):1842-1843, 1966. [24] J. B adia, C.L. Chen, W. Kim bell, and E. Esfandi. C austic spray for sewer crown corrosion control. County Sanitation Districts of Los Angeles, 1992. [25] R.D . Pomeroy. Protection of concrete sewers in the presence of hydrogen sulfide. Water & Sewage Works. 1960. [26] C. Johnson. Numerical Solution o f Partial Differential Equations by the Finite Element Method. Cam bridge U niversity Press, Cam bridge, 1987. [27] F.A .L. Dullen. Porous Media Fluid Transport and Pore Structure. A cadem ic Press. 1 st edition, 1979. [28] Iv. Tazaki, T. Mori, and T . Nonaka. M icrobial jarosite and gypsum from cor rosion of portland cem ent concrete. Canadian Mineralogist, 30:431-444. 1992. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [29] I.G. Rosen. C. W ang, J.S. Devinny, and F.B. M ansfeld. M athem atical model ing of sulfide corrosion of concrete in w astew ater collection system s - analysis, experim ental validation, param eter estim ation and control. National Science Foundation Grant. 1997. [30] H.T. Banks and Iv. Ivunisch. Estimation Techniques fo r Distributed Parameter Systems. B irkhauser, Boston, 1989. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMAGE EVALUATION TEST TARGET (Q A -3 ) ✓ A A IIVI/IGE . Inc 1653 East Main Street Rochester. NY 14609 USA Phone: 716/482-0300 Fax: 716/288-5989 <0 1993. Applied Im age. Inc.. All Rights Reserved Reproduced with permission of the copyright owner. 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Creator Jahani, Fereidoun (author)

Core Title A moving boundary model of concrete sewer pipe corrosion: Theory and simulation

School Graduate School

Degree Master of Science

Degree Program Applied Mathematics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, civil,engineering, sanitary and municipal,Mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rosen, Gary (committee chair), [illegible] (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-30192

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