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Deforming finite element analysis of thermal ablation
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Deforming finite element analysis of thermal ablation
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DEFORMING FINITE ELEMENT ANALYSIS OF THERMAL ABLATION by Sankara Srinivasa Rao Singampalli A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORMA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Engineering) December 1985 UMI Number: DP28396 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI Dlssartation Rjblishing UMI DP28396 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089 This dissertation, written by S ANKA^^ ^ SRINIVASA ^ ^ S ING^PALL^^...... under the direction of h.X^ Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of re quirements for the degree of D O C T O R O F P H ILO S O P H Y ’ a s 56/7 Â/U "Dem of Graduate Studies December 5^ 1985 V D e m of Grc DISSERTATION COMMITTEE 'hairperson To My Parents Alivelu and Hanmaantha Rao 11 ACKNOWLEDGMENT I wish to express my deep sense of gratitude to professor V.I. Weingarten for his supervision and guidance throughout the course of this work. I am indebted to professor H.T. Yang for his encouragement and guidance. I wish to express my thanks to professor L.C. Wellford for serving on my dissertation committee. I would like to express my special thanks to professor S.S. Rao for introducing me to the area of finite elements. I am very thankful to all of my friends who helped me directly or indirectly in accomplishing this work. I wish to acknowledge the financial support provided by the SAP users group. Last, but not the least, I would like to express my thanks to Lori K. Kishi and April M. Taylor for their help in typing the equations. ii TABLE OF CONTENTS DEDICATION ................................. il ACKNOWLEDGMENTS...................................... iii LIST OF FIGURES ........ .........................vii LIST OF TABLES ................................. X NOMENCLATURE........................................xiv ABSTRACT ...... .......................xvii 1. INTRODUCTION............................. 1 2. EQUATIONS OF ABLATION.............................10 2.1 Assumptions..................... .10 2.2 Equations Of Heat Conduction............. . .10 2.3 Equations Of Ablation Boundary Condition . . .14 3. FINITE ELEMENT FORMULATION ......................... 16 3.1 Galerkin Method....................... .16 3.2 Finite Element Formulation ........... 16 3.3 Derivatives Of Interpolation Functions .... 18 3.3.1 Space Derivatives............................ 18 3.3.2 Time Derivatives.............................. 19 3.4 Boundary Nodal Velocity Distribution......... 2 0 3.5 Finite Element Equations Of Ablation......... 24 IV 4. AUTOMATIC FINITE ELEMENT MESH GENERATION ........ 29 4.1 Introduction . . . ............... .29 4.2 Velocity Distribution Of Ablating Boundary Nodes 31 4.3 Unit Outward Normals At The Ablating Boundary Nodes ...................... 32 4.4 Unit Outward Normals At The Ablating Surface Nodes ..... . . . V . . . .33 4.5 Velocity Distribution Of Semi-ablating Nodes .35 4.6 Velocity Distribution Of Interior Nodes. . . .36 5. SOLUTION OF FINITE ELEMENT EQUATIONS ............ 37 5.1 Finite Element Equations...................... 37 5.2 Linearization Of Equations .............. 38 5.3 Equilibrium Iterations . . . . . . .... . .41 5.4 Convergence Criteria .................. 43 6. SAMPLE ANALYSES . . . . . . . . . . . 45 6.1 Ablation In A Semi-infinite Solid With Constant Heat Flux........................................46 6.1.1 Numerical Data..................................46 6.1.2 Finite Element Modelling.......................47 6.1.3 Results And Discussion . .48 6.2 Ablation In A Semi-infinite Solid with Convection Boundary..................... 56 6.2.1 Numerical Data............ 56 6.2.2 Finite Element Modelling..................... 57 6.2.3 Results And Discussion........................58 6.3 Ablation In A Semi-infinite Solid With Radiation Boundary . . . . ............ . 64 6.3.1 Numerical Data...............................64 6.3.2 Finite Element Modelling..................... 65 6.3.3 Results And Discussion........................66 6.4 Ablation In A Finite Solid With Time Variant Heat Fluxes.............. .71 6.4.1 Numerical Data ..........................71 6.4.2 Finite Element Modelling..................... 72 6.4.3 Results And Discussion....................... 72 6.5 Radial Ablation In An Infinite Wide Plate . .90 6.5.1 Numerical Data................................91 6.5.2 Finite Difference Modelling . .............. 92 6.5.3 Finite Element Modelling..................... 93 6.5.4 Results And Discussion....................... 94 6. BIBLIOGRAPHY...................................... 122 7. APPENDIX.......................................... 127 VI LIST OF FIGURES Figure Page 2.1 Body subjected to heat transfer 11 2.2 Ablation boundary conditions 14 4.1 Classification of boundaries 3 0 4.2 Unit normals at the ablating boundary nodes 3 2 4.3 Unit normals at the ablating surface nodes 3 4 6.1.1 Ablation in a semi-infinite solid 52 ( Non dimensional parameter m = 0.2 ) 6.1.2 Ablation in a semi-infinite solid 53 ( Non dimensional parameter m = 1.0 ) 6.1.3 Ablation in a semi-infinite solid 54 ( Non dimensional parameter m = 2.0 ) 6.1.4 Ablation in a semi-infinite solid 55 Comparision of temperature distributions 6.2.1 Finite element modelling of semi-infinite 57 solid with convection boundary 6.2.2 Ablation in a semi-infinite solid 63 with convection Comparision of ablation thicknesses 6.3.1 Finite element modelling of semi-infinite 65 solid with radiation boundary 6.3.2 Ablation in a semi-infinite solid 70 with radiation Comparision of ablation thicknesses vii 6.4.1 Finite element modelling of a finite 72 solid with time variant heat fluxes 6.4.2 Ablation in a finite solid with 82 constant heat flux Comparision of thicknesses 6.4.3 Ablation in a finite solid with 83 linear heat flux Comparision of thicknesses 6.4.4 Ablation in a finite solid with 84 quadratic heat flux Comparision of thicknesses 6.4.5 Ablation in a finite solid with 85 exponential heat flux Comparision of thicknesses 6.4.6 Ablation in a finite solid with 86 constant heat flux Comparision of velocities 6.4.7 Ablation in a finite solid with 87 linear heat flux Comparision of velocities 6.4.8 Ablation in a finite solid with 88 quadratic heat flux Comparision of velocities 6.4.9 Ablation in a finite solid with 89 exponential heat flux Comparision of velocities viii 6.5.1 Radial ablation in an infinite plate 9 0 6.5.2 Finite difference modelling of an infinite plate 92 6.5.3 Finite element modelling of an infinite plate 93 6.5.4 Radial ablation of an infinite plate 114 with constant heat flow/flux Comparision of thicknesses 6.5.5 Radial ablation of an infinite plate 115 with constant heat flow/flux Comparision of velocities 6.5.6 Radial ablation of an infinite plate 116 with linear heat flow/flux Comparision of thicknesses 6.5.7 Radial ablation of an infinite plate 117 with linear heat flow/flux Comparision of velocities 6.5.8 Radial ablation of an infinite plate 118 with quadratic heat flow/flux Comparision of thicknesses 6.5.9 Radial ablation of an infinite plate 119 with quadratic heat flow/flux Comparision of velocities 6.5.10 Radial ablation of an infinite plate 12 0 with exponential heat flow/flux Comparision of thicknesses 6.5.11 Radial ablation of an infinite plate 121 with exponential heat flow/flux Comparision of velocities IX LIST OF TABLES Table Page 6.1.1 Ablation in a semi-infinite solid ( Non dimensional parameter m = 0.2 ) . . 49 6.1.2 Ablation in a semi-infinite solid ( Non dimensional parameter m = 1.0 ) . . 50 6.1.3 Ablation in a semi-infinite solid ( Non dimensional parameter m = 2.0 ) . . 51 6.2.1 Ablation in a semi-infinite solid with convection (Variational approach) .......... .60 6.2.2 Ablation in a semi-infinite solid with convection (Variational approach) ............. 61 6.2.3 Ablation in a semi-infinite solid with convection ( Finite element solution ) 62 6.3.1 Ablation in a semi-infinite solid with radiation ( Variational Approach ) ............. 67 6.3.2 Ablation in a semi-infinite solid with radiation ( Variational Approach ) ............. 68 6.3.3 Ablation in a semi-infinite solid with radiation ( Finite element solution ) 69 6.4.1 Ablation in a finite solid with constant heat flux Comparision of thicknesses .......... 74 6.4.2 Ablation in a finite solid with linear heat flux Comparision of thicknesses .......... 75 6.4.3 Ablation in a finite solid with quadratic heat flux Comparision of thicknesses . . . . . . 7 6 6.4.4 Ablation in a finite solid with exponential heat flux Comparision of thicknesses ...... 77 6.4.5 Ablation in a finite solid with constant heat flux Comparision of velocities .......... 78 6.4.6 Ablation in a finite solid with linear heat flux Comparision of velocities .......... 79 6.4.7 Ablation in a finite solid with quadratic heat flux Comparision of velocities .......... 8 0 6.4.8 Ablation in a finite solid with exponential heat flux Comparision of velocities .......... 81 XI 6.5.1 Radial ablation of an infinite plate with constant heat flow Comparision of thicknesses ........ 98 6.5.2 Radial ablation of an infinite plate with constant heat flow Comparision of velocities ........ 99 6.5.3 Radial ablation of an infinite plate with constant heat flux Comparision of thicknesses . . . . . 100 6.5.4 Radial ablation of an infinite plate with constant heat flux Comparision of velocities ........ 101 6.5.5 Radial ablation of an infinite plate with linear heat flow Comparision of thicknesses ........ 102 6.5.6 Radial ablation of an infinite plate with linear heat flow Comparision of velocities ........ 103 6.5.7 Radial ablation of an infinite plate with linear heat flux Comparision of thicknesses ........ 104 6.5.8 Radial ablation of an infinite plate with linear heat flux Comparision of velocities ........ 105 xii 6.5.9 Radial ablation of an infinite plate with quadratic heat flow Comparision of thicknesses ........ 106 6.5.10 Radial ablation of an infinite plate with quadratic heat flow Comparision of velocities ........ 107 6.5.11 Radial ablation of an infinite plate with quadratic heat flux Comparision of thicknesses . . . . . 108 6.5.12 Radial ablation of an infinite plate with quadratic heat flux Comparision of velocities ..... 109 6.5.13 Radial ablation of an infinite plate with exponential heat flow Comparision of thicknesses ........ 110 6.5.14 Radial ablation of an infinite plate with exponential heat flow Comparision of velocities ..... ill 6.5.15 Radial ablation of an infinite plate with exponential heat flux Comparision of thicknesses ........ 112 6.5.16 Radial ablation of an infinite plate with exponential heat flux Comparision of velocities ........ 113 Xlll NOMENCLATURE X , . . n ^i > ^i Coordinates of node i Unit outward normal "N Unit outward normal at node N "x' V> 'z Components of unit outward normal Temperature T* Prescribed temperature To Initial temperature Tm Melting temperature 0 Dimensionless temperature AT Incremental temperature AAT Correction in incremental temperature t Time f Time derivative Tm Time at which melting starts T Dimensionless time At Increment in time Y Thermal conductivities P Mass density C Specific heat L Latent heat of fusion xiv Heat generation per unit volume Q Heat flux conducting into the body Q Heat flux input b Heat absorbed in the ablation Heat flux due to convection Qa Qc Qy. Heat flux due to radiation Qp Prescribed heat flux Convective heat transfer coefficient Equivalent radiative heat transfer coefficient Convection source temperature Radiation source temperature Q - Stefan-Boltzmann constant E Emissivity of the source H Interpolation functions {V} Vector of ablating boundary nodal velocities {Qg^ } Vector of boundary nodal heat flux absorbed in the ablation u, V, w Components of nodal velocity { } Vector I Norm of a vector I [ ] Matrix [ ■ Transpose of a matrix tj- 2 Matrix evaluated at time t XV [K^] Heat conduction matrix [B] Thermal gradients matrix [Ky] Thermal conductivities matrix Heat convection matrix [K ] Heat radiation matrix r [C] Heat capacity matrix [K^] Ablation matrix [A] Boundary ablation matrix [V] Velocity matrix {q } Internal heat generation load vector {q^} Convective heat transfer load vector {q } Radiative heat transfer load vector (a } Prescribed heat flow load vector {q^} Thermal ablation load vector [K] Equivalent conduction matrix {q} Equivalent thermal load vector XVI ABSTRACT Thermal ablation analysis of finite and infinite solids using the deforming finite element method is presented. It is assumed that the melt is removed immediately on formation and the ablation takes place at a constant temperature. The ablation boundary condition is simulated by the physical movement of the ablating boundaries. The finite element mesh is reformed automatically once in every few time steps using an algorithm which solves the Laplace equation. Time and temperature variant material properties, convection, radiation, and prescribed heat flow boundary conditions have been considered. The effect of shape change due to the ablation is also considered in the analysis. A few sample problems have been solved using one, two, and three dimensional finite elements and the numerical and graphical results are presented. Comparisions of results obtained from the present finite element method with finite differences, heat balance integral, and theta moment integral methods have been presented. Finite difference formulation and solution of a radial ablation problem which demonstrates the effect of shape change is also presented and the results are compared with those of the present finite element method. xvii 1 INTRODUCTION Ablation is a process of the removal of surface material from a body by vaporization, melting, chipping or other erosive processes. Ablation is mainly used to protect the underlying material of re-entry space vehicles due to severe aerodynamic heating. This is achieved by allowing the surface material to melt down by absorbing the most of the incoming heat energy. Since latent heat of fusion of the most.of the materials used in space vehicles is considerbly high, a significant part of the incoming heat energy is absorbed by the melting surface material. The process of ablation of re-entry space vehicles due to the incoming heat energy is only one of the many factors involved. The chemical reaction between the atmosphere and the surface material, shape change due to the factors like thermal expansion and aerodynamic loading may also have significant effects on the ablation. So it is clearly evident that ablation is a highly nonlinear time dependent process. Because of its importance in the thermal shielding of space vehicles, several studies have been undertaken on some problems of ablation. Because of the importance of keeping the internal temperature of re-entry space vehicles within the safe temperature limits, the selection of ablating material and its thickness plays a very important role in the design of thermal shields of these vehicles. The internal temperature of the ablator is proportional to the heat conduction into the body, which in turn depends on the thickness, material properties of the ablator and also the rate of ablation. Since the rate of ablation is proportional to the incoming heat energy, the process of ablation is highly dependent on the surface boundary conditions and the instantaneous shape of the body. Landau [18] was among the first few authors to come up with an analytical solution to some limiting cases of ablation. He considered a semi-infinite solid subjected to constant heat flow. The moving boundary which makes the problem highly nonlinear has been eliminated by proper transformation of the variables. Thus the moving boundary problem has been restated by less complicated non-linear equations with fixed boundaries. He also obtained closed form solutions for the steady state ablation velocity. Sunderland and Grosh [34] extended Landau*s technique to the solution of an ablation problem involving convection boundary conditions. Goodman [15] solved the Landau's problem using the heat balance integral method. He first applied the well known "Momentum and enegy integral methods" of Von Karman and Pohlhausen, developed to deal with momentum and heat transfer problems in fluid mechanics, to a wide class of nonlinear problems of heat conduction in solids [16]. Later he extended this technique to ablation in a semi-infinite solid with constant heat flux, using a second order polynomial approximation for the temperature. Altman [1] extended this technique to the problems involving time dependent heat flux boundary conditions using a fourth order polynomial approximation for the temperature. Vallerani [35] applied the heat balance integral method to a class of ablation problems of a semi-infinite solid subjected to heat flux of the form q m = A t where A and m are positive constants. Assuming an exponential temperature distribution, he obtained closed form solutions. He presented the results in terms of a parameter expressing the ratio of heat capacities between the heat storable in the solid and the latent heat of ablation. Zien [42] introduced the theta moment integral method to solve one dimensional transient ablation with power law and exponential heat flux boundary conditions based on Landau's idealized ablation model. Biot and Daughaday [5] applied the variational and Lagrangian thermodynamics to the Landau * s problem and found a very good agreement in results. Biot and Agrawal [4] later extended this technique to the problems involving variable material properties. Prasad [27] solved the ablation in a semi-infinite solid subjected to radiation boundary conditions. Using a third order polynomial for temperature distribution, he obtained closed form solutions for both the premelt heating regime and the melting regime. He later extended this method to the problem involving convection boundary condition [28]. The application of finite element method to heat conduction analysis has attracted considerable attention since the procedure was found successful in the field of structural mechanics. Bathe and Khoshgoftaar [3] presented a general and effective finite element formulation for analysis of nonlinear steady state and transient heat transfer. They found the results to be in very good agreement with the results available in the literature. Encouraged by the success in the heat conduction analysis, several authors attempted the phase change problems using the finite element method. The phase change problems can be solved using the finite element method by two different ways. i.e., Fixed grid and moving grid formulations. In the fixed grid formulation, the latent heat effects are not included separately but these are approximated by rapid variations of heat capacity within a narrow temperature range in the vicinity of the melting temperature. This formulation is particularly suitable for materials for which the phase change occurs within a wide band of temperatures. In the moving grid formulation, the heat flow balance determines the position of interphase boundary. By allowing continuous deformation as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in the interpolation of parameters and dependent variables across the region where these quantities change abruptly. But the difficulty with this method is the necessity to reform the finite element mesh automatically to preserve the shape of the finite elements. This method is suitable for materials for which the phase change occurs at a constant temperature. Comini et al. [12] used a fixed grid finite element formulation and a finite width transition zone. Latent heat effects in the transition zone were accounted by the apparent heat capacity approach. Morgan et al. [21] have reported improvements of this method. Chung et al. [9] applied this fixed grid formulation to the ablation of one dimensional heat transfer in solids with variable thermal properties and radiation boundary conditions. They assumed that the melt is removed immediately on formation. To simulate the removal of melt, fictitious elements with zero capacity and a large conductivity were employed for the ablated material. Wellford and Ayer [37] solved a one dimensional ablation problem using discontinuous space time finite elements. The method uses a fixed grid of standard space-time finite elements along with certain special two dimensional space-time finite elements including free boundaries. The finite element model on the special elements were defined by discontinuous interpolation. Rubinsky and Cravahlo [32] presented a one dimensional finite élément method in which the grid is stationary and a moving node that coincides with the position of the change of phase interface is tracked continuously in time. Yoo and Rubinsky [40] presented a finite element method for the solution of two dimensional problems of heat transfer with change of phase. Specific to this method is that the energy equation on the moving interface is solved as an independent governing equation using a finite element formulation to yield the position of the interface in time. Roose and Storrer [31] applied the method of concentrated fictitious heat flow to solve the phase change problem in two dimensions. Latent heat effects were simulated by adding a term of fictitious heat flow to the governing energy equation. Hogge and Gerrekens [17] applied the deforming finite element method to the two dimensional ablation problems. They observed that the deforming finite element strategy introduces unsymmetric terms in the resulting matrices. Hence they used special solution methods to solve these equations. As it can be noticed, most of the work in the field of ablation is limited to semi-infinite solids. Only recently, Chung and Hsiao [10] solved the ablation problem in a finite solid and compared the results with those obtained from finite differences, heat balance 8 integral, and theta moment integral methods. In the present work, thermal ablation analysis of finite and infinite solids using the moving grid formulation of the finite element method is presented. Sublimation boundary conditions are assumed. i.e.. The melt is removed immediately on formation. The analysis presented in this work is applicable to the materials for which the phase change takes place at a constant temperature. The instantaneous location of the moving boundary is calculated from the heat flow balance at the ablating boundaries. Time and temperature variant material properties, convection, radiation, and heat flow boundary conditions are also considered. The assumptions made in this analysis, the governing differential equations of heat conduction, and the associated boundary conditions are presented in the chapter 2. Moving grid finite element formulation is presented in the chapter 3. The necessary steps to calculate time and temperature derivatives are also discussed. In the chapter 4, an automatic finite element mesh generation scheme is given. Using an algorithm which solves the Laplace equation, the finite element mesh is reformed once in every few time steps. The procedure used to calculate unit normals at the ablating boundaries and at the ablating surfaces is also discussed. The solution of finite element equations is presented in the chapter 5. The symmetry of finite element matrices is preserved by adding the nonsymmetric terms to the load vector. The resulting equations are solved by the modified Newton-Raphson scheme. A few sample problems have been solved using one, two, and three dimensional finite elements and the numerical and graphical results are presented in the chapter 6. Comparisions of results from the present finite element method with finite differences, heat balance integral, and theta moment integral methods are also presented. To demonstrate the effect of shape change on the ablation, a radial ablation problem is considered. The finite difference formulation and the solution of this problem is presented in the Appendix. 10 2 EQUATIONS OF ABLATION The governing differential equations of heat conduction with ablative boundary conditions are presented in this chapter. 2.1 Assumptions 1. The ablation takes place at a constant temperature. 2. The melt is removed immediately on formation. 3. Properties of the material and the boundary conditions may be functions of temperature. 2.2 Equations Of Heat Conduction Consider the differential equation of heat conduction in a three dimensional body as shown in figure 2.1, which is subjected to both Dirichlet and Neumann boundary conditions. From the conservation of heat energy, Fourier heat conduction equation may be expressed as + ^ ( k y f ) + - Qg = PC | I (2-1) 11 Heat flux \ \ ^ I / / Fig. 2.1 Body subjected to heat transfer The associated boundary conditions are: 1.Dirichlet boundary conditions 1.e.. Prescribed temperature boundary conditions. T = T* (2-2) 2.Neumann boundary conditions i.e.. Prescribed heat flow boundary conditions - kx|^ "x - ny - + Q = 0 (2.3) Where n , n and n are components of unit outward X y ' z normal to the boundary and Q is the heat flux entering the body through the boundary . Note that for a simple case with no phase change boundary conditions, the conservation of heat flow at the boundary may be stated as 12 JTotal heat flux conducting into Total heat flux input = ^ (the body i.e., /Q = (2.4) But once the surface temperature reaches the phase change temperature, a part of the heat input will be absorbed by the phase transformation. Hence the heat conduction into the body is the difference between the heat input and the heat absorbed in the phase transformation. Note that this phase change boundary condition makes the analysis highly nonlinear, since the rate of change of phase depends on the amount of heat flow into the body which in turn depends on the shape and the temperature distribution of the body. The heat balance at an ablative boundary may be expressed as Heat conduction \ ( Heat absorbed } = Heat input - < into the body I (in the ablation Q = Qy - Qa (2.5) Where the heat input ; may consists of the heat flow due to convection, radiation and the prescribed heat flow. 13 The heat flux due to convection : Qc = ^ ( Tp T ) (2.7) The heat flux due to radiation Qr = ce(V-T^) (2.8) where a is the Stefan-Boltzmann constant, € is the emissivity of the source. Note that this boundary condition is highly nonlinear function of temperature. To simplify the equations, this condition may be linearized as follows. Qr = hrdr - T) (2.9) Where the equivalent radiation coefficient is given by hy, = oe(Ty.^ + T^) (Ty. + T) ( 2 .1 0 ) which is a nonlinear function of temperature. Hence the Neumann boundary condition (2.3) including the ablation may be rewritten as follows. - kx|x^x - ky§ÿny - kz^z + hc(Tc - T) (2.11) + hy'(Ty' - T) + Qp - Qa " ( 3 Hence the total heat flux input and the total heat flux conducting into the body can be obtained from the expressions (2.6) and (2.3) respectively. Thus at any 14 given time, the unbalanced heat flux utilized in the phase transformation can be obtained. 2 .3 Ecfuations Of Ablation Boundary Condition As mentioned earlier, the unbalanced heat flow at the boundary is responsible for the ablation. Assuming that the melt is removed immediately on formation, the relation between unbalanced heat flux and the rate of ablation may be derived as follows. Consider the boundary AB as shown in figure 2.2, at which the ablation is taking place. Consider an infinitesimal element of unit surface area. db Fig. 2.2 Ablation boundary conditions 15 Let db be the thickness of the ablated material during a small time dt. The amount of material ablated per unit time \ = -p dt and per unit area ) Hence the amount of heat absorbed in the ablation per unit time and per unit area where L is the latent heat of fusion. The conservation of heat flux at the ablation boundary gives the following equation, Qa = 0 (2.12) from which the rate of ablation can be obtained. at Hence the Neumann boundary condition (2.11) may be rewritten, including the ablation condition as follows. - '^ x f^ x - - k z | ^ 2 + h(-(Tc - T) + h r(T r - T) + Qp + pL ^ = 0 (2.13) Thus the solution of equation (2.1) subject to the boundary conditions (2.2) and (2.13) gives the rate of ablation. 16 3 FINITE ELEMENT FORMULATION 3.1 Galerkin Method Finite element equations of thermal ablation have been derived using the Galerkin method. The principal idea behind this method is the selection of interpolation functions to represent the variation of the field variable and the determination of the unknown coefficients in the field variable approximation. The unknown coefficients are obtained by solving the simultaneous equations resulting from the vanishing of the integral of the weighted residue over the domain. The weights in this method are the same as the interpolation functions. 3.2 Finite Element Formulation In the finite element method, the domain under consideration is divided into finite elements. The variation of temperature in each of these finite elements ^ is assumed as a function of nodal temperatures and the interpolation functions, as follows. T(x,y,z,t) = ]]HjTj (3.1) where H. and T • are interpolation functions and nodal < ] J 17 temperatures respectively. Note that the coordinates X, y and z in the above expression are functions of time. Hence apply the Galerkin method to.the equation (2.1), to get / / / { Green-Gauss theorem states that J lf ^ If = - f f f w If ° - f f f if ^ (3.3) (3.4) and j ' j ' j 'Y dv = 4>dv + J J ' (3.5) where n^, and are x , y and z components of unit outward normal. V i s t h e v o l u m e o f t h e f i n i t e e l e m e n t , s i s t h e b o u n d a r y o f t h e f i n i t e e l e m e n t . Using the Green-Gauss theorem the i^ equation (3.2) may be expressed as 18 X ■ * ■ kygf "y + ^zll "zj [ ds ■ " _ //*{” i{kx|f " - Ü dv = 0 (3.6) Using the equation (2.11), the equation (3.6) can be rearranged as follows. "l) kx-^ + (-%F "() kylf + (|l up k^l^j-dv |I dv +y/'Hi(hc + hr) Ids = JJJ'w^^iqdv +JJ'H,(hcTc + hpTr) ds +^y” H{Qpds - ^^^H^-Qgds (3.7) 3.3 Derivatives Of Interpolation Functions Note that the equation (3.7) involves the derivatives of interpolation functions with respect to both space and time variables. 3.3.1 Space Derivatives - The space derivatives of the temperature obtained from (3.1) are If (3.8, 19 3T _ T. (3.9) âÿ-Z^ay. = (3.10) 3.3.2 Time Derivatives - Now consider the derivatives of temperature with respect to time. £L 3t (3.11) The interpolation functions Hj are explicit functions of space variables and hence the differentiation of Hj with respect to x, y and z can be performed directly. But the functions H^are implicit functions of time. Hence to find the partial derivatives of interpolation functions with respect to time, the following method is employed. Recall the property that the value of the interpolation function Hj at node j is 1 and is 0 at all other nodes. This property is valid for all times, irrespective of the instantaneous location of the nodes. This suggests that the interpolation functions are invariant at any given point with time. 1. (H,).o 20 ^ ^ W & * f 3H'- dx • 3Hg dy 3H,- ^ (3.14) 3t^ = - 3 3 ^ '^ a t • 3ÿ'^ a t • 553 d t - - « # , '=■“ > Hence equation :( 3 .11) may be rewritten as 1 (3.16) The governing equations in a finite element may be obtained by substituting equations (3.8), (3.9), (3.10) and (3.16) into (3.7) and integrating over the volume of the element. The velocity distribution required to form the equation (3.16) may be easily, calculated from the ablation boundary conditions as discussed in. the next section. Then the finite element system of equations may be obtained by assembling the element equations of all finite elements in the domain. 3•4 Boundary Nodal Velocity Distribution As mentioned earlier, the velocity distribution over the entire domain is required to form finite element equations. Assuming isoparametric formulation, the velocity distribution at any point may be obtained from the nodal velocities. Note that of all these nodal 21' points, only the ablating boundary nodes are governed by the heat flow boundary conditions. Hence the ablating nodal velocities are to be solved from the heat flow boundary conditions. The velocities of the remaining nodes may be interpolated using the Laplace equation which is described in the chapter 4. Recall the ablation boundary conditions (2.12) and (2.11) pL ^ + Qa = C (3.17) - kx - g - ky | I ny - k ^ g + h < - (T^ - T) + (T^ - T) + Qp - Qa = 0 Let the normal velocity at any ablating boundary point be V V = A (3.19) dt pLV = - Qa (3.20) Hence for any given time, the total heat flow absorbed in the process of ablation may be obtained from the equation (3.18). So the only unknown in the equation (3.20) is the velocity of ablating boundary point in the outward normal direction which can be calculated as described below. 22 Assuming the same interpolation functions which are used for temperature, the ablating boundary nodal velocities can be interpolated as V =ZHlV] (3.21) where Vj are the boundary nodal velocities. Hence equation (3.20) can be solved by the Galerkin method as (PLV + Qg) ds = 0 (3.22) Assembling all boundary elements, the system of equations which govern the ablating boundary nodal velocities can be obtained as J J ' [H]^ [H] ds Qg ds (3.23) where {V> is a vector of ablating boundary nodal velocities in the outward normal direction. Equation (3.23) can be rewritten in the following form. [A] {V> = - {Qa> (3.24) where [ A ] is boundary ablation matrix {Qg^ } is a vector of boundary nodal heat flows absorbed in the ablation. 23 The order of the system of equations represented by the equation (3.24) is equal to the total number of nodes on the ablating boundary which is usually very small. The x,y and z components of these nodal velocities can be obtained by multiplying with the unit outward normals at those respective nodes. Since the velocity distribution of the whole domain is required to form finite element equations, the velocities of remaining nodes can be obtained by proper interpolation. The step by step procedure to find the velocity distribution is described in the chapter 4. The x,y and z components of velocity in terms of the nodal velocity components can be expressed using the interpolation functions as follows. u (x.y.z) = (3-25) V (X,y,z) = LHlVl (3.26) w (x,y,z) = ^ (3.27) where U , V and W are x, y and z components of nodal velocities respectively. Substitute (3.25), (3.26) and (3.27) into (3.16) to get Il 3t (3.28) = -( ZHlU]) (^7 Hj) Tj - i Z m ) % Hj)Tj - (ZHlWl) Hj) Tj| 3.5 Finite Element Equations Of Ablation Substitute equations (3.1), (3.8), (3.9), (3.10) and (3.28) into (3.7) to get the i^^ equation as follows. dv J H f "x Tj + ky 1^ Tj + 1^ kz Tj ' ' ( Hg + h^) HjTjds + jfyjf)^(pCHjHj "j^) dv pC(Hi(HiUi) | ^ + Hj (HiVi) |üi+ Hj(HiWi) j Tjdv ° U f * I f (hcTc + hpTr) ds + ^ H j Q p d s -j^HjQads (3.29) 25 Equation (3.29), for i = 1 to N , can be expressed in matrix form as [H]T) kx [HD + [H]T) ky (|y [HD + % [H]T) kz % [H])) dv {t} [H]T (hg + hp) [H] ds + ' pC[H]T [H] dvj/|I]. - [_/^ pC[H]^ [H] [|Ü] + v[-g] + w[|M] jdv] {t} + [ <5pds] - [jjlnf Qads] çTç + hpT^} ds ] (3.30) It appears from the equation (3.30) that the thermal ablation introduces the following two terms. PC[H]T [H] I u(|^ [H])'+ v(|y [H]) + w(|^ [H]) ) dv end [ / / [H]T Qads Note that the expression in the square brackets of the first term is an unsymmetric matrix. Hence thermal ablation introduces an unsymmetric matrix which seems to be a major drawback of this formulation. 26 Of course it is possible to retain the symmetry of the matrices by adding this ablation contribution to the right hand side of the equations which makes the system of equations nonlinear even with linear material properties. But on close observation of the ablation problem, it is evident that it is not a serious drawback since the ablation boundary conditions are usually associated with highly nonlinear conditions such as radiation. So this deforming finite element strategy for thermal ablation just makes the problem more nonlinear. Hence the system of equations rearranged in the following form. [K k] + [Kg] + [K ^ ]l { t } + [C ] { f j - = {dg} + {dc} + {dr> + {dp} + {Qg} Heat conduction matrix [Kk] = [ Y [B] dV Thermal gradient matrix = [[-f] [4^] Ff Thermal conductivities matrix (3.3 0) can be [Kg] {T} 0 0 • [Ky] = 0 s 0 . 0 0 (3.31) (3.32) (3.33) (3.34) 27 Heat convection matrix [K^] [H] ds Heat radiation matrix ^ ds Heat capacity matrix [C] ‘JJJpCm [H] dv Ablation matrix [Kg] =J^C[H] [H] [V] [B] dv Velocity matrix (3.35) (3.36) (3.37) (3.38) [V] = [{u} {v} {w}] (3.39) where {u>,{v> and {w> are x,y and z components of velocity vectors respectively. Thermal load vectors Internal heat generation load vector (qg} » _ /) ! /■ [ « ]" Qg dv (3.40) Convective heat transfer load vector (3.41) 28 Radiation heat transfer load vector ds (3.42) Prescribed heat flow load vector Thermal ablation load vector (3.44) 29 4 AUTOMATIC FINITE ELEMENT MESH GENERATION 4.1 Introduction The moving finite element formulation considered in the ablation analysis simulates the ablation boundary condition by physical movement of the ablating boundary nodes. Hence it is required to update the finite element mesh several times during the analysis. So to keep track of these ablating boundaries for different times, an automatic finite element mesh generation program is essential. From the finite element system of equations of ablation, it is obvious that the velocity distribution is required to calculate the ablation matrix at each time step. From the known velocity distribution it is easy to find the displacements and hence the new location of ablating boundaries. It is to be noted that the velocities of nodes on ablating boundaries dictate the velocities of all other nodes. So first consider the velocities of boundary nodes. 30 The boundary of any ablating continuum can be divided into the following three groups. 1. Ablating boundaries 2. Non-ablating boundaries 3. Semi-ablating boundaries Fig. 4.1 Classification of boundaries ( Ablating plane sections of a hollow rocket motor grain ) Ablating boundaries are those from which the ablation of the material may take place. Note that the displacements of nodes on these boundaries are always normal to the boundary. Boundary AB in figure 4.1 is an ablating boundary. Non-ablating boundaries are those for which the nodes are always remain fixed. i.e.. There is no possibility 31 of ablation from these boundaries during the entire analysis. Boundary CD in figure 4.1 is a non-ablating boundary. Semi-ablating boundaries are those which are adjacent to the ablating boundaries, but are not directly affected by ablation. The semi-ablating boundary nodes always move in a direction which is tangent to these boundaries. Boundaries AC and BD in figure 4.1 are semi ablating boundaries. Axis of symmetry of a continuum is a semi-ablating boundary, since the nodes on the axis of symmetry are constrained to move along the axis. 4.2 Velocity Distribution Of Ablating Boundary Nodes Recall the system of equations which govern the ablating boundary nodal velocities. [ A ] { V } = - { Qa ) (4.1) where {V} is a vector of ablating boundary nodal velocities in the outward normal direction to the boundary. {Qg} is a vector of nodal heat fluxes utilized in the ablation. The x,y and z components of these velocities can be 32 obtained by multiplying with the unit outward normals at these boundary points. VNx = "NX (4.2) ' ^ Ny ° "Ny (4.3) ' ' nz ° ' ^ N "nz (4.4) where ,and are x,y, and z components of unit outward normal respectively. 4.3 Unit Outward Normals At The Ablating Boundary Nodes The unit outward normals at the boundary nodes are calculated from the known coordinates of boundary nodes and a reference node selected inside the continuum. The basic idea behind this calculation is as follows. N NM Fig. 4.2 Unit normals at the ablating boundary nodes 33 Choose a reference node R which is well within the body as shown in figure 4.2. Consider the unit normal at the boundary node N. Let M and P be the adjacent boundary nodes. The unit outward normal at the node N is approximated by the average of unit outward normals of line segments NM and NP at node N. i ("NM + ”nP) U("nM + "NP) (4.5) The unit outward normals and can be calculated using the following cross products. "nm and ( {NR} X (NM). ) . X (NM) ( (NR) X (NM) ) X {NM)| (4.6) ( (NR) X (NP) ) X (NP) "NP = I-----------------------r (4-7) ( (NR) X (NP) ) X (NP) 4.4 Unit Outward Normals At The Ablating Surface Nodes The unit outward normals to the ablating surface can be obtained by the cross product of any two nonparallel vectors tangent to the surface. These vectors can be easily obtained form the isoparametric 34 formulation of finite elements as follows. Fig. 4.3 Unit normals at the ablating surface nodes The coordinates of any point on the surface of a finite element are related to the nodal coordinates as X = y = ZH.y. where the interpolation functions natural coordinates r and s. (4.8) (4.9) are functions of Hence the vectors « . > • ' # - j D - ‘ I f and are tangent to the surface. 35 Using expressions (4.8) and (4.9), vectors {V| } and (Vg} can be expressed in the following form. and 3H, . 3H, , , 3H, } = i (23 3s^ j ( Z “â5^yi) + as ^l) (4.11) Hence the unit normal vector to the surface at any point (r,s) is given by {V, } X {Vj } {V ) = (4.12) ^ (V ) X (Vj } 4.5 Velocity Distribution Of Semi-ablating Nodes As mentioned earlier, the velocities of non-ablating boundary nodes are zero during the entire analysis. It is to be noted that at least one end of the semi-ablating boundary is attached to an ablating boundary. Hence the velocity of the node at the intersection of ablating and semi-ablating boundaries is determined by the ablation boundary condition at this node. Of course the direction of the velocity of this node is tangent to the semi-ablating boundary. Assuming 36 the other end of the semi-ablating boundary is joined to a non-ablating boundary, the intermediate nodal velocities are calculated using a linear velocity profile. 4.6 Velocity Distribution Of Interior Nodes. ' It is obvious that the interior nodes are not directly affected by ablation, but it is necessary to redistribute the interior nodal points in the structure in some manner consistent with the original geometry. This distribution process can be generalized by assuming that the nodal velocities satisfy the Laplace equation [36]. 37 5 SOLUTION OF FINITE ELEMENT EQUATIONS 5.1 Finite Element Equations Recall the governing equations of thermal ablation (3.31) + [Kg] + [Kp]] {T} + [C] (f) = {qg} + {q^} + {q^} + {q^} + {q^ } - [K^] {T} Rearrange the above system of equations in the following form [K] {T} + [C] {T} = {q} + {q }-[K 1 {T} (5.2) a a where Equivalent conduction matrix [K] . [K,] + [K;] +[Kr] (5 .3 ) Equivalent thermal load vector {q} = {Ag} +{qc) +{4^} (5.4) It is obvious that the temperature dependent material properties, radiation and ablation boundary conditions make this system of equations highly non-linear. Hence it is appropriate to use incremental solution procedure. i.e., Calculation of temperature distribution at time (t + At) from the known temperature distribution at time t. Hence consider the 38 heat flow equilibrium at time (t + At) of (5.2) . / ( t + At) (t + At) \ / ( t + At) (t + At) . \ ( [K] {T }j+( [C] {T}j /{ t + i t ) \ / ( t + i t ) \ /( t + i t ) (t + i t ) \ = ( { q } j + ( [KJ {T}j (5.5) where the left superscript denotes the time. 5.2 Linearization Of Equations Since the matrices [ K ] and [ C ] are functions of temperature, the solution process can be started with the following linearization. ~ *[K] (5.6) = \C] (5.7) where { AT) is a vector of incremental temperatures between times t and (t + At). (t + At) t {u} ~ {u} + {Au} (5.9) (t + At) t {V} ~ {V} + {AV} (5.10) (t + At) _ t {W} - {W} + {AW} (5.11) where {Au}, { Av}, and { Aw} are incremental nodal 39 velocities during the time, increment At. (t + At) _ t [Ka] (5.12) (t + (t + ^(T) (5.13) The linearized system of equations (5.5) can be represented as /t (t + At) (t + At) [K] (T}, (5.14) (t + At) At + At) \ /(t + At) \ /t t \ Using a forward finite difference scheme, the gradients of temperature and velocities may be expressed as / (t + At) t \ T = \ T - T // At ( (t + At) t ) U = \ U - u // At (5.15) (5.16) / (t + At) _ t \ (5.17) V = \ V V / / At (t + At) , t w = W - W, J / At (5.18) Substitute eg. (5.15) into eg. (5.14) and rearrange the terms to get 40 ft , t 1 (o) (t + At) (t + At) [CK]+~ [C] J{AT} = ■ {q} + {q^} t t t t - [K^] {T} - [K] { T} (5.19) Solution of this system of equations gives the first approximation of incremental temperatures and incremental velocities. Hence the temperatures and the velocities at time (t + At) may be obtained from the equations (5.8), (5.9), (5.10), and (5.11). Since the acceleration of nodal points over time interval 'At is assumed to be constant, the coordinates of nodal points at time (t + r^t) can be obtained as (t +A.t) t , /(t + At) t \ (5.20) ^ ° ="1 + — y /At (t + At) t , /(t + At) t I (5.21) = 4 + T \ ''l + ''i / and Thus the nodal coordinates can be updated to obtain the new location of the ablating boundary. 41 5.3 Equilibrium Iterations Solution of equations (5.19) does not give the correct incremental temperatures and velocities because of the linearization of the equation (5.5). i.e.. The substitution of this solution into the equation (5.5) gives an unbalanced heat flow. , Hence the solution has to be updated to take care of this unbalanced heat flow. This procedure should be repeated until the residual heat flow is within the required tolerance. These iterations which achieve the heat flow equilibrium are called Equilibrium iterations [2]. The temperature correction at the end of the iteration ’i• may be expressed as (t + At) (i) (t t At) (i-1) (i) {T} = {T} + {AAT} (5.23) and {AAT}^‘ ‘^ = {AT}(^^ - {AT} ‘ (5-24) where {AAl} is the correction in incremental temperatures during the iteration ’i•. Consider the heat flow equilibrium of the equation (5.5) at the end of iteration * i *. 42 (t + At) (t + At) ( i ) \ (t + At) / ( t + At) (i) t [K] {T} /+ [C] {T} ' {T} At + At) \ /(t + At) \ / t + At) (t + At) ( i) \ [KJ {T} ) (5 .3 5 ) Substitute the equation (5.23) into the equation (5.25) to get (i) (t + A t) (i - 1) 1 (t + At) (1 - 1) [K] + It [C] {AAT} \ (t + At) /(t + At) (i - l)\it + At) (l - 1)' ‘" A ) t , (t +-At) (i - l)\/(t + At) . (i - 1)\ (t + At) [c] \ (T} 1+ {q } a ' I t + At) (i - lj\/(t + At) (i - 1)\ l\ {T} j (5.26) Correction in incremental temperatures may be obtained by solving the equation (5.26). On close observation it is obvious that this method is similar to Newton-Raphson solution of the equation (5.5). It shows that it involves the triangularization of the coefficient matrix in each iteration which is very expensive. Hence a modified Newton-Raphson scheme is followed in which the coefficient matrix is updated only once in every . few iterations. Thus most of the iterations involve a back substitution only which is inexpensive. 43 The governing equations for modified Newton-Raphson iterations are ft 1 t 1 1 (t + At) (t + At) [ [K] + [C]j ,{AAt} = {q} + {q^} _ /(t + At) (i - 1)\ /(t + At) (i - 1)\ ^ ■ t V / _ /(t + At) (i - l)\/(t + At) . (l - 1)\ I CC] m ) /(t + At) (i - l)\/(t + At) (i - 1)\ ( [Kg] {T} ) (5.27) At the end of each iteration, the updated temperatures and velocities may be obtained. Hence the updated coordinates of all nodes may be obtained from the equations (5.20), (5.21), and (5.22) using this latest velocity distribution. 5.4 Convergence Criteria The following convergence criteria have been used during the equilibrium iterations. (AA (t+ At) (1) { T} < Tolerance of temperature (5.28) 44 {A AU} (i) (t + At) (1) {U} < Tolerance of X component of velocity (5.29) ( 1) {ÂA V} (t + At) (i) {V} < Tolerance of Y component of velocity (5.30) {AAW} ( 1) (t + At) (i) {w> < Tolerance of Z component of velocity (5.31) 45 6 SAMPLE ANALYSES The deforming finite element method presented in this work is tested by the following sample problems involving ablation. Due to the complexity of the ablation phenomenon, solutions of the most of the ablation problems reported in the literature are limited to semi-infinite solids. Since the numerical results are not available in the literature, the following sample problems have been solved in the present work using the finite differences, heat balance integral and theta moment integral methods to compare the results from the present finite element method. The following parameters have been selected for finite element analysis. The number of equilibrium iterations in each time step are limited to 15. The convergence tolerance for temperature is selected as 0.001. The convergence tolerance for velocity is assumed to be 0.1. It is assumed that the surface has reached the melting condition if its temperature is within + 1 °. of the melting temperature. The number of time steps between formation and triangularization of the matrix for Laplacian is 10. 46 6.1 Ablation In A Semi-infinite Solid With Constant Heat Flux A semi-infinite solid is subjected to a constant heat flux at its surface. Assuming that the melt is removed immediately on formation, determine the velocity of the surface as a function of time [18], 6.1.1 _ Numerical Data The following data has been arbitrarily selected to solve this problem and the results are presented in the following non-dimensional quantities. 2 L T_T^ Non-dimensional temperature 0 = - , r~ " — Non-dimensional time t = V — ' • PL ds Non-dimenslonal ablation velocity = — yi Initial temperature = 100.0 % Melting temperature = 256.0 °C Time at which melting starts = 2.0 seconds Applied heat flux = 1000.0 kwatts /m^ Specific heat = 1.0 kj / kg °C 3 Mass density = 1.0 kg / m Thermal conductivity = 104.63836 kj/m s °C 47 6.1.2 Finite Element Modelling The problem has been solved for the following three values of the non-dimensional parameter m. m = 0 .2 , 1 . 0 and 2 . 0 Hence the corresponding latent heats of.fusion are 691.257, 138.251 and 69.1257 respectively. The semi-infinite solid is modelled by a finite solid of length 15.0 and this length is found to be large enough to simulate the semi-infinite solid assumption since the nondimensional temperature of the far end of the solid is less than a small value (5.0E-7) throughout the solution time under consideration. The solid has been divided into 4 00 equal parts. The nondimensional time has been selected as Ax = 0.0005 from the consideration of stability in the explicit finite difference scheme. The same problem has also been solved using the heat balance integral method [15]. The resulting simultaneous differential equations in this method have been solved by a fourth order Runge-Kutta method. 48 6.1.3 Results And Discussion A comparision of the results obtained from these three methods is presented in the tables 6 .1 .1 , 6 .1 .2 , and 6.1.3 and in the figures 6.1.1, 6.1.2, 6.1.3, and 6.1.4. It can be observed that the percentage error is well within the reasonable limits. It may also be noticed that the finite element solution follows more closely with the finite difference solution than the heat balance integral solution. Because of the assumption of temperature profile in the heat balance integral method, the finite difference solution may be considered to be more accurate. Hence the finite element results are found to be satisfactory. ( Non dimensional parameter m = 0 .2 ) Time Thickness Ablated Finite Finite Heat Diff. Element Balance % Error Finite Heat Elem. Balance 1 . 0 0 0.4597 0.4531 0.3368 -1.424 -26.720 1 . 2 0 0.4849 0.4792 0.3736 -1.192 -22.954 1.40 0.5061 0.5009 0.4043 -1.017 -20.117 1 . 60 0.5241 0.5195 0.43 03 -0.879 -17.902 1 . 80 0.5398 0.5356 0.4527 -0.767 -16.123 2 . 0 0 0.5535 0.5497 0.4724 -0.678 -14.659 2 . 2 0 0.5657 0.5623 0.4897 —0.605 -13.434 2.40 0.5766 0.5735 0.5052 —0.544 -12.391 2 . 60 0.5865 0.5836 0.5191 -0.490 -11.492 2 .80 0.5954 0.5928 0.5317 —0.446 -10.708 3 . 00 0.6036 0.6012 0.5431 -0.407 -1 0 . 0 2 0 3.20 0.6111 0.6088 0.5536 -0.374 -9.408 3.40 0.6181 0.6159 0.5633 -0.344 -8.862 3.60 0.6245 0.6225 0.5722 -0.319 -8.370 3.80 0.6304 0.6286 0.5805 -0.296 -7.923 4.00 0.6360 0.6342 0.5882 -0.274 -7.519 4.20 0.6412 0.6395 0.5954 -0.257 -7.148 4.40 0.6461 0.6445 0.6021 -0.239 —6 . 8 08 4. 60 0.6507 0.6492 0.6084 -0.225 —6 .495 4.80 0.6550 0.6536 0.6143 -0 . 2 1 2 -6.204 5. 00 0.6591 0.6578 0.6199 -0.199 -5.935 5.20 0.6629 0.6617 0.6252 -0.188 —5.684 5.40 0 . 6 6 6 6 0.6654 0.6303 -0.178 —5.451 5. 60 0.6701 0.6689 0.6350 -0.169 -5.232 5.80 0.6734 0.6723 0.6395 -0.159 -5.027 6 . 0 0 0.6765 0.6755 0.6438 -0.152 -4.832 6 . 2 0 0.6796 0.6786 0.6479 -0.144 —4.651 6.40 0.6824 0.6815 0.6519 -0.138 -4.479 6 . 60 0.6852 0.6843 0.6556 -0.130 -4.317 6.80 0.6878 0.6870 0.6592 -0.125 -4.163 7. 00 0.6903 0.6895 0.6626 -0.119 -4.017 7.20 0.6928 0.6920 0.6659 -0.114 -3.878 7.40 0.6951 0.6943 0.6691 -0.109 -3.746 7. 60 0.6973 0.6966 0.6721 -0.104 -3.620 50 Table 6.1.2 Ablation in a semi-infinite solid ( Non dimensional parameter m = 1.0 ) Thickness Ablated % Error Time Finite Diff. Finite Element Heat Balance Finite Elem. Heat Balance 1 . 0 0 1 . 2 0 1.40 1 . 60 1.80 0.3369 0.3513 0.3631 0.3730 0.3813 0.3324 0.3482 0.3603 0.3705 0.3791 0.2925 0.3189 0.3399 0.3571 0.3713 -1.311 -0.894 -0.769 —0.664 -0.578 -13.157 -9.230 -6.386 -4.254 -2.620 2 . 0 0 2 . 2 0 2.40 2.60 2 . 80 0.3885 0.3948 0.4002 0.4051 0.4094 0.3866 0.3930 0.3987 0.4037 0.4082 0.3833 0.3935 0.4022 0.4097 0.4162 -0.502 -0.441 -0.390 —0.34 6 -0.3 09 -1.341 -0.330 0.479 1. 132 1 . 660 3.00 3.20 3.40 3.60 3.80 0.4133 0.4168 0.4200 0.4229 0.4255 0.4122 0.4158 0.4191 0.4220 0.4247 0.4219 0.4270 0.4314 0.4353 0.4388 -0.278 -0.250 -0.225 -0.203 -0.185 2.086 2.430 2.708 2.932 3.110 4.00 4.20 4.40 4.60 4.80 0.4279 0.4302 0.4322 0.4341 0.4359 0.4272 0.4295 0.4316 0.4335 0.4353 0.4418 0.4446 0.4471 0.4493 0.4512 -0.169 -0.155 -0.142 -0.131 -0 . 1 2 1 3.249 3.355 3.434 3 .490 3.525 5.00 5.20 5.40 5.60 5.80 0.4375 0.4390 0.4404 0.4418 0.4430 0.4370 0.4386 0.4400 0.4414 0.4426 0.4530 0.4546 0.4560 0.4573 0.4585 -0 . 1 1 1 -0 . 1 0 2 -0.094 -0.088 -0.081 3.545 3.549 3.541 3.524 3.496 6 . 00 6 . 2 0 6.40 6 . 60 6.80 0.4442 0.4453 0.4463 0.4472 0.4482 0.4438 0.4449 0.4460 0.4470 0.4479 0.4595 0.4605 0.4614 0.4621 0.4628 -0.077 -0.071 —0.066 -0.063 -0.057* 3.464 3.423 3.378 3.328 3.276 7. 00 7.20 7.40 7.60 0.4490 0.4498 0.4506 0.4513 0.4488 0.4496 0.4504 0.4511 0.4635 0.4641 0.4 646 0.4651 -0.055 -0.052 -0.048 —0.046 3 .218 3 .160 3.101 3.038 51 Table 6.1.3 Ablation in a semi-infinite solid ( Non dimensional parameter m = 2.0 ) Thickness Ablated % Error Time Finite Diff. Finite Element Heat Balance Finite Elem. Heat Balance 1 . 0 0 1 . 2 0 1.40 1.60 1.80 0.2453 0.2534 0.2599 0.2652 0.2696 0.2431 0.2519 0.2594 0.2647 0.2692 0.2439 0.2596 0.2710 0.2795 0.2859 -0.902 -0.602 -0.204 -0.175 -0.139 -0.597 2.428 4.282 5. 398 6 . 040 2 . 0 0 2 . 2 0 2.40 2.60 2.80 0.2733 0.2765 0.2792 0.2816 0.2837 0.2730 0.2763 0.2791 0.2816 0.2838 0.2907 0.2944 0.2972 0.2994 0.3011 -0 . 1 0 0 -0.063 -0.029 -0 . 0 0 2 0 . 023 6 . 364 6 . 472 6.438 6.311 6 . 1 2 0 3.00 3.20 3.40 3.60 3 .80 0.2856 0.2872 0.2887 0.2900 0.2912 0.2857 0.2874 0.2889 0.2903 0.2915 0.3024 0.3034 0.3042 0.3048 0.3053 0.045 0.062 0.081 0.094 0.106 5.888 5. 632 5.370 5.100 4.834 4.00 4.20 4.40 4.60 4.80 0.2923 0.2933 0.2942 0.2950 0.2957 0.2926 0.2936 0.2946 0.2954 0.2962 0.3057 0.3060 0.3062 0.3064 0.3065 0.117 0.127 0.136 0.145 0.149 4.577 4.327 4. 089 3.863 3.648 5.00 5.20 5.40 5.60 5.80 0.2964 0.2970 0.2976 0.2981 0.2986 0.2969 0.2975 0.2981 0.2987 0.2992 0.3066 0.3067 0.3068 0.3068 0.3069 0.157 0.160 0.166 0.170 0.174 3 .446 3 .257 3.076 2.907 2 .752 6 . 0 0 6 . 2 0 6.40 6 . 60 6 .80 0.2991 0.2995 0.2999 0.3003 0.3007 0.2996 0.3001 0.3005 0.3009 0.3012 0.3069 0.3069 0.3069 0.3070 0.3070 0.177 0.181 0.184 0.187 0.188 2.604 2.464 2.334 2.215 2 . 1 0 0 7.00 7.20 7.40 7.60 0.3010 0.3013 0.3016 0.3018 0.3015 0.3019 0.3021 0.3024 0.3070 0.3070 0.3070 0.3070 0.190 0.192 0.194 0.195 1.993 1.894 1.798 1.712 52 -H 4J c q 0 « H -H 0 C +j w o P -H 1 — t 1 — t 4J O (d P c n p fH O' o o < D Ü 1 O -P c c -p 0) -H c p ( U ( U ( U Ü ( D < N c r—1 r4 (d 0) TJ » H (d 0) Q > A -P -P -H -H 4J c C (d •H -H 0) k k K iH CVJ n a a a < § I —I ( f ) s C M ' 0 • •H O r —1 o I I c n 0 ( U 4 J -H P c ( U -H - P « W Q) c 0 • H ( d 1 P -H ( d e a ( U c n 1 —1 ( d ( d C O c -H -H c n C c ( U o 0 - H-H - P t3 < d 1 —1C A o . — VO O' -H a 53 0) <M iH *H 0) T3 k p L i m ' — ' o • -H rH rH o II 1 0 n 0) -p -H u C 0) -H +> < u G •H (d 1 P -H (d P4 < u i n rH (d (d n o C -H •H 1 0 c c 0) o G -H -H -P T? îd rH c Xi o < % — C N V O I T » -H 54 w ü +J C c 4 - > 0 ) -H 0) (U JQ 4 - > + J -H -H -p C C ( d •H *H Q) k k M < i Q Z CM <ÛQ-J<h->-iOZ >Lü-JOO»-»l->- o T3 • -H C M 1 - 4 O I I U i H 0) +J -H u C < 0 -H +J 44 0) C 6 -H (d 1 U -H fd e A 0) 0 3 1 - 4 < d (d C O c -H -H 0 3 C C (!) O g - r 4- r 4 4 - > " O (d r 4 c A o < % n I —I vo C P - i H 55 m o Q ) < u rpiTTTI’T iT r~ p i y r i-fT*! TTT-r i" ! r i ’T-i’f r I r j r i i i~ *d •H i n c 0 -H 1 •H u rH +J O ( Û 1 0 *H *d 0) ■ p Q ) -S 5 <W « 3 C U •H 0) I •H g t < D -P i n (p ( d o -S g -H C 01 O -H •H P +> « J f d A II vo U» •H 56 6.2 Ablation In A Semi-infinite Solid With Convection Boundary Determine the rate of ablation in a semi-infinite solid with constant material properties, which is subjected to convection boundary condition [28]. As mentioned before, very few closed form solutions are available in the literature for ablation problems. Since the closed form solutions allow the parametric study, several investigators attempted to obtain at least an approximate closed form solution. Prasad [28], using Biot*s variational principle [5], obtained a closed form solution for this problem assuming a cubic temperature profile. Since the numerical results are not available in the literature, this problem has been solved to compare the results of the finite element analysis. 6.2.1 Numerical Data The following numerical data has been arbitrarily selected and the results are presented in the following nondimensional quantities. 2 Nondimensional time t = ^ ^/kpQ Inverse of Stefan Number S. = ^ ^ 'm 57 Initial temperature Melting temperature Ambient temperature Thermal conductivity Specific heat Mass density = 352.7 = 3527.0 C = 4585.1 C 0.25 kj/m s % 2.83527 kj /kg 2000.0 kg /m^ Convective heat transfer coefficient = 50.0 kj /m C s 2.2 Finite Element Modelling 1^0.005 ^ 1 " 0.0075 0.005 1 21 36 4" 2 42 72 82 1 41 71 81 83 143 163 \ 82 \ 142 81 141 161 Fig. 6.2.1 Finite element modelling of semi-infinite solid with convection boundary 58 The semi-infinite solid is idealized by 40 one dimensional elements as shown in the figure 6.2.1. Smaller elements are used near the ablation boundary for better accuracy of the results. Analysis is carried out for a of 50.0. The time step increment is selected a s Ax » 0 . 1 . Various convergence criteria and other required parameters are the same as described in the introduction of this chapter. This problem has been solved for the following two values of inverse of Stefan number. $t = 0.5 and 1.0 This problem has also been solved using two and three dimensional finite elements and the results are found to be in good agreement. 6.2.3 Results And Discussion Results obtained from Prasad’s Variational analysis and from the present finite element method are presented in the tables 6.2.1,6.2.2, and 6.2.3 and in the figure 6.2.2. From these tables and figure , a close agreement of the results may be observed. As already mentioned, Prasad’s analysis is restricted to semi-infinite solids 59 with constant material properties, where as the finite element method presented in this work may be applied to finite solids with variable properties. 60 Table 6.2,1 Ablation in a semi-infinite solid with convection (Variational approach) Inverse of Stefan number = 0.5 Dimensionless Dimensionless time Ablated thickness 5.5 0.04 6.1 0.10 6.8 0.17 7.6 0.27 8.6 0.39 9.6 0.54 10.9 0.72 12.3 0.95 14.0 1.23 16.0 1.58 18.5 2.03 21.7 2.63 25.9 3.46 32.1 4.71 43.3 7.02 61 Table 6.2.2 Ablation in a semi-infinite solid with convection (Variational approach) Inverse of Stefan number = 1.0 Dimensionless Dimensionless time Ablated thickness 5.4 0. 02 6.0 0.05 6.6 0.09 7.3 0.14 8.0 0.20 8.8 0.27 9.7 0.35 10.7 0.44 11.7 0.55 12.9 0.67 14.2 0.81 15.5 0.97 17.1 1.15 18.7 1.36 20.6 1.59 22.7 1.86 25.0 2.17 27.6 2.52 30.6 2.93 34.0 3.41 37.9 3.97 42 . 6 4.65 48.3 5.49 Table 6.2.3 Ablation in a semi-infinite solid with convection ( Finite element solution ) 62 Dimensionless time Dimensionless Ablated thickness Inverse of Stefan number 0.5 1.0 2.00 0. 00 0. 00 4.00 0. 00 0 . 00 6. 00 0.12 0.07 8. 00 0.36 0.22 10. 00 0.64 0.40 12. 00 0.95 0. 61 14. 00 1.29 0.83 16. 00 1. 64 1. 06 18.00 2.00 ^ 1.31 20.00 2.37 1.56 22.00 2.75 1.81 24.00 3.13 2.08 26.00 3.52 2.35 28.00 3.92 2.62 30. 00 4.31 2.89 32.00 4.71 3.17 34.00 5.11 3.45 36. 00 5.52 3.73 38. 00 5.93 4. 02 40. 00 6.33 4.31 42.00 6.74 4.60 44. 00 7.16 4.89 46. 00 7.57 5.18 48. 00 7.98 5.47 50. 00 8.40 5.77 63 0 ) ( d •P ( d -H < d U ) 0 ) '0 U ) 'H i n 1 —1 < D o c i n Ü 0 ) -H P x: -H p -H c c (Wo o c •H tH •H P P 1 u ( d -H Q ) 1 —1 m > P 0 )C < d i no Ü p ( d o X: CPC •H-H o ^ -H c U ) o -H -H p ( d ( d A 1 —1 m A o < u <N <N r I'T i 'i i "i I I I f- i - r r p -i i x j -r 11 i | i i r i- |- i n i j i i i i j i i i i | f i t r * -H < C D -J < I-U J Q h -X ï-iO i^ Z L ü C n ü ) (3 64 6.3 Ablation In A Semi-infinite Solid With Radiation Boundary Determine the rate of ablation in a semi-infinite solid with constant properties, which is subjected to radiative boundary conditions [27]. Assuming a cubic temperature profile, Prasad [27] derived a closed form solution for this problem using Biot's variational principle [5]. From this analysis he obtained analytical relations between the time and the melt removal rate and also between the time and the penetration depth. These closed form relations have been solved for the time values at different penetration depths. Results from the present finite element analysis have been compared with the Prasad's solution using the following numerical data. 6.3.1 Numerical Data The following numerical data has been arbitrarily selected and the results are presented in the following non-dimensional quantities. Non-dimensional time x = t / PCk Inverse of Stefan number S= L/c Tm t Initial temperature = 380.0 °K 65 Ambient temperature = 4940.0 °K Melting temperature = 3800.0 Mass density = 2000.0 kg /m^ Specific heat = 5.263158 kj/kg^K Emissivity = 1.0 S te f an- Bo It z mann constant = 5.66961 *10 kj/ir? sec 6.3.2 Finite Element Modelling ) * — 0.0125 — *^1 "^— 0*014 — ^"1 * — 0.01 — 4 2 6 40 45 Fig. 6.3.1 Finite element modelling of semi—infinite solid with radiation boundary. The semi-infinite solid has been idealized by 44 one dimensional elements as shown in the figure 6.3.1. The total length of the solid is assumed to be 0.0365 m and this length is found to be large enough to simulate the semi-infinite solid assumption. A fine mesh has been chosen near the boundary because of steep temperature gradients. Analysis is carried out for a X _ of 1.16. The time step increment has been max selected as 0.02. 66 This problem has been solved for the following two values of inverse of Stefan number. = 0.1 and 0.5 This problem has also been solved using two and three dimensional finite elements and the results are found to be in good agreement. 6.3.3 Results And Discussion Results obtained from Prasad's variational analysis and from the present finite element method are presented in the tables 6.3.1, 6.3.2, and 6.3.3, and in the figure 6.3.2. From these tables and figure, a close agreement of the results may be observed. 67 Table 6.3.1 Ablation in a semi-infinite solid with radiation ( Variational Approach ) Inverse of Stefan number = 0.5 Dimensionless Time Dimensionless Ablated thickness Dimensionless Penetration depth 0.150 0.003 1.50 0.158 0.007 1.54 0.166 0.011 1.58 0.174 0.016 1.62 0.183 0.022 1.66 0.193 0.028 1.70 0.203 0.036 1.74 0.214 0.044 1.78 0.226 0.053 1.82 0.239 0.063 1.86 0.252 0.074 1.90 0.266 0.087 1.94 0.282 0.101 1.98 0.298 0.116 2.02 0.316 0.133 2.06 0.335 0.152 2.10 0.356 0.173 2.14 0.379 0.197 2.18 0.403 0.224 2 .22 0.431 0.253 2.26 0.461 0.287 2.30 0.495 0.325 2.34 0.532 0.369 2.38 0.575 0.419 2.42 0.625 0.478 2.46 0. 683 0.549 2.50 0.752 0.634 2 .54 0.838 0.742 2.58 0.950 0.884 2.62 1.110 1.089 2.66 68 Table 6.3.2 Ablation in a semi-infinite solid with radiation ( Variational Approach ) Inverse of Stefan number = 1.0 Dimensionless Time Dimensionless Ablated thickness Dimensionless Penetration depth 0.149 0.002 1.50 0.160 0. 005 1.56 0.171 0.009 1.62 0.183 0.014 1.68 0.196 0.019 1.74 0.210 0.025 1.80 0.224 0.032 1.86 0.240 0.040 1.92 0.256 0.050 1.98 0.274 0.060 2.04 0.293 0.071 2 .10 0.313 0.084 2.16 0. 335 0.098 2.22 0.358 0.113 2.28 0.382 0.130 2.34 0.409 0.149 2.40 0.437 0.169 2.46 0.467 0.192 2 .52 0.500 0.217 2 .58 0.536 0.244 2.64 0.574 0.274 2.70 0.615 0.308 2 .76 0. 660 0.345 2 .82 0.710 0.386 2 .88 0.764 0.431 2.94 0. 823 0.482 3.00 0.889 0.540 3.06 0.963 0.604 3.12 1.046 0.678 3.18 1.140 0.764 3.24 1.249 0.863 3.30 1.377 0.981 3.36 1.531 1.124 3 .42 1.723 1.305 3.48 69 Table 6.3.3 Ablation in a semi-infinite solid with radiation ( Finite element solution ) Time Inverse of Stefan number = 0.5 Ablation Inverse of Stefan number = 1.0 Ablation Thickness Velocity Thickness Velocity 0.140 0.0008 0.000069 0.0005 0.000038 0.180 0.0197 0.000186 0.0117 0.000114 0.220 0.0486 0.000235 0.0296 0.000148 0. 260 0.0826 0.000265 0.0512 0.000170 0. 300 0.1199 0.000286 0.0754 0.000186 0.340 0.1598 0.000301 0.1015 0.000198 0. 380 0.2014 0.000313 0.1290 0.000208 0 . 420 0.2445 0.000323 0.1577 0.000216 0. 460 0.2887 0.000331 0.1874 0.000223 0.500 0.3339 0.000337 0.2179 0.000228 0.540 0.3800 0.000343 0.2491 0.000233 0.580 0.4266 0.000347 0.2810 0.000237 0. 620 0.4739 0.000351 . 0.3133 0.000241 0. 660 0.5217 0.000355 0.3461 0.000244 0.700 0.5699 0.000358 0.3794 0.000247 0.740 0.6185 0.000361 0.4130 0.000250 0.780 0.6675 0.000363 0. 4469 0.000252 0.820 0.7167 0.000365 0.4811 0.000254 0. 860 0.7663 0.000367 0.5156 0.000256 0.900 0.8161 0.000369 0.5504 0.000258 0.940 0.8661 0.000370 0.5854 0.000259 0.980 0.9163 0.000372 0.6206 0.000261 1.020 0.9667 0.000373 0.6560 0.000262 1.060 1.0172 0.000374 0.6916 0.000263 1.100 1.0679 0.000375 0.7273 0.000265 1.140 1.1188 0.000376 0.7632 0.000266 1.180 1.1697 0.000377 0.7992 0.000267 70 II* r M 1 I I 0) < 0 ta TJ 0 1 0) 0 1 -H 0 1 i —i 0) O C m 0) ü -H -P XI -H -P C -H c <W c o c o -H -H -H 4J 1 -P r t J -H <0 t —i ë •H A (DT3 r t J 0 1 (0 (0 P m o C 5 c -H •H o c 1 5 -H 0 1 o -H -H P -P ai r t J A r—1 g A O < U OJ m vo t r > -H 71 6.4 Ablation In A Finite Solid With Time Variant Heat Fluxes Determine the rate of ablation in a finite solid subjected to time variant heat fluxes [10]. As already mentioned earlier, most of the available ablation solutions are limited to semi-infinite solids. Only recently Chung and Hsiao [10] presented a deforming finite difference scheme for one dimensional ablation in a finite solid. They compared the results obtained from heat balance integral and theta moment integral methods. Since the numerical results are not available in the literature, this problem has also been solved using heat balance and theta moment integral methods to compare with the solution of the present finite element method. 6.4.1 Numerical Data The following numerical data has been arbitrarily selected to solve this problem. Length of the solid = 1.0 m Initial temperature = 0.0 °C Melting temperature - 100.0 Sc Thermal conductivity = 0.1 kj /m°C s Mass density = 1.0 kg/m^ 72 2 Thermal diffusivity = 0.1 m /s Specific heat = 1.0 kj/Kg°C Latent heat of fusion = 100.0 kj/kg Characteristic time = 10.0 sec Referenced time = 100.0 sec 6.4.2 Finite Element Modelling 10 11 Fig. 6.4.1 Finite element modelling of a finite solid with time variant heat fluxes The one dimensional solid under consideration is idealized by 10 one dimensional elements as shown in the figure 6.4.1. Time variant heat fluxes are applied at node 1 where as the node 11 is assumed to be insulated. 6.4.3 Results And Discussion This problem has been solved using heat balance integral, theta moment integral, and finite element methods for the following four types of heat flux boundary conditions. 73 1. Constant heat flux = q = 20.0 kwatts 2. Linear heat flux = q = 10.0 * t kwatts 3. Quadratic heat flux = q = t * t kwatts 4. Exponential heat flux = q = Exp( t/100.0 ) kwatts Comparisions of the results from the above three methods are presented in the following tables and figures. Tables 6.4.1 - 6.4.4 compare the ablation thicknesses and tables 6.4.5 - 6.4.8 compare the ablation velocities. Graphical representations of these results are given in the figures 6.4.2 to 6.4.9. It may be observed that the finite element solution follows more closely with the theta moment integral solution than that of the heat balance integral. Chung [10] also observed that the theta moment integral solution is superior to the heat balance integral solution. Hence the deforming finite element method is found to be satisfactory in solving ablation problems involving time variant boundary conditions. 74 Table 6.4.1 Ablation in a finite solid with constant heat flux Comparision of thicknesses External heat flux = q = 2 0.0 kwatts Thickness ablated %Error lime Finite Heat Theta Heat Theta element balance moment balance moment method integral integral integral integri 0.20 0.004 0.002 0.007 —50.00 75. 00 0.40 0. Oil 0.009 0. 016 —18.18 45.45 0 .60 0.020 0.018 0. 026 -10.00 30. 00 0.80 0.031 0.029 0.037 -6.45 19 . 35 1.00 0. 043 0. 042 0.049 -2.33 13 .95 1.20 0.056 0. 057 0.062 1.79 10.71 1.40 0.069 0. 073 0.076 5.80 10.14 1.60 0.084 0. 090 0. 091 7.14 8.33 1.80 0.099 0.108 0.107 9.09 8 . 08 2.00 0.115 0.128 0.123 11.30 6. 96 2.20 0.132 0.148 0.141 12.12 6. 82 2.40 0.149 0 .168 0.159 12.75 6.71 2.60 0.167. 0.190 0.178 13.77 6. 59 2.80 0.186 0.212 0.197 13 .98 5.91 3.00 0.206 0.235 0.218 14.08 5.83 3.20 0.226 0.258 0.239 14.16 5.75 3.40 0.247 0.282 0.261 14.17 5. 67 3.60 0.270 0.306 0.284 13 .33 5.19 3.80 0.292 0.332 0.308 13.70 5.48 4.00 0.316 0.357 0.333 12.97 5.38 4.20 0.340 0.384 0.359 12 .94 5.59 4.40 0.366 0.411 0. 386 12.30 5.46 4.60 0.392 0.438 0.413 11.73 5.36 4.70 0.405 0.452 0.428 11.60 5. 68 75 Table 6.4.2 Ablation in a finite solid with linear heat flux Comparision of thicknesses External heat flux q = 10.0 * t kwatts Thickness ablated %Error ?ime Finite element method Heat balance integral Theta moment integral Heat balance integral Theta moment integra: 0.10 0.003 0. 002 0.006 -33.33 100.00 0.20 0.010 0. 006 0.013 -40.00 30. 00 0.30 0.018 0.013 0.022 -27.78 22 .22 0.40 0.028 0.022 0.031 -21.43 10.71 0.50 0.040 0. 032 0. 042 -20.00 5. 00 0.60 0.052 0. 044 0. 054 -15.38 3.85 0.70 0.065 0. 058 0.067 -10.77 3 . 08 0.80 0. 080 0. 074 0.081 -7.50 1. 25 0.90 0.095 0.090 0.096 -5.26 1. 05 1.00 0.112 0.108 0.112 -3 .57 0.00 1.10 0.129 0.127 0.130 -1.55 0.78 1.20 0.147 0.147 0.148 0. 00 0. 68 1.30 0.167 0.169 0.167 1.20 0. 00 1.40 0.187 0.191 0.187 2.14 0.00 1.50 0.208 0.214 0. 208 2 . 88 0 . 00 1.60 0.230 0.239 0.230 3.91 0. 00 1.70 0.252 0.264 0.253 4 .76 0.40 1.80 0.276 0.290 0. 277 5. 07 0.36 1.90 0.301 0.317 0.302 5.32 0.33 2. 00 0.326 0.345 0.328 5.83 0. 61 2.10 0.353 0.374 0. 354 5.95 0.28 2.20 0.380 0.403 0. 382 6.05 0. 53 2.30 0.409 0.434 0.411 6.11 0.49 Table 6.4.3 76 Ablation in a finite solid with quadratic heat flux Comparision of thicknesses External heat fluk = q = t*t kwatts ?ime Finite element method Heat balance integral Theta moment integral Heat balance integral Theta moment integr; 0.10 0.002 0. 001 0. 005 -50.00 150.00 0 .20 0.008 0. 005 0. Oil -37.50 37.50 0.30 0.016 0.011 0.019 -31.25 18.75 0.40 0.025 0. 019 0. 027 -24.00 8 . 00 0.50 0.035 0. 029 0.037 -17.14 5.71 0. 60 0.047 0. 040 0.048 -14.89 2.13 0.70 0.060 0. 053 0.060 -11.67 0. 00 0.80 0.074 0. 068 0.074 -8.11 0. 00 0.90 0. 089 0. 084 0.088 -5. 62 -1.12 1. 00 0.105 0.101 0.103 -3.81 -1.90 1.10 0.122 0.119 0.120 -2.46 -1. 64 1.20 0.140 0.139 0.138 -0.71 -1.43 1.30 0.159 0.160 0.157 0. 63 -1.26 1.40 0.180 0.183 0.177 1.67 -1.67 1.50 0.201 0.206 0.198 2.49 -1.49 1. 60 0.224 0.231 0. 220 3 . 12 -1.79 1.70 0.247 0.257 0.244 4.05 -1.21 1.80 0.272 0.284 0.268 4.41 -1.47 1.90 0.298 0.312 0.294 4.70 -1.34 2 . 00 0.325 0.342 0.321 5.23 -1.23 2 .10 0.354 0. 372 0.350 5. 08 -1. 13 2 .20 0.383 0.403 0. 379 5.22 -1.04 2 . 30 0.414 0.436 0.410 5.31 -0.97 77 Table 6.4.4 Ablation in a finite solid with exponential heat flux Comparision of thicknesses External heat flux = q = EXP( t/100.0 ) kwatts Thickness ablated %Error rime Finite element method Heat balance integral Theta moment integral Heat balance integral Theta moment integra: 2 . 00 0. 008 0.007 0. 001 -12.50 -87.50 3.00 0.018 0. 015 0. 008 -16.67 -55.56 4.00 0.030 0.025 0.019 -16.67 —3 6.67 5.00 0.043 0. 037 0.032 -13.95 -25.58 6 . 00 0.059 0.051 0.047 —13.56 -20.34 7.00 0.075 0. 065 0.064 -13.33 -14.67 8. 00 0.093 0. 081 0.082 -12.90 -11.83 9. 00 0.111 0. 098 0.101 -11.71 - 9.01 10.00 0.130 0.115 0.120 -11.54 -7.69 11.00 0.150 0.134 0.141 -10.67 — 6.00 12. 00 0.171 0.153 0.162 -10.53 -5.26 13. 00 0.192 0.172 0.183 -10.42 -4.69 14. 00 0.213 0.192 0.206 -9.86 -3.29 15.00 0.235 0,212 0. 228 -9.79 -2.98 16.00 0.257 0.233 0.250 -9.34 -2.72 17. 00 0.280 0.254 0.273 -9.29 -2.50 18. 00 0. 303 0,276 0.296 -8.91 -2.31 19. 00 0.326 0.297 0.319 -8.90 -2.15 20.00 0.349 0. 320 0.344 -8.31 -1.43 21.00 0.373 0.342 0.363 -8.31 —2 . 68 22.00 0.397 0.365 0. 387 —8 .06 -2.52 78 Table 6.4.5 Ablation in a finite solid with constant heat flux Comparision of velocities External heat flux = q = 20.0 kwatts Ablation velocity %Error ?ime Finite element method Heat balance integral Theta moment integral Heat balance integral Theta moment integra: 0.20 0. 0309 0.0225 0.0404 -27.18 30.74 0.40 0.0425 0.0389 0.0470 -8.47 10.59 0. 60 0.0504 0.0516 0.0529 2.38 4.96 0.80 0.0564 0.0617 0.0582 i 9.40 3.19 1.00 0.0615 0.0700 0.0631 13 .82 2.60 1. 20 0.0660 0.0771 0.0677 16.82 2.58 1. 40 0.0702 0.0832 0.0721 18.52 2.71 1. 60 0.0742 0.0886 0.0763 19.41 2.83 1.80 0.0780 0.0934 0.0804 19.74 3.08 2 . 00 0.0818 0.0977 0.0844 19.44 3.18 2.20 0.0855 0.1017 0.0884 18.95 3.39 2.40 0.0892 0.1054 0.0924 18.16 3.59 2 . 60 0.0929 0.1089 0.0964 17.22 3.77 2 . 80 0.0966 0.1122 0.1004 16.15 3.93 3.00 0.1004 0.1153 0.1045 14.84 4.08 3.20 0.1042 0.1183 0.1087 13.53 4.32 3.40 0.1081 0.1213 0.1129 12.21 4.44 3 . 60 0.1120 0.1242 0.1173 10.89 4.73 3.80 0.1161 0.1272 0.1218 9.56 4.91 4. 00 0.1202 0.1301 0.1265 8.24 5.24 4.20 0.1245 0.1331 0.1314 6.91 5.54 4.40 0.1289 0.1362 0.1364 5. 66 5.82 4.60 0.1334 0.1393 0.1416 4.42 6.15 4.70 0.1337 0.1410 0.1443 5.46 7.93 Table 6.4.6 79 Ablation in a finite solid with linear heat flux Comparision of velocities External heat flux = q = 10.0 * t kwatts Ablation velocity %Error Time Finite Heat Theta Heat Theta element balance moment balance moment method integral integral integral integral 0.10 0.0554 0.0307 0.0656 -44.58 18.41 0.20 0 . 0760 0.0562 0.0786 -26.05 3.42 0.30 0.0922 0.0781 0.0909 -15.29 -1.41 0.40 0.1057 0.0975 0.1027 -7.76 -2.84 0.50 0.1179 0.1150 0.1141 -2.46 -3.22 0. 60 0.1291 0.1310 0.1251 1.47 -3.10 0.70 0.1397 0.1458 0.1358 4.37 -2.79 0.80 0.1500 0.1596 0.1462 6.40 -2.53 0.90 0.1597 0.1726 0.1564 8 . 08 -2.07 1.00 0.1692 0.1849 0.1665 9.28 -1.60 1. 10 0.1784 0.1966 0.1764 10. 20 -1.12 1.20 0.1875 0.2077 0.1862 10.77 -0.69 1.30 0.1965 0.2184 0.1959 11.15 -0.31 1.40 0.2054 0.2286 0.2055 11. 30 0.05 1.50 0.2143 0.2385 0.2151 11.29 0.37 1. 60 0.2233 0.2480 0.2247 11. 06 0.63 1.70 0.2323 0.2571 0.2342 10. 68 0.82 1.80 0.2414 0.2660 0.2438 10.19 0.99 1.90 0.2506 0.2747 0.2534 9. 62 1.12 2.00 0.2601 0.2831 0.2631 8.84 1.15 2.10 0.2698 0.2914 0.2729 8.01 1.15 2.20 0.2798 0.2996 0.2830 7. 08 1.14 2.30 0.2897 0.3077 0.2933 6.21 1.24 80 Table 6.4.7 Ablation in a finite solid with quadratic heat flux Comparision of velocities External heat flux = q = t*t kwatts Ablation velocity %Error Time Finite Heat Theta Heat Theta element balance moment balance moment method integral integral integral integral 0.10 0.0474 , 0.0266 0.0566 -43.88 19.41 0.20 0.0676 0.0496 0.0689 -26.63 1.92 0.30 0.0838 0.0700 0.0809 -16.47 -3.46 0.40 0.0976 0.0886 0.0926 -9 .22 -5.12 0.50 0.1102 0.1058 0.1041 -3 .99 — 5 .54 0. 60 0.1221 0.1219 0.1155 -0.16 -5.41 0.70 0.1335 0.1371 0.1268 2.70 -5. 02 0.80 0.1445 0.1517 0.1380 4.98 — 4 . 50 0.90 0.1556 0.1657 0.1492 6.49 -4.11 1.00 0.1663 0.1793 0.1605 7.82 -3.49 1.10 0.1770 0.1924 0.1717 8.70 -2.99 1.20 0.1876 0.2052 0.1830 9.38 -2.45 1.30 0.1983 0.2176 0.1944 9.73 -1.97 1.40 0.2090 0.2298 0.2058 9.95 -1.53 1.50 0.2199 0.2418 0.2174 9.96 -1. 14 1. 60 0.2309 0.2535 0.2290 9.79 -0. 82 1.70 0.2421 0.2650 0.2408 9.46 — 0 .54 1.80 0.2535 0.2763 0.2527 8.99 -0.32 1.90 0.2651 0.2874 0.2649 8 . 41 -0 . 08 2.00 0.2771 0.2983 0.2772 7.65 0. 04 2.10 0.2895 0. 3091 0.2898 i 6.77 0.10 2.20 0.3023 0.3198 0.3028 5.79 0.17 2.30 0.3166 0.3305 0.3161 4.39 -0. 16 81 Table 6.4.8 Ablation in a finite solid with exponential heat flux Comparision of velocities External heat flux = EXP( t/100.0 ) kwatts Ablation velocity %Error Time Finite Heat Theta Heat Theta element balance moment balance moment method integral integral integral integral 1.00 0.0047 0.0038 0.0007 -19.15 -85.11 2.00 0.0084 0.0068 0.0054 -19.05 -35.71 3 . 00 0.0109 0.0091 0.0091 -16.51 -16.51 4.00 0.0129 0.0111 0.0119 -13.95 -7.75 5.00 0.0145 0.0127 0.0142 -12.41 -2.07 6.00 0.0159 0.0141 ' 0.0159 -11.32 0. 00 7.00 0.0171 0.0153 0.0174 -10.53 1.75 8 . 00 0.0181 0.0163 0.0185 -9.94 2.21 9.00 0.0189 0.0171 0.0194 -9.52 2 . 65 10.00 0.0196 0.0179 0.0203 -8.67 3 .57 11.00 0.0202 0.0186 0.0208 -7.92 2.97 12.00 0.0208 0.0192 0.0215 -7.69 3 . 37 13.00 0.0212 0.0197 0.0218 -7.08 2.83 14.00 0.0216 0.0202 0.0226 -6. 48 4.63 15.00 0.0220 0.0206 0.0226 -6.36 2.73 16.00 0.0223 0.0229 0.0198 -5.83 2 . 69 17.00 0.0226 0.0213 0.0232 -5.75 2.65 18.00 0.0229 0.0216 0.0232 -5.68 1.31 19.00 0.0232 0.0219 0.0235 -5.60 1.29 20.00 0.0235 0.0237 0.0222 -5.53 1.11 21.00 0.0237 0.0225 0.0238 -5. 06 0.42 22.00 0.0240 0.0228 0.0241 —5 .00 0.42 82 o o C l a m o H I —I 1 - 4 4J f d r t J u u C T » 0 0) 0) t o 4 - > -P . 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Prescribed heat flow per unit length 2. Prescribed heat flux Fig. 6.5.1 Radial ablation in an infinite plate .After satisfying with the results of ablation in semi-infinite solids, the radial ablation from a hole in an infinite plate has been attempted. Since no numerical results are avilable to the best of author's knowledge, this problem has been solved by the explicit 91 finite difference scheme (Appendix) to compare the results from the deforming finite element method. The effect of shape change on the ablation may be observed in this problem. Note that the amount of heat flow entering the plate and the amount of material to be ablated per unit increment in the radius of the hole are governed by the instantaneous radius of the hole. For example, with the constant heat flow boundary condition, the heat flux per unit surface area decreases with an increase in the radius of the hole. Hence the rate of ablation decreases with an increase in the radius. The heat flow due to an electrode placed at the axis of the hole is an example of this boundary condiion. Now consider the constant heat flux boundary condition. Here the amount of the heat flow entering per unit surface area remains constant. Hence the rate of ablation increases with an increase in the'hole radius. The heat flow due to the flow of hot gases through the hole is an example for this type of boundary condition. 6.5.1 Numerical Data The following numerical data has been arbitrarily selected to solve this problem. Radius of the hole = 1.0 m 92 Outer radius of the plate = 6.0 m Initial temperature Melting temperature Heat flow per unit length per unit radian before ablation = 20.0 kwatts 0.0 °C — 100.0 Thermal diffusivity Thermal conductivity Mass density Specific heat Latent heat of fusion Derived quantities Characteristic time Nondimensional heat flow Inverse of Stefan number = 0.1 m /s = 0.1 kwatts /m °C - 1.0 kg/m^ = 1.0 kj/kg °C = 100.0 kj/kg Q = S. = % ' ’C/k ' ' o ^ ’ o / k ( T m - T o ) 6.5.2 Finite Difference Modelling 5.0 51 Nodes Fig. 6.5.2 Finite difference modelling of an infinite plate 93 The infinite wide plate has been idealized by an axisymmetric cylinder with a central hole. The outer radius of the cylinder has been selected such that the temperature of the outer boundary remains at the initial temperature throughout the solution time under consideration. The moving boundary condition due to ablation has been eliminated by proper transformation of the variables. The resulting differential equations have been solved by an explicit finite difference scheme. The complete derivation of these finite difference equations is presented in the Appendix. The radial length of the cylinder has been divided into 50 equal parts. The total number of solution time steps is chosen as 1500 to find the solution at 15.0 seconds. 6.5.3 Finite Element Modelling 20X0.05 15 X 0.01 —^1 " 5 X 0.5 —*1 u n Fig. 6.5.3 Finite element modelling of an infinite plate 94 The infinite wide plate has been idealized by 4 0 axisymmetric elements as shown in the figure 6.5.3. The total number of time steps is selected as 200 for a total solution time of 15.0 seconds. Since the temperature gradients near the hole are steep, a finer finite element mesh is selected at this location. To approach the melting point gradually, a smaller time step has been selected near the melting time. Thus the following variation in time step has been chosen. 0.0 < t < 2.0 time step of 0.1 2.0 < t < 4.0 time step of 0.05 4.0 < t <15.0 time step of 0.1 where the time at the onset of melting is 3.0 seconds. 6.5.4 Results And Discussion Thermal ablation of the surface of a hole located in an infinite wide plate has been analysed using the finite difference scheme and the moving boundary formulation of the finite element method. The results from these analyses are compared in the following tables and figures. A satisfactory agreement between these results may be observed. 95 This problem has been solved for the following two types of boundary conditions. 1. Prescribed heat flow. i.e., The heat flow entering per unit length of the hole is prescribed. 2. Prescribed heat flux. i.e.. The total heat flow per unit surface area is prescribed. For each of the above two types, the following four time variant conditions are considered. 1. Constant heat flow/flux 2. Linear heat flow/flux 3. Quadratic heat flow/flux 4. Exponential heat flow/flux Consider the constant heat flow boundary condition. Since the total heat flow per unit length of the hole is a constant, the heat flow per unit surface area of the hole decreases with an increase in its radius. It may also be noticed that the amount of material to be ablated per unit increment in the radius of the hole increases quadratically. Hence the amount of heat energy available for the ablation of a unit mass of the material decreases. Hence from the physical reasoning. 96 it may be concluded that the ablation velocity approaches zero after some time. It may be observed from figure 6.5,5, the results from these analyses support this conclusion. Note that the ablation velocity reaches a maximum value of 0.039318461 m/s at time t = 7.0 seconds. Then the ablation velocity starts decreasing and finally the ablation stops when its velocity reaches a value of zero. Now consider the constant heat flux boundary condition which is relatively simple. Here the amount of heat flow per unit length and also the amount of material to be ablated per unit increment in the radius of the hole increases with an increase in its radius. Hence the behaviour of ablation in this case may be considered normal. A comparision of results is presented in the figures 6,5.4 to 6.5.11 and in the tables 6.5.1 to 6.5.16 A good agreement of results between the finite difference and the finite element methods can be observed in this problem. As it can be observed from the Appendix, the finite difference formulation is relatively simple for this problem. As said before, the proper transformation of the variables eliminated the moving boundary condition in this problem. Hence it 97 became possible to derive the simple finite difference equations. It may be easily noticed that the moving grid formulation of finite differences, which is essential for solving ablation in finite solids, is difficult. Handling of variable grid spacing is also difficult in the finite difference scheme. But the moving grid formulation of the finite element method can handle the ablation problems in finite solids without special modifications in the formulation. 98 Table 6.5.1 Radial ablation of an infinite plate with constant heat flow Comparision of thicknesses External heat flow q = 20.0 kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.20 0.0024021300 0.0025704871 -6.550 0.40 0. 0066498900 0.0069667478 —4.548 0. 60 0.0118353199 0.0122488178 -3.376 0.80 0.0176328905 0.0181275625 -2.729 1.00 0.0238816794 - 0.0244440883 -2.301 1.50 0. 0408710800 0.0415470339 -1.627 2.00 0.0590866506 0.0598448068 -1.267 2.50 0.0780123100 0.0788335949 -1.042 3.00 0.0973486900 0.0982160345 —0.88 3 3.50 0.1169037372 0.1178033501 -0.764 4.00 0.1365493089 0.1374703646 -0.670 4.50 0.1561976075 0.1571317613 -0.595 5.00 0.1757875681 0.1767283380 -0.532 5.50 0.1952762604 0.1962186694 -0.480 6. 00 0.2146335095 0.2155737132 —0.436 6.50 0.2338381857 0.2347732186 -0.398 7.00 0.2528757453 0.2538032830 -0.365 7.50 0.2717364728 0.2726546228 -0.337 8 . 00 0.2904142141 0.2913213670 -0.311 8.50 0.3089055121 0.3098002672 -0.289 9.00 0.3272089958 0.3280898929 -0.268 9.50 0.3453247547 0.3461902440 -0.250 10. 00 0.3632541299 0.3641023636 -0.233 10.50 0.3809993565 0.3818281293 -0.217 11.00 0.3985633850 0.3993699253 -0.202 11.50 0.4159497321 0.4167306423 -0.187 12 . 00 0.4331623018 0.4339134097 —0.173 12.50 0.4502053857 0.4509215951 -0.159 13.00 0.4670835733 0.4677586854 -0.144 13.50 0.4838016033 0.4844282269 -0.129 14.00 0.5003643632 0.5009338856 -0.114 14.50 0.5167769790 0.5172792077 -0.097 15.00 0.5330445170 0.5334678292 -0.079 99 Table 6.5.2 Radial ablation of an infinite plate with constant heat flow Comparision of velocities External heat flow = q = 2 0.0 kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.20 0.0180751793 0.0189014506 -4.371 0.40 0.0240184907 0.0245298743 -2.085 0. 60 0.0276300609 0.0280689206 -1.564 0.80 0.0302242506 0.0305912420 -1.200 1.00 0.0321831815 0 . 0324903019 -0.945 1.50 0.0354418196 0.0356197022 -0.499 2.00 0.0372679010 0.0374107435 -0.382 2.50 0.0383416191 0.0384482034 -0.277 3.00 0.0389438011 0.0390200280 -0.195 3.50 0.0392361395 0.0392882749 -0.133 4.00 0.0393184610 0.0393518060 -0.085 4.50 0.0392554887 0.0392742977 -0.048 5.00 0.0390907787 0.0390983745 -0.019 5.50 0.0388544612 0.0388534144 0.003 6 . 00 0.0385678485 0.0385601148 0.020 6.50 0.0382462591 0.0382333249 0. 034 7.00 0.0379008986 0.0378838368 0.045 7.50 0.0375400409 0.0375195779 0.055 8. 00 0.0371698514 0.0371464156 0.063 8.50 0.0367949791 0. 0367687196 0.071 9.00 0.0364189707 0.0363897607 0 . 080 9.50 0.0360445194 0.0360119678 0.090 10. 00 0.0356737114 0.0356371589 0.103 10.50 0.0353081487 0.0352666825 0.118 11. 00 0.0349490903 0.0349015258 0.136 11.50 0.0345975086 0.0345424004 0.160 12 . 00 0.0342541784 0 . 0341898054 0.188 12.50 0.0339196995 0.0338440798 0.223 13.00 0.0335945487 0.0335054360 0.266 13.50 0.0332790986 0.0331739858 0.317 14.00 0.0329736397 0.0328497700 0.377 14.50 0.0326783806 0.0325327702 0.448 15.00 0.0323934816 0.0322229229 0.529 100 Table 6.5.3 Radial ablation of an infinite plate with constant heat flux Comparision of thicknesses External heat flux = q = 2 0.0 kwatts/squar meter Time from the onset of melting Finite element solution Finite difference solution %Error 0.20 0.0024350600 0.0026031416 -6.457 0.40 0.0068251798 0.0071507040 —4.552 0.60 0.0123028904 0.0127336122 -3.383 0.80 0.0185586605 0.0190771297 -2.718 1.00 0.0254364200 0.0260305982 -2.283 1.50 0.0447335206 0.0455054045 ,-1.696 2. 00 0.0663926825 0.0673026890 -1.352 2.50 0.0899117514 0.0909069255 -1.095 3.00 0.1149721891 0.1159912497 -0.879 3.50 0.1413028240 0.1423290521 -0.721 4.00 0.1687363535 0.1697537303 -0.599 4. 50 0.1971433163 0.1981375813 -0.502 5.00 0.2264207304 0.2273794711 -0.422 5.50 0.2564848065 ' 0.2573972642 -0.354 6. 00 0.2872660160 0.2881228328 -0.297 6.50 0.3187056482 0.3194987178 -0.248 7.00 0.3507534862 0.3514756858 -0.205 7.50 0.3833660781 0.3840111494 . -0.168 8 . 00 0.4165054858 0.4170677364 -0.135 8.50 0.4501382709 0.4506124258 -0.105 9.00 0.4842348993 0.4846157730 -0.079 9.50 0.5187690258 0.5190513730 —0.054 10. 00 0.5537171364 0.5538952947 -0.032 10.50 0.5890582204 0.5891259909 -0.012 11.00 0.6247735023 0.6247236729 0. 008 11.50 0.6608460546 0.6606701016 0.027 12 . 00 0.6972609758 0.6969487071 0. 045 12.50 0.7340049148 0.7335441113 0. 063 13 . 00 0.7710660100 0.7704420090 0.081 13.50 0.8084340096 0.8076291084 0.100 14. 00 0.8460998535 0.8450931311 0.119 14.50 0.8840560317 0.8828225136 0.140 15. 00 0.9222959876 0.9208065271 0.162 101 Table 6.5.4 Radial ablation of an infinite plate with constant heat flux Comparision of velocities External heat flux = q = 20.0 kwatts /square meter Time from the onset of melting Finite element solution Finite difference solution %Error 0.20 0.0184822306 0.0193218980 -4.346 0.40 0.0249475203 0.0256444551 -2.718 0. 60 0.0295069702 0.0299741626 -1.559 0.80 0.0329363085 0.0333403721 -1.212 1. 00 0.0357648395 0.0361144990 -0.968 1.50 0.0411784090 0.0414905101 -0.752 2.00 0.0452976711 0.0455334336 -0.518 2.50 0.0487160087 0.0487771966 -0.125 3.00 0.0514558591 0.0514858365 -0.058 3.50 0.0538136885 0.0538105145 0. 006 4.00 0.0558793806 0.0558460616 0. 060 4.50 0.0577157103 0.0576558895 0.104 5. 00 0.0593673103 0.0592844523 . 0.140 5.50 0.0608669184 0. 0607641004 0.169 6. 00 0.0622392111 0.0621190928 0. 193 6.50 0.0635032803 0.0633680820 0.213 7.00 0.0646741763 0.0645257160 0.230 7.50 0.0657640621 0.0656037256 0.244 8. 00 0.0667828694 0. 0666116327 0.257 8.50 0.0677388981 0.0675572976 0. 269 9.00 0.0686391890 0.0684472769 0.280 9.50 0.0694897771 0.0692871213 0.292 10.00 0.0702959597 0.0700815544 0.306 10.50 0.0710623786 0.0708346292 0. 322 11.00 0.0717932433 0.0715498924 0. 340 11.50 0.0724923313 0. 0722304210 0.363 12.00 0.0731630996 0.0728789270 0. 390 12.50 0.0738087893 0.0734978244 0.423 13.00 0.0744323730 0.0740892515 0.463 13.50 0.0750366822 0.0746551156 0.511 14.00 0.0756243989 0.0751971528 0.568 14.50 0.0761980712 0.0757169053 0.635 15.00 0.0767601505 0.0762157887 0.714 102 Table 6.5.5 Radial ablation of an infinite plate with linear heat flow Comparision of thicknesses External heat flow = q = 10.0*t kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.0040982701 0.0032109879 27.633 0.40 0.0257181004 0.0249780249 2.963 0.70 0.0560283512 0.0555585474 0. 846 1.00 0.0921746194 0.0918994695 0. 299 1.30 0.1328026354 0.1325559020 0. 186 1.60 0.1769499034 0.1766046435 0. 195 1.90 0.2238570452 0.2233912945 0.208 2.20 0.2730355859 0.2724277079 0.223 2.50 0.3241058886 0.3233385384 0.237 2.80 0.3767687380 0.3758291304 0.250 3.10 0.4307851791 0.4296645522 0.261 3.40 0.4859622419 0.4846550226 0.270 3 . 70 0.5421423912 0.5406454802 0.277 4.00 0.5991958976 0.5975081921 0.282 4.30 0.6570149064 0.6551366448 0 . 287 4.60 0.7155092359 0.7134415507 0.290 4.90 0.7746025920 0.7723475099 0. 292 5.20 0.8342302442 0.8317901492 0.293 5.50 0.8943368793 0.8917143345 0.294 5.80 0.9548748732 0.9520726800 0.294 6.10 1.0158029795 1.0128239393 0.294 6.40 1.0770853758 1.0739320517 0.294 6.70 1.1386909485 1.1353657246 0.293 7.00 1.2005920410 1.1970970631 0. 292 7.30 1.2627649307 1.2591016293 0.291 7.60 1.3251882792 1.3213577271 0.290 7.90 1. 3878434896 1.3838456869 0.289 8.20 1.4507142305 1.4465484619 0.288 8.50 1.5137863159 1.5094501972 0.287 8 .80 1.5770473480 1.5725369453 0.287 9.10 1.6404868364 1.6357959509 0.287 9.40 1.7040961981 1.6992158890 0.287 9.70 1.7678686380 1.7627860308 0.288 10.00 1.8317995071 1.8264969587 0.290 103 Table 6.5.6 Radial ablation of an infinite plate with linear heat flow Comparision of velocities External heat flow = q = 10.0*t kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.0504718497 0.0492934249 2.391 0.40 0.0890163407 0.0898683891 -0.948 0.70 0.1117199510 0.1125635430 -0.749 1.00 0.1284286529 0.1289441139 —0.4 00 1.30 0.1419229507 0.1415980458 0.229 1.60 0.1520817429 0.1517022103 0.250 1.90 0.1603933871 0.1599406898 0.283 2.20 0.1672776341 0.1667621285 0.309 2.50 0.1730443835 0.1724809855 0.327 2,80 0.1779240370 0.1773267239 0.337 3.10 0.1820908785 0.1814713031 0.341 3.40 0.1856787652 0.1850461513 0.342 3.70 0.1887917966 0.1881531924 0.339 4.00 0.1915118694 0.1908725202 0. 335 4.30 0.1939040273 0.1932677925 0. 329 4.60 0.1960204691 0.1953900307 0.323 4.90 0.1979033947 0.1972805709 0. 316 5.20 0.1995872110 0.1989731342 0.309 5.50 0.2011002451 0.2004954666 0.302 5.80 0.2024659067 0.2018705755 0.295 6.10 0.2037037909 0.2031176239 0.289 6.40 0.2048303336 0.2042527199 0.283 6.70 0.2058595270 0.2052894682 0.278 7.00 0.2068032771 0.2062394172 0 . 273 7.30 0.2076719105 0.2071124315 0. 270 7.60 0.2084744573 0.2079169899 0 . 268 7.90 0.2092189044 0.2086604089 0. 268 8.20 0.2099124640 0.2093489915 0.269 8.50 0.2105617970 0.2099882662 0.273 8.80 0.2111731470 0.2105830461 0.280 9.10 0.2117526233 0.2111375630 0.291 9.40 0.2123063952 0.2116555125 0. 308 9.70 0.2128409445 0.2121402174 0.330 10.00 0.2133633643 0.2125945538 0. 362 104 Table 6.5.7 Radial ablation of an infinite plate with linear heat flux Comparision of thicknesses External heat flux = q = 10.0*t kwatts/square meter Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.40 0.70 1.00 1.30 0.0041530202 0.0268165395 0.0602238290 0.1020505577 0.1513928771 0.0032376554 0.0259759668 0.0595543645 0.1015018150 0.1507942230 28.272 3 .236 1.124 0. 541 0.397 1.60 1.90 2.20 2.50 2.80 0.2076528519 0.2702957094 0.3390020430 0.4135101438 0.4936009645 0.2068198323 0.2691528201 0.3374701142 0.4115133584 0.4910688996 0.403 0.425 0.454 0.485 0. 516 3.10 3.40 3.70 4.00 4.30 0.5790882707 0.6698124409 0.7656360865 0.8664411902 0.9721268415 0.5759565234 0.6660224795 0.7611346841 0.8611798882 0.9660610557 0.544 0. 569 0.591 0.611 0. 628 4.60 4.90 5.20 5.50 5.80 1.0826071501 1.1978098154 1.3176749945 1.4421542883 1.5712090731 1.0756957531 1.1900141239 1.3089581728 1.4324800968 1.5605416298 0.643 0. 655 0.666 0. 675 0. 684 6.10 6.40 6.70 7.00 7.30 1.7048109770 1.8429406881 1.9855886698 2.1327559948 2.2844581604 1.6931123734 1.8301696777 1.9716967344 2.1176829338 2.2681217194 0. 691 0. 698 0.705 0.712 0.720 7.60 7.90 8.20 2.4407391548 2.6016838551 2.7674825191 2.4230110645 2.5823519230 2.7461481094 0.732 0.749 0 . 777 8.50 8.80 9.10 2.9387278557 3.1179275513 3.3214669228 2.9144053459 3.0871312618 3.2643344402 0.835 0.998 1.750 105 Table 6.5.8 Radial ablation of an infinite plate with linear heat flux Comparision of velocities External heat flux = g = 10.0*t kwatts/square meter Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.0516564511 0.0500394627 3 .231 0.40 0.0954475179 0.0960987732 -0.678 0.70 0.1261466295 0.1266340911 -0.385 1. 00 0.1521904469 0.1524816304 -0.191 1.30 0.1764925718 0.1758051217 0.391 1.60 0.1983790547 0.1974666864 0.462 1.90 0.2190835029 0.2179091871 0.539 2.20 0.2388302088 0.2373971641 0.604 2 .50 0.2577828169 0.2561064959 0.655 2.80 0.2760643959 0.2741642296 0.693 3.10 0.2937724292 0.2916682661 0.721 3.40 0.3109875619 0.3086980879 0.742 3.70 0.3277786374 0.3253206015 0.756 4.00 0. 3442058563 0.3415937424 0.765 4.30 0.3603226542 0.3575685024 0.770 4.60 0.3761768341 0.3732903600 0.773 4.90 0.3918116093 0.3887999952 0.775 5.20 0.4072661400 0.4041338861 0.775 5.50 0.4225763083 0.4193247855 0.775 5.80 0.4377752841 0.4344018102 0.777 6.10 0.4528945684 0.4493908882 0.780 6.40 0.4679654241 0.4643148184 0.786 6.70 0.4830211997 0.4791933894 0.799 7.00 0.4981018007 0.4940437675 0.821 7.30 0.5132628679 0.5088804364 0.861 7.60 0.5286515355 0.5237156153 0.942 7.90 0.5444000959 0.5385591388 1.085 8 .20 0.5611649156 0.5534188747 1.400 8.50 0.5813859105 0.5683010221 2.302 Table 6.5.9 Radial ablation of an infinite plate with quadratic heat flow Comparision of thicknesses External heat flow = q = t*t kwatts 106 Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.0111702299 0.0084331036 32.457 0.20 0.0260729194 0.0231435895 12.657 0.30 0.0447123796 0.0417792201 7. 021 0.40 0.0664936230 0.0636205077 4.516 0.50 0.0910545811 0.0882557780 3.171 0. 60 0.1181197390 0.1153932661 2 .363 0.70 0.1474679857 0.1448044330 1.839 0.80 0.1789139956 0.1763006002 1. 482 0.90 0.2122986168 0.2097213119 1. 229 1.00 0.2474831939 0.2449278384 1. 043 1.20 0.3227796257 0.3202283084 0.797 1.40 0.4039896727 0.4013961852 0. 646 1.60 0.4904730916 0.4878004789 0.548 1.80 0.5817218423 0.5789420605 0. 480 2.05 0.7021477818 0.6989309788 0. 460 2.35 0.8558722734 0.8510181308 0.570 2.55 0.9627326131 0.9569411874 0. 605 2.75 1.0729260445 1.0662636757 0. 625 2.95 1.1863093376 1.1788264513 0. 635 3.15 1.3027554750 1.2944935560 0. 638 3.35 1.4221531153 1.4131476879 0. 637 3 .55 1.5444045067 1.5346860886 0. 633 3 .75 1.6694232225 1.6590180397 0.627 3.95 1.7971329689 1.7860624790 0. 620 4.15 1.9274660349 1.9157460928 0. 612 4.35 2.0603628159 2.0480022430 0. 604 4.55 2.1957716942 2.1827692986 0.596 4.75 2.3336465359 2.3199908733 0.589 4.95 2.4739556313 2.4596140385 0.583 5.05 2.5450179577 2.5303106308 0.581 107 Table 6.5.10 Radial ablation of an infinite plate with quadratic heat flow Comparision of velocities External heat flow = q = t*t kwatts Time from the onset of melting Finite element solution Finite difference solution (Error 0.10 0.1276565343 0.1230977029 3.703 0.20 0.1692634672 0.1684013009 0.512 0.30 0.2029842138 0.2032260448 -0.119 0.40 0.2323959619 0.2329434305 -0.235 0.50 0.2586856186 0.2592885792 -0.233 0. 60 0.2825399041 0.2830876112 -0.193 0.70 0.3043856323 0.3048263490 -0.145 0.80 0.3245220184 0.3248332739 -0.096 0.90 0.3431775272 0.3433527350 -0.051 1.00 0.3605368137 0.3605783284 -0.012 1.20 0.3919630647 0.3917617202 0.051 1.40 0.4197962284 0.4193958640 0.095 1.60 0.4447892606 0.4442348182 0.125 1.80 0.4675193429 0.4668513536 0.143 2.05 0.4989664853 0.4926543236 1.281 2.25 0.5169238448 0.5117205381 1.017 2.45 0.5343502760 0.5296545625 0.887 2.65 0.5510118604 0.5466457605 0.799 2.85 0.5669560432 0.5628420711 0.731 3.05 0.5822656155 0.5783600211 0.675 3.25 0.5970189571 0.5932918191 0.628 3.45 0.6112837195 0.6077110171 0.588 3.65 0.6251175404 0.6216769218 0.553 3.85 0.6385698318 0.6352374554 0.525 4.05 0.6516839266 0.6484320164 0.502 4.25 0.6644999981 0.6612930298 0.485 4.45 0.6770578027 0.6738476753 0.476 4.65 0.6893810034 0.6861187816 0.475 4.85 0.7015499473 0.6981260777 0.490 5.05 0.7136461139 0.7098864317 0.530 108 Table 6.5.11 Radial ablation of an infinite plate with quadratic heat flux Comparision of thicknesses External heat flux = q = t*t kwatts/square meter Time from Finite element Finite difference the onset of melting solution solution %Error 0.10 0.0113904802 0.0085421968 33.344 0.20 0.0269745104 0.0238191616 13.247 0.30 0.0469737090 0.0437050983 7.479 0.40 0.0709945485 0.0676724762 4.909 0.50 0. 0988431871 0.0954891816 3.512 0.60 0.1304252595 0.1270325482 2.671 0.70 0.1656738967 0.1622326374 2.121 0.80 0.2045490742 0.2010488361 1.741 0.90 0.2470299751 0. 2434585989 1.467 1.00 0.2931065559 0.2894517481 1.263 1.10 0.3427771628 0.3390274346 1.106 1.20 0.3960471749 0.3921926320 0.983 1.30 0.4529284239 0.4489616752 0.884 1.40 0.5134387612 0.5093560219 0. 802 1.50 0.5776044726 0.5734044909 0.732 1.60 0.6454582810 0.6411436200 0.673 1.70 0.7170400023 0.7126180530 0.621 1.80 0.7923969030 0.7878809571 0.573 1.90 0.8715837002 0.8669941425 0.529 2.05 0.9983204007 0.9930406213 0.532 2.15 1.0886099339 1.0821094513 0.601 2.25 1.1829334497 1.1753162146 0.648 2.35 1.2814267874 1.2727632523 0.681 2.45 1.3842090368 1.3745615482 0.702 2.55 1.4913911819 1.4808298349 0.713 2.65 1.6030828953 1.5916944742 0.715 2.75 1.7193998098 1.7072885036 0.709 2.85 1.8404610157 1.8277508020 0.695 2.95 1.9663894176 1.9532254934 0.674 3.05 2.0973150730 2 .0838608742 0.646 3.15 2.2333712578 2.2198085785 0.611 3.25 2.3771157265 2.3612227440 0.673 Table 6.5.12 Radial ablation of an infinite plate with quadratic heat flux Comparision of velocities External heat flux = q = t*t kwatts/square meter 109 Time from the onset of melting Finite element solution Finite difference solution (Error 0.10 0.1318564862 0.1260308027 4.622 0.20 0.1790588945 0.1771504134 1.077 0.30 0.2206931859 0.2197982222 0.407 0.40 0.2597137988 0.2591891587 0.202 0.50 0.2973485589 0.2969506979 0.134 0.60 0.3343655765 0.3338047862 0.168 0.70 0.3707774878 0.3701317012 0.174 0.80 0.4069146216 0.4061558545 0.187 0.90 0.4429036081 0.4420231581 0.199 1.00 0.4788372517 0.4778386652 0.209 1.10 0.5147909522 0.5136865377 0.215 1.20 0.5508326888 0.5496413708 0.217 1.30 0.5870165825 0.5857748389 0.212 1.40 0.6234271526 0.6221593022 0.204 1.50 0.6601310372 0.6588701010 0.191 1.60 0.6971964240 0.6959862709 0.174 1.70 0.7346964478 0.7335905433 0.151 1.80 0.7727066278 0.7717691660 0.121 1.90 0.8113035560 0.8106111288 0.085 2.05 0.8831036091 0.8703160882 1.469 2.15 0.9226877093 0.9112133980 1.259 2.25 0.9637815356 0.9530905485 1.122 2.35 1.0060864687 0.9960342646 1.009 2.45 1.0495576859 1.0401273966 0.907 2.55 1.0940852165 1.0854473114 0.796 2.65 1.1397498846 1.1320650578 0.679 2.75 1.1865882874 1.1800441742 0.555 2.85 1.2346360683 1.2294398546 0.423 2.95 1.2839308977 1.2802987099 0.284 3.05 1.3345823288 ' 1.3326586485 0.144 3.15 1.3865423203 1.3865492344 0.000 3.25 1.4883482456 1.4419929981 3.215 110 Table 6.5.13 Radial ablation of an infinite plate with exponential heat flow Comparision of thicknesses External heat flow = q = Exp(t/100.0) kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0.0015243700 0.0019091971 -20.156 0.40 0.0129213901 0.0139328809 -7.260 0.70 0.0289620999 0.0303257145 -4.497 1.00 0.0478381999 0.0494534783 -3.266 1.40 0.0761994272 0.0780563280 “2.379 1.70 0.0993555635 0.1013457626 -1.964 2 . 00 0.1238631532 0.1259434670 -1.652 2.40 0.1584182978 0.1604979336 -1.296 2.70 0.1855091751 0.1875814795 -1.105 3.00 0.2135123909 0.2155700922 -0.955 3.40 0.2521587908 0.2541852295 -0.797 3.70 0.2820582092 0.2840551436 -0.703 4.00 0.3126991093 0.3146610260 -0.624 4.40 0.3546541035 0.3565624654 -0.535 4.70 0.3869143128 0.3887785375 -0.480 5.00 0.4198337495 0.4216514826 -0.431 5.40 0.4647282958 0.4664797485 -0.375 5.70 0.4991352260 0.5008352995 -0.339 6.00 0.5341641903 0.5358111858 -0.307 6.40 0.5818262696 0.5834008455 -0.270 6.70 0.6182853580 0.6198041439 -0.245 7.00 0.6553524137 0.6568143368 -0.223 7.40 0.7057185769 0.7071031332 -0.196 7.70 0.7442005873 0.7455263138 -0.178 8 .00 0.7832903266 0.7845560312 -0.161 8.40 0.8363576531 0.8375424147 -0.141 8.70 0.8768731952 0.8779955506 -0.128 9.00 0.9180051684 0.9190636873 -0.115 9.40 0.9738132954 0.9747841954 -0.100 9.70 1.0164005756 1.0173023939 -0.089 10.00 1.0596196651 1.0604497194 -0.078 Ill Table 6.5.14 Radial ablation of an infinite plate with exponential heat flow Comparision of velocities External heat flow = q = Exp(t/100.0) kwatts Time from the onset of melting Finite element solution Finite difference solution %Error 0.10 0 . 0253488496 0.0280325115 -9.573 0.40 0.0472845510 0.0488090031 -3.123 0.70 0.0587487184 0.0597174950 -1.622 1.00 0.0667223111 0.0674376711 -1.061 1.40 0.0747122914 0.0752185062 -0.673 1.70 0.0795266107 0.0799183771 -0.490 2.00 0.0839785635 0.0839763582 0.003 2.40 0.0886979401 0.0886828378 0.017 2.70 0.0918650180 0.0918278322 0.040 3.00 0.0947927386 0.0947268456 0. 070 3.40 0.0983916894 0.0983003080 0.093 3.70 0.1009230912 0.1008116677 0. Ill 4 . 00 0.1033381373 0.1032108590 0.123 4.40 0.1064156070 0.1062734351 0.134 4.70 0.1086444110 0.1084905565 0.142 5.00 0.1108154207 0.1106545776 0.145 5.40 0.1136449277 0.1134762242 0.149 5.70 0.1157281920 0.1155562177 0.149 6.00 0.1177917868 0.1176133156 0.152 6.40 0.1205162182 0.1203309149 0.154 6.70 0.1225451231 0.1223567724 0.154 7.00 0.1245655715 0.1243768185 0.152 7.40 0.1272612363 0.1270674318 0.153 7.70 0.1292843372 0.1290873289 0.153 8 . 00 0.1313122958 0.1311118901 0.153 8.40 0.1340280473 0.1338226199 0.154 8.70 0.1360784769 0.1358667910 0.156 9. 00 0.1381368637 0.1379225552 0.155 9.40 0.1409094930 0.1406843662 0.160 9.70 0.1430086344 0.1427731961 0.165 10.00 0.1451241672 0.1448785067 0.170 112 Table 6.5.15 Radial ablation of an infinite plate with exponential heat flux Comparision of thicknesses External heat flux = q = Exp(t/100.0) kwatts/square meter Time from the onset of melting Finite element solution Finite difference solution %Error 0 . 1 0 0.0015370500 0.0019231346 -20.076 0.40 0.0133450702 0.0143934144 -7.283 0.70 0.0306083597 0.0320759229 -4.575 1 . 0 0 0.0517344102 0.0535029173 —3.3 05 1.40 0.0848497078 0.0869319066 -2.395 1.70 0.1130004078 0.1152689904 -1.968 2 . 0 0 0.1438036859 0.1462128907 -1.648 2.40 0.1888784468 0.1913456768 -1.289 2.70 0.2255288064 0.2280141562 -1.090 3.00 0.2645721734 0.2670521438 -0.929 3.40 0.3203131258 0. 3227480352 -0.754 3.70 0.3648635745 0.3672393262 -0.647 4.00 0.4117653370 0.4140594602 -0.554 4.40 0.4779670238 0.4801193178 -0.448 4.70 0.5303817391 0.5324044228 -0.380 5.00 0.5851821899 0.5870568156 -0.319 5.40 0.6619983315 0.6636461020 -0.248 5.70 0.7224521637 0.7239100337 -0 . 2 0 1 6 . 0 0 0.7853735089 0.7866241336 -0.159 6.40 0.8731603622 0.8741092682 -0.109 6.70 0.9419751167 0.9426676035 -0.073 7.00 1.0133814812 1.0137915611 -0.040 7.40 1.1126748323 1.1126842499 -0 . 0 0 1 7.70 1.1902657747 1.1899554729 0.026 8 . 0 0 1.2705848217 1.2699366808 0.051 8.40 1.3820065260 1.3808788061 0.082 8.70 1.4688920975 1.4673787355 0.103 9.00 1.5586870909 1.5567628145 0.124 9.40 1.6830456257 1.6805299520 0.150 9.70 1.7798683643 1,7768771648 0.168 1 0 . 0 0 1.8798357248 1.8763160706 0.188 113 Table 6.5.16 Radial ablation of an infinite plate with exponential heat flux Comparision of velocities External heat flux = q = Exp(t/100.0) kwatts/square meter Time from the onset of melting Finite element solution Finite difference solution %Error 0 . 1 0 0.0256808791 0.0284059681 -9.593 0.40 0.0498824008 0.0515855812 -3.302 0.70 0.0645081773 0.0656264722 -1.704 1 . 0 0 0.0760266930 0.0769180432 -1.159 1.40 0.0892618299 0.0899498388 -0.765 1.70 0.0983104408 0.0988744348 -0.570 2 . 0 0 0.1071780622 0.1073594764 -0.169 2.40 0.1181416884 0.1182395592 -0.083 2.70 0.1261725426 0.1261949539 -0.018 3.00 0.1341070235 0.1340449601 0.046 3.40 0.1445851922 0.1444236934 0 . 1 1 2 3.70 0.1524224281 0.1521843076 0.156 4.00 0.1602631509 0.1599529833 0.194 4.40 0.1707566530 0.1703581810 0.234 4.70 0.1786839068 0.1782185435 0.261 5.00 0.1866683662 0.1861427426 0.282 5.40 0.1974319071 0.1968284100 0.307 5.70 0.2056043744 0.2049466521 0.321 6 . 0 0 0.2138814926 0.2131650299 0.336 6.40 0.2250869572 0.2242946178 0.353 6.70 0.2336979061 0.2327821404 0.393 7.00 0.2423608154 0.2413996160 0.398 7.40 0.2541409433 0.2531066239 0.409 7.70 0.2631528378 0.2620609403 0.417 8 . 0 0 0.2723278105 0.2711742222 0.425 8.40 0.2848331928 0.2835882604 0.439 8.70 0.2944345176 0.2931081653 0.453 9.00 0.3042306006 0.3028182983 0.466 9.40 0.3175836802 0.3160786629 0.476 9.70 0.3279395103 0.3262727559 0.511 1 0 . 0 0 0.3385546803 0.3366924226 0.55: 114 i n u P m k 0) -p (Ü r —i a X P3 10 0 ) i H Q) 4J (W 10 •H \ in c 5 Q) •H O C (W M X C « W Ü •H •H -p x : C (d 4J (0 Q} X : (p <W O O -P c C c (0 O O -P -H • H m 10 4J c -H (Ü o U r—1 o (0 A A < 0 X3 G O i H • H U (Ü 1 5 • H T3 (Ü P< to VO C 7 > -H P m û : < Q i-» < -J Q h^ü )Q K U L U S IU J Z » - 115 rr - i ' T c i o •H C -P • 0 0 •H r H 4J O 0 1 0 r H O ( D C O Ü C 4J ( U C P < U < U g < P 0) < M 1 - 4 * H < D 'd 0) 0 ) + J -P *P *P c c •H * r 4 P m P m .-4 04 (3 in O I SI s U.1 LU | T TT I " | ’I T -l T j i i " f T -|>-| " r -y T y T ~T- | ‘ T T " : T T T - T T T T f y T r ? g § ca Q ) 4J (d 1 -4 Qa X 0 < D r-4 C O +J 44 < u •r4 C ^ -P •H O •H 44 r-4 Ü C 44 O •H 1 — 1 4J 0 ) C (d > (d < u X I 44 44 o O 4J c c C fd o O 4J •H -H C O C O 4J C -H (d O P r-4 Ü (d X* A (d X I e 4J o iH -H Ü (d 3 •H * d ( d in in V O •H P m d 116 m (D d) Q ) PM f c « "r r 11- y - r r r r { T 1 T-w y r r r t j ’IT T T | r t - r r y r i ‘r r | ' r r t T ' p r t > p r i «• g ” d Qi-icna Kouizujzi- 0) 4 - > f d i H A X 0 1 0) 01 d ) d -P iH -H 44 0 1 •H ^ g 44 O X C r4 ü •H 44 -H X3 C -P +J (d ( d d) 44 44 Xî o O b c (d o Q ) -H C 01 •H -H t-4 P f d A X: a (d 4J •H c o -H X> ( d g 1-4^0 ( d •H S V O If) vo en -H P m 117 - H • H I— I CM CM rz 4- > iH Q ) ~ - H « W - H h d Z E - H < W i H -H *H • H * H ü_i C (0 o o 0) - H - H C 10 4J - H * H (d »H M I—I ( d «Se r H > o (d • H ill L l . 3E| tn - H ri TT 118 • f 'TT T 10 o (U (U 0} j T'Tf i-y 1-1 ' TT'^TT r-ry i"rTi ■ j’TT I r-| r i r ii T T-r-r y r r r i-j i i i ^ o : < Q i - « _ i a — (0 0 K U ü J z t iJ Z h - § gS < D 4J ( d X 1 —11 3 A r H (W t o Q) V. Q) 4J ( 0 - H O t o C i H d ) • H « M c tw X C -P ü - H t d - H Q ) 43 C x; 4J ( d ü <w « M - H O O +j t d C c U o o 'ü - H - H t d ( 0 +J - H t d U i H t d A ^ & ( d +J e - H o i H 3 U t d • H TJ ( d « 00 “ i vo t r » -H p L , 119 O Q ) 1 0 U & ( D 0 «H < U (M r— I - H 0) 13 < U Q ) 4J +J - f H ' t H - 0 C - H - H k p L | rH CM . O f r n "rr T »"TT"' « ‘T t"rrT ’i'rT'i-i-’ i r r rT'rt r x & r rrr i rTi i-i r n " i ' i’i"i g 1 0) -p (d X i H 0 C U i H 44 G > \ 1 0 P> > 0) - H O ■ H C rH HJ - H 44 - H < W ü C O - H (d rH 0) Q ) C > (d Ü 44 44 • H O O + J (d c c P o o T 5 - H - H (d c o ■P 0 - H (d c r P « H (d A 43 a (d -P 0 • H o i H 1 5 U (d - H 'd (d « m in vo Ü' - H C9i 120 *H iH -P O CM ( D X CM ■P G ) C P < D ( D g d ) (M I— I * H 0) 0) 0) •H +) C <W f O rX C Û ) o •H ^ *H u. •H <W <W +> O Ï « S t vj *H O à r vo -H 1~T a#-0)0 KOUSUJZ)- 121 C O • H • H r H O 0 01 r H 0 < U 0 1 U Q) Q ) i H * H 0) »o Q ) < U • H “H C c • H * H ( h ( h OJ a a • 1 ( U X ? -p ■ ( üiH » i $ —4«H o O 3 M z O . d -Sx) Q ) -H -P -H «H( dü z C o O • } -H Æ rH • u. Q ) ol C iH> ( Ü( D H* -H«H • I «W 4 - > O C Q ) C d O d o O O O-H — y -HA a i -P X -H « JQ )P h- rH « J A xîAl z « J 4 - ) g ol -H O - m 1 — H /ri 3 ü h U-l T -rT t‘|“ i"r r r |Tf r'T‘| “ i r rT“ |*'i*T’r i I rT 'rT"| ' T'i'T f | ‘T'r r r | i i i' r | r i f g g ; i l d d d in vo tr - H 122 BIBLIOGRAPHY 1. Altman, M., "Some Aspects of the Melting Solution for a Semi- Infinite Slab," Chemical Engineering Progress Symposium series. Vol.57, 1961, pp.16-23 2. Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1982. 3. Bathe, K.J. and Khoshgoftaar, M.R. "Finite Element Formulation and Solution of Nonlinear Heat Transfer," Nuclear Engineering and Design, No.51, 1979, pp.389-401 4. Biot, M.A. and Agrawal, H.C., "Variational Analysis of Ablation for Variable Properties," Transactions of ASME, Journal of Heat Transfer, Series C, Vol.8 6 , 1964, pp.437-442. 5. Biot, M.A. and Daughaday, H., "Variational Analysis of Ablation," Journal of Aerospace Sciences, Vol.29, Feb. 1962, pp.227-229. 6 . Carslaw, H.S. and Jaeger, J.C., Conduction in Solids, Oxford University Press, New York, 1959, P.72. 7. Chung, B.T.F. and Yeh, L.T., "Solidification and Melting of Materials Subject to Convection and Radiation," Journal of Spacecraft and Rockets, Vol.12, June 1975, pp.329-333. 8 . Chung, B.T.F., Chang, T.Y., Hsiao, J.S., and Chang, C.I., "Heat Transfer with Ablation in a Half-Space subjected to Time Variant Heat Fluxes," Transactions of ASME, Journal of Heat Transfer, Vol.105, Feb. 1983, pp.200-203 123 Chung, B.T.F., Chang, T.Y., Hsiao, J.S., and Chang, C.I., "A Finite Element Analysis of Heat Transfer in Solid with Radiation and Ablation," Proceedings of the Seventh International Heat Transfer Conference, Munich, Sep.,6-10, 1982, pp.99-104. 10. Chung, B.T.F., and Hsiao, J.S., "Heat Transfer with Ablation in a Finite Slab Subjected to Time-Variant Heat Fluxes," AIAA Journal, Vol.23, No.l, Jan. 1985. pp 145-150 11. Chin, J.H., "Finite Element Analysis for Conduction and Ablation Moving Boundary," AIAA 15th Thermophysics conference, July, 14-16 1980. 12. Comini, G., Guidice, S.Del., Lewis, R.W., Zienkiewicz, O.C., "Finite Element Solution of Nonlinear Heat Conduction Problems with Special Reference to Phase Change," International Journal for Numerical Methods in Engineering, Vol.8 , 1974, pp.613—624. 13. Cost, T.L. and Weeks, G.E., "Finite Element Analysis of Structures with ablating boundaries," AIAA/SAE 14th Joint Propulsion Conference ,Las Vegas, July, 25-27 1978. 14. Eckert, E.R.G. and Drake,R.M., Analysis of Heat and Mass Transfer, International Student Edition, McGraw-Hill Kogakusha, Ltd., Tokyo, 1972, p.187. 15. Goodman,T.R., "The Heat Balance Integral and its Application to Problems Involving a Change of Phase," Transactions of the ASME, Vol.80, Feb. 1958, pp.335-342 16. Goodman, T.R., "Integral Methods for Nonlinear Heat Transfer," Advances in Heat Transfer, Vol.l, Academic Press, New York, 1964, pp.51-122 124 17. Hogge, M. and Gerrekens, P., "Two Dimensional Deforming Finite Element Methods for Surface Ablation," AIAA 18th Thermophysics conference, Montreal, June 1983. 18. Landau,H.G., "Heat Conduction in a Melting Solid," Quarterly Applied Mathematics, Vol.8 , No.1,1950, pp.312-319 19. Lardner, T.J., "Approximate Solution to Phase Change Problems," AIAA Journal, Vol.5, Nov. 1967, pp.2079-2080. 20. Lynch, D.R. and O'Neill, K., "Continuously Deforming Finite Elements for The Solution of Parabolic Problems, with and without Phase Change," International Journal for Numerical Methods in Engineering, Vol.17, 1981, pp.81-96. 21. Morgan, K., Lewis, R.W. and Zienkiewicz, O.C., "An Improved algorithm for Heat Conduction Problems with Phase Change," International Journal for Numerical Methods in Engineering, Vol.12, 1978, pp.1191-1195. 22. Ozisik, M.N., Heat conduction, John Wiley and Sons, New York, 1980, pp.340-341 23. Polivka, Ronald.M., "Finite Element Analysis of Nonlinear Heat Transfer Problems," Ph.D., Dissertation, University of California, Berkeley, 1976. 24. Prasad, A. and Agrawal, H.C., "Biot's Variational principle for a Stefan Problem," AIAA Journal, Vol.10, march 1972, pp.325-327. 25. Prasad, A. and Agrawal, H.C., "Biot's Variational principle for Aerodynamic Ablation of Melting Solids," AIAA Journal, Vol.12, Feb. 1974, pp.250-252. 125 26. Prasad, A., "Effect of Temperature Dependent Heat capacity on Aerodynamic Ablation of Melting Bodies," AIAA Journal, Vol.16, Sept. 1978, pp.1004-1007. 27. Prasad, A. and Sinha, S.N., "Radiative Ablation of melting Solids," AIAA Journal, Vol.14, Oct. 1976, pp.1494-1497. 28. Prasad, A., "Melting of Solid Bodies Due to Convective Heating with the Removal of Melt," Journal of Spacecraft and Rockets, Vol.16, Nov.-Dec. 1979, pp.445-448. 29. Randall, J.D., "Finite Difference Solution of The Inverse Heat Conduction Problem and Ablation," Proceedings of the 1976 Heat Transfer and Fluid Mechanics Institute, Davis, June 21-23, 1976, pp.257—269. 30. Rao, S.S., The Finite Element Method in Engineering, Pergamon Press, Oxford, England, 1982. 31. Roose, J. and Storrer, O., "Modelization of Phase Changes by Fictitious Heat Flow," International Journal for Numerical Methods in Engineering, Vol.20, 1984, pp.217-225 32. Rubinsky, B. and Cravahlo, E.G., "A Finite Element Method For The Solution of One Dimensional Phase Change Problems," International Journal of Heat and Mass Transfer, Vol.24, No.12, 1981, 1987-1989. 33. Singampalli, Rao,S.S., COSMOST User's Manual, Structural Research and Analysis Corporation, Santa Monica, Ca., Jan 1985. 34. Sunderland,J.E. and Grosh,R.J., "Transient Temperature in a Melting Solid," Transactions of the ASME, Vol.83, Nov. 1961 pp. 409-414 126 35. Vallerani, E., "Integral Technique Solution to a class of Simple Ablation Problems," Meccanica, Vol.9, 1974, pp.94-101. 36. Weeks, G.E. and Cost, T.L., "An Algorithm for Automatically Tracking Ablating Boundaries," International Journal for Numerical Methods in Engineering, Vol.14, 1979, pp.441-449 37. Wellford, L.C. and Ayer, R.M., "A Finite Element Free Boundary Formulation for the Problem of Multiphase Heat Conduction/" International Journal for Numerical Methods In Engineering, Vol.11, 1977, pp.933-943. 38. Yeh, L.T. and Chung, B.T.F., "Transient Heat Conduction in a Finite Medium with Phase Change," ASME Paper 76 WA/HT-3, 1976. 39. Yeh, L.T. and Chung, B.T.F., "Phase Change in a Radiative and Convective Medium with Variable Properties," Journal of Spacecraft and Rockets, Vol.14, March 1977, pp.178-182. 40. Yoo, J. and Rubinsky, B., "Numerical Computation Using Finite Elements for The Moving Interface in Heat Transfer Problems with Phase Transformation," Numerical Heat Transfer, Vol.6 , 1983, pp.209-222. 41. Zien, T.F., "Approximate Solutions of Transient Heat Conduction in a Finite Slab," Heat Transfer and Thermal Control, Progress in Astronautics and Aeronautics, Vol.78, edited by A.Crosbie, AIAA, New York, 1981, pp.229-248. 42. Zien, T.F., "Integral Solution of Ablation Problems with Time Dependent Heat Flux," AIAA Journal, Vol.16, Dec. 1978, pp.1287- 1295. Consider an infinite plate, with a circular hole, initially at a uniform temperature . A heat flow boundary condition has been applied to the surface of the hole. If the temperature is less than its melting temperature, the surface temperature of the hole gradually increases until it reaches the melting temperature. If the heating still continues, the ablation of the material begins. Since the starting of the ablation depends very much on the temperature distribution in the plate, a pre-melt solution at the time of melting is essential. Hence this problem has been solved for both pre-ablation and ablation periods. Now consider the governing equations during these two periods. Pre-ablation Period The governing differential equation of heat conduction in cylindrical coordinates is given by + -1---^ = -L--5I_ (A.l) 3R2 R 3R a 3t where R is the radial coordinate. 12 8 The associated boundary conditions are -K —— = q atR = R^ , t > 0 (A. 2 ) 9K 0 9T _ ^ atR=oo , t > 0 (A. 3) The initial condition is T = Tq Rq < R < oo ,t = 0 (A.4) Equations (A.l) to (A.4) can be expressed in dimensionless form by defining the following dimensionless variables. ( " - :o ) 1. Dimensionless temperature 9 = ______ (A. 5) ( - To) 2. Dimensionless time t = t/t^ (A.6 ) R/ PC 3. Characteristic time " ------- (A.7) ^ K 4. Dimensionless radius ^ (A.8 ) 5. Dimensionless heat input Q = q^R^/K(T^ - T^) (A. 9) Thus non-dimensional governing equations may be written as (A.io) 9r^ 129 The associated boundary conditions are 99 9r = Q at r = 1, T > 0 (A.11) 90 _ r i "af ‘ ^ at r = 00 , T> 0 (A.12) The initial condition is 0 = 0 atx = o, l<r<oo (A.13) Hence the solution of equation (A.10) subjected to the conditions (A.11) to (A.13) gives the temperature distribution during the pre-ablation period. Ablation Period The governing differential equation of heat conduction in dimensionless form is 9r 9^0 1 99 _ 90 (A.14) 2 r 9r 9t The boundary condition at the ablating surface can be derived as follows. The conservation of heat flow at the ablating surface which is at a radial distance S, can be stated as 130 Heat flow input Heat flow conducting into the the plate Heat flow utilized in the ablation q = -2 ttRK —^ + 2 T T R PL (A.15) HR where ^ is the rate of ablation. Define the dimensionless ablation distance as s = Inverse of Stefan number (A.16) (A.17) Using expressions (A.5) to (A.7) and (A.16), equation (A.15) can be restated as 98 9r r=s PL ds "=(Tm - T.) dT (A.18) r=s Now consider the following two types of boundary conditions at the ablating surface. 1. Prescribed heat flow per unit length of the hole. q =2nR q (A.19) where q^is the heat flux input at the time of melting. 131 Using the dimensionless variables defined by (A.9) and (A.17) and the expression (A.19), the boundary condition (A.18) can be simplified as atr = s, T>T^ (A.2 0 ) 2. Prescribed heat flux condition q = 2 T r R q .21) Substitute the expressions (A.9), (A.17) and (A,21) into (A.18) to get _ . Q - S # at r = s, T >T^ (A.2 2 ) 9r ^ t dx The boundary condition at infinite radius is = 0 at r = 0 0 , T > 0 (A. 23 ) dr The initial condition is 0 at X - x^ 1 < Y>< 00 (A. 24) where 9^ is the dimensionless temperature distribution at the time of melting. The ablation condition is (A.25) 9 = 1.0 a t , r 1.0 X > X m 132 Hence the solution of equation (A.14) subjected to the conditions (A.20 or A.22), (A.23), (A.24) and (A.25) gives the rate of ablation and the temperature distribution in the plate. Note that the boundary conditions (A.20) and (A.22) are applied at a moving boundary which makes the problem complicated. This moving boundary condition has been eliminated by the following transformation of variables. Let z = = r - s (A. 26) Y = T ~ (A.27) The required gradients can be derived as 39 = M • + li iz = M 281 3r 3z 3r 3y 3r 3z. = 9^9 (A.29) 3r^ 3 2.2 39 _ 39 . 3y 39 . 32 . 39 39 . dS 3t 3y 3t 32. 3T " 3y 32 dr (A. 30) Note that s is a function of time only. Hence substituting the equations (A.28), (A.29), and (A.30) into (A.14), (A.20), (A.22), (A.23), and (A.24), the following equations of ablation can be obtained. li = li Y >0, 3y 0 < z’ < o o Boundary conditions 133 (A.31) 1. At the surface of the hole a) Prescribed heat flow - w = Q/"- - ^ # b) Prescribed heat flux 2. At the infinite radius 38 _ az = 0 Initial condition 0 = 8, Ablation condition Y %0, Z = 0 Y >0, Z = 0 Y >0, Z = Y = 0, 0 < Z < 0 = 1.0 Y > 0, Z = 0 (A.32) (A.33) (A.34) (A.35) (A.36) Explicit finite difference formulation The infinite plate has been idealized by a circular plate of large radius. The outer radius of the plate has been chosen such that the temperature at this point remains constant throughout the solution time under 134 consideration. Divide the radial distance of the plate into M elements. For stability in the explicit finite difference scheme , choose the time increment such that At 1 — , 1 -? <»■” ) 9 ^ . j = 0(1 A r, j A T ) (A. 38) Pre-ablation Period Use a forward difference scheme for time derivative and a central difference scheme for space derivatives. Hence the differential equation (A.10) can be expressed J+1 - \ , /®N+1. J ' 2®n. J + J / ' \ ar^ + .----- L_ ,—„ , { ®N+1, J - ®N-1, J (ro + (N-1) ar) I 2ar for N = 2 to M (A.39) Using the heat flow input boundary condition, the equation at Z = 0 may be expressed as & 02,0 .Qar(^ - ^ ) \ / \ 4A.40] 135 Using the boundary condition at infinite radius, the equation at Z = oo is The initial condition is given by ®i,J| = 0 (A'42) I j = 0 Hence the solution of simultaneous equations (A.39), (A.40) and (A.41) along with the initial condition (A.42) gives the temperature distribution during the pre-ablation period. Ablation period Finite difference equations of ablation for nodes N == 2 to M can be expressed as ^ GN-l.J ) AZ' , , ds , 1 V ®N+1,J “ PN-l.J , IdT S + (N-I)az H 2az ] (A.4 3 ) The boundary condition at Z = 0 is 0^ J = 1.0 Y è 0 (A.44) The boundary condition at Z = oo is (A.45) i3q Hence the solution of equations (A.43) to (A.45) along with the initial temperature distribution obtained from the pre-melt solution, gives the temperature distribution in the ablation period. Rate of ablation and the Ablation distance Rate of ablation can be obtained from the boundary conditions (A.32) and (A.33). Note that the temperature gradient at the ablating surface is required to find the rate of ablation. An approximate temperature gradient has been obtained by assuming a third order variation for the temperature as Q {Z) = Sg + a^Z + e^Z + a^Z (A.46) where a^, a^, a^, and a^are constants. Hence the finite difference expressions for the rate of ablation can be expressed as - Prescribed heat flow and ®1.0 ®®2,j - 303,0 + I 04,j)^ (A.48) - Prescribed heat flux 137 Assuming constant acceleration between time steps, the increment in ablation distance at any time can be obtained as (A.49) às JAy (J-1)Ay dT Hence the radius of the hole at any time t can be obtained.
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Singampalli, Sankara Srinivasa Rao
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Deforming finite element analysis of thermal ablation
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Doctor of Philosophy
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