CONDUCTION SHAPE FACTOR MODELS FOR 3-D ENCLOSURES

Transcription

1 AIAA--9 CONDUCTION SHAPE FACTOR MODELS FOR -D ENCLOSURES P. Teertstra, M. M. Yovanovich and J. R. Culham? Microelectronics Heat Transfer Laboratory Deartment of Mechanical Engineering University of Waterloo Waterloo, Ontario, Canada NL G htt:// Abstract Analytical models are resented for conduction shae factors for three-dimensional regions formed between an isothermal, arbitrarily-shaed body ants concentric, arbitrarily-shaed surrounding enclosure. The model is basen the exact solution for the concentric sheres and two methods are develoed to redict the effective ga sacing. The models are validated using existing numerical data from the literature and data from simulations erformed using a commercial CFD software ackage. The models are shown to be in excellent agreement with the data for all enclosures with geometrically similar boundary shae, within % RMS. For enclosures formed between different boundary shaes, the models are shown to be accurate within % RMS when the minimum asect ratio, i.e. the smallest outer boundary dimension over the largest inner dimension, is greater than.. Nomenclature a; b; c cuboid side dimensions, m A area, m d diameter, m k thermal conductivity, W mk L general characteristic length, m m combination arameter n outward facing normal vector Q total heat flow rate, W ~r general radial coordinate R thermal resistance, KW s cube side length, m S conduction shae factor, m Graduate Research Assistant, Ph.D. Candidate Fellow AIAA, Distinguished Professor Emeritus? Associate Professor, Director MHTL Coyright cfl by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with ermission. S? L dimensionless conduction shae factor; SLA i T temerature, o C V enclosed volume, m x; y; z Cartesian coordinates Greek Symbols δ ga sacing; ( ), m φ;θ sherical coordinates ψ dimensionless temerature rise ρ local radial osition, m Subscrits e effective i inner o outer full-sace limit Introduction Analytical models for conduction shae factors for threedimensional regions formed between an arbitrarilyshaed, heated body ants surrounding, cooled enclosure are of some interest to thermal designerf sealed enclosures for microelectronics alications. Analytical heat transfer models can be used to quickly and accurately redict oerating temeraturef circuit boards and comonents in the enclosure, roviding a tool for arametric studies and trade-off analyses during the reliminary design rocess. An imortant ste in the formulation of natural convection models for these alications is the develoment of conduction models in the enclosed region. The roblem of interest in the current study involves steady state conduction in the isotroic region bounded by concentric, arbitrarily-shaesothermal boundaries, as shown in Fig.. Geometries examinenclude enclosed regions between similar body shaes with constant or near-constant ga thickness, as well as enclosures with different inner anuter boundary shaes.

2 AIAA--9 T i T i > T o The objective of the current study is to develo analytical models for the conduction shae factor for the general case of enclosures formed between two arbitrarily-shaed, isothermal boundaries. These models will be validated using the available numerical data from the literature, as well as data from a commercial CFD ackage for a variety of different boundary shaes. T o Fig. Schematic of Physical Problem Numerical data are available in the literature for steady state conduction for a limited set of enclosure geometries. Hassani resents data for the concentric circular cylinders and the concentric base-attached cones for a wide range of ga sacing. Warrington et al. resents numerical data for enclosures formed between different concentric boundary shaes for two cases: the cube in a sherical enclosure and the shere in a cubical enclosure. The only analytical model for the conduction shae factor in the literature is resented by Hassani and Hollands for the dimensional shae factor S for -D enclosure geometries with uniform ga sacing: S (S m + S m )m () where S for the shere is used to aroximate all body shaes: S : A i and S is aroximated basen one-dimensional heat transfer: S A i δ where δ is the minimum value of the ga sacing. The blended combination in Eq. () reduces to linear suerosition, m, for the concentric sheres. However, for boundary shaes where corners are resent, such as the concentric cubes, a combination arameter value m > is introduced. The values for m are deendent on geometry and are determined from a correlation of numerical data as a function of the inner body surface area, asect ratio, and the enclosed volume. The model is restricted to enclosed regions formed between geometrically similar boundary shaes, such that the ga sacing is uniform. There are currently no analytical models available in the literature that address the roblem of conduction for an enclosure with a non-uniform ga sacing. Problem Definition The conduction shae factor S is defined by Yovanovich for an isolated, -D body shae basen an area integral of the temerature gradient along an outward facing normal n from the body surface: S ZZ ψ fi fi fi n fi da i () For the articular roblem of the -D enclosure formed between a heatenner and cooleuter boundary, the integral is evaluated at the inner surface, A i, where the dimensionless temerature rise ψ is defined as: ψ T (~r) T o () T i T o The shae factor is non-dimensionalized using the general scale length L as follows: ZZ S? SL L L ψ fi fi fi A i A i A i n fi da i () The dimensionless shae factor is related to the thermal resistance R and total heat transfer rate Q by: S? L QL L ka i R () ka i (T i T o ) The square root of the inner body surface area is selected as the characteristic length, L A i, such that: S? k Q A i R k () A i (T i T o ) Figure resents the range of different inner body shaes examinen this work, and shows the notation used to describe each of the dimensions. Similar notation is used for the dimensionf the outer boundary shaes. Model Develoment The concentric sherical enclosure is a fundamental case of the three-dimensional enclosures, where the thermal resistance can be determined using the exression resented in most heat transfer texts : R (7) k

3 AIAA--9 s s s h z, d d x, b d h c a xs / i Fig. Cube Surface Definition in Sherical Coordinates y Fig. Inner and Outer Boundary Shaes Using Eq. () to relate the resistance to the dimensionless conduction shae factor gives: S? k A i R d () i However, the asymtotic characteristicf the shae factor solution for small and large valuef asect ratio,, can be demonstrated more effectively by introducing the ga thickness, δ, defined as: δ Substituting +δ into the exression for thermal resistance gives: R δ k ( + δ) (9) () The corresonding dimensionless shae factor exression is: S? k A i R k k ( + δ) δ di + () δ This formulation for S? is a linear suerosition of two limiting cases. For small valuef ga sacing with resect to the inner body diameter, δ, the first term in Eq. () is dominant, corresonding to -D, lanar resistance. For δ large, S? tends to a constant value corresonding to the dimensionless conduction shae factor for a shere in a full-sace region. Noting that the numerator of the first term in Eq. () corresonds to A i for the shere, a general model for dimensionless conduction shae factor for threedimensional enclosures is roosed basen the exact solution for the concentric sheres: S? δ e + S? () where S? is the conduction shae factor for the inner body in a full sace region. Models, tabulated values and correlations for S? are available for a variety of body shaes in technical ublicationr handbooks. Effective Ga Sacing The effective ga sacing in Eq. (), δ e, is an exact value in the case of enclosures with uniform ga thickness, such as the concentric sheres. However, for enclosures with different inner anuter boundary shaer boundaries with shar corners, aroximate models are required to redict δ e. In the roosed research study, two different models for the effective ga sacing are develoed. For certain geometries, it issible to calculate the local ga thickness as a function of the angular osition in sherical coordinates, where the origin is concentric with the inner anuter boundaries. For the articular examle of a cube in a sherical enclosure, local ga thickness δ(φ; θ) is determined by: δ(φ;θ) ρ i(φ;θ) () where ρ i (φ;θ) mane quarter of the x s i face of the cube in sherical coordinates, as shown in Fig. : ρ i (φ;θ) (s i) sinφ cosθ» θ» ; tan sec θ» φ» ()

4 AIAA--9 The mean value for the effective ga sacing is determined by an area integration in sherical coordinates over the outer boundary, A o : δ e d o Z Z tan secθ δ(φ;θ) d o sinφdφdθ () The integral is solved numerically to determine δ e for different combinationf and s i. For certain combinationf boundary shaes it may be difficult to define the local ga thickness as a mathematical function that can be integraten the manner described. In this case, an alternate aroximate modelling technique, called the two-rule method, can be used. In this method, the effective ga sacing is determined from the equivalent sherical enclosure in which the surface area of the inner boundary and the volume of the enclosed region are reserved. The diameter of the equivalent inner shere is found by: () The diameter of the equivalent outer sherical boundary is related to the volume of the inner body, V i and the total enclosed volume, V, by: [ (V +V i ) ] (7) The volume of the inner shere is related to its area by: V i d i A i The effective ga sacing is determined by: ψ! δ e V + A i ψ V A + i ()! (9) Conduction Shae Factor Given these two methods for calculating the effective ga sacing, the conduction shae factor can be determined basen Eq. (). For the integral method, the conduction shae factor is: S? + S? () δ e where δ e is determined basen a numerical solution to the integral, where necessary. In the case of the two-rule model, δ e from Eq. (9) can be substitutento Eq. () to yield: S? ψ! + S? + V () where the dimensionless indeendent arameter V A i is selected to match the order of the terms aearing in the general model, Eq. (). Model Alication and Validation In order to validate the conduction shae factor models for a variety of inner anuter boundary configurations, the following test cases are resented. Geometrically-Similar Boundary Shaes Three different configurationf enclosures with geometrically similar boundaries are examined: concentric cubes; concentric circular cylinders; and concentric base-attached double cones. For each configuration, the ratio of the inner to outer boundary dimensions are varied such that the range of the indeendent arameter is: :» V» The conduction shae factor, S? is determined using the two-rule model, Eq. (), as a function of the boundary dimensions definen Fig.. Concentric Cubes V s o s i ; A i s i S? ψ " so #! + S? () + s i where the conduction shae factor for the isolated cube in a full sace domain is S? :9 Concentric Circular Cylinders (hd ) V d o h o di h i ; A i + h i S? ψ " do #! + S? () + where S? is calculated by :7 hi :9 + :77 S? : () + hi

5 AIAA--9 S * Exact Solution, Eq. () Model, Eq. () Hassani Warrington et al. Sheres Cubes Cylinders Double Cones Cylinders Double Cones Cubes two-rule anntegral methods. The ratio of the inner to the outer boundary dimensions are variever a range corresonding to the available data from Warrington et al. :» V» Using the two-rule model, the effective ga sacing, δ e and the dimensionless conduction shae factor, S?, are determined using Eqs. (9) and (): V s i ; A i s i - V / / A i Fig. Geometrically-Similar Boundary Shaes Concentric Base-Attached, Double Cones (hd ) V d o h o di h i ; A i h i S? ψ " do #! + S? () + where S? is determined from the correlation S? :9 + : h i :77 :7 hi + : hi + hi :7 In Fig. the model redictions for each of these enclosure configurations are comared with numerical data from Hassani for the cylinders and base-attached double cones and from Warrington et al. for the concentric cubes, as well as the exact solution for the concentric sheres. Figure demonstrates the excellent agreement between the two-rule model and the data, with an RMS difference of.% for the concentric cubes, and a difference of less than % RMS for both the cylinders and double cones. The use of V as the indeendent arameter and the square root of the inner body surface area as the scale length has effectively collased all the data and models to a single curve. Cube in Sherical Enclosure The conduction shae factor for the region formed between a cube and a concentric, sherical enclosure is modelled using both the δ e s i S? ψ do + s i + " do s i r C A () #! + :9 (7) Since the geometry of both the sherical and cubical boundaries can be exressen sherical coordinates, the integral method can also be usen this case. As described by Eqs. () and (), the local ga thickness can be determined as a function of angular osition, and the effective ga sacing is determined from an area average over the outer, sherical boundary. The integral shown in Eq. () can be artially evaluated to the following exression: δ s» i do e + s i Z s i ln + tan sec θ dθ () cosθ which can be evaluated numerically to give:» δ e s i :7 s i (9) The conduction shae factor for the integral method is determined by substituting Eq. (9) into the general model for S?, Eq. (). Figure comares the redictionf the integral and two-rule models with numerical data from Warrington et al.. The ratio of the diameter of the sherical enclosure and the cube side length, s i, is also lotted versus V A i on the second y-axis. There is good agreement between both models and the data; however, the integral model rovides a better fit of the data than the two rule model, within % RMS difference over the full range of the dimensionless ga sacing.

6 AIAA--9 Warrington et al. Integral Model, Eqs. (,9) Two-rule Model, Eq. (7) Warrington et al. Integral Model, Eqs. (,) Two-rule Model, Eq. () /s i / S * /s i S * / V / / A i Fig. Cube in Sherical Enclosure V / / A i Fig. Shere in Cubical Enclosure Shere in Cubical Enclosure For the shere contained with a concentric cubical enclosure, both the two-rule anntegral methods for redicting δ e and S? are comared with the existing numerical data. Alying the two-rule model to this case gives the following: V s o d i ; S? δ e " A i d i # () " # + () For the integral method analysis, the effective ga sacing is determined basen an area average on the inner sherical surface by interchanging with in Eq. (), where δ(φ; θ) is δ(φ;θ) () d sinφcosθ i» θ» ; tan secθ» φ» Partial evaluation of the integral gives: δ s o e ln + Z tan secθ dθ cosθ Evaluating the integral numerically yields:» δ e :7 () () () Figure comares the S? data of Warrington et al. with both the two-rule anntegral models, as well as the ratio, as a function of V. The agreement between the models and data is not as good as for the cube in a sherical enclosure; however, forv A i >, corresonding to >, the RMS difference between both models and the data is less than %. The error becomes more significant for smaller ga sacing, where the diameter of the inner shere aroaches that of the outer boundary, leading to a local increase in heat transfer that is not accounted for in the model. Cuboin Cubical Enclosure In the next test case, the conduction shae factor for a cuboid with dimensions (a;b :7a;c :7a) in a concentric cubical enclosure is modelled. Due to the comlexity of formulating the limitn the integration for each of the facef the cuboid, the integral method will not be usen this examle. The two-rule model is alied as follows: V s o abc ; V s A i ab + ac + bc s o abc (abc) a + b + c () S? is calculated using Eq. (), where S? is calculated using the model resented by Yovanovich and Culham et al. for the cuboid, which gives S? :9. The model is comared to numerical data from simulations erformed using FLOTHERM 7, a commercial finite volume-based CFD ackage. Full detailf the numerical rocedure, including a grid convergence study,

7 AIAA--9 Two-rule Model, Eq. (,) Data /b Two-rule Model, Eq. (,) Data / S * /b S * / V / / A i V / / A i Fig. 7 Cuboin Cubical Enclosure Fig. Cylinder in Cubical Enclosure as given in the Aendix. The redictionf the tworule model are comared with the numerical data in Fig. 7. As in the revious case, there is good agreement between the model and the data for valuef V A i >, corresonding to b > :, with an RMS difference of less than %. More significant error occurs when V <, where the longest dimension of the cuboid aroaches the side length of the enclosure, b < :. The local increase in the heat transfer caused by the close roximity of these surfaces is not accounted for in the model. Cylinder in Cubical Enclosure The final test case involves a circular cylinder, h i :, in a cubical enclosure. The two rule model formulation for S? is develoed as follows: V s o h i ; A i + h i r V " so hi + # h i () Equation () is used to determine S?, where S? is calculated for h i : using Eq. (). The model redictions are validated using numerical results from simulations erformed using FLOTHERM, as describen the Aendix. Figure demonstrates the good agreement between the model and the data, within % RMS for V A i > :, corresonding to > :. The maximum difference of % occurred when the diameter of the cylinder aroaches the dimensions of the enclosure, :. Summary and Conclusions A general model has been resented for conduction shae factors between an arbitrarily-shaed, isothermal inner body ants surrounding, isothermal enclosure basen the exact solution for the concentric sheres. The model requires an effective value of the ga sacing and two aroaches for determining δ e, the integral method and the two-rule method, have been develoed. The integral method rovides better agreement than the two-rule method with the available numerical data; however, its use is limited to geometries where the local ga thickness anntegration limits can be exressed mathematically. The two-rule methos a general model, alicable in all cases where the enclosed volume annner body surface area can be determined. The agreement between the two-rule model and the numerical data is excellent for all enclosures with geometrically similar boundary shaes, within -% RMS difference in all cases examinen this work. The two rule model is also effective in accurately redicting S? for enclosures with different inner anuter boundary shaes, with an RMS difference of -% inallcases when V A i >. The model is less accurate for certain geometries, such as the shere in a cubical enclosure, and the ratio of the smallest outer body dimension to the largest inner body dimensions, i.e., can be used to indicate the range of alication of the model. In all cases, the accuracy of the two-rule model is significantly reduced when the corresonding ratio of outer annner dimensions is less than.. In these situations, numerical simulation may be requiren order to accurately determine the conduction shae factor. 7

8 AIAA--9 Acknowledgments The authors acknowledge the continued financial suort of Materials and Manufacturing Ontario (MMO) and the Centre for Microelectronics Assembly and Packaging (CMAP). References Hassani, A.V., An Investigation of Free Convective Heat Transfer from Bodief Arbitrary Shae, Ph.D. Thesis, Deartment of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada, 97. Warrington, R.O., Powe, R.E. and Mussulman, R.L., Steady Conduction in Three-Dimensional Shell, Journal of Heat Transfer, Vol.,. 9-9, 9. Hassani, A.V. and Hollands, K.G.T., Conduction Shae Factor for a Region of Uniform Thickness Surrounding a Three Dimensional Body of Arbitrary Shae, Journal of Heat Transfer, Vol.,. 9-9, 99. Yovanovich, M.M., Conduction and Thermal Contact Resistances (Conductances), Handbook of Heat Transfer, rd. ed., eds. W.M. Rohsenow, J.P. Hartnett and Y. Cho, McGraw Hill, New York, Chater,.. -.7, 99. Incroera, F.P. and DeWitt, D.P., Fundamentalf Heat and Mass Transfer, th ed., Wiley, New York,. 9-99, 99. Culham, J.R., Yovanovich, M.M., Teertstra,P., Wang, C.-S., Refai-Ahmed, G. and Tain, R., Simlified Analytical Models for Forced Convection Heat Transfer from Cuboidf Arbitrary Shae, Journal of Electronic Packaging, Vol., No.,. -,. 7 FLOTHERM, Flomerics Inc., Southboro, MA, 77,. Aendix All numerical data for the cuboid-in-cube and cylinderin-cube enclosures were calculated using FLOTHERM 7, a commercial finite volume based CFD software ackage. The CFD modelling is erformen a threedimensional domain, and the mode of heat transfer is limited to conduction only. The software uses an orthogonal grin Cartesian coordinates ans well suited to roblems involving cubes and cuboids; however, FLOTHERM also includes angled lates that are used to create comlex objects, such as the circular cylinder. The models for the cuboid and the cylinder are simlified by assuming / symmetry, with adiabatic boundary conditions imosed at all symmetry (x ;y ;z ) boundaries ansothermal boundary conditions at all outer boundaries, T o o C. The surface of the inner body is treated as an isothermal boundary condition, Q / (W) E+.E+.E+.E+ #CVs Fig. 9 Grid Convergence Results for Cylinder in Cubical Enclosure T i o C. In order to vary the asect ratio of the enclosure, the inner boundary dimensions are changed while the outer boundary was held constant. A constant value for the thermal conductivity, k :WmK, is secified for the region between the source and the outer boundary. The software automatically calculates the total heat flow rate crossing the inner boundary basen an area-weighted sum of the fluxes calculated from the local temerature gradient. Grindeendence was verified for the cylinder-incube configuration for the following test case: mm h i mm mm The total heat transfer rate is lotted as a function of the number of control volumes in Fig. 9. The grid convergence study demonstrates that the use of aroximately, control volumes for the simulation results in a solution for Q that is indeendent of the number of griints. All subsequent numerical simulations were erformed using the same level of mesh refinement. The resultf the FLOTHERM simulations are exressed in termf the total heat transfer rate for the / symmetry roblem, Q. In order to comare these data with the model redictions, the following nondimensionalization is erformed: S? Q k A i T where valuef k and T T i T o are given above and is the square root of the total inner boundary surface area.