Conduction Shape Factor Models for Three-Dimensional Enclosures

Transcription

1 JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 19, No. 4, October December 005 Conducton Shape Factor Models for Three-Dmensonal Enclosures P. Teertstra, M. M. Yovanovch, and J. R. Culham Unversty of Waterloo, Waterloo, Ontaro NL G1, Canada Analytcal models are presented for conducton shape factors for three-dmensonal regons formed between an sothermal, arbtrarly shaped body and ts concentrc, arbtrarly shaped surroundng enclosure. The model s based on the exact soluton for the concentrc spheres, and two methods are developed to predct the effectve gap spacng. The models are valdated usng exstng numercal data from the lterature and data from smulatons performed usng a commercal computatonal flud dynamcs software package. The models are shown to be n excellent agreement wth the data for all enclosures wth geometrcally smlar boundary shape, wthn % rms. For enclosures formed between dfferent boundary shapes, the models are shown to be accurate wthn 5% rms when the mnmum aspect rato, that s, the smallest outer boundary dmenson over the largest nner dmenson, s greater than 1.5. Nomenclature A area, m a, b, c cubod sde dmensons, m d dameter, m k thermal conductvty, W/mK L general characterstc length, m m combnaton parameter n outward facng normal vector Q total heat flow rate, W R thermal resstance, K/W r general radal coordnate S conducton shape factor, m SL dmensonless conducton shape factor, SL/A s cube sde length, m T temperature, C V enclosed volume, m x, y, z Cartesan coordnates δ gap spacng, (d o d )/, m ρ local radal poston, m φ,θ sphercal coordnates ψ dmensonless temperature rse Subscrpts e effectve nner o outer full-space lmt Introducton ANALYTICAL models for conducton shape factors for threedmensonal regons formed between an arbtrarly shaped, Presented as Paper at the 41st Aerospace Scences Meetng and Exhbt, Reno, NV, 6 9 January 00; receved September 004; revson receved December 004; accepted for publcaton 4 December 004. Copyrght c 005 by the Amercan Insttute of Aeronautcs and Astronautcs, Inc. All rghts reserved. Copes of ths paper may be made for personal or nternal use, on condton that the coper pay the $10.00 per-copy fee to the Copyrght Clearance Center, Inc., Rosewood Drve, Danvers, MA 019; nclude the code /05 $10.00 n correspondence wth the CCC. Research Assstant Professor, Mcroelectroncs Heat Transfer Laboratory, Department of Mechancal Engneerng; pmt@mhtl.uwaterloo.ca. Dstngushed Professor Emertus, Mcroelectroncs Heat Transfer Laboratory, Department of Mechancal Engneerng; mmyov@ mhtl.uwaterloo.ca. Fellow AIAA. Assocate Professor, Drector Mcroelectroncs Heat Transfer Laboratory, Department of Mechancal Engneerng; rx@mhtl.uwaterloo.ca. 57 heated body and ts surroundng, cooled enclosure are of nterest to thermal desgners of sealed enclosures for mcroelectroncs applcatons. Analytcal heat transfer models can be used to predct quckly and accurately operatng temperatures of crcut boards and components n the enclosure, provdng a tool for parametrc studes and tradeoff analyses durng the prelmnary desgn process. An mportant step n the formulaton of natural convecton models for these applcatons s the development of conducton models n the enclosed regon. The problem of nterest n the current study nvolves steady-state conducton n the sotropc regon bounded by concentrc, arbtrarly shaped sothermal boundares, as shown n Fg. 1. Geometres examned nclude enclosed regons between smlar body shapes wth constant or near-constant gap thckness, as well as enclosures wth dfferent nner and outer boundary shapes. Numercal data are avalable n the lterature for steady-state conducton for a lmted set of enclosure geometres. Hassan 1 presents data for the concentrc crcular cylnders and the concentrc baseattached cones for a wde range of gap spacng. Warrngton et al. present numercal data for enclosures formed between dfferent concentrc boundary shapes for two cases: the cube n a sphercal enclosure and the sphere n a cubcal enclosure. The only analytcal model for the conducton shape factor n the lterature s presented by Hassan and Hollands for the dmensonal shape factor S for three-dmensonal enclosure geometres wth unform gap spacng, S ( S m 0 + Sm ) 1/m (1) where S for the sphere s used to approxmate all body shapes S.51 A and S 0 s approxmated based on one-dmensonal heat transfer S 0 A /δ where δ s the mnmum value of the gap spacng. The blended combnaton n Eq. (1) reduces to lnear superposton, m 1, for the concentrc spheres. However, for boundary shapes where corners are present, such as the concentrc cubes, a combnaton parameter value m > 1sntroduced. The values for m are dependent on geometry and are determned from a correlaton of numercal data as a functon of the nner body surface area, aspect rato, and the enclosed volume. The model s restrcted to enclosed regons formed between geometrcally smlar boundary shapes, such that the gap spacng s unform. There are currently no analytcal models avalable n the lterature that address the problem of conducton for an enclosure wth a nonunform gap spacng.

2 58 TEERTSTRA, YOVANOVICH, AND CULHAM Fg. 1 Schematc of physcal problem. The objectve of the current study s to develop analytcal models for the conducton shape factor for the general case of enclosures formed between two arbtrarly shaped, sothermal boundares. These models wll be valdated usng the avalable numercal data from the lterature, as well as data from a commercal computatonal flud dynamcs (CFD) package for a varety of dfferent boundary shapes. Problem Defnton The conducton shape factor S s defned by Yovanovch 4 for an solated, three-dmensonal body shape based on an area ntegral of the temperature gradent along an outward facng normal n from the body surface, S A ψ n da () A For the partcular problem of the three-dmensonal enclosure formed between a heated nner and cooled outer boundary, the ntegral s evaluated at the nner surface A where the dmensonless temperature rse ψ s defned as ψ [T (r) T o ]/(T T o ) () The shape factor s nondmensonalzed usng the general scale length L, asfollows: S L SL L ψ da (4) A A n A A The dmensonless shape factor s related to the thermal resstance R and total heat transfer rate Q by S L L ka R QL (5) ka (T T o ) The square root of the nner body surface area s selected as the characterstc length, L A, such that 1 / k A R Q /[ k A (T T o ) ] (6) Fgure presents the range of dfferent nner body shapes examned n ths work and shows the notaton used to descrbe each of the dmensons. Smlar notaton s used for the dmensons of the outer boundary shapes. Model Development The concentrc sphercal enclosure s a fundamental case of threedmensonal enclosures, where the thermal resstance can be determned usng the expresson presented n most heat transfer texts, 5 R (1/k)(1/d /d o ) (7) Usng Eq. (6) to relate the resstance to the dmensonless conducton shape factor gves S 1 k A R (8) (1 d /d o ) Fg. Inner and outer boundary shapes. However, the asymptotc characterstcs of the shape factor soluton for small and large values of aspect rato d /d o can be demonstrated more effectvely by ntroducng the gap thckness δ, defned as δ (d o d )/ (9) Substtutng d o d + δ nto the expresson for thermal resstance gves R δ/kd (d + δ) (10) The correspondng dmensonless shape factor expresson s S 1 / k A R 1 / k d kd (d + δ)/δ / d δ + (11) Ths formulaton for S s a lnear superposton of two lmtng cases. For small values of gap spacng wth respect to the nner body dameter, δ/d, the frst term n Eq. (11) s domnant, correspondng to one-dmensonal planar resstance. For large δ/d, S tends to a constant value correspondng to the dmensonless conducton shape factor for a sphere n a full-space regon. When t s noted that the numerator of the frst term n Eq. (11) corresponds to A for the sphere, a general model for dmensonless conducton shape factor for three-dmensonal enclosures s proposed based on the exact soluton for the concentrc spheres, ( / δe ) + S (1) where S s the conducton shape factor for the nner body n a full-space regon. Models, tabulated values, and correlatons for S are avalable for a varety of body shapes n techncal publcatons or handbooks. 4 Effectve Gap Spacng The effectve gap spacng n Eq. (1), δ e,sanexact value n the case of enclosures wth unform gap thckness, such as the concentrc spheres. However, for enclosures wth dfferent nner and outer boundary shapes or boundares wth sharp corners, approxmate models are requred to predct δ e.inthe proposed research study, two dfferent models for the effectve gap spacng are developed. For certan geometres, t s possble to calculate the local gap thckness as a functon of the angular poston n sphercal coordnates, where the orgn s concentrc wth the nner and outer boundares. For the partcular example of a cube n a sphercal enclosure, local gap thckness δ(φ,θ) s determned by δ(φ,θ) (d o /) ρ (φ, θ) (1) where ρ (φ, θ) maps one-quarter of the x s /face of the cube n sphercal coordnates, as shown n Fg., ρ (φ, θ) (s /) sn φ cos θ 0 θ 4, tan 1 sec θ φ (14)

3 TEERTSTRA, YOVANOVICH, AND CULHAM 59 Fg. Cube surface defnton n sphercal coordnates. shapes, such as the concentrc cubes or cylnders, the lower lmt on the ndependent parameter s V 1/ / A 0, where the conducton shape factor approaches nfnty as the nner and outer boundares come nto contact. However, when the nner and outer boundary shapes are dfferent, the lower lmt on V 1/ / A s a fnte, nonzero value. For the example of a sphere n a cubcal enclosure when the dameter of the nner sphere d equals the sde length of the cube s o, the total enclosed volume and ndependent parameter values are V s o (/6)d 0.476s o, V 1 / The mean value for the effectve gap spacng s determned by an area ntegraton n sphercal coordnates over the outer boundary A o : δ e 4 d o /4 / 0 tan 1 sec θ δ(φ,θ) d o 4 sn φ dφ dθ (15) The ntegral s solved numercally to determne δ e for dfferent combnatons of d o and s. For certan combnatons of boundary shapes, t may be dffcult to defne the local gap thckness as a mathematcal functon that can be ntegrated n the manner descrbed. In ths case, an alternate approxmate modelng technque called the two-rule method has been developed by the authors. In ths method, the effectve gap spacng s determned from the equvalent sphercal enclosure n whch the surface area of the nner boundary and the volume of the enclosed regon are preserved. The dameter of the equvalent nner sphere s found by d A / (16) The dameter of the equvalent outer sphercal boundary s related to the volume of the nner body V and the total enclosed volume V by d o [6(V + V )/] 1 (17) The volume of the nner sphere s related to ts area by V (/6)d A / 6 (18) The effectve gap spacng s determned by δ e d o d 1 [( [( 6V 6 V A + A ) 1 + 1) 1 ] ( ) 1 ] (19) Conducton Shape Factor Gven these two methods for calculatng the effectve gap spacng, the conducton shape factor can be determned based on Eq. (1). For the ntegral method, the conducton shape factor s ( / δe ) + S (0) where δ e s determned based on a numercal soluton to the ntegral, where necessary. In the case of the two-rule model, δ e from Eq. (19) can be substtuted nto Eq. (1) to yeld [ ( V 1 / ) ] 1 + S (1) where the dmensonless ndependent parameter V 1/ / A s selected to match the order of the terms appearng n the general model, Eq. (1). In cases nvolvng geometrcally smlar boundary As a result, the lmt of occurs when V 1 / A for the sphere n a cubcal enclosure. Model Applcaton and Valdaton To valdate the conducton shape factor models for a varety of nner and outer boundary confguratons, the followng test cases are presented. Geometrcally Smlar Boundary Shapes Three dfferent confguratons of enclosures wth geometrcally smlar boundares are examned: concentrc cubes, concentrc crcular cylnders, and concentrc base-attached double cones. For each confguraton, the rato of the nner to outer boundary dmensons are vared such that the range of the ndependent parameter s 0. V 1 / 5 The conducton shape factor s determned usng the two-rule model, Eq. (1), as a functon of the boundary dmensons defned n Fg.. Concentrc Cubes V s o s, A 6s ( /6 )[ (so /s ) ]} 1 + S () where the conducton shape factor for the solated cube n a fullspace doman s 4 S.91 Concentrc Crcular Cylnders, h/d 1 V 4 ( ) ( d o h o d h ), 4 d + d h where S s calculated by4 ( / 6 )[ (do /d ) ]} 1 + S () S (h /d ) h /d.44 (4) Concentrc Base-Attached, Double Cones, h/d 1 V 1( [ ] d o h o d h ), 4 d h [(do /d ) ] } 1 + S (5)

4 50 TEERTSTRA, YOVANOVICH, AND CULHAM Fg. 4 Geometrcally smlar boundary shapes. where S s determned from the correlaton,4 S h d h d h d h (6) d In Fg. 4, the model predctons for each of these enclosure confguratons are compared wth numercal data from Hassan 1 for the cylnders and base-attached double cones and from Warrngton et al. for the concentrc cubes, as well as for the exact soluton for the concentrc spheres. Fgure 4 demonstrates the excellent agreement between the two-rule model and the data, wth an rms dfference of.8% for the concentrc cubes and a dfference of less than 1% rms for both the cylnders and double cones. The use of V 1/ / A as the ndependent parameter and the square root of the nner body surface area as the scale length has effectvely collapsed all of the data and models to a sngle curve. Cube n Sphercal Enclosure The conducton shape factor for the regon formed between a cube and a concentrc, sphercal enclosure s modeled usng both the two-rule and ntegral methods. The rato of the nner to the outer boundary dmensons are vared over a range correspondng to the avalable data from Warrngton et al., 0.5 V 1 / When the two-rule model s used, the effectve gap spacng δ e and the dmensonless conducton shape factor S are determned usng Eqs. (19) and (0), δ e s V 6 d o s, A 6s {[( ) do + 6( 6 ) ] 1 6/} s (7) ( / 6 6 )[ (do /s ) 6/ ]} (8) Because the geometry of both the sphercal and cubcal boundares can be expressed n sphercal coordnates, the ntegral method can also be used n ths case. As descrbed by Eqs. (1) and (14), the local gap thckness can be determned as a functon of angular poston, and the effectve gap spacng s determned from an area Fg. 5 Cube n sphercal enclosure. average over the outer, sphercal boundary. The ntegral shown n Eq. (15) can be partally evaluated to the followng expresson: δ e s [ do ln ( 1 + ) ] + s /4 tan 1 sec θ dθ (9) s 0 cos θ whch can be evaluated numercally to gve [ δ e s 1 (d o/s ) ] (0) The conducton shape factor for the ntegral method s determned by substtutng Eq. (0) nto the general model for S, Eq. (0). Fgure 5 compares the predctons of the ntegral and two-rule models wth numercal data from Warrngton et al. The rato of the dameter of the sphercal enclosure and the cube sde length, d o /s, s also plotted vs V 1/ / A on the second y axs. There s good agreement between both models and the data; however, the ntegral model provdes a better ft of the data than the two-rule model, wthn % rms dfference over the full range of the dmensonless gap spacng. Sphere n Cubcal Enclosure For the sphere contaned wth a concentrc cubcal enclosure, both the two-rule and ntegral methods for predctng δ e and are compared wth the exstng numercal data. Applyng the tworule model to ths case gves the followng: V s o 6 d, δ e d [( 6 A d ) 1 ( ) ] so d (1) [ (6/) 1 s o / d ] + () For the ntegral method analyss, the effectve gap spacng s determned based on an area average on the nner sphercal surface by nterchangng d o wth d n Eq. (15), where δ(φ,θ) s δ(φ,θ) s o / sn φ cos θ d, 0 θ 4 tan 1 sec θ φ Partal evaluaton of the ntegral gves δ e s o ln ( 1 + ) s o /4 0 tan 1 sec θ cos θ dθ d () (4)

5 TEERTSTRA, YOVANOVICH, AND CULHAM 51 Fg. 7 Cubod n cubcal enclosure. Fg. 6 Sphere n cubcal enclosure. Evaluatng the ntegral numercally yelds [ ] δ e d (so /d ) (5) Fgure 6 compares the S data of Warrngton et al. wth both the two-rule and ntegral models, as well as the rato s o /d,asa functon of V 1/ / A. The agreement between the models and data s not as good as for the cube n a sphercal enclosure; however, for V 1/ A > 1, correspondng to s o /d >, the rms dfference between both models and the data s less than 5%. The error becomes more sgnfcant for smaller gap spacng, where the dameter of the nner sphere approaches that of the outer boundary, leadng to a local ncrease n heat transfer that s not accounted for n the model. Cubod n Cubcal Enclosure In the next test case, the conducton shape factor for a cubod wth dmensons a, b.785a, and c.175a n a concentrc cubcal enclosure s modeled. Because of the complexty of formulatng the lmts on the ntegraton for each of the faces of the cubod, the ntegral method wll not be used n ths example. The two-rule model s appled as follows: V s o abc, V 1 A ab + ac + bc ( / ) 1 s o abc (abc) 1 (1/a + 1/b + 1/c) (6) s calculated usng Eq. (0), where S s calculated usng the model presented by Yovanovch 4 and Culham et al. 6 for the cubod, whch gves S.469. The model s compared to numercal data from smulatons performed usng FLOTHERM, 7 a commercal fnte volume based CFD package. Full detals of the numercal procedure, ncludng a grd convergence study, are gven n the Appendx. The predctons of the two-rule model are compared wth the numercal data n Fg. 7. As n the precedng case, there s good agreement between the model and the data for values of V 1/ / A > 1, correspondng to s o /b > 1.5, wth an rms dfference of less than %. A more sgnfcant error occurs when V 1/ / A < 1, where the longest dmenson of the cubod approaches the sde length of the enclosure, s o /b < 1.4. The local ncrease n the heat transfer caused by the close proxmty of these surfaces s not accounted for n the model. Cylnder n Cubcal Enclosure The fnal test case nvolves a crcular cylnder, h /d 0.5, n a cubcal enclosure. The two-rule model formulaton for S s Fg. 8 Cylnder n cubcal enclosure. developed as follows: V s o ( ) 4 d h, A + d h 4 d [ V 1 (so /d o ) (/4)(h /d ) ] 1 (7) h /d + 1 Equaton (0) s used to determne S, where S s calculated for h /d 0.5 usng Eq. (4). The model predctons are valdated usng numercal results from smulatons performed usng FLOTHERM, as descrbed n the Appendx. Fgure 8 demonstrates the good agreement between the model and the data, wthn % rms for V 1/ / A > 0.8, correspondng to s o /d > 1.5. The maxmum dfference of 10% occurred when the dameter of the cylnder approaches the dmensons of the enclosure, s o /d 1.. Summary A general model has been presented for conducton shape factors between an arbtrarly shaped, sothermal nner body and ts surroundng, sothermal enclosure based on the exact soluton for the concentrc spheres. The model requres an effectve value of the gap spacng, and two approaches for determnng δ e, the ntegral method and the two-rule method, have been developed. The ntegral method provdes better agreement than the two-rule method wth the avalable numercal data; however, ts use s lmted to geometres where the local gap thckness and ntegraton lmts can be expressed mathematcally. The two-rule method s a general model, applcable n all cases where the enclosed volume and nner body surface area can be determned. The agreement between the two-rule model and the numercal data s excellent for all enclosures wth geometrcally smlar boundary shapes, wthn 1 % rms dfference n all cases examned n ths work. The two-rule model s also effectve n accurately predctng

6 5 TEERTSTRA, YOVANOVICH, AND CULHAM for enclosures wth dfferent nner and outer boundary shapes, wth an rms dfference of 5% n all cases when V 1/ / A > 1. The model s less accurate for certan geometres, such as the sphere n a cubcal enclosure, and the rato of the smallest outer body dmenson to the largest nner body dmensons, that s, s o /d, can be used to ndcate the range of applcaton of the model. In all cases, the accuracy of the two-rule model s sgnfcantly reduced when the correspondng rato of outer and nner dmensons s less than 1.5. In these stuatons, numercal smulaton may be requred to determne accurately the conducton shape factor. Appendx: Numercal Smulatons All numercal data for the cubod-n-cube and cylnder-n-cube enclosures were calculated usng FLOTHERM, 7 a commercal fnte volume-based CFD software package. The CFD modelng s performed n a three-dmensonal doman, and the mode of heat transfer s lmted to conducton only. The software uses an orthogonal grd n Cartesan coordnates and s well suted to problems nvolvng cubes and cubods; however, FLOTHERM also ncludes angled plates that are used to create complex objects, such as the crcular cylnder. The models for the cubod and the cylnder are smplfed by assumng one-eghth symmetry, wth adabatc boundary condtons mposed at all symmetry (x 0, y 0, and z 0) boundares and sothermal boundary condtons at all outer boundares, T o 5 C. The surface of the nner body s treated as an sothermal boundary condton, T 50 C. To vary the aspect rato of the enclosure, the Fg. A1 Grd convergence results for cylnder n cubcal enclosure. nner boundary dmensons are changed, whereas the outer boundary s held constant. A constant value for the thermal conductvty, k W/mK, s specfed for the regon between the source and the outer boundary. The software automatcally calculates the total heat flow rate crossng the nner boundary based on an area-weghted sum of the fluxes calculated from the local temperature gradent. Grd ndependence was verfed for the cylnder-n-cube confguraton for the test case d 60 mm, h 0 mm, and s o 14 mm. The total heat transfer rate s plotted as a functon of the number of control volumes n Fg. A1. The grd convergence study demonstrates that the use of approxmately 00,000 control volumes for the smulaton results n a soluton for Q that s ndependent of the number of grd ponts. All subsequent numercal smulatons were performed usng the same level of mesh refnement. The results of the FLOTHERM smulatons are expressed n terms of the total heat transfer rate for the one-eghth symmetry problem, Q 1/8.Tocompare these data wth the model predctons, the followng nondmensonalzaton s performed: S 8(Q 1/8 ) / k A T where values of k and T T T o are gven earler and A s the square root of the total nner boundary surface area. Acknowledgments The authors acknowledge the contnued fnancal support of Materals and Manufacturng Ontaro and the Centre for Mcroelectroncs Assembly and Packagng. References 1 Hassan, A. V., An Investgaton of Free Convectve Heat Transfer from Bodes of Arbtrary Shape, Ph.D. Dssertaton, Dept. of Mechancal Engneerng, Unv. of Waterloo, Waterloo, ON, Canada, Warrngton, R. O., Powe, R. E., and Mussulman, R. L., Steady Conducton n Three-Dmensonal Shell, Journal of Heat Transfer, Vol. 104, May 198, pp. 9, 94. Hassan, A. V., and Hollands, K. G. T., Conducton Shape Factor for a Regon of Unform Thckness Surroundng a Three Dmensonal Body of Arbtrary Shape, Journal of Heat Transfer, Vol. 11, May 1990, pp Yovanovch, M. M., Conducton and Thermal Contact Resstances (Conductances), Handbook of Heat Transfer, rd ed., edted by W. M. Rohsenow, J. P. Hartnett, and Y. Cho, McGraw Hll, New York, 1998, Chap., pp Incropera, F. P., and DeWtt, D. P., Fundamentals of Heat and Mass Transfer, 4th ed., Wley, New York, 1996, pp Culham, J. R., Yovanovch, M. M., Teertstra, P., Wang, C.-S., Refa- Ahmed, G., and Tan, R., Smplfed Analytcal Models for Forced Convecton Heat Transfer from Cubods of Arbtrary Shape, Journal of Electronc Packagng, Vol. 1, No., 001, pp FLOTHERM, Flomercs, Inc., Southboro, MA, 00.