SUMMARY.TEX SUMMARY OF CONVECTION CORREATION EQUATIONS ME 353 Heat Transfer 1 Department of Mecanical Engineering University ofwaterloo M.M. Yovanovic November 10, 1997 Te attaced material is a summary of some of te important results for forced and natural convection eat transfer from isotermal or isoæu surfaces. Correlation equations for local and area-average eat transfer for eternal and internal æow are given. Many empirical and analytic correlation equations ave been developed for te local and area-average values of te Nusselt number for limited ranges of te forced and buoyancyinduced æow parameters: Reynolds, Peclet, Grasof and Rayleig numbers; for laminar or turbulent æows; and for various æuids wic are caracterized by te Prandtl number. One sould consult te course tet for te deænitions of te various dependent and independent parameters and te basis for te æuid properties evaluation for te particular correlation equation. Tis summary does not cover te numerous forced and buoyancy-induced internal æows troug and witin comple conægurations. One sould consult te course tet or te several andboos wic deal wit tese topics. 1. aminar and Turbulent Forced Eternal Flow Deænitions of ocal and Area-average Values @Tè; y 0è, @y, èt w, T 1 è 1 Z 0 èè d è è, Nu ç, Re U 1 ç Re U 1 ç 1
Flat Plate, aminar Boundary ayer Flow Correlation imits Conditions æ 5Re,1 100 ére é 500; 000 ocal C f; 0:664Re,1 100 ére é 500; 000 ocal C f; 1:38Re,1 100 ére é 500; 000 Area-Average ææpr,13 100 ére é 500; 000 ocal Re 1 Re 1 Re 1 Nu Nu 0:3387Pr 13 100 ére é 500; 000 ocal, UWT, Pr!1 0:564Pr 1 100 ére é 500; 000 ocal, UWT, Pr! 0 0:3387Pr 13 è0:0468p rè 3 i 14 100 ére é 500; 000 ocal, UWT, 0 épré1 0:4637Pr 13 100 ére é 500; 000 ocal, UWF, Pr!1 0:886Pr 1 100 ére é 500; 000 ocal, UWF, Pr! 0 0:4637Pr 13 è0:005p rè 3 i 14 100 ére é 500; 000 ocal, UWF, 0 épré1 0:6774Pr 13 è0:0468p rè 3 i 14 100 ére é 500; 000 Average, UWT, 0 épré1 0:974Pr 13 è0:005p rè 3 i 14 100 ére é 500; 000 Average, UWF, 0 épré1 Flat Plate, Turbulent Boundary ayer Flow Correlation imits Conditions æ 0:37Re,15 5 æ 10 5 ére é 10 8 ocal C f; 0:059Re,15 5 æ 10 5 ére é 10 8 ocal C f; 0:074Re,15, 174Re,1 Re ;c 5æ 10 5 Mied-Average ææpr,13 100 ére é 500; 000 ocal 0:096Re 45 Pr 13 100 ére é 500; 000 ocal, UWT, 0:6 épré60 Nu ç 0:037Re 45, 871 ç Pr 13 100 ére é 500; 000 Average, UWT, 0:6éPré60
Cross Flow Over Circular Cylinders Correlation imits Conditions ç è 58 45 Nu D S D+? 0:6Re1 D Pr13 ReD Average; UWT; ç ç 0:4 3 è 14 ç 8; 000 100éRe D é10 7 0 épré1 Pr è! S D? ç 4 0:869èDè 0:76 0 ç D ç 8 Re 0:5+D D! 0 0 1 S D? p 4 @ q 1 A 1 ç 0:5D lnèdè Flow Over Speres D ç 8 Re D! 0; Asymptote Correlation imits Conditions C D 0:4+ 4 Re D + 6 0çRe D çæ10 qre 5 Total Drag, æ10è D Nu D + 0:6Re1 D Pr 13 ç è 58 45 ReD Average; UWT; ç ç 0:4 3 è 14 ç 8; 000 100éRe D é10 7 0 épré1 Pr. Eternal Flow Over Isotermal Oblate and Prolate Speroids Te following universal correlation equation: è! 3 1 Nu p A Nu0 p Pp A + 40:15 p A A +0:35Rep 0:566 5 A Pr 13 was developed by Yovanovic è1988è from two accurate correlation equations proposed by Yuge è1960è for air cooling of isotermal speres, and te correlation equations developed by several researcers for convection eat and mass transfer from isotermal oblate and prolate speroids. In te above correlation equation te Nusselt and Reynolds numbers are bot based on te lengt scale p A. Te diæusive limit Nu 0 p A corresponding to Re! 0iste dimensionless sape factor S? p A. Yovanovic è1988è blended te two Yuge equations and introduced te parameter P p A wic accounts for te blocage of te body as te æuid æows around it. Also te Yuge 3
correlation equations wicwere developed for air were etended to account for large Prandtl number æuids, i.e. Pr é 0:7. Te correlation equation is valid for te wide Reynolds number range: 0 ç Re p A é æ 105. Te general correlation equation is in very good agreement wit numerous analytical and eperimental correlation equations over various ranges of te Reynolds number for Pr 0:7. It is also in good agreement wit te empirical correlation equation of Pasterna and Gauvin è1960è wic was developed from 0 diæerent conve body sapes to account for bot body sape and orientation. Te body lengt scale wic tey proposed was based on te ratio of te total surface area of te body divided by te maimum projected area of te body perpendicular to te air æow. Tey acieved good correlation of teir eat and mass transfer data wit a single power-law equation wic was converted to te body scale lengt p A Nu p A 0:914Re0:514 p A Pr 13 886 ç Re p A ç 8860 Tis equation correlated data for speres, ænite circular cylinders wit aes parallel and perpendicular to te æow, prisms, cubes in various orientations, and emisperes positioned wit te æat section at te rear. Te turbulence intensity was reported to be in te range: 9 to 10 è in all teir eperiments. Te single equation correlated all data wit a deviation of only æ15 è in te speciæed range. Te general correlation equation agrees wit te Pasterna-Gauvin correlation equation witin te given Reynolds number range to witin 15 è. Terefore, te general equation of Yovanovic can be used for arbitrary conve isotermal bodies over a muc wider range of te Reynolds number. 3. aminar Forced Internal Flow Deænitions and Notation Reynolds number: Re D UD ç aminar Flow: Re D é 300 Hydraulic Diameter: D 4 P A Cross Section Area Wetted Perimeter Dimensionless aial distance:? D Pe D ReP r ç 4 Gz 1 q ocal, Isotermal Wall, Nusselt number: ;UW T w èè ët w, T m èèë D Mean-value, Isotermal Wall, Nusselt number: Nu m;uw T q w ët w, T m èèë D ocal, Isoæu Wall, Nusselt number: ;UW F q w ët w èè, T m èèë D q Mean-value, Isoæu Wall, Nusselt number: Nu m;uw F w ët w èè, T m èèë D 4
ocal Nusselt Number for Termally Developing Flow Curcill and Ozoe è1973è propose te following correlation equations for te local Nusselt number for te developing termal æeld for te UWT and UWF cases: ;UW T +1:7 5:357 ç ç 388,89 è 38 ç? æ 5è ;UW F +1 5:364 ç ç è 0,109 310 ç? æ 5è Tey developed tese epressions based on asymptotic solutions valid for small and large values of?. Area-Average Nusselt Number for Fully Developed Hydraulic, Termally Developing Flow Te following approimations of Sa è1975è for fully developed ydraulic æow and termally developing in an isotermal èuwtè or an isoæu èuwfè circular pipe are based on te analytic solutions of te Graetz-type problems. Epressions for area-mean Nu m;uw T ;Nu m;uw F versus te local dimensionless position? èd èèrep rè are given below. Te approimations are quite accurate over te entire range: 0:005 é? ç 1: Te maimum diæerence wit respect to accurate analytic results is less tan 4:4è. For very small values? é 0:005 te approimate epressions approac te eveque asymptotes wic were obtained by te metod of similarity transformation. For large values? ç 0:5, te approimations go to te fully-developed ydraulic and termal solutions: Nu UWT 3:656 and Nu UWF 4:354 wic were obtained by te metod of separation of variables wic leads to a diæerential equation of te Sturm-iouville type. Te solution is presented as an inænite series epansion of eigenfunctions and corresponding eigenvalues. 8 é Nu m;uw T é: 8 é Nu m;uw F é: 1:615 è? è 13, 0:; 0:005 é? é 0:03 3:656 + 0:0499? ;? ç 0:03 1:953 è? è 13;? ç 0:03 4:354 + 0:07? ;? é 0:03 Te circular cylinder results may be used to ænd approimate values for isotermal and isoæu tubes aving oter cross-sections èe.g. square or triangular pipesè by te use of te ydraulic diameter in te Nusselt and Reynolds numbers. 5
4. aminar and Turbulent Natural Eternal Flow Deænitions of ocal and Area-average Values y @Tè; y 0è, @y èt w, T 1 è, ç 4è è 3, Nu ç y Gr gæ èt w, T 1 è 3 ç, Ra gæ èt w, T 1 è 3 æç, Ra Gr Pr y Gr gæ èt w, T 1 è 3 ç, Ra gæ èt w, T 1 è 3 æç, Ra Gr Pr y For UWF cases, use te midpoint temperature diæerence: èt w è è, T 1 è Flat Plate, Buoyancy-Induced aminar Boundary ayer Flow Ra 14 Ra 14 Ra 14 Ra 14 Ra 14 Ra 14 Nu Ra 14 Nu Ra 14 0:507 10 4 égr é 10 9 ocal, UWT, Pr!1 0:6004Pr 14 10 4 égr é 10 9 ocal, UWT, Pr! 0 0:507 è0:49p rè 916 i 49 10 4 égr é 10 9 ocal, UWT, 0 épré1 0:567 10 4 égr? é 10 9 ocal, UWF, Pr!1 0:69Pr 14 10 4 égr? é 109 ocal, UWF, Pr! 0 0:567 è0:437p rè 916 i 49 10 4 égr? é 109 ocal, UWF, 0 épré1 0:6703 è0:49p rè 916 i 49 10 4 égr é 10 9 Average, UWT, 0 épré1 0:7503 è0:437p rè 916 i 49 10 4 égr? é 10 9 Average, UWF, 0 épré1 6
Flat Plate, Buoyancy-Induced Turbulent Boundary ayer Flow Nu 0:150Ra 13 Average; UWT; i 10 9 égr 167 é 10 1 è0:49p rè 916 0 épré1 8 9 é Nu é: 0:85 + 0:387Ra 16 é i 87 è0:49p rè 916 é; 10,1 éra é 10 1 Average; UWT; 0 épré1 ong Horizontal Isotermal Circular Cylinders, aminar and Turbulent Flow 3 0:387Ra Nu D 40:60 16 D + 5 ëè0:559p rè 916 ë 87 0 épré1; 10,5 éra D ç 10 1 Finite Horizontal Isotermal Circular Cylinders, aminar Flow Nu D S? D + 0:518Ra 14 D ëè0:559p rè 916 ë 49 0 épré1; 0 éra D ç 10 9 Isotermal Speres, aminar Flow Nu D + 0:589Ra 14 D ëè0:469p rè 916 ë 49; 0 épré1; 0 ç Ra D é 10 11 5. General Correlation Equation for Arbitrary Isotermal Conve Bodies Nu Nu 0 + F èprèg Ra 14 ; 0 épré1 0 ç Ra ç 10 11 were te caracteristic body lengt is p A and A is te total active orwetted surface area. Te universal Prandtl number function valid for all isotermal conve bodies is given by: 0:670 F èprè ëè0:5p rè 916 ë 49 wic for air èpr 0:71è as te value F èpr 0:71è0:513. Te diæusive limit Nu 0 or sape factor S? wit p A is a wea function of body sape and its aspect ratio. For 7
eample, its range is 3:0 ç S p? A é 7:55 for a solid circular cylinder wose lengt-to-diameter ratio varies from 0 èa circular disè to 100 èvery long cylinderè. For long aisymmetric bodies èe.g. circular cylinder and long square cuboidè te sape factor can be accurately approimated by S p A 4q D lnèdè were D is te diameter of te circular cylinder and it is equal to te geometric-mean of te diameters of te inscribed and circumscribed circular cylinders respectively, and is te lengt. Te body-gravity function G accounts for te buoyancy-induced æow over te conve body. It is a relatively wea function of te body sape and its orientation wit respect to te gravity vector wen p A is used and te comple conve body èe.g. a cuboidè as dimensions èh; W;è wic do not go to zero H 6 0 in te direction parallel to te gravity vector èe.g. a orizontal rectangular plateè, and te oter dimensions èw;è wic are perpendicular to te gravity vector do not go to 1. Oterwise te body-gravity function will lie in te narrow range 0:9 ég p A é 1:1. For eample te body-gravity function for a spere, orizontal cube, and a sort circular cylinder D 1 wit active sides and ends in te orizontal, inclined at 45, degrees and vertical orientations ave te empirical values: G p A 1:03; 0:951; 1:019; 1:004; 0:967 respectively. Tese empirical values are witin 3 è of te teoretical values. Tere are many oter comple conve bodies wic ave body-gravity functions close to unity. Te body-gravity function for a orizontal cuboid èh; W;è were te H,side is parallel to g and te oter two sides èw;è are perpendicular to g may be used to estimate G p A for conve bodies wic ave surfaces wic are parallel and perpendicular to g. Te bodygravity function for cuboids is given by: è 0:65 43 G p W + Hè + W è 43 34 A 18 èhw + H + W è 76 Te above analytic-empirical relationsip reduces to several important special cases. Horizontal Cube, All Surfaces Active: H W 1 G p A 0:984 Horizontal Rectangular Plates, Bot Sides Active: H 0; ç W G p A 0:7665èW è18 ; W ç 1 Horizontal Square Plates, Bot Sides Active: H 0; W G p A 0:7665 8
Vertical Rectangular Plates, Bot Sides Active: 0 G p A 18 èwhè 18 ; 0 é W H é 1 If te vertical plate as one side active, omit te factor 18. Vertical Square Plate, Bot Sides Active: H W; 0 G p A 18 1:0905 If te vertical square plate as one side active, omit te factor 18 and G p A 1. ong Horizontal Square Prisms wit Active Ends: H Wéé G p A 0:856 ç H ç 18 ; H é 10 ong Vertical Square Prisms wit Active Ends: W; 0 ç HW é 1 G p A 14è0:50 + HWè34 è0:500 + HWè 78 Heated Horizontal Rectangular Plates Facing Upward or Downward: H 0; W ç 1 G p A 18 ç W ç 18 facing upward G p A 18 ç W ç 18 facing downward Oter Body Sapes Te body-gravity function for oter body sapes suc as te ænite circular cylinder wit active sides and ends in te orizontal and vertical orientations can be accurately approimated by using te results for te square prism wit active ends in te orizontal or vertical orientations. 9