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A theoretical model for predicting rough pipe heat transfer. Kiss, Mart 1963

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A THEORETICAL MODEL FOR PREDICTING ROUGH PIPE HEAT TRANSFER by MART KISS 3.A.Sc., The U n i v e r s i t y of B r i t i s h C o l u m b i a 1 9 5 9 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of ' ' CHEMICAL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH December, 1963 COLUMBIA I n 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 , of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives,, I t i s understood that, copying, or• p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of f^#6fifSf*<q>/ (- t^g/U^d'S/ The U n i v e r s i t y of B r i t i s h Columbia,, Vancouver 8, Canada. Date . . t>7 ABSTRACT A model has been developed f o r p r e d i c t i n g t u r b u l e n t heat t r a n s f e r c o e f f i c i e n t s and a s s o c i a t e d temperature p r o f i l e s i n rough pipes from a knowledge of the f l u i d mechanics. The proposed method employs the Lyon heat t r a n s f e r equation together with the v e l o c i t y p r o f i l e equations of Rouse and von Karman. Nus s e l t numbers were c a l c u l a t e d by the proposed method f o r the f o l l o w i n g range of v a r i a b l e s : f = f to 0 .0 20 Re = 4 x 1 0 3 to 1 0 7 Pr = 0 . 0 0 1 to 1 ,00 0 Temperature p r o f i l e s were c a l c u l a t e d f o r a l l combinations of the above extreme c o n d i t i o n s , as w e l l as f o r Pr = 1 . 0 . The v a l i d i t y of the proposed model was t e s t e d by comparison of the p r e d i c t e d r e s u l t s with the experimental data of Nunner, Smith and E p s t e i n and Dipprey. A s i m i l a r t e s t was made of Nunner's t h e o r e t i c a l e q uation. I t i s concluded t h a t , except f o r f l u i d s with very low P r a n d t l numbers, e.g. l i q u i d m etals, the proposed model gives no b e t t e r p r e d i c t i o n of N u s s e l t number than Nunner's e q u a t i o n , which i s l e s s cumbersome to apply. i i I n t h e e x i s t i n g f o r m , t h e p r o p o s e d model i s n o t a d e q u a t e . C e r t a i n c o m b i n a t i o n s o f t h e i n d e p e n d e n t v a r i a b l e s g i v e r i s e t o a d i s c o n t i n u i t y i n t h e p r e d i c t e d v a l u e o f N u s s e l t number. T h i s i s i n c o n c e i v a b l e i n t h e p h y s i c a l l y r e a l s i t u a t i o n . Beyond t h e d i s c o n t i n u i t y a p p e a r s a p r e d i c t e d r e g i o n o f z e r o n e t f l o w i n t h e p i p e . Two l i m i t i n g a s s u m p t i o n s c an be made r e g a r d i n g t h e method o f h e a t t r a n s p o r t t h r o u g h t h i s l a y e r - v i z . by m o l e c u l a r c o n d u c t i o n o n l y , o r by an i n f i n i t e c o n d u c t i v i t y eddy mechanism. Both a s s u m p t i o n s have been made, and v a l u e s o f Nu c a l c u l a t e d f o r e a c h , whenever t h e s i t u a t i o n a r o s e . The agreement betv;een t h e p r e d i c t e d and t h e e x p e r i m e n t a l t e m p e r a t u r e p r o f i l e s i s i n g e n e r a l good. However, n o t enough e x p e r i m e n t a l d a t a a r e a v a i l a b l e t o s a t i s f a c t o r i l y d e f i n e t h e e f f e c t o f Re and f , and t o s u b s t a n t i a t e t h e c a l c u l a t e d r e s u l t s . v i A C K N O W L E D G E M E N T S I t h a s b e e n a p l e a s u r e w o r k i n g w i t h D r , N . E p s t e i n . I w i s h t o t h a n k h i m f o r t h e i n v a l u a b l e a d v i c e a n d c o o p e r a t i o n h e h a s g i v e n me i n s u p e r v i s i n g t h e w o r k o n t h i s p r o j e c t . I am i n d e p t e d t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a a n d t o t h e D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g 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 f o r t h e g e n e r o u s f i n a n c i a l a s s i s t a n c e t h a t h a s b e e n e x t e n d e d t o m e . L a s t l y , I w o u l d l i k e t o e x t e n d my a p p r e c i a t i o n t o t h e s t a f f o f t h e C o m p u t i n g C e n t r e f o r t h e i r a s s i s t a n c e i n o v e r c o m i n g p r o g r a m m i n g d i f f i c u l t i e s , a n d f o r m a k i n g a v a i l a b l e t h e i r c o m p u t i n g f a c i l i t i e s . i i i TABLE OF CONTENTS Page LIST OF FIGURES v ACKNOWLEDGEMENTS v i INTRODUCTION 1 DESCRIPTION OF THE PROPOSED METHOD FOR PREDICTING 6 HEAT TRANSFER IM ROUGH PIPES 1. Heat T r a n s f e r Equation 6 2. E v a l u a t i o n of R a d i a l C o n d u c t i v i t y R a t i o 8 3. V e l o c i t y P r o f i l e 10 Table 1. Summary of Equations Required f o r 13 S o l u t i o n of Equations 2 and 3 4. Boundaries of Flow Regions i n Rough Pipes 16 5. Thermal C o n d u c t i v i t y i n the '//all Layer 19 NUNNER'S EQUATION FOR PREDICTION OF ROUGH PIPE HEAT 2 2 TRANSFER RESULTS 2 8 1. Heat T r a n s f e r C o e f f i c i e n t and Temperature 28 P r o f i l e R e s u l t s C a l c u l a t e d a c c o r d i n g to Proposed Method 2. Comparison of P r e d i c t e d R e s u l t s with 29 Experimental Data and with Nunner's Theory a) Comparison with the Work of Nunner 29 b) Comparison with the Work of Smith 40 and E p s t e i n c) Copparison with the Work of Dipprey 5 2 Table 2. Tabular Comparison of P r e d i c t e d 56 R e s u l t s with Dipprey's Experimental Data and with Nunner's Equation 3. E v a l u a t i o n of Experimental C o n d u c t i v i t y R a t i o 59 i n the Wall Layer DISCUSSION OF RESULTS 1. General D i s c u s s i o n of Proposed Method 2. Comparison of R e s u l t s as P r e d i c t e d from the Proposed Method with Experimental Data 3. E v a l u a t i o n o f Experimental C o n d u c t i v i t y R a t i o i n the Wall Layer U, D i s c u s s i o n of Temperature P r o f i l e Data CONCLUSIONS AND RECOMMENDATIONS NOMENCLATURE LITERATURE CITED APPENDIX 1. Computational D e t a i l s 2. T e s t of Computer Program Table 3, Summary of Data used to check Correctness of Computer Program 3. Computer Program t V LIST OF FIGURES Figu r e 1 Fig u r e 2 Fig u r e 3 F i g u r e 4a to 4e Figure 5 Fig u r e 6a to 6d Figure 7 a to 7 d F i g u r e 8a to 8 c Fig u r e 9a to 9d Fig u r e 10a to 10c F i g u r e 11 Figure 12 Sketch of Various P o s s i b l e V e l o c i t y P r o f i l e s i n Rough Pipes P l o t Showing the Range of A p p l i c a t i o n of CASES 1, 2 and 3 on f - Re Coordinates Sketch of Nunner's Model Showing Flow C o n d i t i o n s i n Rough Pipes B a s i c Heat T r a n s f e r R e s u l t s C a l c u l a t e d A c c o r d i n g to Proposed Method: Nu versus Re I n f l u e n c e of Pr on the Basic Heat T r a n s f e r R e s u l t s B a s i c Temperature P r o f i l e s C a l c u l a t e d A c c o r d i n g t o Proposed Method: 9 and V vs. y / r Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Munner Comparison of P r e d i c t e d Temperature P r o f i l e s with Experimental R e s u l t s of Nunner Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental Results, of Smith and E p s t e i n Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Dipprey Average K/k i n Wall Layer as Computed from Experimental Data Flowsheet of the Computer Program w Page 15 20 24 30 35 36 41 45 4 8 53 6 0 86 1 INTRODUCTION The p r e d i c t i o n of heat t r a n s f e r from a knowledge of f l u i d p r o p e r t i e s and a s s o c i a t e d flow c o n d i t i o n s has been the t o p i c of study f o r many i n v e s t i g a t o r s . As e a r l y as 1 8 7 H Reynolds r e c o g n i z e d the e x i s t e n c e of a r e l a t i o n s h i p between s k i n f r i c t i o n and heat t r a n s f e r , and i n i t i a t e d development of the analogy now known by h i s name ( 1 ) . Since t h a t time, a multitude of both experimental and t h e o r e t i c a l i n v e s t i g a t i o n s have been conducted i n the f i e l d . S e v e r a l s u c c e s s f u l methods are now a v a i l a b l e i n standard t e x t s , e.g. Knudsen and Katz ( 1 ) , f o r p r e d i c t i n g or c o r r e l a t i n g heat and/or momentum t r a n s f e r data i n smooth pi p e s . However, there are no methods that s u c c e s s f u l l y c o r r e l a t e heat and momentum t r a n s f e r i n rough p i p e s , f o r a l l flow c o n d i t i o n s and roughness geometries. Many t h e o r e t i c a l and semi-e m p i r i c a l heat t r a n s f e r equations have been p o s t u l a t e d , but most of these are r e s t r i c t e d i n theory to c e r t a i n ranges of f l u i d p r o p e r t i e s and i n p r a c t i c e to s p e c i a l types of w a l l roughness. The more notable are those due to Lyon ( 2 ) , Nunner ( 3 ) , M a r t i n e l l i ( 4 ) and M a t t i o l i ( 5 ) . Most equipment i n which heat exchange takes place c o n t a i n c o n duits t h a t are rough e i t h e r by manufacture or through chemical d e p o s i t i o n or c o r r o s i o n . In d e s i g n i n g such u n i t s , i t i s p r e s e n t l y i mpossible to a c c u r a t e l y p r e d i c t the 2 heat t r a n s f e r i n the prototype. Often i t i s d i f f i c u l t to measure such data i n e x i s t i n g equipment. On the other hand, a vast amount of work has been c a r r i e d out on the o v e r a l l f r i c t i o n a l pressure drop c h a r a c t e r i s t i c s of rough p i p e s , r e s u l t i n g i n w e l l d e f i n e d c o r r e l a t i o n s found i n most books on f l u i d mechanics, e.g. Knudsen and Katz ( 1 ) . From these i t i s p o s s i b l e to make reasonably accurate estimates about the f l u i d f r i c t i o n c h a r a c t e r i s t i c s of rough pipes f o r e s s e n t i a l l y a l l flow c o n d i t i o n s . A method f o r p r e d i c t i n g rough pipe heat t r a n s f e r from f l u i d mechanics has t h e r e f o r e great p r a c t i c a l importance. A c a d e m i c a l l y , a s u c c e s s f u l heat t r a n s f e r equation f o r rough pipes would s u b s t a n t i a t e the theory u n d e r l y i n g the analogy, and thence shed some l i g h t on the mechanism of heat and momentum t r a n s f e r i n t u r b u l e n t f l u i d flow. Thus, the purpose of t h i s work i s to i n v e s t i g a t e a t h e o r e t i c a l method f o r p r e d i c t i n g heat t r a n s f e r data from a knowledge of f r i c t i o n a l pressure drop data f o r t u r b u l e n t f l u i d flow i n rough p i p e s , and to compare the c a l c u l a t e d r e s u l t s with experimental data. The i n v e s t i g a t i o n i s based on the fundamental heat t r a n s f e r equation d e r i v e d by Lyon ( 2 ) , which i s used i n c o n j u n c t i o n with v e l o c i t y p r o f i l e s developed by Rouse ( 6 ) and von Karman ( 7 ) . Required values of the t o t a l thermal c o n d u c t i v i t y are c a l c u l a t e d on the assumption that eddy d i f f u s i v i t i e s of heat and momentum are eq u a l . 3 In the d e r i v a t i o n of the Lyon heat t r a n s f e r equation, an e x p r e s s i o n i s obtained f o r the temperature d i s t r i b u t i o n i n the pipe. Normalized temperature p r o f i l e s have been computed and compared to some experimental p r o f i l e s measured by Nunner ( 3 ) . The Lyon heat t r a n s f e r equation was chosen f o r s e v e r a l reasons. I t i s the only equation known that i s d e r i v e d from f i r s t p r i n c i p l e s and does not embody r e s t r i c t i o n s which w i l l l i m i t i t s a p p l i c a t i o n to c e r t a i n ranges of f l u i d p r o p e r t i e s , flow c o n d i t i o n s and types of roughness. From p r i o r work by Smith and E p s t e i n ( 8 ) and by McAndrew ( 9 ) , i t i s known t h a t the equation of Lyon ( 2 ) , Nunner ( 3 ) and M a r t i n e l l i (4) y i e l d e s s e n t i a l l y s i m i l a r r e s u l t s f o r a smooth pipe. The heat t r a n s f e r equation developed by Nunner ( 3 ) has proven to be q u i t e s u c c e s s f u l i n p r e d i c t i n g and c o r r e l a t i n g rough pipe heat t r a n s f e r data f o r common gases and l i q u i d s . The u s e f u l P r a n d t l number range i s , however, l i m i t e d to values c l o s e to u n i t y . R e s u l t s based on Nunner's equation have been c a l c u l a t e d and p l o t t e d with the r e s u l t s from the proposed method and with some experimental data. Only through d i r e c t comparison i s i t p o s s i b l e to eva l u a t e the r e l a t i v e merits of va r i o u s p r o p o s a l s . Despite the i n t e r e s t i n the s u b j e c t , there have been r e l a t i v e l y few experimental i n v e s t i g a t i o n s conducted on heat t r a n s f e r i n rough p i p e s . Some e a r l y work was c a r r i e d out by 4 Pohl (10) and by Cope (11), u s i n g water f o r t h e i r i n v e s t i g a t i o n s . U n f o r t u n a t e l y t h e i r experimental procedures were not w e l l developed, and hence t h e i r r e s u l t s and c o n c l u s i o n s remain somewhat i n q u e s t i o n . Since the e a r l y 1950"s, however, a number of w e l l designed and evaluated experimental data have become a v a i l a b l e . Sams (12, 13), Nunner ( 3 ) , and Smith and E p s t e i n . ( 8 ) conducted experiments u s i n g a i r i n t h e i r p i p e s . B r o u i l l e t t e (14) and n o t a b l y Dipprey (15) used water. R e c e n t l y , the use of l i q u i d metals f o r c o o l i n g of atomic r e a c t o r s has evoked some i n t e r e s t i n t h i s s p e c i a l i z e d f i e l d . S e v e r a l s t u d i e s have been i n i t i a t e d on l i q u i d metal heat t r a n s f e r from rough s u r f a c e s , but no q u a n t i t a t i v e data are a v a i l a b l e f o r c i r c u l a r pipes ( 9 ) . In g e n e r a l , the above i n v e s t i g a t o r s agree t h a t f o r f l u i d f l o w i n g t u r b u l e n t l y i n a p i p e , the e f f e c t of rough-ness elements i s to i n c r e a s e the r a t e of heat t r a n s f e r and the u n i t pressure drop above the corresponding values f o r a smooth pipe of the same diameter. Savage and Myers (16) i n v e s t i g a t e d the behaviour of the f l u i d near l a r g e a r t i f i c i a l roughness elements, and concluded that the i n c r e a s e i n heat t r a n s f e r i s mostly due to an i n c r e a s e i n s u r f a c e area. Though the f i n d i n g s of r e f e r e n c e (16) may be a b s o l u t e l y true f o r l a r g e roughness elements, i t i s d o u b t f u l that the same mechanism accounts f o r the i n c r e a s e i n r a t e of heat t r a n s f e r 5 f o r m i c r o s c o p i c a l l y . r o u g h p i p e s . In any case, the a d d i t i o n a l s u r f a c e area of such pipes cannot be determined with any meaningful accuracy. 6 DESCRIPTION OF THE PROPOSED METHOD FOR PREDICTING HEAT TRANSFER IN ROUGH PIPES 1. Heat T r a n s f e r Equation The proposed model i s based on a heat t r a n s f e r equation developed by R.II. Lyon (2) from p u r e l y t h e o r e t i c a l c o n s i d e r a t i o n s . I t i s v a l i d f o r a l l flov; r e g i o n s and w a l l c o n d i t i o n s . The assumptions u n d e r l y i n g the d e r i v a t i o n a re: a) The v e l o c i t y and temperature p r o f i l e s are f u l l y developed - there are no end e f f e c t s . b) Macroscopic steady s t a t e e x i s t s ; c) The f l u i d p r o p e r t i e s are independent of temperature. d) The system i s symmetrical about the tube a x i s . e) N e g l i g i b l e heat t r a n s f e r by the process of thermal r a d i a t i o n , f ) No i n t e r n a l heat generation w i t h i n the f l u i d . g) Constant heat f l u x at the w a l l of the tube along the Lyon's d e r i v a t i o n begins with the d e f i n i t i o n of the heat t r a n s f e r c o e f f i c i e n t : tube a x i s . It f o l l o w s t h a t • ( — ) • \ 2 z lr constant. Qw = h Aw ( *w " *b } (1) Separate e x p r e s s i o n s are then d e r i v e d f o r the heat e n t e r i n g the f l u i d q w , and the temperature d r i v i n g f o r c e ( t w - t ^ ) . When these'are s u b s t i t u t e d i n t o equation 1, one obt a i n s an ex p r e s s i o n f o r the heat t r a n s f e r c o e f f i c i e n t h i n terms of the r a d i u s , the time-average p o i n t v e l o c i t y , and the t o t a l r a d i a l c o n d u c t i v i t y . T h i s e x p r e s s i o n i s rendered dimensionless and s i m p l i f i e d to y i e l d the heat t r a n s f e r e quation: Nu = (2) rR - 2 RV dR ] 7) dR o R (K/k) In the d e r i v a t i o n of equation (2) one a l s o o b t a i n s an ex p r e s s i o n f o r the temperature d i f f e r e n c e between the pipe w a l l and any p o i n t i n the f l u i d . T h i s e x p r e s s i o n may be normalized by making use of known boundary c o n d i t i o n s : t = t at r = 0 c Subsequent s i m p l i f i c a t i o n gives J RV dR t„ - t dR . R (K/k) e = = — (3) t. - t -w " L c pi. rR o J RV dR ) R (K/k) o dR To e v a l u a t e equations 2 and 3, i t i s necessary to know values o f V and K/k i n terms of the dimensionless r a d i u s . 8 2 . E v a l u a t i o n o f R a d i a l C o n d u c t i v i t y R a t i o T h e t o t a l t i m e - a v e r a g e s h e a r s t r e s s a c t i n g o n a f l u i d i n t u r b u l e n t f l o w i s g i v e n b y T g c = < / * + E M ) ^ ( 4 ) w h e r e E ^ = i s d e f i n e d a s t h e e d d y v i s c o s i t y . A c c o r d i n g t o t h e l a w o f l i n e a r s t r e s s d i s t r i b u t i o n , T = r w . ( Z . ) = r ^ R = r w ( i _ y ) (5 > w S u b s t i t u t i n g e q u a t i o n 4 i n t o 5, a n d r e a r r a n g i n g t h e v a r i a b l e s g i v e s T w g , , , , d u ( 1 - y ) = ( if + £ M ) ' ( 6 ) _ - . v _ - y , - v - ~ M / d y I t i s p o s s i b l e t o s i m p l i f y t h i s e q u a t i o n t h r o u g h t h e u s e o f t h e e x p r e s s i o n f o r f r i c t i o n v e l o c i t y , d e f i n e d b y ( 7 ) t o g e t h e r w i t h t h e d e f i n i t i o n o f f r i c t i o n f a c t o r , T w g . = 1 / 2 f ^ U 2 ( 8 ) 9 The combination of equations 7 and 8 y i e l d s an ex p r e s s i o n f o r the f r i c t i o n v e l o c i t y i n terms of the f r i c t i o n f a c t o r and the average bulk v e l o c i t y ( 1 ) . u- U yffn (7_) Now by s u b s t i t u t i n g equation 7a and 8 i n t o equation 6, and r e n d e r i n g the r e s u l t dimensionless by us i n g the dimensionless v a r i a b l e s u + and y + , one obt a i n s _____ _ 1 - y + / [ ( Re/2 ) Jf/T ] >^ d u + / d y + ; " (9) The t o t a l r a d i a l c o n d u c t i v i t y of heat i n t o a pipe i s the sum of the molecular c o n d u c t i v i t y and the eddy c o n d u c t i v i t y : k + E H (10) where EJ.J i s d e f i n e d by E ^ = Cp ^ £ H arranged and rendered d i m e n s i o n l e s s . Equation 10 may be r e -K/k = 1 + Pr ( 1 + Pr ( 4*- )( «__ e, ( i i ) ( l l a ) I t i s now p o s s i b l e t o eva l u a t e the thermal c o n d u c t i v i t y r a t i o K/k u s i n g equations 9 and l l a i n c o n j u n c t i o n with a v e l o c i t y p r o f i l e , provided one can make some assumption concerning the 10 r a t i o of the eddy d i f f u s i v i t i e s of heat and momentum, - j r - - . Experimental data on smooth pipes by numerous i n v e s t i g a t o r s appear to i n d i c a t e t h a t i s a f u n c t i o n of Pr, Re and R. I t seems reasonable t h a t w a l l roughness should be i n c l u d e d as a v a r i a b l e i n the case of rough p i p e s . A r e c e n t review by K e s t i n and Richardson (17) of experimental work i n smooth tubes r e p o r t s values f o r the d i f f u s i v i t y r a t i o r a n g i n g almost anywhere between 0,5 and 1.6. However, the experimental data do not i n d i c a t e a c l e a r t r e n d i n regard to any of the above v a r i a b l e s . In view of the c o n f l i c t i n g i n f o r m a t i o n , i t i s assumed f o r the purpose of t h i s work t h a t the d i f f u s i v i t y r a t i o i s equal to u n i t y f o r a l l s i t u a t i o n s . T h i s i s a convenient assumption which the above reviewers i n d i c a t e w i l l i n t r o d u c e very l i t t l e e r r o r when a p p l i e d t o p r e d i c t i n g o v e r a l l heat t r a n s f e r . 3. V e l o c i t y P r o f i l e f u n c t i o n of the p o s i t i o n i s r e q u i r e d to allow the s o l u t i o n of Lyon's heat t r a n s f e r equations. For the t u r b u l e n t c o r e , Rouse (6) d e r i v e d an e x p r e s s i o n based on Mikuradse's e x t e n s i v e experimental r e s u l t s (18) v a l i d f o r both rough and smooth p i p e s : F i n a l l y , a knowledge of the p o i n t v e l o c i t y as a u + = 2.5 l n ( - | - ) + 3.75 + - „ w (12) 11 I n t r o d u c i n g dimensionless d i s t a n c e y + i n t o equation 12 and r e a r r a n g i n g , we get u + = 2.5 In y + - [2.5 l n ( \/f/2) - . — - 3.75] (12a) 1 1 {Til For the r e g i o n between the l i m i t o f the t u r b u l e n t core and the pipe w a l l , there are no rough pipe v e l o c i t y p r o f i l e data a v a i l a b l e . T h e r e f o r e , von Karman's ( 7 ) smooth pipe c o r r e l a t i o n s are used. In the l a m i n a l s u b l a y e r u_ = y_ » Vs ^ 5 < 1 3> and f o r the b u f f e r zone u| = 5.0 In y | - 3.05 , 5 as y_ sr. 30 (14) By d e f i n i t i o n , u + and y + i n equations 13 and 14 denote smooth pipe v a r i a b l e s , whereas the same q u a n t i t i e s i n equation 12a denote e i t h e r smooth or rough pipe v a r i a b l e s . S u b s c r i p t s w i l l be used t o i d e n t i f y a smooth pipe q u a n t i t y , and the unsubscribed v a r i a b l e s w i l l stand f o r the ge n e r a l case. The v e l o c i t y term i n equations 2 and 3 i s d e f i n e d by v a r i a b l e V, and i s r e q u i r e d as a f u n c t i o n of R. I t i s t h e r e f o r e necessary to tra n s f o r m u + and y + i n equations 12a, 13 and 14 12 to the given format. From equation 7a f o r a smooth w a l l + u U v ' S i m i l a r l y f o r a rough w a l l U = {171 (16) In a manner analogous to the p r e c e d i n g , y + can be expressed i n terms of the dimensionless r a d i u s R. Moting that y = r w - r the dimensionless d i s t a n c e y + becomes through simple r e -arrangement f o r a smooth w a l l , *t -^{h^ ( i - R ) d 7 ) and f o r a rough w a l l , y + = -|£ ([777 ( 1 - R ) (18) In t h i s i n v e s t i g a t i o n i t i s assumed t h a t the f l u i d near a rough w a l l behaves s i m i l a r l y to t h a t near a smooth w a l l at the same Reynolds number , Equations 15 and 17, t h e r e f o r e , are used t o g e t h e r with equations 13 and 14 to d e r i v e working ex p r e s s i o n s f o r the v e l o c i t y p r o f i l e i n the laminar s u b l a y e r and b u f f e r zone of a rough pipe. Equations 16 and 18 are used together with equation 12a to o b t a i n the corresponding equation i n the Table 1. SUMMARY OF EQUATIONS REQUIRED FOR SOLUTION OF EQUATIONS 2 AND 3 13 Turbulent Core V = ^ " [ 2 . 5 ln(l - R) + 3.75] + I (22) k = l + P r [ ^ v / f ( , ~ R , R " '] + { 2 3 ) Buffer Zone Lominar Sublayer V = - | Red - R) 4=1.0 + t k (26) (27) 1 I t i s p o s s i b l e t h a t the combination of the v a r i a b l e s i n the range 0 «=. R < 1.0 gives values f o r K/k which are s m a l l e r than u n i t y . T h i s i s impossible i n the p h y s i c a l l y r e a l s i t u a t i o n . The r e s t r i c t i v e c o n d i t i o n t h a t K/k _2s 1.0 has t h e r e f o r e been imposed. T T The d e r i v e d equation i s K/k = 1 + Pr ( R - 1 ). In t h i s r e g i o n the f l u i d i s i n . l a m i n a r flow, and t h e r e f o r e by d e f i n i t i o n K/k = 1.0. 14 t u r b u l e n t core, These working equations are summarized i n Table 1. In order to show the v e l o c i t y p r o f i l e f o r a l l three flow r e g i o n s on a s i n g l e p l o t , and to c a l c u l a t e the r e s p e c t i v e boundaries, i t i s necessary to be able to express smooth pipe v a r i a b l e s i n terms of rough pipe v a r i a b l e s . T h i s i s p o s s i b l e through use of the modified d e f i n i t i o n f o r f r i c t i o n v e l o c i t y , equation 7a u* = U \jf17 E l i m i n a t i o n of the average bulk v e l o c i t y between the two e x p r e s s i o n s y i e l d s , u* = u* J f / f _ (19) Through a simple t r a n s f o r m a t i o n of the v a r i a b l e s u s i n g equation 19, i t becomes now p o s s i b l e to express the rough pipe v e l o c i t y p r o f i l e i n the t u r b u l e n t core, equation 12a, on a smooth pipe c o - o r d i n a t e system: (20) . I — ; — \ + _ j _ ( \ Re I u* = 2.5 J f / f s l n y s + ^ J^- J f / f s C 2 . 5 In (— J f _ / 2 ) - 3 . 7 5 ] The choice f o r use of smooth pipe v a r i a b l e s was made p u r e l y on the b a s i s of convenience. On a p l o t such as Figure 1, where Figure 1. Sketch of Various P o s s i b l e V e l o c i t y P r o f i l e s i n Rough Pipes 16 i s p l o t t e d a g a i n s t y* , the laminar s u b l a y e r and b u f f e r zone v e l o c i t y p r o f i l e s remain f i x e d , and only the t u r b u l e n t core p r o f i l e changes f o r each combination of f and Re. When f = f s i n equation 20, i t can be shown t h a t the equation reduces to P r a n d t l ' s smooth pipe v e l o c i t y p r o f i l e ; f o r the t u r b u l e n t core ( 1 ) . The term ^f / f s * m o d i f i e s both the slope and the negative p o r t i o n of the i n t e r c e p t i n equation 20. By d e f i n i t i o n f'=s f , and i t f o l l o w s t h a t y f / f s ^ !• T h e r e f o r e , the slope of the t u r b u l e n t core v e l o c i t y p r o f i l e f o r a rough pipe on the given c o - o r d i n a t e s i s always g r e a t e r than t h a t f o r a smooth pipe and the i n t e r c e p t i s always l e s s . N e c e s s a r i l y the rough pipe t u r b u l e n t core v e l o c i t y p r o f i l e i n t e r s e c t s the smooth pipe b u f f e r zone v e l o c i t y p r o f i l e always below y^= 30t. : 4 • Boundaries of Flow Regions i n Rough Pipes I t was shown i n the preceding s e c t i o n t h a t the boundaries of rough pipe flow regions are f u n c t i o n s of f and Re. The method f o r c a l c u l a t i n g the l i m i t of each zone w i l l now be e x p l a i n e d b r i e f l y . The APPENDIX co n t a i n s a more complete d i s c u s s i o n of the procedure. Three d i s t i n c t s i t u a t i o n s a r i s e , each of which has to be t r e a t e d s e p a r a t e l y . F i g u r e 1 i s a ^ A c t u a l l y , the p o i n t of i n t e r s e c t i o n as given by the simultaneous s o l u t i o n of equations 14 and 20 i s a t y_ = 30.9. 17 sketch of some r e p r e s e n t a t i v e v e l o c i t y p r o f i l e s . CASE 1. The t u r b u l e n t core v e l o c i t y p r o f i l e i n t e r s e c t s the b u f f e r zone p r o f i l e i n the range 5 _s y|-£;30. The t r a n s i t i o n to the laminar s u b l a y e r takes place at y_i= 5. Lines (a) and (b) i n F i g u r e 1 are examples of t h i s case. To f i n d the p o i n t of i n t e r s e c t i o n , equation 14 and 20 are s o l v e d s i m u l t a n e o u s l y f o r y|, e l i m i n a t i n g u_j between them. The r e s u l t i n g equation allows c a l c u l a t i o n of y§ d i r e c t l y from a knowledge of f and Re. The dimensionless r a d i u s may be c a l c u l a t e d from the d e f i n i t i o n of y|. Upon r e a r r a n g i n g , we get: (21) In c o r r e s p o n d i n g terms, the boundary between the t u r b u l e n t core and the b u f f e r zone i s at R = R-^ , and between the b u f f e r zone and the laminar s u b l a y e r at R = R 2 . CASE 2. The t u r b u l e n t core v e l o c i t y p r o f i l e i n t e r s e c t s the laminar s u b l a y e r p r o f i l e i n the range 2.5 as: y_j < 5. Line (c) i n F i g u r e 1 i s an example of t h i s s i t u a t i o n . To f i n d the p o i n t of i n t e r s e c t i o n , equations 13 and 20 are s o l v e d s i m u l t a n e o u s l y , e l i m i n a t i n g us between them. 18 The r e s u l t i n g equation cannot be so l v e d e x p l i c i t l y f o r y_j f o r given values of f and Re. A t r i a l and e r r o r s o l u t i o n i s r e q u i r e d . In ge n e r a l the two p r o f i l e s i n t e r s e c t a t two p o i n t s . Only the r o o t l o c a t e d between 2.5 _s y | <c 5 and nearer yg = 5 i s of i n t e r e s t . The dimensionless r a d i u s i s again c a l c u l a t e d from equation 21. Correspondingly the boundary i s at R = R_. CASE 3. The t u r b u l e n t core v e l o c i t y p r o f i l e misses both the b u f f e r zone and laminar s u b l a y e r p r o f i l e s , and i n t e r s e c t s the u| = 0 a x i s at some p o s i t i v e f i n i t e value of y|. Line (d) i n Fi g u r e 1 i s an example. The r e g i o n of zero net v e l o c i t y between the w a l l and the t u r b u l e n t core w i l l be c a l l e d i n t h i s work the w a l l l a y e r . In t h i s case equation 20 i s s o l v e d f o r u| = 0. From the r e s u l t i n g equation and a knowledge of Re and f, one can c a l c u l a t e values of y|, and u s i n g equation 21, convert these i n t o values of the dimensionless r a d i u s denoted by R = R_. I t i s i n t e r e s t i n g to note i n Fi g u r e 1 the e f f e c t of f r i c t i o n f a c t o r and Re on the t u r b u l e n t core v e l o c i t y p r o f i l e . An. i n c r e a s e i n e i t h e r w i l l r a i s e the slope above that f o r the smooth pipe, and t r a n s l a t e the l i n e to the r i g h t . The e f f e c t of Re at constant f r i c t i o n f a c t o r i s seen by comparison of l i n e s . ( a ) and ( d ) , and the e f f e c t of f r i c t i o n f a c t o r at 19 constant Re, by l i n e s ( b ) , ( c ) , (d) and ( e ) . Figu r e 2 was c o n s t r u c t e d t o give a p i c t u r e of the range oyer which the three cases apply. The s o l i d l i n e s r e p r e s e n t l i m i t i n g combinations of f r i c t i o n f a c t o r and Re. The s o l i d l i n e nearest the f = 0 a x i s d e f i n e s Nikuradse's smooth pipe f r i c t i o n f a c t o r (19) f o r t u r b u l e n t flow, and gives the bottom l i m i t f o r the rough pipe theory. Values of f s used throughout t h i s work have been c a l c u l a t e d from equation 34 a c c o r d i n g to a method d e s c r i b e d i n the APPENDIX. Throughout the preceding development of the model.no separate assumptions have been made concerning the shape and d i s t r i b u t i o n of the roughness elements. The f r i c t i o n f a c t o r at the given Re i s t h e r e f o r e assumed to f u l l y c h a r a c t e r i z e the e f f e c t of the w a l l roughness. Nunner (3) was able to show t h i s to be true at l e a s t f o r h i s own data. 5. Thermal C o n d u c t i v i t y i n the Wall Layer Some hypothesis about the method of heat t r a n s f e r through the w a l l l a y e r i s necess»ary. Two l i m i t i n g cases are obvious: the l a y e r i s t r u l y stagnant and molecular conduction p r e v a i l s ; or the l a y e r i s v i o l e n t l y t u r b u l e n t and o f f e r s no e f f e c t i v e r e s i s t a n c e to heat t r a n s f e r . In the former case the c o n d u c t i v i t y r a t i o K/k i s u n i t y , and i n the l a t t e r i s i n f i n i t y . Undoubtedly each of these extremeties 70 Re Figure 2. P l o t Showing the Range of A p p l i c a t i o n of CASES 1, 2 and 3 on f - Re Coordinates 21 i s u n l i k e l y , and the r e a l s i t u a t i o n i s somewhere i n between. S e v e r a l i n v e s t i g a t o r s have proposed models f o r the behaviour of the f l u i d near the s o l i d boundary. Dipprey (15) d e s c r i b e s the f l u i d motion i n terms of a l t e r n a t i n g p o i n t s of s t a g n a t i o n and high t u r b u l e n c e , g i v i n g r i s e to a p a t t e r n of s t a n d i n g v o r t i c e s around the roughness elements. Owen and Thomson (20) d e s c r i b e a system of horseshoe eddies "which wrap themselves around the i n d i v i d u a l excrescences and t r a i l down-stream". Nunner (3) proposes a "roughness zone" of high t u r b u l e n c e caused by the t u r b u l e n t wakes behind the roughness elements. However, i f one assumes no s l i p at the s o l i d boundary, one i m p l i e s the presence of a laminar s u b l a y e r of some t h i c k n e s s i n a l l s i t u a t i o n s . C l e a r l y the s u b j e c t i s s t i l l wide open. In the present study, t h e r e f o r e , both l i m i t i n g assumptions w i l l be made about the thermal c o n d u c t i v i t y i n the w a l l l a y e r , and two values f o r the Nusselt number c a l c u l a t e d f o r each example of CASE 3. NUNNER'S EQUATION FOR PREDICTION OF ROUGH PIPE HEAT TRANSFER 22 W. Nunner's (3) e x t e n s i v e experimental measurements and t h e o r e t i c a l c o n s i d e r a t i o n s i n the f i e l d of t u r b u l e n t heat and momentum t r a n s f e r were p u b l i s h e d i n 1956 i n Germany. He d e r i v e d a heat t r a n s f e r equation which i s of p a r t i c u l a r i n t e r e s t because i t i s r e l a t i v e l y simple, yet s u r p r i s i n g l y e f f e c t i v e . The range of a p p l i c a t i o n of the e q u a t i o n , however, i s l i m i t e d to systems which have P r a n d t l numbers c l o s e to u n i t y . At the present time Nunner's heat t r a n s f e r equation i s probably most s u i t a b l e f o r the m a j o r i t y of a p p l i c a t i o n s . T herefore i n order to j u s t i f y the use of more complicated e q u a t i o n s , such as Lyon's, these must cover a wider range of Pr v a l u e s , and/or be more a c c u r a t e . In t h i s work, Nunner's equation i s used t o g e t h e r with experimental data as the b a s i s of comparison f o r the proposed method. An o u t l i n e of Nunner's rough pipe theory f o l l o w s . The heat t r a n s f e r equation i s b a s i c a l l y an e x t e n s i o n of P r a n d t l ' s (21) smooth pipe equation. S i m i l a r s i m p l i f y i n g assumptions are made: a) M o l e c u l a r t r a n s p o r t c o n t r o l s i n the laminar s u b l a y e r ; eddy t r a n s p o r t c o n t r o l s i n the t u r b u l e n t core. 23 b) Reynolds analogy holds i n the t u r b u l e n t core. T h i s assumption i m p l i e s t h a t there i s s i m i l a r i t y between temperature and v e l o c i t y d i s t r i b u t i o n i n the t u r b u l e n t core and a l s o t h a t , i n view of the p r e v i o u s l y assumed predominance of eddy over molecular t r a n s p o r t i n t h i s r e g i o n , € H i s equal to or at l e a s t p r o p o r t i o n a l to € M . c) f = T w and -3- = ( —T— ) i n the laminar s u b l a y e r . w A A w A b r i e f d e s c r i p t i o n of Nunner's model w i l l be h e l p -f u l . The pipe c r o s s - s e c t i o n i s d i v i d e d i n t o three flow r e g i o n s . Outermost i s the laminar s u b l a y e r , followed f i r s t by the roughness zone and then by the t u r b u l e n t core. Nunner c o n s i d e r s the f r i c t i o n a l r e s i s t a n c e of the rough pipe w a l l to be the r e s u l t of two separate processes - the s k i n f r i c t i o n of the laminar s u b l a y e r , and the shape r e s i s t a n c e of the roughness elements i n the roughness zone. He f u r t h e r assumes t h a t the rough pipe laminar s u b l a y e r i s e s s e n t i a l l y s i m i l a r to the smooth pipe cas:e, and i s of the same t h i c k n e s s . The s h e a r i n g s t r e s s next to the w a l l i s t h e r e f o r e c h a r a c t e r i s t i c of the smooth pipe. The roughness zone i s a h y p o t h e t i c a l l a y e r e x i s t i n g below the t i p s of the roughness elements. A form drag r e s u l t s when f l u i d passes roughness elements which protrude through the laminar s u b l a y e r , and accounts f o r the jump i n the s h e a r i n g s t r e s s t o the rough pipe v a l u e . The t u r b u l e n t wakes behind the roughness elements give r i s e to l a r g e eddy d i f f u s i v i t y i n the r e g i o n . F i g u r e 3 i s a sketch of Nunner's model t o g e t h e r with the r e s u l t i n g s h e a r i n g s t r e s s d i s t r i b u t i o n . ^ T U R B U L E N T CORE F i g u r e 3. Sketch of Nunner's r e p r e s e n t a t i o n of the flow process i n rough pipes An a d d i t i o n a l assumption i s necessary concerning the nature of the temperature and v e l o c i t y p r o f i l e s i n the roughness zone. These were assumed to be f l a t , due to the high r a t e s of eddy t r a n s p o r t w i t h i n the t u r b u l e n t wakes. Then by f o l l o w i n g a procedure s i m i l a r to that of P r a n d t l (21), Nunner was able to d e r i v e the f o l l o w i n g heat t r a n s f e r equation f o r t u r b u l e n t flow i n rough p i p e s : Nu = (f/2) Re Pr  1 + _» ( Pr JL - 1 ) <2 l J f s Equation 28 reduces to the P r a n d t l smooth pipe equation when f = f s . Before equation 28 has any p r a c t i c a l a p p l i c a t i o n however, the r a t i o u^/ U must be e v a l u a t e d . Nunner argued t h a t the value i s determined by the t h i c k n e s s of the thermal 25 ' laminar s u b l a y e r , which seems to be only s l i g h t l y dependent on the roughness ( 3 ) . He t h e r e f o r e assumed that smooth pipe data may be used. An e m p i r i c a l e x p r e s s i o n due to E. Hoffman ( 3 ) , f= 1.5 R e " 1 / 8 P r " 1 / 6 , (29) was thus s u b s t i t u t e d i n t o equation 28 to o b t a i n Nunner's f i n a l heat t r a n s f e r equation f o r rough or smooth p i p e s . A d i f f e r e n t e x p r e s s i o n can be d e r i v e d f o r the r a t i o U_/U from the u n i v e r s a l v e l o c i t y d i s t r i b u t i o n . A c c o r d i n g to von Karman, the momentum laminar s u b l a y e r i n a smooth pipe extends to a value of y | = 5 , a f t e r which a b u f f e r zone i n t e r -cedes between the s u b l a y e r and the t u r b u l e n t core. As shown i n r e f e r e n c e ( 1 ) , i t f o l l o w s t h a t the d e s i r e d r a t i o i s given by -g> = 5 jf 17 ' (30) I t i s t h i s r a t i o which i s i n f a c t used i n the present work. Though Nunner c a r r i e d out e x t e n s i v e temperature p r o f i l e measurements art three Reynolds numbers and f o r s e v e r a l pipe roughnesses, he d i d not d e r i v e a t h e o r e t i c a l e x p r e s s i o n to c o r r e l a t e these data. Instead, he s t a t e d that the experimental p r o f i l e s were,, w i t h i n the experimental e r r o r , the same as the corre s p o n d i n g smooth pipe r e s u l t s . He t h e r e f o r e concluded t h a t temperature p r o f i l e i s e s s e n t i a l l y independent 26 of pipe roughness. However, i t i s p o s s i b l e t o d e r i v e an e x p r e s s i o n f o r the normalized temperature p r o f i l e based on Nunner's t u r b u l e n t flow model and the assumptions u n d e r l y i n g t h i s model. This was done by M.A. McAndrew ( 9 ) , with the r e s u l t : Tw " T c = 1 - 2.5 if/2 l n ( r w / y ) • " y 1 . 5 R e ~ 1 / 8 P r ~ 1 / 6 ( P r f / f g - 1) + 3.75 j f / 2 " + 1 T h e o r e t i c a l l i n e s based on equation 31 have been p l o t t e d i n Fig u r e s 8a t o 8c and compared to Nunner's experimental data. Comments are made on t h i s p o i n t i n the D i s c u s s i o n o f R e s u l t s . The main weakness of Nunner's theory a r i s e s from the use of Reynolds', analogy i n the t u r b u l e n t core. In e f f e c t t h i s analogy as used by P r a n d t l s t a t e s t h a t e H = ^ + e„ ( 3 2 ) Assuming as bef o r e that £-H = C M , equation 32 i s e x p l i c i t l y t r u e only when k / C P ^ = / K / ^ f t h a t i s when Pr i s u n i t y . When the theory i s a p p l i e d to l i q u i d metals - very s m a l l Pr, or t o very v i s c o u s l i q u i d s - very l a r g e Pr, i t p r e d i c t s erraneous r e s u l t s . I t i s euan^mathematically .'possible ' f o r s m a l l 27 v a l u e s o f P r t o r e n d e r t h e d e n o m i n a t o r o f e q u a t i o n 2 8 n e g a t i v e . T h i s o f c o u r s e , w o u l d y i e l d n e g a t i v e v a l u e s f o r N u w h i c h a r e i n c o n c e i v a b l e i n r e a l i t y . F o r P r g r e a t e r t h a n a p p r o x i m a t e l y 5 o r R , p r e d i c t e d r e s u l t s c o m e i n t o g r o s s e r r o r , e s p e c i a l l y f o r l a r g e p i p e r o u g h n e s s . 28 RESULTS 1. Heat T r a n s f e r C o e f f i c i e n t and Temperature P r o f i l e R e s u l t s C a l c u l a t e d a c c o r d i n g t o Proposed Method  Values of Nu were c a l c u l a t e d i n terms of f , Re and Pr. A range wide enough to take i n most p r a c t i c a l a p p l i c a t i o n s was covered: f = f s , 0.004, 0,006, , 0.020 T Re = ( 4, 7, 10, 20 ) x 10 3, , 10 7 Pr = 0 . 001 , 0 . 010 , . . . . , 1 ,000 S e l e c t e d r e s u l t s have been p l o t t e d on the f o l l o w i n g pages. F i g u r e s 4a to 4e c o n t a i n p l o t s of Nu as a f u n c t i o n of Re a t constant Pr, u s i n g f as the parameter. For f u l l y rough flow, f i s a unique measure of e q u i v a l e n t sand roughness, but other-wise i s an i n d i c a t i o n of the pipe w a l l c o n d i t i o n only when Re i s a l s o s p e c i f i e d . To b r i n g out the e f f e c t of Pr, Nu was p l o t t e d a g a i n s t Re i n Figure 5 a t a constant f r i c t i o n f a c t o r , u s i n g Pr as the parameter. Normalized temperature p r o f i l e s were c a l c u l a t e d f o r a l l combinations of the f o l l o w i n g values of the independent v a r i a b l e s : T T h e smooth pipe f r i c t i o n f a c t o r i s a f u n c t i o n of Re. I t forms a bottom l i m i t to the range of f covered f o r any value of Re. f = fs • 0.020 Re = U,000 , 10,000,000 Pr = 0,001 , 1 , 1,000 29 T y p i c a l r e s u l t s are p l o t t e d t o g e t h e r with corresponding v e l o c i t y p r o f i l e s i n F igures 6a to 6d. 2. Comparison of P r e d i c t e d R e s u l t s with Experimental Data and with Nunner's Theory.  a ^ Comparison with the Work of Nunner (3) Nunner c a r r i e d out an e x t e n s i v e experimental i n v e s t i g a t i o n f o r a i r f l o w i n g through rough p i p e s , c o v e r i n g a Reynolds number range from laminar flow to Re = 80,000. He used both a r t i f i c i a l l y roughened and n a t u r a l l y rough p i p e s . The former were produced by jamming ring-shaped roughness elements of v a r y i n g shape i n t o a smooth p i p e . Using d i f f e r e n t s e p a r a t i o n between the r i n g s , Nunner was able to o b t a i n f r i c t i o n f a c t o r s r a n g i n g from approximately 0.008 to 0.075. The average Pr i n the h e a t i n g runs was 0.72. Some of Nunner's heat t r a n s f e r data are p l o t t e d t o g e t h e r with the r e s u l t s from the proposed method i n F i g u r e s 7a to 7d. I t i s i n t e r e s t i n g to note how w e l l Nunner's heat t r a n s f e r equation c o r r e l a t e s h i s experimental data. Nunner a l s o r e p o r t e d normalized temperature p r o f i l e s F i g u r e 4a. B a s i c Heat T r a n s f e r R e s u l t s C a l c u l a t e d According to Proposed Method: Nu versus Re F i g u r e 4b. B a s i c Heat T r a n s f e r R e s u l t s C a l c u l a t e d According to Proposed Method: Nu versus Re 10 6 I- 1 I 1 I 10' I O 4 Nu I O 3 10' 1 0 10s — o 00 k b f = .020 f = .010 Pr= 10 1 • I I 1 1 MM • I I I O 4 Re I O 5 I O 6 1111 I O 7 F i g u r e Ud. B a s i c Heat T r a n s f e r R e s u l t s C a l c u l a t e d According to Proposed Method: Nu versus Re I i l i l I 1 1 MM i l i l I I I I 111 • I • i i i i n u IO3 IO4 _ IO8 IO6 10 Re F i g u r e 4e. Basic Heat T r a n s f e r R e s u l t s C a l c u l a t e d A c c o r d i n g to Proposed Method: Nu versus Re Figure 5. I n f l u e n c e of Pr on the Basic Heat T r a n s f e r Results F i g u r e 6 a . B a s i c T e m p e r a t u r e P r o f i l e s C a l c u l a t e d A c c o r d i n g t o P r o p o s e d Method: 6 and V v e r s u s y / r w 37 F i g u r e 6 b . B a s i c T e m p e r a t u r e P r o f i l e s C a l c u l a t e d A c c o r d i n g t o P r o p o s e d Method: 6 and V v s . y / r w 38 39 f o r most of h i s a r t i f i c i a l l y roughened pipes a t three values of Re: 10,000 , 30,000 and 60,000. To our knowledge he i s the only i n v e s t i g a t o r who has made such measurements. Some of h i s experimental data, as s c a l e d o f f from, h i s smoothed curves, are shown i n F i g u r e s 8a to 8c, together with the r e s u l t s p r e d i c t e d by the proposed method, For comparison, the co r r e s p o n d i n g smooth pipe p r o f i l e and the p r e d i c t i o n based on Nunner's theory, equation 31, are i n c l u d e d i n each case. b^ Comparison with the Work of Smith and E p s t e i n (8) Smith and E p s t e i n measured heat t r a n s f e r co-e f f i c i e n t s f o r a i r f l o w i n g through one smooth copper p i p e , and s i x commercial rough pipes. The Reynolds number span i n v e s t i g a t e d ranged from 10,000 to 80,000. The experimental r e s u l t s were c o r r e l a t e d by a simple e m p i r i c a l power equation r e l a t i n g Re and j ^ , both of which were evaluated at the average f i l m temperature. The P r a n d t l number was assumed constant at 0.69. For the purpose of t h i s work, Smith and E p s t e i n data were c a l c u l a t e d from t h e i r e m p i r i c a l e q u a t i o n s , and p l o t t e d t ogether with the r e s u l t s from the proposed method i n Figures 9a to 9d. The r e s u l t s c a l c u l a t e d from the t h e o r a t i c a l equations of Nunner (3) and of M a r t i n e l l i (4) are copied from McAndrew's (9) work. For the sake of i n t e r e s t , the p r e d i c t i o n s based on McAndrew's hypothesis are a l s o i n c l u d e d . 41 4 3 4 \ -10 £ = 0 0 , P R O P O S E D k = I , P R O P O S E D N U N N E R ' S D A T A N U N N E R Pr = .72 f =.0302 10 Re 8 10 5 Figure 7 C . Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental Results of Nunner 10 ii_a> k — y P R O P O S E D / / / / / / £ = I , P R O P O S E D N U N N E R N U N N E R ' S D A T A Pr = .72 f = .0545 44 IO4 Re 8 io! Figure 7 d. Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Nunner 45 46 F i g u r e 3 b . C o m p a r i s o n o f P r e d i c t e d T e m p e r a t u r e P r o f i l e s w i t h E x p e r i m e n t a l R e s u l t s o f Nunner 0 0 .2 .4 .6 .8 l.O y / r w 0 48 - | " C O P P E R Figure 9 a . Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Smith and E p s t e i n 49 10' 8 6 Nu T" STANDARD S T E E L McANDREW NUNNER -PROPOSED SMITH - EPSTEIN DATA MARTINELLI f, i o IO4 Figure 9b. Re f 10 s i o 5 r3 Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental Re s u l t s of Smith and E p s t e i n 5 0 Nu 10 8 6 T K A R B A T E McANDREW NUNNER MARTINELLI PROPOSED SMITH 8 EPSTEIN DATA ft i o i o 4 Re Figure 9 c . 8 10 10 5 - 3 Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Smith and E p s t e i n 51 Figure 9d. Comparison of P r e d i c t e d Heat T r a n s f e r R e s u l t s with Experimental R e s u l t s of Smith and E p s t e i n h 7 G A L V A N I Z E D I O 2 8 Nu 10 T I O 4 N U N N E R P R O P O S E D M A R T I N E L L I S M I T H - E P S T E I N D A T A I O " 2 8 f , R e f 8 10 10 s -5 5? c) Comparison with the Work of Dipprey (15) Some very f i n e experimental work was conducted r e c e n t l y by Dipprey u s i n g water as the working f l u i d . A p r e s s u r i z e d system allowed him to cover a wide temperature range, and so o b t a i n heat t r a n s f e r data f o r s e v e r a l values of Pr, v i z . 5.94, 4.38, 2.79 and 1.20. The Reynolds number covered v a r i e d somewhat between the i n d i v i d u a l s e r i e s of runs, but i n g e n e r a l ranged from 14,000 to 52,000. Dipprey used three a r t i f i c i a l l y roughened pipes and one smooth pipe i n h i s i n v e s t i g a t i o n . The rough pipes were prepared by e l e c t r o p l a t i n g n i c k e l over aluminum mandrels coated with c l o s e l y graded sand. The mandrels were sub-sequently d i s s o l v e d l e a v i n g a p a t t e r n of roughness elements t h a t was b a s i c a l l y the negative of sand g r a i n roughness. The smooth pipe was prepared u s i n g a s i m i l a r method but without sand-coating the mandrel. Some of Dipprey's data are p l o t t e d i n F i g u r e s 10a to 10c, t o g e t h e r with comparative r e s u l t s from the proposed method and from Nunner's heat t r a n s f e r e q uation. A l l the data together with comparative r e s u l t s are l i s t e d i n Table 2. Dipprey's data have been c o r r e c t e d by him f o r r a d i a l temperature g r a d i e n t s accompanying f i n i t e heat f l u x v a l u e s , to the zero temperature d i f f e r e n c e c o n d i t i o n . Figure 10b. T a b l e 2. TABULAR COMPARISON OF PREDICTED RESULTS WITH DIPPREY fS EXPERIMENTAL DATA AND WITH NUNNER'S EQUATION  56 Pr Re X I O " 4 f X 103 N u s s e l t Number D i p p r e y T h i s work Nunner K/k = 1 t K/k = co t 5 .94 1.40 7 .85 106 99 113 5. 94 2.41 6.88 173 156 178 5.94 4.10 6 . 14 285 244 279 5.94 6.82 5.72 4 54 380 437 5.94 11.9 5.61 806 627 727 4.38 1.85 7 . 37 118 113 128 4.38 3.16 6.45 192 176 199 4 . 33 5.40 5 .90 312 277 316 4.33 8.98 5.63 511 434 497 4 .38 15 . 7 5 .77 901 737 842 2. 79 2.76 6.66 136 131 147 2 . 79 4 . 7 7 6.00 217 207 233 2.79 8.15 5.68 357 330 373 2 . 79 13 . 6 5.69 573 531 601 2 . 79 23.5 5.90 1023 940 1018 1.20 6.05 5 .80 174 169 184 1.20 10 . 6 5 . 60 290 276 303 1. 20 17 . 8 5. 81 466 463 , 501 1.20 29 . 8 5 .96 751 809 8 24 1.20 51. 4 6.00 1277 1417 1586 13 84 1" These h e a d i n g s d e f i n e t h e t h e r m a l c o n d u c t i v i t y r a t i o i n t h e w a l l l a y e r when CASE 3 a p p l i e s . The d a t a f o r CASES 1 and 2 a r e l i s t e d u n d e r t h e h e a d i n g K/k = 1, d e s p i t e t h e a b s e n c e o f a w a l l l a y e r f o r t h e s e c a s e s . T a b l e 2 ( c o n t i n u e d ) 57 P r Re X I O " 4 f X 1 0 3 N u s s e l t Number D i p p r e y T h i s work Nunner K/k = 1 K/k = oo 5.94 2 . 37 9 . 50 282 159 188 5.94 4.01 10.0 512 269 305 5.94 6.68 10 . 5 829 844 1356 489 5.9 4 11.6 10 . 7 1337 12 39 2977 816 4.38 3.10 9.75 330 188 218 4 . 38 5.2 3 10 . 3 566 350 355 4 . 38 8 . 80 10 . 7 90 2 907 1582 576 4 . 38 15.3 10 . 8 1454 12 37 3077 962 2. 79 4 .68 10 .?. 505 260 273 2.79 7.92 10 . 6 643 64 3 913 447 2 . 79 13.3 10.7 1017 941 1691 723 2.79 22.9 11.2 1597 1178 3165 1214 1. 20 10 . 2 10 .7 468 4 51 588 399 1.20 17. 3 11. 0 749 678 1034 658 1.20 29 . 2 11. 3 1198 908 1790 1086 1.20 50 .1 11.4 1882 1138 3121 1812 Table 2 (continued) 58 Pr Re X i o - 4 f X 10 3 N u s s e l t Number Dipprey T h i s work Nunner K/k = 1 K/k = co 5 .94 1.35 17.2 249 112 125 5.94 2. 32 18.0 418 364 916 203 5.94 3.94 18.0 672 445 1689 326 5.94 6 . 56 18 . 0 1013 501 2919 518 5.94 11. 5 18.0 1538 542 5226 867 4 . 38 1.78 17.8 276 270 478 148 4. 38 3.05 18.0 458 369 945 239 4.38 5 .19 18.0 721 445 1687 387 4. 38 8.64 18.0 1071 500 2877 616 4.38 15.1 18.1 1654 , 535 5126 1032 2.79 2.68 18.0 310 283 532 188 2.79 4 . 59 18.0 506 370 954 305 2 . 79 7 .85 18.0 784 444 1670 497 2 . 79 13.0 18 . 0 1146 497 2799 790 2.79 22.6 18. 3 1778 523 4975 1327 1 . 20 5.85 18.0 353 287 546 286 1. 20 10.1 18.0 562 369 946 472 1.20 17 .2 18.2 869 434 1631 775 1. 20 28.6 18.5 1335 473 2754 1251 1. 20 49 . 3 18 . 8 2017 495 4819 2099 59 3. E v a l u a t i o n of Experimental C o n d u c t i v i t y R a t i o i n the Wall Layer  For each combination of f and Re where the w a l l l a y e r i s pr e s e n t , two values o f Nu have been c a l c u l a t e d . One assumes that the thermal c o n d u c t i v i t y r a t i o , K/k, i n the w a l l l a y e r i s u n i t y and the oth e r assumes that i t i s i n f i n i t y . When experimental data f a l l between t h i s range, i t i s p o s s i b l e to c a l c u l a t e an average value f o r the r a t i o K/k th a t w i l l make the r e s u l t s as p r e d i c t e d by the proposed method c o i n c i d e with the experimental p o i n t s . I t i s i n t e r e s t i n g to note here that i n the great m a j o r i t y o f cases i t was p o s s i b l e to evaluate an experimental c o n d u c t i v i t y r a t i o . These are p l o t t e d i n Fig u r e 11 as a f u n c t i o n o f Re, f and Pr. • NUNNER'S DATA , f = .0388 © " " f= 0545 o " » f = .0758 o DIPPREY'S DATA , Pr= 1.20 4 6 8 IO4 2 4 6 8 IO5 2 4 6 8 10 Re F i g u r e 11. Average K/k i n Wall Layer as Computed from Experimental Data 61 DISCUSSION OF RESULTS 1. General D i s c u s s i o n of Proposed Method The measure of success any model achieves has to be based u l t i m a t e l y on how w e l l i t d e s c r i b e s or p r e d i c t s p h y s i c a l r e a l i t y . In most cases, however, p h y s i c a l r e a l i t y can only be d e s c r i b e d q u a n t i t a t i v e l y by experimental f a c t s , and t h e r e f o r e c o n t a i n a multitude of experimental e r r o r s . Even so, experimental r e s u l t s w i l l have to remain the b a s i s of our knowledge. On t h i s b a s i s , the proposed method f o r p r e d i c t i n g values of Mu i s not a t o t a l s uccess. The most s e r i o u s o b j e c t i o n to the model i s the d i s c o n t i n u i t y which appears at c e r t a i n combinations of Re and f . The f a u l t can be best seen i n F i g u r e s Ua to 4e, where the computed values of Nu have been p l o t t e d a g a i n s t Re. The d i s c o n t i n u i t y a r i s e s from the v e l o c i t y p r o f i l e . I n c r e a s i n g values of Re and/or f i n c r e a s e the slope of the t u r b u l e n t core v e l o c i t y p r o f i l e and lower the p o i n t of i n t e r s e c t i o n with the b u f f e r zone p r o f i l e , F i g u r e 1 . I t i s obvious that combinations of Re and f e x i s t which y i e l d t u r b u l e n t core p r o f i l e s t h a t j u s t touch the laminar s u b l a y e r p r o f i l e . In f a c t , these have been c a l c u l a t e d and p l o t t e d i n Figure 2 . When t h i s o c c u r s , the 62 thermal c o n d u c t i v i t y r a t i o , K/k, i s c a l c u l a t e d from the t u r b u l e n t core v e l o c i t y p r o f i l e to the po i n t o f tangency, below which the laminar s u b l a y e r p r o f i l e i s used. By d e f i n i t i o n , K/k becomes u n i t y i n the laminar s u b l a y e r . Now c o n s i d e r a t u r b u l e n t core v e l o c i t y p r o f i l e e valuated from approximately s i m i l a r values of Re and f , but which j u s t misses the laminar s u b l a y e r p r o f i l e . I t then i n t e r s e c t s the Ug = 0 a x i s at some value of y* l e s s than yjt at the point of tangency, and give s r i s e to the w a l l l a y e r . Two values f o r Nu can now be c a l c u l a t e d based on the two d i f f e r e n t assumptions made f o r the thermal c o n d u c t i v i t y i n the w a l l l a y e r . However, i n the r e g i o n between the po i n t of tangency and the outer l i m i t o f the w a l l l a y e r , the thermal c o n d u c t i v i t y r a t i o i s d i f f e r e n t f o r the two v e l o c i t y p r o f i l e s . In the former case i t i s u n i t y , and i n the l a t t e r i t i s evaluated u s i n g the t u r b u l e n t core v e l o c i t y p r o f i l e , which u s u a l l y r e s u l t s i n a value c o n s i d e r a b l y g r e a t e r than u n i t y . Obviously on i n t e g r a t i o n , three d i f f e r e n t Nu are obtained from approximately s i m i l a r values of Re and f . By t a k i n g the d e s c r i b e d s i t u a t i o n t o i t s p h y s i c a l l i m i t , one obt a i n s a d i s c o n t i n u i t y i n the Nu versus Re p l o t at constant f and Pr. The assumption of complete stagnancy i n the w a l l l a y e r i s inadequate. T h i s i s obvious from F i g u r e s 4a to 4e. The branch i n which the assumption i s made approaches a constan t , l i m i t i n g v alue. Figure 5 shows t h i s value to be 63 i n d e p e n d e n t o f P r . An e x p l a n a t i o n f o r t h i s phenomenon i s a v a i l a b l e . M a t h e m a t i c a l l y i t i s p o s s i b l e t o show t h a t t h e w a l l l a y e r i s a f u n c t i o n o n l y o f t h e f r i c t i o n f a c t o r , and t h a t i t s t h i c k n e s s i s a p p r o x i m a t e l y 1/30 o f t h e h e i g h t o f t h e e q u i v a l e n t s a n d r o u g h n e s s . The e q u a t i o n f o r t h e t u r b u l e n t c o r e v e l o c i t y p r o f i l e on t h e smooth p i p e c o o r d i n a t e s c a l e , e q u a t i o n 20, i s s o l v e d f o r t h e c o n d i t i o n Ug r o. Upon s i m p l i f y i n g and r e a r r a n g i n g t h e r e s u l t i n g e x p r e s s i o n , one o b t a i n s an e q u a t i o n w h i c h l o c a t e s t h e b o u n d a r y between the t u r b u l e n t c o r e and t h e w a l l l a y e r i n t e r m s o f f : I r -j== 4.07 l o g i w - 2 . 65 ( 33 ) iff y E q u a t i o n 33 i s i d e n t i c a l i n form t o an e x p r e s s i o n d e v e l o p e d by N i k u r a d s e (18) f o r t h e f r i c t i o n f a c t o r i n t e r m s o f t h e h e i g h t o f t h e sand g r a i n r o u g h n e s s e l e m e n t s , e s , i n f u l l y , t u r b u l e n t f l o w . When t h e two e x p r e s s i o n s a r e compared, an a l m o s t c o n s t a n t r a t i o i s o b t a i n e d between y and e s e q u a l t o a p p r o x i m a t e l y 1/30. Now, f o r a g i v e n P r and f r i c t i o n f a c t o r , t h e r e s i s t a n c e t o h e a t t r a n s f e r i n t h e t u r b u l e n t c o r e depends o n l y on Re. I n c r e a s i n g t h e v a l u e o f Re w i l l d e c r e a s e t h e r e s i s t a n c e i n t h e t u r b u l e n t c o r e , b u t w i l l l e a v e t h e r e s i s t a n c e i n t h e w a l l l a y e r u n a f f e c t e d . U l t i m a t e l y t h e w a l l l a y e r r e s i s t a n c e becomes l a r g e compared t o t h e t u r b u l e n t c o r e r e s i s t a n c e , and w i l l i n f a c t c o n t r o l t h e r a t e 64 of heat t r a n s f e r . F u r t h e r change i n e i t h e r Re or Pr, s i n c e n e i t h e r of these a f f e c t s the w a l l l a y e r r e s i s t a n c e , w i l l t h e r e f o r e not change the p r e d i c t e d value of Nu. A c c o r d i n g to F i g u r e 5, the magnitude of Pr determines the r a t e a t which Nu approaches the l i m i t i n g v a l u e . Equation 23 shows the reason f o r t h i s . For the t u r b u l e n t c o r e , the thermal c o n d u c t i v i t y r a t i o i s the sum of a c o n s t a n t , equal to u n i t y , and a f a c t o r which i s a f u n c t i o n of Re, f and Pr. Thus, the e f f e c t of small Pr i s to depress the r e s u l t of any change i n Re, and hence to depress the r e l a t i v e c o n t r i b u t i o n of eddy t r a n s p o r t to the t o t a l t r a n s p o r t of heat. In the w a l l l a y e r , the assumption of complete stagnancy r e s t r i c t s the method of heat t r a n s f e r to m olecular t r a n s p o r t , which i s independent of the value of Re. Consequently f o r constant w a l l roughness and s m a l l Pr, l a r g e changes i n Re are r e q u i r e d i n order to c o n t r i b u t e s i g n i f i c a n t eddy t r a n s p o r t to the predominant molecular t r a n s p o r t . Conversely, the e f f e c t of l a r g e Pr i s to magnify any change i n Re. In g e n e r a l , a l a r g e value of Pr i s a s s o c i a t e d with a s m a l l value of molecular c o n d u c t i v i t y . Therefore i n r e l a t i v e terms, a small change i n Re has a c o n s i d e r a b l e e f f e c t on the r e s i s t a n c e i n the t u r b u l e n t core. Since the w a l l l a y e r r e s i s t a n c e remains f i x e d , i t becomes q u i c k l y c o n t r o l l i n g and r e s u l t s i n the f i x e d value of the Musselt number. This i s born out i n Figure 5. The preceding argument may be extended. In view of equation 33, the thermal r e s i s t a n c e of the stagnant w a l l l a y e r f o r f i x e d values of Pr and Re becomes l a r g e r with i n c r e a s i n g w a l l roughness. How t h i s a f f e c t s the value of Nu depends on the r e l a t i v e magnitude of the thermal r e s i s t a n c e i n the t u r b u l e n t core and i n the w a l l l a y e r - hence on the value of Re and Pr. For s m a l l f and constant Pr, values of Nu w i l l i n c r e a s e with both Re and f as long as CASES 1 or 2 apply. However, when CASE 3 comes i n t o b e i n g , the w a l l l a y e r becomes with i n c r e a s i n g f the dominating r e s i s t a n c e . When t h i s o c c u r s , the value of Nu w i l l o b v i o u s l y decrease with i n c r e a s i n g w a l l roughness. A p l o t o f Nu versus f f o r some constant Pr and Re w i l l t h e r e f o r e show a maximum i n the curve. The value of f at which t h i s takes place i s ob v i o u s l y a f u n c t i o n of Re and Pr a c c o r d i n g t o the preceding argument about the r a t e at which Nu reaches the l i m i t i n g v alue. The model used by McAndrew (9) i s s i m i l a r t o , though more r e s t r i c t i v e , than the one proposed i n t h i s work. In h i s model CASES 1 and 2 cover a narrower range of f and Re. He furthermore a r b i t r a r i l y l i m i t e d h i s model (except f o r some i s o l a t e d p o i n t s ) to assuming u n i t y f o r the thermal c o n d u c t i v i t y r a t i o i n the w a l l r e g i o n . The formal d i f f e r e n c e a r i s e s from the f r i c t i o n f a c t o r used to d e r i v e the working equations l i s t e d i n Table 1. McAndrew used the rough pipe f r i c t i o n f a c t o r i n the corresponding 66 places i n equation 24, 25 and 76 where t h i s model uses the smooth pipe f r i c t i o n f a c t o r . Equation 22 and 23 are i d e n t i c a l f o r the two models. T h e r e f o r e , f o r values of Re and f that give r i s e to CASE 3, the two models p r e d i c t i d e n t i c a l values of Nu and e x h i b i t s i m i l a r t r e n d s , i n c l u d i n g the occurrance of maxima i n the r e s u l t s with i n c r e a s i n g Re and f . When i n f i n i t y i s assumed f o r the value of thermal c o n d u c t i v i t y r a t i o i n the w a l l l a y e r , the p r e d i c t e d r e s u l t s vary e s s e n t i a l l y l i n e a r l y with Re f o r l a r g e Pr. For s m a l l values of Pr, the r e s u l t s f o l l o w along c l o s e to those c a l c u l a t e d u s i n g the a l t e r n a t e assumption. These f a c t s are seen i n F i g u r e s 4a to 4e and i n F i g u r e 5. At low P r a n d t l number, the c o i n c i d e n c e of the two l i n e s a r i s e s from the f a c t t h at even i f t h i s l a y e r i s assumed stagnant, i t s thermal r e s i s t a n c e i s n e g l i g a b l e r e l a t i v e to that of the t u r b u l e n t c o r e , as i s a l s o the case when K/k = o° . At high P r a n d t l number, i n c r e a s i n g divergence of the two l i n e s i s caused by the f a c t t h a t the stagnant w a l l l a y e r , u n l i k e the i n f i n i t y c o n d u c t i v i t y w a l l l a y e r , assumes an i n c r e a s i n g l y l a r g e r p o r t i o n of the t o t a l thermal r e s i s t a n c e with i n c r e a s i n g Re and f . The d i f f e r e n c e between the c a l c u l a t e d r e s u l t s u s i n g the two l i m i t i n g assumptions t h e r e f o r e r e p r e s e n t s a measure of the thermal r e s i s t a n c e i n the w a l l l a y e r . Q u a l i t a t i v e l y , the trends p r e d i c t e d i n t h i s case are more r e a l i s t i c , even though the values of Nu are high compared to 67 experimental d a t a , e s p e c i a l l y f o r l a r g e Pr. Of more importance i s the f a c t t h a t the c a l c u l a t e d r e s u l t s show a dependence on Pr and f f o r a l l values of Re. The dotted l i n e i n Fi g u r e 2 d e f i n e s the l i m i t beyond which the rough pipe f r i c t i o n f a c t o r i n ge n e r a l becomes independent of Re, i . e . the f l u i d i s i n f u l l y rough flow. In the r e g i o n between the dotted l i n e and the smooth pipe f r i c t i o n f a c t o r , S c h l i c h t i n g (23) has shown t h a t the c o e f f i c i e n t B i n the rough pipe u n i v e r s a l v e l o c i t y d i s t r i b u t i o n equation f o r t u r b u l e n t flow, u + = 2.5 In _Y_ + B( > <12b) e V i s i n f a c t a f u n c t i o n of w a l l roughness. I t can be seen i n F i g u r e 2 t h a t the dotted l i n e p r a c t i c a l l y c o i n c i d e s with the l i m i t f o r CASE 3. Therefore p r a c t i c a l l y a l l the r e s u l t s f o r CASES 1 and 2 l i e w i t h i n the r e g i o n where c o e f f i c i e n t B i n equation (12b) i s a v a r i a b l e . Since a l l the equations used i n the c a l c u l a t i o n of the presented r e s u l t s were developed assuming f u l l y rough flow, some doubt could a r i s e about t h e i r v a l i d i t y f o r CASES 1 and 2. Rouse (6) was able to show t h a t the t u r b u l e n t core v e l o c i t y p r o f i l e f o r both f u l l y rough and smooth pipes can be expressed by a s i n g l e r e l a t i o n s h i p , equation 12. His d e r i v a t i o n was repeated here u s i n g any number f o r the value of B i n equation 12b. The r e s u l t was again equation 12. This proves that equation 12 68 i s a p p l i c a b l e a l s o i n the t r a n s i t i o n r e g i o n . Quite a s i d e from the t h e o r e t i c a l aspects of the proposed model, c e r t a i n problems arose with the mechanics of computing the r e s u l t s . The r a t e of convergence of the numerical i n t e g r a t i o n was found to be exceedingly slow, and i n some cases i t was impossible to achieve the d e s i r e d l e v e l of accuracy even though a l a r g e number of increments was used to compute the area. ' The r a t e of convergence was t e s t e d s e p a r a t e l y i n each flov; r e g i o n . When two co n s e c u t i v e e v a l u a t i o n s of the area u s i n g a d i f f e r e n t number of increments agreed w i t h i n 1 0.6%, the c r i t e r i o n f o r convergence was s a t i s f i e d . I t f o l l o w s t h a t two e v a l u a t i o n s of Nu w i l l a l s o agree at l e a s t to w i t h i n the s p e c i f i e d l i m i t . In t h i s way, the d e s i r e d l e v e l of accuracy was achieved or exceeded i n the maximum number of cases. For c e r t a i n combinations of Re, f and Pr, i t was not p o s s i b l e to achieve the d e s i r e d accuracy, although up to 160 increments were used to compute the area w i t h i n the given flow r e g i o n . I n v e s t i g a t i o n pointed out two zones of d i f f i c u l t y . I t was imp o s s i b l e to o b t a i n convergence w i t h i n the s p e c i f i e d l i m i t s i n the b u f f e r zone f o r CASE 1 when Pr = 1,000, and d i f f i c u l t when Pr was between 10 and 100. The maximum di s c r e p e n c y was approximately 40% f o r Pr = 1,000, a t Re = 700,000. However t h i s value i s 69 a b n o r m a l l y h i g h . A l l t h e r e m a i n i n g d e v i a t i o n s i n t h e b u f f e r zone r a n g e down f r o m 3.5%. In any c a s e , f o r l a r g e Re, t h e c o n t r i b u t i o n o f t h e b u f f e r zone i s i n t h e o r d e r o f 1 t o 3% o f t h e t o t a l a r e a f o r a l l t h r e e r e g i o n s , and t h e r e f o r e q u i t e i n s i g n i f i c a n t . The o t h e r zone o f d i f f i c u l t y was t h e t u r b u l e n t c o r e , e s p e c i a l l y f o r l a r g e v a l u e s o f t h e t h r e e i n d e p e n d e n t v a r i a b l e s . The p r o b l e m was overcome by d i v i d i n g t h e r e g i o n i n t o two s u b - r e g i o n s w i t h t h e b o u n d a r y a t R = 0.925. The computer program made p r o v i s i o n s f o r s i t u a t i o n s where t h e t u r b u l e n t c o r e b o u n d a r y o c c u r r e d b e f o r e R = 0.92 5. Each o f t h e two s u b - r e g i o n s t h e n becomes a s e p a r a t e e n t i t y w i t h an i n d e p e n d e n t number o f i n c r e m e n t s and i t s own t e s t f o r c o n v e r g e n c e . S u b s e q u e n t l y , the r e q u i r e d l e v e l o f a c c u r a c y was met i n t h e m a j o r i t y o f c a s e s . The r e a s o n f o r t h e s l o w c o n v e r g e n c e o f t h e n u m e r i c a l i n t e g r a t i o n i s u n d o u b t e d l y t h e m a t h e m a t i c a l a l g o r i t h m u s e d . Simpson's r u l e i s e x a c t o n l y f o r a t h i r d d e g r e e p o l y n o m i a l . The m a j o r p r o b l e m r e g i o n s a r e c h a r a c t e r i z e d by l a r g e c u r v a t u r e , where t h e o r d i n a t e c hanges o v e r a s h o r t d i s t a n c e by s e v e r a l f a c t o r s o f m a g n i t u d e . The t r u n c a t i o n e r r o r s a s s o c i a t e d w i t h t h e m a t h e m a t i c a l method, and t h e r o u n d o f f e r r o r s a s s o c i a t e d w i t h t h e c o m p u t e r , combine and s e e m i n g l y r e n d e r t h e modus o p e r a n d i i n a d e q u a t e f o r c e r t a i n v a l u e s o f t h e v a r i a b l e s . 70 2. C o m p a r i s o n o f R e s u l t s as P r e d i c t e d f r o m t h e P r o p o s e d Method w i t h E x p e r i m e n t a l D a t a  C o m p a r i s o n o f p r e d i c t e d r e s u l t s w i t h e x p e r i m e n t a l d a t a i s made d i f f i c u l t b e c a u s e o f t h e d i s c o n t i n u i t y i n t h e m odel. However, i n g e n e r a l t h e r e s u l t s a r e i n good agreement w i t h t h e e x p e r i m e n t a l d a t a up t o t h e p o i n t o f d i s c o n t i n u i t y . In t h e r e g i o n b e y o n d , i t i s i n t e r e s t i n g t h a t , w i t h v e r y few e x c e p t i o n s , t h e e x p e r i m e n t a l d a t a l i e between t h e two s e t s o f c a l c u l a t e d r e s u l t s o b t a i n e d by a s s u m i n g t h e two l i m i t i n g v a l u e s f o r t h e t h e r m a l c o n d u c t i v i t y r a t i o i n t h e w a l l l a y e r . Nunner's d a t a p l o t t e d i n F i g u r e s 7a t o 7d c o v e r a v e r y wide r a n g e o f f r i c t i o n f a c t o r and R e y n o l d s number. U n f o r t u n a t e l y h i s r e s u l t s a r e o f r e l a t i v e l y l i t t l e v a l u e i n t e s t i n g t h e p r o p o s e d model. In a l l b u t two e x p e r i m e n t a l p i p e s , h i s r e s u l t s l i e i n t h e r a n g e o f CASE 3, where a w a l l l a y e r e x i s t s and where t h e p r o p o s e d model i s c a p a b l e o n l y o f d e f i n i n g l o w e r and u p p e r l i m i t s o f N u s s e l t number. I t i s e n c o u r a g i n g t o n o t e , however, t h a t f o r o n l y one example o f CASE 3 does Nunner's d a t a l i e o u t s i d e t h e two l i m i t i n g v a l u e s o f Mu. In F i g u r e s 7a and 7b t h e p r o p o s e d method y i e l d s v a l u e s f o r Nu t h a t compare f a v o u r a b l y w i t h t h e e x p e r i m e n t a l d a t a . Though Nunner's t h e o r y as d e f i n e d by e q u a t i o n 2 8 p r e d i c t s a l l h i s e x p e r i m e n t a l r e s u l t s w i t h i n a p p r o x i m a t e l y 15 t o 20%, i t i s h a r d l y s u p e r i o r t o t h e p r o p o s e d method i n t h o s e 71 cases covered by F i g u r e s 7 a and 7 b . The Smith and E p s t e i n data are p l o t t e d i n Figures 9a to 9d. Values of Nu c a l c u l a t e d from the proposed method compare w e l l with the experimental data. The maximum d i f f e r e n c e i s approximately 40% i n the case of the 1/8" Galvanized pipe. However, the average d e v i a t i o n c o n s i d e r i n g a l l the data would be c l o s e r t o 10 - 15%. Both the Nunner and the M a r t i n e l l i equations give good p r e d i c t i o n of the data, though i n g e n e r a l no b e t t e r than the proposed method. The p r e d i c t i o n based on McAndrew's model (9) compares p o o r l y . Dipprey's work o f f e r s a means to compare p r e d i c t e d r e s u l t s from the proposed method with experimental data of P r a n d t l numbers other than that f o r a i r ( Pr — 0.7 ). These r e s u l t s are l i s t e d i n Table 2. Both the proposed method and Nunner's theory c o r r e l a t e the experimental data w e l l f o r a l l values of Pr when the pipe roughness i s s m a l l , F i g u r e 10a and Table 2. In each case Nunner's theory p r e d i c t s somewhat hi g h e r values of Nu than the proposed theory. The r e s u l t s f o r the two rougher pipes l i e l a r g e l y i n the range where the w a l l l a y e r e x i s t s (CASE 3 ) , Again, the experimental r e s u l t s l i e almost e x c l u s i v e l y between the two l i m i t i n g p r e d i c t i o n s . I t i s i n t e r e s t i n g to note i n F i g u r e s 10a to 10c how Nunner's theory,equation 28, gives an i n c r e a s i n g l y poorer 72 r e p r e s e n t a t i o n of the experimental data with i n c r e a s i n g Pr. T h i s i s e s p e c i a l l y obvious with the two rougher tubes, where h i s theory p r e d i c t s w e l l f o r Pr = 1.20, but i s low by a f a c t o r of n e a r l y two at Pr = 5.9 '4. Why the theory p r e d i c t s w e l l f o r the smoother p i p e , Table 2, e s p e c i a l l y f o r the h i g h e r values of Pr, i s somewhat dubious i n view of the b a s i c assumptions that were made. Only a c o n j e c t u r e i s p o s s i b l e . Because Nunner's theory i s b a s i c a l l y an e x t e n s i o n of P r a n d t l ' s smooth pipe theory, i t i s p o s s i b l e t h a t h i s equation a p p l i e s over a l a r g e r Pr range f o r near-smooth pipes than f o r very rough ones. Apart from the reasons a l r e a d y d i s c u s s e d f o r the l a c k of c o r r e l a t i o n between the p r e d i c t e d r e s u l t s from the proposed method and the experimental data, the e f f e c t of the i n i t i a l assumptions o u t l i n e d on page 6 have not yet been c o n s i d e r e d . Assumptions ( a ) , ( b ) , ( d ) , (e) and ( f ) are probably q u i t e w e l l s a t i s f i e d i n any experimental i n v e s t i g a t i o n . However, assumptions (c) and (g) w i l l i n general i n t r o d u c e some disc r e p a n c y between the t h e o r e t i c a l and the experimental r e s u l t s . The magnitude of the e r r o r depends on the measures taken by the i n v e s t i g a t o r to f u l f i l l the assumptions e x p e r i m e n t a l l y , and on the p r o p e r t i e s of the f l u i d . For example,, assumption (c) i s q u i t e sound f o r moderate h e a t i n g of the common gases. The experimental c o n d i t i o n s of a l l three s t u d i e s compared here were such t h a t 7 3 d e v i a t i o n s from both of the major assumptions were i n -s u f f i c i e n t l y l a r g e to account f o r more than minor d i s c r e p a n c i e s from- the theory. 3 . E v a l u a t i o n of Experimental C o n d u c t i v i t y R a t i o i n the Wall Layer  The c a l c u l a t e d values of the average thermal c o n d u c t i v i t y r a t i o , K/k, based on the experimental data of Nunner and Dipprey have been p l o t t e d i n Figure 11. The c a l c u l a t e d p o i n t s l i n e up w e l l , and i n d i c a t e a d e f i n i t e t r e n d . To f i n d a c o r r e l a t i n g f u n c t i o n , however, i s next to impossible i n view of the r e l a t i v e l y narrow range of data a v a i l a b l e . S u f f i c e i t to suggest t h a t , i n view of the a v a i l a b l e i n f o r m a t i o n , (K/k)^ = <^)(Re, Pr, f, roughness geometry). 4. D i s c u s s i o n of T e m D e r a t u r e P r o f i l e Data Some g e n e r a l i z e d temperature p r o f i l e s have been p l o t t e d i n Figures 6a to 6 d, together with the corresponding v e l o c i t y p r o f i l e s . For small values of Re (e.g. 4 , 0 0 0 ) , and f o r very s m a l l or very l a r g e values of Pr, the w a l l roughness appears to have no s i g n i f i c a n t e f f e c t on the temperature p r o f i l e . However, f o r i n t e r m e d i a t e values of Pr ( a range about u n i t y ) , i n c r e a s i n g w a l l roughness f l a t t e n s the temperature p r o f i l e . On the other hand, f o r constant values of Re, a l a r g e r roughness i n c r e a s e s the form drag at the w a l l , with the w e l l known r e s u l t that the v e l o c i t y p r o f i l e becomes more pointed i n the main stream. The v e l o c i t y p r o f i l e i s of course independent of Pr when f l u i d p r o p e r t i e s are u n a f f e c t e d by temperature v a r i a t i o n s . Nunner (3) s t a t e d on the b a s i s of h i s experimental data t h a t roughness has a r e l a t i v e l y s m a l l e f f e c t on the temperature p r o f i l e . T h i s i s c e r t a i n l y born out f o r the extreme values of Pr by the r e s u l t s from the proposed method. For Pr near u n i t y , however, the p r e d i c t e d r e s u l t s d e f i n e a f l a t t e r temperature p r o f i l e with i n c r e a s i n g w a l l roughness. This i s not unreasonable. Wall roughness promotes wake t u r b u l e n c e , which i n c r e a s e s the eddy mechanism of heat t r a n s f e r i n the r e g i o n . In e f f e c t , the f l u i d i s more mixed up near the w a l l . T h i s should y i e l d a high e r heat t r a n s f e r c o e f f i c i e n t and a more even temperature d i s t r i b u t i o n , t h a t i s , a f l a t t e r temperature p r o f i l e . For small values of Pr, molecular conduction predominates and completely masks any e f f e c t due to an i n c r e a s e i n tu r b u l e n c e . For l a r g e Pr, eddy conduction predominates to such an extent that the temperature p r o f i l e i s e s s e n t i a l l y f l a t f o r a l l Re and f . U n f o r t u n a t e l y Nunner's experimental data are too i n c o n c l u s i v e to e i t h e r support or r e j e c t the h y p o t h e s i s . The temperature and v e l o c i t y p r o f i l e s show s i m i l a r c h a r a c t e r i s t i c s to the above at very l a r g e Re 75 ( R e = 10,000,000). H o w e v e r , m a n y o f t h e t r e n d s s e e n a t l o w Re a r e m a s k e d i n t h i s c a s e , a s t h e t e m p e r a t u r e p r o f i l e b e c o m e s e s s e n t i a l l y f l a t f o r Re g r e a t e r t h a n u n i t y a n d f o r n e a r l y a l l w a l l r o u g h n e s s e s g r e a t e r t h a n f _ . F o r c o m p a r i s o n , r e p r e s e n t a t i v e p r o f i l e s h a v e b e e n p l o t t e d a t P r = 1 i n F i g u r e 6d. A l l o f N u n n e r ' s e x p e r i m e n t a l t e m p e r a t u r e p r o f i l e s w e r e c o m p a r e d t o c o r r e s p o n d i n g p r e d i c t i o n s f r o m t h e p r o p o s e d m e t h o d . S o m e o f t h e s e a r e p l o t t e d i n F i g u r e s 8 a t o 8 c . T h o u g h i n g e n e r a l t h e a g r e e m e n t i s r e a s o n a b l y g o o d , s o m e p u z z l i n g t r e n d s c a m e t o l i g h t . F o r c o n s t a n t v a l u e s o f R e , i n c r e a s i n g w a l l r o u g h n e s s a b o v e f s w i l l f i r s t f l a t t e n t h e p r o f i l e a s p r e d i c t e d . B u t a t s t i l l l a r g e r v a l u e s o f w a l l r o u g h n e s s , t h e v e l o c i t y p r o f i l e f a l l s b e l o w t h a t f o r a c o r r e s p o n d i n g s m o o t h p i p e a n d b e c o m e s m o r e p o i n t e d . T h e t e m p e r a t u r e p r o f i l e c h a r a c t e r i s t i c s a r e f u r t h e r c o n f u s e d b y t h e p r e s e n c e o f t h e d i s c o n t i n u i t y i n t h e m o d e l , t h e s u b s e q u e n t a p p e a r a n c e o f t h e w a l l l a y e r , a n d t h e t w o l i m i t i n g a s s u m p t i o n s f o r t h e t h e r m a l c o n d u c t i v i t y r a t i o . T h e i r e f f e c t i s d i f f i c u l t t o j u d g e f r o m t h e l i m i t e d r e s u l t s a v a i l a b l e . T h e t e m p e r a t u r e p r o f i l e a s s u m i n g i n f i n i t y f o r t h e t h e r m a l c o n d u c t i v i t y r a t i o i n t h e w a l l l a y e r c o m p a r e s p o o r l y w i t h N u n n e r ' s e x p e r i m e n t a l d a t a i n a l l c a s e s , F i g u r e s 8b a n d c . T h e o t h e r a s s u m p t i o n , t h a t o f s t a g n a n t w a l l l a y e r 76 (K/k = 1), y i e l d s r e s u l t s which compare reasonably with the experimental data. However any trends are r a t h e r u n c l e a r and i m p o s s i b l e to assess r a t i o n a l l y f o r CASE 3, because of the indeterminacy of K/k i n the w a l l l a y e r . The temperature p r o f i l e s c a l c u l a t e d from equation 31 based on Nunner's assumptions have been p l o t t e d i n F i g u res 8a to 8c. I t i s obvious t h a t Nunner's theory does not d e s c r i b e the p h y s i c a l phenomena p a r t i c u l a r l y w e l l . Some b a s i c c h a r a c t e r i s t i c s of equation 31 are worth n o t i n g . I n c r e a s i n g w a l l roughness at constant Re and Fr y i e l d s i n c r e a s i n g l y f l a t t e r temperature p r o f i l e s . Though t h i s trend i s i n agreement with the hypothesis put f o r t h i n the present work, i t does not agree with Nunner's own e m p i r i c a l c o n c l u s i o n s ( 3 ) . Furthermore, equation 31 s u f f e r s from the same b a s i c f a u l t as Nunner's heat t r a n s f e r e q u ation. For c e r t a i n values of Re, f and Pr the denominator of equation 31 becomes n e g a t i v e . This i s , of course, impossible i n r e a l i t y and shows a s e r i o u s l i m i t a t i o n i n Nunner's theory. 77 CONCLUSIONS AND RECOMMENDATIONS 1. The proposed method f o r p r e d i c t i n g heat t r a n s f e r i n rough pipes i s not adequate i n the present form. B a s i c a l l y the f a u l t l i e s with the manner of a p p l y i n g the v e l o c i t y p r o f i l e , which gives r i s e to the d i s c o n t i n u i t y . 2. The proposed method d e s c r i b e s the experimental f a c t s reasonably w e l l i n the absence of the w a l l l a y e r ; t h a t i s , i n e f f e c t before f u l l y rough flow has developed. However, once the w a l l l a y e r appears, only heat t r a n s f e r l i m i t s can be d e f i n e d . 3. Except f o r s p e c i a l s i t u a t i o n s , such as l i q u i d metals, there i s l i t t l e j u s t i f i c a t i o n f o r u s i n g the proposed method i n the present form over Nunner's heat t r a n s f e r equation, i n the c a l c u l a t i o n of heat t r a n s f e r c o e f f i c i e n t . The l a t t e r i s much si m p l e r to use, and gives at l e a s t as good r e s u l t s i n the range .5 < Pr < 5. 4. The temperature p r o f i l e s c a l c u l a t e d from the proposed model are i n reasonable agreement with a v a i l a b l e experimental data. More experimental and computed data are r e q u i r e d to a s c e r t a i n the e f f e c t of Re and f . 5. Comparison of experimental and p r e d i c t e d heat t r a n s f e r 7 8 r e s u l t s i n d i c a t e t h a t our understanding of the behaviour of the f l u i d near the s o l i d boundary i s inadequate. Depending on the magnitude of Re and f , there are probably r e g i o n s of laminar flow and high t u r b u l e n c e present of v a r y i n g r e l a t i v e t h i c k n e s s f o r a l l flow c o n d i t i o n s . A b e t t e r comparison i s p o s s i b l e between Nunner's experimental temperature p r o f i l e s and the p r e d i c t e d p r o f i l e s f o r h i s data that give r i s e to CASE 3. From the c a l c u l a t i o n of Nu, one o b t a i n s an experimental value f o r the average thermal c o n d u c t i v i t y r a t i o i n the w a l l l a y e r . Using t h i s value of K/k, one can compute a normalized temperature p r o f i l e f o r the same value of f and Re and compare the r e s u l t to the experimental data. The proposed model should be improved by e l i m i n a t i n g the d i s c o n t i n u i t y before any a d d i t i o n a l r e s u l t s are obtained. This i s p o s s i b l e i n the f o l l o w i n g manner: (a) Define the presence of a thermal laminar s u b l a y e r near the rough w a l l , f o r a l l Re and f , of t h i c k -ness y * = 5 . By d e f i n i t i o n , the thermal c o n d u c t i v i t y r a t i o K/k i n t h i s r e g i o n i s always equal to u n i t y . In-the presence of a w a l l l a y e r , the value of K/k i s put equal to i n f i n i t y i n the r e g i o n 5 $ V g ^  ( i n t e r s e c t i o n of t u r b u l e n t v e l o c i t y p r o f i l e with u* = 0 a x i s ) . I f i n t h i s 79 r e g i o n the value of K/k = 1 , the r e s u l t would, of course, be e q u i v a l e n t to the bottom l i m i t f o r CASE 3 i n the present model. The experimental data used i n t h i s t h e s i s should then be compared with the improved model, and the p r e d i c t e d r e s u l t s r e v a l u a t e d i n l i g h t of t h i s comparison. 8. The model might a l s o be improved by i n c l u d i n g some of the f o l l o w i n g , about which there i s u n f o r t u n a t e l y l i t t l e r e l i a b l e i n f o r m a t i o n a v a i l a b l e at present: (b) A d i f f e r e n t v e l o c i t y p r o f i l e t h a t b e t t e r d e s c r i b e s the f l u i d behaviour near the rough w a l l . (c) Values of - r ^ - as f u n c t i o n s of Re, f , Pr and R. (d) An e m p i r i c a l r e l a t i o n t h at d e s c r i b e s the average t o t a l thermal c o n d u c t i v i t y of the w a l l l a y e r . 80 NOMENCLATURE A - area, f t ^ B - c o e f f i c i e n t i n equation 12b, dimensionless Cp - s p e c i f i c heat at constant p r e s s u r e , Btu/(lb)(°F) d - i n s i d e diameter of pi p e , f t e - average height of roughness elements, f t e s - e q u i v a l e n t sand-roughness height of roughness elements, f t E]J; - r a d i a l eddy c o n d u c t i v i t y , (Btu)(ft)/(hr)(ft 2)(°F) Fy, - eddy v i s c o s i t y , l b M / ( h r ) ( f t ) f - Fanning f r i c t i o n f a c t o r , dimensionless g - denotes f u n c t i o n , dimensionless g' - denotes f i r s t d e r i v a t i v e of f u n c t i o n , dimensionless g c - conver s i o n f a c t o r = 4.17 x 10 8 (lt> M) ( f t ) / (lb ? ) ( h r ? ) h - c o e f f i c i e n t of heat t r a n s f e r , Btu/(hr)(ft 2)(°F) 2/3 ; j H - heat t r a n s f e r f a c t o r = S t ( P r ) , dimensionless k - molecular thermal c o n d u c t i v i t y , ( B t u ) ( f t ) / ( h r ) ( f t )(°F) K - t o t a l r a d i a l c o n d u c t i v i t y = k + E^, (Btu)(ft)/(hr)(ft 2)(°F) Nu - Nu s s e l t number = hd/k, dimensionless Pr - P r a n d t l number C p y M / k , dimensionless q - heat t r a n s f e r r a t e , Btu/hr r - point r a d i u s , f t r w - r a d i u s o f pi p e , f t R - dimensionless r a d i u s = r/r„ 81 r a d i u s which d e f i n e s the outer boundary of t u r b u l e n t c o r e , dimensionless r a d i u s which d e f i n e s boundary between b u f f e r r e g i o n and laminar s u b l a y e r , dimensionless Reynolds number = -^<£, dimensionless Stanton number = n , dimensionless ? P temperature a t r a d i u s r , °F bulk temperature, °F temperature at r = 0 , °F temperature at a d i s t a n c e y, °F po i n t v e l o c i t y along the pi p e , f t / h r maximum f l u i d v e l o c i t y ( v e l o c i t y at r = 0 ) , f t / h r f r i c t i o n v e l o c i t y = ^ , f t / h r dimensionless v e l o c i t y = u/u average bulk v e l o c i t y , f t / h r dimensionless v e l o c i t y = u/u„_ J max dimensionless v e l o c i t y = u/U denotes any independent v a r i a b l e i n equation 35 d i s t a n c e from w a l l of p i p e , f t + dimensionless d i s t a n c e from w a l l of pipe = ^ coor d i n a t e i n d i r e c t i o n of pipe a x i s or d i r e c t i o n of f l u i d flow, dimensionless 32 Greek L e t t e r s & - t h i c k n e s s o f momentum or thermal laminar s u b l a y e r , f t C H - eddy d i f f u s i v i t y of heat = E H / ^ C D , f t 2 / h r - eddy d i f f u s i v i t y of momentum = E ^ / ^ , f t ' / h r © - dimensionless temperature = ~ *w ~ ^ c yU. - molecular v i s c o s i t y , I b ^ / C f t K h r ) V - kinematic v i s c o s i t y , f t ~ / h r ~> ^ - d e n s i t y , l b ^ / f t T - shear s t r e s s , l b / f t " - denotes f u c t i o n or f u n c t i o n a l r e l a t i o n s h i p , dimens i o n l e s s S u b s c r i p t s i - denotes the i 1 t h e v a l u a t i o n o f the v a r i a b l e f - denotes f l u i d p r o p e r t i e s evaluated at the f i l m temperature = 1/2 ( t + t ) w >avg b,avg s - denotes smooth pipe v a r i a b l e s w - denotes c o n d i t i o n s at the inn e r w a l l o f the pipe h - denotes c o n d i t i o n s a t y = & 83 LITERATURE CITED (1) Knudsen, J.G., and Katz, D.L., " F l u i d Dynamics and Heat T r a n s f e r " , McGraw-Hill Book Company, Inc., 1958. (2) Lyon, R.H. Chem. Eng. Progr. , 4_7_, (2 ), 75 (19 51). (3) Nunner, W. , VDI - Forschungsheft Li55, S e r i e s B, 2_2, 5 (1956); A.E.R.E. L i b / T r a n s . 786, (1958). (i+) M a r t i n e l l i , R.C., Trans. A.S.M.E., 69 , 947 (1949 ). (5) M a t t i o l i , G.D., Forschung auf dem Gebiete des Ingenieurwesens, 11, ( 4 ) , 149 (1940); T r a n s l a t i o n , B o e l t e r , L.M.K., NACA TM No. 10 37, (1942). (6) Rouse, H., "Elementary Mechanics of F l u i d s " , John Wiley & Sons, Inc., New York, 1948. ( 7 ) von Karman, T,, J . A e r o n a u t i c a l S c i . , 1, 1 (1934) - as found i n Knudsen and Katz ( 1 ) . (8) Smith, J.W., and E p s t e i n , N., A.I. Ch. E. J o u r n a l , 3_» (2), 242 (19 57). (9) McAndrew, M.A., "The P r e d i c t i o n of Heat T r a n s f e r i n Rough Pi p e s " , M.A. Sc. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C., 1962. (10) Pohl, W., Forsch. Ing. Wes., 4, 230 (1933). (11) Cope, W.F., Proc. I n s t . Mech. Engrs. (London), 145, 99 (19 41). (12) Sams, E.W., NACA RM No. E52D17, (1952). 84 (13) Sams, E.W., United States Atomic Energy Commission T e c h n i c a l Information S e r v i c e E x t e n s i o n , Oak Ridge, Tenn., TID - 7529 (Pt. 1) Book 2, Physics and Mathematics, 390 (1957). (14) B r o u i l l e t t e , E.C. M i f f l i n , T.R., Myers, J.E., Paper presented at the ASME Annual Meeting, New York, N.Y; P r e p r i n t p u b l i s h e d by the American S o c i e t y of Mechanical Engineers, New York, M.Y., 57-A-47', (1957 ). (15) Dipprey, D.F., "An Experimental I n v e s t i g a t i o n of Heat and Momentum T r a n s f e r i n Smooth and Rough Tubes at Various P r a n d t l Numbers", Ph. D. T h e s i s , C a l i f o r n i a I n s t i t u t e of Technology, 1961; Dipprey, D.F., Sabersky, R.H., J e t P r o p u l s i o n Laboratory, C a l i f o r n i a I n s t i t u t e of Technologv, Pasadena, C a l i f . , T e c h n i c a l Report Ho. 32 - 269, (1962). (16) Savage, D.W., Myers, J.E., A.E.Ch.E. J o u r n a l , 9_, ( 5 ) , 694 (1963). (17) K e s t i n , J . , Richardson, P.D., I n t . J . Heat Mass T r a n s f e r , 6 , 147 (1963 ). (18) Nikuradse, J . , VDI-Forschungsheft, 361 (19 33); T r a n s l a t i o n , The Petroleum Engineer, 6, 164 (1940 ); 8 , 75 (1940 ); 9 , 124 (1940 ); T l , 38 (1940); T2, 83 (1940). (19) Nikuradse, J . , VDI-Forschungsheft 356 (1932). (20) Owen, P.R., Thomson, W.R., J . F l u i d Mech., 15, 321 (1963). (21) P r a n d t l , L., Phys. Z., 11, 1072 (1910). (22) N a t i o n a l P h y s i c a l Laboratory, "Modern Computing Methods", Her Majesty's S t a t i o n e r y O f f i c e , 1962. (23) S c h l i c h t i n g , H,, NACA TM Mo. 1718, (1949). APPENDIX 85 1, Computational D e t a i l s The equations used f o r c a l c u l a t i n g values of N u s s e l t number and temperature p r o f i l e a c c o r d i n g to the proposed method are b a s i c a l l y s i m i l a r . Both use double i n t e g r a l s of almost the same format, which have to be e v a l u a t e d f o r i d e n t i c a l boundary c o n d i t i o n s i n order to get values of the dependent v a r i a b l e . T h e r e f o r e , once a method has been devised f o r e v a l u a t i n g the i n t e g r a l , l i t t l e e x t r a e f f o r t i s r e q u i r e d to get values f o r both the N u s s e l t number and the temperature p r o f i l e over the work r e q u i r e d f o r s o l u t i o n of e i t h e r one alone. The complexity of the i n v o l v e d equations r u l e d out -any hope of o b t a i n i n g a n a l y t i c a l s o l u t i o n s to the i n t e g r a l s . The advent of high speed computing d e v i c e s , however, has rendered numerical i n t e g r a t i o n techniques i n t o powerful mathematical t o o l s . Thus, a FORTRAN language program was w r i t t e n u s i n g Simpson's r u l e (22) f o r n u m e r i c a l l y i n t e g r a t i n g the necessary equations l i s t e d i n Table 1. Values of Nu and the temperature p r o f i l e were subsequently computed f o r the given range of Re, f and Pr, u s i n g the IBM 1620 d i g i t a l computer at the U n i v e r s i t y of B.C., and the IBM 7090 d i g i t a l computer at the U n i v e r s i t y of Toronto. F i g u r e 12 gives a s i m p l i f i e d flowsheet of the program. A d e s c r i p t i o n of the main computing technique and of any l o g i c not immediately obvious to the reader f o l l o w s . 86 Input Data : Re , f , Pr Calculate : fs Calculate LIMITS Of FLOW REGIONS Define : CASE 1 CASE 2 CASE 3 / / (turbulent core ) Conv. Test CASE 3 CASE 2 " C A S E I ™ 2 / /(buffer zone ) Conv. Test f /(laminar sublayer) Conv. Test - [ /RVO*RJ -In R, Calculate Nu F i g u r e 12. Flowsheet of the Computer Program Calculate : TEMPERATURE PROFILE Output 87 The three independent v a r i a b l e s f o r which a s o l u t i o n i s r e q u i r e d are read i n t o the machine o f f punched data cards. The f i r s t q u a n t i t y c a l c u l a t e d by the program i s the smooth pipe f r i c t i o n f a c t o r . The e x p r e s s i o n due to Nikuradse (19), r e l a t i n g Re to the smooth pipe f r i c t i o n f a c t o r f o r t u r b u l e n t flow, i s used f o r t h i s purpose: — = 4 l o g (Re \[fs) _ 0.40 (34) Equation 34 cannot be s o l v e d e x p l i c i t l y f o r f s . An i t e r a t i v e method due to Newton-Raphson (22) i s t h e r e f o r e used. This method has the f o l l o w i n g general form, and i s s u i t a b l e f o r f i n d i n g roots of polynomials i n a systematic manner. Y v g(Xj) X i + 1 = X i " 1 TTX77 (35) Equation 34 i s s u i t a b l y rearranged and s u b s t i t u t e d i n t o equation 35. A f i r s t approximation f o r the value of f s i s given i n the program. The i t e r a t i o n i s then s t a r t e d and allowed to continue u n t i l two consecutive e v a l u a t i o n s of f s agree w i t h i n i, 1 0 , The flow p a t t e r n and the boundaries of the r e s p e c t i v e regions are c a l c u l a t e d next. I t has a l r e a d y been d e s c r i b e d how the t u r b u l e n t core v e l o c i t y p r o f i l e i n a rough pipe can i n t e r s e c t e i t h e r the b u f f e r zone or the laminar s u b l a y e r v e l o c i t y p r o f i l e , 88 or the u| = 0 a x i s , thus g i v i n g r i s e to the three cases. A means to t e s t as to which case a p p l i e s f o r any combination of Re and f i s necessary. The t u r b u l e n t core v e l o c i t y p r o f i l e equation i s f i r s t s o l v e d s i m u l t a n e o u s l y with the b u f f e r zone equation. I f the p o i n t of i n t e r s e c t i o n l i e s i n the range 5 < ys < 31, CASE 1 a p p l i e s and a l l three regions are present. The boundary between the b u f f e r zone and the laminar s u b l a y e r i s by d e f i n i t i o n at y | = 5. However, i f the p r o f i l e s do not i n t e r s e c t i n the given range, the t u r b u l e n t core equation i s s o l v e d together with the laminar s u b l a y e r equation, and the r e s u l t i n g e x p r e s s i o n t e s t e d f o r i n t e r s e c t i o n i n the range 2.5 <J yg ^ 5, A p o s i t i v e t e s t gives r i s e to CASE 2. In t h i s case the point of i n t e r s e c t i o n cannot be obtained e x p l i c i t l y . The Newton-Raphson i t e r a t i v e method d e f i n e d by equation 35 i s again used. The i t e r a t i o n i s stopped when co n s e c u t i v e e v a l u a t i o n s of the r o o t agree w i t h i n ± 0 .0002. In general the mathematical e x p r e s s i o n gives no r o o t s , or e l s e at lease two r o o t s . Only the one i n the given range and c l o s e r to : 5 _ s of i n t e r e s t . F i n a l l y , i f n e i t h e r CASE 1 nor 2 apply, the t u r b u l e n t core v e l o c i t y p r o f i l e n e c e s s a r i l y i n t e r s e c t s the u| = 0 axis-and d e f i n e s CASE 3. With a knowledge of the flow regions t h a t are present and t h e i r boundaries, i t i s now p o s s i b l e to proceed with the formal i n t e g r e t i o n of the r e q u i r e d i n t e g r a l s . As a f i r s t 89 approximation, the t u r b u l e n t core i s d i v i d e d i n t o a s p e c i f i e d number of increments (e.g. l i 0 ) , and the area of the f i r s t i n t e g r a l determined through the a p p l i c a t i o n of Simpson's r u l e . The accumulated area at a l t e r n a t e increments i s s t o r e d i n the computer memory as s u b s c r i b e d v a r i a b l e s . On completion of the f i r s t i n t e g r a t i o n , the s t o r e d values become the o r d i n a t e s i n the second i n t e g r a t i o n which i s c a r r i e d out by two subroutines u s i n g Simpson's r u l e . The r e s p e c t i v e area i s used f o r the u l t i m a t e c a l c u l a t i o n . o f N u sselt number and temperature p r o f i l e . The number of increments i n the r e g i o n i s then i n c r e a s e d by e i g h t and the preceding c a l c u l a t i o n repeated. The two areas are compared, and should agree w i t h i n i 0.6%. I f the t e s t i s ne g a t i v e , the number of increments i s in c r e a s e d by a f u r t h e r e i g h t , and the i n t e g r a t i o n i s repeated. I f convergence i s not achieved i n the t u r b u l e n t core when 56 increments are used, the r e g i o n i s d i v i d e d i n t o two sub-regions with a f i x e d boundary at R = 0.925. Each sub-re g i o n becomes a separate zone with an i n d i v i d u a l number of increments and i t s own convergence t e s t . In a s i t u a t i o n where the boundary between the t u r b u l e n t core and the b u f f e r zone takes place at values of R l e s s than 0.925, the a u x i l i a r y zone does not come i n t o e x i s t e n c e . The number of increments are then i n c r e a s e d i n the r e g i o n as a whole u n t i l convergence i s achieved. A maximum of 160 increments can be used i n each of the sub-regions of the t u r b u l e n t core. With proper choice of the i n i t i a l number of increments, convergence i s u s u a l l y 90 obtained a f t e r only one or two t r i a l s . The numerical i n t e g r a t i o n of the area i n the b u f f e r zone and i n the laminar s u b l a y e r i s b a s i c a l l y s i m i l a r to t h a t a l r e a d y d e s c r i b e d f o r the t u r b u l e n t core. The maximum all o w -able number of increments i s 160 i n the b u f f e r zone and 64 i n the laminar s u b l a y e r , g i v i n g a grand t o t a l of 544 increments t h a t can be used f o r e v a l u a t i o n of the o v e r a l l i n t e g r a l . When CASE 3 a p p l i e s , the area i n the w a l l l a y e r can be evaluated a n a l y t i c a l l y . The mathematical e x p r e s s i o n that r e s u l t s upon i n t e g r a t i o n between the l i m i t s i s given i n the a p p r o p r i a t e box of the flowsheet, F i g u r e l ? . The area c o n t r i b u t e d by each flow r e g i o n that i s present at any given combination of the independent v a r i a b l e s i s f i n a l l y summed. The t o t a l area i s used to c a l c u l a t e a value o f Nu a c c o r d i n g to equation 2, and the normalized temperature p r o f i l e a c c o r d i n g to equation 3. The r e s u l t s are p r i n t e d on the computer output d e v i c e , and the system i s ready to repeat the c y c l e from the beginning. 2. Test of Computer Program The program was t e s t e d f o r proper o p e r a t i o n and accuracy of r e s u l t s u s i n g two worked out examples i n Knudsen 91 and Katz (1) f o r a smooth pipe. Thus, f o r Example 15 - 2, p.447, i n which Pr = 7.74, R e = 30,000 and f = 0.00 585, the value of Nu i s r e p o r t e d to be 185; and f o r Example 1 5 - 1 , p.460, i n which Pr = 0.0038, Re = 30,000 and f = 0.00585, the value of Nu i s c a l c u l a t e d by the authors to be 8,80. The corresponding values c a l c u l a t e d by the computer are r e s p e c t i v e l y 202 and 7,95. The comparison i s only f a i r . However, our r e s u l t s were s u b s t a n t i a t e d by the work of McAndrew (9) who, i n a d d i t i o n to a s i m i l a r check with the above examples, a l s o computed the same by g r a p h i c a l i n t e g r a t i o n u s i n g a desk c a l c u l a t o r . A l l the r e s u l t s c a l c u l a t e d by v a r i o u s i n v e s t i g a t o r s are summarized i n Table 3. As can be seen, the s c a t t e r between the r e s u l t s c a l c u l a t e d by d i f f e r e n t means i s c o n s i d e r a b l e . However c e r t a i n evidence i s a v a i l a b l e that supports the c o r r e c t n e s s of the program. F i r s t , the r e s u l t s c a l c u l a t e d by d i g i t a l computation, u s i n g two completely independent programs and machines, agree w e l l w i t h i n the e r r o r a s s o c i a t e d with the convergence. Second, some computer computations c a r r i e d out with d i f f e r e n t number: of increments i n the three flow regions gave r e s u l t s f o r Example 1 5 - 2 which v a r i e d between 191.9 and 202.1. The l a t t e r f i g u r e i s obtained u s i n g the r e g u l a r convergence c r i t e r i o n f o r t h i s work. T h i r d , the data obtained by g r a p h i c a l i n t e g r a t i o n are s u b j e c t to e r r o r s a r i s i n g from both the method used to measure the areas under the c u r v e s , and from the f i n i t e number of increments used. 9 ? Table 3. SUMMARY OF DATA USED TO CHECK CORRECTNESS OF COMPUTER PROGRAM Conditions Method of C a l c u l a t i o n Nu Reported by Numerical i n t e g r a t i o n (IBM - 1520 computer) 202 This work Pr = 7.74 Numerical i n t e g r a t i o n (Alwac 111 -E Computer) 201 McAndrew (9) Re = 30,000 G r a p h i c a l i n t e g r a t i o n ( t r a p e z o i d a l r u l e ) 217 i t f=f s=0.00585 Same as previous method, but on a l a r g e r c o - o r d i n a t e s c a l e 196 I I G r a p h i c a l i n t e g r a t i o n (counting squares) 208 I I G r a p h i c a l i n t e g r a t i o n 185 Knudsen and Katz (1) Pr = 0.0038 Numerical i n t e g r a t i o n (IBM - 1620 computer) 7.95 This work Re = 30 ,000 Numerical i n t e g r a t i o n (Alwac 111 - E computer) 7.94 McAndrew (9) f=f s=0.00585 G r a p h i c a l i n t e g r a t i o n 8. 80 Knudsen and Katz (1) 3. Computer Program 93 a o CO O X 2 L L L L ! CD o cc _ J oc CO D C — o o • a o >- I—i •> Lt_ _ i C L Q C >• Q_ o C O *. a >- L U 1—1 LTl Q C L U C O L U L T \ •— or CNJ L O o L L D C O C o rn — U _ L L • • L L o CNJ 1— L O «\ CN] Q C L L L L \ O LTV — O J C O L O o r H r - l r-H -4" O o LO L L O 1— • DC * I + L U QC CO — L L O r -O QC t O O Z r - l — o < O L U _ J U QC Q C O L L t— O < s: II QC O L L L L Z . ^ II II It II I— I— C Q a x o z: z z : Q C U _ ' II L O L L C-J r—( LU QC * * < < O * o O C O • — <r II it L O L O L L L L L U \ QC L U L O L L I Q _ L O n L L CO CO » Q C L U + ~ '-0 r - i L L O — O o * o O LT I r-i » o L O U_ II o o O QC O L L O O e O L U QC I «x L O Q C U . r— QC * O o » a r\J f\J \ \ L U Q C C C L L L U * i n I o QC L O L U L L L O i n CQ r H II II — a. LU oe LO D Q L U L L L L L O - < L L L L QC r - t 0-< QC O L L QC O L O II QC L L I— QC r co C M LC I LU I O e co L L II \ co C O QC L O o L L * <j" r H L L ' Q X m II Lf\ 1 r— i n CNJ QC o cn » 1 L L r H « o r H L O + — L L L L '• X II 1— QC * QC TF L U L U * '•^  QC L O m e + •— X L L r H O — i n * «• _ CNJ O i n o \ CN) L U o o « O r H II r H • CNJ r H L O — L L o oc L L — L U L U — * O L U cn ac ° a C M L O II II Z CQ _ J Q . L U >-C O L L L U L O L J O CNJ O •> C O o » CNJ C O »- CNJ O '• -t CNJ _ J CO a-ca >- a co > - C L >-i i o » o a x L U II - H O co i n co — — C L L L L L y~ _—« •—< a x 10 r- • C O " * QC O L U r H QC " » — <!-\ r H L U L L CO Q . * - II >- CQ * co a O C L > * >-CNJ X - i n I i n — O CNJ « h~ <-t r H i — < O II Z 5 " l l 11 w K _ J CO QC O CO O 0_ QC C L L L Z L O > -L U i n *o o m CNi Q I i n L U — CNJ C O —I * -Q. co >- rci L U O — <f O • o —; * C L L U > -# i n | O CNJ CNI CNJ m CNJ o C O CNJ U • r— i n II C L U Z > I! h- — _ J C L L L C L D — ' > -CNJ 0 1 o o s ^ 1 3 4 U P Y L = Y P L - 2 . 5 * E * L 0 G ( Y P L ) - E * ( 5 . 5 + 2 . 5 * E L N - D ) D E Y L = 1.0 - 2.5 * E / Y P L ' ER= U P Y L / D E Y L ' '  ' Y P L = Y P L - E R I F ( A B S ( E R ) - 0 . 0 0 0 2 ) 3 3 . 3 4 , 3 4 2 3 Y P L = Y P B ( 3 3 R L = 1 . 0 - 2 . 0 * Y P L / ( R E * S O R T ( F S / 2 . 0 ) ) P R I N T 3 5 » Y P L » E > - RL 3 5 FORMAT ( 5 H Y P L = F 1 4 . 1 0 » 7 H C A S E 2 NB = 2 GO TO 5 6 4 0 Y P O L = ( D - 5 . 5 J / 2 . 5 - E L N RO = 1 . 0 - 2 . 0 * Y P 0 / < R E * S Q R T ( F S / 2 . 0 ) ) i P R I N T 4 5 >YPO » E > RO 4 5 FORMAT ( 5H YPO= F 1 4 . 1 0 > 7 H C A S E 3 / 5 H E = F 1 0 . 5 \ / 5 H R 0 = F 1 2 . 8 / / ) NB = 3 56 C A L L L I N K ( l ) END S F O R T R A N L I N K ( 1 ) D I M E N S I O N O R D ( 1 6 0 ) D I M E N S I O N TEMPI 1 6 0 ) > T E M T 1 4 0 ) » T E M B ( 4 0 ) »' T E X T < 4 0 ) » T E M L ( 1 6 ) • DIMENSI-ON R A ( 4 0 ) » R M ( 4 0 ) » R X ( 4 0 ) > R C ( 1 6 > COMMON F S » F R > R E » Y P B » Y P L > Y P 0 » R 6 J R L > R 0 » N B » N E J N F J M 0 T I N X T » N 0 B J N O L , NC = 1 GO TO 1 0 2 1 0 8 NC = 2 1 0 2 N A = 1 GO TO 1 0 3 1 0 4 NOT = NOT + 2 GO TO ( 1 0 6 J 1 0 7 ) » NC 1 0 6 I F ( N O T - 1 4 ) 1 6 6 , 1 6 6 , 1 0 9 1 0 7 I F ( N O T - 4 0 ) 1 6 6 » 1 6 6 » 1 2 9 1 0 9 GO TO ( 1 3 8 , 1 3 9 > 1 4 0 ) > NB 1 3 0 I F ( R b - 0 . 9 2 5 ) 1 0 7 , 1 0 7 , 1 0 8 1 3 9 1F (R L • .- 0 . 9 2 5 ) 1 0 7 , 1 0 7 , 1 0 8 YPO = E X P ( Y P O L ) 95 CO o co Lc. ft O r-o r -o cc m o z CO — LU o CC CO — z u z r—1 \ 1 r -i — i AC 1 i—i ( M AC i cn CM AC o o t—i i—i CL r-H r - a. r-H r- a o r— CO r— CO r— 00 CM Z o o + U o + u b + *\ u o — z \ < z \ < z \ < o o oD r— vO a O CL r— CM CL o o O O o 1 CO r—l m o C O o a , 1' CO CM LTV o o CO (XI lT\ O CO o e Q < f\! o o < CM o o < CM o o o o o 1 o O _ l — r—l \ rH o o <j" _ l —• r—l \ r-l o o — o 0 o © II II CM o o o o o ft * o o o II O o i—t o II ll o O O o o o II II O o o _ J O o o II ll o o O O z CC r- II h- r— u r— OC r— t— u r— r— Oi r— r— 1— OC II II II 1—1 — CO II II CO < II II II CO < i: II II CO < II 1—1 o o LL LU < o _ J o < CL o o II o CO o < a O O II o CO _ J < CL o O II _ ) I—1 cc o r— —« r— Z e> DC DC CO X CD X a cc OC _ J C O C D X CD X cD cc cr cr; CD X CD X cO cc > CO CO o o • e o o o II II o II L L Z II UJ •—i •—• r- > ~> —• O vO CO O o r-l O O (\J (XI CO CO CM o CM ( M O I— " I— I J J = 0 R3 = 2.0*H - • • • - - - -RTFR= S Q R T I F R * . 0 . 5 ) ' REM = RE * 0.2 .* R T F R HF = H / 3.0 - . . . ... . PRINT 125 • SPAC 125 FORMAT (26H TURBULENT CORE - SPACES =F6.1) DO 120 1 = 1 » LAST , 2 - - • • . F I = I ' ' • J = J + 1 R2 - H * F I VBAR2= RTFR * ( 2 . 5 * L O G ( 1 . 0 - R2) + 3.75) + 1.0' V'BAR3= RTFR % ( 2 . 5 * L 0 G ( 1 . 0 - R3) + 3.75) + 1.0 VR2 = R2 *VBAR2 . . . VR3 = R3 * VBAR3 SUM = SUM + HF •* (VR1 + 4.0*VR2 + VR3 ) SUMSQ= SUM ** 2 RCON = R3 * (PR * (REM * R3 * (1.0 - R3) - 1.0) + 1.0) IFt R C O N - R3) 360 » 361 , 361 360 RCON •- R3 • 361 CONTINUE ' I F ( I I ! 1 2 1 » 1 2 2 , 1 2 1 122 I I - 1 . . . . . . . GO TO 128 121 J J = J J + 1 R A ( J J ) = H * ( F I + 1.0) I I = 0 128 O R D ( J ) = SUMSQ / RCON TEM P ( J ) = SUM/RCON ' . . . . . R 3 = R 3 + 2 . 0 * H 120 VR1= VR3 CALL I G R A L (0 R D * J » VIN » . HF •» DOBIN) GO TO (104 > 163) > NA 163 DEVN =' (DOBIN - TEST) / DOBIN PRINT 62 5 > DEVN 625 FORMAT ( E 2 0 . 7 / ) I F ( A B S ( D E V N ) - 0.006) 129 , 129 • 104 129 SDOIN '= SDOIN + DOBIN ^ CALL ' TEMPROtTEMP , J , TEF , STUM , HF » TEMT » JA) - °> GO TO (132 » 133) 133 VIM =• ORD( J) ••• TEF = TEMP(J) TESUM = SUM N A = 1 GO TO 150 151 NXT = NXT + 2 IF(NXT - 40 ) 152 152 TEST = DOB IN SUM = TESUM N A = 2 150 LAST = 4 * NXT -SPAC= LAST + 1 GO TO (134 , 135 13A- H = ( RB - 0.92 5) GO TO 13 7. 135 H = (RL -GO TO 137 136 H = (RO -137 R l = 0.92 5 • H F = H / 3.0 VBAR1= RTF R * (2 VR1= V3AR1 R3 = 0.925 J = 0 I I - 0 JX = 0 PRINT 155 > 155 FORMAT (26H DO 199 I= F l = I J = J + 1 R2 = 0 .925- + VEAR2= RTFR VL3AR3= RTFR V R2 = R2 *VBAR2 VR3 = R3 * VBAR3 SUM = SUM + HF -* SUMSQ-- SUM *-* 2 NC 0.925) 0.925) Rl-2 , 0 152 158 136 ) SPAC SPAC SPAC N B 5*LOG(1.0 * H R l ) + 3.75) + 1.0 SPAC TURBo 1 CORE LAST > AUX 2 SPACES =F6.1) H * F I ( 2 . 5--LOG (1.0 (2.5*LOG< 1.0 R2 ) R3 ) . 75 ) 75 ) 0 0 (VR1 + 4.0*VR2 + VR3) LO R C O N = R 3 * ( P R * ( R E M * R 3 * ( 1 . 0 - R 3 ) - 1 . 0 ) + 1 . 0 ) I F I R C O N - R 3 ) 3 7 0 » 3 7 1 , 3 7 1 • 3 7 0 R C O N = R 3 3 7 1 C O N T I N U E ' I F ( I I ) 1 5 6 , 1 5 7 , 1 5 6 1 5 7 I I = 1 GO TO 1 6 0 1 5 6 J X = J X + 1 - • •• R X ( J X ) = 0 . 9 2 5 + H * ( F I + 1 . 0 ) I I = 0 1 6 0 O R D ( J ) = S U M S Q / R C O N T E M P ( J ) = S U M / R C O N R 3 = R 3 + 2 . 0 * H 1 9 9 V R 1 = V R 3 C A L L I G R A L t O R D » J > V I N , HF » DOB I N ) GO TO ( 1 5 1 » 1 5 9 ) 5 NA 1 5 9 D E V N = ( D O B I N - T E S T ) / DOB I N P R I N T 6 2 5 » D E V N I F ( A B S ( D E V N ) - 0 . 0 0 6 ) 1 5 8 » 1 5 8 , 1 5 1 1 5 8 S D O I N = S D O I N + D O B I N C A L L T E M P R O ( T EMP » J . T E F , S T U M , H F » T E X T , J D ) 1 3 2 I F ( R b ) 1 3 0 , 1 1 1 , 1 3 0 1 1 1 I F ( R L ) 1 1 2 , . 1 1 3 , 1 1 2 -1 3 0 V I N = O R D ( J ) " . T E F = T E M P ( J ) T E S U M = S U M ' - • • ' • • NA = 1 GO TO 1 1 0 . 1 1 9 NOB = NOB + 2 I F J N O Q - 4 0 ) , 1 7 7 , 1 7 7 , 1 7 8 111 T E S T = D O B I N S U M = T E S U M / . ' * • • . NA = 2 1 1 0 L A S T = 4 * NOB - 1 S P A C = L A S T + 1 . H= ( R L - R B ) / S P A C R l = RB H F = K / 3 . 0 R T F S = S O R T ( F S * 0 . 5 ) 1 4 5 R.EM = RE * 0 . 5 * R E F •= R E * 0 . 1 * V B A R 1 = R T F S * (5 V R 1 = V.BAR 1 * R l R 3 = R B •+ 2 . 0 * H J = 0 11 = 0 K K = 0 P R I N T 1 4 5 '•> S P A C F O R M A T 1 2 6 H B U F F E R 3 50 3 5 1 1 4 2 1 4 1 R T F S R T F S 0 - L O G ( R E M * ( 1 . 0 R l - 3 . 0 5 ) Z O N E SPACE'S = F 6 . 1 ) 1= 1 » L A S T -DO 1 7 0 F l "= I J = J + 1 •R2 = RB + H * F l V B A R 2 = R T F S * ( 5 . V B A R 3 = R T F S *- (5 V R 2 -= R 2 * V B A R 2 0 * L O G ( R E M 0 * L O G ( R E M ( 1. ( 1 - sR2 ) ) -\^3 ) ) 0 5 ) 0 5 ) V R 3 = S U M = 5 U M S Q ; R C O N : I F ( R O R C O N : R 3 * S U M = SUf-' = R3 ) M -= R 3 V B A R 3 + HF * * * 2 * ( PR • V R 1 + ' 4 . 0 * V R 2 + V R 3 ) 0 - R 3 ) ( R E F R3 3 5 0 - 3 5 1 R 3 * (1 •> 3 5 1. - 1 . 0 1 . 0 1 4 8 1 7 0 1 3 1 C O N T I N U E • I F ( I I ) 1 4 1 » 1 4 2 > 1 4 1 I I = 1 GO TO 1 4 8 K K = KK + 1 R M ( K K ) • = R B + H * ( F I I.I •= 0 O R D ( J ) = S U M S Q / R C O N T E M P ( J ) = S U M / R C O N R3= R 3 + 2 . 0 * H V R 1 = V R 3 C A L L I G R A L t O R D » G O T O ( 1 1 9 , 1 3 1 ) , + 1 . 0 ) J MA V I N > HF » D O B ' I N ) ( D O B I N - T E S T ) / D O B I N 7 8 , 1 7 8 , 1 1 9 D E V N P R I N T 6 2 5 » D E V N I F ( A B S ( D E V N ) - 0 . 0 0 6 ! CO CO 1 7 8 S D O I N = S D O I N + D O B I N C A L L T E M P R O ( T E M P > J J T E F > S T U M » H F » T E M B > J B ) 1 1 2 V I N = O R D ( J ) T E F = T E M P ( J ) T E S U M = S U M N A = 1 GO TO 7 1 0 1 8 1 N O L --- N O L + 2 . . . . . . . . I F ( N O L - 1 6 ) 1 8 3 » 1 8 3 » 1 8 4 1 8 3 T E S T = D O B I N S U M = T E S U M N A -2 7 1 0 L A S T = 4 * N O L - 1 S P A C = L A S T + . 1 . • • • H= ( 1 . 0 - R L ) / S P A C H F = H / 3 . 0 Rl = R L F S M = 0 . 2 5 - ' F S V B A R 1 = F SM * RE * ( 1 . 0 - R l ) V R 1 = R l * V B A R 1 R 3 = R L + 2 . 0 * H J = 0 1 1 = 0 L L = 0 , P R I N T 1 6 5 > S P A C 1 6 5 F O R M A T ( 2 6 H L A M I N A R S U B L A Y E R - S P A C E S = F 6 . DO 1 9 0 1= 1 J L A S T , 2 F l = r . J = J + 1 R 2 = R L + H * F I V B A R 2 = F S M * RE * ( 1 . 0 - R 2 ) • V 3 A R 3 = F S M * R E * ( 1 . 0 - R 3 ) V R 2 = R 2 * V B A R 2 V R 3 = R 3 * V B A R 3 S U M = S U M + H F * ( V R 1 + 4 . 0 * V R 2 + V R 3 ) S U M S Q = S U M 2 I F ( I I ) 1 9 1 , 1 9 2 , 1 9 1 1 9 2 I I = 1 G O TO 1 9 8 1 9 1 L L = L L + 1 R C ( ' L L ) = R L + H • ( F l + 1 . 0 ) 11 = 0 1 9 3 O R D ( J ) = S U M S Q / R 3 T E M P ( J ) = SUM /• R 3 R 3 = R 3 + 2 . 0 * H 1 9 0 V R 1 = V R 3 C A L L IGR A L ( O R D , J > V I M > H F » D O B I N ) GO TO ( 1 8 1 ? 1 8 2 ) , NA 1 8 2 D E V N = ( D O B I N - T E S T ) / D O B I N P R I N T 6 2 5 , D E V N I F ( A S S ( D E V N ) - 0 . 0 0 6 ) 1 8 4 , 1 8 4 ', 18 1 1 8 4 S D O I N = S D O I N + D O B I N C A L L T E M P R O ( T E M P , J , T E F , S T U M >HF , T E M L , J C ) P R O D = 0 . 5 / S D O I N : P R I N T 1 8 5 , S D O I N 1 8 5 F O R M A T ( 1 7 H D O U B L E I N T E G R A L - E 2 0 . 7 - / ) P R I N T 2 3 5 » P R O D •> PR > R E 2 3 5 F O R M A T ( 1 2 H N U S S E L T N O = F 1 4 . 5 , 9 H . PR N 0 = F 1 0 . 4 , 9 H RE - N F •= 2 GO TO 2 4 0 1 1 3 N F = NF + 1 GO TO ( 7 50 , 7 5 1 ) , NF 7 51 S D O I N = S D O I N - S U M S Q * L O G ( R O ) S T U M = S T U M - SUM * L O G ( R O ) 7 5 0 P R O D = 0 . 5 / S D O I N P R I N T 1 8 5 » S D O I N P R I N T 2 35 , P R O D > P R » R E • P R I N T 2 45 2 4 5 F O R M A T ( 1 4 H Z E R O N E T FLOW / / ) 2 4 0 - P R I N T 5 2 5 5 2 5 F O R M A T ( 2 1 H T E M P E R A T U R E P R O F I L E / / ) • • •• P R I N T 5 3 5 5 3 5 F O R M A T ( 3 0 H T E M P Y / ) DO 5 4 0 1 = 1? J J • P R O F = 1 . 0 - T E M T ( I ) / S T U M Y = 1 . 0 - RA < I ) j P R I N T 54 5 , P R O F ' , Y .545 F O R M A T ( 2 E 1 8 . 7 ) ; . 0 = E 1 5 . 3 / / 5 40 C O N T I N U E GO TO ( 5 7 1 » 578 ' ) » NC '578 DO 5 7 0 1 = 2 , J X » 4 P R O F = 1 . 0 - T E X T ( I ) / S T U M • Y = 1 . 0 - R X ( I ) • - . P R I N T 5 4 5 ? P R O F , Y 5 70 C O N T I N U E 5 7 1 GO TO ( 5 7 2 , 5 5 2 , 5 6 2 ) , NB 5 7 2 DO 5 5 0 1=2 > KK , 3 P R O F = 1 . 0 - T E M B ( I ) / S T U M • Y = 1 . 0 - R M( I ) P R I N T 5 4 5 , P R O F , Y 5 50 C O N T I N U E 5 5 2 DO 5 6 0 I .= l . , L L P R O F = 1 . 0 - T E M L ( I ) / S T U M Y = 1 , 0 - R C ( I ) P R I N T 5 4 5 , P R O F , Y 5 6 0 C O N T I N U E 5 6 2 C O N T I N U E I F ( NF - 2 ) 1 1 3 , -58 1 - ,• 5 8 1 • 5 8 1 C O N T I N U E P R I N T 5 7 5 5 7 5 F O R M A T ( / 3 2 H E N D OF T H I S S E T OF C A L L L I N K ( O ) E N D S * , E V A L U A T I O N OF D O U B L E I N T E G R A L S F C R T R A N S U B R O U T I N E . I G R A L ( A R R A Y , I N T D I M E N S I O N A R R A Y ( 8 0 ) TOT = 0 . 0 L L 1 = I N T - 1 ' DO 4 0 0 M= .1 , 2 DO 4 10 N= M , L L 1 , 2 4 1 0 T O T = T O T + A R R A Y ( N ) I O N S / / / ) F I R S T , H F , D O B I N ) a 4 0 0 TOT= 2 . 0 * TOT DOS I N = ( F I R S T +-.TOT + A R R A Y ( I N T ) ) * HF * 2 . 0 P R I N T 4 0 5 > DOS I N 4 0 5 F O R M A T ( 2 3 H D O U B L E I N T E G R A L C O M P = E 1 4 . 7 ) R E T U R N • E N D $ * T E M P E R A T U R E P R O F I L E C A L C U L A T I O N S ' S F G R T R A N S U B R O U T I N E T E M P R O ( C O L » I N P U T , F R O N T > T U M > HF , TOR » J M ) D I M E N S I O N - C O L ( 8 0 ) » T O R ( 4 0 ) LMO = I N P U T - . 1 J M = 1 TUM = TUM + 2 . 0 * HF * ( F R O N T + 4 . 0 * C 0 L ( D + C O L ( 2 ) ) T O R ( l ) = TUM DO 5 1 0 J S •= 3 » L M O - , 2 TUM = TUM + 2 . 0 * HF * ( C O L ( J S - l ) + 4 . 0 * C O L ( J S j + C O L ( J S + l ) ) J M = J M + 1 5 1 0 T O R ( J M ) = TUM R E T U R N END r-1 O O J Figure 2. P l o t Showing the Range of A p p l i c a t i o n of CASES 1, 2 and 3 on f - Re Coordinates Figure 5. Influence of Pr on the Basic Heat T r a n s f e r R e s u l t s Figure 4 e . Basic Heat T r a n s f e r Results C a l c u l a t e d According to Proposed Method: Nu versus Re 

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