Illlllllllllllllllllllllllllllllllll!lilllllllllllllllllllllll 3 1293 01691 8918 This is to certify that the thesis entitled EMPIRICAL TIME STEP EQUATIONS FOR THE RADIAL FIELD PROBLEMS presented by Wie Tjung Tan has been accepted towards fulfillment of the requirements for M.S. degree in Agricultural Engineering Date 8" q’9( 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINE return on or before date due. MTE DUE MTE DUE DATE DUE 1/” WWW.“ EMPIRICAL TINIE STEP EQUATIONS FOR THE RADIAL FIELD PROBLEMS By Wie Tjung Tan A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Agricultural Engineering Department of Agricultural Engineering 1995 ABSTRACT EMPIRICAL TIME STEP EQUATIONS FOR THE RADIAL FIELD PROBLEM By Wie Tjung Tan A variety of time-dependent problems in science and engineering are governed by a class of partial differential equations called parabolic. Applying a numerical procedure changes the Space and time-dependent partial differential equation into a time-dependent system of ordinary differential equations. That can be solved using numerical integration methods in time. This process requires a time step value that produces an accurate result. Empirical time step equations were developed for the radial field problem using numerical experimentation. The study was limited to radial problems consisting of a solid disk, solved using linear elements, a lumped formulation for the capacitance matrix and the Single step integration methods in time. The empirical time step equations developed in this study were Forward difference method: -2.13 At = 3.91” Central difference method: 4.66 At = 5.28” Backward difference method: N—l.81 I At = 0.12 The time step equations were validated using eight different problems involving a combination of materials and boundary conditions. Copyright by ‘WHIETULHQCETVUN 1995 Approved: \‘ery J. Segerlind Major Professor Robert von Bernuth Chair Person To my mom Jut Kioe vi ACKNOWLEDGMENT A very Special thank to Dr. Segerlind for the time spent guiding and correcting my thesis. Everything that he has done to me, I greatly appreciated. Thank to Dr. Bralts and Dr. Martin as my committee members. A great appreciation to all of my friends at Michigan State University and the Agricultural Engineering Department especially, that help me finish my work and study at Michigan State University. vii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTER 1 . Introduction 2. Review of Literature 3. Objectives 4. The Radial Field Equation 4.1. The Step Change Problem 4.2. Numerical Solution in Space 5. Methodology and Empirical Equations 6. Evaluation of the Time Step Estimates 7. Discussion and Conclusion 7.1. Future Study viii 10 11 13 14 17 32 42 42 LIST OF TABLES TABLE 5.1. The Reference values for the Step Change Problem 5.2. Calculated Time Step Values for the Forward Difference Method 5.3. Calculated Time Step Values for the Central Difference Method 5.4. Calculated Time Step Values for the Backward Difference Method 5.5. The Time Step Values Used in the Best Fit Equation for Each Integration Method 6.1. Time Step values for the Three Different Integration Methods 6.2. Time Step Values for Different Materials, 11 Node Grid 6.3. Time Step Values for Different Materials, 21 Node Grid 6.4. Regression Results between At and At/2, 2At, and 4At ix Page 19 29 33 35 38 41 LIST OF FIGURES FIGURE 4.1. Mathematical Description for the Step Change Problem 5.1. The Error Ratio for the Forward Difference Method 5.2. The Error Ratio for the Central Difference Method 5.3. The Error Ratio for the Backward Difference Method 5.4. Time Step Estimates for the Three Difference Integration Methods 31 CHAPTER ONE INTRODUCTION A variety of time-dependent problems in science and engineering are governed by a class of partial differential equations called parabolic. These equations are also referred to as the diffusion equations. These equations have the general form c .35? = k V.(VU) + Q (H) where c and k are material coefficients and U is the unknown variable: temperature, moisture content, pressure head, and so forth. The diffusion equation governs several biosystem processes including the heating and cooling of food products, grain drying, water infiltration through the soil, and salt movement into and through the soil. The movement of odors from production and waste facilities through the air is also governed by the diffusion equation. The derivation of (1-1) is included in almost every engineering and mathematics book dealing with the solution of the diffusion equations. Powers (1987), Ozisik (1980), Pakantar (1980), and Churchill and Brown (1987) are a few examples. Applying a numerical procedure such as the finite element or the finite difference method to (1-1), changes the space and time-dependent partial differential equation into a time—dependent system of ordinary differential equations. This conversion is discussed in books dealing with the numerical solution of partial differential equations (Narasimhan, 1978, Segerlind, 1984, and Smith, 1985). The system of ordinary differential equations has the general form [q%g+mm—m=m am where [C] is the global capacitance matrix, [K] is the global stiffness matrix, and { F ) is the forcing function. Finite element or finite difference methods can be used to solve (1-2) in the time domain. Although the finite element method has clear advantages over the finite difference method in the space domain, (1-1), this advantage does not extend to the time domain, Segerlind (1984). There are many schemes available for solving (1-2). Many criteria are used to test these schemes including stability, oscillation, and accuracy. Despite the fact that many authors have presented and discussed solution procedures for the system of ordinary differential equations in (1-2), there is a lot of art and experience involved in selecting the proper scheme and time step to obtain an accurate solution, particularly in two- and three-dimensional problems. Mohtar (1994) has pioneered the development of empirical equations that can be used to estimate the time step required to solve (1-2) accurately when using any one of the three single step methods; Euler’s forward difference method, the central difference method and the backward difference method. Mohtar developed equations that included the lowest eigenvalue as a parameter. Equations were developed for the one-dimensional form of (1-1) and the two-dimensional case when the grid was restricted to square elements. Each of the equations given by Mohtar had the general form AlAt = C N” (1-3) where A, is the lowest eigenvalue for the problem being solved, N is the number of nodes in the region, C and b are empirically determined coefficients and At is the time step. The general objective of this study was to extend the work started by Mohtar (1994) to the diffusion equation in radial coordinates DaZU + 0'3” will (14) rarz r 3r 'at This study can be considered a preliminary step to a study of axisymrnetric shapes. CHAPTER TWO REVIEW OF LITERATURE A complete literature search of published work on the numerical solution of partial differential equations and their implementation is probably impossible. Among the extensive published resources on the subject, there are at least two questions that remain unanswered. These questions are: "How did the investigators who developed the methods come up with the time step value they used in their numerical solution?" and "Do the stability, and oscillation criteria for selecting a time step ensure an accurate solution?" Although the numerical solution of (1-1) and (1-2) are presented in numerous books and technical articles, Mohtar (1994) presented a review of literature that shows these two questions have not been answered. Mohtar’s research was directed at answering these questions for specific applications of the diffusion equation. This research continues by considering the radial field problem. The review of literature given here discusses a few general publications related to solving (1-1) and (1—2) and the work done by Mohtar (1994). The numerical solution of the system of ordinary differential equations have been discussed for several years, Zienkiewicz (1971) and Segerlind (1976). The finite element and finite difference methods for solving time-dependent field problems have been explained in finite element books or technical papers. Allaire (1985) discussed solving the one-dimensional problem using Euler’s single step method. Euler’s method was also used to solve a non-dimensional form of the one-dimensional diffusion equation by Smith 5 (1985). Three node linear triangular elements were used to solve heat transfer problems, Jaluria and Torrance (1986). Segerlind (1984) discusses using the four node quadrilateral elements. The error bounds for different orders of a finite element method solution are given by Shih (1984). Some authors have used a mathematical approached to define the time Step value (Gear, 1971, Stoer and Bulivsch, 1980, Myers, 1977 and Reddy, 1984), and/or computer programs (Pakantar, 1991). Ortega (1990), Rushton and Tomlinson (1971), and Henrici (1977) developed various types of error procedures which were associated with the time step values. Williams (1980) and Fried (1979) discussed stability requirements for the numerical solution of partial differential equations. Haghighi and Segerlind (1988) used Maadooliat’s (1983) non-oscillation criteria to solve the coupled heat and mass transfer equations in finite element method. Irudayaraj (1991) and Irudayaraj et al., (1990) used a stability criteria for selecting the time step value in a coupled heat and mass transfer problem. Segerlind and Scott ( 1988) defined the time step estimate using a non-oscillation criteria. Maadoliat (1983) and Segerlind (1984) discussed numerical solutions for the physical reality and oscillations in finite element problems. In fluid flow, Peraire et al., (1988) defined a Courant stability criteria. Cleland and Earle (1984) solved food freezing problems using six different finite difference methods. They reduced the time step and resolved the problem until the calculated results did not change. Mohtar (1994) was among the first researchers to define the time step values based on an experimental accuracy criteria. He investigated the one-dimensional problem and a two-dimensional grid with square elements. The general procedure developed by Mohtar was to (a) define a measure of the error, (b) solve a problem using several different 6 values of the time step (At) and several subdivisions of the problem in space, (c) plot the error value against the number of nodes and select the time step value, At, that produced a specified error, (d) empirically fit an equation to the time step data using the lowest eigenvalue as a basic parameter, and (e) checked the equations by solving a different set of problems. In one-dimensional problems, Mohtar (1994) defined the accuracy ratio as: n m 2 X |N0DEij-APDEU| e = f" "1 (2-1) 2": 2m: |AODEij-APDEij| jel i-l where NODE is the numerical solution for the system of ordinary differential equations, APDE is the analytical solution for the partial differential equation, AODE is the analytical solution for the system of ordinary differential equations, and n and m are the number of sampling points in the space and time domain. The dynamic time step equations developed by Mohtar for three single step methods are Forward Difference: N—l.6 At = 0.27 (2-2) Central Difference: N—HS At 1.13 (2-3) Backward Difference: -3.9l At = 30.6 N)» (24) In each equation, At is the time step value, N is the number of free nodes and A] is the lowest eigenvalue for the system. The time step estimates were validated using four different problems; a Sine wave variation and a linear variation with the boundary temperatures known and two problems with derivative boundary conditions. The problems were solved using fractions or multiples of the calculated time step. Time step values of one-half, two, and three times At were used along with the ratio ENODE, e = f"— (2-5) X APDE, is! The accuracy ratio for At/2 and At were equivalent. The results for multiples of two and greater were less accurate than the results for At. In two dimensional problems, Mohtar (1994) defined the accuracy ratio as: Z": i [NODEU-APDEUI NODEU (2-6) j-l i=1 mn where NODE is the numerical solution for the system of ordinary differential equations, APDE is the analytical solution for the partial differential equation, m is the number of sampling points in the space domain, and n is the number of sampling points in the time domain. The sampling points in the time domain were at 9.5, 19, 28.6, 38.1, 47.6, 57.1, 8 66.7, 76.2, 85.7 and 95.2 percent of the time to steady state. The accuracy ratio used with the two-dimensional problem was different from the accuracy ratio, (2-1), used with one-dimensional problems because the analytical solution of the system of ordinary differential equations, AODE, became too difficult to evaluate. Mohtar (1994) restricted the two-dimensional study to square elements and square grids because he wanted to compare the finite element and finite difference formulations in space. Using (2-5) and a five percent error in the calculated values when compared to the analytical solution of the partial differential equation, Mohtar developed the empirical time step estimates for a two-dimensional square grid given below. Equations (2-7) through (2—9) are for the finite difference formulation in space while the next three are for the finite element method in space. Forward Finite Difference: -l.01 At - 1.19” (2-7) A'l Central Finite Difference: -0.55 At = 1.6N (2-8) A'1 Backward Finite Difference: N -0 1 At = 0.05 __'_ 2-9 x ( ) Galerkin Forward Difference: Galerkin Central Difference: Galerkin Backward Difference: At N—l.04 -0.55 1.6 N -0.l 0.05L 7. l (2- 10) (2-11) (2-12) The equations are valid when the number of nodes used with the finite difference formulation in space is equal to or greater than nine. The equations are valid for the finite element method in Space, when the number of nodes is equal to or greater than twenty five. CHAPTER THREE OBJECTIVES The general objective of this study was to developed empirical equations for calculating the time step required to numen'cal solve the system of ordinary differential equations related to the time-dependent radial field problem. The specific objectives in this study were to develop an empirical time step estimate for the three single step integration methods that satisfy an accuracy criteria and the stability requirements and validate the time step equations by solving a different set of problems. This study was limited to -the linear radial element -three single step integration methods: forward difference, central difference, and backward difference -using the finite element method in space -use a lumped formulation for the capacitance matrix. The time step equations were placed in the same form as used by Mohtar (1994) At = C (34) AN“ where At is the time step value, A, is the smallest eigenvalue of the system, N is the number of nodes used to solve the problem, and a and C are parameters to be determined. 10 CHAPTER FOUR THE RADIAL FIELD EQUATION The one-dimensional time dependent field equation for radial coordinates is 03;" + Drfl = 0% (4-1) 'ar2 r 3r '8! where U is the pressure in flow problems, temperature in heat conduction problems, and concentration in solute transport problems, D, and DI are material properties, and r is the radial distance. The analytical solution of the left side of (4-1) is an infinite series solution involving Bessel functions (Myers, 1971, Boyce and Diprima, 1986, Ozisik, 1980, Kreyszig, 1988, and Zill, 1993). The general form of the steady state differential equation in radial coordinates is d 2u 1 du (4_2) and a series solution is obtained by assuming u(r) = a0 + alr + azr2 + = 2a r" (4'3) Using the method of Frobenius gives u(r) = r ‘2”: anr " (4'4) n-O where s is a parameter to be determined. The first and the second derivatives of (4-4) are 11 12 Eli = 2 (n + s)a rm"1 (4'5) dr " and d2“ = Z (n + s)(n + s — l)a r"""2 (4'6) drz n-O Substituting (4-4), (4-5), and (4-6) into (4-2) gives 2 (n+s)(n+s-l)anr""‘l + 2 (n+s)anr"’*"'l + E:cznr"’s+1 = 0 (4‘7) n-O n=0 n-O Equation (4-7) combines to give a Bessel function of first kind of order zero (Myers, 1971) _ (-1)"r _ r’ + r" _ (4—8) Jo(r) g— 22"(n!)2= _2_2_.-1_ W and a Bessel function of second kind of order zero co (_1)n-lh (4-9) Y(r) = J(r) 1n. + ___"r2" 012:1[0 ( 2+7] g; 22"(n!)2 J where h = O, 1, 2, and y = 0.57722 is called the Euler-Mascheroni constant. The solution of (4-2) is a linear combination of (4-8) and (4-9) giving u = A100) + BYo(r) (4-10) where A and B are parameters to be determinate using the boundary conditions. Bessel functions of integer order can also be developed for the first and the second kinds of order k 13 Jk(r) = i: ('4er (4-11) n=0 22""‘n!(n +k)! ,. °° <-1>"" = EJ,(r)[ln 57- +7] + r" b! (k-n-l)!r2,, (1_12) 7t n=0 22’"an where k is an integer. The general solution in integer order gives u = AJk(r) + BYk(r) (4-13) The solution of equation (4-2) can also be written using the general Bessel equation U = your) + BYOOJ) (4-14) where A is a positive number. The solution of (4-1) that includes the transient term is more involved than discussed above. The solution has eigenvalues as well as the Bessel functions. Since the Bessel functions are defined by infinite series, the analytical solution to (4-1) is really a numerical solution because of the need to evaluate the Bessel functions and the summation of an infinite series. 4.1 THE STEP CHANGE PROBLEM The step change problem in radial coordinates contains all the frequency components and has the shortest time to steady state. The boundary and initial conditions for the step change problem are U(r0,t) = U0, and U(r,0) = Ui, Figure 4.1. Equation (4-1) is usually written as am 13v 13v (“5) where a = Dr/Dt. The boundary conditions in Figure 4.1 are not homogeneous, therefore, a new variable T = U — UO can be defined and (4-15) becomes 822‘ + laT _ 1 31 (4-16) a—ri r 3r 0!. a: The boundary and initial condition change to T(O,t) < 00, T(r0,t) = 0, and T(r,0) = Ui - U0. The analytical solution to (4-15) for the step change problem, Myers (1971), is U(r,t) -U0 _ °° 0(7tmr)e “‘-’°’"°“’"°’ __ 2 Ui-Uo 2 m1 (1J0)! 101nm (4-17) where Ui is the initial temperature, U0 is outside surface temperature, 01 is the thermal diffusivity for thermal problems, t is time, and the (Aura) satisfy J0(Mro) = 0. 4.2 NUMERICAL SOLUTION IN SPACE Equation (4-1) can be solved in space using the finite element or finite difference methods to produce a system of ordinary differential equations in time [0% + [K]{U} — {F} = {0} (4-18) where [C] is the global capacitance matrix, [K] is the global stiffness matrix, and {F} is the forcing function (Segerlind, 1984). The finite element procedure was used in this study for two reasons. First, the global matrix [K] remains symmetric. Second, the Singularity that occurs at r = O for the finite difference method does not occur for the 15 finite element method. The finite element matrices for one-dimensional radial element are -Lumped capacitance matrix: R.+2— O [Cm] __. 2MB: ' r - (4-19) 6 0 Rj+2r -Stiffness matrix: [K m] = ZRFDr l -1 (4-20) L _-l 1 where L is the element length, Ri is the radial distance of the left node of the element, R,- is the radial distance to the right node, and 27 = R: + R,- The general form of the solution of the system of ordinary differential equations for the single step methods is ([C]+6At[K]){U}2 = ([C]—(l-6)AI[K]){U}l + mem,+(1—e){F},) (4-21) where 0 = 0 gives the Euler’s forward difference method, 9 = 0.5 gives the central difference method, and 9 = 1 gives the backward difference method, Segerlind (1984). U—N=6 —o—N=11—o—N=16 +N ”_LLrlF Ta 4 1 4L _ d 2586 . ccmood 1. 8390 1 covood . comood - 8:56 t 8256 - omoood % ovoocd 4 omoood t Booed t wooood t 3255 oooood 76543210 33M Bum Time Step, Seconds Figure 5.1. The error ratio for the forward difference method 21 +N=6 +N=11+N=l6 +N=21 1 l I l L l l w oovod I oomod t 0206 I owood I omood w omood Time Step, Seconds Figure 5.2. The error ratio for the central difference method 22 +N=6+N=11+N=16+N=21 _ A _ _ c _ . _ _ _ S 4 3 2 1 O ovum Loam ] l l I I. I I 1 00v00.0 r 0050.0 1 0m000.0 I 0300.0 1 0N000.0 1 0800.0 1 00000.0 1 m00000 00000.0 Time Step, Seconds Figure 5.3. The error ratio for the backward difference method 23 20883 0000 0 0.0 0000000 0000 0 0.0 00000000 0000010 ‘“00000 .0 0 0 0000.0 0000000 0000000 30000 000000; 000000." 0000000 20.333 00.000 .0 00-00000 00020.0 000000 .0 0 000000 V000_ 0.0 0000000 000000 .0 0 0 0000.0 000.000 00 0 000.0 0000004 000000 00508 3:80.06 0.33.80 05 .80 829 08m 080 082330 . 000.0~ T 300.0? 0000.0 0 - 0_00v.0- 00000. 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08m 080 0030330 . 00000.0 0000 00.0 0000 00.0 000000 .0 0000000 0 000000 0000010 0000000 0000 00.0 0000000 0000000 00000.0 00000.0 0000000 00000.0 000000 .0 0000 0 0.0 000 000.0 0000 0 0.0 00000 0 .0 0000000 0000000 0000000 0000000 0000000 00000.0 00000 .0 00000 .0 000000 .0 0000000 000000 .0 0 00000.0 000000 .0 000 0 00.0 0000000 0000000 00000.0 00000.0 00000.0 00000.0 000000 .0 0000000 0000000 0000 010 00000 0.0 0000000 0000000 0000000 0000000 0000000 00000.0 00000 .0 00000 .0 000000 .0 0 0 0000.0 000 0 00.0 0000010 000000 .0 0000000 0000000 0000 0 0.0 0 00000.0 00.000 .0 00000.0 00000.0 000000 .0 000 0 0 0.0 0000000 0000 0 0.0 000 00 0 .0 0000000 0000000 0000000 00 0000.0 0000000 00000.0 00000.0 00000.0 000000 .0 0000000 000 0 00.0 0000010 00 0000.0 0000000 00 0000.0 0000000 000 000.0 00000.0 00000.0 00000.0 000000 .0 0000 0 0.0 0000000 00000 0.0 00000 0 .0 00 00000 0000000 0000000 0000000 0000000 00 0000.0 00000 .0 00-000 .0 000000 .0 0000000 0000 00.0 0000010 00 0000.0 0000000 0000000 0000 0 0.0 00 0 000.0 00000.0 00000.0 00 0 000.0 000000 .0 00000 0.0 0000000 00000 0.0 00000 0 .0 0000000 0000000 0000000 0000000 0000000 00000.0 00000 .0 00000 .0 000000 .0 0000000 000 0 00.0 0 00000.0 0000000 0000000 0000000 00 00 0 0.0 0000000 00000.0 00000.0 00000.0 00000 0 .0 0000000 0000000 0 0 000 0.0 0000 0 0 .0 000 000.0 0000000 0000000 0000000 0000000 0.0 0090.0. 00000.0 00000 .0 00000 .0 000000 .0 0000000 000 0 00.0 0000010 0000000 0000000 0000000 0000 0 0.0 0000000 00000.0 00000.0 00000.0 000000 .0 0000 0 0.0 0000000 00000 0.0 0000 0 0 .0 000 000.0 0000000 0000000 0000000 0000000 382 : 382 0 com .8 28 00000.0 00000.0 00000.0 000000 .0 0000000 0000000 0000010 00 0000.0 0000000 0000000 0000 00.0 000 0000 00000.0 00000.0 00000.0 000000 .0 00 0000.0 00 0000.0 0000010 0000000 0000000 0000000 0000000 0000000 0000000 00000. 0 00000.0 00000.0 000000 .0 000 0 00.0 00000 0.0 000 0 00.0 0000000 00 0 000.0 0000000 0000 0 0.0 0000000 00000.0 00-00 0.0 00000.0 000000 .0 0000000 0 000000 0000010 000 000.0 0000000 0000000 0000 0 0.0 0 0 0000.0 0000000 0000 00.0 00000.0 0000000 000000 .0 000 0 00.0 0000000 0000010 0000000 0000000 0000000 0000000 000 000.0 00000.0 00000.0 00000.0 000000 .0 0000000 0000000 00 0 000.0 0000000 00 0000.0 0000000 0000 0 0.0 0000000 0000000 00000 .0 00000.0 00000.0 000000 .0 0000000 0000000 0000010 0000000 0000000 0000000 0000000 00 0000.0 00000.0 00000.0 00000.0 000000 .0 0000000 00000 0.0 0 000000 0000000 0000000 0000000 0000000 0000000 0000000 00000 .0 00000.0 00000.0 000000 .0 0000000 0000 0 0.0 0000010 000 000.0 0000000 0000000 0 0000 0.0 0000000 00000.0 00000.0 00000.0 000000 .0 0000000 00000 0.0 000 0 00.0 0000000 0000000 0000000 0000 0 0.0 000 000.0 0000000 00000.0 0000000 00000.0 000000 .0 0000000 0 0000 0.0 0000010 0000000 0 00000.0 0000000 0 0 00 0 0.0 0000 00.0 00000 .0 00000.0 00000.0 00 0 000 .0 0 00 0 00.0 00000 0.0 000 0 00.0 0000000 0000000 0000000 0000 0 0.0 0000000 0000000 00000 .0 0000000 00000.0 000000 .0 0 0 0000.0 0000 0 0 .0 0000010 000000 .0 0000000 0000000 0000 0 0.0 0000000 00000 .0 00000.0 00000.0 0 0 0 000 .0 000 0 00.0 00000 0.0 0000 00.0 0000000 0 00000.0 0000000 0000 0 0.0 0000000 0000000 0000 00.0 00000.0 00000.0 000000 .0 0000000 00000 0 .0 0000010 0000000 00 0 000.0 0000000 0000000 0000000 00000 . 0 00000. 0 00000.0 000 000 .0 000 0 00.0 00000 0.0 000 0 00.0 0000000 0000000 0000000 0000 0 0.0 0000000 0000000 0020:0080 02008 300000.000 0.830080 20 8.0 8:05, 08m 080 0030:0000 .10 000.00. 0000 00.0 00000.0 00000.0 000000 .0 0000000 00000 0.0 0 00000.0 0 00000 .0 00 0 000.0 0000000 00000 0.0 0000000 00000 .0 00000.0 00 0 000.0 000000 .0 0000000 000 00 0.0 000 0010 0000000 0000000 0000000 0000 0 0.0 0 00000.0 0000000 80070 00 80070 00 com .000 29 At values used in the best fit equation (5-3) for each integration method. Table 5.5. The time step values used in the best fit equation for each integration method Number Forward Difference Central Difference Backward Difference 0f NOdes At At * A, At At * 9., At At * 2., 6 0.015 0.08640 0.050 0.2880 0.00100 0.005760 F l 1 0.004 0.02304 0.015 0.0864 0.00020 0.001 152 16 0.002 0.01 152 0.010 0.0576 0.00015 0.000864 21 0.001 0.00576 0.006 0.0346 0.00010 0.000576 F The empirical equation for each integration method can be calculated using the linear programming tool-solver function included in Microsoft Excel for Windows, Version 5.0, or a scientific calculator. The results are given below as equations (54) through (5-6) for the three integration methods. These equations can be used to predicted an appropriate time step value to accurately integrate a system of ordinary differential equations. -Forward difference method: At = 3.91 N-2.l3 (5-4) 30 -Central difference method: -l.66 At = 5.28N (5-5) 1 -Backward difference method: -l.8l At = 0.12” (5-6) A plot of the time step estimates is presented in Figure 5.4. This figure shows that the central difference method has the largest time step while the backward difference method has the lowest time step value for the particular grid. 31 + Forward difference + Central difference + Backward difference 0.30 -— 0.25 0 0.20 -- 0.15 0 0.10 *- Deltat * lambda 1 0.05 *- o.oo 40 t *3 Number of nodes Figure 5.4. Time step estimates for the three different integration methods CHAPTER SIX EVALUATION OF THE TIME STEP ESTIMATES The time step estimates presented in the previous chapter were developed using the step change problem on a solid disk of radius one and assuming Dr = D, = 1. The ability of these equations to predict the time step for other problems was evaluated using different materials and other boundary conditions. The comparison problems consisted of two different materials, steel and copper, individually and in combination using grids of 11 and 21 nodes, respectively. Eight different material combinations were used. The combinations were steel, copper, half steel and half copper, and half copper and half steel, each with prescribed temperatures at the boundary. Half steel and half copper with a derivative boundary condition of M equal to 10, half copper and half steel with M = 10, half steel and half copper with M = 50, and half copper and half steel with M = 50. The eight combinations are summarized in Table 6.1. The value of M corresponds to the product of a convection coefficient and surface area. The fluid temperature was taken as zero. The procedure for evaluating the time step equations was as follows. 1. The smallest and largest eigenvalues were calculated and were used to estimate the time step for integrating the system of ordinary differential equations. These time step values are summarized in Table 6.1. 2. Time steps corresponding to At/2, 2At, and 4At were determined. 3. Each of the eight comparison problems were solved using At/2, At, 2At, and 4At 32 33 Table 6.1. Time step values for the three different integration methods At, sec Materials Nodes M Ami“ Am F D C D B D Steel 1 1 - 0.401 * 27.30 0.0732 0.2465 0.00391 (1 “1:14.29) 21 - 0404* 108.57 0.0184 0.0843 0.00121 Copper 1 1 - 6.666* 453.57 0.0044 0.0147 0.00023 (1 “1:0. 86) 21 - 6.709* 1804.01 0.001 1 0.0050 0.00007 Steel & 11 - 0.565 421.68 0.0047 0.1730 0.00274 Copper 21 - 0.566 1802.06 0.0011 0.0591 0.00085 Copper 1 1 - 0.844 477.30 0.0042 0.1 160 0.00184 & Steel 21 - 0.853 1764.90 0.001 1 0.0397 0.00057 Steel & 11 10 0.333 463.87 0.0043 0.2961 0.00470 Copper 21 10 0.333 1853.15 0.0011 1.1012 0.00146 Copper 1 1 10 0.228 476.83 0.0042 0.4325 0.00686 & Steel 21 10 0.228 1756.87 0.001 1 0.1478 0.00213 Steel & 11 50 0.498 533.72 0.0037 0.1980 0.00314 Copper 21 50 0.499 1934.00 0.0010 0.0676 0.00097 Copper 1 1 50 0.570 476.81 0.0042 0.1729 0.00274 & Steel 21 50 0.573 1756.78 0.0011 0.0589 0.00085 * The km was used to calculated the time step values for the forward difference method (FD) otherwise Am, governed the time step. CD is the central difference method and BD is the backward difference method. Am was used to calculate the time step for the central and backward difference methods 34 for each of the three integration schemes. The solution values for At/2, 2At, and 4At were compared with the solution values for At at six different points. The specific comparison points in time varied with the size of the time step and the integration method. The comparison points in time were at 4, 8, 12, 16, 20, and 24 At. The six values obtained using each integration method for each comparison problem are summarized in Tables 6.2 and 6.3. The solution values at At/2, 2At, and 4At were compared to the solution values for At using regression equations. The regression information is summarized in Table 6.4. The data from the eight comparison problems were lumped together for the central and backward difference methods. 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Regression results between At and At/2, 2At, and 4At. (31mg) At/2 2At 4At Combination X Coefficient 1.02343 Unstable Unstable 7»... and 71...... for Forward Std Err of Coef 0.00353 Unstable Unstable D'ff l erence R Squared 0.99888 Unstable Unstable Method Forward X Coefficient 0.50525 Unstable Unstable D’fference Std Err of Coef 0.03871 Unstable Unstable Method (31m) R Squared 0.64445 Unstable Unstable Central X Coefficient 0.99961 1.00149 1.00637 D‘ffe‘ence Std Err of Coef 0.00021 0.00143 0.00408 Method (km) R Squared 1.00000 0.99981 0.99846 Backward X Coefficient 0.88975 1.21677 1.62889 D‘ffe‘ence Std Err of Coef 0.00270 0.00628 0.02132 Method R Squared 0.99914 0.99750 0.98415 CHAPTER SEVEN DISCUSSION AND CONCLUSION The empirical equations for calculating the time step required to solve the system of ordinary differential equations for radial field problems have been successfully obtained. Based on an accuracy criteria or the stability requirements (Table 6.4), these equations define a usable time step. The forward difference method is conditionally stable, therefore, both A, and km must be evaluated. The time step value should be the lowest of the two possible values. The R squared value in Table 6.4 indicates that the integration is accurate when the procedure is used. Solutions in time using a time step based only on the stability criteria are not accurate. The R squared value is 0.64 in this case. The time step equation for the central difference method appears to be conservative since the R squared value for 4At is still 0.998. Time step value for the backward difference method gives accurate results at 2At with greater error at 4At. Time step value calculated for the forward difference method should be rounded down because of the stability criteria. 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