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LIBRARY Michigan State University; This is to certify that the dissertation entitled ESTIMATION OF TRANSIENT HEAT TRANSFER COEFFICIENTS IN MULTI-DIMENSIONAL PROBLEMS BY USING INVERSE HEAT TRANSFER METHODS presented by Arafa Mohamed Osman has been accepted towards fulfillment of the requirements for Ph.D. Mechanical Engineering degree in 7" .4 . '52,: , LL T I Major professér ‘ Date 40‘;- ’7/1 /9‘€[ MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771 IVIESI_J RETURNING MATERIALS: Piace in book drop to LJBRARJES remove this checkout from n. your record. FINES will be charged if book is returned after the date stamped below. ESTIMATION OF TRANSIENT HEAT TRANSFER COEFFICIENTS IN HULTI—DIHENSIONAL PROBLEMS BY USING INVERSE HEAT TRANSFER.HETHODS BY Arafa Mohamed Osman A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering Michigan State University East Lansing, Michigan 48824 1987 Copyright by ARAEA MOHAMED OSMAN 1987 ABSHACT ESTIMATION OF RANSIENT HEAT TRANSFER COEFFICIENTS IN HULTI-DIHENSIONAL PROBLEMS BY USING INVERSE HEAT TRANSFER METHODS By Araf a Mohamed Osman The inverse heat transfer problem is one of considerable practi- cal interest in the analysis and design of experimental heat transfer investigations. The present dissertation is concerned with the analyti- cal and experimental investigation of the inverse heat transfer coefficient problem for the estimation of transient heat transfer coef- ficients in multi-dimensional convective heat transfer applications. An application considered is the sudden quenching of a hot solid in a cold liquid. Other applications includes thermal analysis of forced convec- tion over impulsively started solid bodies and investigation of short duration wind tunnel experiments. The primary aim in the analytical part of the thesis is to describe methods and algorithms for the solution of the "ill-posed" inverse heat transfer coefficient problem. The solution method used is an extension of the sequential future-information method of Beck. The finite-difference method, based on the control volume approach, is used for the discretization of the direct heat conduction problem. The Crank-Nicolson and the alternating-direction implicit schemes are used for the numerical solution of the one-and two-dimensional direct problems, respectively. Numerical experiments are conducted for a systematic investiga- tion of the developed algorithms on selected heat transfer coefficient test.cases. The numerical results show that the sequential future- information method is accurate, efficient, and capable of estimating sharp and large variations (in time and space) in the heat transfer coefficient functions. The overall objective of the experimental work is to investigate the early transients in the heat transfer coefficients from spheres in one-and two-dimensional quenching experiments. Several experiments were performed by plunging hollow spheres in either ethylene glycol or water. The transient thermocouple data was acquired with a multi-channel data acquisition system. The developed methods are used for the analysis of the quenching experiments for the estimation of the transient heat transfer coefficients. The estimated results show that the early tran- sient values are about 60%-80% higher compared to the well-known, empirical correlations. The later values are in good agreement with known.steady-state convection correlation. Analysis of the results indicate that the transient inverse technique has the capability of estimating early transients and subsequent quasi-steady state values of the heat transfer coefficient in a single transient experiment. To my wife, Hanan; Children, Dina and Dalia ACKNOWLEDGMENTS The author wishes to express his sincere appreciation and gratitude to his major adviser Professor James V. Beck for his continued support in the form of knowledge, enthusiasm, and guidance during the course of this research. Professor James V. Beck deserves much credit for his contributions during my graduate study and also for his friendship and painstaking review of my dissertation. The author is also grateful to his other committee members, Professor Merle Potter, Professor David Yen, and professor Craig Somerton for their guidance and valuable discussion. Professor Merle Potter made valuable comments and criticisms of the first draft copy. Thanks is also due to Dr. Kevin Cole, Dr. Saleem Shakir, and Dr. Paul Zang for their inputs. My wife, Hanan, and my children, Dina and Dalia, merits gratitude for their limitless patience and moral support throughout my graduate study and research. I thank them from the bottom of my heart. vi TABLE OF CONTENTS LIST OF TABLES .......................................... LIST OF FIGURES ......................................... NOMENCLATURE ............................................ INTRODUCTION ............................................ 1.1 Objective of Present Study ......................... 1.2 Scope of Dissertation .............................. 2.4 Review of the Two-dimensional Inverse Procedures 2.5 Summary ............................................ -ERIHENTAL FACILITIES AND PROCEDURES .................. 3.1 Introduction ....................................... 3.2 Test Specimens ..................................... 3.2.1 Test Sphere for One-Dimensional Problem ..... 3.2.2 Test Sphere for Two-Dimensional Problem ..... 3.3 Quenching Apparatus ................................ 3.4 Multi-Channel Data Acquisition System .............. 3.4.1 Hardware Devices Description ................ 3.4.2 Software System Description ................. 3.5 Experimental Procedures ............................ ESTIMATION OF TRANSIENT HEAT TRANSFER COEFFICIENTS FOR I-D PROBLEMS ........................................ 4.1 Introduction ....................................... 10 11 12 16 16 20 20 22 24 4.2 Statement of the One-Dimensional IHTCP for the Estimation of h(t) ................................. 27 4.3 Sequential, "Future-Information" Method for Solving the IHTCP .................................. 30 4.4 Formulation of the Finite-Difference Equations for the Direct Problem ............................. 34 4.4.1 Finite Control Volume Technique ............. 35 4.4.2 Time integration ............................ 40 4.5 Calculation of Sensitivity Coefficients ............ 41 4.6 Overall Computational Algorithm of the IHTCP ....... 42 4.7 Test Cases ......................................... 42 4.7.1 Sensitivity Coefficient for h(t)= Const ...... 44 4.7.2 Step-Increase Test Case ..................... 45 4.7.3 Triangular-Pulse Test Case .................. 56 4.7.4 Comparison of the Direct and Quotient Approaches .................................. 64 THO-DIMENSIONAL.IHTCP FOR.THE ESTIMATION OF TIME-AND-SPACE DEPENDENT HEAT TRANSFER COEFFICIENTS .................... 68 5.1 Introduction ....................................... 68 5.2 Statement and Formulation of the 2-D Inverse Problem ............................................ 70 5.3 The Sequential, Future-Information Method .......... 74 5.3.1 Sensitivity Coefficients .................... 81 5.3.2 Solution of Equations ....................... 82 5.4 Test Cases ......................................... 85 5.4.1 Numerical Results and Discussion ............ 89 5.5 Conclusions ........................................ 95 EHPERIMENTAL'RESUETS AND DISCUSSION ..................... 96 6.1 Introduction ....................................... 96 6.2 Estimated Heat Transfer Coefficient from One-Dimensional Experiments ........................ 97 6.2.1 Results for Ethylene Glycol ................. 98 6.2.2 Results for Water ...: ....................... lll 6.3 Comparison of Experimental Results with Empirical Convection Correlation ............................. 117 6.4 Estimated h(t,0) from Two-Dimensional Experiments .. 125 6.4.1 Results for Ethylene Glycol ................. 125 6.4.2 Results for water ........................... 137 7. SUMMARX'AND CONCLUSIONS ................................. 149 APPENDIX A. NUMERICAL SOLUTION OF THE 2-D DIRECT PROBLEM . . . 155 A.1 Introduction ....................................... 155 A.2 Derivation of FCV Equations ........................ 156 A.2.1 FCV Equation for a Typical Interior Node (i,j) .................................. 158 A.2.2 FCV Equation for Convective Boundary Node (I,j) .................................. 160 A.3 Alternating-Direction Implicit Scheme .............. 164 A.3.1 ADI Equations for the First-Half Time Step ................................... 165 A.3.2 ADI Equations for the Second-Half Time Step ................................... 170 A.4 Solution of ADI Equations .......................... 172 LIST OF REFERENCES .......................................... 175 ix .II Ill. Table Table Table Table 6. 6 6. 6. 1 .2 1 2 A subset l-D test A subset 1-D test A subset 2-D test A subset 2-D test of the sphere of the sphere of the sphere of the sphere LIST OF TABLES measured quenched measured quenched measured quenched measured quenched temperature histories from in ethylene glycol ............. temperature histories in distilled water ............. temperature histories from in ethylene glycol ............. temperature histories in distilled water PAGE 101 112 126 139 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure .10 LIST OF FIGURES PAGE Schematic diagram of the test sphere assembly for the estimation of time-dependent heat transfer coefficients in one-dimensional IHTCP ................ 13 A photograph of the one-dimensional test sphere assembly ............................................. 14 Schematic diagram of the test sphere assembly for the estimation of time-and-space dependent heat transfer coefficients in two-dimensional IHTCP ................ 17 A photograph of the quenching apparatus and the control panel ........................................ 18 A block diagram of the temperature data acquisition system ............................................... 21 Schematic diagram of the l-D model for a hollow sphere ............................................... 29 Layout of the grid points and control volumes in 1-D spherical body ....................................... 37 Sensitivity coefficients, zBi(t)’ for constant Bi(t) - Bi ........................................... 46 Heat transfer coefficient function test cases (a) step-increase in Bi(t), (b) triangular-pulse in Bi(t) ............................................. 47 Simulated temperature history at thermocouple location for step-increase test case (At - 0.05 s, E -0.006 m) .......................................... 48 Estimated Biot number, Bi(t), for the step-increase in Bi(t) (r - 1, a - 0 c, and At; - 0.156) .......... 50 Estimated Biot number, Bi(t), for the step-increase in Bi(t) (r - 1, a - 0.005 c, and At; - 0.156) ...... 51 Sensitivity coefficient, Z(t,Bi(t)), for the step-increase test case .............................. 54 Comparison of the Sensitivity coefficients for different values of the step-increase in Bi .......... 55 Estimated Biot number, Bi(t), for the step-increase in Bi(t) (r - 4, a - 0.25 c, and At+ - 0.156) ....... 57 E xi Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure U1 U‘l U1 U‘I U1 0‘ 0‘1 .11 .12 .13 14 .15 .16 17 Simulated temperature history at thermocouple location for the triangular-pulse test case (At - 0.05 s, E -0.006 m) ............................ 58 Estimated Biot number, Bi(t), for the triangular- pulse in Bi(t) (r - 1, a - 0 c, and Ac; - 0.156) .... 59 Estimated Biot number, Bi(t), for the triangular-pulse in Bi(t) (r - 1, a - 0.005 c, and At; - 0.156) ...... 61 Sensitivity coefficient, Z(t,Bi(t)), for the triangular-pulse case ................................ 62 Estimated Biot number, Bi(t), for the triangular-pulse in Bi(t) (r - a, a - 0.25 c, and At; - 0 156) ....... 63 Comparison between Bi(t) data estimated by using the direct and quotient methods (r = 1, a - 0, and At; - 0.56); (a) direct method, (b) quotient method .. 66 Comparison between Bi(t) data estimated by using the direct and quotient methods (r - 4, a - 0.25 C, and At; - 0.14); (a) direct method, (b) quotient method .. 67 Solution region for the two-dimensional convective heat transfer problem in hollow sphere ............... 71 (a) Approximation of hm(0) by linear segments; (b) Linear basis functions ........................... 77 Layout of the test sphere showing locations of the thermocouples (*) and the parameters flu m(-) ......... 87 Temperature histories at the thermocouple locations for the two-dimensional inverse problem .............. 90 Estimated Bi(t,0) function obtained by using the SFI method (r - 1, a - 0, and At; - 0.156) ............... 91 Estimated Bi(t,9) at (a) 9 - w, (b) 0 - n/2, and (c) a - 0 (r =1, 0 - 0 005 c, and AtE= 0 156) ...... 92 Estimated Bi(t,9) function (r = 4, a = 0.25 C, and AtE- 0 156) .......................................... 94 Locations of the thermocouples inside the one-dimensional test sphere .......................... 99 Experimental temperature histories from the 1-D sphere quenched in ethylene glycol .......................... xii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 0‘ 0‘ 0‘ O" .10 .11 .12 .13a .13b .14 .15 .16 Estimated heat transfer coefficient, h(t), from the sphere quenched in ethylene glycol (r - 4, Sr- 0.1 5, At - 0.05 s, and Ar - 0.00261 m) ..................... 105 Comparison of the estimated results from three replicates of the quenching experiments in ethylene glycol ..................................... 106 Sensitivity coefficient for the sphere quenched in ethylene glycol (r - 4, At - 0.05 s, and Ar - 0.00261 m) .................................. 108 Residuals em for the sphere quenched in ethylene glycol ...................................... 110 Experimental temperature histories from the 1-D sphere quenched in distilled water .......................... 115 Estimated heat transfer coefficient, h(t), from the sphere quenched in distilled water (r - 4, Sr- 0.1 5, At - 0.05 s, and Ar - 0.00261m) ........... 116 Sensitivity coefficient for the sphere quenched in distilled water (r-4, At - 0.05 s, and Ar - 0.00261 m) 118 Residuals em for the 1-D test sphere quenched in distilled water ...................................... 119 Comparison of the estimated results from the transient quenching experiments and the empirical correlation results (ethylene glycol) ................ 123 Comparison of the estimated results from the transient quenching experiments and the empirical correlation results (distilled water) ................ 124 Estimated heat transfer coefficient function from the 2-D test sphere quenched in ethylene glycol (r - 3, Sr- 0.1 5, At - 0.05 s, Ar - 0.00203 m and A9 - 0.3927 radians) ............................. 129 Estimated heat transfer coefficient function from the 2-D test sphere quenched in ethylene glycol (r - 3, Sr- 0.1 5, At = 0.05 s, Ar = 0.00203 m and A0 - 0.3927 radians) ............................. 130 Residuals e1 m at TC # 1 for the 2-D test sphere quenched in ethylene glycol .......................... 132 Residuals e2 m at TC # 2 for the 2-D test sphere quenched in ethylene glycol .......................... 133 Residuals e3 m at TC # 3 for the 2-D test sphere quenched in ethylene glycol .......................... 134 xiii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 6. 6. .17 .18 19a .19b .20 21 .22 .23 .24 .25 Residuals e4 m at TC # 4 for the 2-D test sphere quenched in ethylene glycol .......................... Residuals e4 m at TC # 5 for the 2-D test sphere quenched in ethylene glycol .......................... Estimated heat transfer coefficient function from the 2-D test sphere quenched in distilled water (r - 3, Sr- 0.1 5, At - 0.05 s, Ar - 0.00203 m and A0 - 0.3927 radians) ............................. Estimated heat transfer coefficient function from the 2-D test sphere quenched in distilled water (r - 3, Sr- 0.1 3, At - 0.05 s, Ar - 0.00203 m and A9 - 0.3927 radians) ............................. Residuals e1 m at TC # 1 for the 2-D test sphere quenched in distilled water .......................... Residuals e2 m at TC # 2 for the 2-D test sphere quenched in distilled water .......................... Residuals e3 m at TC # 3 for the 2—D test sphere ’ quenched in distilled water .......................... Residuals e4 m at TC # 4 for the 2-D test sphere quenched in distilled water .......................... Residuals eh m at TC # 5 for the 2-D test sphere quenched in distilled water .......................... Comparison of the residuals at TC # 1 for the original and adjusted temperature data from sphere quenched in water .................................... Computational grid points and the corresponding finite control volumes for two-dimensional IHTCP ..... A typical interior node (i,j) and the associated heat fluxes ............................................... A typical convective node (I,j) and the associated heat fluxes ......................................... xiv 135 136 147 148 Bi(t) th EI-l F(r,0) h(t,0) t E T(r,o,t) Tact) w m,i Y’ NOMENCLATURE Biot number, defined by Eq. (4.30) depth of thermocouples, m expected value operator initial temperature distribution, °C heat transfer coefficient, W/m2°C thermal conductivity, W/m °C number of thermocouples characteristic length, m time index number of time steps number of parameters heat flux, W/m2 number of future time steps inner radius of sphere,m outer radius of sphere, m least-squares function time coordinate for inverse problem, 5 total time, 3 temperature, °C fluid temperature, °C weight factor measurement vector, °C IN ¢V(0) 14 Symbols FCV IHCP IHTCP SFI sensitivity matrix, °02 m2/W parameters vector, W/m2°C error vector, °C basis functions, Eq. (5.11) standard deviation, °C thermal diffusivity, m2/s gradient operator alternating-direction implicit finite control volume inverse heat conduction problem inverse heat transfer coefficient problem sequential future-information xvi CHAPTER 1 INTRODUCTION In convective heat transfer studies of advanced technological applications with sudden and large changes in convective heat transfer boundary conditions, values of the transient heat transfer coefficient, h(t), are of primary importance. Typical examples where h(t) data are needed include: thermal analysis of rocket nozzles and gas turbine blades, safety analysis of nuclear reactor elements for the loss-of- coolant accidents, thermal protection of space shuttles, analysis of quenching processes, and development of new materials and composites for high temperatures applications. To obtain the required h(t) data, one has to experimentally measure transient temperatures in an instrumented prototype or scaled model. For bodies with high surface temperatures which are subjected to an abrupt change in the convective boundary conditions, it is very difficult to measure directly the surface heat fluxes and/or the surface temperatures which are required for the estimation of the h(t). In such circumstances, the h(t) can be conveniently estimated from transient temperature measurements at appropriate points inside a heat conducting solid. This estimation technique depends on solving an inverse problem, the inverse heat transfer goefficient problem (IHTCP). In the inverse heat transfer problem, a boundary condition is estimated when the initial condition is known and the temperature his- tories at certain local positions inside the solid are known. In the direct heat transfer problem (sometimes called the initial/boundarv value problem), the temperature distribution in a region is calculated when the boundary and initial conditions are known. Thus, the IHTCP is concerned with the estimation of transient heat transfer coefficients in convective heat transfer experiments by utilizing measured interior temperature histories. 1.1 Objectives of Present Study The present research is concerned with the analytical and ex- perimental investigation of the estimation of transient heat transfer coefficients in multi-dimensional convective heat transfer problems by using inverse heat conduction methods. There are several objectives regarding the estimation of h(t). These objectives are outlined below. 1. The first objective is to develop methods and algorithms for solving the one-and two-dimensional IHTCP. The desired charac- teristics of the algorithms are: (a) the ability to produce accurate estimates for the case of small nondimensional time steps, (b) the capability to handle different ranges of the heat transfer coefficients, and (c) the reduced sensitivity to measure- ment errors. 2. The second objective is to apply the developed algorithms to estimate the transient heat transfer coefficients for actual convective heat transfer experiments with sudden imposition of convective boundary condition. 3. The third objective is to compare the relative merits of the quotient and direct approachs for the estimation of the transient heat transfer coefficients. In the quotient approach, the solu- tion of IHTCP is reduced to the solution of the inverse heat ~~yfi gonduction problem (IHCP); in the direct approach, the solution proceeds as follows: (1) the surface heat flux, q, and the surface temperature are individually estimated, and (2) the heat transfer coefficient is determined by the quotient q/AT (Newton’s law of cooling) where AT stands for the difference between the surface temperature and the fluid bulk temperature. In the direct ap- proach, the heat transfer'coefficient is estimated directly without the estimation of the intermediate values of the heat flux and the surface temperature histories. An application considered here, which approximates the conditions of many physical problems, is the quenching of a solid in a liquid. The application is directly applicable to a sphere immersed in a liquid and to many other problems. The IHTCP is in a class of the ill-posed problems in the sense that arbitrarily small errors in the temperature input histories may lead to arbitrarily large errors in the estimated values of the heat transfer coefficients. In other words, the solution of the IHTCP is characterized by its strong sensitivity to errors in the input data. Thus, the IHTCP cannot be solved efficiently without some information concerning the physics of the problem and the structure of the desired solution. An efficient method for solving the IHTCP is one which can "suppress" the sensitivity of the IHTCP by "offsetting" the effect of the errors in the input data, and at the same time maintaining the desired accuracy. To alleviate the strong sensitivity of the estimated results on the errors in the input data, special approximate methods are introduced. Several methods have been applied to the solution of the inverse heat conduction problems. Two well-known solution methods that can be used for stablizing the solution of the IHTCP are the sequential, future-information method of Beck [1962,1965,1970,1982] and the regularization method of Tikhonov (Tikhonov and Arsenin [1977]). The solution method used in this dissertation is an extension of the sequential future-information (SFI) method. The SFI method has been selected because it is more efficient than the regularization method and can be implemented on microcomputers. 1.2 Scope of Dissertation This thesis is divided into seven chapters. Chapter 1 is the introduction. Chapter 2 includes a survey of the previous work. Chapter 3 describes the experimental facilities and the experimental procedures for the investigation of transient heat transfer coefficients in quenching experiments of hollow spheres. The SFI method for solving the nonlinear one-dimensional IHTCP is described in Chapter 4. Chapter 5 extends the SFI procedure to the solution of the nonlinear two- dimensional IHTCP. Chapter 6 contains the experimental heat transfer coefficient results from one-and two-dimensional quenching experiments obtained by using the SFI methods. Finally, Chapter 7 contains the conclusions and some recommendations. CHAPTER2 LITERATURE REVIEW 2 . 1 Introduction In recent years, the inverse heat transfer problem has received increasing attention in the investigation of advanced heat transfer processes. The inverse heat transfer problem‘is one of considerable practical interest in the design and analysis of the heat transfer experiments. This chapter contains a review of the previous work con- cerning the estimation of transient heat transfer coefficients in one- dimensional and two-dimensional problems. In this review, experimental procedures for estimating IHTCP using Beck's method [1970] receive particular emphasis. A more comprehensive review of the solution methods for the inverse heat conduction problem is available in Beck et a1. [1985], and Hensel [1986]. 2.2 Review of Solution Methods There are several methods that have been applied to the solution of the inverse heat ponduction problem (IHCP). They include the future- information method of Beck [1970,1982], the regularization method of Tikhonov (Tikhonov and Arsenin [1977]), the iterative regularization method (Alifanov [1974,1984]), and the integral equation method (Beck [1962,1968] and Sparrow and Haji-Sheikh [1964]). A variety of other numerical methods have been developed (see e.g., Beck et a1. [1985]). 2.3 Review of One-Dimensional Procedures Stolz [1960] introduced a simple numerical procedure for solving linear one-dimensional inverse heat conduction problems. Stolz's method is a sequential method and is based on the Duhamel's integral formula- tion. Stolz applied his method for the investigation of the quenching of a sphere into light oil. The surface heat flux and the heat transfer coefficient were estimated as a function of time. In this paper, exact matching of the experimental and calculated temperatures was used. Consequently, Stolz’s method turns out to be restricted to large time steps. The method is very sensitive to measurement errors and is con- ditionably stable. Powell [1973] presented a graphical procedure for solving the inverse heat conduction problem. Powell's methodology was based on solution charts obtained for the direct heat conduction problem. The charts in Powell's paper contain a family of curves for the nondimen- sional temperature distribution plotted versus nondimensional parameters which include the Fourier number, the Biot number, and a nondimensional distance from the surface. The analysis and methodology were applied for the experimental determination of the surface heat flux, hot gas temperature, and gas heat transfer coefficient in a rocket motor. Powell's method is restricted to large values of Fourier number ( > 0.6) and Biot number ( > 0.5). The uncertainty in the estimated values of Biot number were several times the uncertainty of the temperature input data. Muzzy et a1. [1975] applied a variant of Beck's method for the analysis of the blow-down heat transfer data from the BWR reactor sys- tem. In their work, the transient heat transfer coefficient was "directly" estimated, and the estimated heat transfer coefficient values were used to determine the resulting surface heat flux history. Berkovich et al. [1977] proposed a method for the reconstruction of transient heat transfer coefficients in jet cooling of metal plates. This method uses a finite difference formulation of the IHTCP and util- izes the minimization of a least squares objective function. A gradient method is used to minimize the objective function. An iterative proce- dure is used in the minimization until the sum of squares of the errors is within a prescribed tolerance which depends on a certain measure of the accuracy of the input temperature data. Mehta [1981] presented an iterative scheme for analyzing the experimental data form a typical divergent rocket nozzle. His scheme is based on an implicit finite difference formulation of the inverse heat conduction problem. Mehta estimated various parameters including sur- face heat flux, wall temperature, combustion gas temperature, and heat transfer coefficient. Bass [1980a] used Beck’s, Beck [1970], for the estimation of surface heat flux and surface temperature in a one-dimensional problem expressed in cylindrical coordiantes. Bass’ technique is based on a finite element formulation of the IHCP. Both linear and quadratic elements are used in this analysis. Bass applied the finite element method to test cases in an electrically heated composite cylinder. In the first test case "exact" temperature data with time step of 0.05 sec are used as input data for the inverse code. In the second test case Bass used actual experimental data taken from the ORNL thermal- hydraulics facility. The reconstructions of the surface heat flux and surface temperature histories in both test cases are good. 2.4 Review of the Two-dimensional Procedures The material presented next includes a review of some previous studies on the two-dimensional IHCP which are related to the present investigation. Bass et al. [1980b] applied Beck’s method, with some modifica- tions, to the solution of a two-dimensional inverse heat conduction problem. The surface heat flux is estimated as a function of angle and time. This procedure is based on the finite element formulation of the direct heat conduction problem. Bass et a1. tested their algorithm using "exact" data as well as actual experimental data taken from a heated cylinder under high surface heat flux conditions. Their results appear to be very good. In their study, six interior thermocouples were located at the same radius and at equal angular locations around the test cylinder. The number of parameters describing the surface heat flux distribution around the cylinder was six, chosen to be exactly at the thermocouples locations. Their algorithm utilizes a set of trigonometric basis functions to approximate the angular dependency of the unknown surface heat flux function. Alifanov and Kerov [1980] presented a method for solving the two- dimensional IHTCP in the cylindrical coordinates. The transient surface heat flux was assumed to be function of the angular position. The solution method is based on the finite difference formulation of the direct heat conduction problem. Alifanov and Kerov used an iterative regularized procedure for solving the IHCP by minimizing the RMS (Loot hean pquare) residual functional. The RMS functional is the integral, over space and time, of the square of the deviation between the calcu- lated and measured temperatures. They used the method of conjugate gradient in the minimization of the RMS functional. A preliminary smoothing of the input data was performed using a procedure based on the second-order smoothing of Tikhnov [1977]. Alifanov and Egorov [1984] investigated the effectiveness of the "iterative" regularization for solving the two-dimensional IHTCP with constant thermophysical properties. The solution method is based on the integral formulation (using Green's function) of the direct problem. The study dealt with the analysis of the linear IHCP in a rectangular shape. The heat flux was applied to one surface while the other sur- faces were assumed to be thermally insulated. The termination of the iteration process, or the maximum number of iterations, is based on an estimate of the standard deviation of the input data. Irving and Westwater [1986] used a variant of Beck's method for obtaining the boiling curves from quenching experiments with hollow spheres. Recently, Hensel [1986] investigated two-dimensional IHCP’s in rectangular and cylindrical coordinates. His analysis is based on the finite element formulation of the direct (forward) heat conduction problem. The solution method used is similar in many aspects to that introduced by Alifanov and Egorov [1984]. Hensel used a first order regularizer to introduce spatial discretization for the two-dimensional IHTCP. A low pass smoothing filter has been used over the time domain to reduce the sensitivity of the estimates to high frequency measurement errors. The use of the regularization method allows the solution to underdetermined IHCP's where the number of temperature sensors is less than the number of heat flux unknown parameters. 10 2.5 Su-ary The two well-known solution methods that have been successfully applied to many practical IHCP problems and industrial applications are the temperature future-information method and the regularization method. The solution method used here is an extension of the sequential future- information method. This method has been selected for a number of important reasons. First, it is more efficient than the regularization method. Second, it can be implemented on microcomputers. Further, the temperature future-information method offers conceptual simplicity and applicability to a wide range of nonlinear inverse problems. There are two general forms in which the estimation of transient heat transfer coefficients can be accomplished. The first form is called the whole domain estimation procedure. The unknown function is estimated based on the entire set of the measured data. This method is an off-line estimation procedure. The second form is called the sequen- tial estimation procedure or the on-line estimation form. The unknown function is recursively estimated over time, from a data set at the present time plus a few future times. Most of the previous work dealt with the estimation of surface heat flux and surface temperature histories. (ho the other hand, however, the problem of estimating the transient heat transfer coeffi- cients is rarely treated in the literature. The review of the literature in the previous sections indicates that the analysis of the one-dimensional IHTCP is inadequate. Further, the nonlinear two- dimensional IHTCP has not been treated in the published literature. The present study presents the first treatment of the nonlinear two- dimensional IHTCP for the estimation of time-and-space dependent heat transfer coefficients. CHAPTER3 .ERIMENTAL FACILITIES AND FROG-URES 3 . 1 Introduction The experimental facilities and procedures for the estimation of transient heat transfer coefficients in one-and two-dimensional quench- ing experiments are described in this chapter. The experiments involve plunging (quick immersion) of hot hollow spheres into two cooling fluids: distilled water and ethylene glycol. This chapter is divided into four major sections. Section 3.2 describes the test spheres for one-and two-dimensional problems. In Section 3.3 the experimental quenching apparatus is briefly described. Section 3.4 deals with the set-up of the multi-channel data acquisition system (both hardware devices and software programs). The experimental procedures are outlined in Section 3.5. 3 . 2 Test Specimens Two hollow test spheres are used in the present investigation. The first one is used for the one-dimensional quenching experiments and the second one is used for the two-dimensional quenching experiments. Both test specimens were made from OFHC copper. The dimensions of each specimen and the details of thermocouple locations are given below. 12 3.2.1 Test Sphere for One-Dimensional Problem Figure 3.1 shows a schematic diagram of the test sphere assembly for the one-dimensional IHTCP. The test specimen assembly consists of the following parts: (a) OFHC hollow sphere, (b) thermocouple sensors, and (c) supporting steel tube. The hollow sphere, with 2.92 inch (7.416 X 10'2 m) outer diameter and 0.867 inch (2.202 x 10'2 m) inner diameter, was machined as two hemispheres. The two hollow hemispheres were firmly joined together with four small stainless steel screws. A photograph of the l-D test sphere is shown in Figure 3.2. Six thermocouples, labeled TC #1 - TC #6, are positioned in a horizontal diametral plane (flat surface of the lower hemisphere in Figure 3.1). Three thermocouples, TC #1, TC #2, and TC #3, are located at the same depth, 0.66 inch (1.67 x 10'2 m) from the outer surface, and x/3 radians apart. The other three thermocouples, TC #4, TC #5, and TC #6, are located at the same depth, 0.103 inch (2.61 x 10'3m), and n/3 radians apart. The thermocouples were made from 30 gauge thermocouple wire type "E": Nickel-10 percent chromium (+) versus constantan (-). A type E thermocouple was selected primarily because it develops the highest Seebeck coefficient (emf/degree) of all the commonly used types. Approximately 4 inches of the thermocouple fibergals insulation was stripped out and a 30 gauge Teflon insulating tube was used to insulate each thermocouple lead, except for 0.15 inch at the end. Each thermocouple assembly was laid into radial grooves, 0.05 inch in width and 0.03 inch in depth, machined on the lower hollow hemisphere from the inner surface to thermocouple location. Two separated slots, each 0.01 inch in width, 0.02 inch in depth, and 0.2 inch in length, were machined at each thermocouple location. The measuring (hot) junction was made by Copper hollow sphere Dimensions in inches Figure 3.1 Schematic diagram of the test sphere assembly for the es- timation of time-dependent heat transfer coefficients in one - dimens ional IHTCP. 14 Figure 3.2 A photograph of the one-dimensional test sphere assembly. 15 inserting each lead into the separated slots and carefully peening 0.15 inch of the lead into each slot. A measuring junction of this type is called a separated junction or intrinsic junction. This installation method was selected because it provides rapid response and good thermal contact between the thermocouples and the test specimen. The output of such a separated junction is a weighted mean of the two individual junction temperatures plus some error (see ASTM [1974]). This error depends on the difference between the two junction temperatures and the material of the test specimen. Since the leads are installed close together and the OFHC sphere has relatively high conduc- tivity, the uncertainty in the measured temperatures are very small. After the thermocouples were installed, the surface of each hemisphere was coated with thin layer of silicone heat sink compound (Dow Corning 340) to prevent leakage of the fluids into the inner sur- face of the hollow sphere. The thermocouple wires were led through an axially drilled hole in the upper hemisphere. The support steel tube was used to attach the test specimen vertically to the piston of the hydraulic dropping system. One end of the tube is soldered into the top of the sphere and is used for bringing the thermocouple wires out of the sphere. The test sphere assembly is insulated from the piston by a Transite disk to minimize the heat losses from the specimen. The thermocouple leads from the specimen are con- nected to the backplane of the signal conditioning unit, as described below in Section 3.4. —W 16 3.2.2 Test Sphere for Two-Dimensional Problem Figure 3.3 shows schematically the test hollow sphere assembly for the two-dimensional IHTCP. The hollow OFHC sphere, 3.0 inch (7.62 x 10'2 m) outer diameter and 1.4 inch (3.56 x 10'2 m) inner diameter, was machined in two hollow hemispheres. Five type E thermocouples of 30 gauge were positioned in a vertical diametral plane (surface of the right-hand hemisphere in Figure 3.3). These thermocouples are labeled TC #1 - TC #5. All the thermocouples are located at the same depth, 0.16 inch (4.06 x 10-3 m), from the outer surface and at 9 -0, w/4, n/2, 3r/4, and n radians, respectively. The details of thermocouples' installation are the same as described in subsection 3.2.1. 3.3 Quenching Apparatus A photograph of the quenching apparatus and the control panel is shown in Figure 3.4. Some parts of the quenching apparatus used in the present investigation are modifications of that described in Ebrahimzdeh [1983]. The other parts were developed to support the investigation of transient heat transfer coefficients in the quenching experiments. The quenching apparatus consists of the following parts: (1) dropping hydraulic system, (2) heating furnace, (3) fluid vessel, and (4) steel frame holder. The dropping hydraulic system is operated by a hydraulic pump which operates the cylinder-piston assembly. This hydraulic system is used to quickly immerse the test sphere specimens into the fluid. The speed of the piston can be adjusted by a hand operated valve. The hydraulic cylinder piston assembly was selected in the present study 17 .mOHzH HoconcoEfit-o3u CH notewowmwooo Hometown use: acetaoeop modem-pco-oaau mo sowuoEHu -mo ozu pom hang—omen chosen umou 0.3 no Emuwmwo owumsozom m.m 0.4:me motocH cw mconCoEHn chosen soHHot Monaco l8 Figure 3.4 A photograph of the quenching apparatus and the control panel. 19 because it facilitates the repeatability of the quenching experiments. The heating furnace is a split type (clam shell) tubular heater. The tubular heating elements utilize a sturdy ceramic holder. A heli- cally coiled nickel-crane wire is embedded and cemented in longitudinal grooves in the ceramic holder. The nominal dimensions of each tubular half section are 3.75 inch inner diameter and 6 inch long. Each half is wrapped with a thin steel sheet. The heating furnace is mounted on automatically controlled arms designed to bring the furnace to the test specimen and wrap it around the sphere during the heating period and to move it away before dropping the sphere. A 110 V alternating current was used to provide the power input to the heating furnace. The power supply was controlled by two voltage regulators. A large cylindrical vessel, 220 mm in diameter and 150 mm deep, was used to hold the quenching fluid bath. A thermocouple was attached to the piston to record the moment the center of the sphere touches the fluid surface. The bulk temperature of the fluid was measured at a depth of 50 mm, and about 80 mm from the center of the vessel. The vessel was mounted on the steel frame holder such that the vibrations during the experiments were minimal. The steel frame holder assembly consists of two flat plates connected.with four adjustable connecting rods. It is used to hold the cylinder-piston assembly, the heating furnace controlling arms, and the quenching vessel. The cylinder assembly is attached to the upper steel plate while the frame itself is installed on a steel stand through four holes in the lower flat plate. 20 3.4 Hulti-Channel Data Acquisition System In this section the hardware devices and the software programs of the transient temperature data acquisition system in the Dynamic Heat Transfer Laboratory are described. 3.4.1 Hardware Devices Description The transient temperature data acquisition hardware is based on a PDP 11/03 microcomputer (Plessey Peripheral Systems) working under an RT-ll/V4 operating system. The PDP 11/03 is interfaced with the follow- ing units: (a) Signal conditioning unit, (2) Analog-to-digital unit model DT2764, and (c) Real-time clock/counter unit model DT2769. These units were provided by Data Translation, Inc. A block diagram of the data acquisition system is given in Figure 3.5. The Data Translation signal conditioning unit, Data Translation, Inc. [1981] , consists of a 1'15 V power supply model DT7692, a backplane connecting board model DT750, and eight amplifiers model DT6705E. Each amplifier has an insulated input module which provides transient input insulation, low pass filtering, and cold junction compensation for type "E" thermocouples. Interfacing to the thermocouples is made by wiring the thermocouple to the channel barrier strip on the DT750 backplane. The DT6705E amplifier module has a zero and scale calibration screws on the front panel. The amplifier gain is 103. 21 Amplifier Multiplexer 12-bit A/D RT Clock/Counter 64 K Memory Disk drives Thermocouples from test sphere Control signals PDP 11/03 Computer Signal conditioning unit ‘ System control Decwriter Results Figure 3.5 A block diagram of the temperature data acquisition system. 22 The DT2764 unit is a wide range, 12-bit resolution, analog-to digital (A/D) converter with 16 single-ended (SI) input channels or 8 differential-ended (DI) input channels, Data Translation, Inc. [1979a]. The full scale input voltage is 10 V (unipolar) or i 5 V (bipolar). At the present the A/D is configured to operate with 8 DI input channels, a unipolar input voltage, and gain equal to 1. This configuration was selected to improve the accuracy of the collected data. The DT2769 unit is a programmable clock/counter combination that determines the intervals of count events, Data Translation Inc. , [1979b]. Data Translation provided a software package SP0023 for the calibration and testing of the A/D and the clock units. 3.4.2 Software System Description Programs necessary to initiate and control different aspects of transient temperature data collection and processing are briefly described below. 1. QTLIBZRT Libragy A real-time software package DTLIB/RT is provided by Data Translation Inc. [1981] to support data acquisition process. It con- sists of a single library of subroutines that is called from the user main Fortran program. DTLIB/RT allows the user to initiate and control different aspects of the data collection process using the A/D converter and the Real-time clock/counter units. The subroutines are written in macro assembly language to allow for high speed data collection. The following subroutines from the DTLIB/RT are used to support the data acquisition process: 23 i. W. This subroutine sets the operating rate and the mode of the clock/counter unit. The repeated interval mode was selected. In this mode the clock counts "preset" intervals, overflow, reload "preset" and count again. ii. W. This subroutine initiates and controls the Real- time sampling of the analog input channels of the A/D unit. The Real- time input is used to trigger the A/D. The TTl trigger source was connected to the Real-time clock input (RTC INl). At each trigger, the RTS routine "sweeps" the input channels starting from channel 0 and sampling 7 consecutive channels. This causes a "data point" (eight thermocouple readings) to be collected. The minimum sampling rate between successive "data points" is 0.002 seconds. iii. LWAIT suhroutihe. This subroutine causes the main program to wait for the Real-time sampling to complete. 2. DATACQ Erggrfl; pain dgta pollection progpam The DATACQ program is the main program for utilizing the A/D and the Real-time clock/counter units to perform the data acquisition process. This program is written in Fortran and utilizes the DTLIB/RT subroutines to collect "large" numbers of "data points" and store them sequentially in a main buffer divided into sub-buffers. The data is stored in compact binary code. The main program utilizes a "completion routine" to take data from a filled sub-buffer and write it out to an output file on a floppy disk. The SAVEDA completion routine is called by RTS routine each time that a buffer is filled with data. The data collection process continues even while the "completion routine" is executed, which allows a fast data collection process. The maximum 24 number of the data points collected depends on the available space on the floppy disk. The DATACQ program also utilizes several system subroutines SYSLIB of the RT-ll to support different aspects of the real-time data logging . The DATFIT program is designed to convert the A/D output counts into experimental temperature measurements and store them on a floppy disk in a form convenient for the introduction into the IHTCP codes. First, the counts are converted into voltages by multiplying them by the scale factor (10/4096) and by a correction factor (1/0.996). Second, the temperature range 0-150 °C in ASTM [1974] was divided into 15 equal segments and first order interpolating polynomials were used to convert voltages into degree Celsius. A primary data processing of the temperature data can be done with the DATFIT program. The arithmetic mean (average) of a few read- ings from each channel can be calculated and stored. In the present study, only the raw data (no primary smoothing) is used as input to the IHTCP codes. 3.5 Experimental Procedures In preparation of the quenching experiments the spheres were carefully polished and cleaned with acetone before every experiment. Between experiments, the zero and scale setting of the eight amplifiers were adjusted. The A/D converter was tested and calibrated with the 25 present sphere assembly and the results showed that the accuracy of the temperature data to be within i 0.2 0C. The sphere is heated to the desired temperature (which is less than the boiling point of the cooling fluid) and then the power is turned off. The two halves of the furnace were opened and the tempera- tures inside the sphere were monitored with a DVM on the control panel. When a uniform temperature is reached the data acquisition process starts using the DATACQ program and then the sphere is plunged quickly in the cooling bath. The sampling rate was set to 0.1 seconds in the experiments. The duration of each experiment was 50 seconds. During this period of time, 500 "data points" (500 x 8 - 4000 readings) are collected and written into an output file on a floppy disk. The DATAFIT program is used to convert the A/D counts into tem- peratures and store them in form convenient for the IHTCP codes. CHAPTER4 ESTIMATION OF TRANSIENT HEAT TRANSFER COEFFICIENTS FOR ONE-DIMENSIONAL PROBLEMS 4 . 1 Introduction In this chapter the one-dimensional IHTCP (inverse heat pransfer poefficient problem) is treated for the estimation of time-dependent heat transfer coefficients, h(t), in one—dimensional convective heat transfer problems. An application considered in this study is the quenching of a hot hollow sphere into a cold fluid. The heat transfer coefficient is assumed to be uniform over the outer heat transfer sur- face of the sphere. The SFI (pequential iuture—information) method introduced by Beck [1962,1970] is used in the solution of the IHTCP. The solution method is based on a finite difference (FD) method. Two general inverse approaches for the estimation of transient heat transfer coefficients are investigated. The first approach is the "quotient" approach in which the solution of the IHTCP is reduced to the solution of the IHCP (inverse heat ponduction problem). The surface heat flux and the surface temperature are individually estimated and then the heat transfer coefficient is determined by the quotient q/AT (Newton's law of cooling). The second approach is the "direct" approach in which the heat transfer coefficient is found directly without the estimation of the intermediate values of the heat flux and surface 26 27 temperature. The relative merits of the direct and quotient approachs are discussed in a latter section of this chapter. The outline of this chapter is briefly given below. The state- ment and the mathematical formulation of one-dimensional IHTCP are given in Section 4.2. Section 4.3 develops the SFI method for solving the one-dimensional IHTCP. Section 4.4 describes the numerical solution of the direct convective heat transfer problem. Section 4.5 describes the calculations of the sensitivity coefficients. Section 4.6 outlines an overall computational algorithm for the IHTCP. The efficiency and accuracy of the SFI method are discussed in Section 4.7. 4.2 Statement of the One-Dimensional IHTCP for the Estimation of h(t) The hollow spherical body is sketched in Figure 4.1. The hollow sphere is assumed to be exposed to an external time-dependent convective boundary condition starting at time t - 0. In the experiments, however, the quenching process always starts after 0.5 or 1 second. The tran- sient convective boundary condition may be due to sudden quenching or a sudden change in convective heat transfer. The temperature history at an appropriate point inside the solid body is measured as a function of time. The inner surface of the hollow sphere is thermally insulated. The IHTCP is to estimate h(t) from measured transient temperature data. In the one-dimensional model of Figure 4.1, the temperature distribution inside the solid body is governed by the heat conduction equation, assuming constant thermal properties, %(r211‘_(_,_1rt)=;l_fl_i_t_lTrt 0-§ rpyw) pm, (5.11) u-l where the coefficients fly m’ v-1,2, . . . ,p, are the unknown parameters at time tm’ and the (pu(6) are a set of linear expansion (basis) functions. A general expression for the basis function may be written as I 0 - 9 I u 1 - | o - a | 5A6 «9(9) - ' My V V (5.12) 0 elsewhere Here the basis functions, cpy(6), are non-zero only for those elements that contain or are adjacent to the node u; see Figure 5.2b. The SFI method for solving the 2-D IHTCP is formulated as the problem of estimating, sequentially over time, the optimal values (in the least squares sense) of the set of parameters flu m which minimizes the residuals in the least squares objective function, 2 L r r Sm(ém) - E E [Yp,m+i-l - Tp,m+i-l2 --§§-w“' M“ p i a‘62,m p i afiZ,m a’Bp,m 93- . (5.28a) symmetric L r aTn ( g,m+i-l)2 L p-l i-l a’Bp,m j _ L r aTn , [ Y _ Tn ](_2.i.ull'.i;l) p,m+i-l p,m+i-l 65 p-l i-l l,m n . gm- . (5.28b) L r n 6T . [ Y _ Tn ](_11.m1_-1) p,m+i-l p,m+i-l as .u-l 1'1 p.111 . There are several ways in which the system of equations in Eq. (5.24) may be solved for the correction ES. The standard Gauss elimina- tion with partial pivoting is used for solving the above algebraic system, Conte and de Boor [1981] and Golub and Loan [1983]. 84 At time index m, m - 1,2, . . . ,M. the iterative solution starts by selecting some initial value E: and then calculating the corresponding temperature vector Inm+1_1. Next the step-function sensitivity coeffi- cients z: ”(m+i-l) are calculated. Then the algebraic system in Eq. (5.24) is solved for 23. If the Euclidean norm 13“ b? 2 b3 2 b 2 1/2 7- -[ . |+ . l+. + -xP—| ] (5.29) e“ 13“ 13“ 19“ _ m 2 l,m 2,m p,m is not less than the tolerance, 1017 say, one can obtain a new estimate A for 2;. The iteration process continues until the closeness criterion is satisfied. In case the numerical solution of Eq. (5.24) for the parameters m’ . . , 6 p Incauses trouble, the regularization method on space could ‘91 be used, Beck et al. [1985]. The "zero-order" regularization procedure can be implemented easily by adding the regularization parameter, a, to the diagonal elements of the 9; matrix. The SFI method described above is an efficient algorithm for solving the IHTCP and can be implemented on microcomputers with small memories. In the SFI method the computations of the (Mxp) unknowns are reduced to that of solving a set of M algebraic system of equations each 85 system of p-gh order which result in a substantial reduction in computa- tion time and storage. 5.4 Test Cases This section contains the numerical results of the systematic investigation of the SFI algorithm for specially selected test cases for the 2-D IHTCP. The numerical experiments were conducted with a copper sphere in a quenching problem. The following data are used in the numerical modeling: Inner radius R - 0.01 m, Outer radius R - 0.03 m, in out 0 Initial temperature F(r,6) - T0 - 99.0 C, 0 Thermal conductivity k - 379.0 W/m C, Thermal diffusivity a - 11.234 x 10'5 m2/s, Fluid temperature Tm(t) - Tdo - 20.0 C. Five thermocouples are located at five internal points having the same * depth r -'0.024 m and the angular locations 6 - O, «/4, n/Z, 3x/4, and 1r radians, respectively. The number of unknown parameters describing the space variation of hm is 3; at the angular locations 6 - O, «/2, and x radians; see Figure 5.3. The transient two-dimensional direct heat conduction problem in a hollow sphere was solved by using the finite difference method. The formulation of the finite-difference equations are based on the finite control volume technique, Patankar [1980]. The spatial domain, 1n’ out 86 was discretized into mesh points having the discretization coordinates (ri,6j), with r1 - i Ar + Rin’ i - O,l,2,...,I, and 6 - 3 A6 J J’ j - O,l,2,...,J. The Ar and A6j values are given by M _M M __7r_ I ’ j J The time interval 0 S r S '7' is divided into equal sub-intervals each of ~ length Ar - It? with the discrete time coordinate rn- n Ar, n -l,2, . . . ,N. The finite difference equations are solved by using the ADI scheme, Peaceman and Rachford [1965]. Details of the formulation of the finite control volume equations and the ADI. scheme are given in Appendix A. The accuracy of the computational algorithm for the IHTCP depends, in part, on the accuracyof the solution method for the direct problem. The accuracy of the numerical solution of the direct problem depends on the numerical values of the discretization parameters A1, A6, and Ar. The computational time step Ar for the direct problem may have to be made smaller than the experimental time step (sampling rate) At in order to obtain the required accuracy. The computational time step Ar can be any fraction, such as 1/2, 1/3, . . . ,l/n, of the measured time step At. The computational spacing A6 and the thermocouple spacing A6“ can J be related by , A6 -I¢A9. p 5% where n is an even positive integer. The radial location r of the 87 Figure 5.3 Layout of the test sphere showing locations of the ther- mocouples (*) and the parameters 13V m(O). 88 thermocouples are matched exactly with any of the ris for Rin s ri< Rout ' A triangular-pulse test case is numerically investigated. The heat transfer coefficient varies in time in a triangular shape. The values of the heat transfer coefficients considered cover a wide range of the Biot number, Bi, from lumped body to boiling heat transfer. The Bi is defined by * 131 - h L (5.30a) k where 3 3 * vo ume o h e Rout ' Rin L _ - (5.30b) outer surface area 2 3 R out is a characteristic length for the sphere geometry and k is the thermal conductivity. Two FORTRAN V computer programs were written to study the 2-D IHTCP test cases. Program NUMTWO was written.to solve the unsteady 2-D direct heat conduction problem for the quenching of a spherical body with a specified heat transfer coefficient function. This program implements the ADI scheme. Program NLINVl was written to solve the 2-D IHTCP. The NLINVl program utilizes the SFI method described in Section 5.3 and uses NUMTWO as a subroutine. 89 55.4.1 Numerical Results and Discussion The direct heat conduction problem with the triangular-pulse heat transfer coefficient was solved numerically to generate temperature test data for the IHTCP method. Five temperature profiles at the ther- mocouple locations and for the time interval 0 s r s 2.5 s were computed numerically using 11 nodes in the r-direction, 9 nodes in the 6-direction and A? - 0.05 s. The five temperature profiles are shown in Figure 5.4. Figure 5.5 shows both the assumed Bi and estimated Bi functions obtained by using the SFI algorithm for the triangular-pulse test case with errorless data (a - 0) and for no additional future information (i.e. , r - l). The nondimensional time based on the thermocouples depth, Eth’ At+ _ m ’ E E2 th is 0.156. The reconstruction of the Bi function is excellent in this test case since it is nearly indistinguishable from the assumed func- tion. Figure 5.6 demonstrates the accuracy and variability of the estimated functions when a - 0.005 and r - 1. Here the three plots, a, b, and c, depict both the assumed function and the estimated function at the bottom (6 - 1r), equator (6 - 1r/2), and top (6 - 0) of the sphere, respectively. Although a is very small, the recovery of the assumed function is very poor at all the three locations. In part (a), for Bi at the bottom of the sphere, the oscillations start at the 15th time step. However, the solution is stable in the time range considered. 90 Temperature, °C 105. u I . I . I . I r T f 95.1 ' .. 85m _ 75.1 _ 1 6=0 , 65.n _ 55.4 9=W/4 ._ - 1 45:5 6=1r/2 .J 6=1r 35. 0=3fl/4 .. ‘1 4 25. r r f If r r T r F I r O 10 2O 30 4O 50 60 Time index, m Figure 5.4 Simulated temperature histories at the thermocouple loca- tions for the two-dimensional inverse problem. 91 5 IIIrrIFITIIIIUFTITITrrTIIIITI[rtIIIIFT e—e Est. function, 6=1r ‘ a—EJ Est. function, 6=1r/2 5 A--A Est. function, 6=0. -—- Ass.funcfion Biot number, Bi O UFUIIIfFIITTTIFITUTrT—TTTIIITrIIIIIl’IIrf O 5 10 15 20 25 3O 35 40 Time index, m Figure 5.5 Estimated nondimensional heat transfer coefficient obtained by using the SFI method (r - l, a - 0, At - 0.05 s, and + AtE - 0.156). 6. TY'F V'IVI‘T'V' , ,....,H. ,reeq....r.q. . o—o Est. functon, 6-1r . 5,4 — Ass. function . 15 if Q) .0 E 3 C in! .9 CD ‘1 ""1'IITI'"'IrfitrwIfiI'IfifitIIFIIIII O 5 10 15 20 25 30 35 40 Time index, m 4. vvvrwffrrfivvfirvaTI—ruuv'Ivrv'vvrvlvrv , a—a Est. functon, 0-1r/2 3 q -- Ass. function _ ii 1 - ‘ ‘ 4 . 2- 6 ‘ “ q 5 , 1 . y ..D I ‘ 1 E 1.. , i [ ~ - 3 . V _ . c it ‘1 \_,;" ‘ T (b) :§ Om V V - m 5 . -15 '-2. ....,.r..r....,-...,...rr....,-r-.,--r11 0 5 10 15 2O 25 30 35 40 Time index, m 3. .-..,-...,-...,...,,....,....,--..,..- H Est. functon. 6-0. — Ass. function 1 ._ 2w 1 (D ‘ 1 ’5 I .8 1 ‘ ' ' 1‘ E 1 ,," 1‘ ' 1 ‘ 1 3 C O .1 v I — (C) .5 . 1 .9 1 ‘ fl \ 1 CD . -1 i _ -2 1 , ewe -rr..q, .. r.,. .-.,.. e.,.....,.-.., o 5 10 "1‘5” 20 25 30 35 40 Time index, m Figure 5.6 Estimated Bi(t) at (a) 6 - n, (b) 6 - n/2, and (c) 61- 0 + (r -l, a - 0.005 C, At - 0.05 s, and AtE 0.156). 93 In part (b) the oscillations start as early as the lO-_t_h_ time step, and after the 34th time step the solution becomes unstable. In part (c) the oscillations are even worse and the instability starts at the ZS-Q time step. The conclusion is that for no additional future temperatures the estimation of very small Bi values may cause some difficulties espe- cially with small dimensionless time step and "large" errors in the input data. Figure 5.7 illustrates the effects of the temperature future- information on the accuracy and stability of the solution of the IHTCP. Four future time steps (r - 4) are used in this case with a - 0.25. This value of a simulates the accuracy of actual data acquisition sys- tems: In Figure 5.7 there are no oscillations or instability observed and the agreement between the assumed function and the estimated one is very good. Large values of r greatly reduce the sensitivity to errors. The estimated results are very accurate except in the neighborhood of the triangle‘corners where the effects of excess smoothing are visible over two time steps (0.1 s). The SFI method gives a biased estimate of the heat transfer function while substantially reducing the sensitivity to errors . 94 50 FIT!IIIIITIlIrlIfTTrIrIrjIIIIIIIII‘IIII e—e Est. function, 6=1r ‘ a—a Est. function, 6=1r/2 4.4 A—A Est. function, 6=0. _ -—- Ass.funcfion q .3.-5 4 2.4 Biot number, Bi -1. I I I III I II I I I ITITT‘Ij] I I I I I I I I I I I I rI I I I I I O 5 10 15 20 25 30 35 40 Time index, m Figure 5.7 Estimated Bi(6,t) function (r -4, a - 0.25 C, At - 0.05 s, and Ac; - 0.156). 95 5 . 5 Conclusions , An accurate and efficient method is described for the estimation of the time-and-space dependent heat transfer coefficients when the transient temperature histories at appropriate interior points inside the heat conducting solid are available. The numerical results of the IHTCP test cases show that the SFI method is able to handle abrupt, and large steep changes in the heat transfer coefficients in the two- dimensional problems. The SFI estimation procedure described here is quite general and can be extended easily to cylindrical and rectangular coordinate sys- tems, as well as three-dimensional problems. It is recommended for the experimental estimation of the transient heat transfer coefficients in applications such as quenching processes and short duration wind tunnel experiments . CHAPTERG EXPERIMENTAL RESULTS AND DISCUSSION 6 . 1 Introduction Time-dependent heat transfer coefficients, estimated using the experimental data (temperature measurements) from quenching spheres, are presented in this chapter. Two quenching fluids are used in the experi- ments: ethylene glycol and water. The experiments were performed for an approximate one-dimensional model configuration and for the more realis- tic two-dimensional configuration of the quenching experiments. For the one-dimensional model, the surface heat transfer coeffi- cient is assumed to be uniform over the whole outer surface of the test sphere. The one-dimensional inverse program NUMER2, which implements the SFI method described in Chapter 4, is used for the estimation of the transient heat transfer coefficient, h(t), in the quenching experiments. In the quenching experiments, the variation of the heat transfer coefficient around the sphere (from the bottom to top) can be large. As a result, the temperature field inside the sphere is expected to depend on the angular coordinate 6 and the problem is actually represented by the two-dimensional model. The two-dimensional inverse code, program NLINV2, is used to estimate the time-and-space dependent heat transfer coefficients, h(t, 6), in the quenching experiments. Program NLINV2 implements the SFI method described previously in Chapter 5. 96 97 The strategy of the present study is to investigate the early transients in the heat transfer coefficient due to sudden imposition of convection heat transfer. However, the SFI method is not restricted to the above condition. Test cases show that the SFI method is capable of handling the whole boiling curve under severe transient conditions. Two test spheres are used in the experiments. Test sphere 1 (see Figure 3.1) is for the approximate one-dimensional configuration of the quenching experiments. Test sphere 2 (see Figure 3.3) is for the inves- tigation of the two-dimensional quenching experiments. This chapter is divided into four sections. Section 6.2 presents the estimated h(t) values from one-dimensional quenching experiments using both ethylene glycol and water. In Section 6.3, the estimated h(t) values are compared with the steady-state empirical correlations for free convection. The estimated h(t,6) values from two-dimensional experiments are given in Section 6.4. 6.2 Estimated Heat Transfer Coefficients from 1-D Experiments In this section, the estimated results from quenching experiments using the approximate one-dimensional model of the IHTCP, are presented and discussed. Several quenching experiments were performed using both ethylene glycol and water as quenching baths. The dimensions of the test sphere and thermocouple configuration for the estimation of h(t) are given in Figure 3.1 of Chapter 3. From the viewpoint of the mathematical formulation of the 1-D IHTCP it is sufficient to use only one interior thermocouple in order to estimate the unknown.heat transfer coefficient, h(t). However, due to the sensitivity of the solution of the IHTCP to errors in the measured values of temperature, it is useful to use more than one thermocouple in 98 order to minimize the effect of errors on the estimated values and to estimate the precision of the measurements. Therefore, three ther- mocouples, TC #4, TC #5, and TC #6 were embedded inside the sphere ,on the flat surface of the lower hemisphere, at the same depth of 0.103 inch (0.00261 m ) from the outer surface as shown in Figure 6.1. See also Figure 3.1. (The data from thermocouples TC #1, TC #2, and TC #3 have not been used at the present study.) The average of the three thermocouples is used as an input to the inverse code. Temperature data from the thermocouples were acquired at a constant rate of 0.1 s in the time interval from 0 to 50 s. Details of temperature data acquisition system are given in Chapter 3. The finite difference calculations for the sensitivity coeffi- cients and the predicted temperatures were performed using 11 nodes in the r-direction (Ar - 0.00261 m) and a time step of 0.05 5. Numerical testing of Crank-Nicolson scheme shows .that the above values are ap- propriate for the range of heat transfer coefficient, h(t), considered. 6.2.1 Results for Ethylene Glycol O Ethylene glycol E-l78 (boiling range 196-199 C, Fisher Scientific Company). initially at room temperature of about 27.5 C, was used. The test sphere was heated in a cylindrical furnace to a tempera- ture of 170 C. The furnace was then removed and the sphere allowed to reach a uniform temperature of about 150 C before it was quenched in the ethylene glycol bath. The thermal properties of the copper test sphere were assumed to be constant with the following values: k = 377 W/m 00 and a - 11.234 x 10'5 m2/s (Touloukian [1967]). 99 Dimensions in inches Figure 6.1 Locations of the thermocouples inside the one-dimensional test sphere. 100 Figure 6.2 displays the temperature histories at the three thermocouples, TC #1, TC #2, and TC #3. The measured temperature of the fluid is also shown. A subset of the measured temperature histories in the time period 0-12 5 is given in Table 6.1. Notice that the tempera- ture plots for the three thermocouples are almost identical, indicating that the heat transfer around the sphere is axisymmetric. The bulk temperature of the fluid , TF, remained constant during the experiment. The arithmetic mean over the three thermocouples was used as the input data for the inverse code. An estimate of the standard deviation of the mean measurement is investigated below. At any time m, m -1,2,...,M, there is a set of three measurement Y , Y , and Y . The estimated mean value, Y , of this set is, ml m2 m3 m n Y - 1} Y., m-l,2,...,M (6.1) m n mi i-l where n - 3, and the summation index i is over the thermocouples. The estimated variance of the measurements Ymi is given by n sz-lE [Y.-Y]2, m-l,2,...,M (6.2) m U. ml m i=1 For normal, equal variances, and uncorrelated errors, the es- timated standard deviation of the mean Ym is, Beck and Arnold [1977], s - ————' ; m - 1,2, ...... ,M (6.3) lOl TECCSPF IIIIIIIIIIIIl[ITIIIIIlllIIIIIIIIIITTTTITTTTTTTIT — TC #456 —*k304$>0103\JGJEDC)—*k3bd$>010) Y (00) 99999999999999999 Fluid Temp. lllllllLLLLlllillililll[[1 ILlJJlllllilllllllll 1111111111 1111 IIIIIIIIIIIIIITIIIIIIIIIITIIIIIIIIIIITTIIIITTIIII O. 10. 20. 30. 40. 50. t (s) Figure 6.2 Experimental temperature histories from the l-D sphere quenched in ethylene glycol. .._ 102 Table 6.1 A subset of the measured temperature histories from 1nd) test Time 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000 1.1000 1.2000 1.3000 1.4000 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000 2.1000 2.2000 2.3000 2.4000 2.5000 2.6000 2.7000 2.8000 2.9000 3.0000 3.1000 3.2000 3.3000 3.4000 3.5000 3.6000 3.7000 3.8000 3.9000 4.0000 4.1000 4.2000 4.3000 4 . 4000 4 . 5000 4 . 6000 4 . 7000 4 - 8000 4 . 9000 5 . 0000 53.1000 5 . 2000 5 . 3000 5 . 4000 5 . 5000 5 . 6000 sphere quenched in ethylene glycol (time in seconds and temperature in C). TF 27.52 27.47 27.47 27.43 27.52 27.43 27.43 27.43 27.43 27.43 27.43 ‘27.43 27.39 27.47 27.39 27.35 27.39 27.35 27.43 27.43 27.43 27.39 27.39 27.35 27.43 27.43 27.35 27.39 27.43 27.35 27.35 27.35 27.39 27.43 27.39 27.43 27.43 27.47 27.43 27.43 27.43 27.43 27.47 27.43 27.43 27.43 27.43 27.47 27.47 27.47 27.52 27.43 27.47 27.43 27.60 27.56 27.47 Temperatures et Thermocouple Locetione. 7,0 :1 150.49 150.49 150.56 150.49 150.42 150.42 150.42 150.42 150.42 150.53 150.42 150.42 150.42 150.29 150.18 150.11 150.01 149.84 149.59 149.46 149.32 149.11 148.97 148.77 148.59 148.42 148.32 148.14 148.04 147.80 147.73 147.49 147.45 147.24 147.11 146.93 146.83 146.69 146.55 146.38 146.28 146.17 145.93 145.86 145.72 145.65 145.45 145.38 145.17 145.10 144.95 144.79 144.72 144.58 144.41 144.34 144.20 TC '2 150.42 150.49 150.42 150.42 150.42 150.39 150.42 150.39 150.42 150.42 150.42 150.42 150.25 150.29 150.08 150.08 149.94 149.73 149.59 149.28 149.28 149.04 148.90 148.77 148.56 148.42 148.07 148.14 147.87 147.76 147.62 147.35 147.45 147.00 147.07 146.90 146.79 146.62 146.34 146.41 146.03 146.07 145.93 145.83 145.79 145.34 145.52 145.10 145.17 145.00 144.93 144.86 144.51 144.62 144.34 144.31 144.13 TC ’3 150.49 150.53 150.42 150.53 150.42 150.42 150.42 150.42 150.49 150.49 150.42 150.42 150.42 150.39 150.15 150.15 149.94 149.73 149.59 149.46 149.25 149.11 148.90 148.77 148.59 148.42 148.21 148.07 147.94 147.73 147.66 147.49 147.38 147.11 146.97 146.90 146.83 146.55 146.52 146.34 146.21 146.00 145.83 145.72 145.65 145.45 145.31 145.17 145.10 144.89 144.89 144.72 144.55 144.41 144.31 144.06 144.06 TC'I4 150325 150.25 150.25 150.22 150.25 150.22 150.25 150.25 150.08 149.70 149.32 148.90 148.49 148.18 147.87 147.62 147.45 147.31 147.21 147.11 146.93 146.90 146.76 146.62 146.52 146.34 146.24 146.07 146.00 145.83 145.69 145.55 145.45 145.27 145.17 144.96 144.86 144.79 144.62 144.48 144.34 144.24 144.06 143.99 143.86 143.75 143.58 143.51 143.41 143.23 143.10 143.03 142.92 142.78 142.65 142.51 142.40 TC 15 150.39 150.29 150.42 150.42; 150.39 150.35 150.35 150.32 150.22 149.87 149.53 149.11 148.83 148.49 148.21 148.04 147.94 147.76 147.62 147.45 147.38 147.21 147.07 146.93 146.83 146.62 146.52 146.41 146.28 146.00 145.86 145.79 145.72 145.52 145.45 145.17 145.03 144.89 144.86 144.72 144.58 144.48 144.24 144.17 144.06 143.92 143.79 143.65 143.51 143.41 143.30 143.20 143.10 142.96 142.82 142.68 142.58 TC '6 150.56 150.56 150.56 150.53 150.56 150.49 150.56 150.56 150.42 150.08 149.66 149.25 148.87 148.59 148.21 148.00 147.80 147.66 147.52 147.42 147.31 147.17 147.11 146.93 146.83 146.76 146.59 146.52 146.38 146.24 146.14 146.00 145.83 145.69 145.52 145.41 145.27 145.13 145.00 144.86 144.72 144.69 144.48 144.34 144.24 144.06 143.92 143.79 143.68 143.61 143.51 143.41 143.23 143.10 143.03 142.92 142.78 Table 6.1 5.7000 5.8000 5.9000 6.0000 6.1000 6.2000 6.3000 6.4000 6.5000 6.6000 6.7000 6.8000 6.9000 7.0000 7.1000 7.2000 7.3000 7.4000 7.5000 7.6000 7.7000 7.8000 7.9000 8.0000 8.1000 8.2000 8.3000 8.4000 8.5000 8.6000 8.7000 8.8000 8.9000 9.0000 9.1000 9.2000 9.3000 9.4000 9.5000 9.6000 9.7000 9.8000 9.9000 10.0000 10.1000 10.2000 10.3000 10.4000 10.5000 10.6000 10.7000 10.8000 10.9000 11.0000 (Continued) 27.52 144.03 27.52 143.92 27.47 143.86 27.52 143.65 27.52 143.58 27.52 143.51 27.56 143.34 27.52 143.23 27.56 143.10 27.60 142.96 27.56 142.82 27.60 142.68 27.60 142.61 27.56 142.51 27.60 142.40 27.60 142.27 27.56 142.13 27.60 142.02 27.68 141.95 27.56 141.78 27.60 141.71 27.60 141.57 27.68 141.40 27.60 141.30 27.60 141.19 27.60 141.12 27.60 141.02 27.64 140.88 27.60 140.81 27.60 140.71 27.60 140.54 27.60 140.43 27.64 140.33 27.64 140.19 27.60 140.16 27.64 139.91 27.64 139.91 27.68 139.77 27.60 139.67 27.60 139.56 27.60 139.46 27.56 139.42 27.60 139.21 27.60 139.07 27.64 139.04 27.64 138.93 27.64 138.83 27.60 138.69 27.60 138.58 27.68 138.48 27.64 138.37 27.60 138.24 27.60 138.20 27.64 136.10 27.68 137.92 27.64 137.85 27.64 137.68 27.68 137.64 27.64 137.54 27.66 137.47 27.66 137.33 27.68 137.19 27.60 137.05 27.60 136.98 143.99 143.99 143.61 143.79 143.51 143.41 143.27 142.99 142.68 142.89 142.68 142.58 142.47 142.20 142.33 141.92 142.02 141.85 141.82 141.68 141.37 141.54 141.09 141.23 141.09 141.02 140.92 140.57 140.71 140.54 140.47 140.26 140.19 140.16 139.88 139.91 139.74 139.63 139.60 139.35 139.35 139.07 139.14 139.04 138.93 138.79 138.51 138.62 138.30 138.37 138.24 138.24 138.06 137.82 137.89 137.68 137.64 137.50 137.40 137.29 137.05 137.12 136.94 103 143.92 143.66 143.65 143.51 143.44 143.30 143.20 143.06 142.96 142.62 142.66 142.54 142.51 142.40 142.27 142.13 141.99 141.65 141.76 141.66 141.57 141.47 141.33 141.19 141.09 140.95 140.66 140.74 140.61 140.54 140.43 140.26 140.19 140.12 139.91 139.66 139.77 139.63 139.56 139.49 139.32 139.21 139.14 139.07 136.66 136.79 136.69 136.62 136.41 136.30 136.24 136.10 136.06 137.96 137.62 137.66 137.57 137.50 137.36 137.26 137.12 137.12 136.96 136.64 142.33 142.20 142.13 141.95 141.85 141.71 141.57 141.54 141.40 141.30 141.16 141.02 141.02 140.81 140.74 140.61 140.47 140.36 140.26 140.19 140.09 139.98 139.88 139.74 139.63 139.49 139.46 139.28 139.21 139.07 139.00 138.86 138.79 138.65 138.58 138.44 138.30 138.34 138.13 138.03 137.96 137.82 137.68 137.64 137.47 137.36 137.22 137.15 137.01 136.94 136.84 136.70 136.66 136.56 136.38 136.35 136.21 136.07 135.96 135.86 135.79 135.72 135.58 135.51 142.47 142.33 142.30 142.13 141.99 141.85 141.82 141.64 141.57 141.44 141.30 141.23 141.09 140.99 140.85 140.74 140.68 140.47 140.43 140.26 140.19 140.05 139.98 139.91 139.77 139.63 139.49 139.42 139.32 139.18 139.14 138.97 138.90 138.76 138.65 138.58 138.44 138.30 138.24 138.10 138.03 137.96 137.82 137.68 137.64 137.50 137.40 137.26 137.15 137.08 136.91 136.84 136.70 136.59 136.56 136.42 136.35 136.21 136.07 136.00 135.86 135.79 135.72 135.51 142.68 142.54 142.47 142.33 142.20 142.13 141.99 141.85 141.78 141.64 141.57 141.44 141.26 141.19 141.02 141.02 140.88 140.74 140.61 140.57 140.47 140.33 140.19 140.12 139.98 139.88 139.77 139.63 139.60 139.46 139.35 139.25 139.07 139.00 138.90 138.79 138.62 138.58 138.41 138.34 138.24 138.17 138.03 137.96 137.82 137.68 137.54 137.47 137.40 137.26 137.19 137.05 136.91 136.84 136.73 136.59 136.42 136.35 136.28 136.17 136.07 136.00 135.82 135.79 104 Values of s_ are calculated using Eq. (6.3) for each time index m. The Y 111 maximum s_ value is 0.11 C (occurring at t - 1.7 s), and the minimum Y m value of s_ is 0.05 0C. The time average of s_ is 0.07 OC. Y111 Ym Figure 6.3(a) shows the estimated heat transfer coefficient function for the test sphere quenched in ethylene glycol, for the entire time of the experiment (from 0 to 50 s). Figure 6.3(b) shows the es- timated heat transfer coefficient function in an expanded scale from 0 to 10 s. The instant at which the sphere is completely immersed into the fluid is indicated by a vertical arrow in Figure 6.3(b). The time of complete immersion of the sphere is about 0.2 s. The parameter values used in the SFI method are Sr - 0.1 s, and r - £1. The quenching experiments were repeated several times to ascer- tain the repeatability of the results. The same initial and fluid temperatures are used in each experiment. The estimated values from three replicates of the quenching experiments are compared in Figure 6.4(a) . Figure 6.4(b) displays the estimated values plotted as a func- tion of AT. The important feature of the plots in Figures 6.3 and 6.4 is that the shape of the heat transfer coefficient has three distinct regions. These three regions can be seen more clearly on the expanded scale of Figure 6.3(b). During early-time transients, h(t) increases in ap- proximately a linear manner to a maximum value. The estimated h(t) function has its maximum of about 700 111/m2 C at 1.2 s; then the heat transfer coefficient decreases with time. During this early transient, the heat transfer from the sphere to fluid seems to take place mainly 105 HECCR4.SPF (I) O O dTilleIrTIlIIIITTIIITI’FIITFTIIIIIIITTIIIIIIIIIIIq 700.5 5 600.: .2 23‘ ~ = o 500.5 5 N d d E : 1 .1 .- ‘w/ :fl30.1 ‘ -C 1 1 200.5 1 100.5 3 o. ‘ ITIjTII’lIITTFI‘IIrirtFrITrTITrIIITITTTIFVTIIIIIYT O. 10. 20. 30. 40. 50. t (s) HECCRSPF 8000-1IITIIIIIIII‘IIFTITIIIIITIIIIIIITTTITITII- 700.5 5 500.5 5 6‘ . . a 500.5 5 N .1 E 1 ‘ 400.4 g 3 3 (b) .c: j 1 200.5 31 100.5 5 o.‘ ‘ T1T'll'[IUIIIIIIVIIIIUIrr1IrIllrIIerrj 0. 1. 2. 3. 4. 5. 6. 7. 8. 9.10. t (s) Figure 6.3 Estimated heat transfer coefficient, h(t), from the sphere quenched in ethylene glycol (r =- 4, Srs 0.1 s, At - 0.05 s, and Ar - 0.00261 m). 105 HECCR4.SPF 800.dTIIIIIIIIIIIIIIIIIIIIIIIIIITIITITIITIIIIIIIIIIIIq « 4 700.5 .3 600.5 5 8 « : . 500.5 5 NE 1 : § 400.5 5 v 300.5 5 4: : : 2005 .1 100.5 5 1 : 0 IT! I rrr l IIITTITIIIIII 111—TTT—T 0. 10. 20. t (s) HECCRSPF IlrIIIIIIIIIIITTTIIIIIfII1|lIIYjIIIl—Trr 800. 700. llllJJllJ O) O O h (W/m2 °C) .6 8 J l lJllllllllllll lLlllllllllllll Figure 6.3 Estimated heat transfer coefficient, h(t), 30. 40. 50. /'\ O V (b) from the sphere quenched in ethylene glycol (r = 4, Sr= 0.1 5, At - 0.05 s, and Ar - 0.00261 m). 106 HECR4.SPF 800.IFIITTT'IIITTIIIIIY'IIIIIII’FIFIYIYII ‘ -—- RUhl1 700. 600. 500. V . V j o O FflJN 2 ‘, A RUN 3 A O D 1 1. , 1 /"\ C) v 400. 300. 200. ° 8 h (W/m2 °C) 100. ‘ llllLllLlLlllll :5 1'3?- 06 II—TTIIIITIJTIIIIUIFTIITITITrTIIUTIIUI O. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. t (s) 800. . 1 . .HEC,R4T.'SP:: — RUN 1 o RUN 2 g A RUN 3 1 I I 700. 600. 500. 400. 300. 200. 100. LlllJlllllJJlllll D h (W/m2 °C) 141111141111111111111114111111 A 0‘ v 06 r r h r I T r ' T 75. 85. 95. 105. 115. 125. AT (°C) Figure 6.4 Comparison of the estimated results from three replicates of the quenching experiments in ethylene glycol. 107 by conduction. The plot in Figure 6.3(b) shows a distinct change of the slope at about 2 s, which apparently is due to transition of heat trans- fer from conduction to convection, and the establishment of free convection heat transfer from the sphere. For time t > 2 5, heat trans- fer from.sphere is mainly by free convection, and the heat transfer coefficient decreases monotonically with time. More discussions regard- ing the transition from conduction to convection heat transfer are given below in connection with Figures 6.11 and 6.12. The observed early-time trends (smaller values of heat transfer coefficient) 1J110(t) are slightly different than one would expect. It was expected that the surface heat transfer coefficient would have very large values at the moment the sphere comes into contact with the liquid in the quenching bath. The mathematical justification of this point is based on the analysis of the problem of two semi-infinite bodies, initially at different temperatures, and brought into perfect contact instantaneously; (See, e.g., Carslaw and Jaeger [1959].) The observed experimental trends are probably due to imperfect thermal contact be- tween the surface of the sphere and the quenching fluid due to a thin air film. During the course of quenching, after the sphere was com- pletely immersed in fluid, air bubbles were seen to detach from the sphere's surface. Figure 6.5 shows the sensitivity coefficient as a function of time. The sensitivity coefficient was calculated at the converged value A of the estimated heat transfer coefficient, hm, and is given by A A 6T(hm) A T(hm+6hm) - T(hm) Z (h) -h ——A—=h ' A , m-l,2,...,M (6.4) m m m 6hm m 611m 108 SEN ECCR4.SPF 0.0 1 I 1 I 1 I F1 1 In I [VI r I r I [j i 3 —O.1-4 41 A z (a) i V —0.2-4 —' .C CD \ 1— —0.35 5 CO 13 l —o.4— -+' ! "-05 T I fir T r FI r r r fit I r I fI 1 I O. 1. 2. 3. 4 5. 6. 7 8. 9. 1O t(s) Figure 6.5 Sensitivity coefficient for the sphere quenched in ethylene glycol. 109 For early times, the absolute value of the sensitivity coefficient increases to a maximum value of 0.45 C and then decreases. A distinct change of slope occurs at 2 s. The sensitivity coefficient reaches a ’ 0 value of 0.22 C at 10 s. The corresponding drop in measured tempera- ture is about 12 C. Hence, the sensitivity coefficient is considered to be small (0.22 C compared to 12 C) and several future time steps are therefore necessary for estimating a reasonably smooth curve for the heat transfer coefficient, h(t). SM of Residuals The residuals em (difference between measured temperature history and the calculated temperature history), are estimates of the random measurement errors 6 , e-£=’?-T(h) (6.5) where 1.11“ is the average of the three thermocouples TC #4, TC #5, and TC #6. The residuals are plotted in Figure 6.6 as a function of time. The residuals are randomly scattered about the zero line with zero mean. It appears that the statistical assumptions regarding the measurement errors (see Section 4.3) are approximately valid. The residuals seem to be additive and uncorrelated. The important feature of the plot in Figure 6.5 is that the residuals are very small, indicating that the estimates are accurate. In the time interval from t =- 0.5 s to t = 2 s, the residuals fall above the zero line at the beginning and below it for the rest of the time. The residuals in this interval are relatively 110 RESECCR4.SPF 0.50 r r I I l I I I I I 1 I l I T I 1 I 0.254 ° 0.00— e(C) L —O.25# I _O.50 rIrIrITITfTIrffTrff O. 1. 2. 3. 4. 5. 5. 7. 8. 9. 1(8) Figure 6.6 Residuals em for the sphere quenched in ethylene glycol. 10. lll larger than the residuals in the rest of the plot. In this region, the SFI method deliberately introduces small biases to the estimated values to achieve smaller mean square error (variance + biasz) than the variance obtained with exact matching unbiased methods. For more infor- mation see Beck et al. [1985]. Another estimate of the standard deviation of the temperature measurements can be found by calculating the estimated standard devia- tion of em, see Beck and Arnold [1977], s - a - M-l e (6.6) where M is the total time steps. The estimated standard deviation calculated using Eq. (6.6) is 0.032 C. This value is about 1/2 of the estimated value using Eq. (6.3). 6.2.2 Results for Water Results are given for the quenching experiments of the test sphere in water. Distilled water initially at room temperature of 22.8 C was used as the second quenching bath. The test sphere was heated in O the cylindrical furnace to about 99 C (which is below the boiling point of water). Thermal properties of the test sphere were assumed to be constant with the following values: k - 379 W/m C, a - 11.234 x 10-5 m2/s. 112 A subset of the measured temperature data in the time interval from 0 to 12 s is shown in Table 6.2. Figure 6.7 displays the tempera- ture histories at the three thermocouples, TC #4, TC #5, and TC #6. The measured temperature of the fluid is also shown. Notice that the tem- perature plots- for the three thermocouples are almost identical, indicating that the heat transfer around the sphere is axisymmetric. The bulk temperature of the fluid remained constant during the experi- ment. The arithmetic mean over the three thermocouples was used as the input data for the inverse code. The estimated standard deviation of the average of the three thermocouples data, s_ , is calculated using Y In Eq. (6.3). The maximum s_ value is 0.15 C (at t = 2.4 s) and the Y m minimum value of s_ is 0.06 c’C. The time average of s_ is 0.09 0C. Ym m The estimated results for quenching the sphere in water are shown in Figures 6.8 through 6.10. Figure 6.8(a) shows the estimated heat transfer coefficient function for sphere quenched in water, for the entire time of the experiment. Figure 6.7(b) shows the estimated heat transfer coefficient function in an expanded scale (from O to 10 s). The instant at which the sphere is completely immersed into the fluid is indicated by a vertical arrow in Figure 6.8(b). (The time of complete immersion of the sphere is in the vicinity of 0.2 s.) The parameter values used in the SFI method are Sr - 0.1 s, and r = 4. The heat transfer coefficient has its maximum of about 1750 1.1/m2 C in the vicinity of 1.2 s. The magnitude of h(t) function is greater for water than for ethylene glycol. The observed trends and the discussion given in connection of Figure 6.3 for the case of ethylene glycol is also valid for the case of water. 113 Table 6.2 A subset of the measured temperature history from l-D test Time 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.6000 0.9000 1.0000 1.1000 1.2000 1.3000 1.4000 1.5000 1.6000 1.7000 1.6000 1.9000 2.0000 2.1000 2.2000 2.3000 2.4000 2.5000 2.6000 2.7000 2.6000 2.9000 3.0000 3.1000 3.2000 3.3000 3.4000 3.5000 3.6000 3.7000 3.6000 3.9000 4.0000 4.1000 4.2000 4.3000 4.4000 4.5000 4.6000 4.7000 4.6000 4.9000 5.0000 5.1000 5.2000 5.3000 5.4000 5.5000 5.6000 sphere quenched in distilled water (time in seconds and temperatures in C). TF 22.97 22.93 22.89 22.93 23.01 22.93 22.93 22.85 22.89 22.89 22.85 22.89 22.89 22.85 22.85 22.93 22.85 22.85 22.85 22.81 22.85 22.85 22.85 22.85 22.81 22.85 22.85 22.85 22.85 22.85 22.85 22.77 22.77 22.85 22.77 22.85 22.77 22.77 22.81 22.81 22.85 22.77 22.77 22.77 22.77 22.85 22.77 22.77 22.77 22.77 22.77 22.77 22.81 22.73 22.77 22.81 22.77 Temperatures at Thermocouple Locations. TC '1 98.32 98.39 98.32 98.32 98.39 98.32 98.24 98.24 98.39 98.32 98.17 98.02 97.80 97.59 97.29 97.07 96.78 96.56 96.31 96.05 95.83 95.69 95.32 95.17 94.88 94.81 94.59 94.44 94.26 94.15 93.86 93.71 93.57 93.27 93.20 92.98 92.84 92.65 92.40 92.25 92.11 91.92 91.81 91.67 91.52 91.38 91.16 91.01 90.86 90.75 90.50 90.35 90.17 90.06 89.91 89.77 89.62 TC '2 98.06 98.10 98.10 98.02 98.02 98.10 98.02 98.02 98.10 98.02 97.80 97.59 97.37 97.22 96.93 96.64 96.38 96.12 95.91 95.61 95.39 95.17 94.96 94.81 94.66 94.37 94.26 94.01 93.86 93.71 93.57 93.35 93.13 92.91 92.80 92.65 92.54 92.33 92.11 91.96 91.81 91.67 91.45 91.38 91.16 91.01 90.86 90.72 90.54 90.35 90.21 90.06 89.91 89.77 89.62 89.47 89.32 TC #3 99.12 99.08 99.05 99.08 99.05 99.08 99.12 99.05 99.12 99.01 98.97 98.90 98.83 98.61 98.39 98.17 97.88 97.62 97.44 97.15 96.93 96.71 96.49 96.27 95.98 95.83 95.69 95.50 95.28 95.03 94.88 94.70 94.52 94.30 94.15 93.93 93.79 93.57 93.42 93.20 93.09 92.84 92.69 92.51 92.33 92.18 91.96 91.81 91.67 91.45 91.38 91.16 90.90 90.86 90.64 90.50 90.35 TC #4 98.79 98.83 98.83 98.83 98.79 98.83 98.72 98.24 97.62 96.93 96.34 95.83 95.47 95.10 94.88 94.63 94.52 94.37 94.30 94.15 94.08 93.93 93.79 93.64 93.49 93.42 93.20 93.06 92.84 92.65 92.54 92.33 92.18 92.03 91.89 91.67 91.45 91.30 91.16 91.01 90.79 90.64 90.43 90.28 90.17 89.99 89.84 89.65 89.54 89.32 89.14 89.03 88.88 88.73 88.58 88.40 88.25 TC '5 98.83 98.90 98.94 98.79 98.83 98.90 98.83 98.32 97.73 97.00 96.49 95.98 95.69 95.32 95.17 94.88 94.81 94.70 94.52 94.37 94.30 94.12 94.04 93.86 93.71 . 93.57 93.42 93.20 93.13 92.91 92.76 92.54 92.40 92.18 92.03 91.89 91.74 91.59 91.34 91.12 90.94 90.79 90.72 90.46 90.28 90.21 90.06 89.84 89.69 89.47 89.43 89.17 89.10 88.88 88.73 88.58 88.43 TC #6 99.12 99.12 99.05 99.19 99.12 99.05 98.97 98.54 97.91 97.29 96.71 96.20 95.83 95.54 95.28 95.10 94.96 94.88 94.81 94.66 94.59 94.37 94.30 94.15 94.08 93.86 93.68 93.57 93.42 93.20 92.98 92.91 92.69 92.54 92.33 92.18 92.03 91.81 91.59 91.45 91.30 91.16 90.94 90.79 90.64 90.43 90.28 90.21 89.99 89.84 89.62 89.54 89.40 89.28 89.03 88.88 88.66 114 Table 6.2 (Continued) 5.7000 22.77 89.47 89.10 90.21 88.06 88.29 88.58 5.8000 22.73 89.25 88.95 90.06 87.99 88.14 88.43 5.9000 22.77 89.03 88.80 89.91 87.84 87.92 88.25 6.0000 22.77 88.95 88.66 89.69 87.69 87.84 88.06 6.1000 22.77 88.73 88.47 89.62 87.55 87.69 87.99 6.2000 22.77 88.66 88.36 89.40 87.36 87.55 87.77 6.3000 22.77 88.43 88.21 89.32 87.33 87.33 87.69 6.4000 22.77 88.29 88.06 89.10 87.07 87.25 87.55 6.5000 22.77 88.21 87.95 88.88 86.96 87.10 87.36 6.6000 22.73 88.03 87.71’ 88.73 86.73 86.96 87.25 6.7000 22.77 87.92 87.62 88.66 86.66 86.88 87.07 6.8000 22.77 87.69 87.47 88.51 86.51 86.66 86.96 6.9000 22.81 87.62 87.33 88.36 86.36 86.59 86.81 7.0000 22.73 87.47 87.25 88.14 86.29 86.36 86.66 7.1000 22.77 87.25 87.07 88.14 86.11 86.29 86.44 7.2000 22.77 87.18 86.88 87.84 85.99 86.07 86.44 7.3000 22.73 87.03 86.73 87.77 85.85 85.99 86.22 7.4000 22.77 86.81 86.66 87.66 85.70 85.85 86.14 7.5000 22.77 86.73 86.47 87.47 85.55 85.70 85.99 7.6000 22.73 86.55 86.29 87.33 85.40 85.55 85.85 7.7000 22.77 86.44 86.14 87.18 85.25 85.40 85.70 7.8000 22.77 86.29 86.07 87.10 ' 85.11 85.25 85.55 7.9000 22.81 86.14 85.92 86.88 85.03 85.18 85.40 8.0000 22.77 85.99 85.77 86.73 84.96 85.00 85.33 8.1000 22.77 85.88 85.62 86.59 84.74 84.81 85.11 8.2000 22.77 85.85 85.48 86.44 84.59 84.74 85.00 8.3000 22.77 85.55 85.25 86.36 84.48 84.59 84.88 8.4000 22.81 85.40 85.18 86.14 84.29 84.44 84.74 8.5000 22.77 85.33 85.07 86.07 84.22 84.29 84.66 8.6000 22.77 85.18 84.96 85.81 84.07 84.22 84.44 8.7000 22.81 85.03 84.81 85.77 84.00 84.07 84.37 8.8000 22.77 84.88 84.59 85.70 83.81 83.92 84.22 8.9000 22.81 84.88 84.48 85.48 83.70 83.78 84.15 9.0000 22.81 84.59 84.44 85.33 ' 83.63 83.70 83.92 9.1000 22.85 84.59 84.22 85.25 83.33 83.55 83.85 9.2000 22.85 84.40 84.15 85.11 83.26 83.41 83.78 9.3000 22.85 84.22 84.00 84.96 83.18 83.26 83.55 9.4000 22.85 84.07 83.85 84.77 83.04 83.11 83.48 9.5000 22.85 83.92 83.78 84.70 82.89 83.04 83.26 9.6000 22.89 83.85 83.63 84.52 82.81 82.85 83.18 9.7000 22.85 83.70 83.48 84.37 82.67 82.74 83.11 9.8000 22.85 83.63 83.33 84.29 82.59 82.59 82.93 9.9000 22.93 83.41 83.18 84.22 82.52 82.59 82.81 10.0000 22.93 83.33 83.04 84.07 82.37 82.37 82.67‘ 10.1000 22.89 83.18 82.93 83.85 82.22 82.33 82.56 10.2000 22.89 83.11 82.81 83.78 82.08 82.15 82.44 10.3000 22.89 82.96 82.74 83.63 81.93 82.00 82.37 10.4000 22.93 82.74 82.52 83.44 81.78 81.93 - 82.15 10.5000 22.93 82.70 82.44 83.37 81.71 81.85 82.08 10.6000 22.93 82.52 82.30 83.18 81.63 81.63 81.85 10.7000 22.93 82.52 82.15 83.18 81.48 81.56 81.85 10.8000 22.89 82.37 82.00 83.04 81.30 81.41 81.71 10 9000 22.93 82.15 81.93 82.89 81.26 81.34 81.63 11.0000 22.93 82.00 81.82 82.74 81.11 81.19 81.41 11.1000 22.89 82.00 81.71 82.59 80.97 81.04 81.34 11.2000 22.93 81.78 81.56 82.44 80.82 80.97 81.19 11.3000 22.93 81.71 81.41 82.37 80.74 80.82 81.11 11.4000 22.89 81.56 81.34 82.22 80.52 80.67 80.97 11.5000 22.97 81.41 81.19 82.11 80.52 80.56 80.82 11.6000 22.93 81.34 81.04 81.93 80.45 80.41 80.74 11.7000 22.97 81.26 80.97 81.85 80.30 80.34 80.67 11.8000 22.93 81.04 80.82 81.78 80.15 80.23 80.52 11.9000 22.97 80.97 80.67 81.63 80.08 80.12 80.37 12.0000 23.01 80.82 80.60 81.48 79.93 80.00 80.30 Table 6.2 5.7000 5.8000 5.9000 6.0000 6.1000 6.2000 6.3000 6.4000 6.5000 6.6000 6.7000 6.8000 6.9000 7.0000 7.1000 7.2000 7.3000 7.4000 7.5000 7.6000 7.7000 7.8000 7.9000 8.0000 8.1000 8.2000 8.3000 8.4000 8.5000 8.6000 8.7000 8.8000 8.9000 9.0000 9.1000 9.2000 10.0000 10.1000 10.2000 10.3000 10.4000 10.5000 10.6000 10.7000 10.8000 10.9000 11.0000 11.1000 11.2000 11.3000 11.4000 11.5000 11.6000 11.7000 11.8000 11.9000 12.0000 (Continued) 22.77 89.47 22.73 89.25 22.77 89.03 22.77 88.95 22.77 88.73 22.77 88.66 22.77 88.43 22.77 88.29 22.77 88.21 22.73 88.03 22.77 87.92 22.77 87.69 22.81 87.62 22.73 87.47 22.77 87.25 22.77 87.18 22.73 87.03 22.77 86.81 22.77 86.73 22.73 86.55 22.77 86.44 22.77 86.29 22.81 86.14 22.77 85.99 22.77 85.88 22.77 85.85 22.77 85.55 22.81 85.40 22.77 85.33 22.77 85.18 22.81 85.03 22.77 64.66 22.81 84.88 22.81 84.59 22.85 84.59 22.85 84.40 22.85 84.22 22.65 64.07 22.85 83.92 22.89 83.85 22.85 83.70 22.85 83.63 22.93 83.41 22.93 83.33 22.89 83.18 22.89 83.11 22.89 82.96 22.93 82.74 22.93 82.70 22.93 82.52 22.93 82.52 22.89 82.37 22.93 82.15 22.93 82.00 22.89 82.00 22.93 81.78 22.93 81.71 22.89 81.56 22.97 81.41 22.93 81.34 22.97 81.26 22.93 81.04 22.97 80.97 23.01 80.82 89.10 88.95 88.80 88.66 88.47 88.36 88.21 88.06 87.95 87.71 87.62 87.47 87.33 87.25 87.07 86.88 86.73 86.66 86.47 86.29 86.14 86.07 85.92 85.77 85.62 85.48 85.25 85.18 85.07 84.96 84.81 84.59 84.48 84.44 84.22 84.15 84.00 83.85 83.78 83.63 83.48 83.33 83.18 83.04 82.93 82.81 82.74 82.52 82.44 82.30 82.15 82.00 81.93 81.82 81.71 81.56 81.41 81.34 81.19 81.04 80.97 80.82 80.67 80.60 114 90.21 90.06 89.91 89.69 89.62 89.40 89.32 89.10 88.88 88.73 88.66 88.51 88.36 88.14 88.14 87.84 87.77 87.66 87.47 87.33 87.18 87.10 86.88 86.73 86.59 86.44 86.36 86.14 86.07 85.81 85.77 85.70 85.48 85.33 85.25 85.11 84.96 84.77 84.70 84.52 84.37 84.29 84.22 84.07 83.85 83.78 83.63 83.44 83.37 83.18 83.18 83.04 82.89 82.74 82.59 82.44 82.37 82.22 82.11 81.93 81.85 81.78 81.63 81.48 88.06 87.99 87.84 87.69 87.55 87.36 87.33 87.07 86.96 86.73 86.66 86.51 86.36 86.29 86.11 85.99 85.85 85.70 85.55 85.40 85.25 85.11 85.03 84.96 84.74 84.59 84.48 84.29 84.22 84.07 84.00 83.81 83.70 83.63 83.33 83.26 83.18 83.04 82.89 82.81 82.67 82.59 82.52 82.37 82.22 82.08 81.93 81.78 81.71 81.63 81.48 81.30 81.26 81.11 80.97 80.82 80.74 80.52 80.52 80.45 80.30 80.15 80.08 79.93 88.29 88.14 87.92 87.84 87.69 87.55 87.33 87.25 87.10 86.96 86.88 86.66 86.59 86.36 86.29 86.07 85.99 85.85 85.70 85.55 85.40 85.25 85.18 85.00 84.81 84.74 84.59 84.44 84.29 84.22 84.07 83.92 83.78 83.70 83.55 83.41 83.26 83.11 83.04 82.85 82.74 82.59 82.59 82.37 82.33 82.15 82.00 81.93 81.85 81.63 81.56 81.41 81.34 81.19 81.04 80.97 80.82 80.67 80.56 80.41 80.34 80.23 80.12 80.00 88.58 88.43 88.25 88.06 87.99 87.77 87.69 87.55 87.36 87.25 87.07 86.96 86.81 86.66 86.44 86.44 86.22 86.14 85.99 85.85 85.70 85.55 85.40 85.33 85.11 85.00 84.88 84.74 84.66 84.44 84.37 84.22 84.15 83.92 83.85 83.78 83.55 83.48 83.26 83.18 83.11 82.93 82.81 82.67‘ 82.56 82.44 82.37 82.15 82.08 81.85 81.85 81.71 81.63 81.41 81.34 81.19 81.11 80.97 80.82 80.74 80.67 80.52 80.37 80.30 115 TWCASPF 11o. IjrrlitrlIlIITIllllIllllllIIIIITIIIIIIIITIIIITI 100. 90. 80. 70. 60. 50. 40:5 -1 30. 11111. Y(°C) 1L1L1l l l f Fluid Temp. _. 20.“ __ 10. O. TTIIVTTFFIIIIIWTIIIITI[IITIIIrrrrlTITIITrTIIITIF O. 10. 20. 30. 40. 50. 1(5) Figure 6.7 Experimental temperature histories from the l-D sphere L l l quenched in distilled water. 116 HWCAR4.SPF 20000 ‘ 11111[TITFTI[FITTIIrrTIF—IFTIIIIIIIWIII[IIFIT O) O .0 11111 h (W/m’°C) 11111111114 11111LLL11111111111I1111111 /'\ D V O. rtttttrrrrlrrtrtrrtltilIIIrrIIrrIIIIIIIIrrrrrrrrr 0. 10. 20. 30. 40. 50. t (s) HWAR4.SPF 20000:TWTTFFFFTIIIIIIIrrrrtfilriIrITIlllIlrfi: 1800:] .3 ’9‘ ‘2 NE .3 \ e (b) 5 600.5 1 .C 600.5 .5 400.5 .3 200.5 3: 1 1 1. 2. 3. 4. 5. 6. 7. 8. 9.10. t (s) 9 0e “frlrITIIrTIrT—FTFITl—VTIFIFIIIrII—rTlIII! Figure 6.8 Estimated heat transfer coefficient, h(t), from the sphere quenched in distilled water (r - 4, Sr- 0.1 s, At - 0.05 s, and Ar - 0.0026lm). 117 Figure 6.9 shows the sensitivity coefficient as a function of time. The sensitivity coefficient is calculated using Eq. (6.4). For early times, the absolute value of the sensitivity coefficient increases to a maximum value of 0.65 C and then decreases. The sensitivity 0 coefficient reaches a value of 0.27 C at 10 s. The corresponding drop 0 in measured temperature is about 17 C. Several future time steps are necessary for estimating a reasonably smooth curve for the heat transfer coefficient, h(t). Stud! of Residuals The residuals em are calculated using Eq. (6.5). Values of e are plotted in Figure 6.10 as a function of time. The estimated stan- dard deviation of the residuals calculated using Eq. (6.6) is 0.048 C. This value is less than the estimated value 0f 0.09 C using Eq. (6.3). 6.3 Comparison of Experimental Results with Empirical Correlation Results Unfortunately, there are no published results on the transient heat transfer coefficients for the free convection in quenching experi- ments. In order to evaluate the accuracy of the propoSed SFI method, the obtained experimental results are compared with the steady-state published empirical correlations for free convection. The Churchill and Churchill [1975] and Amato and Tien [1972] correlations are used to compare the experimental data. 118 SENWCAR4.SPF 0.1 T I 1 I I I I I I I T I 1 I I I r I I ”0.7 frrIfIlIfTFITIrIIIr O. 1. 2. 3. 4. 5 6. 7. 8. 9.10. 1(5) Figure 6.9 Sensitivity coefficient for the sphere quenched in distilled water . -.__—" 119 RESWCAR4SPF 1.0 l I 1 I r I 1 I 1 I 1 I T If I r I I 0.54 5 -. .. C.) i/ 0.0-1 CD .1 _ —o.5-1 — n -1 _1.0 r I W I I I r I r I 1 I T I I I T I I Figure 6.10 Residuals em for the 1-D test sphere quenched in distilled water . 120 Churchill and Churchill obtained the following correlations between the Nusselt number and Rayleigh number: -1/4 16/9 1/4 0 5 9/16 4 9 Nu - Nuo + (Ra/5) 1 + I—Ee—I , 10 < Ra < 10 r (6.7a) and 9/16 16/9 '1/6 Nu1/2 - Nué/Z + (Ra/300)1/6 I 1 + [—%+§] I , R > 109 r a (6.7b) 3 where Nu (- k D/kf) is the Nusselt number, Ra (- g_fl_AI;Q ) is the V0 Rayleigh number, Pr (- v/a) is the Prandtl number, and Nu0-2. Equation (6.7) is a comprehensive correlation for free convection from vertical plates, cylinders, and spheres. Amato and Tien obtained the following experimental correlation for laminar free convection from isothermal spheres immersed in water: Eu - 2 + c Ral/a, 3 x 105 s Ra s 8 x 108 (6.8) where C - 0.5 i 0.009. They reported a mean standard deviation of less than 11% for the range of Ra considered. The physical properties in Eqs. (6.7) and (6.8) were obtained from Holman [1976] and were evaluated at the mean film temperature. Figure 6.11 shows the estimated results, obtained by using the experimental data form quenching the test sphere in ethylene glycol, 121 plotted against corresponding values of temperature difference (AT - Ts- Tf) between surface temperature and fluid bulk temperature. The results obtained from the above correlations are also shown in the graph (denoted by the solid circles and triangles). The range of Rayleigh number, Ra, from 5.7 x 108 to 1.35 x 10 9was covered in the quenching experiments using ethylene glycol. As shown in Figure 6.11, Eq. (6.8) underestimates the surface heat transfer coefficient with a maximum difference of less than 15%, while Eq. (6.7) gives estimates which compare well with the estimated values after the early-transient region which is poorly predicted by the empirical correlations. Figure 6.12 shows the estimated results for the case of water. Also shown are the data from empirical correlations for comparison purposes. The range of Ra number from 5.4 x 108 to 1.92 x 10 9was covered in the quenching experiments using water. Notice that the data points obtained by using Eq. (6.7) lie slightly above the estimated values with a maximum deviation of less than 20% occuring at lower values of AT. Equation (6.8) compares very well with the estimated values, especially at the middle and lower values of AT. The conclusion is that this transient inverse estimation tech- nique has the capability of determining the early transients as well as the steady-state results in a single transient experiment. Further insight into the mechanisms of heat transfer during the quenching processes can be obtained by carefully examining the variation of heat transfer coefficient in Figures 6.11 and 6.12. The data in Figure 6.11 for ethylene glycol shows that the transition from conduc- 0 tion to convection occurs at AT = 119 C. This AT value corresponds to a Rayleigh number of about 1.2 x 109 which is close to the transition Rayleigh number. For water (Figure 6.12), the transition occurs at AT - 122 71 C, which corresponds to a Rayleigh number of about 1.7 x 109. Thus, within the accuracy of the estimated values, the transition from conduc- tion to convection appears to depend weakly on the quenching fluid. It is also clear from Figures 6.11 and 6.12 that when the conduction mode is no longer dominant, convective heat transfer coefficient tends to rise slightly for some time and then decreases monotonically as ex- pected. The convective heat transfer initiates in turbulent free convection and slowly progresses to laminar free convection. The estimated results for transient heat transfer coefficient in quenching experiments have not been previously reported in the litera- ture. Early transient and transition from conduction to convection heat transfer in quenching experiments need more investigation. 123 HECCR4T.SPF 800- llllIllllllllll:lllllllrrllllllIlllllIllllIllllII —— Quenching o Eq.(6.7) A Eq. (6.8) 700. 600. 500. 400. 300. h (W/m2 °C) 200. 100. llllllllllllllllllllllllllllLlL lllllllllllllllillllllLlllllll O. rrTlrlTlliIrllIrTrIIlIIIIIIIIITIIIIIIIIIIVFFTTIIT 75. 85. 95. 105. 115. 125. AT (°C) Figure 6.11 Comparison of the estimated results from the transient quenching experiments and the empirical correlation results (ethylene glycol). 124 2000. lIlITIITIfiIIlIflWICIAIIRIfl-IolslIPrlFIlIIIIIIIIIIIII]II Quenching :1 1800' . Eq. (5.7) 1600. 3 A Eq. (6.8) 1400. 1200. 1000. 800. 600. 400. 200. h (W/m2 °C) -¢ 4 _4 .— _ _. fl _.1 lJJlllllllllLllLLLLlllLlllLlLLLLLLLlLlI O .0 IIIITTIIIIIIIIIrIIIIIIIIIIIII[IITTIIIIIIFITII III 30. 40. 5o. 50. 7o. 80. AT (°C) FiguretilJ Comparison of the estimated results from the transient quenching experiments and the empirical correlation results (distilled water). 125 6.4 Estimated h(t,” from filo-Dimensional Experiments The estimated results of quenching experiments treated as a two- dimensional IHTCP are presented. Both ethylene glycol and water are used as quenching baths. The dimensions of the test sphere and loca- tions of the thermocouples TC #1 - TC #5 for the estimation of the time- and-space dependent heat transfer coefficient are given in Figure 3.3 of Chapter 3. The thermocouples TC #1, TC #2, . . . ,TC #5 are located at the same depth, 0.16 inch (4.06. x 10.3 m) from the outer surface, and at 0 - 0 (top), n/Q, 3/2, 3w/4, and a (bottom) radians, respectively. The finite difference calculations for the sensitivity coeffi- cients and the predicted temperatures were performed using 11 nodes in the r-direction (Ar - 0.00203 m), 9 nodes in the 0-direction (A0 - 0.3927 radians), and a time step of 0.05 3. Numerical testing of the ADI scheme shows that the above values are appropriate for the range of the heat transfer coefficient considered. 6.4.1 Results for ethylene glycol The time-and-space dependent heat transfer coefficient estimated from quenching the second test sphere into ethylene glycol is presented and discussed. The test sphere at uniform temperature of 140 C was quenched in an ethylene glycol bath at 24.42 C. A subset of the measured temperature histories in the time interval 0-12 3 is given in Table 6.3. Table 6.3 A subset of the measured temperature histories from 2-1) test Time 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 1.0000 1.1000 1.2000 1.3000 1.4000 1.5000 1.0000 1.7000 1.0000 1.9000 2.0000 2.1000 2.2000 2.3000 2.4000 2.5000 2.0000 2.7000 2.0000 2.9000 3.0000 3.1000 3.2000 3.3000 3.4000 3.5000 3.0000 3.7000 3.0000 3.9000 4.0000 4.1000 4.2000 4.3000 4.4000 4.5000 4.0000 4.7000 4.0000 4.9000 5.0000 5.1000 5.2000 5.3000 5.4000 5.5000 5.0000 1! 24.42 24.42 24.34 24.42 24.42 24.42 24.42 24.42 24.50 24.42 24.42‘ 24.42 24.34 24.34 24.42 24.42 24.42 24.42 24.34 24.34 24.34 24.34 24.42 24.42 24.30 24.34 24.30 24.34 24.34 24.30 24.30 24.42 24.42 24.42 24.42 24.34 24.42 24.42 24.34 24.34 24.42 24.42 24.30 24.34 24.34 24.42 24.34 24.42 24.42 24.34 24.42 24.34 24.42 24.42 24.34 24.42 24.42 temperatures in C). 126 Temp. at ThermocoupIc Locations. TC '1 130.70 130.70 130.70 130.70 130.70 130.70 130.70 130.70 130.09 130.40 130.20 137.05 137.50 137.22 130.94 130.73 130.52 130.30 130.17 130.10 135.90 135.02 135.75 .135.00 135.00 135.54 135.47 135.40 135.40 135.20 135.12 134.98 184.04 134.01 134.70 134.50 134.42 134.20 134.20 134.00 134.00 133.79 133.72 133.51 133.44 133.30 133.10 133.02 132.95 132.09 132.75 132.00 132.01 132.54 132.40 132.33 132.20 TC '2 139.04 130.70 130.97 139.04 130.70 130.90 130.90 130.02 130.70 ‘130.44 130.40 130.34 130.20 130.00 137.99 137.57 137.04 137.50 137.22 137.30 137.00 130.07 130.94 130.73 130.00 130.52 130.35 130.24 135.02 130.03 135.02 135.00 135.00 135.54 135.20 135.40 135.20 135.12 135.05 134.70 134.70 134.42 134.50 134.42 134.14 134.20 134.07 133.97 133.93 133.72 133.72 133.50 133.44 133.44 133.30 TC I3 130.90 130.90 130.90 130.97 130.90 130.90 130.03 130.02 130.34 137.99 137.01 137.30 137.05 130.00 130.00 130.49 130.31 130.24 130.10 135.90 135.09 135.09 135.00 135.01 135.54 135.40 135.20 135.19 135.05 134.91 134.04 134.70 134.03 134.50 134.35 134.20 134.21 134.07 134.00 133.00 133.09 133.05 133.50 133.44 133.30 133.10 133.13 133.02 132.09 132.75 132.00 132.01 132.47 132.33 132.33 132.19 132.00 TC '4 139.00 139.00 139.00 139.04 139.04 130.00 130.55 130.20 137.92 137.04 137.30 137.00 130.91 130.73 130.59 130.52 130.31 130.10 130.10 135.90 135.02 135.75 135.54 135.40 135.33 135.20 135.12 134.90 134.04 134.70 134.50 134.49 134.35 134.21 134.07 134.00 133.00 133.72 133.50 133.44 133.34 133.10 133.10 133.02 132.09 132.02 132.01 132.01 132.40 132.29 132.12 132.00 131.91 131.77 131.70 131.50 sphere quenched in ethylene glycol (time in seconds and TC '5 130.90 130.90 130.97 130.90 130.00 130.90 130.70 130.34 130.00 137.50 137.22 130.94 130.73 130.52 130.31 130.24 130.10 135.90 135.02 135.75 135.01 135.47 135.37 135.23 135.12 135.05 134.00 134.04 134.03 134.50 134.42 134.20 134.14 134.07 134.00 133.00 133.72 133.50 133.40 133.30 133.10 133.09 132.99 132.09 132.75 132.01 132.47 132.33 132.20 132.12 132.05 131.07 131.77 131.03 131.49 131.42 131.20 Table 6.3 5.7000 5.0000 5.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 7.0000 7.1000 7.2000 7.3000 7.4000 7.5000 7.0000 7.7000 7.0000 7.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 .0000 .1000 .2000 .3000 .4000 .5000 .0000 .7000 .0000 .9000 10.0000 10.1000 10.2000 10.3000 10.4000 10.5000 10.0000 10.7000 10.0000 10.9000 11.0000 11.1000 11.2000 11.3000 11.4000 11.5000 11.0000 11.7000 11.0000 11.9000 12.0000 0000000000 (Continued) 24.42 24.42 24.42 24.34 24.30 24.34 24.30 24.42 24.42 24.42 24.42 24.42 24.42 24.42 24.42 24.42 24.42 24.30 24.42 24.42 24.40 24.40 24.42 24.30 24.42 24.42 24.42 24.42 24.42 24.42 24.40 24.42 24.42 24.30 24.40 24.42 24.42 24.42 24.42 24.42 24.42 24.50 24.42 24.42 24.42 24.50 24.42 24.42 24.42 24.40 24.42 24.40 24.42 24.42 24.42 24.50 24.42 24.42 24.42 24.42 24.50 24.42 24.50 24.42 132.19 132.12 132.05 131.91 131.04 131.77 131.70 131.50 131.49 131.20 131.20 131.07 131.00 130.93 130.72 130.72 130.50 130.51 130.33 130.30 130.23 130.09 129.95 129.00 129.01 129.77 129.00 129.59 129.52 129.45 129.30 129.10 129.10 129.03 120.09 120.02 120.75 120.00 120.00 120.53 120.50 120.25 120.25 120.00 120.00 120.11 127.97 127.03 127.70 127.09 127.55 127.55 127.40 127.40 127.20 127.19 127.05 120.90 120.91 120.77 120.77 120.63 120.03 127 133.10 133.13 133.02 132.09 132.02 132.00 132.01 132.01 132.40 132.33 132.20 132.12 132.05 132.05 131.77 131.77 131.70 131.50 131.49 131.35 131.20 131.21 131.14 130.93 130.93 130.79 130.72 130.05 130.50 130.44 130.37 130.23 130.10 130.09 129.95 129.00 129.01 129.00 129.59 “129.52 129.30 129.21 129.17 129.10 120.90 120.09 120.75 120.00 120.01 120.53 120.39 120.32 120.25 120.11 120.04 127.97 127.03 127.03 127.02 127.55 127.55 127.33 127.20 127.23 132.05 131.91 131.77 131.03 131.50 131.49 131.35 131.20 131.21 131.07 130.93 130.93 130.79 130.05 130.05 130.51 130.37 130.37 130.23 130.12 130.09 129.91 129.01 129.01 129.59 129.52 129.30 129.24 129.21 129.10 129.03 120.90 120.05 120.75 120.00 120.53 120.40 120.39 120.32 120.10 120.11 127.97 127.90 127.03 127.09 127.02 127.51 127.40 127.33 127.19 127.12 127.05 120.95 120.91 120.77 120.70 120.50 120.50 120.42 120.34 120.20 120.13 120.03 125.92 131.49 131.20 131.21 131.21 131.00 130.93 130.79 130.79 130.05 130.51 130.37 130.37 130.23 130.02 130.02 129.77 129.01 129.00 129.52 129.45 129.30 129.24 129.10 120.90 120.90 120.09 120.04 120.53 120.40 120.39 120.25 120.00 120.11 127.97 127.03 127.70 127.02 127.51 127.47 127.33 127.20 127.12 127.12 120.04 120.00 120.70 120.50 120.49 120.49 120.34 120.20 120.10 125.99 125.99 125.01 125.70 125.04 125.57 125.50 125.30 125.25 125.14 131.17 131.07 130.93 130.00 130.72 130.05 130.50 130.51 130.37 130.10 130.12 129.00 129.01 129.00 129.59 129.52 129.45 129.31 129.20 129.10 120.90 120.09 120.02 120.71 120.01 120.40 120.39 120.25 120.25 120.04 127.97 127.90 127.03 127.70 127.55 127.55 127.40 127.20 127.12 127.05 120.95 120.04 120.70 120.70 120.03 120.50 120.42 120.34 120.20 120.13 120.00 125.92 125.05 125.71 125.07 125.57 125.43 125.43 125.25 125.21 125.07 124.97 124.00 124.00 128 Figure 6.13(a) shows the estimated heat transfer coefficient function, h(t,0), for the sphere quenched in ethylene glycol, for the entire time of the experiment (from 0 to 50 s) . Figure 6.13 (b) shows the estimated heat transfer coefficient function in an expanded scale from 0 to 10 s. The instant at which the sphere was completely immersed into the fluid is indicated by a vertical arrow in Figure 6.13 (b). (The time of complete immersion of the sphere is in the vicinity of 0.2 s.) The parameter values used in the SFI method are sr - 0.1 s and r - 3. The results obtained from the empirical correlations in Eqs. (6.7) and (6.8) are also shown in Figure 6.13(a) and denoted by the solid circles and triangles. The physical properties are evaluated at the average surface temperature of the sphere. The estimated values for 0 - 0 and 0 - 1r/2 are very close. Notice that the solid circles and triangles lie closer to the estimated values for 0 - 0 and 0 - 1r/2 than for 0 - 1r. The results obtained from the one-dimensional quenching experiment are shown in Figure 6.13(b) for comparison purposes. Two important observations are given regarding the estimated values shown in Figure 6.13. First, there are three distinct regions in the time domain. These regions can be seen more clearly on the expanded scale of Figure 6.13(b). During early transients, all the heat transfer coefficients vary with time in approximately a linear manner to the maximum values. Then the heat transfer coefficients decrease with time. The largest maximum value occurs at the bottom of the sphere. The maximum ratio of the value of the heat transfer coefficient at the bottom and top of the sphere is quite large, about 1.5 times. These changes should be taken into account in analyzing thermal systems with 129 .Amcmemu nmmm.o 1 m< pcm E moNoo.o n u< .m no.0 1 04 .m H.o Hum .m 1 My Hooham mamaxcum CH pmsosmaa chosen umou Q-N as» scum coauocsw acoHOammooo nommcmuu 0mm: pmumsaumm wma.o ouswwm E U. .om .0.V .om .ON .0— .0 __.._..’. ”can...“ 5...... at.“ “1... . a ... wufin . . . / . . .. ..., l. . .__. ...,... ..._...__._._._. as..." .5... 1...... ...,..__._......____.. .. .. ._.. ..., . . . w . ..3 . a. ...,. . v I. s I a. .. f . : ... I, . x. . . 1.... . .._.. N . . . . ‘ In...“ .4...“ .sn .... . . N\ a ,. w n .. I I: ll r 3.8 5m 4 A58 .3 . mchcozo PI-P._-L1—-r—P_p—___b.b-ph—F————-Lrb._Ph0—___——_- hnbbbb-LhFh—bbh_-PLL-L—__-th|—__.——hbbb—Lh-_hhhth .OO—‘ll ..Emhm $.41 (oozw/AA) q 1000. 130 BIAF1R3.SPF 900.4 800.4 700.4 500-1 500-4 400a 500+ 200:1 h (W/m2 °C) 100.4~ —100. HIIWIIIHWHIIqnnTlrnrrlnrnTlprwnlnlln1rrIrITrnIrnnlnnluannnunrnnn H 9=1r r—a 0=7r/2 '1 —~- 9=O - ---- 1D Experim. ' 1. 2. 3. 4. 5 6. 7. 8. 9. 10. t(s) Figure 6.13b Estimated heat transfer coefficient function from the 2-D test sphere quenched in ethylene glycol (r = 3, Sr= 0.1 5, At - 0.05 s, Ar = 0.00203 m and A0 = 0.3927 radians). 131 sudden imposition of convective boundary conditions. In such applica- tions the usually-used average steady-state values are not appropriate and may lead to large errors in the thermal calculations, especially'iJI the early-transient time period. The data of Figure 6.13(b) shows a distinct change of the slope at 2 s which apparently corresponds to the transition of heat transfer from conduction to convection and the estab- lishment of a free convection mode. And finally, for t > 2 3, each heat transfer coefficient decreases monotonically with time and it is ex- pected that as t becomes large the difference between the three curves will diminish and then approach a constant value. Our second observation has to do with the space dependence of the heat transfer coefficient function. The results in Figure 6.13 show that the heat transfer coefficient function undergoes substantial varia- tions in the space coordinate, especially at the early transient time period. These data shows that the SFI method is capable of estimating sharp and large variations (with respect to time and space) in.the surface heat transfer coefficient function, h(t,9). The residuals for the five‘thermocouples are plotted as a func- tion of time in Figures 6.14 through 6.18. The residuals for thermocouples TC #1 - TC #3 are larger than the residuals for TC #4 and TC #5. There are biases in the above thermocouples. These biases will be investigated at the end of this chapter after the data for water is presented. For TC #4 and TC #5 the residuals are nearly scattered about the zero line with relatively small biases. 132 RES1 A1 R3.SPF 0.50 I I j T T I I r I I I I I I I I r I I 0.25- .1 0.004 .— .025A I mIW “-0.50 FITII Residuals, 0 (°C) Figure 6.14 Residuals e1 m at TC # l for the 2-D test sphere quenched in ethylene glycol. 133 RE82A1R3SPF 1.00 I I I I T I I I I I fiI I I r I I I I A 0.75-1 —1 9 V 0.50-1 — CD .2“: 0.25-4 .. O 3 2 0.004 - U) 0‘2 —0.25J — _O.50 1* T f I I r I I r I Ti I I T I f I 0. 1. 2. 3. I 4. 5. 6. 7. 8. 9. 10. t(s) Figure 6.15 Residuals e2 m at TC # 2 for the 2-D test sphere quenched in ethylene glycol. 134 RE83A1R3.SPF 0.50 I I I I I I I I I I I I I j I I I j I g) 0.25-1 - v (D .9: O'OO‘JI/ — U 3 . :9 U3 E12 —0.25- v —O.50 IIIIIIIIIIIFIIIIIfiT 0. 1. 2. 3. 4. 5 6. 7. 8. 9. 10. t(s) Figure 6.16 Residuals e3 m at TC # 3 for the 2-D test sphere quenched in ethylene glycol. 135 RES4A1R3.SPF O-SO'I'II'III'IIIInvIfiI' /'\ (0.3 0.25- — V (D 2 0.004 - O 3 .9 3’3 025 _ _1 .. D: . —O.50 VrIrII—ITIITIIITIIIF Figure 6.17 Residuals e at TC # 4 for the 2-D test sphere quenched 4,m in ethylene glycol. 136 RE85A1RI’JSPF 0.50 ITI I rI I I I—1 I I I I T I I I I A 9 0.25-4 - 0) 1’1 MW .0“ 0.00- O 3 E U) (D -0.25-1 - 0: ..0050 r I I IT I I FI I I I I f I I I I t(s) Figure 6.18 Residuals e5 m at TC # 5 for the 2-D test sphere quenched in ethylene glycol. 137 16.4.2 Results for Water Estimated results for the case of water being the quenching bath are given. A subset of the measured temperature data in the time inter- val from 0 to 10 s is given in Table 6.4. The symbol TF in Table 6.4 denotes the bulk temperature of the fluid. Figure 6.19(a) shows the estimated heat transfer coefficient function, h(t,0), for sphere quenched in water, for the entire time of the experiment (from 0 to 50 s). Figure 6.19(b) shows the estimated heat transfer coefficient function in an expanded scale from 0 to 10 s. The instant at which the sphere was completely immersed into the fluid is indicated by the vertical arrow in Figure 6.19(b). (The time of complete immersion of the sphere is in the vicinity of 0.2 s.) The parameter values used in the SFI method are Sr - 0.1 s and r = 3. The discussion given above in connection with Figure 6.13 for the case of ethylene is also valid for Figure 6.19. The residuals for the five thermocouples are plotted as a func- tion of time in Figures 6.20 through 6.24. The residuals for the case of water follow the same trend as for the case of ethylene glycol. There are biases in thermocouples TC #1 through TC #3. To investigate the effect of these biases, constant values of - 0.3, 0.65, - 0.2, -0.15, and 0.1 C were subtracted from ther- mocouples TC #1 through TC #5, respectively. The new set of temperature data are used as input to the inverse code. The estimated results using the adjusted data does not change more than 0.2%. All the biases in the thermocouples are eliminated. The residual for the adjusted data and the original data for thermocouple TC #1 is shown in Figure 6.25. The other four thermocouples demonstrate the same trend. This indicates that the biases are not due to the SFI method. In fact the biases are 138 due to small systematic errors (offset) in the calibration and adjust- ment of the amplifiers and/or the analog-to-digital converter unit. 139 Table 6.4 A subset of the measured temperature histories from 2-1) test sphere quenched in distilled water (time in seconds and temperatures in C). Time TF Temp. et Thermocouple Locations. TC '1 TC '2 TC '3 TC #4 TC '5 0.0000 20.71 97.44 97.51 97.51 97.40 97.37 0.1000 20.71 97.44 97.40 97.51 97.37 97.44 0.2000 20.71 97.40 97.51 97.51 97.44 97.40 0.3000 20.07 97.44 97.51 97.40 97.44 97.33 0.4000 20.07 97.37 97.51 97.51 97.37 97.37 0.5000 20.03 97.37 97.51 97.51 97.37 97.37 0.0000 20.71 97.44 97.40 97.37 97.15 97.15 0.7000 20.71 97.00 97.22 90.93 90.42 90.04 0.0000 20.07 90.27 97.00 90.45 90.20 95.90 0.9000 20.03 95.05 90.04 95.91 95.70 95.47 1.0000 20.55 94.00 90.34 95.47 95.20 95.00 1.1000 20.55 94.22 90.12 95.03 94.90 94.74 1.2000 20.51 93.04 95.70 94.74 94.59 94.44 1.3000 20.59 93.13 95.25 94.37 94.44 94.22 1.4000 20.55 92.04 95.25 94.15 94.22 93.93 1.5000 20.47 92.54 94.90 93.93 94.01 93.71 1.0000 20.55 92.25 94.70 93.00 93.79 93.04 1.7000 20.47 92.03 94.52 93.49 93.04 93.42 1.0000 20.47 91.01 94.30 93.35 93.49 93.27 1.9000 20.47 91.07 94.00 93.20 93.35 93.09 2.0000 20.47 91.45 93.00 92.91 93.13 93.00 2.1000 20.47 91.30 93.04 92.07 92.90 92.04 2.2000 20.47 91.30 93.42 92.04 92.04 92.09 2.3000 20.47 91.10 93.27 92.02 92.05 92.54 2.4000 20.47 91.10 93.00 92.54 92.54 92.33 2.5000 20.47 91.00 92.04 92.30 92.25 92.10 2.0000 20.39 91.01 92.09 92.25 92.10 92.03 2.7000 20.43 90.00 92.54 92.11 92.03 91.74 2.0000 20.47 90.72 92.33 91.90 91.01 91.70 2.9000 20.39 90.04 92.10 91.70 91.07 91.03 3.0000 20.47 90.40 91.90 91.59 91.45 91.30 3.1000 20.47 90.35 91.74 91.45 91.30 91.23 3.2000 20.43 90.20 91.03 91.30 91.19 91.01 3.3000 20.47 90.13 91.40 91.10 91.05 90.00 3.4000 20.39 90.00 91.30 91.01 90.79 90.72 3.5000 20.47 09.95 91.10 90.57 90.04 90.57 3.0000 20.43 09.04 91.01 90.72 90.50 90.43 3.7000 20.47 09.40 90.00 90.57 90.35 90.21 3.0000 20.31 09.54 90.57 90.43 90.13 90.00 3.9000 20.39 09.30 90.57 90.20 90.00 09.91 4.0000 20.39 09.25 90.43 90.13 09.04 09.77 4.1000 20.39 09.10 90.20 09.99 09.09 09.40 4.2000 20.47 09.03 90.13 09.04 09.54 09.40 4.3000 20.47 00.00 90.00 09.05 09.43 09.32 4.4000 20.43 00.73 09.91 09.47 09.25 09.17 4.5000 20.39 00.55 09.04 09.40 09.10 00.95 4.0000 20.47 00.43 09.40 09.25 00.00 00.00 4.7000 20.43 00.29 09.40 09.10 00.00 00.73 4.0000 20.47 00.21 09.47 00.99 00.50 00.51 4.9000 20.47 00.03 09.32 00.00 00.43 00.30 5.0000 20.43 07.92 09.17 00.02 00.29 00.21 5.1000 20.39 07.01 09.10 00.51 00.00 00.00 5.2000 20.43 07.00 00.95 00.21 00.00 07.04 5.3000 20.39 07.55 00.00 00.21 07.92 07.09 5.4000 20.43 07.47 00.73 00.00 07.02 07.55 5.5000 20.43 07.40 00.50 07.92 07.02 07.40 5.0000 20.47 07.25 00.21 07.04 07.40 07.25 Table 6 . 4 5.7000 5.0000 5.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 7.0000 7.1000 7.2000 7.3000 7.4000 7.5000 7.0000 7.7000 7.0000 7.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 9.0000 9.1000 9.2000 9.3000 9.4000 9.5000 9.0000' 9.7000 9.0000 9.9000 10.0000 10.1000 10.2000 10.3000 10.4000 10.5000 10.0000 10.7000 10.0000 10.9000 11.0000 11.1000 11.2000 11.3000 11.4000 11.5000 11.0000 11.7000 11.0000 11.9000 12.0000 (Continued) 20.47 20.47 20.47 20.39 20.39 20.47 20.47 20.47 20.43 20.47 20.43 20.47 20.39 20.43 20.47 20.47 20.43 20.39 20.47 20.47 20.39 20.47 20.47 20.39 20.47 20.47 20.47 20.47 20.47 20.43 20.47 20.47 20.47 20.47 20.39 20.51 20.43 20.43 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.51 20.47 20.43 20.51 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.55 20.47 20.51 20.55 87.18 87.03 00.90 88.77 00.00 88.58 06.51 88.38 88.38 88.22 00.07 85.92 85.92 05.70 05.00 05.55 05.40 85.33 85.33 05.11 05.03 04.00 84.83 04.00 84.83 84.44 84.37 04.29 84.11 84.07 83.92 03.05 83.70 83.70 83.48 03.41 83.33 83.18 83.04 82.88 82.81 82.87 02.59 02.52 82.37 02.22 82.22 82.15 82.08 81.83 01.05 81.78 81.59 81.48 81.41 81.34 81 . 19 01.04 80.97 80.82 80.82 80.74 00.52 00.45 00.30 00.21 00.14 07.99 07.92 07.73 07.02 07.47 07.40 07.25 07.10 00.90 00.00 00.01 00.00 00.51 00.44 00.29 00.14 00.07 05.05 05.77 05.70 05.02 05.40 05.33 05.25 05.10 05.03 04.00 04.01 04.00 04.59 04.40 04.29 04.22 04.15 04.00 03.05 03.70 03.03 03.52 03.41 03.33 03.22 03.11 02.90 02.09 02.74 02.59 02.52 02.30 02.30 02.22 02.00 02.00 01.05 01.70 01.03 01.50 01.40 01.34 01.11 01.11 140 07.09 07.55 07.40 07.25 07.10 07.03 00.00 00.77 00.59 00.44 00.33 00.10 00.07 05.99 05.05 05.70 05.55 05.40 05.29 05.10 05.11 04.90 04.01 04.74 04.59 04.44 04.29 04.10 04.15 04.00 03.05 03.70 03.03 03.55 03.41 03.20 03.11 03.11 02.90 02.01 02.07 02.59 02.44 02.37 02.22 02.15 02.00 01.93 01.70 01.03 01.59 01.41 01.11 01.20 01.11 01.04 00.09 00.02 00.74 00.00 00.45 00.37 00.23 79.93 07.25 07.10 00.90 00.01 00.00 00.59 00.44 00.22 00.14 05.99 05.05 05.70 05.55 05.40 05.25 05.10 04.90 04.00 84981 04.03 04.55 04.37 04.29 04.15 04.00 03.92 03.70 03.03 03.40 03.37 03.10 03.11 02.90 02.09 8207‘ 02.07 02.52 02.44 02.37 02.11 02.15 02.00 01.02 01.71 01.50 01.41 01.34 01.19 01.19 01.00 00.02 00.74 00.07 00.00 00.49 00.37 00.23 79.93 00.00 79.05 79.70 79.03 79.50 79.40 07.10 07.03 00.04 00.73 00.59 00.44 00.25 00.14 00.07 05.05 05.70 05.55 05.44 05.25 05.10 05.03 04.90 04.01 04.00 04.52 04.29 04.20 04.15 04.00 03.92 03.70 03.03 03.44 03.33 03.10 03.11 02.09 02.01 02.74 02.03 02.52 02.30 02.22 02.15 02.00 01.05 01.71 01.03 01.40 01.34 01.11 01.11 01.04 00.97 00.02 00.71 00.50 00.52 00.45 00.30 00.15 00.00 79.93 79.02 79.70 79.07 79.41 79.33 79.29 Table 6 . 4 5.7000 5.0000 5.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 7.0000 7.1000 7.2000 7.3000 7.4000 7.5000 7.0000 7.7000 7.0000 7.9000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.0000 0.7000 0.0000 0.9000 9.0000 9.1000 9.2000 9.3000 9.4000 9.5000 9.0000 9.7000 9.0000 9.9000 10.0000 10.1000 10.2000 10.3000 10.4000 10.5000 10.0000 10.7000 10.0000 10.9000 11.0000 11.1000 11.2000 11.3000 11.4000 11.5000 11.0000 11.7000 11.0000 11.9000 12.0000 (Continued) 20.47 20.47 20.47 20.39 20.39 20.47 20.47 20.47 20.43 20.47 20.43 20.47 20.39 20.43 20.47 20.47 20.43 20.39 20.47 20.47 20.39 20.47 20.47 20.39 20.47 20.47 20.47 20.47 20.47 20.43 20.47 20.47 20.47 20.47 20.39 20.51 20.43 20.43 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.51 20.47 20.43 20.51 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.47 20.55 20.47 20.51 20.55 07.10 07.03 00.90 00.77 00.00 00.59 00.51 00.30 00.30 00.22 00.07 05.92 05.92 05.70 05.00 05.55 05.40 05.33 05.33 05.11 05.03 04.00 04.03 04.00 04.03 04.44 04.37 04.29 04.11 04.07 03.92 03.05 03.70 03.70 03.40 03.41 03.33 03.10 03.04 02.90 02.01 02.07 02.59 02.52 02.37 02.22 02.22 02.15 02.00 01.93 01.05 01.70 01.59 01.40 01.41 01.34 01.19 01.04 00.97 00.02 00.02 00.74 00.52 00.45 00.30 8802‘ 00.14 07.99 07.92 07.73 07.02 07.47 07.40 07.25 07.10 00.90 00.00 00.01 00.00 00.51 00.44 00.29 00.14 00.07 05.05 05.77 05.70 05.02 05.40 05.33 05.25 05.10 05.03 04.00 04.01 04.00 04.59 04.40 04.29 04.22 04.15 04.00 03.05 03.70 03.03 03.52 03.41 03.33 03.22 03.11 02.90 02.09 02.74 02.59 02.52 02.30 02.30 02.22 02.00 02.00 01.05 01.70 01.03 01.50 01.40 01.34 01.11 01.11 140 07.09 07.55 07.40 07.25 07.10 07.03 00.00 00.77 00.59 00.44 00.33 00.10 00.07 05.99 05.05 05.70 05.55 05.40 05.29 05.10 05.11 04.90 04.01 04.74 04.59 04.44 04.29 04.10 04.15 04.00 03.05 03.70 03.03 03.55 03.41 03.20 03.11 03.11 02.90 02.01 02.07 02.59 02.44 02.37 02.22 02.15 02.00 01.93 01.70 01.03 01.59 01.41 01.11 01.20 01.11 01.04 00.09 00.02 00.74 00.00 00.45 00.37 00.23 79.93 07.25 07.10 00.90 00.01 00.00 00.59 00.44 00.22 00.14 05.99 05.05 05.70 05.55 05.40 05.25 05.10 04.90 04.00 04.01 04.03 04.55 04.37 04.29 04.15 04.00 03.92 03.70 03.03 03.40 03.37 03.10 03.11 02.90 02.09 02.74 02.07 02.52 02.44 02.37 02.11 02.15 02.00 01.02 01.71 01.50 01.41 01.34 01.19 01.19 01.00 00.02 00.74 00.07 00.00 00.49 00.37 00.23 79.93 00.00 79.05 79.70 79.03 79.50 79.40 07.10 07.03 00.04 00.73 00.59 00.44 00.25 00.14 00.07 05.05 05.70 05.55 05.44 05.25 05.10 05.03 04.90 04.01 04.00 04.52 04.29 04.20 04.15 04.00 03.92 03.70 03.03 03.44 03.33 03.10 03.11 02.09 02.01 02.74 02.03 02.52 02.30 02.22 02.15 02.00 01.05 01.71 01.03 01.40 01.34 01.11 01.11 01.04 00.97 00.02 00.71 00.50 00.52 00.45 00.30 00.15 00.00 79.93 79.02 79.70 79.07 79.41 79.33 79.29 141 .Om .Amcmwvmu mammé." §< 6cm .5 moNoo.o 1. HQ .8. no.0 I 04 .m HAY-um .ml ..C .0583 UoHHfiumfip a. monocosv ouosmm umou Q-N Eoum cowuocsm ucofioammooo umMmCmuu umm: poumsfiumm 05H.o oudmfim Am. . .0... .0m .ON .0? -o’ prhbru..P-.L..a.phb..LhLL_L._.........h.p.#0...# . o a I 22.. I .e ..s... I T.............. 11...“ a II. 1" _,. T ._ I I. T I. 1 w 1 1 I .- .I. I II- .1 r a. a. a I 1 I I 1 r r a 1 1 I 1. I. I TI I I I. I ._.....4._P.~.p._.._..L...._FPL._LLP..__...?FLP.. mdmfimmmBI d o 1 c5 CD sf 0° 0 01 (Oozw/M) L1 142 .Amsmfiomu mummén .3 0:0 .E momood n ..5 .m mo.0 1 HQ .8. H.01um .m1 .3 young... 9:3“.me CH 00:0:050 whoa—am umou 0-N Scum soHuocsm ucofiofimmooo nommcmuu use: pmumawumm an.o ousmwm ”m0 . a: .0 .0 .m .0 .0 .4 .0 .N .— 0 .00—l 0 0 fi— fiIIIIr’ V‘ x 0 0 0 0 0 <— II is 3 IIIIIIIjIIIIIjIITjII—IIIIIIfiIVII H. r .82 n m\.=.um 41.4 T the eie . pEPP:PP—I—E—E—phprPPP.—_::_.:_—_~h::___:_::b-——Pb:::_P-—~_::—_.:;P-—p:::. 00mm mamhmmBI (30 zLU/M) L1 143 RES1W2R3.SPF 1‘0 . I . I . I I . I . I I I . I . I I § 0.54 .1 CD " . E 0.0-W .. O .3 . '5 cm —0.54 _ CE. I . _1.0 I I I I r I I I I IT I r I—FI I 0 1 2. 3 4 5 6 7. 8. 9. 10. Figure 6.20 Residuals e:L m at TC # l for the 2-D test sphere quenched in distilled water. 144 RE82W2R3.SPF 1‘5'I'I‘I'Tfil'fr1 A O o \./ (D 2.. O 3 ‘ ~ .0 ”(r-3 0.5 __ CD . _ 01 -1.0-—1 _ l L 1 UI __, '1 .1 .1 A .7 1 O I" (A 4s I 5. 6. 7. 8. 9. 10. ( Figure 6.21 Residuals 02 m at TC # 2 for the 2-D test sphere quenched 1 in distilled water. 145 RESBW2R3SPF 1.0 I I I I I I I I I I II I I II I I T /‘\ 9 0.5— 4 (D ‘ .. .9“ 00- .. O .3 . - '8 05 0: - ' 7 7 “1.0 I I—i—I I IIIT FTTI TI r I r I I 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1(8) Figure 6.22 Residuals e3 m at TC # 3 for the 2-D test sphere quenched 9 in distilled water. 146 RES4W2R3.SPF 1.0 I Ifi I I I I r1” I I In I I I r I I /'\ g.) O.5~ 4 v (D ‘ 7 £9. 0 O— ._ O 3 9 1 d U) (D —0.5-1 . — m . -1.0 I I I I I IfiI .If I I 1—1 I’m I I I I 0. 1. 2. 3. 4. 5. 6. 7. 8. 9.10. t (5) Figure 6.23 Residuals 64 m at TC # 4 for the 2-D test sphere quenched in distilled water. 147 RESSW2R3.SPF 1.0 I I I I I j fl I T I I I I— I l I T T I . 4 /'\ g.) 0.5- - V G) 7 .1 £5 0 0-4 _. o 3 ‘0 i 7 ’63 0) -054 _ D: _1.0 IrrrlgfIFIfifI—IffTrr 4. 5. 6. 7. 8. 9. 10. 1(8) Figure 6.24 Residuals e5 01 at TC # 5 for the 2-D test sphere quenched in distilled water. 14.8 1 O RES1W2R3BSPF . I I I I I I I I I I I I VI r I I I I — Original ‘ e—o Adjusted ‘ A ~ g) 0.5-4 s V G.) .- 33“ 0.0-J O 3 E -I (I) <0 -o.54 _ D: "—1.0 I r I f r I F I I I r I I I FT Figure 6.25 Comparison of the residuals at TC # 1 for the original and adjusted temperature data from sphere quenched in water. CHAPTER 7 SUMMARY AND CONCLUSIONS The multi-dimensional inverse heat transfer coefficient problem has been treated. Solution methods and algorithms were developed for the estimation of the transient heat transfer coefficient in multi- dimensional convective heat transfer applications from transient temperature measurements inside the the heat conducting solid and the bulk temperature of the fluid. This inverse estimation problem, called herein the inverse heat transfer coefficient problem (IHTCP), considers only the governing equations in the solid side of the convective heat transfer problem. An application considered is the plunging of a hot solid in a cooling bath which also simulates many practical applica- tions with sudden imposition of time-dependent convective boundary conditions. The developed methods are directly applicable to a sphere quenched in a liquid. The IHTCP is an ill-posed problem and cannot be solved effi- ciently without some information about the physics of the problem and the structure of the desired solution. The sequential future- information (SFI) method of Beck was used for the solution of the IHTCP. The parameters describing the heat transfer coefficient function were estimated under the condition of minimizing the residuals between the calculated and measured temperatures in the "future-information" least- squares objective function. The finite-difference (FD) method was used for the discretization of the direct heat conduction problem in hollow spheres. The FD equations were obtained by the finite-control yolume 149 150 (FCV) approach. The Qrank-Hicolson (CN) scheme and the alternating- direction implicit scheme were used for the numerical solution of the one-and two-dimensional direct problem in hollow spheres, respectively. The one-and two-dimensional nonlinear IHTCP is treated for the case of quenching a solid in a fluid. The iterative SFI method is developed for the solution of the nonlinear one-dimensional IHTCP. Numerical experiments were conducted for the systematic investigation of the developed SFI method. A FORTRAN inverse code NUMER2 was‘written to study the IHTCP test cases. A direct problem solver NUMERl was written to solve the transient heat conduction problem in a hollow sphere. This program implements the CN scheme and is used as a subroutine for the main inverse code. Numerical results are presented for h(t) test cases which simulate a sudden imposition of large and sharp variations in convective boundary conditions. Careful analysis of the numerical results show that the SFI method is accurate and efficient. Two inverse procedures for the estimation of h(t) in one- dimensional problems were investigated: the direct and quotient procedures. In the direct procedure, the solution is accomplished with. the direct solution of the IHTCP for the unknown heat transfer coeffi- cient. In the quotient procedure, the solution of the IHTCP is reduced to the solution of the corresponding inverse heat conduction problem (IHCP). The surface heat flux and surface temperature were individually estimated and the heat transfer coefficient was determined from the quotient q/AT. The performance of the two approachs was compared for the estimation of a known square-pulse heat transfer coefficient. The results show that the direct approach is more appropriate for the inves- tigation of the IHTCP with rapidly changing convective boundary conditions especially for cases which permit large time steps. 151 The two-dimensional nonlinear IHTCP for the case of quenching a solid in a fluid is treated using the SFI method. The two-dimensional SFI method is described for the estimation of the time-and-space depend- ent heat transfer coefficients, h(t,0), when the transient temperature histories at appropriate interior points inside the heat conducting solid are available along with the bulk temperature of the fluid. An inverse code NLINVl is developed for the numerical investigation of selected test cases using the two-dimensional SFI method. The direct problem solver is NUMTWO which uses the ADI scheme. The numerical results show that the SFI method is accurate, efficient, and capable of handling abrupt and large changes (in time and space) in the heat transfer coefficients. It should be mentioned that the inverse codes NUMER2 and NLINV2 can be used for flat plate geometries by setting Rout/Ring l in the direct problem solvers. This can be done since hollow sphere geometry approachs flat plate for Rout and Rin approach large values. The developed methods are used for the analysis of the experimen- tal results from the quenching experiments with hollow spheres. Several quenching experiments were performed using either ethylene glycol or water. The analysis is carried out for an approximate one-dimensional model configuration of the quenching experiments and for the more realistic two-dimensional configuration. The thermocouple temperatures were acquired with a multi-channel data acquisition system. The system hardware is based in the PDP 11/03 microcomputer which is interfaced with a signal conditioning unit, an analog-to-digital unit, and a real- time clock/counter unit. Programs necessary to initiate and control different aspects of temperature collection and processing are developed using a real-time package DTLIB/RT provided by Data Translation, Inc. [1981b]. 152 The estimated heat transfer coefficient functional form in the one-dimensional experiments has a large early-time transient followed by a distinct change of the slope which corresponds to the establishment of free convection. The same trend occurs for the two-dimensional con- figuration of the quenching experiments. Furthermore, the two- dimensional results show that the heat transfer coefficient around the sphere undergoes a substantial angular variation. (The angle is the azimuthal angle which varies from 0 to It radians). The early-time values of the obtained experimental results using the SFI method are 60% - 80% higher than the results obtained using the well-known quasi-steady state empirical convection correlations. After early transients, the obtained results correlate with high degree of accuracy with the empiri- cal correlations results. The following conclusions are drawn from the results of the analytical investigation and experimental validation of the SFI methods. 1. Accurate, efficient, and easy-to-implement SFI methods are described for the estimation of the time or time-and-space dependent heat transfer coefficients. The SFI methods require transient tempera- ture histories at appropriate interior points inside the heat conducting solid and the bulk temperature of the fluid. 2. The SFI estimation procedure described here for a spherical geometry is quite general and can be extended easily to cylindrical and rectangular coordinate systems, as well as to three-dimensional problems . 153 The direct estimation procedure for treating the IHTCP is more appropriate for the estimation of transient heat transfer coeffi- cient in convective heat transfer applications with abrupt and large temperature variations in the convective boundary conditions. The experimentally estimated results were compared with the well- known quasi-steady state empirical correlations for free convection. The obtained early-time transients (immediately after the immersion of the sphere in the fluid) values are about 60%-80% higher than the values predicted by the empirical correlations. For the rest of the time, the obtained data correlates well with the empirical rela- tions. The conclusion is that the usually-used quasi-steady state values are not accurate for early transients and may lead to large errors in the thermal analysis for early times. The developed transient inverse technique has the capability of estimating the early transients and the subsequent quasi-steady and spatial variations of the heat transfer coefficient function in a single transient experiment. The following are some recommendations for future work. The SFI method for solving the IHTCP is recommended for the ex- perimental estimation of the transient heat transfer coefficients in heat transfer applications such as quenching processes with change of phase, forced convection over impulsively started solid bodies, and short duration wind tunnel experiments. 154 The estimated results for the transient heat transfer coefficients in the present quenching experiments have not been previously studied in the literature. Early transients in time-dependent heat transfer coefficients should be studied in other complex geometries such as gas turbine blades and disks. APPENDIX A APPENDIXHA NUMERICAL SOLUTION OF THE 2-D DIRECT PROBLEM A,1 Introduction The numerical solutiOn of the transient 2-D direct heat problem in a hollow sphere by using FD (finite difference) method is presented here. The formulation of the FD equations is based on the FCV (finite control xolume) approach (see, e.g., Patankar [1980]). As mentioned in Section 5.3, the solution of the IHTCP by SFI method requires the solu- tion of the associated direct heat conduction problem. This solution provides approximate values for the model temperature function and the sensitivity coefficients. The governing equations of the continuous direct heat conduction problem are given by Eqs. (5.1) - (5.5). The time variable,t, in the statement of IHTCP is replaced by r in the direct problem for con— venience. The convective heat transfer coefficient h(0,r) is assumed to be known and it is required to determine the transient temperature * * , history T(r ’op’fn)’ where (r ,6“) are the coordinates of the tempera- ture sensors inside the body of the hollow sphere and n is the time index. 155 156 .A.2 Derivation of FCV Equations Approximate FCV equations for the direct problem are obtained by discretizing the spatial solution domain, Ds:{r€[Rin’Rout]’ 0e[0,« ] } , (A.la) into mesh points having the discretization coordinates (ri,flj), with r.- iAr + R. , i-O,l,2,...,I, and 0.- jAfl., j - O,l,2,...,J. The Ar and i in J J A0 are, respectively, given by Rout - Rin « Ar - I and A0j- ‘3—' (A.lb) The temperature value at a spatial grid point (ri,0j) is denoted by TJ.- j' The computational grid points and the corresponding finite con— trol volumes are shown in Figure A.l. Control volume surfaces in Figure A.l are located midway between the grid points. The integral form of conservation of energy applied to a control volume element surrounding any node is given by [I - q;n dA - JJJ pc 3: dV (A-Z) c.s c.v at where the left-hand integral is carried over the control volume surfaces while the right-hand integral is carried over the control volume ele- ment. To obtain the semi-discrete equations a simple energy balance, 157 [I . 4% 45‘ \‘ .x‘ as; § § I i j-J Figure A.l Computational grid points and the corresponding finite -eontrol volumes for the two-dimensional IHTCP. 158 using Eq. (A.l), is carried over each control volume surrounding the grid points. A.2.1 FCV Equation.for a Typical Interior Node (i,j) A typical interior node (i,j) and the associated FCV are shown in Figure A.2. The simplest possible approximation of Eq. (A.2) is the lumped approximation given by 1. q A. + q A - q.+ A + - q + A.+ - p c V j (A.3) where the q’s are the mean values over the FCV surfaces, and are given by T - T T - T 1-1, 1, 1. 1+1,' q_-k J .1, q+-k J J ’ i Ar i Ar T - T. . T - T . q - _ k i,j-l 1,3 ’ q + _ k i,j i,J+1 . (A.4) j ri A9 j ri A9 The exact expression for the A’s and V are given by A - 2 n r 2 (cos 0 - cos 0 ) i- i-l/2 j-l/2 j+l/2 A - 2 n r 2 (cos 9 - cos 0 ) i+ i+l/2 j-1/2 j+1/2 A._ - 2 n ri Ar Sln 0j_1/2 J Aj+ - 2 n ri Ar Sln 0j+l/2 V - 2 fl rz A0 Ar (A.5) 159 Interior FCV element Centriod Figure A.2 A typical interior node (i,j) and the associated heat fluxes. 160 where rz is the distance from the z-axis to the centroid of the control volume element: 2 2 r 2 (r1+1/2 ' ri:l[2 )(°°s 9 1-1/2_' °°S 9 i+1/2) (A 6) z 3 A0 ( r2 2 1+1/2 ' r1.1/2) substituting Eqs. (A.4) - (A.6) into Eq. (A.3) and simplifying yields 3 a r2 i-l/2 ( T . _ T. . ) Ar (r3 -r3 ) i-l,3 1,3 i+1/2 i-l/2 2 3 “ r1+1/2 - (T..-T. .) Ar (r3 .-r3 ) 1,3 1+l,3 i+l/2 i-l/2 a sin 01-1/2 + - 2 (T1 3-1 ' T1 3) r2 r1 (A0) - a sin 014-1/2 (T - T ) - dTi . (A 7) 2 1,3 i,j+1 dt ° r2 r1 (A0) which is the FCV equation (semi-discreate equation) for a typical inte- rior node (i,j). An equation of the above form holds for each of the (I-l)x(J-l) interior grid-points. A.2.2 FCV Equation for Convective Boundary Node (I,j) An arbitrary convective boundary node (1,3), O rz ri (A0) n+1 n+1/2 _ a 51“ 93+1/2 ( Tn+1 _ Tn+l ) _ Ti.i - Ti.1 (A 20) 2 1,3 i,3+1 A; ' r2 r1 (A0) 2 Collecting coefficients of similar temperatures, the following five- point difference equation is found: 171 a Ar sin 0j_1/2 2 r2 ri(A0) 1,3 r ri(A0) r ri(A9) “ A' 51“ ”i+1/2 n+1 + 2 Ti 3+1 rz ri(A0) 3 a Ar r2 _ i-1/2 Tn+1/2 Ar (r3 - r3 ) i'l’j i+l/2 i-l/2 2 2 2 3 a Ar ri-1/2 3 a Ar ri+l/2 - - - T Ar (r3 - r3 ) Ar (r3 - r3 ) i+1/2 i-l/2 i+1/2 i-l/2 3 0 Ar r2 i+1/2 n+1/2 + 3 3 Ti+l ° Ar (r ’3 1+1/2 ' r1-1/2> Eq. (A.21) holds for each of the (I-l)x(J-l) interior nodes. n+1/2 (A.21) Applying the ADI scheme to each of the semi-discrete FCV equa- tions for the boundary nodes, the following i-th tridiagonal system of grid-equations for the second half time-step is obtained: + n+1 + n+1 n+1/2 ( 2 + 10 ) T ’0 - A9 Ti,1 DO n+1 n+1 n+1 n+1/2 - A0 Ti,3-l + ( 2 + A + A6 ) Ti,3 - 0 Ti,3+l DJ - n+1 - n+1 n+1/2 - A0 Ti,J-l + ( 2 + A9 ) Ti,J DJ 00000000 00000000 172 n+1/2 :1 where the quantities D , 3-0,l,...,J, are defined as: a)For 0 < i < I n+1/2 _ - n+1/2 _ - _ + n+1/2 + n+1/2 ._ Dj Ar Ti-l,j + ( 2 Ar Ar ) Ti,3 + Ar Ti+1,j’ 3 0,1,.. b) For i-O n+1/2 _ _ + n+1/2 + n+1/2 ._ Dj ( 2 Ar ) T0,j + Ar T1,j , 3 0,1,... c) For i-I n+1/2 _ - n+1/2 _ - _ * n+1/2 A; n+1/2 Dj Ar $1-1 j + ( 2 Ar Ar 31j * ) TI’j + A: H“+1/2( 9% )T:+1/2, 3-0,1,...,J. L . . * - + The quantities A , A , A r r r 0 A,4 Solution of ADI Equations , A5, and A+ are defined before by Eq. (A.18). For the first half-time step, the 3-gh system of equations given by Eq. (A.16) (one system of equations for every 3, 3=0,l,2,...,J) can be expressed in matrix form as -A T n+1/2 . n+1/2 C T n+1/2 1 1-1 + Bi T1 ' i 1+1 1 i=0,1,...,l, (A.24) 173 where the subscripts j are omitted for convenience. The coefficients n i are defined by Eq. (A.16). A., B., and C.; and the vector D 1 1 1 The matrix of coefficients Ai’ B and C1 in Eq. (A.24) is i! diagonally dominant and the set of equations is readily solved by Thomas algorithm using the recursion relations, 50 ' B0 A 0 pl - Bi - .15—1:1’ i-1,2,....,I 1-1 H _ 3Q 10 30 n D - A. 1 _ 7 - i L i 1, i-l,2,...,I. (A.25) 1 01 Finally, the solution of the system of equations, Eq. (A.24), is given by n+1/2 TI 71 C Tn+1/2 T?+l/2 - 71 - -l——lil——-, i-I-1,I—2,...,l,0. (A.26) 51 n+1/2 The computation for T1 advances from the outer surface to the inner surface of the hollow sphere transforming the convective boundary condition information to the interior of the sphere. Eq. (A.24) is solved for all j, 3-0,l,2,...,J, until all the grid temperatures at the (n+1/2)-§h time level are obtained. 174 The computations in the second phase, from (n+l)-§h to (n+l)-§h, is performed is a similar manner. The computations advances from the bottom of the sphere, j-J, to the top of the sphere, 3-0. The system of equations (A.24) is solved for all i, i-0,l,2,...,I, until all the grid temperatures at the (n+l)-§h time level are obtained. LIST OF REFERENCES LIST OF REFERENCES Alifanov, 0.M. [1974] , "Solution of Inverse Heat-Conduction Problems by Iteration Methods," 4, of Eng. Phy,, Vol. 26, No. 4, PP.*682-689. Alifanov, 0.M. and Kerov, N.V. 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