I: Ill I ‘ " l -ll‘ I'I \ll \lllllllalllllllzlllll 10111658136 will \2 .7 ‘ -- «Mg ,, is c 1.1811115. RY M cmgan Sm ‘ ‘menity This is to certify that the thesis entitled A NUMERICAL HEAT TRANSFER ANALYSIS OF A CRYOMICROSCOPE CONDUCTION STAGE presented by Vorapot Khompis has been accepted towards fulfillment of the requirements for "aSter'S—dggree in Mechanical Engineering We (2 fimufiwg’ Major professor Date/$24 .‘,//7X§ 0-7639 ate-1W“ II ‘VI. 1‘ "~.n‘ OVERDUE FINES: 25¢ per day per item W Place in book return to remove charge from circulation records A NUMERICAL HEAT TRANSFER ANALYSIS OF A CRYOMICROSCOPE CONDUCTION STAGE By Vorapot Khompis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1980 Rem eke ABSTRACT A NUMERICAL HEAT TRANSFER.ANALYSIS OF A CRYOMICROSCOPE CONDUCTION STAGE By Vorapot Khompis Numerical heat transfer analyses of a Cryomicroscope Conduction Stage have been performed for the two-dimensional steady-state and transient cases. The Gauss-Seidel finite difference method has been applied for the steady-state solution and the explicit finite difference method has been applied for the transient solution. Both computer programs are written in general form to include variable boundary conditions, non-uniform grid size, and composite materials. Experimental tests were performed on an existing Cryomicroscope Conduction Stage and compared with computer predictions. The computer models yielded results accurate to within 5% of the actual temperatures observed. Potential modifications for improvement of the Cryomicroscope Conduction Stage design were considered by performing a parametric study using the computer model developed. Factors studied included: a) variable convection boundary condition; b) heater dissipation area; c) heater thickness; d) heater material (quartz versus glass). The work has demonstrated that the Cryomicroscope Conduction Stage performance is mostly effected by a change in the natural convective heat transfer coefficient of the surrounding air. The heater dissipation Vorapot Khompis area plays a major role in the Cryomicroscope Conduction Stage perfor- mance. In general, modifications yielding an isothermal temperature will yield a slower transient response for the Cryomicroscope Conduction Stage. Therefore an ideal design is impossible and a choice must be made between the desirable characteristics of uniform temperature and rapid response. To - My Father, My Mother, Teachers and the Royal Thai Air Force ii ACKNOWLEDGEMENTS To all the individuals who have helped and encourage me during the research and writing of this dissertation, I offer my sincere thanks. Foremost appreciation is extended to my thesis advisor, Dr. John J. McGrath, who supplied the original motivation and continued to give generously of his time and valuable consultations during the course of this work. Other thesis guidance committee members, Drs. Ronald C. Rosen- berg, James V. Beck, and John R. Thome, are also warmly thanked for their concern and interest in this work. A special note of thanks is due to my academic advisor, Dr. Ronald C. Rosenberg, for his guidance and numerous suggestions throughout my study. Grateful thanks are extended to the Royal Thai Air Force for providing funds to finance my graduate program. TABLE OF CONTENTS LIST or TABLES .......................... iii LIST OF FIGURES .......................... vi NOMENCLATURE ........................... ix SUBSCRIPT NOMENCLATURE ...................... x GREEK NOMENCLATURE ........................ xi Chapter 1. INTRODUCTION ........................ l The Cryomicroscope Conduction Stage ........... l 2. CONDUCTION STAGE DESIGN CONSIDERATIONS. . ....... . . 7 3. THE THEORY OF FINITE DIFFERENCE ...... . ....... l0 3.l Introduction to the Finite Difference Technique. . . l0 3.2 Mathematical Model and Governing Equations ..... ll 3.3 Two-Dimensional Steady-State Conduction.‘ ...... 14 3.4 Two-Dimensional Transient Conduction ........ 23 3.5 Finite Difference Formulation in Explicit Method . . 26 4. APPLICATION OF THE FINITE DIFFERENCE TECHNIQUE TO HEAT TRANSFER OF THE CRYOMICROSCOPE CONDUCTION STAGE ..... 34 4.l Assumptions ..................... 34 4.2 Numerical Analysis for Two-Dimensional Steady State Conduction .................... 37 4.3 Numerical Analysis for Two-Dimensional Transient Conduction .................... 42 5. EXPERIMENTAL RESULTS FOR THE CRYOMICROSCOPE CONDUCTION STAGE .......................... 52 5.1 Description of the Experimental Equipment ...... 52 5.2 Experimental Technique for Steady State Conduction . 55 ' 5.3 Experimental Technique from Transient Conduction . . 56 6. COMPARISON OF THE COMPUTER MODEL RESULTS WITH EXPERIMENTAL RESULTS ......................... 61 iv 7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ......... 69 7.1 Conclusions ...................... 69 7.2 Suggestions for Future Work .............. 82 APPENDICES A: Nodal Equations for Two-Dimensional Steady-State Conduction . 84 B Nodal Equations for Two-Dimensional Transient Conduction. . . 88 C: Assumptions of the Cryomicroscope Conduction Stage ...... 9l D The Convective Heat Transfer Coefficient for the Refrigerant and Surrounding Air .................... 93 REFERENCES ............................. 96 h-h (JON 00000) (JON phwwwww (”UT-9% (”Nam-h LIST OF FIGURES Top and Cross-sectional View of the Copper Stage ....... Half Sectional View of the Cryomicroscope Conduction Stage . . The Conduction Stage ..................... Differential Element ..................... Interior Node and Grid Network ................ The Example of Different Locations of Nodes to be Used in Node Equations for the Cryomicroscope Conduction Stage . . . Interior Node for Composite Materials ............. Surface Node ......................... Corner Node .......................... Interior Corner Node for Composite Materials ......... Space-Time Grid Network .................... Three-Dimensional Conduction Stage Temperature Profile . . . . Approximation Grid Network of the Cryomicroscope Conduction Stage ............................ Flow Diagram for the Steady-State Computer Program ...... Two-Dimensional Steady-State Conduction of the Cryomicroscope Conduction Stage ...................... Flow Diagram for the Transient Conduction Program ....... Computer Modeling Results for the Transient Conduction . . . . Experimental Schematic of the Cryomicroscope Conduction Stage. Temperature Measurement with the Thermocouples at Various Radial Positions ...................... The Change of Temperature with Time Plotted from the Strip Chart Recorder ....................... vi 2 4 5 l2 15 T8 20 20 22 22 28 35 36 39 43 48 5O 53 54 58 Comparison of Non-Dimensional Temperature Between the Computer Modeling Results and the Experimental Results for the Steady-State Conduction of the Cryomicroscope Conduction Stage ...................... 62 Comparison of the Change of Temperature with Time at the Centerline ............. . ........... 63 Comparison of the Change of Temperature with Time at = 0.047 in. from the Centerline; T = T2 .......... 64 Comparison of the Change of Temperature with Time at = 0.08 in . from the Centerline; T 2 T3 .......... 65 Comparison of the Change of Temperature with Time at = 0.18 in. from the Centerline; T = T4 .......... 66 Comparison of the Change of Temperature with Time at = 0.24 in. from the Centerline; T = T5 .......... 67 Typical Heater Dimensions ................... 71 The Temperature Difference Between Nodes (T -T ) by Fixing the Diameters, Materials and Thickness ans Vgrying the Width W ........................... 72 Effects of the Potential Cryomicroscope Conduction Stage . Modification on Transient Response ............. 74 The Temperature Difference Between Nodes (TC -T ) by Fixing the Diameters, Materials, the Width W andCV aPying the Thickness .......................... 76 The Temperature Difference Between Nodes (Tc- T) by Changing Materials from Quartz to Glass ............... 78 Comparison of the Change of Temeprature with Time by Changing Materials from Quartz to Glass ............... 79 Comparison of Non-dimensional Temperature Distribution on the Top Surface of the Cryomicroscope Conduction Stage . . . 80 Comparison of Non-dimensional Temperature Distribution on the Top Surface of the Cryomicroscope Conduction Stage . . . 8l vii 4.1 4.2 5.1 5.2 7.1 7.2 7.3 LIST OF TABLES Variables for the Steady-State Conduction Program ...... Variables for the Transient Conduction Program ........ Numerical Experimental Results for Steady-State Conduction of the Cryomicroscope Conduction Stage ........... Numerical Experimental Results for Transient Conduction (at the Top Surface) of the Cryomicroscope Conduction Stage The Temperature Differences Expected Between Nodes (Tc-TD) by Varying Width W ........... ‘ .......... The Temperature Difference Expected Between Nodes (T -T ) by Varying the Quartz Thickness .............. The Temperature Differences Expected by Changing the Materials from Quartz to Glass .................... viii 4T 47 57 59 7T 75 76 Bi alazorzflfi-I-non 0 21:1: Nl,N2,N' Nu p Pr —1 —1 d. -<|><|<|<-o NOMENCLATURE Area Biot number Specific heat at cOnstant pressure Hydraulic diameter Convergence criterion Fourier number Average convective heat transfer coefficient Thermal conductivity Length Proportional of AX and AY Mass Non-dimensional heat generation Nusselt number Power Prandtl number Uniform heat generation Thermal resistance Electrical resistance Reynolds number Temperature Initial temperature Time Thickness Internal energy Velocity Voltage Non-dimensional length in X—direction Non—dimensional length in Y-direction Pressure difference Grid size in X-direction Grid size in Y-direction ix -'° m U 0 on X -I W T" Li- SUBSCRIPT NOMENCLATURE Boundary or refrigerant Centerline Viewing hole End surface Increment in a spatial variable in Y-direction Increment in a spatial variable in X-direction Left surface Right surface Top surface X-direction of rectangular coordinate Y-direction of rectangular coordinate Z-direction of rectangular coordinate SUPERSCRIPT NOMENCLATURE Time interval GREEK NOMENCLATURE Thermal diffusivity. Non-dimensional temperature Density Dynamic viscosity Angular distance lO'6 meter xi CHAPTER I INTRODUCTION Cryomicroscope is a tool designed to yield visual evidence of the changes that occur in and around cells (or small samples of living material) as they are frozen and thawed. The Cryomicroscope Conduction Stage The conduction stage under consideration was designed by John J. McGrath as a modification of the Diller-Cravalho conduction stage. (M.S.ME. Thesis, MIT. Feb, l974) The purpose of this modification was to allow fluorescence optics to be employed as a means of detecting of cell viability on the microscope. The top and side view of the copper conduction stage are shown in Figure l.l. The design is essentially a sheet of copper with a tiny viewing hole in the center. The top surface of the stage is completely flat for a convenient working area and allows room for auxiliary parts such as clamping or measuring devices that might be added at a later time. The circular heater is attached to the top surface with its center coincident with the center of the viewing hole. The large center hole is cut out of the stage to allow the oil immersion condenser to be positioned close enough to the specimen to be properly focused. Channels are also cut in the bottom of the copper plate for 2 J a. _ _ 1 _ d 1.! R m _ n a.“ _ o . . e5 nun 5.“ ' F... 5.. 1. _ H _ - vi. 1 1 Av . a a a .4 w... II... a * 4' n. 111 .H. . u u.nw . “HUN .5, x. M m .... .. \ r:. . \\\e|\\\ 0 Ii .0 0“, I III o4 .& ... .s x . \4» I, / ,o 3 .. . a x a .. . u. u ‘5 m a L. n . 11 . . .. . ./ “x. .. . o \ s I ail III I ‘\ \\s \s gs or IIIIIIUI tun/MA \ss. elicit! fi l\\ \\ IOOfiOeu-I 1:. In.“ . . I. a 1 .\\\‘\\\\\\\\\‘ Top and Cross-sectional View of the Copper Stage. Figure l.l. refrigerant to flow circumferentially around the heater. The refrigerant enters and exits the stage through the elbows soldered through the top. A plug is inserted in the channel between the entrance and exit elbows to prevent short-circuit flows.. A bottom plate with a center condenser hole cut out is soldered on the bottom of the stage to seal the bottom of the flow passage. The top surface of the copper is covered with a thin (0.0025 in.) layer of mylar tape to electrically insulate the resistance heater from the stage. Dynaloy silver paint is used as both an adhesive and as an electrical conductor. A layer of this paint between the heater and the tape bonds the heater to the top of the stage. Each side of the heater is attached to the stage in this manner with no paint in the center part of the heater so that the current passes through the resistive coating and dissipates joule heating. The paint layer on each side of the heater is extended laterally beyond the edges of the heater so that electrical leads can be attached. The silver paint attachment is not as permanent as epoxy which was used previously but this is advantageous. New heater configura- tions can be applied and be ready for use within two hours after deciding the previous unit undesirable. Acetone dissolves the paint bond without damaging the old heater which can be saved for future applications. There are also indications that the use of epoxy restrains the heater too rigidly such that thermally induced stresses will crack the heaters. Further details of the Cryomicroscope conduction are discussed in Ref. 1. The work that follows presents a numerical heat transfer analysis Glass Silver epoxy Quartz Copper J— 1 . Polyester 0.0» l ' 0.35 i tape deaf—I II T 0' 024. . : Refrigerant .1— I 0.037' I T |-a——a " .u——:-1 I“ 1.22" —1=-1 | Figure 1.2 Half Sectional View of the Cryomicroscope Conduction Stage. 0t .5 —. u N 2‘. 8 so a: MILDl.l.lTlHllllLLfiIILLmlTlnlHill?“uluml° ' I l llllll‘lll' ir‘lll'l‘ ‘ . I‘llllll l i, l‘llll ‘lllll'llll Figure 1.3 The Conduction Stage. of the Cryomicroscope Conduction Stage designed by Dr. J. J. McGrath. The technique of finite difference is chosen to apply and the computer model is generated to investigate the temperature distribution of the conduction stage for both steady-state and transient situations. Results of the computer simulations are compared to experimental results performed as part of this thesis. Close agreement between the analyti- cal and experimental results indicates that the models are accurate to within 5%. The models have been used to study potential improvements of the cryomicroscope design. Preliminary parametric studies are presented. CHAPTER 2 CONDUCTION STAGE DESIGN CONSIDERATIONS The two most important measurements for the Cryomicroscope Conduction Stage are temperature and its time rate of change. It is important to be able to determine these two variables accurately and in a reproducible fashion. The ideal Cryomicroscope Conduction Stage should be an easily Operated tool for the cryobiologist, which allows him to visually observe small samples of living material and to control precisely the temperature and its time rate of change during observation. On the basis of experience with the present system, the ideal stage should possess the following list of features: 1) The temperature and the rate of change of temperature should be measurable and controllable 2) The operational temperature range should be specified 3) The stage should be capable of being used with the highest possible magnification and the maximum mechanical flexibility 4) It should be of sturdy construction and easily mounted on a standard light microscope 5) It should be designed for convenient experimental procedures One factor that should not be overlooked is that generally, even subminiature thermocouples yield bulk temperature measurements on the cellular scale. This is obvious when one compares the average human red blood cell diameter of 8 u with the thermocouple junction width of 125 u presently used in the system. However, this is not a critical shortcoming since thermodynamic models which use informa- tion generated from the cryomicrosc0pe data will often assume iso- thermal conditions immediately surrounding the cell. Hence an acceptable temperature is the bulk measurement if the thermocouple junction is not placed in a severe thermal gradient. If it is assumed that the bulk measurement is valid for the moment, the question becomes one of locating the thermocouple to eliminate handling of the miniature thermocouple. One of the designs considered involves embedding the thermocouple in either the glass heater or in the top coverslip covering the sample, such that it is displaced from the sample by some vertical distances. The designer must be extremely careful in selecting the path over which the thermocouple leads are routed from the point of tempera— ture measurement at the junction to the point where ambient temperature exists on the lead wires. The thermocouple leads act as fins, con- ducting along their length when placed in a thermal gradient. Many times during the course of the experiments the thermocouple was obser- ved to be in error depending on the routing of the leads from the site of temperature measurement. This error was noted by microscopically observing the phase change during ice formation in a sample around the thermocouple junction. The solution to this problem is to remove the leads as much as possible from the areas of steepest gradients and to reduce the cross- sectional area of the leads so that longitudinal conduction is minimized. For the conduction stage a foil copper-constantan thermocouple junction is placed at the center of the viewing hole in the copper plate. This places the junction at the very center of the stage. This center temperature is the temperature being controlled since this thermocouple is interfaced with the analog control unit. CHAPTER 3 THE THEORY OF FINITE-DIFFERENCES 3.1 Introduction to the Finite Difference Technique The finite difference method is used to approximate differential increments in temperature and space coordinates. The smaller we choose these finite increments, the more closely the true temperature distri- bution will be approximated. The problem of steady-state two-dimensional conduction, which is the first type of problem considered in this thesis, is a special case of the general three-dimensional, transient conduction problem. The conservation of energy principle is applied to a properly chosen differential element, and Fourier's Law of conduction is used to derive the governing equation for the temperature distribution. Since the dependent variable, temperature, is a function of more than one independent variable, we expect the temperature distribution to be specified by a partial differential equation. The resulting partial differential equation, along with associated boundary conditions, forms the mathematical model which predicts the temperature field and heat transfer rates. The solution to this partial differential equation can be obtained by analytical or numerical means. For the present work an analytical solution technique was chosen. IO 11 3.2 Mathematical Model and Governing Equations Consider the governing partial differential equation for three-dimensional transient heat conduction in rectangular coordinates. A differential element describing this case is shown in Figure 3.1. This element, which is a closed thermodynamic system is located entirely with- in a solid material for the present work. Thus, the energy crossing the boundaries of this system is a result of conduction due to temperature gradients and internal heat generation. Assuming a stationary, homogeneous, isotropic material, the con- duction of energy across the differential element can be accounted for by applying the First Law of Thermodynamics. For the net heat conduction across each of the orthogonal surfaces normal to the X, Y, Z directions shown in Figure 3.l, we will obtain: . 3 (qx+ M" d")-( +5.39%» “x + dx/2 ’ 0x - dx/2 = ax 2 = _a_q_x_ dx In a similar manner: 34 . _ _ - = _.Y. qy + dy/Z qy - dy/Z 3y dy ._ an dZ q2 + dz/2 ’ 92 - dz/2 ' 32 For the heat generation rate within the differential system, the internal heat generation per unit volume Q is multiplied by the differen- tial volume dxdydz. The internal heat generation is treated as a uni- form source from which energy enters the system. The conservation of energy principle requires that the internal heat generation rate equal the net heat transfer rate crossing the surfaces 12 T=T(X,Y,Z) /qz+d2/ 2 q'x-dx/Z qz-dz/z qy-dy/ 2 Figure 3.1 Differential Element. 13 by conduction plus the rate of change of internal energy within the system. The rate of change of internal energy can be expressed as dUYdt = mcp(dT/dt)==dedydch(aT/at). Thus we can write q q q = 3.5. £31. 3.l. .31 dedydz ax dx + 3y dy + 32 dz + pdxdydsz at Using Fourier's Law of Conduction to relate heat flux and temperatures: qx -kAx(aT/aX) where Ax dydz Also, qy = -kdzdx (aT/aY) qz Then we can obtain the Energy Equation for Conduction which -kdxdy(aT/ 32) applies to the present case: §_. §1_ a gl_ 3 31. = BI. - 3x(k3X I Sylkav) + Silkaz) I Q pcPat (3 I) If we assume constant thermal conductivity and two-dimensional conduction in the X and Y directions, the energy equation reduces to 2 2 3 T .§;I = 31. - “(5;2'I 3Y2) I pcP at (3 2) where a is the thermal diffusivity: a = k/pCi,“ For steady-state, two-dimensional conduction the temperature dis- tribution is not a function of time, and the governing equation can be written as: '14 2 2 _a_'2T_+_a_;_+_Q =0. (3'3) 3X BY k 3.3 Two-Dimensional Steady-State Conduction For a finite difference analysis of two-dimensional conduction, an element of unit depth and dimensions AX and AV on the sides is chosen as the system to which the conservation of energy principle is applied. A node is defined at the center of each finite volume. A tempera— ture is assigned to each node, and it is assumed that this one value gives the temperature of the entire element. The sum of finite elements forms a grid network as shown in Figure 3.2(a). An interior, finite element of unit depth with width AX and height AV is shown shaded in Figure 3.2(a). The curved line represents the temperature variation through the solid material. In a finite difference formulation the temperature gradient is calculated as though a linear profile exists between nodes as shown in Figure 3.2(b). At the same time, the nodal representation of the temperature field implies a linear change in temperature across the interface between two adjacent elements. This formulation is obviously more accurate as the distance between nodes decreases. The principle of conservation of energy for steady state heat transfer is now applied to an interior node. The temperature gradient between nodes is assumed constant. For the heat transfer crossing the boundaries of the system using Fourier's Law , _ 3T qx ' 'kAx'SX' 15 (a) 1.3‘ 7 x «C. s ,//./d. .. / ’37 $6,... /// a /’ l 7.x, f/ I ,///. 7,13,, I ’3" .LI, FCL V’LV i-l .- iv |V_ 1+1 3+1 j-l (b) Interior Node and Grid Network. Figure 3.2 16 If the temperature of the central node T4 4 is larger than T4 4_4. the heat transfer between nodes i, j and i,j-l is approximated as follows: _T.. qx = kAY(]) Ti21:] Ax 193 If one uses similar equations for the other three surrounding nodes and accounts for internal heat generation, application of the conservation of energy principle will lead to the following equation: 14,444 - T1,: AX T- ._ _ T- . T-_ . _ T- . kAY(l) "3 1Ax l’3 + kAX(l) ‘ 1’44 1’3 + kAY(l) T- . T- . + kAX(l) 1+‘:JA; l’3 + QAXAY(l) = The application considered in this thesis required cases where AX and AV were unequal. Hence a general formulation is developed where AY = mAX. k mk(T4 . _4 T4 4) + —(T4_ 4 4T 4) + mk(T4 444 T4 4) k 2 - +—(T444 4- T444) + mQ(AX) m2(T - T. .)+ (1. -1- ) + m2(T - 1 ) isj‘] 19j 1' 1 sj isj isj+1 isj m2 + (1.4.4.1 ,j ' Tinj) + F Q(AX)2 (2 2 + 2)T - 44424,. + sz + T + T + '-“—ZQ(AX)2 m 1,3 i,j-l i,j+l i-l,j i+l,j k The following variables are defined in order to non-dimensionalize the governing equatibns: 17 0= 8 Tc'TB V=l 2 7:1 IL where TC’ TB, and 2 are chosen reference temperatures and a reference length, respectively. Then the above equation can be written as ‘ 2 2 _ 2 2 2 2 t ‘2'" I “91.: ‘ "‘ Gui-1 " "‘ 94,444 *9 14.3” 9141.: + '" (79:) k Tc-TB) Define: ”‘ = H4944 ‘ 2 023 where N1 is a non-dimensional heat generation term. The numerator is a heat transfer rate resulting from internal generation in a reference volume 13. The dominatoris a reference heat 2 conduction term across a reference area 2 and along a reference tempera- ture gradient (Tc7T3)/t. Thus Nl can be thought of as a ratio of energy internally generated to energy conducted. Thus, the final form of the finite difference equation fbr all interior nodes for steady-state, two dimensional conduction becomes 2 2 4 2 9 = m Gigi-l + m ei,j+l-+91-1’j.494+1’j + NZFEX) NI . 2 (3'4) 1:3 (2m + 2) It is necessary to use the different node equations for the different locations of node in the Cryomicroscope Conduction Stage as shown in Figure 3.3 (details in Figure 3.1 through Figure 3.7). l8 .mmmum cowuuaccou maoumocowaozcu mg» cow mcowpmacm ovoz cw won: on op mmuoz mo mcowuaoog acmcmwero mo mpasmxm ugh m.m wcamvu Aampc.c;\=pm omuxv caaaou Q @ sh J: m eh em Adeflfimau-m n-1w. Ads“ 5; 3-}: .u m we.-. . ." .m..qm:\us_ M_an\m«#T ..o....ml.i.. ._r\ \1... burn... . .. . __ _.L .. . 1 . . _ // "mrvwm__ .u _” n . . ”u .m . . .xv.. m ”We... _ u m Tomes: m ”4.. 3.: n .2” . . 4.. 0.5322 My . my} ” Z 19....«Hfl: . e W \ \ mu...P a. . mpawcmpmz muwmoaeou mg» mo Ago up muoz coccou cowcmucm .cg\yum ¢¢¢.ouxv Nbcazo . muoz .cwcuou muoz mumycam mmecmumz ouwmoasou mga mo muoz cowcmucu 669636369 muoz caveman” 19 In the same manner (details in Appendix A), the interior node equation for composite materials is given as: = 91+] ’j + ei-l 21 + 21112(k161 “j" + k2 ei,j+1j1k3 + 'm2(fi)2N2/k3 ei.j (2m2+2) (3-5) where kl = The thermal conductivity of material 1. k2 = The thermal conductivity of material 2. k3 = k1 + k2. In convection boundary conditions, for the general situation, we consider the environment temperatures at the different sides of the Cryomicroscope Conduction Stage are not equal to the refrigerant temperature TB(Tm,Lf Tm’R f T444,E f Tm’T f TB). We also con51der the Biot number 318 f BiLf 81R f BiE f BiT. The node equation for convection at the right vertical surface: . 2 , 2-—-. 2-—— 2 e = ei+1.jI 2'” OiJ-l I 91-1.3' I 2‘” AXBIROR I '“ (Ax) "‘ (3444 i.j ___ (2m2 + 2 + 2m2AXBiR) The node equation for convection at the left vertical surface: 2 2——-. 2 ——-2 e . ._ 9mg I 2'“ 91.31" 91-1.:‘I 2m AXBILGL I "‘ (AX) ”I (344 1.3 ___ (2m2 + 2 + 2m2 AX BiL) where: Bi = Biot Number = hi/k. The node equation for convection at the top surface: For the left corner node 4 0' + mze + mBi AXO + szi AYO + 93(AX)2N1 0 = i+l,j i,j+l T T L L 2 (3-8) . . 2 “5' (m2+1 + "1314347 + m 314m 20 R T.m 0 :k2 \. 4w 41 .1 am .1 T....l VI R nnTIIIAHIIIW. _\\ rn H o o 10‘ 1* 4 ¢ .J .III/IIII I III IIHIIII IIII I.“ A a . 2 III/III IIJ X , u. I . ”(IOIVI I UNIIIIIIK M A” :Amcc.z .o\y _\\\ \ \s \\\\\\ ‘0. * Interior Node for Composite Materials j-1 i-l T AX I 1+1 Figure 3 4 1- 74* AY. L 1+1 k1' Figure 3.5 Surface Node 21 For the right corner node 2 2 . ——- 2 . ——- m -—-2 + O. 4 = 0444 j m 04 .24 + mB1TAXeT + m Bil-AxeL + §~(AX) N1 (3-9) ‘13 (m2+1 + mBiTAx + sziLAX) For the remaining top surface nodes 2 2 m . m -—-2 + ——- + - + X + ——{ "3 (m2+l + mBiTAX) The node equation for convection at the end surface: For the left corner node 2 2 . 2 . m ——-2 O: = O4_4 j + m 04,444 + mBiEAXOE + m BiLAXOL + 2 (AX) Nl (3-14) 1.1 ___ ___ (m2+l + mBiE Ax + sziL AX) For the right corner 2 . ——- 2 . ——- m2 ——-2 oi_4,4 + m 04 4_4 + mBlEAXOE + m BTRAXGR + 7—(Ax) N1 (3-12) 9. . = I I’3 (m2+1 + mBiEAX + szikZX) For the remaining end surface nodes 9 + m3-(e +0 ) + mBi are + EEIAX)2N1 O. . = i-1,j 2 i,j+1 i,j-l E E 2 (3_43) "3 (m?+ 1 + mBiEAX) where 0L = (T L - TB)/(TC - TB),OR w, (Tw.R ‘ TBIIITC ' TB) 0 = (Tm,T ' TB)/(TC ' TB)’GE 1 (Te E ‘ TBI/(Tc ‘ TB)' The node equations for convection at an interior corner node with composite materials are given as: For the top right interior corner node -= 2” k491.14 1-1.j 1,3 ( (2m2+1)k4 + m + k4e + (m2+m)BiAXoR + m2k50. +9 1.j:J 1+1.j I 2 O 2 .r—- k5 + (m +m)B1 Ax + 1) (3-14) 22 (€- «a—AX—a’ 3‘ V 1 \N \ ‘\ \ ‘1 P ‘1 \V C \ 4\ CA \9 Figure 3.6 Corner Node k1 0‘\N: 3L LINE-I FL Xe o_ u ><. LF< 37 Conduction Stage that will be used to approximate the temperature and heat transfer with the finite difference method. The quartz on the top of the copper plate has a radius of 0.64 in. and is 0.0l6 in. thick. The copper plate is 1.28 in. long and 0.024 in. thick with a viewing hole radius of 0.08 in. at the centerline. All of the surfaces in Figure 4.2 are in contact with air where h'= l.06 Btu/hr ft2 0 F (see Appendix 0-2). However, at a dis- tance of 0.64 in. from the bottom end of the copper plate, the plate is cooled by the refrigerant (air) at 20 psig, 89°F with 5': 36.85 Btu/ hr ft2 oF (see Appendix D-l) and the centerline grids are at an adiabatic surface. The heat is assumed to be generated between the interface of quartz and copper within a distance of 0.24 in. from the centerline. It is assumed to dissipate a power of 5 watts, which corresponds to a heat generation of 8.5 X 105 Btu/hr ft3. Assuming AX = 0.008 in. and AY = 0.08 in., 82 nodes are created to approximate the temperature diStribution for both steady- state and transient conduction cases. 4.2 Numerical Analysis for Two-Dimensional Steady State Conduction The solution of two-dimensional steady-state conduction prob- lems requires the determination of the nodal temperatures in the finite difference grid. The temperature of each node must satisfy the appropriate node equation. In order to calculate the temperature at each node in the grid, a scheme must be devised so that calculations at each node are made in a sequential manner by means of the proper 38 nodal equations until all nodal temperature are known. All nodes must be assigned an initial value to begin the calculations. These initial values should be based on known boundary conditions and estimated internal temperatures. Consider a two-dimensional grid network of the conduction stage shown in Figure 4.2. The ambient temperature and the air refrigerant temperature are approximated as 89°F. Assume the initial temperature at the centerline to be T'»90°F. c Then at = TC --T8 = l.0 Tc'TB 93‘ TB'TB_= o T-T c B Thus the non-dimensional temperature distribution will vary from 0 to l. Now the computer model for steady-state conduction developed in Reference 2 will be applied. This computer program is written to solve the temperature distribution in the rectangular region of the Cryomicroscope Conduction stage. The temperature of each node must satisfy the appr0priate nodal equations from Equations (3-4) through (3-l5). The variables and constants have been assigned in order to make the computer modeling simple and easy to use. A list of the variables used in this program is given in Table 4.1. The rows and columns in the computer program are noted by i's and j's, respectively. The computer modeling program for the two-dimensional steady-state case for the Cryomicroscope Conduction Stage is shown in the form of a flow diagram (Figure 4.3). Throughout the calculations, X(I,J) represents @9 Enter data Assign initial control cons ants Assi n ini ial tem erature g d$stri utgon l Let N ‘II’I ' Let U Investigate all nodes and locations l Apply suitable nodal equations II C II C Calculate the difference between nodal values and previous nodal values l Lhas_conve:gence—been achieved? Yes No Let U = U+1 ) Redefine current nodal values as previous nodal values Figure 4.3 Flow Diagram for the Steady-State Computer Program. 40 Let N=N+l Yes N >= M? Yes Is U= 0? No Yes Is P >0? No Print Comments Let P=P+l Pring N,U r Is U f 0? No Print N,U Print nodal values from last iteration ll Print some assign control variables 41 TABLE 4.] Variables for the Steady-State Conduction Program Variables Definition T(I,J) Current nodal temperature X(I,J) Newly calculated nodal temperature Y(I,J) Difference in nodal temperature Z(I,J) Truncated nodal temperature Bl BlL = hLll/kl = 0.118 82 BiE = hEBl/kl = 0.118 B3 BiR = hR22/k2 = 0.002 B4 BiT = hTBZ/kZ = 0.002 BS Biref' = href.£2/k2 = 0.079 B6 Bi = hLBl/(kl+k2) = 0.001 01 H1 = X/Bl DZ 372 = X/BZ El Convergence criterion = 0.001 11, 12 Number of insulated nodes on the left and right surfaces M,N,P Counting variable 2 _ N1 021 /k1(Tc-TB) — 0 N2 sz/(khkzmc-TB) = 24.2 2 _ = N3 022 /k2(TC TB) 0 U Number of nodes-not satisfying con- vergence criterion Xl Number of grid points along 7'= 0 Y1 Number of grid points along Y’= 0 the current calculated values in the nodal matrix, and T(I,J) repre- sents the previously calculated values in the nodal matrix. Y(I,J) is the difference between the current nodal value and the previously calculated nodal value. The convergence of the calculation is indica- ted by the value of Y(I,J). As the calculation proceeds, the difference between the newly calculated nodal values and the previously calculated values becomes smaller, and the correct temperature distribution is approached. The allowable difference is a function of the accuracy required in the calculations. This allowable difference is determined by the programmer and is read into the computer as the variable El(in this case E1 = 0.001). After one complete iteration of all nodes, the value of N is increased by an increment N = N+1 to account for the number of iterations completed. If convergence has not been achieved after a maximum number of iterations (M = 300), provision is made to terminate the iteration procedure. The program concludes by printing Out the final non-dimensional temperature along with parameters. Typical results are shown in graphical form in Figure 4.4 and these will be compared to the experimental data and the results discussed in the next chapter. 4.3 Numerical Analysis for Two-Dimensional Transient Conduction Consider the analysis of transient, two-dimensional conduction of the cryomicroscope Cbnduction Stage with constant properties and internal heat generation terms shown in Figure 4.2. In chapter 3 we obtained the node equations to use in the transient conduction case for various boundary conditions. Recall that in the finite difference 43 .wmmpm :pruaucoo maoumoguwsoxgu mgu mo cowuuzucou mumpm-»ummpm chowmcmswo-ozh ¢.e mgamwm wo.o eN.o me.o om.o -.o mm.o no.F ~._ m~._ Aeo=Fv> o L (I, n w, u +1 w a a 1. . c I m m " mv.- \NV _ 4 ..ta\ 1: \& ID .All 22. 9%; xx 44 approximations to the governing equations, that it is necessary to initially assign arbitrary temperatures to all nodes within the grid under consideration. In the analysis of transient conduction we must accurately specify the initial condition of each node within the grid. There- fore it is not necessary to continually recalculate the initially assigned temperature values to reach the solution. Once the proper step size AX, AY and a time increment At are chosen, we apply the finite difference equations to each nodal point for each successive time increment. A thermal disturbance propagates only at times greater than the initial disturbance which occurs at time t = 0. In our case, we assumed that at time t = 0, the initial temperature of all nodes T?,j are equal to the ambient temperature (90°F) and the refrigerant temperature is 10F lower than the initial nodal temperature with zero internal heat generation. For _ T ‘ TB P"???— i B 0 ei,j=1.0 0. 98-0 After increasing the time increment, and maintaining the refrigerant temperature at the boundaries, the heater is turned on which causes the temperature of all nodes to be increased by the heat generation term. In order to obtain the solution of the transient conduction problem, the temperature of each node at a given time must satisfy the appropriate nodal equation. The necessary nodal equations shown in Chapter 3 are applied to each node to calculate the new values of the nodal temperature 45 at the particular time increment. For the explicit finite difference approximation method, it is necessary to insure convergence in order to calculate the temperature distribution with the node equations developed. According to Chapter 3 of Reference 4 and Chapter 4 of Reference 6, the limiting values of the convergence criterion aAt/(AX)2 when using the Central difference approximation for convective boundary conditionicases is given by: F = :éf2 + 2 < -——1gL——- (AX) (A?) “’ 2+hAX/k 3| In our case AY = 10 AX, therefore: F = 191. aAt 1 100 (AX)‘2 — 2+FAX/k (AX)2 —'101(2+BjoA7) 111 where Bio= k Applying this criterion to the cryomicroscope conduction stage where the average h'= 9.93 Btu/hr.ft2°F (see Appendix 0-3) (AX)2 = 4.44 x 10'5 ftz. At = 0.25 sec. Bio §_0.028 Substituting into Equation (4-1) F = (7.8x10'5 ftZ/sec)(0.25 sec) < 100 (4.44x10'5 ft?) —' 101(2 + (0.025)(0.028)) F = 0.437 5_o.495 46 This value gives acceptable accuracy for the finite difference approximation (Reference 4 and Reference 6). The program discussed here has been written to obtain the solution to the transient temperature response for the boundary condi- tions and dimensionsindicated in Figure 4.1. The variables, assigned constants and their physical meaning are given in Table 4.2 with the same Biot number and heat generation term as was used in the steady- state case. The discussion above relates to the transient conduction pro- gram of the Cryomicroscope Conduction Stage given in the form of a flow diagram (Figure 4.4) and variables for the program in Table 4.2. The rows and columns in this computer program are denoted by i's and j's, respectively. The increment nondimensional time (F3+1 - F3 = (AX)2F3) where a time interval is denoted by a superscript n (Reference 2), As the final steady-state temperature distribution is reached, the change in the nodal temperatures during each successive time interval approaches zero. For the boundary conditions being considered the values of 0?+} - 0? j approach zero, the steady-state value. The computation is terminated when the nondimensional temperature of a node is close to zero, (o?+} §_0? j)SEE, where E is a small value (0.001). An alternate way to terminate the calculations is to specify a maximum Fourier number. During the integration, the nondimensional temperatures for all nodes are changed to the actual temperatures(°F) and printed at the specific time intervals expressed as both dimensional and nondi- mensional time. At the end of the program the values of kl, k2, B], B2. 83, B4 and 85 are printed out after reaching the steady-state. The 47 TABLE 4.2 Variables for the Transient Conduction Program Variables Definition T(I,J) Nondimensional temperature at time n U(I,J) Nondimensional temperature at time n+l D(I,J) Difference U(I,J) - T(I,J) Z(I,J) Nondimensional truncated temperatures X(I,J) Temperatures in Degree Fahrenheit used in the printout = Z(I,J)(Ti-TB) = TB 80 'Bio = 0.028 Counting variable Minimum value of D(I,J) used to terminate the computation F Convergence criterion = aAt/(AX)2 = 0.437 F0 Fourier number = at/22 Fl Convergence criterion of double material = a*At/(AX)2 = 0.371 G Time to be print out = RZFO/o M Number of grid points along V'= 0 N Number of grid points along X'= 0 P Print interval x1 50 = X/u x2 AXZ = X/22 . "Investigate all nodes and locations 48 Enter data l Assign control constants Assign initial temperature distribution 1 Check stability No convergence criterion ll TIC Let F5 Let C Apply suitable nodal equations Calculate future nodal temperature l Increment Fourier number l Print Convergence Criterion Figure 4.5 .Flow Diagram for the Transient Conduction Program. 49 Yes Is Fo <9? No Print Fourier number Print time in second 0 Integerize Z(I,J), Calculate X(I,J) I Print X(I,J) l Increment print interval C = C + P 1 Calculate difference D(I,J) r ‘ Yes Is D(I,J)< E? *4- Let T(I,J)=U(I,J) 50 .copuozucou ucmmmcmgh msu Lo$ mupzmmm mcwpmcoz gmpaasou o.¢ mesa?“ o. .. mmmo .orwo . fiw.o Nm.o 04.0. mm.o eo.o Acucwv> 4. . 4x . 4‘ o 0 0 m u v _ _ _ A““ “ .umm _w.e Ilmv ?nw|||| m o mcpzwp p :_ . .> .umm Nw.m ow .. _ .umm mw.m_ l-Il OH“ — 1 ll llllll |+ llllll vllllnhfllllllj llllll D4IIIID ov— mcwmumgucH vo.n_ m¢.NN .uwm oe.o~ Ampaum-»ummpmv Amumccam nop 8:6 u X\ \\ Emcmoxa 1. .- 53032.8 mandala—Baum e pcmswgmaxm A, u x .uummcam gob map ul Figure 7.1 Typical Heater Dimensions TABLE 7.1 The Temperature Differences Expected Between Nodes (T -TD) by Varying width w Width W' AT' 0 Error 0.32 in 0.67 6.83°F 0.0764 0.48 in. 1.0 5.36°F 0.0589 0.64 in. 1.33 4.37°F 0.0481 0.80 in. 1.67 3.34°F 0.0367 72 w<.O\3 ".8 . .3 g»u»: mg».mg»>gm> can mmmg up mgm»mEm»o mg» mcwxwm mg thtupv mmgoz gmm3»mm mugmgmwmwo m»:»mgmasm» wgwghmmmmmW%Mme»mz A\ :OE 73 For Table 7.1 and Figure 7.2 the original configuration was W = 0.48 inch. Let W'= W/0.48 Increasing the width is a beneficial effect in that it makes the temperature at the top of the quartz more isothermal. Increasing the width of this dissipation region effects the transient response of the Cryomicroscope Conduction Stage by making the Cryomicroscope Conduction Stage respond more slowly. This effect is not desirable. The results are shown in Table 7.1, Figure 7.2, and Figure 7.3. Thicknesscf the Quartz Heater Quartz discs used as heaters are available in various thick- nesses from the manufacturer. The effect of changing the existing heater from a thickness of 0.016 inch was examined. In Table 7.2 and Figure 7.4, for the original configuration, the thickness of quartz was t. = 0.016 inch. Let t7= t'/0.0l6. Table 7.2, Figure 7.3 and Figure 7.4 shows that an increased heater thickness is expected to decrease the temperature gradient at the top surface of the quartz (where the sample would be). This is a desired result. But the effect on the transient response of the Cryomicroscope Conduction Stage would be slower for increasing the heater thickness. This is an undesired result. Using of Glass Heaters Glass heaters were considered as a possibility since glass coverships are readily available and economical. The conclusions of the results ware shown in Table 7.3, Figure 7.5 and Figure 7.6. 74 .mmgoammm »gm»mgmg» go go»»mU»m»coz mom»m go»»u:ocou maommogopso>go Pm»»gm»og mg» mo m»umm$m m.» mesa»; Amngommmv ms»» mm «N cu m» N» w e — — p — p — 0 Ali “ . _ » - _ 1q 4 cm Amg»_gm»gmu mg» » new 2 suumz mgu .mpmwcmpmz v mmvoz :wwzumm mocmcmmewo mczumgmaEmh on» ¢.~ mczmwu ...:: In . w . A_.. o N m P o F m o o m“ 1 m 41 i- m lj N 11 G EOE 77 For the thermal conductivity of quartz, k = 0.44 Btu/hr ft 0F, let k'= k/0.444. From Table 7.3 and Figure 7.5 show that there is not much difference in the temperature gradient (AT andc)error) in replacing the quartz heater with a glass heater. Figure 7.6 shows the comparison of the transient response at the centerline of the Cryomicroscope Conduction Stage. From this modification, we can see that it is possible to use glass, instead of quartz, as a heater for the Cryomicro- scope Conduction Stage. This conclusion is based on the fact that there was not much difference in the temperature gradient and tran- sient response by using glass instead of quartz as the heater. The time difference before reaching steady-state was about 6.6 seconds or approximately 25% slower for glass (k = 0.l Btu/hr ft OF). The last part of this thesis is to show the comparison of the non-dimensional temperature distribution at the top surface of the Cryomicroscope Conduction Stage as shown in Figure 7.7. For curve (3) in Figure 7.7, we applied the Transient Conduction Program with all the same data used in curve (2) except we use glass (k = 0.1 Btu/ hr ft oF, diameter = 0.84 inch and thickness = 0.012 inch) as the heater of the Cryomicroscope Conduction Stage. The results show that it reached the steady-state rapidly (about 22 seconds) but that its temperature gradient was high. In the case when we change the convective heat transfer co- efficient for the surrounding air, Figure 7.8 shows that the tempera- 20 ture difference (at the same location of nodes) using F'= 5 Btu/hr ft F and F = 1.06 Btu/hr ft2°F is about 10%. 78 A.a .pc.c;\=om m_.o 6» No.0 ozone mmme $0 xuw>wuuzvcou Fascmgh wcpv mmm_w op Nummao sage mmecmpmz mcwmcmsu An thuupv mmcoz :mm3pmm mocmgmwmwo mczpmcmaem» use m.n mgzmwu 8.3. u m o.~ m.o m-.o o hum» “ at 1+ A .1 m -1. m AV11. e. N mmoFQ 11.nu 3.820 11.0 1? 0 EE 79 soc» mFmacmumz msp mcwmcmsu an wave saw: mczumcmnEmk mo .mmmpu op Npcmzo mucosa msa we comwcmano m.~ bosom; Accoummv 62.? Nm mm am om o_ NP m a 1.11 . w u w _ n n w 111 AmCPFcaocau as» o o .. L. n a w u ._ t 1.V Amucoumm mm H a u o n m “ m m h u # lV . N6 #1 _ I-W _AT1 2% meta; o 0 ..\.. ed 1. ..d‘ 0 CL = o I'l \ o. “.8 t i am 8 F m - a ....\4\ 1r A 6 ..\fl I. I. .lll-ilb ..\ \‘oID‘I 0 ..MV \p\\ 0 o .6‘ \D“ 1.. C\ . \R wmvx m o - .v domfigifim m u m o 11 6. \ .._ 3.2:; on u m. \n\\ o om o._ m u cv< mcwuczoccam as» com.m mcwacu> Am F - k m u e mupamma Engmocm macaw-xummpm mcu soc» pop; AF . h - p 82 In summary the optimal Cryomicroscope Conduction Stage based on the alternatives considered above could not be specified exactly because if we want a uniform temperature gradient for the Cryomicro- scope Conduction Stage, it will yield a slow transient response. It depends on a choice between the desirable characteristics of uniform temperature and rapid response. But in terms of its effectiveness, the width of heater dissipation (heater dissipation area) plays a major role in the Cryomicroscope Conduction Stage performance. 7.2 Suggestions for Future Work Future work should be aimed at using these programs to study the following: a) Simulation using actual refrigerant conditions used in Cryomicroscopy. b) (Substitute alternate backing materials for copper (i.e. sapphire, aluminum alloys, etc.) c) Study the effect of non-rectangular heater dissipation area. d) Study the effect of non-uniform energy dissipation. e) Study the effect of viewing hole size. The program would also be a useful reference for developing a similar study of an existing Cryomicroscope Convection Stage. This analysis will be useful when coupled with thermal stress analyses and feedback control analyses. Owing to time constraints and also because the materials pre- sented here satisfy the purpose of this research, we were not able to 83 modify and develop the Cryomicroscope Conduction Stage more than what has been done. We do hope that this thesis could be useful for future work that will lead to the modification and development of a specific and practical model for the Cryomicroscope Conduction Stage. APPENDIX A Nodal Equations for Two-Dimensional Steady-State Conduction Interior Node Equation for the Composite Materials (EquatTOn’B-S) From Figure 3.3(a) and the application of the conservation of energy principle k. 01. (71+1.1 ' 71.1) . k2 ax.(71+1.1 ' T1.1) 2 AV 2 AX + klAY (T113 1 115) + kl A5- ( ""J Tl’l) AX 2 AV + k2 AX (TM’J ‘33 + kZAY ”‘33” TM) 2 AY AX + QAXAY = 0 For AY = mAX and let k3 = kl + k2 k3 2 k3 2m'(Ti+l,j ‘ Ti,j) T mk‘ (Ti,j-l ’ Ti,j) T 25'(T1-1,j - Ti,j) 2 _ + mk2 (Ti,j+l - Ti,j) + mQ(AX) - 0 Then we can get Equation (3-5) 2 2 2 01.+1 j + 01,1 j + 2m (k191.j-1 + kZOiljf])/k3 + m (37) N2/k3 , (2m2+2) 84 85 The Sides Node Equations with the Boundary Condition Consider the surface node as shown in Figure 3.4. Energy is conducted across three surfaces of the element, and convection occurs across the exposed surface. (T° ' ‘ T~ -) (1. .-T. .) m 12H 1.1 2 k A22" H’J "3 + FRAY (Too R - T.- .) AX AY ’ 33 Al. _ Al. = + k 2 (T144 aj T1,,j) + Q 2 AY 0. For AY = mAX _- h AX 2 2 R 2 2 1. . = TI'LJ + Ti+l.j + 2'" Ti,j-l + 2'" 'k_ Too,R T m QMX) /k 1,3 _— (2m2 + 2 + ZmthAX/k) Then the node equation for convection at the right vertical surface (Equation 3-6) 2 ; 2——-. 2 2 e. = 01+] + 2m ei,j-l+'ei-l,j + 2m AXB‘RCh + m.(ZX) Nl 2 (21112 + 2 + 2m Z1? BiR) In the same manner, we can get the Equation (3-7) for the left ver— tical surface. The Top and End Node Equation with the boundary condition Consider the left end corner node as shown in Figure 3.6. We can see that conduction occurs across the two interior surfaces and convection occurs across the left side and the lower exposed surface. The conservation of energy principle is applied to the end corner node, the result is 86 T. . - 1. .) (1. . - 1. .) Al( 1"];11 193 AY 193+] 19;, _ __A__x_ k 2 AY T k ‘2' AX * hE 2 (Tw,E ' Ti.1) —' AY AX AY _ + hL “2“Tm,L‘ Ti,j) + Q (”29(‘29 ‘ 0 Again AY = max. KYI= AX/£ and solving for T. j lip. 1111 2 2 2 E 2 L m 02 T . = Ti-l,j + m Ti,j+l +m k AX Tm”E + m k AXT% +§—--E—-(AN i’J Fit h). (m---E 37+m2—R—A—X+m2+l) k k 0 + mze‘ + mBi 370 f mZBi 370’ + m~2--(-'z§7)2 NT 0. = i-l,j i,j+l E E L L 2 1.0 ___ .__ (m2 + 1 + mBiEAX + sziLAX) (The above equation is Equation 3-ll). In the same manner, we can get Equation (3-8) (3-9), (3-l0), (3-12) and (3-l3). The Node Equations for the Convection at an Interior Corner Node with Composite Materials From Figure 3.7 for the left and interior corner node, we can write (T. T. .) 1.4-1 ' Tm) . 11... 1.21 9*— (Ti-"1' ' ...- AX 2 AY AY kl'-§ + kZAY (Ti’5T‘ - T ‘23) + k2 Ax (T‘Tl’J Ti’j) AX 2 AV + El(é§_§_él_) (Tm,L - Ti J.) + Q (— 3) AXAY= 0 For AY= max, kl =k2 + k3 and solving for T1 j 87 11. j(m2kl/k3 + 1 + 2m2k2/k3 + k2/k3 + (m2+m)T1'LAX/k3) 2 - 2 . + 2m k2 Ti,j+l/k3 + kZT 1,3 /k3 i+l,j 2 - 2 2 2 + (m +m)hLAXTm L/k3 + 3m Q(AX) /2k3 In non-dimensional form we can get Equation (3-l5) + ma. 2 ExiN j+l 1+l,j + (m 2 .g— 2 3 = 2m kSOi’ +m)B1uXe£ + m k49i,j-l + ei-l,j + 5m OW. 2 2 2 '— ((2m +1)k5 + m k4 + (m +m)BiAX + l) The Equation (3-l4) can be obtained in the same manner as Equation (3-l5). APPENDIX B Nodal Equations for Two-Dimensional Transient Conduction Interior Node Equation for the Composite Materials From Figures 3.4 and the general energy equation a .El 8T 87(kax)+8— +33“ 7)”): QCPat We can write . ‘2 k1(1? 1 - T? .) - k2 (1? . - 1? .+1) + 1 2 (AX) ~1sj 19J 193 1’3 (Av) (kl+k2) (Tn _ Tn _(kl+k2) (T1) + Q 1'- -1 11.1) 1+1..1' Tn+l n + . . _ (plCPl p'chz )(Tl TlaL) 2 dAt In the same manner as we used for Equation (3-18), AY = mAX and k3 = kl + k2. The non-dimensional equation becomes 2 kl n n k2 n n n n 2'" E (91.14 ' 91.1) ' E (91.1 ' 91.141) T 91-1..“ ’ 291.1 + o" + 2m2N2(ZX')2 = m-2—-(o" - o" ) 1+1 1 Fl i,j i,j Then Equation (3-l9) can be obtained. 88 89 n+l. - _ n ___ - e11.1' 2F1_fl(91.j-l 91.1) 2F1W(e ein.j+l)1’.j + L] (9" - 29" + .) + 2F1N2(AX)2 m2 i-l,j i,j O1+1,j The Sides Node Equations with the 'Boundary Condition Nhen convective occurs at the boundary the energy conducted to the surface is equal to the energy leaving the surface by convection. Using a forward difference approximation for aT/ax and aT/aY at the left surface. This method yields '1 n n n n “fin-m1 T131) +k11x ”1+le Tm) +k_A_)$(Ti-l,j 71,1) AX 2 AY 2 AY 1 Q (Q25) AY =BLAY(T?’J. - 12¢) To write a central finite difference approximation for a convec- tive boundary condition at the left surface (j = I), it is necessary to use a fictitious node outside the solid at j-l. Likewise, at the right surface (j = N) a fictitious node at j+l is used. n n n n n 1111 ”1.10 ' T1314) . 1. 11): ”141.1 ' 1m), 1- 0: --‘-T1°-1..1' ' T1.1) 2AX 2 AY 2 41 n + Q A2 AY =lHAY(T: j -Tm,L) For AY = mAX, non-dimensionalized and solving for O? j-l n = l 2 n n n _ n 2 ——-2 9131-1 FL" 91.141 9111,. +91-1 .1 291.1 ‘2 "‘ WAX) 2 n 2m 3119*(9'1'J -0L)] ‘90 Substituting this equation into Equation (3-18) to eliminate n i,j-l’ right vertical surface can be solved in the same manner. 6 then we can get Equation (3-20). Equation (3-2l) for the The Top and End Node Equations with the Boundary Condition Consider the left end corner node as shown in Figure (3-6). At time n, we can write '1 n n ,1'+1' Tm) + k _A_X_ (Ti-1,1" T133) 2 AX AY n (Ti AY k7 = E' 91-(T” - T" ) + E' A5-(1'? . - T" ) L 2 i,j m,L E 2 1,3 w,E Consider AY = mAX, Tm,E = Tm,L, hE = hL, Tm,T = Tm,R and hT and hR. Write in non-dimensional form. m2(en - o" ) + o" - o" + m2N1(ZX)2 = 2m(m+l) i,j+l i,j-l i-l,j i+l,j ———. n n AXB1E(ei,j ' 9T) n = 2 n _ n n 2 1—— 2_ Oi+1 j m (ei,j+l 9i,j-l) + Gi-l,j + m Nl(AX) 2m(m+l) -—-. n n AXB1E(Gi,j ' 9T) Substituting 99 1+] j in Equation (3-l7), we obtain Equation (3-22). In the same manner, we can get Equations(3-23), (3-24), (3-25), (3-26) and (3-27). APPENDIX C The Assumptions of the Cryomicroscope Conduction Stage By running the steady-state program with the quartz and the glass properties, in the same boundary conditions, there was not much in the temperature difference between the quartz and the glass. Then we assumed that the temperaturesin the glass are the same as the tempera- tures in the quartz and so neglected the glass by adding the thickness of the quartz from 0.014 in. to 0.016 in. Possible errors in the predicted temperature distribution resulting from the omission of the tape and silver epoxy were approxima- ted by examining the respective thermal resistances in the X and Y directions (see Figure 4.2). In X-direction Quartz R = 0.056 hr-OF/Btu Tape R = 0.1 hr-OF/Btu Silver Epoxy R = 3.7le5 hr-OF/Btu Copper R = 7.5X10"4 hr-OF/Btu Air R = 24 hr-OF/Btu In Y-direction Quartz R = l0 hr-OF/Btu Tape Rw= 22,000 hr-OF/Btu Silver Epoxy R = 0.52 hr-OF/Btu Copper R = 0.8 hr-OF/Btu Refrigerant R = 0.55 hr-OF/Btu 92 From these resistance values the following conclusions were made: a) The dominant in X-direction resistance is the natural convection on the top and bottom surfaces of the Cryomicroscope Conduc- tion Stage. Hence omission of the tape and silver epoxy will have an insignificant effect on the overall temperature profile in the X-direction. Also the tape and silver epoxy are placed away from the centerline. Thus, the changes in the temperature distribution would tend to be localized away from the centerline temperature of primary interest. b) The least resistance path in the Y-direction would be through the copper plate. A comparison of a representative path resistance through the copper lone (R = 0.8) is not altered significantly by including the resistance of the tape. The effect of omitting the tape is therefore considered negligible, especially at the centerline tempera- ture which is removed from the local effects close to the tape. Note that the absence or presence of the tape will have a negligible effect for the transient results because the thermal capaci- tance of the tape is extremely small due to its tiny volume. APPENDIX D The Convective Heat Transfer Coefficient for the Refrigerant and Surrounding Air Convective Heat Transfer Coefficient for the Refrigerant In this experiment using Ap = 20 psig = 2880 psig. Assume the fully-developed turbulent flow occurs in the refrigerant tube. Ap = 0.046 From f = 2 0 2 (Ref. l6) F(2/DH)(pV /2) (pVDH/u) ' _ (mm/pop“ ”‘3 The" V ‘ 0.184(R/DH)(p/2) ' ‘(‘) where DH = 0.207 in. = 0.0l73 ft. 2 0.47 ft. The properties of air at 20 psig 1.08 x10‘5 lb/ft.sec. u = p = 0.075 lb/ft.3 Pr = 0.7 k = 0.0l6 Btu/hr.ft.°F Substituting the above properties into Equation (l), we get V = 126 ft./sec. To check the Reynolds number Re = pVDH/u = 15,100 DH This value of ReD shows the turbulent flow and valid for the H assumption. For the turbulent flow in tube or duct 93 94 0.5<:Pr.< l.0 (Gas) (Ref. 7) 0.6 0.8 e Wu '-' 0.022 Pr 9H R DH = 39.15 h' = Rh k/D H H D href (air at 20 psig.) = 36.85 Btu/hr.ft.2°F Convective Heat Transfer Coefficient for the Surrounding Air For free convection from horizontal surfaces to air at atmos- pheric pressure, Assume for laminar case h = 1.32 (%I)°'25 (Ref.16) In our case,£ = 1.28 in. = 0.036 m. Assume the average of the temperature difference AT = Tw-Tco 2 8.500. Then _ 8 5 0.25 _ 20 11le “1.32 ( m) - 5.01 N/m C For the constant heat rate, h'= ghX=£ (Ref. l6) E'= 6.26 w/m2°c fi'= l.06 Btu/hr.ft.2°F 95 The Approximation of the Average Value of h Between hair and “ref —- 20 For hair 1.06 Btu/hr.ft. F 20 href 36.85 Btu/hr.ft. F 0n the basis of the area contacted for the Cryomicroscope Conduction Stage Area in contact with air/Area in contact with refrigerant: 3:1 + (—ref) 4 The average h'= 3(hair) 20 = 9.93 Btu/hr.ft. F 10. 11. 12. 13. 14. REFERENCES . J. McGrath, The Dynamics of Freezing and Thawing Mammalian Cells: The Hilla Cell, M.S.M.E. Thesis, MIT.. pp. 48-64, 1974. . A. Adams, and D. F. Rogers, Computer-Aided Heat Transfer Analysis, McGraw-Hill, Inc., 1973. . N. Ozisik, Boundary Value Problems of Heat Conduction, Inter- national Textbook, Scranton, 1966. . D. Smith, Numerical Solution of Partial Differential Equations, Oxford, London, 1965. . E. Myers, Analytical Methods in Conduction Heat Transfer, Mc- Graw Hill, Inc., 1971. . Z. Barakat, and J. A. Clark, 0n the Solution of the Diffusion Equations by Numerical Network, J. Heat Transfer, Vol. 88, ser C. pp. 421-427, 1966. . M. Kays, Convective Heat and Mass Transfer, McGraw-Hill, Inc., 1966. . E. Coorperatate Research and Development, Heat Transfer Data Book, 1975. . A. Luther, B. Cannahan, and J. 0. Hikes, Applied Numerical Methods, Wiley, New York, 1969. . S. Touloukian, Thermophysical Properties, Plenum Press, New York, 1967. . Crank and P. Nicolson, A Practical Method for Numerical Solu- tions of Partial Differential Equation of the Heat Conduction Type, Proc. Cambridge Phil. Soc., Vol. 43, pp. 50-67, 1967. . S. Carslaw and J. G. Jaeger, Conduction of Heat in Solids, Claredon Press, Oxford, 1959. . M. Dusenberre, Heat Transfer Calculations by the Finite Differ- ences, International Textbook, Scranton, Pa., 1961. . L. James, G. M. Smith and J. C. Holford, Analog and Digital Computer Methods in Engineering Analysis, International Textbook, Scranton, Pa., 1965. 96 15. 16. 17. 97 . Richtmeyer, Difference Methods for Initial Value Problems, Interscience, New York, 1967. . Holman, Heat Transfer, McGraw-Hill, Inc., 1976. . Rohsenow and H. Y. Choi, Heat, Mass and Momentum Transfer, McGraw Hill, Inc., 1968. A nICHIGAN STATE UNIV. LIBRARIES 1|l1|W11“1111111111111I111”1111111111111"111111 31293104658632