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UBC Theses and Dissertations

Combined free and forced convection through vertical noncircular ducts and passages Ansari, Saghir A. 1969

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COMBINED FREE AND FORCED CONVECTION THROUGH VERTICAL NONCIRCULAR DUCTS AND PASSAGES by SAGHIR A. ANSARI B.Sc. Eng.(Mech.), Aligarh Muslim University, Aligarh, India, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. in the department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date Jan. 12, 1970 ABSTRACT i i The problem of laminar combined free and forced convection through vertical noncircular ducts and passages i n the f u l l y developed region has been treated. The f l u i d properties are considered to be constant, except the variation of the density i n the buoyancy term of the momentum equation. Pressure work and viscous dissipation terms of the energy equation have been neglected. Heat flux has been considered to be constant i n the flow direction. A general solution to the problem has been obtained i n "the form of i n f i n i t e series containing modified Bessel functions. Two possible thermal boundary conditions on the c i r -cumference of the heated wall have been analyzed, Case 1 - uniform circumferential wall temperature, and Case 2 - uniform circumferential wall heat flux. Information of engineering interest l i k e Nusselt number, heat flux, r a t i o , shear stress r a t i o , temperature distribution on the' wall, velocity and temperature distributions i n the flow f i e l d have been ob-tained for two sets of geometries, namely, (i) flow through regular polygonal ducts, and ( i i ) flow between cylinders arranged i n regular arrays. For flow through regular polygonal ducts, the case of uniform circumferential wall temperature results i n higher values of Nusselt num-bers as compared to the case of uniform circumferential wall heat flux. This difference i n Nusselt number values decreases as the number of sides of the regular polygon i s increased, u n t i l for a c i r c l e i t completely dis-appears. For both the cases, at higher values of Rayleigh number, the Nusselt number i s less sensitive to the number of sides of the polygon. Also, at higher values of Rayleigh number, both the cases tend to produce the same results. For low sided polygons, an increase i n Rayleigh number tends to s h i f t the maximum value of shear stress from the centre of the duct wall towards the apex of the duct. For flow between cylinders arranged i n regular arrays, Case 1 results i n higher values of Nusselt number compared to Case 2, for low spacing ratios. However, as the spacing ratio i s increased, the two cases tend to produce the same results. Cylinders arranged i n equilat-eral triangular arrays produce higher values of Nusselt number compared to those i n square arrays. This difference i n Nusselt number values decreases when the spacing ratio i s high. For higher values of Rayleigh number, however, the results are less sensitive to the type of arrays. Also, at higher values of Rayleigh number, both the cases tend to pro-duce the same results. iv TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWI_DGEMENTS ix NOMENCLATURE l INTRODUCTION 3 FORMULATION OF THE PROBLEM. 9 GENERAL SOLUTION 13 Section I - Regular Polygonal Ducts 17 Section II - Flow Between Cylinders,.... 24 DISCUSSIONS 31 CONSLUSION 46 BIBLIOGRAPHY 84 APPENDIX A - Derivation of Heat Flux Expression 88 APPENDIX B - Derivation of Nusselt Number Expressions 89 V LIST OF TABLES Table Page i 1. Improvement of Nusselt Numbers with Number of Points on the Wall for an Equilateral Triangular Duct and Compari-son with an Exact Solution, for the Case of Uniform Circumferential Wall Temperature... 4 7 2. Improvement of Nusselt Numbers with Number of Points on the Wall for a Square Duct and Comparison with an Exact Solution, for the Case of Uniform Circumferential Wall Temperature.... 4 8 3. Nusselt Numbers Against Rayleigh Numbers for Various Poly-gons Under Uniform Circumferential Wall Temperature.... 49 4. Nusselt Numbers Against Rayleigh Numbers for Various Poly-gons Under Uniform Circumferential Wall Heat Flux 50 5 . Numerical Values of the Function &4>/d N at Various Points on the Wall of an Equilateral Triangular Duct Under Unif-orm Circumferential Wall Heat Flux. 51 6. Numerical Values of the Function d<l>/dN at Various Points on the Wall of a Square Duct Under Uniform Circumferential Wall Heat Flux 5 2 7. Pressure Drop Parameters L Against Rayleigh Numbers for Various Polygons for Both the Boundary Conditions... 53 v i LIST OF FIGURES Figure Page 1. Flow Through Vertical Noncircular Duct 10 2. Coordinate System for Regular Polygons... 19 3. Flow Configuration and Coordinate System for Flow Between Vertical Cylinders 25 4. Cylinders Arranged i n Equilateral Triangular and Square Arrays 26 5. Nusselt Number Against Rayleigh Number For Various Polygons Under Uniform Circumferential Wall Temperature 54 6. Nusselt Number Against Number of Sides for Various Rayleigh Numbers Under Uniform Circumferential Wall Temperature 55 7. Nusselt Number Against Rayleigh Number for Various Polygons Under Uniform Circumferential Wall Heat Flux.... 56 8. Nusselt Number Against Number of Sides for Various Rayleigh Numbers Under Uniform Circumferential Wall Heat Flux 57 9. Nusselt Number Against Number of Sides for Various Rayleigh Numbers Under Both The Boundary Conditions 58 10. Velocity Distribution i n an Equilateral Triangular Duct Under Both The Boundary Conditions 59 11. Velocity Distribution i n a Square Duct Under Both The Boundary Conditions 60 12. Temperature Distribution i n an Equilateral Triangular Duct Under Both The Boundary Conditions 61 13. Temperature Distribution i n a Square Duct Under Both The Boundary Conditions 62 14. Local to Average Heat Flux Ratio for an Equilateral Triangular Duct Under Uniform Circumferential Wall Temperature 63 15. Local to Average Heat Flux Ratio for Hexagonal Duct Under Uniform Circumferential Wall Temperature 64 16. Local to Average Shear Stress Ratio for an Equilateral Triangular Duct Under Both The Boundary Conditions 65 V l l Figure Page 17. Local Wall Temperature Difference For Square and Octagonal Ducts Under Uniform Circumferential Wall Heat Flux 66 18. Pressure Drop Parameter Against Rayleigh Number for Equilateral Triangular Duct Under Both The Boundary Conditions 67 19. Nusselt Number Values for Cylinders Arranged i n Triangular Arrays Under Uniform Circumferential Wall Temperature...... 68 20. Nusselt Number Against Rayleigh Number for Cylinders Arranged i n Square and Triangular Arrays Under Uniform Circumferential Wall Temperature 9^ 21. Nusselt Number Against Rayleigh Number for Cylinders Arranged i n Square and Triangular Arrays Under Uniform Circumferential Wall Heat Flux 7 0 22. Nusselt Number Against Rayleigh Number for Cylinders Arranged i n Square Arrays Under Both The Boundary Conditions 7 1 23. Velocity Profiles Along DC for Cylinders Arranged In Square Arrays Under Uniform Circumferential Wall Temperature. 7 2 24. Velocity Profiles Along AB for Cylinders Arranged i n Square Arrays Under Uniform Circumferential Wall Temperature 7 3 25. Velocity Profiles Along BC for Cylinders Arranged i n Square Arrays Under Uniform Circumferential Wall Temperature 7^ 26. Temperature Profiles along DC, for Cylinders Arranged i n Square Arrays Under Uniform Circumfer-ential Wall Temperature 7 5 27. Temperature Profiles along AB, for Cylinders Arranged i n Square Arrays Under Uniform Circumfer-ential Wall Temperature 28. Temperature Profiles along BC, for Cylinders Arranged i n Square Arrays Under Uniform Circumfer-ent i a l Wall Temperature. 77 29. Local to Average Heat Flux Ratio for Cylinders Arranged i n Triangular Arrays Under Uniform Circum-ferential Wall Temperature 78 Figure Page V 1 1 1 30. Local to Average Heat Flux Ratio for Cylinders Arranged i n Square Arrays Under Uniform Circumfer-ential Wall Temperature 79 31. Local Wall Temperature Difference for Cylinders Arranged i n Square Arrays Under Uniform Circumfer-e n t i a l Wall Heat Flux 80 32. Local Wall Temperature Difference for Cylinders Arranged i n Triangular Arrays Under Uniform Circumferential Wall Heat Flux. 81 33. Local to Average Shear Stress Ratio for Cylinders Arranged in Square Arrays Under Uniform Circumfer-e n t i a l Wall Temperature 82 34. Local to Average Shear Stress Ratio for Cylinders Arranged i n Triangular Arrays Under Uniform Circum-ferential Wall Temperature 83 ACKNOWIJ_<_?ENTS The author would l i k e to express his sincere gratitude to Dr. M. Iqbal, who devoted considerable time and gave invaluable advice and guidance throughout a l l stages of the present work. Sincere thanks are also extended to Dr. B.D. Aggarwala of the Mathematics Department, University of Calgary, for his valuable suggestions. Use of the Computing Centre f a c i l i t i e s at the University of B r i t i s h Columbia and the financial support of the National Research Council of Canada are gratefully acknowledged. NOMENCLATURE A = area of cross-section Cp = specific heat of the f l u i d at constant pressure C| = temperature gradient in flow direction, 3T/3Z = hydraulic diameter = (4 cross-sectional area)/(heat transfer perimeter) F = Q/fCpC-jU, heat generation parameter, dimensionless F = F/L, dimensionless g = gravitational acceleration h Q V = average peripheral heat transfer coefficient L = pressure drop parameter, dimensionless Nu hD^/K, Nusselt number, dimensionless 2 P = dimensionless tube spacing 2 *\ Ra = (p gC C-|3D^^(cu, Rayleigh number, dimensionless Q = heat generation rate R = r/Dn> dimensionless radius s = number of sides of a regular polygon = 4 for square array and 6 for triangular array q a v = average surface heat flux T = temperature u = axial velocity V = u/U, dimensionless axial velocity V = V/L, dimensionless R, 6 = coordinates R = R Cos 6 , dimensionless Z = axial coordinate in flow direction R-j = radius of cylinder P / R - j = s p a c i n g r a t i o * = ( T ' V o r ( T - T a p e x ) pUC C,Dn2 < J > = X / [ 2 - — ^ ] , d i m e n s i o n l e s s t e m p e r a t u r e f u n c t i o n 4 > = L " ' d i m e n s i o n l e s s T = l o c a l w a l l s h e a r s t r e s s Tg v = a v e r a g e w a l l s h e a r s t r e s s M M 1 ' 4 3 , p , K , y = t h e f l u i d p r o p e r t i e s i n s t a n d a r d n o t a t i o n 3 INTRODUCTION In a convective heat transfer process, the density differences arising due to temperature differences give r i s e to free convection effects under a gravitational force f i e l d . In situations where the forces and momentum transport rates are very large, the free convection effects may not be very important, and could be ignored. Such a situation would then be called forced convection. In cases where the buoyancy forces arising due to temperature (or equivalently density) differences are very large, the forced convection effects could be neglected, and the situation could be treated as free convection. However, i n many cases of practical inter-est, both the effects of forced and free convection can be of comparable order. Such a situation i s known as combined free and forced convection. In laminar flows associated with comparatively small i n e r t i a for-ces, the free convection effects are generally important. It i s known that the free convection effects can considerably alter the heat transfer rates. In such situations, i n addition to Grashoff, Reynolds and Prandtl numbers, the parameter describing flow orientation with respect to the gravitational force f i e l d i s also important. One can divide the studies of combined free and forced convection into two broad catagories, (i) external flows, and ( i i ) internal flows. In the present study we w i l l r e s t r i c t our attention to laminar flows through vertical ducts and passages. For flow through v e r t i c a l circular ducts, when the wall temperature varies linearly i n the direction of flow, the circumferential wall temper-ature and heat flux at any section of the duct remain constant. 4 For flow through noncircular ducts, the f l u i d slows down near the corners and has a tendency to carry less heat through these regions. This could result i n higher temperatures near the corners« Conse-quently, neither the wall temperature nor the wall heat flux may be circumferentially uniform at any section of the duct. This rotational asymmetry w i l l depend upon duct configuration and the ratio of c i r -cumferential wall conduction to that of normal conduction to the f l u i d at any point on the wall. Depending upon the relative magnitudes of these parameters, there could be two extreme situations on the boundary, Case 1 - uniform circumferential wall temperature, resulting from large circumferential wall conduction, Case 2 - uniform circumferential wall heat flux, resulting from negligible circumferential wall conduction. The true situation, however, would l i e somewhere in between these two extremes. In the proceeding pages, a brief literature survey i s pre-sented for forced and combined free and forced convection through ve r t i c a l noncircular ducts and passages. As far as flow through pas-sages i s concerned, we w i l l be mainly concerned with cylinders arranged in regular arrays. The survey i s restricted to laminar f u l l y developed flow with constant properties and uniform heat input per unit length i n the flow direction. The survey i s divided under the two boundary con-ditions stated e a r l i e r . Case 1 - UNIFORM CIRCUMFERENTIAL WALL TEMPERATURE We f i r s t treat the forced convection case. 5 Forced Convection Clark and Kays [1]1 used numerical relaxation method to solve the problem for rectangular and triangular ducts. They confirmed their results experimentally for a rectangular duct. Tao [2] has shown that the exact solution of a class of lami-nar forced convection problems with heat sources can be approached by complex variable method. He presented the results for equilateral t r i -angular and e l l i p t i c a l ducts. Tao [3] approached the same problem by using the conformal mapping method, where the particular configuration under investigation i s transformed onto a unit c i r c l e . Cheng [4] by method of point matching solved the problem for regular polygonal ducts. He pointed out that the various points chosen on the boundary to satisfy the boundary conditions could either be equally spaced on the boundary or at equal angular intervals. Siegel and Savino [5] studied the effect of peripheral heat conduction within the heated walls on the wall temperature distributions of a rectangular channel. They pointed out that, as the wall-to-fluid conduction ratio increases, the peripheral heat conduction increases and the wall temperatures become considerably more uniform with a substantial reduction i n peak temperatures. Marco and Han [ 6] pointed out that the equation describing deflection of a thin plate under uniform l a t e r a l load, and simply sup-ported along i t s edges i s similar to the equation describing temperature distribution i n a laminar f u l l y developed pure forced convection problem •Numbers in brackets designate references at the end of the thesis. 6 through a duct of the same cross section as that of the plate. As such, one can borrow the plate theory solutions (whenever possible) to solve for the temperature distribution of the heat transfer problems. Direct application of this solution i s permissible only when the corresponding boundary conditions are also identical. The boundary condition of uni-form circumferential wall temperature has a correspondence with that of a plate which i s simply supported along i t s edges. For flow outside cylinders arranged i n equilateral triangular arrays, Sparrow et al.[7] calculated Nusselt number values as a fun-ction of the pitch-to-diameter r a t i o . They observed that when the pitch-to-diameter ratio was greater than 1,5, the heat flux ratio was uniform over a substantial portion of the circumference of the heated cylinders. On this basis, they pointed out that when the pitch-to-diameter ratio i s greater than 1.5, both the conditions of uniform circumferential wall temperature and uniform circumferential heat flux would be simultaneously achieved. Combined Free and Forced Convection Han L"8J solved the problem of v e r t i c a l rectangular ducts by using double Fourier series. Tao [9,10]suggested a method to solve such problems by introducing a complex function which i s directly re-lated to the velocity and temperature f i e l d s . Exact solutions were then established i n terms of Bessel and associated functions. Agarwal [11] u t i l i z e d Tao's formulation and converted i t into a variational expression. This variational equation was then solved by assuming a suitable polynomial for the temperature function. 7 Aggarwala and Iqbal [12] u t i l i z e d the solutions of membrane vibration to solve the problem of a set of straight v e r t i c a l triangular ducts. They obtained exact analytical expressions i n form of i n f i n i t e series for velocity and temperature. Iqbal et al.[13] approached the same problem for a set of geometries by variational method. Lu [14] pointed out, that for the case of combined free and forced convection also, an analogy exists between the equations des-cribing deflection of a thin plate resting on an e l a s t i c foundation with no deflection along i t s edges, and the equation describing temp-erature distribution i n a laminar f u l l y developed combined free and forced convection problem, Rayleigh number, which determines the relative effect of buoyancy i n the temperature equation i s equivalent to the pressure exerted by the e l a s t i c foundation i n the deflection equation. Using this analogy he solved the problem for concentric annulii and rectangular ducts. There does not seem to be any information available for combined free and forced convection for flow between v e r t i c a l cylinders arranged i n regular arrays. Case 2 - UNIFORM CIRCUMFERENTIAL WALL HEAT FLUX  Forced Convection Unlike the condition of Case 1, very l i t t l e i s known about the condition of uniform circumferential heat flux i n duct flow problems. Eckert et al.ClS] and Yen [16] obtained an exact solution for a circular sector. They noted that the two circumferential boundary conditions (Case 1 and Case 2) could result i n average heat transfer coefficients which may d i f f e r by an order of magnitude. Sparrow and Siegel [17] used variational method to solve for rectangular ducts, 8 while Cheng [18] by using point matching method obtained the results for regular polygons. For flow between cylinders arranged i n regular arrays, there does not seem to be any available reference for this class of boun-dary condition. Combined Free and Forced Convection It seems that there i s no available information for combined free and forced convection through noncircular ducts or passages with uniform circumferential wall heat flux. In the next section, formulation of combined free and forced convection problem i s presented. The problem i s then solved for two sets of geometries (i) flow through regular polygons and ( i i ) flow out-side cylinders arranged i n equilateral triangular and square arrays. FORMULATION OF THE PROBLEM 9 Consider a vertical straight noncircular duct as shown i n Fig. 1. The flow i s considered to be laminar and f u l l y developed (both hydrodynamically and thermally) i n the vertical upwards d i r -ection along the positive Z - axis. Uniform heat flux per unit length i s assumed i n the direction of flow. The viscous dissipation and pressure work terms have been neglected. The f l u i d properties are considered to be constant, except variation of density in the buoy-ancy term of the equation of motion. The f l u i d may contain uniform volume heat source. Under the above mentioned conditions, the di f f e r e n t i a l form of the continuity equation i s identically equal to zero. The momentum and energy equations can be written as, K ^ f l « k ( t L ^ | ^ + ^ *L) . (2) For uniform heat input i n the flow direction, the wall and f l u i d temperature gradients a r e ^ = C i , where Cn i s a constant. In the above equations, density i s being considered variable only i n the buoyancy term of equation (1). This assumption i s known C19] to be valid as long as the density variations i n the flow f i e l d are small. Under this r e s t r i c t i o n , the equation of state for density in the linear form can be written as, o q M FIGURE 1 - Flow Through Vertical Noncircular Duct 11 where suffix 'ref* denotes condition of f l u i d at a reference point on the duct wall. For the case of uniform circumferential wall temperature, this reference point could be anywhere on the duct wall. For the case of uniform circumferential heat flux, the apex of the duct was chosen as a reference point. The reference wall temperature T^f^ can be defined as, Tre<. = To + z . where T Q i s the reference temperature of the duct at Z = 0. By choosing the following dimensionless variables, R= r/D h , V = U ./U , 4> = C T - Tr e f .)/CfUC t >C,D^/At; , and inserting equation (3) i n (1), the momentum and energy equations can be nondimensionalized as, Vz V + - R a <$> = - L , ( „ ) v 2 4> - V = - F • • ( 5 ) where, £ R 2 " R 2>R ^e^ In equations (4) and (5), the Rayleigh number Ra, and the heat generation parameter F are prescribed quantities, while V, <^ >, and L are the three unknowns. In order to solve equations (4) and (5) we therefore need another equation, which i s obtained from the continuity consideration i n the duct. For constant properties, this continuity equation i s , Jj u oU « / / U oLA Jj VcLA = ffjL II x, . . IT . . (6) 1 2 Although the equations (4) to (6) have been developed for flow through a ve r t i c a l noncircular duct, these equations w i l l remain unchanged for flow outside cylinders with negligible side effects of the shell wall. In the present analysis, equations (4) to (6) have to be solved for the following two sets of boundary conditions on the circumference, Case 1 - Uniform Circumferential wall temperature V =4>= 0 at the boundary . (7) The above thermal boundary condition results from large circumferential wall conduction. Case 2 - Uniform circumferential wall heat flux ^ V = 0 at the boundary , • = 0.25C1-F) at the boundary , 4* = 0 at the apex . ^ (8) 4^> This thermal boundary condition - r — = 0,25(1-F) results from small circumferential wall conduction and i s derived i n Appendix A. A general solution to equations (4) to (6) w i l l be obtained in the next section. 13 GENERAL SOLUTION In equations (4) and (5), pressure drop parameter, L i s an unknown constant. Since L i s independent of the coordinate system, equations (4) and (5) can be divided by L to obtain, V 2*? - V • , (10> where ?  \j - , <$> - <*>/L ^ P= FA- • *, V* Define = V - F and / = (Ra) , and combine equations (9) and (10) to give, A general solution to this biharmonic equation (11) can be written from McLachlan [2 C] as, o o + C ^ ^ O l M * t>M K ^ ^ r O ] S i » me. (12) Once V]_is known, the nondimensional temperature difference can be obtained from equation (9) as, (13) where, V V = \ i YH~ A ~ ^ • ^ W ^ W ^ R ) - (V K „ l m + D m kir, C^)J Sin This completes the general solution of equations (4) to (6), and the problem reduces to determining the unknown coefficients Am, Cm and Dm and pressure drop parameter L. These coefficients are determined by point matching method, where the boundary conditions are sa t i s f i e d exactly at a prechosen number of points. The method of point-matching would be discussed i n detail while dealing with 1he particular configuration. Once these coefficients are known, the problem i s completely solved and the nondimensional velocity and temperature at any point can be evaluated as, \j = C N 1 + F ) L , I ( I S ) $ - <*> u J Having obtained the velocity and temperature functions, one can obtain the following information of engineering interest. Nusselt Numbers Nusselt number, which signifies energy convected from a surface can be written as, N u = = — — , (16) h k TM-Tb 15 where T w i s the average wall temperature at any section of the duct and T]j i s the bulk temperature of the f l u i d . For uniform circumferential wall temperature, Case 1, equation (16) in the nondimensional form can be written^ as, 1 » mx where, - . » J J 4 >VCLA <p - ii . (18) For the case of uniform circumferential wall heat flux, Case 2, the nondimensional form of equation (16) i s ^ , I - F Nu -TT/S . ' (19) o iwau. 1 mx J Local Heat Flux Ratio For Case 1, since the heat flux varies circumferentially, the local to average heat flux ratio i n the nondimensional form can be ex-pressed as, Local Shear Stress Ratio The local to average shear stress ratio can be written as, 2For details please see appendix B, Equations (12) to (21) are the general solutions to the prob-lem of combined free and forced convection through a v e r t i c a l arbitrary shaped duct, under the assumptions made. In the next two sections, I and I I , we w i l l specialize i n two sets of geometries, namely, regular polygonal ducts and flow outside cylinders. SECTION I Regular Polygonal REGULAR POLYGONAL DUCTS 18 By taking the coordinate system as shown i n Fig. 2, only "the area OCB needs to be considered, because of the symmetry of the regu-l a r polygons. Since the problem i s even i n 6 , the terms i n equation (12) containing Sin n S w i l l vanish. In addition, since the velocity i s f i n i t e at the centre 0, the ker and kei functions w i l l not exist. Hence equation (12) reduces to V, = £ [ A m t«-JnC"|R) + Bm\»elm('l«]CoS m6. (22) From physical considerations, i^L- - 4-^" - 0 a t 0^0 and 0 =7 T/ s (23) where s i s the number of sides of a regular polygon. The condition expressed by equation (23) can only be sati s -f i e d by equation (22), i f m = ns, where n i s a summation index. Hence, equation (22) can be written as, V, - V -F -I A n s W*- Co S » s e K» = © and oo C O 25) 19 20 The point-matching method used to evaluate the unknown coef-ficients A n s and B and the pressure drop parameter L, i s explained below for the two boundary conditions. SOLUTION FOR F = 0 For the time being take F = 0, which i s true for most of the f l u i d s . Case 1 Uniform Circumferential Wall Temperature The boundary conditions (7), can be rewritten as, \y Q at the boundary t (26) <$> — 0 at the boundary, which results i n , 2 _ \/ - - 1 at the boundary . (27) Inserting equation5(24) and (25) i n equations (26) and (27) one obtains, 0 = JZ. A „ s t > e r „ i C ' V O Cos ns© n = o o — o C*3 + ^ 1 E > n & ^ e - ^ s C ^ R ) C o s « s e . For n number of points on the boundary BC ( f i g . 2), equations (29) (28) and (29) w i l l result i n 2n number of equations i n 2n number of un-knowns. These equations are of linear algebraic type and were solved on a d i g i t a l computer by using Gauss elimination method. Once these equations 21 are solved, the coefficients Ans and Bns are known, and the pressure drop parameter, L can then be evaluated from the integral form of con-tinuity equation (6), , I U A L_ =. • (30) i i V d U Case 2 Uniform Circumferential Heat Flux The boundary conditions (8) can be rewritten as , \J — O at the boundary , (31) ^ 4> _ 0'2~5 (|_p*) at the boundary . (32) a n L For the coordinate system shown i n Fig. 2, = Cos© J ^inl (33) 6u c>H R In equation (32), since the unknown L appears simultaneously, n number of points on the boundary BC w i l l result i n (2n + 1) number of unknowns with only 2n number of equations. Therefore, one more equation i s needed, which i s given by the definition of the nondimen-sional temperature difference, <^> — 0 at the apex, which results i n , 2. ^7 \j - - \ at the apex . (34) Now equations (31) to (34) w i l l result i n (2n + 1) number of linear algebraic equations i n (2n + 1) number of unknowns, and can be solved. It may be noted here, that the continuity equation i s not ex-p l i c i t l y required to evaluate L. The reason for this i s that the con-tinuity requirement has been u t i l i z e d i n deriving equation (32). 22 SOLUTION FOR F # 0 In the event, results are required for a f i n i t e value of F other than zero, F i n equation (24) becomes an unknown constant. For Case 1, the point matching method described above can be followed. In this situation n number of points on the boundary w i l l result i n 2 n number of equations i n (2n + 1) number of unknowns. In order to solve these equations, one therefore needs one more equation. This (2n + 1) equation i s provided by the integral form of the continuity equation, For Case 2, however, the procedure explained for F = 0 can be followed exactly. S E C T I O N n Flow Between Cy l inders Arranged in Regular A r r a y s FLOW OUTSIDE CYLINDERS 24 Fig. 3(a) shows the system under study. The cylinders between which the f l u i d flows are considered to be ve r t i c a l and arranged i n equi-l a t e r a l triangular or square arrays (Fig. 4). The side effects of the shel l wall on the flow pattern i s considered to be negligible. As such, the flow can be considered perfectly symmetrical about each cylinder. Due to the symmetry of flow configuration only the shaded area of typical element shown i n Fig. 3(b) need to be considered. This typical element i s a four or six sided regular polygon with a central c i r -cular hole, s being four or six depending upon whether we are dealing with a square or a triangular array. For this flow configuration, the governing equations (4) to (6) and the general solution (12) to (14) remain the same. We now specialize the solution for this configuration. Since the problem i s even i n 0 , the terms of equation (12) con-taining Sin n © w i l l vanish. Also from physical considerations, -=.0 at 0 = 0 and Q = fr/ s . (36) 2>e The conditions of (36) would be satis f i e d by equation (12) only when m = ns, n being summation index. Hence the solution for V, , for this configuration can be written as. 25 ^3 K3 t Ul a) Flow Configuration b) Typical Element FIGURE 3 - Flow Configuration and Coordinate System for Cylinders Arranged i n Regular Arrays 26 a) Tr iangular Array b) Square Ar ray FIGURE 4 - Cylinders Arranged i n Regular Arrays The temperature difference, c £ > c a n again be obtained from equation (13) which i s where, " C n s K % S ^ R ) + D » S K e r C^R)]C« n S 6 } • (38) Equations (37) and (3 8) are the solutions of the problem under consideration. The unknown coefficients of these equations have to be determined by satisfying the boundary conditions on the sides AD and BC of the typical element. From the geometrical and physical considerations of the problem, we have, ^ — — - 0 along side BC of the element For the typical element shown i n Fig. 3(b) _ Cos © ^ S\n9 £ (39) The condition of no s l i p on the wall results i n , V - 0 at R = R x The two thermal boundary conditions considered are, Case 1 - Uniform circumferential wall temperature, <^> — O at R = Ri Case 2 - Uniform circumferential wall heat flux, (40) (41) at R = % (42) £ R _ 28 and ,+'=0 at R = and G - Vf/s - (43) The condition (43) simply states that the temperature difference i s zero at the chosen reference point. The unknown coefficients A ^ , B n s , and and the pres-sure drop parameter L, are evaluated by point matching procedure. This procedure could be followed i n two different ways. DIRECTION I This procedure i s similar to the one described for Regular Poly-gons, except that the boundary conditions (39) to (43) have to be s a t i s -f i e d at two sides AD and BC of the typical element under consideration. Solution for F = 0 By taking n and m number of points on the boundaries AD and BC respectively, the conditions of Case 1 (39), (40) and (41) would result i n 2(n+m) linear algebraic equations i n 2(n+m) unknowns. The pressure drop parameter L, can then be evaluated from the integral form of continuity equation (6), |j (44) //VctA For Case 2, however, since L appears i n (42) as an unknown, n and m number of points on the boundaries AD and BC would result i n 2(n+m) equations in[2(n+m)+llunknowns. The [2(n+m)+l]^1 equation i s given by (43). The equations are again of linear algebraic type and can be solved. For point matching, one could take either n = m or n i m. The tot a l number of equations and unknowns would be i n any case equal to 2(n+m) or 2(n+m)+l depending upon Case 1 or Case 2. However, i n the present anal-y s i s , n was taken to be equal to m. 29 Solution for F t 0 In case solution i s required for a f i n i t e nonzero value of F, the basic approach outlined for Regular Polygons could be followed i n conjunc-tion with the one described above. DIRECTION II It i s however, possible to render the analysis mathematically more accurate by satisfying the thermal boundary conditions (41) or (42) exactly at the cylinder wall, R = R^ , and point matching only at the boundary BC of the element. This method i s demonstrated below for the case of uniform circumferential wall temperature with no internal heat generation. F o r V = 4» = 0 at R = R^  , equations (37) and (38) can be written as, for n = 0 A 0 b e r OlR,) + B 0 be.coC^R.,)4-C^kelrD0l*«> -t- L\KeleCl«,) = O , (45) -ADfc>*lo0iO+ B > e i i O i « . ) -C^.Ove.) + T>.^c^*,>l=.o, (46) similarly for n = 1 A ster s C7<e,)+B s^eo sC^0 + C^r^i) + D s K e c $ C ^ . ^ O , (47) - A s b e S CT*,) + B s ber^R.) - C£K<U^Rt) * t> sKerC^R^O , (48) and similar set of equations can be written for higher values of n. From (45) and (46) AQ and B Q can be expressed i n terms of C Q and D Q . Similarly, from (47) and (48), Ag and B s can be expressed i n terms of C s and D s . Therefore, i n general, A n s and B n s can be expressed i n terms of C^g and D^ . The expressions of A ^ and B n s i n terms of C n s and D when inserted i n (37) and (38) give velocity and temperature functions 30 in terms of only two sets of unknown coefficients C n s and D n S . These coef-ficients can then be obtained by point matching only at the remaining boundary BC of the element shown. For Case 1, the point matching procedure, for n number of points on the boundary BC satisfying condition (39) w i l l result i n 2n number of equations i n 2n number of unknowns. For Case 2, however, n number of points on the boundary BC would result i n 2n number of equations i n (2n+l) unknowns. The additional un-known would be L. One therefore needs another equation to solve the sys-tem. This (2n+l)*^ equation i s given by (43). Now there are as many equations as number of unknowns. This procedure can also be used for nonzero values of F. In such a situation, the basic approach outlined e a r l i e r for F i 0 could be f o l -lowed. In the proceeding section we w i l l discuss the results of the numerical computations. 31 DISCUSSIONS The results and discussions of the present work have been pre-sented in the proceeding pages under three major subsections. The f i r s t deals with the computational details, while i n the second and third sub-sections, results are discussed for the two sets of geometries, regular polygonal ducts and flow outside cylinders. SOLUTION DETAILS The accuracy of the point-matching method w i l l depend upon the number of points taken on the boundary to satisfy the boundary conditions. In general, more number of points give better results. In the present work, however, this i s not quite true. Since i n the velocity and temp-erature expressions, (12) and (13), each term of these equations i s an i n f i n i t e series, the accuracy of the results w i l l also depend upon the convergence of these series. These terms, which are i n the form of modi-fi e d Bessel functions, ber n s(7R), b e i ^ C l R ) , etc., can be evaluated from expressions i n the form of i n f i n i t e series given i n McLachlan [20], There are, however, following two factors i n the evaluation of these functions, which affect their convergence. (a) value of the argument 1R, and (b) value of the suffix ns. These functions were evaluated on an IBM d i g i t a l computer. When the suffix ns was ^ 14, the convergence was found satisfactory for values of argument ^ 10. For most of the values of arguments and suffix, 15 to 20 terms of the i n f i n i t e series were found satisfactory to achieve desir-able convergence. 32 The numerical value of the argument % R imposes a restriction on the value of Rayleigh number, since L= (Ra) . In this study R was nondimensionalized i n such a way that R ^ l . As such, i t was possible to take the maximum value of 1 as 10, which corresponds to Ra = 10000, In the suffix ns, n refers to the number of points taken on the boundary and s to the number of sides of the polygon. Therefore, for example, for higher sided polygons, the number of points that can be taken on the boundary have to be relatively less, as ns has to be less than or equal to 14. In the present work, a study was made on the effect of number of points and their distribution on the boundary. It was observed that the results improved as the number of points were increased (Tables 1 and 2), so long as the numerical values of argument and suffix did not exceed the limits mentioned ear l i e r . The present results were found to be re l a t i v e l y close to the available results (wherever applicable), when the points on the boundary were evenly distributed with respect to their angular position. In the next subsection, results are presented for regular polygonal ducts. 33 REGULAR POLYGONAL DUCTS Nusselt Numbers Under Uniform Circumferential Wall Temperature The accuracy of the present point-matching method could be det-ermined by comparing the present results with those available i n the l i t -erature, wherever i t i s possible. Table 1 presents the Nusselt numbers obtained by the present analysis with different number of points, as com-pared to those of an exact solution by Aggarwala and Iqbal [12] for an equilateral triangular duct. This table shows that as we increase the number of points on the wall BC,the results improve. This table clear-l y shows that with only four points the results are very close to those of the exact solution. However, due to the limitations of the suffix ns, for an equilateral triangular duct, one cannot take more than five points on the boundary. A similar comparison i s given i n Table 2 for a square duct. The results were compared with an exact solution by Han [ 8 ] 3 . This table i gives additional evidence that the present point-matching method i s quite accurate. Figure 5 presents the variation of Nusselt number against Rayleigh number for different number of sides. This figure shows that as the Rayleigh number increases, the effect of number of sides on the Nusselt number d i -minishes. Figure 6 gives the plots of variation of Nusselt numbers against number of sides for some values of the Rayleigh number. From this figure one can clearly note how the Nusselt numbers attain asymptotic values for 'In Table 3, of Han, these Nusselt numbers ard given as 3.69, 4.27 and 9.46 respectively. A recalculation of Han's expressions shows that some of the values i n his Table 3 were somewhat i n error. 34 various amounts of buoyancy effects. Table 3 l i s t s Nusselt number values. It may be added here that i n this table the values of Nusselt number at Ra = 1 should closely correspond to the results of pure forced convection through regular pol-ygonal ducts. Comparison with Cheng's [4] values indicates that this indeed i s the case. Nusselt Numbers Under Constant Circumferential Wall Heat Flux As indicated e a r l i e r , for combined free and forced convection through v e r t i c a l non-circular ducts, there does not appear to be a v a i l -able any study corresponding to the case of uniform peripheral wall heat flux. As such, the accuracy of the present results i s d i f f i c u l t to estimate. However, at Ra = 1, the Nusselt number values given i n Table 4 agree closely with the corresponding values given by references [17] and [18] for pure forced convection case. In the point-matching method, the unknown coefficients of equations (31) to (34) are obtained by satisfying the boundary conditions exactly, at a prechosen number of points on the boundary. Since the boundary conditions are not satisfied on the entire boundary, the numer-i c a l values of these functions V and e>4>/^ N, when evaluated at different points on the boundary could be different from each other. As such, to establish the accuracy of the present results for combined free and forced convection, the numerical values of these functions were evaluated at various points on the wall BC. The numerical values of the -thermal bound-ary condition 0«25 are presented i n Tables 5 and 6 for equilateral triangular and square ducts respectively. These values are obtained by taking 4 points on the boundary BC to satisfy the boundary conditions, and 35 are evaluated at 10 points on the wall BC, which are equally spaced with respect to their angular positions. These tables show that at low values of Rayleigh number, the fluctuations i n the numerical values from 0.25 (exact value) are negligible. At higher values of Rayleigh numbers, at some points there are some fluctuations. These fluctuations, however, do not increase with Rayleigh number. I t was also observed that these fluctu-ations were considerably reduced, when more points were taken to satisfy the boundary conditions. In a similar fashion, the satisfaction of the condition V = 0 on the wall was also studied and was found very satisfactory. Therefore, i t appears that the results obtained from the present point-matching method for case of uniform circumferential wall heat flux are also quite accurate. The Nusselt numbers are plotted as a function of Rayleigh num-ber i n Fig. 7. This figure shows that at higher values of Rayleigh numbers, the Nusselt numbers are less sensitive to the number of sides of the reg-ular polygon. The variation of Nusselt numbers against number of sides for various values of Rayleigh numbers i s plotted i n Fig. 8. Comparison of Nusselt Numbers for the Two Boundary Conditions For laminar forced convection through regular polygonal ducts, Cheng [18] obtained lower values of Nusselt numbers for the case of uniform circumferential wall heat flux compared to the case of uniform circumferential wall temperature [4], Yen [16] in a study of forced convection through wedge shaped passages also observed that Case 2 re-sults i n lower values of Nusselt numbers compared to Case 1. 36 Fig. 9 presents the variation of Nusselt numbers against num-ber of sides for various values of Rayleigh numbers. This figure clear-l y shows that the case of uniform circumferential wall heat flux results i n lower values of Nusselt numbers, as compared to the case of uniform circumferential wall temperature. It i s also observed from this figure that for a given value of Rayleigh number, the influence of the increas-ing number of sides on the Nusselt number values diminishes e a r l i e r for Case 1 as compared to Case 2, Velocity Distribution Fig. 10 presents the velocity distribution i n an equilateral triangular duct for both the boundary conditions, for various buoyancy effects. Fig. 11 presents the same for a square duct. For the case of Uniform circumferential wall temperature, these profiles were confirmed from the available exact solutions. For the case of uniform circumfer-ential heat flux, however, no verification could be made. From both these figures i t i s noted that as Rayleigh number i s increased, the f l u i d near the wall accelerates, and to satisfy the con-tinuity condition, the f l u i d near the centre of the duct slows down. In most cases, a flow reversal was observed at the centre of the duct when the Rayleigh number was increased to about 7000, although the net flow remains in the upward direction. Fig, 10 also shows that for equilateral triangular duct, as the Rayleigh number increases, the f l u i d near the corner C of the duct accelerates more than that near the centre, B, of the duct wall. Also, for the same buoyancy effect, Case 1 gives higher velocity gradients near B compared to Case 2, Whereas, near the corner of the duct, C, Case 2 gives higher velocity gradients compared to Case 1. Temperature Distribution Figures 12 and 13 present, for both the Cases, the temperature distribution i n equilateral triangular and square ducts for different values of Rayleigh number. For Case 1, these profiles were compared with those of the available exact solutions, [ 12] and [ 8]. This comparison has shown that the temperature distribution obtained by the point-matching method agrees very closely with those obtained from the exact solutions men-tioned. Due to this closeness of the results, no plots are presented showing the comparison of the data. For Case 2, however, no such com-parison could be made, due to unavailability of any other solution. Figures 12 and 13 show that for both the Cases, as the Ray-leigh number increases, the temperature distribution i n the duct tends to become uniform. These figures also show that for Case 1, higher values of Rayleigh numbers produce higher temperature gradients at the wall. However, for Case 2, these gradients remain constant. Local Heat Flux Ratio For the case of uniform circumferential wall temperature, the lo c a l heat flux distribution according to equation (20) has been eval-uated and for two particular geometries are plotted i n Figs. 14 and 15. Figure 14 i s for the equilateral triangle and shows that while at low values of the Rayleigh number, the maximum value of lo c a l heat flux occurs at point B; at a high value of the Rayleigh number, the heat flux becomes uniform over substantial portion of the wall BC. Figure 15 i s for an hexagonal duct and shows that for this geometry, the local heat 38 flux ratio i s relatively less sensitive to the free convection effects. It can be noted from both the Figs. 14 and 15 that the buoy-ancy effects increase the local heat flux ratio at the apex C while they reduce the same at point B. According to expectations, i t has been observed that as the number of sides i s increased, the differences i n local heat flux ratio are reduced, u n t i l for a c i r c l e they disappear completely. Local Shear Stress Ratio The local shear stress distribution has been evaluated from equation (21). Both the Cases 1 and 2 have been analyzed. Figure 16 presents, for an equilateral triangle, the local shear stress ratio for both the boundary conditions. This figure shows that while for low values of the Rayleigh number, the shear stress r a t i o i s maximum at point B, at high values of the buoyancy effects this maxima shifts to-wards the apex C. This rather unexpected result i s borne out by c a l -culations made from the exact solution given i n [8]. Figure 16 also presents a comparison of shear stress ratios between the two cases. I t i s noted from this diagram that at high values of the buoyancy effects, the condition of uniform circumferential wall heat flux produces lower shear stress ratios at the point B while, i t produces higher values of the same near the apex C. As the number of sides i s increased, the variations i n local shear stress ratios are reduced whether or not there are buoyancy ef-fects present. This applies to both the circumferential boundary con-ditions under consideration. The differences i n local values reduce very rapidly as the number of sides are increased from three. Local Wall Temperature For the case of uniform circumferential wall heat flux, the variation of circumferential wall temperature difference <|> has been studied. With or without buoyancy effects, the maximum wall tempera-ture difference occurs at point B. As the number of sides i s increased, the effect of buoyancy on lo c a l wall temperature variation diminishes, u n t i l for the c i r c l e i t completely disappears. Local wall temperature distribution for four and eight sided regular polygonal ducts are shown in Fig. 17. Pressure Drop Parameter Pressure drop parameter L as a function of the Rayleigh number and number of sides has been evaluated for the two wall boundary condi-tions. It i s observed that when the number of sides i s small and the Rayleigh number i s high, the uniform circumferential heat flux condition results i n higher values of the pressure drop parameter compared to the case of uniform peripheral wall temperature. Table 7 l i s t s some values of the pressure drop parameter for a number of polygons under the two wall boundary conditions. Figure 18 presents the pressure drop behav-iour for an equilateral duct for the two cases, and clearly shows that the uniform wall heat flux condition results i n higher pressure drops. A brief presentation of the results of the regular polygonal ducts i s available i n reference [21], In the next subsection, results are discussed for flow outside ve r t i c a l cylinders arranged i n regular arrays. 40 FLOW OUTSIDE CYLINDERS Nusselt Numbers Under Uniform Circumferential Wall Temperature From the results of the present analysis, we f i r s t of a l l present the Nusselt number results and compare them with those of reference [7] for the limiting situation when Ra -* 0. In the present analysis, we can compute the results for Ra as low as 0.01 but not for Ra=0. However, i t i s known that at very small values of the Rayleigh number, of the order of unity, the heat transfer results remain the same as those for pure forced convec-tion. In reference [7], the forced convection results presented are only for triangular arrays and for the condition of uniform circumferential wall temperature. Figure 19 presents the Nusselt number plots against the spacing ra t i o with the Rayleigh number as a variable. For the purpose of comparing the present result with those of reference [7, Fig. 3], two types of Nusselt numbers have been plotted in this figure. One with the hydraulic diameter as a base and the other with the tube diameter as a base. The significance of these two types of Nusselt numbers i s that when the Nusselt numbers are based on hydraulic diameter, then an increase i n the tube spacing ratio causes an increase i n the Nusselt number values. This i s because of the fact that the hydraulic diameter increases as the spacing ratio i s increased. In order to study the heat transfer rate, i t would probably be worthwhile looking at the Nusselt number values based on the tube diameter. This figure shows that an increase i n the spacing ratio causes a decrease i n the Nusselt number values, i f i t was based on tube diameter. In figure 19, attention has been directed on the spacing range of 1.3 to 2.5. In tubular exchangers, spacing r a t i o of 1.3 i s generally used. 41 Now comparing Figure 19 with Figure 3 of reference [7], i t i s noted that Ra = 1, the results of the present analysis agree with those of I"?]. We now present a detailed discussion of Nusselt number variation under various parameters. In the following discussions, Nusselt numbers have been based on the hydraulic diameter only, as this i s a common prac-t i c e i n engineering. For Case 1, figure 20 presents the Nusselt number results for square and triangular arrays with Rayleigh number and spacing ratio as parameters. This figure shows that as the spacing ratio increases, the effect of free convection on Nusselt number decreases. However, the free convection effects influence the Nusselt number substantially at lower spacing ratios, a fact, which i s also apparent from Figure 19. It i s also noted that the effect of increased buoyancy i s to diminish the d i f -ferences i n Nusselt number between the square and the triangular arrays. However, at lower values of the Rayleigh number, triangular array results i n higher values of the Nusselt number. Nusselt Numbers Under Uniform Circumferential Wall Heat Flux For the boundary condition of Case 2, the Nusselt number plots are presented i n Figure 21 and are essentially of the same nature as those for Case 1, Figure 20. Comparison of Nusselt Number Values for the Two Boundary Conditions Figure 22 presents for a square array, the differences between the Nusselt number values for the two boundary conditions, Case 1 and Case 2. As predicted (but not evaluated) by reference [7], the uniform circum-ferential wall temperature condition results i n higher values of the Nusselt 42 number. This i s true when spacing ratio i s low. However, for higher spacing ratios, the results become almost identical. At higher values of the Rayleigh number, the differences i n the Nusselt number for the two boundary conditions diminish and this result i s i n accordance with the study of heat transfer through the regular polygonal ducts. The present analysis shows that for triangular arrays, the differences between the Nusselt number values for Case 1 and Case 2 are much smaller as compared to those i n square arrays. Velocity Distribution To gain some insight into the flow behaviour and eventually the shear stress distribution at the wall, velocity profiles are presented i n Figures 23 to 25. These profiles are along lines DC, AB and BC respectively of the element, Figure 3(b). The velocity profiles are presented for only two spacing ratios, 1.3 and 2. Figure 23 shows velocity profiles along DC. As the buoyancy effects are increased, the velocity at the centre of the system of tubes, point C, i s reduced and to satisfy continuity, the velocity near the wall increases. For spacing r a t i o of 1.3, i f the Rayleigh number i s increased enough, say Ra 6000, i t i s theoretically possible to attain flow reversal (negative velocity) at point C. However, i t i s doubtful whether at such high values of the Rayleigh number, laminar flow w i l l exist any longer. Figure 24 presents the velocity profiles along AB. From this diagram, an interesting observation i s that the increasing values of the Rayleigh number increase the velocity at point B. As spacing ratio increases, the influence of buoyancy on the velocity p r o f i l e along AB diminishes. 43 The velocity profiles along BC are plotted i n Figure 25. These velocity profiles are i n accordance with the observations made regarding Figures 23 and 24, Temperature Distribution For Case 1, temperature profiles are presented i n Figures 26 to 28 for a square array. These profiles are along sides DC, AB and BC of the typical element shown i n Fig, 3(b). These profiles are again only for two spacing ratios, 1.3 and 2. Figure 26 presents the temperature distribution along DC of the element. This figure shows that for small spacing ratios, an increase i n Rayleigh number causes the temperatures to become uniform along the side DC. This figure also shows that for higher spacing ratios, the temperature profiles along DC are less sensitive to Rayleigh number. Figure 27 presents the temperature profiles along the side AB. This figure shows that for small spacing ratios, an increase i n Rayleigh number increases the temperature difference at point B, This figure also shows that the temperature distribution along AB, for higher spacing ratios, i s less sensitive to Rayleigh number. The temperature profiles along BC are presented i n Fig. 28, These profiles are i n accordance with the observations made regarding Figures 26 and 27. Wall Heat Flux Ratio For Case 1, the variation of heat flux ratio on the tube wall AD i s presented i n Figures 29 and 30, for square and equilateral triangular 44 arrays respectively. For Ra=l, the values of loc a l to average heat-flux ra t i o for an equilateral triangular array were confirmed from reference C7]. As expected, the lower spacing ratio of 1,3 gives larger v a r i -ations of the circumferential heat flux. Again the increasing buoyancy effects reduce these circumferential variations. At the higher spacing rat i o of 2, Figures 29 and 30 indicate that, the circumferential heat flux has almost become uniform and that, there i s very l i t t l e added i n -fluence of the buoyancy effects. As expected, the minimum value of heat flux occurs at point A. When figure 29 i s compared with 30, i t i s noted that the varia-tion i n the heat flux r a t i o i s higher for a square array compared with that of an equilateral triangular array. Wall Temperature Distribution For Case 2, i t i s important to determine the temperature d i s t r i -bution on the tube wall. This i s shown i n Figures 31 and 32 for square and triangular arrays respectively, for two spacing ratios, 1.3 and 2. When these two figures are compared with each other, i t i s noticed that the square array gives large temperature variations of the tube wall AD, as compared to a triangular array. These figures also show that at the lower spacing ratios, there i s a wide temperature variation. However, this temperature variation i s reduced when the buoyancy effects are increased. In addition, at higher spacing ratios, of the order of 2, temperature be-comes almost uniform over the tube circumference and. i s not perceptibly influenced by the buoyancy effects. As expected, the maximum temperature occurs at point A of the tube wall. 45 Wall Shear Stress Ratio The wall shear stress distributions for Case 1 are shown i n Figures 33 and 34 for square and triangular arrays respectively, for two spacing rati o s , 1.3 and 2. When these two figures are compared, i t i s found that the triangular array results i n smaller shear stress variations compared to a square array. Also at smaller spacing ratios, 1.3, and for situations approaching forced convection, the shear stress i s minimum at point A and maximum at D. As the Rayleigh number i s i n -creased, the shear stress distribution becomes uniform. When the Ray-leigh numbers are increased further, the situation gets reversed and now the maximum value of the wall shear stress occurs at A and the minimum at D. For higher spacing ratios, the shear stress i s uniform over the portion AD of the element and i s less sensitive to the Rayleigh number. For Case 2 boundary condition, the behaviour of the wall shear stree remains essentially similar to the one explained above. A br i e f presentation of the results of the flow outside c y l i n -ders i s available i n reference [22], CONCLUSION 46 A general solution to the problem of laminar combined free and forced convection through vertical noncircular ducts and passages has been obtained i n form of i n f i n i t e series containing modified Bessel functions. Results have been obtained for two sets of geometries, (i) regular polygonal ducts, and ( i i ) flow between cylinders arranged i n regular arrays. Two possible thermal boundary conditions have been analyzed, Case 1 - uniform circumferential wall temperature, and Case 2 - uniform circumferential wall heat flux. In general ,the case of uniform circumferential wall tempera-ture results i n higher values of Nusselt number as compared to the case of uniform circumferential wall heat flux. However, at higher values of the Rayleigh number, both the boundary conditions tend to produce the same results, TABLE 1. Improvement of Nusselt Numbers with Number of Points on the Wall for an Equilateral Triangular Duct and Comparison with an Exact Solution, for the Case of Uniform Circumferential Wall Temperature. •'•I NUSSELT NUMBERS Ra Number of Points on the Wall BC, Present Analysis Exact 2 3 4 oo i U L i o n [12] 10 3.1250 3.1248 3.1248 3.1249 100 . 3.2390 3.2472 3.2475 3.2475 500 3.5547 3.7489 3.7537 3.7537 ,1000 3.6921 4.2846 4.3029 4.3029 5000 1.6686 6.4505 6.7896 6.7971 TABLE 2. Improvement of Nusselt Numbers with Number of Points on the Wall for a Square Duct and Comparison with an Exact Solution, for Uniform Circumferential Wall Temperature. Ra NUSSELT NUMBERS Number of Points on the Wall BC, Present Analysis Exact Solution [*] 2 3 4 3.7702 3.7092 3.7004 3.6962 10ir4 4.5891 4.4722" 4.4492 4.4372 IOOTT 4 7.4134 8.3815 8.3342 8.2716 00 TABLE 3. Nusselt Numbers Against Rayleigh Numbers for Various Polygons for Uniform Circumferential Wall Temperature. Ra ; NUSSELT NUMBERS Number of Sides of Polygon 3 4 5 6 7 8 12 Ci rcl e (Exact Solution) 3.11 3.61 3.87 4.01 4.10 4.16 4.27 4.36 100 3.25 3.70 3.95 4.08 4.17 4.23 4.34 4.43 500 3.75 4.Q6 .4.26 4.37 4.45 4.51 4.60 4.69 1000 . 4.30 4.47 4.63 4.72 4.78 4.85 4.92 4.99 2000 5.18 5.20 5.30 5.35 5.38 5.46 5.50 5.56 5000 6.79 6.80 6.88 6.89 6.90 6.91 6.91 6.94 TABLE 4. Nusselt Numbers Against Rayleigh Numbers for Various Polygons for Uniform Circumferential Heat Flux. Ra NUSSELT NUMBERS Number of Sides of Polygon / 3 4 5 6 7 8 12 Ci rcle (Exact Solution) V 1.90 3.23 3.65 3.88 4.02 4.13 4.28 4.36 100 2.53 3.32 3.72 3.96 4.09 4.17 4.34 4.43 500 3.35 3.69 4.02 4.24 4.36 4.44 4.61 4.69 '1000 . 3.61 4.14 4.37 4.57 4.69 4.77 4.93 4.99 2000 4.69 4.95 5.00 5.18 5.28 5.34 5.52 5.56 5000 6.01 6.34 6.47 6.61 6.70 6.73 6.93 6.94 O Table 5. Numerical Values of the Function - at Various Points on the Wall BC of an Equilateral Triangular Duct Under Uniform Circumferential Heat Flux Ra Numerical Values of the Function c ^ / ^ N Location of Points on Wall BC, 0 i n Degrees 0.00* 6.66 13.33 20.00* 26.66 33.33 40.00* 46.66 53.33 60.00* 1 .2499 ,2499 .2499 .2499 .2499 .2499 .2500 .2500 .2500 .2500 100 .2499 .2499 .2499 .2499 .2501 .2502 .2499 .2481 .2433 .2500 1000 .2499 .2497 .2493 .2499 .2525 .2554 .2500 .2129 .1179 .2500 2000 .2499 .2498 .2496 .2499 .2514 .2531 .2499 .2261 .1589 .2499 5000 .2500 .2501 .2502 .2499 .2491 .2485 .2499 .2599 .2127 .2499 *These were the prechosen points. Table 6. Numerical Values of the Function — 3 — at an Various Points on the Wall BC of a Square Duct Under Uniform Circumferential Heat Flux Ra Numerical Values of the Function £4?/2>N Location of Points on Wall BC, & i n Degrees .0* 5 10 15* 20 25 30* 35 40 45* 1 .2499 .2499 .2499 .2499 .2499 .2499 .2500 .2500 .2500 .2500 100 .2499 .2499 .2499 .2499 .2500 .2500 .2499 .2499 .2499 .2500 1000 .2500 .2499 .2499 .2499 .2502 .2505 .2499 .2477 .2448 .2501 2000 .•2500 .2499 .2499 .2500 .2501 .2502 .2499 .2489 .2475 .2501 5000 .2499 .2500 .2500 .2499 .2498 .2496 .2499 .2514 ,2531 .2499 *These were the prechosen points. TABLE 7. Pressure Drop Parameters L Against Rayleigh. Numbers f o r .. Various Polygons for Both the Boundary Condit ions. PRESSURE DROP PARAMETER L Ra Uniform Circumferential Wall Temperature Uniform Circumferential Heat Flux Number of Sides. Number of Sides 3 4 6 8 16 C i r c l e 3 4 6 8 16 i Ci rc le l 26.75 28.50 30.04 30.17 31.98 32.06 27.03 28.56 30.06 30.17. . 32.00 32.06 TOO 34.53 35.26 36.16 36.86 37.16 37.69 53.39 41.90 38.07 36.87 37.32 37.69 500 63.10 61.03 59.81 59.81 59.72 59.63 129.02 94.67 68.73 62.94 60.15 59.63 . 1000 94.13 90.35 87.30 86.67 85.40 85.45 202.39 158.35 104.99 93.58 87.01 85.45 2000 146.80 142.01 136.97 135.57 133.36 132.81 347.16 278.30 171.04 .149.73 136.29 132.81 5000 271.27 266.35 259.92 257.95 253.31 252.56 810.05 570.88 341.22 292.42 261.21 252.56 1 U N I F O R M C I R C U M F E R E N T I A L W A L L T E M P E R A T U R E 9 , •• •  • • NUMBER OF SIDES ^ L_ : — L _ :—I , _ J I I i i i 1 1 1 1 10 10 1 0 . l i R A Y L E I G H N U M B E R R a FIGURE'5..- Nusselt Number_Agsiinst_. Polygons • ^ 10 s 8 to GO 2 6 4 T U N I F O R M " T — — 1 — — — T — 1 ~ C I R C U M F E R E N T I A L W A L L T E M P E R A T U R E Ra= 5x10" 3 10' 1 »»» • • • • • • • • • • • • • • • • • • • • • • a M I 9 11 13 15 N U M B E R OF S IDES FIGURE 6 - Nusselt Number Against Number of Sides for Various Rayleigh Numbers I I - — T " ——I 1 —I— U N I F O R M C I R C U M F E R E N T I A L H E A T F L U X 9 Ra = &x10' 10 3 1 1 7 9 11 13 1.5 NUMBER O F S I D E S 1 FIGURE 8 - Nusselt Number Against Number of Sides for Various Rayleigh Numbers or LU CD I Z CO CO 10 8 0 1 ~ 1 Ra= 5 0 0 0 2 0 0 0 UNIFORM CIRCUMFERENTIAL WALL TEMPERATURE UNIFORM CIRCUMFERENTIAL HEAT FLUX I I I 1 I 5 7 9 11 NUMBER OF SIDES 13 15 17 FIGURE 9 - Nusselt Number Against Number of Sides for Both The Boundary Conditions 00 FIGURE 10 - Velocity Distribution i n Equilateral Triangular Duct 2 . 5 L SQUARE D U C T Uniform Circumferential Wall Temperature .......Uniform Circumferential Heat Flux - 0 . 2 5 0.0 DISTANCE R* FIGURE 11 - Velocity Distribution i n Square Duct 0.25 R Cos 0 o FIGURE 12 - Temperature Distribution i n Equilateral Triangular Duct TEMPERATURE DIFFERENCE - * 3 9 1.5 E Q U I L A T E R A L T R I A N G L E FIGURE 14 - Local To Average Heat Flux Ratio f o r E q u i l a t e r a l Triangular Duct 1.5 CO r -< X h -< 1.0 0.5 H E X A G O N B U N I F O R M C I R C U M F E R E N T I A L W A L L T E M P E R A T U R E T H E W A L L B C FIGURE 15 •- Local to Average Heat Flux Ratio for Hexagonal Duct ON 1.5 > o < co CO *-00 01 < LU CO 1 .0 0.5 0.0 E Q U I L A T E R A L T R I A N G L E Ra= 10 •UNIFORM CIRCUMFERENTIAL HEAT FLUX • UNIFORM CIRCUMFERENTIAL W A L L TEMPERATURE B T H E W A L L B C FIGURE 16 - Local to Average Shear Stress Ratio for Equilateral Triangular Duct 0.20 i UNIFORM C I R C U M F E R E N T I A L H E A T F . L U X FIGURE 17 - Local Wall Temperature Difference for Square and Octagonal Ducts 5x ICT LU LLJ < < O or Q LU ai co CO LU 10 o 2 10 1— I I I 11 U N I F O R M C I R C U M F E R E N T I A L W A L L T E M P E R A T U R E U N I F O R M C I R C U M F E R E N T I A L H E A T F L U X E Q U I L A T E R A L T R I A N G L E 1 r 10 10 10 _1 t I I II II 3 -A 10 RAYLEIGH NUMBER Ra FIGURE 18 - Pressure Drop Parameter Against Rayleigh Number for Equilateral Triangular Duct ON 68 r i i T [ - — i — i — i — i — | — i — i — i — r 1.0 1.5 2.0 2.5 SPACING RATIO FIGURE 19 - Nusselt Number Values for Cylinders Arranged i n Triangular Array 2 0 L U cn 15 LU CO CO Z) 2 5 0 1 1 — r i i r r r ~l i 1—i—rr-rrj r ~i i — r — i i T T T ~» i j j—rrnr Uniform Circumferential Wall Temperature Spacing Ratio J I 1 L i I i i I J I I t i i Square Array Triangular Array i i i i i 1111 i i i i 1 1 1 1 10 100 1000 RAYLEIGH NUMBER Ra FIGURE 20 - Nusselt Number Against Rayleigh Number for Various Spacing Ratios 1 0 0 0 0 2 0 U J CD 1 5 \— 10 UJ co CO 3 0 Uniform Circumferential 1 1 1 1 » i » | 1 1 1 \ \ VTT Heat Flux Spacing Ratio = 2 . 0 j u u w w i . v u y u J i M J i R W W ^ J ^ ^ *jjf«»mf . . . . . . . •-. . . . - ^ _ _ . inn • 1 A 1 - 1 1 » I 1 1 1 . 1 1 1 1 i 1 ( t f i Square Array .... Triangular Array i - i I_I .i111 i i i i \ I , i 10 100 1000 RAYLEIGH NUMBER Ra FIGURE 21 - Nusselt Number Against Rayleigh Number for Various Spacing Ratios 1 0 0 0 0 o 2 5 LU CQ 2 0 => 15 h-10 CO CO ZD - Uniform Circumferential Wall Temperature .. Uniform Circumferential Heat Flux i 1 1—I I I I I I ' ' I I L J I 1 » ' ' ' ' I I i i i i i i • 10 100 1000 RAYLEIGH NUMBER Ra 10000 FIGURE 22 - Nusselt Number Values for Cylinders Arranged i n Square Arrays S Q U A R E A R R A Y Uniform Circumferential Wall Temperature FIGURE 23 - Velocity Profiles Along DC of the Typical Element 2.0 1.5 SQUARE ARRAY Uniform Circumferential Wall Temperature Spacing Ratio 1000 6000 THE SIDE A B B FIGURE 24 - Velocity Profiles Along AB of the Typical Element u> 2.5 O o L U > 0.0 B SQUARE A R R A Y Uniform Circumferential Wall Temperature Spacing Ratio 1.3 1000 6000 THE SIDE BC FIGURE 25 - Velocity Profiles Along BC of the*j.Typical Element 4> TRIANGULAR ARRAY ' I i I • ! • I 0 10 2 0 3 0 ANGLE 9 , Degrees FIGURE 29 - Local to Average Heat Flux Ratio at the Wall for Cylinders Arranged i n Triangular Arrays SQUARE ARRAY' 0 10 20 30 40 45 A N G L E 0, Degrees FIGURE 30 - Local To Average Heat Flux Ratio at the Wall for Cylinders "° Arranged i n Square Arrays. TEMPERATURE DIFFERENCE cfrxlOO o -* *o CO ^ 1 1 I r T R I A N G U L A R A R R A Y Uniform Circumferential Wall Temperature A N G L E 9 , Degrees 00 FIGURE 32 - Local Wall Temperature Difference for Cylinders Arranged M i n Triangular Arrays SJ-3 J*5 ^ i 1 r S Q U A R E A R R A Y Uniform Circumferential Wall Temperature Spacing Ratio 6000 0.7 JL 10 20 30 A N G L E 9, Degrees 40 FIGURE 33 - Local to Average Shear Stress Ratio at the Wall for Cylinders Arranged i n Square Arrays 45 T R I A N G U L A R ARRAY 0 10 20 30 A N G L E 9, Degrees 00 FIGURE 34 - Local to Average Shear Stress Ratio at the Wall for Cylinders W Arranged i n Triangular Arrays BIBLIOGRAPHY 84 1. Clark, S.H. and Kays, W.M., ''Laminar Flow Forced Convection in Rectangular Tubes", Trans. ASME, Vol. 75, 1953, pp. 859-866. 2. Tao, L.N., "On Some Laminar Forced Convection Problems", Trans. ASME, Vol. 83, Series C. J . Heat Transfer, 1961, pp. 466-472. 3. Tao, L.N., "Method of Conformal Mapping i n Forced Convection Problems", International Developments i n Heat Transfer, ASME, 1961, pp. 598-606. 4. Cheng, K.C., "Laminar Flow and Heat Transfer Characteristics i n Regular Polygonal Ducts", Proceedings of the Third International Heat Transfer Conference, A.I.Ch.E., Vol. 1, 1966, pp. 64-76. 5. Siegel, R. and Savino, J.M., "An Analytical Solution of the Effect of Peripheral Wall Conduction on Laminar Forced Convection i n Rectangular Channels", ASME Paper No. 64 - HT - 24. 6. Marco, S.M. and Han, L.S., "A Note on Limiting Laminar Nusselt Number i n Ducts with Constant Temperature Gradient by Analogy to Thin-Plate Theory", Trans. ASME, Vol. 77, 1955, pp. 625-630. 7. Sparrow, E.M., Loeffler, A.L. and Hubbard, H.A., "Heat Transfer to Longitudinal Laminar Flow Between Cylinders", Trans. ASME, Vol. 83, Series C, J. Heat Transfer, 1961, pp. 415-422. 8. Han, L.S., "Laminar Heat Transfer i n Rectangular Channels", Trans. ASME Vol. 81, Series C, J. Heat Transfer, 1959, pp. 121-128. 9. Tao, L.N., "On Combined Free and Forced Convection i n Channels", Trans. ASME, Vol. 82, Series C, J. Heat Transfer, 1960, pp. 233-238. 10. Tao, L.N., "On Combined Free and Forced Convection i n Circular and Sector Tubes", Applied S c i e n t i f i c Research, Section A, Vol. 9, No. 5, 1960, pp. 357-368. 11. Agrawal, H.C., "Variational Method for Combined Free and Forced Convection i n Channels", Int. J . Heat and Mass Transfer, Vol. 5, 1962, pp. 439-444. 12. Aggarwala, B.D. and Iqbal, M., "On Limiting Nusselt Number From Membrane Analogy for Combined Free and Forced Convection Through Vertical Ducts", Int. J . Heat Mass Transfer, Vol. 12, 1969, pp. 437-748. 13. Iqbal, M., Aggarwala, B.D, and Fowler, A.G., "Laminar Combined Free and Forced Convection i n Vertical Noncircular Ducts Under Uniform Heat Flux", Int. J . Heat and Mass Transfer, Vol. 12, 1969, pp. 1123-1139. 14. Lu, P.C., "A Theoretical Investigation of Combined Free and Forced Convection Heat Generating Laminar Flow Inside Vertical Pipes With Prescribed Wall Temperatures", M.S. Thesis Kansas State College, Manhattan, Kansas, 1959. 15. Eckert, E.R.G., Irvine, T,F., J r . , and Yen J.T., "Local Laminar Heat Transfer i n Wedge-Shaped Passages", Trans. ASME, Vol. 80, 1958, pp. 1433-1438. 16. Yen, J.T., "Exact Solution of I^aminar Heat Transfer i n Wedge-Shaped Passages with Various Boundary Conditions", Wright A i r Development Centre, Technical Report 57-224, July, 1957, 17. Sparrow, E.M. and Siegel, R., "A Variational Method for Fully Developed Laminar Heat Transfer i n Ducts", Trans. ASME, Vol. 81, Series C, J. Heat Transfer, 1959, pp. 157-167. 18. Cheng, K.C., "Laminar Forced Convection i n REgular Polygonal Ducts with Uniform Peripheral Heat Flux", Trans. ASME, Vol. 91, Series C, J. Heat Transfer, 1969, pp. 156-157. 19. Iqbal, M., "Effect of Tube Orientation in I^aminar Convective Heat Transfer", Ph.D. Thesis, McGill University, 1965. 20. McLachlan, N.W., "Bessel Functions for Engineers", Oxford University Press, England, 1934. 21. Iqbal, M., Ansari, S.A., and Aggarwala, B.D., "Effect of Buoyancy on Forced Convection i n Vertical Regular Polygonal Ducts", Trans. ASME, J. Heat Transfer (in Press). 22. Iqbal, M., Ansari, S.A. and Aggarwala, B.D., "Buoyancy Effects on Longitudinal Laminar Flow Between Vertical Cylinders Arranged i n Regular Arrays", Proceedings of Fourth International Heat Transfer Conference, Paris, August 31 to September 5, 1970, (to be published). A P PENDI CES 88 APPENDIX A For the case of uniform circumferential wall heat flux, the heat input, q = c~= constant, expression i s obtained i n the nondimen-sional form i n this appendix. Ji_i i i * f H — f — r H A Z — Consider the f l u i d flowing between sections 1 and 2, of a duct i n the figure shown above. By making a simple energy balance, we get, f U C ^ A (T.-TO - ^  P AX + a A &TL , (A-l) where, Tj_ and T2 are the bulk temperatures at the two sections, and P i s the heat transfer perimeter of the duct. Since, C± a~L "* q = h t^T equation (A-l) reduces to, From the dimensionless variables,<^> and N, we get, and also, (A-2) c> n = D h c> N , A l b - . *~P ~ 4-Substituting the above variables i n equation (A-2), we obtain a n = 0.25(1-F) (32) 89 APPENDIX B Nusselt number expressions are obtained i n terms of the dimensionless variables in this appendix for both the boundary conditions. Nusselt number, which signifies the energy converted from a surface, can be written as, N u ^ , (B-l) where, T w i s the average wall temperature, Tjj i s the bulk temperature of the f l u i d and can be written as, I / T L L dob Tb = • (B-2) il a da Substituting (B-2) i n (B-l), we get, N u = — • (B.3) As i n Appendix A, by making a simple energy balance between sections 1 and 2, we obtain, f oCjjC, A - ^ p + a A > or 9- - i U L ( f U C b C . - Q ) . Substituting (B-4) i n (B-3) we get , ih 1 (fUC^C.-Q) * k X . - (//"Tu. oU/f/u.«U) (B-5) F i r s t of a l l we present the nondimensional form of Nusselt number for Case 1, 90 Case 1 - Uniform Circumferential Wall Temperature For this boundary condition, T = T w = constant, and from the dimensionless v a r i a b l e s , a n d V, we get, u , - u V . Substituting (B-6) i n (B-5), we obtain I - F mx Now we consider the boundary condition of Case 2. Case 2 - Uniform Circumferential Wall Heat Flux For this boundary condition, the dimensionless variables > <^ > and V result i n , T = T ^ e x + Cf U C y . C ^ / f e ) 4 , ix = VVJ . Substituting (B-7) i n (B-5), we obtain, (B-6) (17) "here, J/4>VclA <p — • (18) J J v a A (B-7) N a = ^ — — • ^ 

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