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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.), A l i g a r h Muslim U n i v e r s i t y , A l i g a r h , India, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc.  i n the department of Mechanical Engineering  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 1969  In  presenting  this  an a d v a n c e d d e g r e e the L i b r a r y I  further  for  of  at  agree  tha  written  for  It  fulfilment of of  gain  Mechanical  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  shall  Jan. 12,  1970  Engineering Columbia  the  requirements  Columbia, reference  copying  of  I agree and this  that  not  copying  or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  financial  for  for extensive  permission.  of  British  available  p u r p o s e s may be g r a n t e d  thesis  Department  Date  freely  permission  representatives.  this  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t  scholarly  by h i s  thesis  or  publication  be a l l o w e d w i t h o u t  my  ii ABSTRACT The problem o f laminar combined free and forced convection through v e r t i c a l n o n c i r c u l a r 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 t o be  constant, except the v a r i a t i o n o f the density i n the buoyancy term o f the momentum equation.  Pressure work and viscous d i s s i p a t i o n terms of  the energy equation have been neglected. to be constant i n the flow d i r e c t i o n .  Heat f l u x has been considered  A general s o l u t i o n t o the problem  has been obtained i n "the form o f i n f i n i t e series containing modified Bessel functions. Two p o s s i b l e thermal boundary conditions on the c i r cumference o f the heated w a l l have been analyzed, Case 1 - uniform circumferential w a l l temperature, and Case 2 - uniform circumferential w a l l heat f l u x . Information o f engineering i n t e r e s t l i k e Nusselt number, heat f l u x , r a t i o , shear stress r a t i o , temperature v e l o c i t y and temperature  d i s t r i b u t i o n on the' w a l l ,  d i s t r i b u t i o n s i n the flow f i e l d have been ob-  tained f o r two sets o f geometries, namely, ( i ) flow through regular polygonal ducts, and ( i i ) flow between cylinders arranged i n r e g u l a r arrays. For flow through regular polygonal ducts, the case o f uniform circumferential w a l l temperature r e s u l t s i n higher values o f Nusselt numbers as compared t o the case o f uniform circumferential w a l l heat f l u x . This difference i n Nusselt number values decreases as the number o f sides o f the regular polygon i s increased, u n t i l f o r a c i r c l e appears.  i t completely d i s -  For both the cases, a t higher values o f Rayleigh number, the  Nusselt number i s l e s s s e n s i t i v e t o the number o f sides o f the polygon. A l s o , a t higher values o f Rayleigh number, both the cases tend t o produce the same r e s u l t s .  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 o f the duct wall towards the apex o f the duct.  For flow between cylinders arranged i n r e g u l a r arrays, Case 1 r e s u l t s i n higher values o f Nusselt number compared t o Case 2, f o r low spacing r a t i o s .  However, as the spacing r a t i o i s increased, the two  cases tend t o produce the same r e s u l t s .  Cylinders arranged i n e q u i l a t -  e r a l t r i a n g u l a r arrays produce higher values of Nusselt number compared t o those i n square arrays.  This difference i n Nusselt number values  decreases when the spacing r a t i o i s high.  For higher values o f Rayleigh  number, however, the r e s u l t s are l e s s s e n s i t i v e t o the type o f arrays. Also, at higher values o f Rayleigh number, both the cases tend t o produce the same r e s u l t s .  iv TABLE OF CONTENTS Page  ABSTRACT  i i  LIST OF TABLES  v  LIST OF FIGURES  vi  ACKNOWI_DGEMENTS  ix  NOMENCLATURE  l  INTRODUCTION  3  FORMULATION OF THE PROBLEM.  9  GENERAL SOLUTION  13  Section I  - Regular Polygonal Ducts  Section I I - Flow Between Cylinders,....  17 24  DISCUSSIONS  31  CONSLUSION  46  BIBLIOGRAPHY  84  APPENDIX A - Derivation o f Heat Flux Expression  88  APPENDIX B - Derivation o f Nusselt Number Expressions  89  V  LIST OF TABLES Table  Page  i  1.  Improvement of Nusselt Numbers with Number of Points on the Wall f o r an E q u i l a t e r a l Triangular Duct and Comparison with an Exact S o l u t i o n , f o r the Case of Uniform Circumferential Wall Temperature...  4  Improvement o f Nusselt Numbers with Number o f Points on the Wall f o r a Square Duct and Comparison with an Exact S o l u t i o n , f o r the Case of Uniform Circumferential Wall Temperature....  4  Nusselt Numbers Against Rayleigh Numbers f o r Various Polygons Under Uniform Circumferential Wall Temperature....  49  4.  Nusselt Numbers Against Rayleigh Numbers f o r Various Polygons Under Uniform Circumferential Wall Heat Flux  50  5.  Numerical Values o f the Function &4>/d N at Various Points on the Wall of an E q u i l a t e r a l Triangular Duct Under Uniform Circumferential Wall Heat Flux.  51  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  2.  3.  6.  7.  Pressure Drop Parameters L Against Rayleigh Numbers f o r Various Polygons f o r Both the Boundary Conditions...  7  8  53  vi LIST OF FIGURES Figure  Page  1.  Flow Through V e r t i c a l Noncircular Duct  10  2.  Coordinate System f o r Regular Polygons...  19  3.  Flow Configuration and Coordinate System f o r Flow Between V e r t i c a l Cylinders  25  Cylinders Arranged i n E q u i l a t e r a l Triangular and Square Arrays  26  Nusselt Number Against Rayleigh Number For Various Polygons Under Uniform Circumferential Wall Temperature  54  Nusselt Number Against Number o f Sides f o r Various Rayleigh Numbers Under Uniform Circumferential Wall Temperature  55  Nusselt Number Against Rayleigh Number f o r Various Polygons Under Uniform Circumferential Wall Heat Flux....  56  Nusselt Number Against Number o f Sides f o r Various Rayleigh Numbers Under Uniform Circumferential Wall Heat Flux  57  Nusselt Number Against Number o f Sides f o r Various Rayleigh Numbers Under Both The Boundary Conditions  58  V e l o c i t y D i s t r i b u t i o n i n an E q u i l a t e r a l Triangular Duct Under Both The Boundary Conditions  59  V e l o c i t y D i s t r i b u t i o n i n a Square Duct Under Both The Boundary Conditions  60  Temperature D i s t r i b u t i o n i n an E q u i l a t e r a l Triangular Duct Under Both The Boundary Conditions  61  Temperature D i s t r i b u t i o n i n a Square Duct Under Both The Boundary Conditions  62  Local t o Average Heat Flux Ratio f o r an E q u i l a t e r a l Triangular Duct Under Uniform Circumferential Wall Temperature  63  Local t o Average Heat Flux Ratio f o r Hexagonal Duct Under Uniform Circumferential Wall Temperature  64  Local t o Average Shear Stress Ratio f o r an E q u i l a t e r a l Triangular Duct Under Both The Boundary Conditions  65  4.  5.  6.  7.  8.  9. 10. 11. 12.  13. 14.  15.  16.  V l l  Figure 17.  18.  19.  20.  21.  22.  23.  24.  25.  26.  Page Local Wall Temperature Difference For Square and Octagonal Ducts Under Uniform Circumferential Wall Heat Flux  66  Pressure Drop Parameter Against Rayleigh Number f o r E q u i l a t e r a l Triangular Duct Under Both The Boundary Conditions  67  Nusselt Number Values f o r Cylinders Arranged i n Triangular Arrays Under Uniform Circumferential Wall Temperature......  68  Nusselt Number Against Rayleigh Number f o r Cylinders Arranged i n Square and Triangular Arrays Under Uniform Circumferential Wall Temperature  ^9  Nusselt Number Against Rayleigh Number f o r Cylinders Arranged i n Square and Triangular Arrays Under Uniform Circumferential Wall Heat Flux Nusselt Number Against Rayleigh Number f o r Cylinders Arranged i n Square Arrays Under Both The Boundary Conditions V e l o c i t y P r o f i l e s Along DC f o r Cylinders Arranged In Square Arrays Under Uniform Circumferential Wall Temperature. V e l o c i t y P r o f i l e s Along AB f o r Cylinders Arranged i n Square Arrays Under Uniform Circumferential Wall Temperature V e l o c i t y P r o f i l e s Along BC f o r Cylinders Arranged i n Square Arrays Under Uniform Circumferential Wall Temperature Temperature P r o f i l e s along DC, f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Temperature  7  0  7  1  7  2  7  3  ^  7  7  5  27.  Temperature P r o f i l e s along AB, f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Temperature  28.  Temperature P r o f i l e s along BC, f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Temperature.  77  Local t o Average Heat Flux Ratio f o r Cylinders Arranged i n Triangular Arrays Under Uniform Circumf e r e n t i a l Wall Temperature  78  29.  Figure 30.  31.  32.  33.  34.  Page Local t o Average Heat Flux Ratio f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Temperature  79  Local Wall Temperature Difference f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Heat Flux  80  Local Wall Temperature Difference f o r Cylinders Arranged i n Triangular Arrays Under Uniform Circumferential Wall Heat Flux.  81  Local t o Average Shear Stress Ratio f o r Cylinders Arranged i n Square Arrays Under Uniform Circumfere n t i a l Wall Temperature  82  Local to Average Shear Stress Ratio f o r Cylinders Arranged i n Triangular Arrays Under Uniform Circumf e r e n t i a l Wall Temperature  83  V  1  1  1  ACKNOWIJ_<_?ENTS The author would l i k e t o express h i s sincere gratitude to Dr. M. Iqbal, who devoted considerable time and gave invaluable advice and guidance throughout a l l stages o f the present work. Sincere thanks are also extended t o Dr. B.D. Aggarwala o f the Mathematics Department, University o f Calgary, f o r h i s valuable suggestions.  Use o f the Computing Centre f a c i l i t i e s a t the U n i v e r s i t y o f B r i t i s h Columbia and the f i n a n c i a l support o f the National Research Council o f Canada are g r a t e f u l l y  acknowledged.  NOMENCLATURE  A  = area of cross-section  Cp  = s p e c i f i c heat of the f l u i d at constant pressure  C|  = temperature  gradient in flow d i r e c t i o n , 3 T / 3 Z  = 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  = average peripheral heat transfer c o e f f i c i e n t  Q V  L  = pressure drop parameter, dimensionless  Nu  hD^/K, Nusselt number, dimensionless  2  *\  Ra  = (p gC C-|3D^^(cu, Rayleigh number, dimensionless  2P  = dimensionless tube spacing  Q  = heat generation rate  R  =  r  /D > dimensionless radius n  s  = number of sides of a regular polygon = 4 f o r square array and 6 f o r triangular array  q  a v  = average surface heat flux  T  = temperature  u  = axial velocity  V  = u/U, dimensionless axial v e l o c i t y  V  = V/L, dimensionless  R, 6  = coordinates  R  = R Cos 6 , dimensionless  Z  = axial coordinate i n flow direction  R-j  = radius of cylinder  P / R - j  = s p a c i n g r a t i o  *  =  < J > 4> T g  v  T  o  = X / [  r  (  T  -  T  a p e x )  2 n  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  L" ' dimensionless  =  l o c a l w a l l s h e a r s t r e s s  = a v e r a g e w a l l s h e a r s t r e s s  M M 3,p,K,y  ' V  pUC C,D  =  T  (  1  '  4  = 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 t r a n s f e r process, the density  differences  a r i s i n g due t o temperature differences give r i s e t o free convection e f f e c t s under a g r a v i t a t i o n a l force f i e l d .  In s i t u a t i o n s where the forces and  momentum transport rates are very l a r g e , the free convection e f f e c t s may not be very important, and could be ignored. be c a l l e d forced convection.  Such a s i t u a t i o n would then  In cases where the buoyancy forces a r i s i n g  due t o temperature (or equivalently density) differences are very l a r g e , the forced convection e f f e c t s could be neglected, and the s i t u a t i o n could be treated as free convection.  However, i n many cases o f p r a c t i c a l i n t e r -  e s t , both the e f f e c t s o f forced and free convection can be o f comparable order.  Such a s i t u a t i o n i s known as combined free and forced convection. In laminar flows associated with comparatively small i n e r t i a f o r -  ces, the free convection e f f e c t s are generally important. the free convection e f f e c t s can considerably  I t i s known that  a l t e r the heat t r a n s f e r rates.  In such s i t u a t i o n s , i n addition t o Grashoff, Reynolds and Prandtl numbers, the parameter describing flow o r i e n t a t i o n with respect t o the g r a v i t a t i o n a l force f i e l d i s also important.  One can divide the studies o f combined free and forced convection i n t o two broad catagories, ( i ) external flows, and ( i i ) i n t e r n a l flows. In the present study we w i l l r e s t r i c t our attention t o laminar flows through v e r t i c a l ducts and passages. For flow through v e r t i c a l c i r c u l a r ducts, when the w a l l temperature varies l i n e a r l y i n the d i r e c t i o n o f flow, the circumferential wall temperature and heat f l u x a t any section o f 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 t o carry l e s s heat through these regions. This could r e s u l t i n higher temperatures near the corners«  Conse-  quently, neither the wall temperature nor the w a l l heat f l u x may c i r c u m f e r e n t i a l l y uniform at any section o f the duct.  be  This r o t a t i o n a l  asymmetry w i l l depend upon duct configuration and the r a t i o of c i r cumferential wall conduction t o that of normal conduction to the f l u i d at any point on the w a l l .  Depending upon the r e l a t i v e magnitudes  of these parameters, there could be two extreme s i t u a t i o n s on the boundary, Case 1 - uniform circumferential wall temperature, r e s u l t i n g from large c i r c u m f e r e n t i a l w a l l conduction, Case 2 - uniform c i r c u m f e r e n t i a l w a l l heat f l u x , r e s u l t i n g from n e g l i g i b l e c i r c u m f e r e n t i a l w a l l conduction. The true s i t u a t i o n , however, would l i e somewhere i n between these two extremes. In the proceeding pages, a b r i e f l i t e r a t u r e survey i s presented f o r forced and combined free and forced convection through v e r t i c a l n o n c i r c u l a r ducts and passages.  As f a r as flow through pas-  sages i s concerned, we w i l l be mainly concerned with c y l i n d e r s arranged i n regular arrays.  The survey i s r e s t r i c t e d to laminar f u l l y developed  flow with constant properties and uniform heat input per unit length i n the flow d i r e c t i o n . The survey i s divided under the two boundary cond i t i o n s stated e a r l i e r .  Case 1 - UNIFORM CIRCUMFERENTIAL WALL TEMPERATURE  We  f i r s t t r e a t the forced convection  case.  5 Forced Convection Clark and Kays [1]1 used numerical r e l a x a t i o n method t o solve the problem f o r rectangular and t r i a n g u l a r ducts.  They confirmed t h e i r  r e s u l t s experimentally f o r a rectangular duct.  Tao [2] has shown that the exact s o l u t i o n of a c l a s s of laminar forced convection problems with heat sources can be approached by complex v a r i a b l e method.  He presented the r e s u l t s f o r e q u i l a t e r a l 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 p a r t i c u l a r configuration under i n v e s t i g a t i o n i s transformed onto a u n i t c i r c l e . Cheng [4] by method of point matching solved the problem f o r r e g u l a r polygonal ducts.  He pointed out that the various points chosen  on the boundary t o s a t i s f y the boundary conditions could e i t h e r be equally spaced on the boundary o r at equal angular i n t e r v a l s . Siegel and Savino [5] studied the e f f e c t o f p e r i p h e r a l heat conduction within the heated walls on the w a l l temperature o f a rectangular channel.  distributions  They pointed out t h a t , as the w a l l - t o - f l u i d  conduction r a t i o increases, the p e r i p h e r a l heat conduction increases and the w a l l temperatures become considerably more uniform with a s u b s t a n t i a l reduction i n peak temperatures.  Marco and Han  [ 6]  pointed out that the equation describing  d e f l e c t i o n o f a t h i n plate under uniform l a t e r a l load, and simply supported along i t s edges i s s i m i l a r t o the equation d e s c r i b i n g temperature d i s t r i b u t i o n i n a laminar f u l l y developed pure forced convection problem  •Numbers i n brackets designate references at the end o f the t h e s i s .  6 through a duct o f the same cross section as that o f the p l a t e .  As such,  one can borrow the plate theory solutions (whenever possible) t o solve f o r the temperature d i s t r i b u t i o n o f the heat t r a n s f e r problems.  Direct  a p p l i c a t i o n o f t h i s s o l u t i o n i s permissible only when the corresponding boundary conditions are also i d e n t i c a l .  The boundary condition o f u n i -  form c i r c u m f e r e n t i a l wall temperature has a correspondence with that o f a p l a t e which i s simply supported along i t s edges.  For flow outside c y l i n d e r s arranged i n e q u i l a t e r a l t r i a n g u l a r arrays, Sparrow e t a l . [ 7 ] calculated Nusselt number values as a func t i o n o f the pitch-to-diameter pitch-to-diameter  ratio.  They observed that when the  r a t i o was greater than 1,5, the heat f l u x r a t i o was  uniform over a s u b s t a n t i a l p o r t i o n o f the circumference o f the heated cylinders.  On t h i s b a s i s , they pointed out that when the p i t c h - t o -  diameter r a t i o i s greater than 1.5, both the conditions o f uniform circumferential wall temperature and uniform c i r c u m f e r e n t i a l heat f l u x would be simultaneously  achieved.  Combined Free and Forced Convection Han  L"8J solved the problem o f v e r t i c a l rectangular ducts by  using double Fourier s e r i e s .  Tao [9,10]suggested a method t o solve  such problems by introducing a complex function which i s d i r e c t l y r e l a t e d t o the v e l o c i t y and temperature f i e l d s .  Exact solutions were  then established i n terms o f Bessel and associated functions.  Agarwal  [11] u t i l i z e d Tao's formulation and converted i t i n t o a v a r i a t i o n a l expression.  This v a r i a t i o n a l equation was then solved by assuming a  s u i t a b l e polynomial f o r the temperature function.  7 Aggarwala and Iqbal [12] u t i l i z e d the solutions o f membrane v i b r a t i o n to solve the problem o f a set o f s t r a i g h t v e r t i c a l t r i a n g u l a r ducts.  They obtained exact a n a l y t i c a l expressions i n form o f i n f i n i t e  s e r i e s f o r v e l o c i t y and temperature.  Iqbal e t al.[13] approached the  same problem f o r a set of geometries by v a r i a t i o n a l method.  Lu [14] pointed out, that f o r the case of combined free and forced convection a l s o , an analogy e x i s t s between the equations describing deflection of a thin plate  r e s t i n g on an e l a s t i c foundation  with no d e f l e c t i o n along i t s edges, and the equation describing temperature d i s t r i b u t i o n i n a laminar f u l l y developed combined free and forced convection problem,  Rayleigh number, which determines  the  r e l a t i v e e f f e c t o f buoyancy i n the temperature equation i s equivalent t o the pressure exerted by the e l a s t i c foundation i n the d e f l e c t i o n equation.  Using t h i s analogy he solved the problem f o r concentric  a n n u l i i and rectangular ducts.  There does not seem to be any information a v a i l a b l e f o r combined free and forced convection f o r flow between v e r t i c a l c y l i n d e r s arranged i n r e g u l a r 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 c i r c u m f e r e n t i a l heat f l u x i n duct flow problems. Eckert e t a l . C l S ] and Yen [16] obtained an exact solution f o r a c i r c u l a r sector.  They noted that the two circumferential boundary  conditions (Case 1 and Case 2) could r e s u l t i n average heat t r a n s f e r c o e f f i c i e n t s which may d i f f e r by an order o f magnitude.  Sparrow and  Siegel [17] used v a r i a t i o n a l method to solve f o r rectangular ducts,  8 while Cheng [18] by using point matching method obtained the r e s u l t s f o r regular polygons.  For flow between c y l i n d e r s arranged i n regular arrays, there does not seem to be any a v a i l a b l e reference f o r t h i s c l a s s o f boundary condition.  Combined Free and Forced Convection I t seems that there i s no a v a i l a b l e information f o r combined free and forced convection through noncircular ducts o r passages with uniform circumferential wall heat f l u x .  In the next s e c t i o n , formulation of combined free and convection problem i s presented.  forced  The problem i s then solved f o r two  sets of geometries ( i ) flow through regular polygons and ( i i ) flow outside c y l i n d e r s arranged i n e q u i l a t e r a l t r i a n g u l a r and square arrays.  9 FORMULATION OF THE PROBLEM  Consider a v e r t i c a l s t r a i g h t n o n c i r c u l a r duct as shown i n F i g . 1.  The flow i s considered to be laminar and f u l l y  developed  (both hydrodynamically and thermally) i n the v e r t i c a l upwards d i r e c t i o n along the p o s i t i v e Z - a x i s .  Uniform heat f l u x per unit length  i s assumed i n the d i r e c t i o n o f flow.  The viscous d i s s i p a t i o n and  pressure work terms have been neglected.  The f l u i d properties are  considered t o be constant, except v a r i a t i o n o f density i n the buoyancy term o f the equation o f motion.  The f l u i d may contain uniform  volume heat source.  Under the above mentioned conditions, the d i f f e r e n t i a l form of the continuity equation i s i d e n t i c a l l y equal t o zero.  The momentum  and energy equations can be written as,  «  K^fl  k (tL^  | ^  +  ^ *L)  .  (2)  For uniform heat input i n the flow d i r e c t i o n , the w a l l 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 v a r i a b l e only i n the buoyancy term o f equation (1).  This assumption  i s known  C19] t o be v a l i d as long as the density v a r i a t i o n s i n the flow f i e l d are small.  Under t h i s r e s t r i c t i o n , the equation o f state f o r density  i n the l i n e a r form can be written as,  o  q M  FIGURE 1 - Flow Through V e r t i c a l Noncircular Duct  11 where s u f f i x 'ref* denotes condition o f f l u i d a t a reference point on the duct w a l l .  For the case o f uniform circumferential w a l l temperature,  t h i s reference point could be anywhere on the duct w a l l . For the case of uniform circumferential heat f l u x , the apex o f the duct was chosen as a reference point.  The reference w a l l temperature  Tre<. = To + z where T  Q  T^f^ can be defined as,  .  i s the reference temperature o f the duct a t Z = 0.  By choosing the following dimensionless v a r i a b l e s ,  R = r/D  , V = U./U , 4> = C T - Tr .)/CfUC C,D^/At; h  ef  ,  t>  and i n s e r t i n g equation (3) i n ( 1 ) , the momentum and energy equations can be nondimensionalized as,  V  V  z  + - R a <$>  v 4> 2  where,  V £R "  = = R  2  -  L  („)  ,  - F •• 2>R  ( 5 )  ^e^  In equations (4) and ( 5 ) , the Rayleigh number Ra, and the heat generation parameter F are prescribed q u a n t i t i e s , while V, <^>, and L are the three unknowns.  In order t o solve equations (4) and (5) we therefore  need another equation, which i s obtained from the continuity consideration i n the duct.  For constant properties, t h i s continuity equation i s ,  Jj u o U « II  x,  . .  Jj VcLA  / / U oLA =  IT . . ffjL  (6)  12  Although the equations (4) to (6) have been developed f o r flow through a v e r t i c a l noncircular duct, these equations w i l l remain unchanged f o r flow outside cylinders with n e g l i g i b l e side e f f e c t s of the s h e l l w a l l . In the present a n a l y s i s , equations (4) to (6) have t o be f o r the following two sets of boundary conditions on the  solved  circumference,  Case 1 - Uniform Circumferential w a l l temperature  V  =4>=  0 at the boundary  .  (7)  The above thermal boundary condition r e s u l t s from large circumferential w a l l conduction.  Case 2 - Uniform c i r c u m f e r e n t i a l w a l l heat f l u x V = 0  at the boundary  • = 0.25C1-F) 4*  = 0  ^  ,  at the boundary ,  ^  (8)  at the apex .  ^4> This thermal boundary condition  - r — = 0,25(1-F) r e s u l t s  from small  circumferential w a l l conduction and i s derived i n Appendix A. A general s o l u t i o n to equations (4) to (6) w i l l be i n the next s e c t i o n .  obtained  13 GENERAL SOLUTION  In equations (4) and ( 5 ) , pressure drop parameter, L i s an unknown constant.  Since L i s independent o f the coordinate system,  equations (4) and (5) can be divided by L t o obtain,  V *?  -  2  where  V  • ,  >  (10  ?  \j -  ,  <$> -  *, Define  = V - F and  <*>/L  ^  P= A- • F  *  V  / = (Ra) , and combine equations  (9) and (10) t o give,  A general s o l u t i o n t o t h i s biharmonic equation (11) can be written from McLachlan [2 C] as,  o o  + C^^OlM  * t> K ^ ^ r O ] S i » me. (12) M  Once V ] _ i s known, the nondimensional temperature difference can be obtained from equation (9) as,  (13)  where,  V  V =  \ i YH~ ~ ^ • ^ W A  - (V K „ l  ^ W ^ R )  + D kir, C^)J Sin m  m  This completes the general s o l u t i o n o f equations (4) to ( 6 ) , and the problem reduces t o determining the unknown c o e f f i c i e n t s Cm and Dm and pressure drop parameter L.  Am,  These c o e f f i c i e n t s are  determined by point matching method, where the boundary conditions are s a t i s f i e d exactly at a prechosen number o f points.  The method  o f point-matching would be discussed i n d e t a i l while dealing with 1he p a r t i c u l a r configuration. Once these c o e f f i c i e n t s are known, the problem i s completely solved and the nondimensional v e l o c i t y and temperature at any point can be evaluated as,  \j = $  -  CN <*>  1 +  F) L  ,  u  I  (IS)  J  Having obtained the v e l o c i t y and temperature functions, one can obtain the following information o f engineering i n t e r e s t . Nusselt Numbers Nusselt number, which s i g n i f i e s energy convected from a surface can be written as,  N u  =  =  h  —  k  ,  —  T -T M  b  (16)  15 where T  w  i s the average w a l l 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) i n the nondimensional form can be written^ as,  » mx  1  where,  -.  J J 4>VCLA  »  <p  -  ii  .  (18)  For the case o f uniform circumferential w a l l heat f l u x , Case 2, the nondimensional form o f equation (16) i s ^ ,  Nu -  I - F TT/S  o  .  iwau.  ' 1  mx  (19)  J  Local Heat Flux Ratio For Case 1, since the heat f l u x varies c i r c u m f e r e n t i a l l y , the l o c a l t o average heat f l u x r a t i o i n the nondimensional form can be expressed as,  Local Shear Stress Ratio The l o c a l t o average shear stress r a t i o can be written as,  2For d e t a i l s please see appendix B,  Equations (12) t o (21) are the general solutions t o the problem of combined free and forced convection through a v e r t i c a l a r b i t r a r y shaped duct, under the assumptions made.  In the next two sections, I and I I , we w i l l s p e c i a l i z e i n two sets o f geometries, namely, r e g u l a r polygonal ducts and flow outside cylinders.  SECTION  Regular  I  Polygonal  18 REGULAR POLYGONAL DUCTS  By taking the coordinate system as shown i n F i g . 2, only "the area OCB needs t o be considered, because o f the symmetry o f the regul a r polygons.  Since the problem i s even i n 6 , the terms i n equation  (12) containing S i n n S w i l l vanish.  In a d d i t i o n , since the v e l o c i t y i s  f i n i t e a t the centre 0, the k e r and k e i functions w i l l not e x i s t .  Hence  equation (12) reduces t o  V, = £ [ A  m  t«- C"|R) + B \»el ('l«]Co m6. (22) Jn  m  m  S  From p h y s i c a l considerations,  i^L-  -  4-^"  -  0  a  t  0^0  and  0  =7T/  (23)  s  where s i s the number o f sides o f a r e g u l a r polygon.  The condition expressed by equation (23) can only be s a t i 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 t o evaluate the unknown coefficients A  n s  and B  and the pressure drop parameter L, i s explained  below f o r the two boundary conditions.  SOLUTION FOR F = 0 For the time being take F = 0, which i s true f o r most o f the fluids.  Case 1  Uniform Circumferential Wall Temperature The boundary conditions ( 7 ) , can be rewritten as,  \y  Q  < $ >  at the boundary  — 0  (26)  t  at the boundary, which r e s u l t s i n ,  2 _ \/ - - 1  a t the boundary  .  (27)  Inserting equation5(24) and (25) i n equations (26) and (27) one obtains,  0 = JZ. A „ t > e r „ C ' V O Cos ns© s  i  n =o o  —  o  C*3  + ^ 1 E > ^ e - ^ C ^ R ) Cos «se . n &  s  (29)  For n number o f points on the boundary BC ( f i g . 2 ) , equations (28) and (29) w i l l r e s u l t i n 2n number o f equations i n 2n number o f unknowns.  These equations are o f l i n e a r 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 c o e f f i c i e n t s Ans and Bns are known, and the pressure drop parameter, L can then be evaluated from the i n t e g r a l form o f cont i n u i t y equation ( 6 ) ,  ,  L_  Case 2  =.  IUA  iiVdU  •  (30)  Uniform Circumferential Heat Flux The boundary conditions (8) can be rewritten as ,  \J  —  an  ^ 4>  O  a t the boundary ,  _ '2~5  (|_p*)  0  a t the boundary  .  (31) (32)  L  For the coordinate system shown i n F i g . 2,  = Cos© J  6u  c>H  ^inl  (33)  R  In equation (32), since the unknown L appears simultaneously, n  number o f points on the boundary BC w i l l r e s u l t i n (2n + 1) number  o f unknowns with only 2n  number o f equations.  Therefore, one more  equation i s needed, which i s given by the d e f i n i t i o n o f the nondimens i o n a l temperature  difference,  <^> — 0  a t the apex, which r e s u l t s i n ,  2. ^7  \j -  -\  at the apex .  (34)  Now equations (31) t o (34) w i l l r e s u l t i n (2n + 1) number o f l i n e a r algebraic equations i n (2n + 1) number o f unknowns, and can be solved. I t may be noted here, that the continuity equation i s not exp l i c i t l y required t o evaluate L.  The reason f o r t h i s i s that the con-  t i n u i t y requirement has been u t i l i z e d i n d e r i v i n g equation (32).  22 SOLUTION  FOR F #  0  In the event, r e s u l t s are required f o r a f i n i t e value o f F other than zero, F i n equation (24) becomes an unknown constant. Case 1, the point matching method described above can be followed. this situation  n  For In  number o f points on the boundary w i l l r e s u l t i n 2 n  number o f 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 i n t e g r a l form o f the c o n t i n u i t y equation,  For Case 2, however, the procedure explained f o r F = 0 can be followed exactly.  SECTION  Flow  Arranged  n  Between  in  Regular  Cylinders  Arrays  24 FLOW OUTSIDE CYLINDERS  F i g . 3(a) shows the system under study.  The c y l i n d e r s between  which the f l u i d flows are considered t o be v e r t i c a l and arranged i n equil a t e r a l t r i a n g u l a r o r square arrays ( F i g . 4 ) . The side e f f e c t s o f the s h e l l w a l l on the flow pattern i s considered t o be n e g l i g i b l e . the flow can be considered p e r f e c t l y symmetrical  As such,  about each c y l i n d e r .  Due t o the symmetry o f flow configuration only the shaded area o f t y p i c a l element shown i n F i g . 3(b) need t o be considered. element i s  a  This t y p i c a l  four o r s i x sided r e g u l a r polygon with a c e n t r a l c i r -  c u l a r h o l e , s being four o r s i x depending upon whether we are dealing with a square o r a t r i a n g u l a r array.  For t h i s flow c o n f i g u r a t i o n , the governing equations (4) t o (6) and the general s o l u t i o n (12) t o (14) remain the same.  We now s p e c i a l i z e  the s o l u t i o n f o r t h i s configuration. Since the problem i s even i n 0 , the terms o f equation (12) cont a i n i n g S i n n © w i l l vanish.  2>e  -=.0 a t 0 = 0  Also from p h y s i c a l considerations,  and Q = f r / .  (36)  s  The conditions o f (36) would be s a t i s f i e d by equation (12) only when m = ns, n being summation index. Hence the s o l u t i o n f o r V, , f o r t h i s configuration can be written as.  25  K3 ^3  t  Ul  a)  Flow  b)  Typical  Configuration  Element  FIGURE 3 - Flow C o n f i g u r a t i o n and Coordinate System f o r C y l i n d e r s Arranged i n Regular Arrays  26  a)  Triangular  b)  Square  Array  Array  FIGURE 4 - Cylinders Arranged i n Regular Arrays  The temperature d i f f e r e n c e , c £ > c a n again be obtained from equation (13) which i s  where,  " ns % C  K  ^  S  R  ) + »S D  Ker  C^R)]C«  n  S 6  }  • (38)  Equations (37) and (3 8) are the solutions o f the problem under consideration.  The unknown c o e f f i c i e n t s o f these equations have t o be  determined by s a t i s f y i n g the boundary conditions on the sides AD and BC o f the t y p i c a l element.  From the geometrical and p h y s i c a l considerations o f the problem, we have, ^ —  —  - 0  along side BC o f the element  (39)  For the t y p i c a l element shown i n F i g . 3(b)  _  Cos ©  ^  S\n9  £  The condition o f no s l i p on the wall r e s u l t s i n ,  V -0  at  R = R  (40)  x  The two thermal boundary conditions considered are, Case 1 - Uniform c i r c u m f e r e n t i a l w a l l temperature, <^> — O  at R =  Ri  (41)  Case 2 - Uniform circumferential w a l l heat f l u x ,  at R = %  £R  (42)  _ and  28  ,+'=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 c o e f f i c i e n t s 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 d i f f e r e n t ways.  DIRECTION I This procedure i s s i m i l a r t o the one described f o r Regular Polygons, except that the boundary conditions (39) t o (43) have t o be s a t i s f i e d at two sides AD and BC o f the t y p i c a l element under consideration.  Solution f o r F = 0 By taking n and m number o f points on the boundaries AD and BC r e s p e c t i v e l y , the conditions o f Case 1 (39), (40) and (41) would r e s u l t i n 2(n+m) l i n e a r algebraic equations i n 2(n+m) unknowns. parameter equation  The pressure drop  L, can then be evaluated from the i n t e g r a l form o f c o n t i n u i t y (6),  |j (44)  //VctA  For Case 2, however, since L appears i n (42) as an unknown, n and m number o f points on the boundaries AD and BC would r e s u l t i n 2(n+m) equations in[2(n+m)+llunknowns.  The [2(n+m)+l]^ equation i s given by (43). 1  The equations are again o f l i n e a r algebraic type and can be solved. For point matching, one could take e i t h e r n = m  o r n i m. The  t o t a l number o f equations and unknowns would be i n any case equal t o 2(n+m) or 2(n+m)+l depending upon Case 1 o r Case 2. y s i s , n was taken t o be equal t o m.  However, i n the present anal-  29 Solution f o r F t 0 In case solution i s required f o r a f i n i t e nonzero value o f F, the basic approach outlined f o r Regular Polygons could be followed i n conjunct i o n with the one described above. DIRECTION I I I t i s however, possible t o render the analysis mathematically more accurate by s a t i s f y i n g the thermal boundary conditions (41) o r (42) exactly at the c y l i n d e r w a l l , R = R^ , and point matching only a t the boundary BC o f the element.  This method i s demonstrated below f o r the case o f uniform  c i r c u m f e r e n t i a l w a l l temperature with no i n t e r n a l heat generation. F o r V = 4» = 0  at R = R^ , equations (37) and (38) can be written  as, for n = 0  A b e r OlR,) + B be.c C^R.,)4-C^kelr 0l*«> -t- L\Kel Cl«,) = O , (45) -A fc>*l 0iO B > e i i O i « . ) - C ^ . O v e . ) + T>.^c^*,>l=.o, (46) 0  0  D  o  e  D  o  +  similarly for n = 1  A ter C7<e,)+B ^eo C^0 s  s  - s A  s  S CT*,) + B  b e  s  s  + C^r^i)  D Kec C^.^O,  +  ber^R.) - C K<U^R ) £  t  s  *  (47)  $  t > K e r C ^ R ^ O , (48) s  and s i m i l a r set o f equations can be written f o r higher values o f n. From (45) and (46) AQ and B D  Q  Q  can be expressed i n terms o f C  . S i m i l a r l y , from (47) and (48), Ag and B  of C  s  and D  s  . Therefore, i n general, A  n s  s  and  can be expressed i n terms  and B  n s  can be expressed i n  terms o f C^g and D ^ . The expressions o f A ^ and B D  Q  n s  i n terms o f C  n s  and  when inserted i n (37) and (38) give v e l o c i t y and temperature functions  30 i n terms o f only two sets o f unknown c o e f f i c i e n t s C  n s  and D  n S  . These coef-  f i c i e n t s can then be obtained by point matching only at the remaining boundary BC o f the element shown.  For Case 1, the point matching procedure, f o r n number o f points on the boundary BC s a t i s f y i n g condition (39) w i l l r e s u l t i n 2n number o f equations i n 2n number o f unknowns.  For Case 2, however, n number o f points on the boundary BC would r e s u l t i n 2n number o f equations i n (2n+l) unknowns. known would be L. tem.  The a d d i t i o n a l un-  One therefore needs another equation t o solve the sys-  This (2n+l)*^ equation i s given by (43).  Now there are as many  equations as number o f unknowns.  This procedure can a l s o be used f o r nonzero values o f F.  In such  a s i t u a t i o n , the b a s i c approach o u t l i n e d e a r l i e r f o r F i 0 could be f o l lowed. In the proceeding numerical  computations.  section we w i l l discuss the r e s u l t s o f the  31 DISCUSSIONS  The r e s u l t s and discussions o f the present work have been presented i n the proceeding pages under three major subsections.  The  first  deals with the computational d e t a i l s , while i n the second and t h i r d subsections, r e s u l t s are discussed f o r the two sets of geometries, r e g u l a r polygonal ducts and flow outside c y l i n d e r s .  SOLUTION DETAILS  The accuracy o f the point-matching method w i l l depend upon the number o f points taken on the boundary t o s a t i s f y the boundary conditions. In general, more number o f points give b e t t e r r e s u l t s . work, however, t h i s i s not quite true.  In the present  Since i n the v e l o c i t y and temp-  erature expressions, (12) and (13), each term o f these equations i s an i n f i n i t e s e r i e s , the accuracy o f the r e s u l t s w i l l a l s o depend upon the convergence o f these s e r i e s .  These terms, which are i n the form o f modi-  f i e d Bessel functions, b e r ( 7 R ) , b e i ^ C l R ) , e t c . , can be evaluated from ns  expressions i n the form o f i n f i n i t e s e r i e s given i n McLachlan [20],  There  are, however, following two f a c t o r s i n the evaluation o f these functions, which a f f e c t t h e i r convergence. (a) value of the argument 1R, (b) value o f the s u f f i x  and  ns.  These functions were evaluated on an IBM d i g i t a l computer.  When  the s u f f i x ns was ^ 14, the convergence was found s a t i s f a c t o r y f o r values o f argument ^ 10.  For most o f the values o f arguments and s u f f i x , 15 to  20 terms o f the i n f i n i t e series were found s a t i s f a c t o r y t o achieve d e s i r able convergence.  32 The numerical value o f the argument % R imposes a r e s t r i c t i o n on the value o f Rayleigh number, since  L= (Ra) .  nondimensionalized i n such a way that R ^ l .  In t h i s study R was  As such, i t was p o s s i b l e  to take the maximum value o f 1 as 10, which corresponds t o Ra = 10000,  In the s u f f i x ns, n r e f e r s t o the number o f points taken on the boundary and s t o the number o f sides o f the polygon.  Therefore,  f o r example, f o r higher sided polygons, the number o f points that can be taken on the boundary have t o be r e l a t i v e l y l e s s , as ns has t o be l e s s than o r equal t o 14.  In the present work, a study was made on the e f f e c t o f number o f points and t h e i r d i s t r i b u t i o n on the boundary.  I t was observed that  the r e s u l t s improved as the number o f points were increased (Tables 1 and 2 ) , so long as the numerical values o f argument and s u f f i x d i d not exceed the l i m i t s mentioned e a r l i e r .  The present r e s u l t s were found t o  be r e l a t i v e l y close t o the a v a i l a b l e r e s u l t s (wherever a p p l i c a b l e ) , when the points on the boundary were evenly d i s t r i b u t e d with respect t o t h e i r angular p o s i t i o n .  In the next subsection, r e s u l t s are presented f o r regular polygonal ducts.  33 REGULAR POLYGONAL DUCTS Nusselt Numbers Under Uniform Circumferential Wall Temperature  The accuracy o f the present point-matching method could be determined by comparing the present r e s u l t s with those a v a i l a b l e i n the l i t erature, wherever i t i s p o s s i b l e .  Table 1 presents the Nusselt numbers  obtained by the present analysis with d i f f e r e n t number o f p o i n t s , as compared t o those o f an exact s o l u t i o n by Aggarwala and Iqbal [12] f o r an e q u i l a t e r a l t r i a n g u l a r duct.  This t a b l e shows that as we increase  the number o f points on the w a l l BC,the r e s u l t s improve.  This table c l e a r -  l y shows that with only four points the r e s u l t s are very close t o those o f the exact s o l u t i o n .  However, due t o the l i m i t a t i o n s o f the s u f f i x ns,  f o r an e q u i l a t e r a l t r i a n g u l a r duct, one cannot take more than f i v e points on the boundary.  A s i m i l a r comparison i s given i n Table 2 f o r a square duct.  The  r e s u l t s were compared with an exact s o l u t i o n by Han [ 8 ] . This t a b l e 3  i gives a d d i t i o n a l evidence that the present point-matching method i s quite accurate. Figure 5 presents the v a r i a t i o n o f Nusselt number against Rayleigh number f o r d i f f e r e n t number o f sides.  This f i g u r e shows that as the Rayleigh  number increases, the e f f e c t o f number o f sides on the Nusselt number d i minishes.  Figure 6 gives the p l o t s o f v a r i a t i o n o f Nusselt numbers against  number o f sides f o r some values o f the Rayleigh number.  From t h i s f i g u r e  one can c l e a r l y note how the Nusselt numbers a t t a i n asymptotic values f o r  'In Table 3, o f Han, these Nusselt numbers ard given as 3.69, 4.27 and 9.46 r e s p e c t i v e l y . A r e c a l c u l a t i o n o f Han's expressions shows that some of the values i n h i s Table 3 were somewhat i n e r r o r .  34 various amounts o f buoyancy e f f e c t s .  Table 3 l i s t s Nusselt number values.  I t may be added here  that i n t h i s table the values of Nusselt number at Ra = 1 should c l o s e l y correspond t o the r e s u l t s o f pure forced convection through r e g u l a r p o l ygonal ducts.  Comparison with Cheng's [4] values indicates that t h i s  indeed i s the case.  Nusselt Numbers Under Constant Circumferential Wall Heat Flux As indicated e a r l i e r , f o r combined free and forced convection through v e r t i c a l n o n - c i r c u l a r ducts, there does not appear to be a v a i l able any study corresponding t o the case o f uniform p e r i p h e r a l w a l l heat f l u x .  As such, the accuracy o f the present r e s u l t s i s d i f f i c u l t  to estimate.  However, a t Ra = 1, the Nusselt number values given i n  Table 4 agree c l o s e l y with the corresponding values given by references [17] and [18] f o r pure forced convection case. In the point-matching method, the unknown c o e f f i c i e n t s o f equations (31) t o (34) are obtained by s a t i s f y i n g the boundary conditions e x a c t l y , at a prechosen number o f points on the boundary.  Since the  boundary conditions are not s a t i s f i e d on the e n t i r e boundary, the numeri c a l values o f these functions V and  e>4>/^N, when evaluated at d i f f e r e n t  points on the boundary could be d i f f e r e n t from each other.  As such, t o  e s t a b l i s h the accuracy o f the present r e s u l t s f o r combined free and forced convection, the numerical values of these functions were evaluated at various points on the w a l l BC. ary  condition  The numerical values o f the -thermal bound-  0«25 are presented i n Tables 5 and 6 f o r e q u i l a t e r a l  t r i a n g u l a r and square ducts r e s p e c t i v e l y .  These values are obtained by  taking 4 points on the boundary BC t o s a t i s f y the boundary conditions, and  35 are evaluated at 10 points on the w a l l BC, which are equally spaced with respect to t h e i r angular p o s i t i o n s .  These tables show that at low values  o f Rayleigh number, the fluctuations i n the numerical values from (exact value) are n e g l i g i b l e .  0.25  At higher values of Rayleigh numbers, a t  some points there are some f l u c t u a t i o n s . not increase with Rayleigh number.  These f l u c t u a t i o n s , however, do  I t was a l s o observed that these f l u c t u -  ations were considerably reduced, when more points were taken to s a t i s f y the boundary conditions.  In a s i m i l a r fashion, the s a t i s f a c t i o n o f the condition V = 0 on the w a l l was a l s o studied and was  found very s a t i s f a c t o r y .  Therefore, i t appears that the r e s u l t s obtained from the present point-matching method f o r case of uniform circumferential w a l l heat f l u x are a l s o quite accurate. The Nusselt ber i n F i g . 7.  numbers are p l o t t e d as a function o f Rayleigh num-  This figure shows that a t higher values o f Rayleigh numbers,  the Nusselt numbers are l e s s s e n s i t i v e to the number o f sides o f the u l a r polygon.  reg-  The v a r i a t i o n o f Nusselt numbers against number o f sides  f o r various values o f Rayleigh numbers i s p l o t t e d i n F i g . 8. Comparison o f Nusselt Numbers f o r the Two Boundary Conditions For laminar forced convection through regular polygonal ducts, Cheng [18] obtained lower values o f Nusselt numbers f o r the case of uniform circumferential w a l l heat f l u x compared to the case of uniform circumferential wall temperature [ 4 ] ,  Yen [16] i n a study o f forced  convection through wedge shaped passages a l s o observed that Case 2 r e s u l t s i n lower values o f Nusselt numbers compared to Case 1.  36 F i g . 9 presents the v a r i a t i o n of Nusselt numbers against number o f sides f o r various values o f Rayleigh numbers.  This figure c l e a r -  l y shows t h a t the case o f uniform circumferential w a l l heat f l u x r e s u l t s i n lower values o f Nusselt numbers, as compared t o the case of uniform circumferential w a l l temperature.  I t i s also observed from t h i s figure  that f o r a given value o f Rayleigh number, the influence o f the i n c r e a s ing number o f sides on the Nusselt number values diminishes e a r l i e r f o r Case 1 as compared t o Case 2,  Velocity Distribution Fig.  10 presents the v e l o c i t y d i s t r i b u t i o n i n an e q u i l a t e r a l  t r i a n g u l a r duct f o r both the boundary c o n d i t i o n s , f o r various buoyancy effects.  F i g . 11 presents the same f o r a square duct.  For the case o f  Uniform circumferential w a l l temperature, these p r o f i l e s were confirmed from the a v a i l a b l e exact s o l u t i o n s .  For the case of uniform circumfer-  e n t i a l heat f l u x , however, no v e r i f i c a t i o n 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 w a l l a c c e l e r a t e s , and t o s a t i s f y the cont i n u i t y c o n d i t i o n , the f l u i d near the centre o f the duct slows down.  In  most cases, a flow r e v e r s a l was observed at the centre o f the duct when the Rayleigh number was increased t o about 7000, although the net flow remains  i n the upward d i r e c t i o n .  Fig,  10 also shows that f o r e q u i l a t e r a l t r i a n g u l a r duct, as  the Rayleigh number increases, the f l u i d near the corner C o f the duct accelerates more than that near the centre, B, o f the duct w a l l .  Also,  f o r the same buoyancy e f f e c t , Case 1 gives higher v e l o c i t y gradients near B compared t o Case 2,  Whereas, near the corner o f the duct, C, Case 2  gives higher v e l o c i t y gradients compared t o Case 1.  Temperature D i s t r i b u t i o n Figures 12 and 13 present, f o r both the Cases, the temperature d i s t r i b u t i o n i n e q u i l a t e r a l t r i a n g u l a r and square ducts f o r d i f f e r e n t values o f Rayleigh number.  For Case 1, these p r o f i l e s were compared with those o f the a v a i l a b l e exact solutions, [ 12] and [ 8]. that the temperature  This comparison has shown  d i s t r i b u t i o n obtained by the point-matching method  agrees very c l o s e l y with those obtained from the exact solutions mentioned.  Due t o t h i s closeness o f the r e s u l t s , no p l o t s are presented  showing the comparison o f the data.  For Case 2, however, no such com-  parison could be made, due t o u n a v a i l a b i l i t y o f any other s o l u t i o n . Figures 12 and 13 show that f o r both the Cases, as the Rayl e i g h number increases, the temperature d i s t r i b u t i o n i n the duct tends t o become uniform. These figures also show that f o r Case 1, higher values o f Rayleigh numbers produce higher temperature wall.  gradients a t the  However, f o r Case 2, these gradients remain constant.  Local Heat Flux Ratio For the case o f uniform c i r c u m f e r e n t i a l wall temperature, the l o c a l heat f l u x d i s t r i b u t i o n according t o equation (20) has been e v a l uated and f o r two p a r t i c u l a r geometries are p l o t t e d i n F i g s . 14 and 15. Figure 14 i s f o r the e q u i l a t e r a l t r i a n g l e and shows that while at low values o f the Rayleigh number, the maximum value o f l o c a l heat f l u x occurs a t point B; at a high value o f the Rayleigh number, the heat f l u x becomes uniform over s u b s t a n t i a l p o r t i o n o f the w a l l BC.  Figure 15 i s  f o r an hexagonal duct and shows that f o r t h i s geometry, the l o c a l heat  38 f l u x r a t i o i s r e l a t i v e l y l e s s s e n s i t i v e t o the free convection e f f e c t s .  I t can be noted from both the F i g s . 14 and 15 that the buoyancy e f f e c t s increase the l o c a l heat f l u x r a t i o a t the apex C while they reduce the same a t point B.  According  t o expectations, i t has been observed that as the  number o f sides i s increased, the differences i n l o c a l heat f l u x r a t i o are reduced, u n t i l f o r a c i r c l e they disappear completely.  Local Shear Stress Ratio The l o c a l shear stress d i s t r i b u t i o n has been evaluated equation (21).  Both the Cases 1 and 2 have been analyzed.  from  Figure 16  presents, f o r an e q u i l a t e r a l t r i a n g l e , the l o c a l shear stress r a t i o f o r both the boundary conditions.  This figure shows that while f o r  low values o f the Rayleigh number, the shear stress r a t i o i s maximum at point B, a t high values o f the buoyancy e f f e c t s t h i s maxima s h i f t s t o wards the apex C.  This rather unexpected r e s u l t i s borne out by c a l -  culations made from the exact s o l u t i o n given i n [ 8 ] .  Figure 16 also  presents a comparison o f shear s t r e s s r a t i o s between the two cases.  It  i s noted from t h i s diagram that at high values o f the buoyancy e f f e c t s , the condition o f uniform c i r c u m f e r e n t i a l wall heat f l u x produces lower shear stress r a t i o s a t the point B while, i t produces higher values o f the same near the apex C.  As the number o f sides i s increased, the v a r i a t i o n s i n l o c a l shear s t r e s s r a t i o s are reduced whether o r not there are buoyancy e f f e c t s present.  This applies t o both the circumferential boundary con-  d i t i o n s under consideration.  The differences i n l o c a l values reduce  very r a p i d l y as the number o f sides are increased from three.  Local Wall Temperature For the case o f uniform c i r c u m f e r e n t i a l w a l l heat f l u x , the v a r i a t i o n o f c i r c u m f e r e n t i a l w a l l temperature d i f f e r e n c e <|> has been studied.  With o r without buoyancy e f f e c t s , the maximum w a l l tempera-  ture d i f f e r e n c e occurs a t point B.  As the number o f sides i s increased,  the e f f e c t o f buoyancy on l o c a l w a l l temperature v a r i a t i o n diminishes, u n t i l f o r the c i r c l e i t completely disappears.  Local w a l l temperature  d i s t r i b u t i o n f o r four and eight sided regular polygonal ducts are shown i n F i g . 17.  Pressure Drop Parameter Pressure drop parameter L as a function o f the Rayleigh number and number o f sides has been evaluated f o r the two w a l l boundary conditions.  I t i s observed that when the number o f sides i s small and the  Rayleigh number i s high, the uniform c i r c u m f e r e n t i a l heat f l u x condition r e s u l t s i n higher values o f the pressure drop parameter compared t o the case o f uniform p e r i p h e r a l wall temperature.  Table 7 l i s t s some values  o f the pressure drop parameter f o r a number o f polygons under the two wall boundary conditions.  Figure 18 presents the pressure drop behav-  i o u r f o r an e q u i l a t e r a l duct f o r the two cases, and c l e a r l y shows that the uniform wall heat f l u x condition r e s u l t s i n higher pressure drops.  A b r i e f presentation o f the r e s u l t s o f the regular  polygonal  ducts i s a v a i l a b l e i n reference [21],  In the next subsection, r e s u l t s are discussed f o r flow outside v e r t i c a l c y l i n d e r s arranged i n r e g u l a r arrays.  40 FLOW OUTSIDE CYLINDERS  Nusselt Numbers Under Uniform Circumferential Wall Temperature  From the r e s u l t s o f the present a n a l y s i s , we f i r s t o f a l l present the Nusselt number r e s u l t s and compare them with those o f reference [7] f o r the l i m i t i n g s i t u a t i o n when Ra -* 0. In the present a n a l y s i s , we can compute the r e s u l t s f o r Ra as low as 0.01 but not f o r Ra=0.  However, i t i s known  that a t very small values o f the Rayleigh number, o f the order o f u n i t y , the heat t r a n s f e r r e s u l t s remain the same as those f o r pure forced convection.  In reference [ 7 ] , the forced convection r e s u l t s presented are only  f o r t r i a n g u l a r arrays and f o r the condition o f uniform c i r c u m f e r e n t i a l w a l l temperature.  Figure 19 presents the Nusselt number p l o t s against the spacing r a t i o with the Rayleigh number as a v a r i a b l e . For the purpose o f comparing the present r e s u l t with those o f reference [7, F i g . 3], two types o f Nusselt numbers have been p l o t t e d i n t h i s f i g u r e .  One with the h y d r a u l i c diameter  as a base and the other with the tube diameter as a base.  The s i g n i f i c a n c e  o f these two types o f Nusselt numbers i s that when the Nusselt numbers are based on hydraulic diameter, then an increase i n the tube spacing r a t i o causes an increase i n the Nusselt number values.  This i s because o f the  f a c t that the h y d r a u l i c diameter increases as the spacing r a t i o i s increased. In order t o study the heat t r a n s f e r r a t e , i t would probably be worthwhile looking at the Nusselt number values based on the tube diameter.  This  f i g u r e shows that an increase i n the spacing r a t i o causes a decrease i n the Nusselt number values, i f i t was based on tube diameter.  In figure 19, a t t e n t i o n has been d i r e c t e d on the spacing range o f 1.3 t o 2.5.  In tubular exchangers, spacing r a t i o o f 1.3 i s generally used.  41 Now comparing Figure 19 with Figure 3 o f reference [ 7 ] , i t i s noted that Ra = 1, the r e s u l t s o f the present a n a l y s i s agree with those o f I"?]. We now present a d e t a i l e d discussion o f Nusselt number v a r i a t i o n under various parameters.  In the following discussions, Nusselt numbers  have been based on the hydraulic diameter only, as t h i s i s a common pract i c e i n engineering.  For Case 1, figure 20 presents the Nusselt number r e s u l t s f o r square and t r i a n g u l a r arrays with Rayleigh number and spacing r a t i o as parameters.  This figure shows that as the spacing r a t i o increases, the  e f f e c t o f free convection on Nusselt number decreases.  However, the free  convection e f f e c t s influence the Nusselt number s u b s t a n t i a l l y a t lower spacing r a t i o s , a f a c t , which i s a l s o apparent from Figure 19.  It i s  a l s o noted that the e f f e c t o f increased buoyancy i s t o diminish the d i f ferences i n Nusselt number between the square and the t r i a n g u l a r arrays. However, a t lower values o f the Rayleigh number, t r i a n g u l a r array r e s u l t s i n higher values o f the Nusselt number. Nusselt Numbers Under Uniform Circumferential Wall Heat Flux For the boundary condition o f Case 2, the Nusselt number p l o t s are presented i n Figure 21 and are e s s e n t i a l l y o f the same nature as those f o r Case 1, Figure 20. Comparison o f Nusselt Number Values f o r the Two Boundary Conditions Figure 22 presents f o r a square array, the differences between the Nusselt number values f o r the two boundary conditions, Case 1 and Case 2.  As predicted (but not evaluated) by reference [ 7 ] , the uniform  circum-  f e r e n t i a l w a l l temperature condition r e s u l t s i n higher values o f the Nusselt  42 number.  This i s true when spacing r a t i o i s low.  However, f o r higher  spacing r a t i o s , the r e s u l t s become almost i d e n t i c a l .  At higher values o f  the Rayleigh number, the differences i n the Nusselt number f o r the  two  boundary conditions diminish and t h i s r e s u l t i s i n accordance with the study o f heat t r a n s f e r through the regular polygonal ducts.  The  present  analysis shows that f o r t r i a n g u l a r arrays, the differences between the Nusselt number values f o r Case 1 and Case 2 are much smaller as compared t o those i n square arrays.  Velocity Distribution  To gain some i n s i g h t i n t o the flow behaviour and eventually the shear s t r e s s d i s t r i b u t i o n at the w a l l , v e l o c i t y p r o f i l e s are presented i n Figures 23 to 25.  These p r o f i l e s are along l i n e s DC, AB and BC r e s p e c t i v e l y  o f the element, Figure 3(b). two spacing r a t i o s , 1.3 and  The v e l o c i t y p r o f i l e s are presented f o r only 2.  Figure 23 shows v e l o c i t y p r o f i l e s along DC.  As the buoyancy e f f e c t s  are increased, the v e l o c i t y at the centre of the system o f tubes, point C, i s reduced and to s a t i s f y c o n t i n u i t y , the v e l o c i t y near the w a l l increases. For spacing r a t i o o f 1.3, i f the Rayleigh number i s increased enough, say Ra 6000, i t i s t h e o r e t i c a l l y p o s s i b l e to a t t a i n flow r e v e r s a l (negative v e l o c i t y ) at point C.  However, i t i s doubtful whether at such high values  o f the Rayleigh number, laminar flow w i l l e x i s t any longer.  Figure 24 presents the v e l o c i t y p r o f i l e s along AB.  From t h i s  diagram, an i n t e r e s t i n g observation i s that the increasing values of the Rayleigh number increase the v e l o c i t y at point B.  As spacing r a t i o increases,  the influence of buoyancy on the v e l o c i t y p r o f i l e along AB diminishes.  43 The v e l o c i t y p r o f i l e s along BC are p l o t t e d i n Figure 25. These v e l o c i t y p r o f i l e s are i n accordance with the observations made regarding Figures 23 and 24,  Temperature D i s t r i b u t i o n  For Case 1, temperature p r o f i l e s are presented i n Figures 26 t o 28 f o r a square array.  These p r o f i l e s are along sides DC, AB and BC o f  the t y p i c a l element shown i n F i g , 3(b).  These p r o f i l e s are again only f o r  two spacing r a t i o s , 1.3 and 2.  Figure 26 presents the temperature d i s t r i b u t i o n along DC o f the element.  This f i g u r e shows that f o r small spacing r a t i o s , an increase i n  Rayleigh number causes the temperatures DC.  t o become uniform along the side  T h i s f i g u r e also shows that f o r higher spacing r a t i o s , the temperature  p r o f i l e s along DC are less s e n s i t i v e t o Rayleigh number.  Figure 27 presents the temperature p r o f i l e s along the side AB. This figure shows that f o r small spacing r a t i o s , an increase i n Rayleigh number increases the temperature d i f f e r e n c e a t point B,  This figure also  shows that the temperature d i s t r i b u t i o n along AB, f o r higher spacing r a t i o s , i s l e s s s e n s i t i v e t o Rayleigh number.  The temperature p r o f i l e s along BC are presented i n F i g . 28, These p r o f i l e s are i n accordance with the observations made regarding Figures 26 and 27. Wall Heat Flux Ratio  For Case 1, the v a r i a t i o n o f heat f l u x r a t i o on the tube w a l l AD i s presented i n Figures 29 and 30, f o r square and e q u i l a t e r a l t r i a n g u l a r  44 arrays r e s p e c t i v e l y .  For Ra=l, the values o f l o c a l t o average heat-flux  r a t i o f o r an e q u i l a t e r a l t r i a n g u l a r array were confirmed from reference C7].  As expected, the lower spacing r a t i o o f 1,3 gives l a r g e r v a r i ations o f the circumferential heat f l u x .  Again the increasing buoyancy  e f f e c t s reduce these circumferential v a r i a t i o n s .  At the higher spacing  r a t i o o f 2, Figures 29 and 30 i n d i c a t e t h a t , the circumferential heat f l u x has almost become uniform and t h a t , there i s very l i t t l e added i n fluence o f the buoyancy e f f e c t s .  As expected, the minimum value o f heat  f l u x occurs a t point A.  When f i g u r e 29 i s compared with 30, i t i s noted that the v a r i a t i o n i n the heat f l u x r a t i o i s higher f o r a square array compared with that o f an e q u i l a t e r a l t r i a n g u l a r array.  Wall Temperature  Distribution  For Case 2, i t i s important t o determine the temperature bution on the tube w a l l .  distri-  This i s shown i n Figures 31 and 32 f o r square  and t r i a n g u l a r arrays r e s p e c t i v e l y , f o r two spacing r a t i o s , 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 v a r i a t i o n s o f the tube wall AD, as compared t o a t r i a n g u l a r array.  These figures also show that a t the  lower spacing r a t i o s , there i s a wide temperature v a r i a t i o n . However, t h i s temperature v a r i a t i o n i s reduced when the buoyancy e f f e c t s are increased. In a d d i t i o n , a t higher spacing r a t i o s , o f the order o f 2, temperature becomes almost uniform over the tube circumference and. i s not p e r c e p t i b l y influenced by the buoyancy e f f e c t s . occurs at point A o f the tube w a l l .  As expected, the maximum temperature  45 Wall Shear Stress Ratio The w a l l shear stress d i s t r i b u t i o n s f o r Case 1 are shown i n Figures 33 and 34 f o r square and t r i a n g u l a r arrays r e s p e c t i v e l y , f o r two spacing r a t i o s , 1.3 and 2.  When these two figures are compared,  i t i s found that the t r i a n g u l a r array r e s u l t s i n smaller shear s t r e s s v a r i a t i o n s compared t o a square array.  Also at smaller spacing r a t i o s ,  1.3, and f o r s i t u a t i o n s approaching forced convection, the shear s t r e s s i s minimum at point A and maximum at D.  As the Rayleigh number i s i n -  creased, the shear s t r e s s d i s t r i b u t i o n becomes uniform.  When the Ray-  l e i g h numbers are increased f u r t h e r , the s i t u a t i o n gets reversed and now the maximum value o f the w a l l shear s t r e s s occurs a t A and the minimum a t D.  For higher spacing r a t i o s , the shear s t r e s s i s uniform over the portion AD o f the element and i s l e s s s e n s i t i v e t o the Rayleigh number.  For Case 2 boundary condition, the behaviour o f the w a l l shear  stree remains e s s e n t i a l l y s i m i l a r t o the one explained above.  A b r i e f presentation o f the r e s u l t s o f the flow outside c y l i n ders i s a v a i l a b l e i n reference [22],  46 CONCLUSION  A general s o l u t i o n t o the problem o f laminar combined free and forced convection through v e r t i c a l n o n c i r c u l a r 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 f o r two sets o f geometries,  ( i ) regular polygonal ducts, and ( i i ) flow between cylinders arranged i n regular arrays.  Two p o s s i b l e thermal boundary conditions have been  analyzed, Case 1 - uniform circumferential w a l l temperature,  and  Case 2 - uniform circumferential w a l l heat f l u x . In general ,the case o f uniform circumferential w a l l temperature r e s u l t s i n higher values o f Nusselt number as compared t o the case o f uniform circumferential w a l l heat f l u x .  However, at higher values  o f the Rayleigh number, both the boundary conditions tend t o produce the same r e s u l t s ,  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  Ra  NUSSELT NUMBERS Number of Points on the Wall BC, Present Analysis 2  3  4  Exact oo i U L i o n  [12]  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  10  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.  NUSSELT NUMBERS  Ra  Number of Points on the Wall BC, Present Analysis 2  10ir  4  IOOTT  4  3  4  Exact Solution [*]  3.7702  3.7092  3.7004  3.6962  4.5891  4.4722"  4.4492  4.4372  7.4134  8.3815  8.3342  8.2716  00  TABLE 3. Nusselt Numbers Against Rayleigh Numbers for Various Polygons for Uniform Circumferential Wall Temperature.  ;  NUSSELT NUMBERS  Number of Sides of Polygon  Ra  Ci rcl e (Exact Solution)  3  4  5  6  7  8  12  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  4.37  4.45  4.51  4.60  4.69  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  1000  .  .4.26  TABLE 4. Nusselt Numbers Against Rayleigh Numbers for Various Polygons for Uniform Circumferential Heat Flux.  NUSSELT NUMBERS  Ra  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 o f the Function  - at  Various Points on the Wall BC o f an E q u i l a t e r a l Triangular Duct Under Uniform Circumferential Heat Flux  Numerical Values o f the Function  c^/^N  Location o f Points on Wall BC, 0 i n Degrees  Ra 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 o f the Function  —3—  at  an Various Points on the Wall BC o f a Square Duct Under Uniform Circumferential Heat Flux Numerical Values o f the Function Location o f Points on Wall BC, &  £4?/2>N  i n Degrees  Ra .0*  5  10  1  .2499  .2499  .2499  100  .2499  .2499  1000  .2500  2000  5000  15*  20  25  .2499  .2499  .2499  .2500  .2500  .2500  .2500  .2499  .2499  .2500  .2500  .2499  .2499  .2499  .2500  .2499  .2499  .2499  .2502  .2505  .2499  .2477  .2448  .2501  .•2500  .2499  .2499  .2500  .2501  .2502  .2499  .2489  .2475  .2501  .2499  .2500  .2500  .2499  .2498  .2496  .2499  .2514  ,2531  .2499  *These were the prechosen points.  30*  35  40  45*  TABLE 7.  Pressure Drop Parameters L Against Rayleigh. Numbers f o r .. Various Polygons f o r Both the Boundary Conditions.  PRESSURE DROP PARAMETER L Uniform Circumferential Wall Temperature Ra  Uniform Circumferential Heat Flux  Number of Sides. 8  Number of Sides 16  4  6  32.06  27.03  28.56  30.06  30.17. . 32.00  32.06  37.16  37.69  53.39  41.90  38.07  36.87  37.32  37.69  59.81  59.72  59.63  129.02  94.67  68.73  62.94  60.15  59.63 .  86.67  85.40  85.45  202.39  158.35  104.99  93.58  87.01  85.45  136.97  135.57 133.36  132.81  347.16  278.30  171.04  .149.73  136.29  132.81  259.92  257.95 253.31  252.56  810.05  570.88  341.22  292.42  261.21  252.56  4  6  l  26.75  28.50  30.04  30.17  31.98  TOO  34.53  35.26  36.16  36.86  500  63.10  61.03  59.81  1000  94.13  90.35  87.30  2000  146.80  142.01  5000  271.27  266.35  1  Circle  8  i  3  3  16  Circle  UNIFORM  9  ••  ,  CIRCUMFERENTIAL  L_  1  :  —  TEMPERATURE  •• • • NUMBER  ^  WALL  L _  10  RAYLEIGH FIGURE'5..- Nusselt Number_Agsiinst_.  OF  :—I  SIDES  _J  ,  1  10  NUMBER R  I  0  i i i  I  .  l  111  i  a Polygons  •  ^  "  10  s  UNIFORM  T  —  —  1  —  —  —  T  CIRCUMFERENTIAL  —  1  ~  WALL  T  TEMPERATURE  8  3 Ra= 5x10" 6  4  10' 1  •  • • • • • • • • • • • • • • • • • • • • •  »»»  to  GO 2  11  9 NUMBER  OF  13  15  SIDES  FIGURE 6 - Nusselt Number Against Number o f Sides f o r Various Rayleigh Numbers  a  M  I  I  I —T" -  UNIFORM  1  ——I  CIRCUMFERENTIAL  HEAT  —I— FLUX  9  Ra = &x10'  10 1  1  7 9 NUMBER OF  3  11 13 1.5 SIDES 1  FIGURE 8 - Nusselt Number Against Number o f Sides f o r Various Rayleigh Numbers  10  1  ~ 1 Ra=  5 0 0 0  8  or  2000  LU CD I Z  CO CO  UNIFORM CIRCUMFERENTIAL WALL TEMPERATURE UNIFORM 0  CIRCUMFERENTIAL  HEAT  FLUX  I  I  I  1  I  5  7  9  11  13  NUMBER  15  17  OF SIDES  FIGURE 9 - Nusselt Number Against Number o f Sides f o r Both The Boundary Conditions  00  FIGURE 10 - Velocity D i s t r i b u t i o n i n E q u i l a t e r a l Triangular Duct  2.5L  SQUARE DUCT Circumferential Wall  Uniform .......Uniform  Circumferential  -0.25  Heat  0.0  DISTANCE  Temperature Flux  0.25  R*  FIGURE 11 - V e l o c i t y D i s t r i b u t i o n i n Square Duct  R Cos 0  o  FIGURE 12 - Temperature D i s t r i b u t i o n i n E q u i l a t e r a l Triangular Duct  TEMPERATURE  39  DIFFERENCE  - *  1.5  EQUILATERAL  TRIANGLE  FIGURE 14 - Local To Average Heat Flux Ratio f o r E q u i l a t e r a l T r i a n g u l a r Duct  1.5  HEXAGON  CO  r-  <  1.0  X  0.5 h-  <  UNIFORM  B  CIRCUMFERENTIAL  WALL  TEMPERATURE  THE WALL B C FIGURE 15 •- Local to Average Heat Flux Ratio f o r Hexagonal Duct ON  1.5 EQUILATERAL  > o  TRIANGLE  1 .0 <  co  CO *00 0.5  Ra=  01  10  <  LU  •UNIFORM CIRCUMFERENTIAL HEAT  CO  • UNIFORM CIRCUMFERENTIAL 0.0  B  THE  WALL  FLUX  WALL  TEMPERATURE  BC  FIGURE 16 - Local t o Average Shear Stress Ratio f o r E q u i l a t e r a l Triangular Duct  0.20 i  UNIFORM  CIRCUMFERENTIAL  HEAT  F.LUX  FIGURE 17 - Local Wall Temperature Difference f o r Square and Octagonal Ducts  5x ICT LU  LLJ  < <  1—I  UNIFORM  CIRCUMFERENTIAL  UNIFORM  CIRCUMFERENTIAL  WALL  I I 11  TEMPERATURE  HEAT  FLUX  10  O  or  o  2  Q LU r  ai  EQUILATERAL  co  CO LU  10  TRIANGLE  1  10  RAYLEIGH  10  NUMBER  3 10  _1  t I I II II -A 10  Ra  FIGURE 18 - Pressure Drop Parameter Against Rayleigh Number f o r E q u i l a t e r a l Triangular Duct  ON  68  r  1.0  i  i  T  [ - — i — i — i — i — | — i — i — i — r  1.5  SPACING  2.0  2.5  RATIO  FIGURE 19 - Nusselt Number Values f o r Cylinders Arranged i n Triangular Array  1  1—r i i r r r  Uniform 2 0  r  i  ~l  1—i—rr-rrj  Circumferential  Spacing  ~i  i—r—i  Wall  i  TTT  ~»  i  j  j—rrnr  Temperature  Ratio  LU  cn  15  LU CO CO Z) 2  5  Square  Array  Triangular 0  J  I  1  L i  I i i  I  10  J  I  I  t  i i  i  i  i  100  RAYLEIGH  Array i  i  1111  i  i  i  1000  NUMBER  Ra  FIGURE 20 - Nusselt Number Against Rayleigh Number f o r Various Spacing Ratios  i  1111  1 0 0 0 0  1  Uniform 2 0  Circumferential  Spacing  Ratio  1  1  1  » i » |  Heat  1  1  1  \  \  VTT  Flux  =2.0  UJ CD 15  *jjf«»mf  juuwwi.vuyuJiMJiRWW^J^^  . . . . . . . •-  ....-^  \— 10 UJ  co  CO 3  inn  _ _ .  • 1  Square  0  Array  .... Triangular A  1 - 1  1  » I 1 1  1  .1  1  1  10 RAYLEIGH  1  i  1 ( t  f  i  i  - i  I_I  100 NUMBER  .i111  Array i  i  i  1000  i  \  I , i  1 0 0 0 0  Ra  FIGURE 21 - Nusselt Number Against Rayleigh Number for Various Spacing Ratios  o  25  20  LU CQ =>  h-  15  10  CO CO ZD  - Uniform  Circumferential  Wall  Temperature  .. Uniform  Circumferential  Heat  Flux  i  1  1—I  I  I I I  I  '  '  I  I  L  10  J  I  1  100  RAYLEIGH  »  '  ' ' ' I  I  i  1000  NUMBER  Ra  FIGURE 22 - Nusselt Number Values f o r Cylinders Arranged i n Square Arrays  i  i i i i •  10000  SQUARE Uniform Circumferential  ARRAY Wall  Temperature  FIGURE 23 - Velocity P r o f i l e s Along DC o f the T y p i c a l Element  2.0  SQUARE Uniform 1.5  Spacing  ARRAY  Circumferential Wall  Temperature  Ratio  1000 6000  THE  SIDE  AB  B  FIGURE 24 - Velocity P r o f i l e s Along AB o f the T y p i c a l Element u>  SQUARE  2.5  Uniform  ARRAY  Circumferential  Spacing  Wall  Temperature  Ratio  1.3  O o LU >  1000 6000 0.0  B  THE  SIDE  BC 4>  FIGURE 25 - V e l o c i t y P r o f i l e s Along BC o f the*j.Typical Element  TRIANGULAR  I 0  i  I  10  ANGLE  •  ARRAY  '  !  • 2 0  9 , Degrees  FIGURE 29 - Local to Average Heat Flux Ratio at the Wall f o r Cylinders Arranged i n Triangular Arrays  I 30  SQUARE  0  10  20 ANGLE  ARRAY'  30  40  45  0, Degrees  FIGURE 30 - Local To Average Heat Flux Ratio a t the Wall f o r Cylinders Arranged i n Square Arrays.  "°  TEMPERATURE o  -*  DIFFERENCE cfrxlOO *o  CO  ^  1  1  TRIANGULAR Uniform  Circumferential  ANGLE  9,  I  r  ARRAY Wall  Temperature  Degrees 00  FIGURE 32 - Local Wall Temperature Difference f o r Cylinders Arranged i n Triangular Arrays  M  ^  SJ-3  J*  5  Uniform Spacing  i  1  r  SQUARE ARRAY Circumferential Wall Temperature Ratio  6000 0.7  JL  10  20  ANGLE  30  40  9 , Degrees  FIGURE 33 - Local t o Average Shear Stress Ratio a t the Wall f o r Cylinders Arranged i n Square Arrays  45  TRIANGULAR  0  10  ANGLE  9,  ARRAY  20  Degrees  30 00  FIGURE 34 - Local to Average Shear Stress Ratio at the Wall f o r Cylinders Arranged i n Triangular Arrays  W  84  BIBLIOGRAPHY 1.  Clark, S.H. and Kays, W.M.,  ''Laminar Flow Forced Convection  i n 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 o f 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 C h a r a c t e r i s t i c s i n Regular Polygonal Ducts", Proceedings o f the Third International Heat Transfer Conference, A.I.Ch.E., V o l . 1, 1966, pp. 64-76.  5.  S i e g e l , R. and Savino, J.M., "An A n a l y t i c a l Solution o f the E f f e c t o f 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., L o e f f l e r , A.L. and Hubbard, H.A., "Heat Transfer t o Longitudinal Laminar Flow Between Cylinders", Trans. ASME, V o l . 83, Series C, J . Heat Transfer, 1961, pp. 415-422.  8. Han, L.S., "Laminar Heat Transfer i n Rectangular Channels", Trans. ASME V o l . 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 C i r c u l a r and Sector Tubes", Applied S c i e n t i f i c Research, Section A, V o l . 9, No. 5, 1960, pp. 357-368.  11.  Agrawal, H.C.,  " V a r i a t i o n a l Method f o r Combined Free and Forced  Convection i n  Channels", Int. J . Heat and Mass Transfer, V o l . 5,  1962, pp. 439-444.  12.  Aggarwala, B.D. and Iqbal, M., "On Limiting Nusselt Number From Membrane Analogy f o r Combined Free and Forced Convection Through V e r t i c a l Ducts", I n t . J . Heat Mass Transfer, V o l . 12, 1969, pp. 437-748.  13.  Iqbal, M., Aggarwala, B.D, and Fowler, A.G., "Laminar Combined Free and Forced Convection i n V e r t i c a l Noncircular Ducts Under Uniform Heat Flux", I n t . J . Heat and Mass Transfer, V o l . 12, 1969, pp. 1123-1139.  14.  Lu, P.C., "A Theoretical Investigation o f Combined Free and Forced Convection Heat Generating Laminar Flow Inside V e r t i c a l Pipes With Prescribed Wall Temperatures", M.S. Thesis Kansas State College, Manhattan, Kansas, 1959.  15.  Eckert, E.R.G., I r v i n e , T,F., J r . , and Yen J.T., "Local Laminar Heat Transfer i n Wedge-Shaped Passages", Trans. ASME, V o l . 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, J u l y , 1957,  17.  Sparrow, E.M. and S i e g e l , R., "A V a r i a t i o n a l Method f o r F u l l y Developed Laminar Heat Transfer i n Ducts", Trans. ASME, V o l . 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., " E f f e c t o f Tube Orientation i n I^aminar Convective Heat Transfer", Ph.D. Thesis, M c G i l l U n i v e r s i t y , 1965.  20.  McLachlan, N.W.,  "Bessel Functions  f o r Engineers", Oxford U n i v e r s i t y  Press, England, 1934. 21.  Iqbal, M., A n s a r i , S.A., and Aggarwala, B.D., " E f f e c t o f Buoyancy on Forced Convection i n V e r t i c a l Regular Polygonal Ducts", Trans. ASME, J . Heat Transfer ( i n Press).  22.  Iqbal, M., A n s a r i , S.A. and Aggarwala, B.D., "Buoyancy E f f e c t s on Longitudinal Laminar Flow Between V e r t i c a l Cylinders Arranged i n Regular Arrays", Proceedings of Fourth International Heat Transfer Conference, P a r i s , August 31 t o September 5, 1970, (to be published).  A P PENDI C E S  88 APPENDIX A  For the case o f uniform circumferential wall heat f l u x , the heat input, q = c ~ = constant, expression i s obtained i n the nondimens i o n a l form i n t h i s  appendix.  i  Ji_i  f  i *  H — f — r  H  Z —  A  Consider the f l u i d flowing between sections 1 and 2, o f a duct i n the figure shown above.  By making a simple energy balance, we get,  f U C ^ A (T.-TO - ^ P A X + a A &TL ,  (A-l)  where, Tj_ and T2 are the bulk temperatures  a t the two s e c t i o n s , and  P i s the heat t r a n s f e r perimeter o f the duct. Since,  a~L  C  "*  ±  q = h  t^T  equation (A-l) reduces t o ,  (A-2)  From the dimensionless variables,<^> and N, we get,  and  c> n A *~P  also,  D  =  c> N  h  ,  lb- .  ~  4-  Substituting the  above variables i n equation (A-2), we obtain  = 0.25(1-F)  (32)  a n  89 APPENDIX B  Nusselt number expressions are obtained i n terms o f the dimensionless variables i n t h i s appendix f o r both the boundary conditions.  Nusselt number, which s i g n i f i e s the energy converted from a surface, can be written as, Nu  ^  ,  (B-l)  where, T  w  i s the average w a l l temperature,  Tjj i s the bulk temperature o f the f l u i d and can be written as,  I/TLL  dob  Tb =  •  (B-2)  il a da Substituting (B-2) i n ( B - l ) , we g e t ,  Nu=—  •  (B  .  3)  As i n Appendix A, by making a simple energy balance between sections 1 and 2, we obtain,  f oCjjC, A - ^ or  9-  -  p +  a A  iUL ( f U C b C . - Q )  > .  Substituting (B-4) i n (B-3) we get ,  ih  1  * k  (fUC^C.-Q)  X . - (//"Tu. oU/f/u.«U)  (B-5)  90  F i r s t o f a l l we present the nondimensional  form o f Nusselt number f o r  Case 1,  Case 1 - Uniform Circumferential Wall Temperature For t h i s boundary c o n d i t i o n , 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  (B-6)  .  Substituting (B-6) i n (B-5), we obtain  I-F (17)  mx  J/4>VclA  "here,  <p  —  • J J  (18)  vaA  Now we consider the boundary condition of Case 2.  Case 2 - Uniform Circumferential Wall Heat Flux For t h i s boundary c o n d i t i o n , the dimensionless variables <^>  >  and V r e s u l t i n ,  T  = T^  +  e x  ix = VVJ  Cf U C y . C ^ / f e ) 4 , (B-7)  .  Substituting (B-7) i n (B-5), we obtain,  Na =  ^ — —  •  ^  

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