THE PREDICTION OF HEAT TRANSFER IN ROUGH PIPES by MURRAY ALEXANDER Mc ANDREW B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, I960 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department of CHEMICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department ofr MVHJ^JQ^ C%f£^nPJ2U*i{; The University of British Columbia, Vancouver 8, Canada. Date vi ABSTRACT An e v a l u a t i o n o f methods f o r p r e d i c t i n g t u r b u l e n t h e a t t r a n s f e r rough p i p e s has been made w i t h t h e i n t e n t i o n o f o b t a i n i n g a b e t t e r in under- s t a n d i n g o f the t r a n s f e r p r o c e s s e s i n v o l v e d and o f p r o v i d i n g a g e n e r a l d e s i g n e q u a t i o n , v a l i d f o r a l l t y p e s o f roughness shapes and d i s t r i b u t i o n s . The e q u a t i o n s o f M a r t i n e l l i , Nunner, and M a t t i o l i , a l o n g w i t h an e m p i r i c a l method suggested b y E p s t e i n , have 'been t e s t e d u s i n g t h e a v a i l a b l e data. experimental I n a d d i t i o n , p a r t i c u l a r a t t e n t i o n has been g i v e n t o a p r o p o s e d method w h i c h makes use o f t h e v e l o c i t y p r o f i l e e q u a t i o n s o f Rouse and von Karman i n L y o n ' s fundamental e q u a t i o n , f o r the N u s s e l t number. The r e s u l t s i n d i c a t e t h a t t h e p r o p o s e d method i s n o t s u c c e s s f u l , largely because o f i g n o r a n c e o f v e l o c i t y c o n d i t i o n s near t h e w a l l s o f rough p i p e s . M a t t i o l i ' B e q u a t i o n a l s o does n o t g i v e a s a t i s f a c t o r y experimental r e s u l t s . c o r r e l a t i o n of E p s t e i n ' s e m p i r i c a l method, w h i c h , i n t h e d i m e n s i o n l e s s g r o u p s , uses f r i c t i o n v e l o c i t y and e q u i v a l e n t pertinent sand-roughness h e i g h t o f t h e roughness elements i n p l a c e o f the average f l u i d v e l o c i t y and the pipe diameter, r e s p e c t i v e l y , investigation. shows p r o m i s e b u t r e q u i r e s further N u n n e r ' s e q u a t i o n and M a r t i n e l l i ' s ( s i m p l i f i e d ) e q u a t i o n g i v e good p r e d i c t i o n o f t h e e x p e r i m e n t a l r e s u l t s and are recommended f o r use a t p r e s e n t , p r o v i d i n g 0.5 < Tr < J..0. The s u c c e s s o f t h e s e latter e q u a t i o n s g i v e s s u p p o r t t o t h e h y p o t h e s i s t h a t the f l u i d a d j a c e n t t o a rough w a l l i s probably i n laminar motion. U s i n g N u n n e r ' s model o f t h e f l o w c o n d i t i o n s i n rough p i p e s , equations have been d e r i v e d f o r p r e d i c t i n g t e m p e r a t u r e p r o f i l e s from v e l o c i t y data. profile G e n e r a l l y , t h e a b s o l u t e agreement between p r e d i c t e d p r o f i l e s and N u n n e r s e x p e r i m e n t a l p r o f i l e s i s good, b u t the i n f l u e n c e s o f Re and e s p e c i a l l y 1 f are n o t t o o w e l l a c c o u n t e d f o r . N u n n e r ' s c o n c l u s i o n t h a t t e m p e r a t u r e and v e l o c i t y p r o f i l e s i n rough p i p e s are n o t s i m i l a r i s s u b s t a n t i a t e d by t h e results. V ACKHOWLDEGEMENT It has been a privilege for me to have worked with Dr. Norman Epstein and I wish to thank .'.him for the assistance he has given in supervising this work. I am very grateful for a Bursary from the National Research Council of Canada and for the. generous financial assistance of the Department of Chemical Engineering. Lastly, I greatly appreciate the cooperation of the U.B.C. Computing Centre in providing the use of the computing f a c i l i t i e s for the calculations involved in this investigation. TABLE OF CONTENTS ' List of Figures' iv Acknowledgement v Abstract "" vi 1 Introduction Description of the Proposed Method for Predicting Heat Transfer in Rough Pipe8 k 1. k Development of the Method Table 1. Summary of Equations Required for Calculating 9 Values of Nu by Equation (2) 2. Computational Details Table 2. 12 Check on Correctness of Computer Calculation of Values of Nu by Equation (2) 17 Description of Other Methods for Predicting Heat Transfer in Rough Pipes 1. Equation of Martinelli l8 2. Equation of Nunner 18 3. Equation of Mattioli 19 k. Empirical Method of Epstein 20 Results 22 1. 2. Basic Results Obtained from the Proposed Method 22 Comparison of Experimental Results with Results Predicted from the Proposed Method and from the Equations of Martinelli and Nunner 22 (a) Comparison with the Results of Smith and Epstein 22 (b) Comparison with the Results of Sama 3^ (c) Comparison with the Results of Edwards and Sheriff 3^ (d) Comparison with the Results of Lancet 3^ (e) Comparison with the Results of Nunner 1+1 (f) Comparison with the Results of Cope kl iii 3. Test of the Effectiveness of Mattioli's Method for Correlating Rough-Pipe Heat Transfer Results k. Test of the Effectiveness 42 of Epstein's Empirical Method for Correlating Rough-Pipe Heat Transfer Results Discussion of Results 1. Values of Nu Calculated by the Proposed Method 2. Comparison of Other Methods of Prediction with the Experimental Results 42 48 48 55 Conclusions and Recommendations 60 Nomenclature 6l Literature Cited 65 Appendix: 68 Temperature Profiles in Rough Pipes LIST OF FIGURES Figure 1 Sketch of Various Possible V e l o c i t y P r o f i l e s i n Rough Pipes 10 Figure 2 Plot of the Function Y on (Re,f) Coordinates 13 Figure 3 Figure h ' Flowsheet Showing Logic f o r Computer Program f o r Calculating Values of Nu According to the Proposed Method 15 Epstein's Empirical Plots of the Results of Smith and Epstein 21 Figures 5a to 5e Figure 6 Figure 7 Figures 10a Figure 11 28 Some Further C h a r a c t e r i s t i c s of the Proposed Prediction Method: Occurrence of Maxima with Increasing f and Re 29 Comparison of Predicted Results with Results of Smith and Epstein Experimental Comparison of Predicted Results with Experimental of Sams, Square-Thread-Type-Roughness and 10b 30 Results 35 Comparison of Predicted Results with Experimental Results of Sams, Wire-Coil-Type-Roughness Comparison of Predicted Results with Experimental of Edwards and S h e r i f f Results Comparison of Predicted Results with Experimental of Lancet Results Figures 13a Figure lh Figure 15 to 13c Comparison of Predicted Results with Results of Cope h0 Experimental 43 Plot of Rough-Pipe Heat Transfer Results on M a t t i o l i ' s Coordinates 1+6 Plot of Rough-Pipe Heat Transfer Results on Epstein's Coordinates / \ ^7 2 RVdR J IJ Figure l6 36 39 r Figure 12 23 Basic Results Calculated from the Proposed Prediction Method: Influence of Pr Figures 8a to 8d Figure 9 Basic Results Calculated from the Proposed Prediction Method: Nu as a Function of Re Sketch of several the Influence of Re — — R ^ ' -versus-R^ Curves Showing 50 k Figure 17 Results of More Detailed Calculations Based on the Proposed Prediction Method: Location of Points of I n f l e c t i o n Figure l8! - Sketch of''Nunner's Model of "the Flow Conditions -in-Rough Pipes Figures 19a Figure 20 to 19e Comparison of Predicted and Experimental and V e l o c i t y P r o f i l e s i n Rough Pipes Sketch of C h a r a c t e r i s t i c s Exhibited by Equation Temperature •"' (7I+) 52 71 ',, 75 79 1. INTRODUCTION In the design of heat exchange equipment and piping systems for conveying .1 J . .1 fluids i t is commonly found that for various duties (e.g. high pressure, high temperature, corrosive fluids, etc.) and/or for economic reasons, i t is necessary to use piping which, due to the method of fabrication, has rough walls. For the case of a fluid flowing turbulently through such pipes, the effect, of the roughness elements is to increase both the pressure drop per unit length and the rate of heat transfer above the corresponding values for smooth pipes of the same diameter, using the same average fluid velocity. A vast amount of work on the overall pressure drop characteristic of smooth and rough pipes has resulted in methods, well summarized by Knudsen and Katz. (l), p.171-179^ able accuracy. from' which these characteristics can be predicted with reasonMethods for correlating or predicting heat transfer results for turbulent flow in smooth tubes are given in several works, including those of McAdams (2) and Knudsen and Katz ( l ) ; however, no methods are given which successfully account for the influence of a l l types of wall roughness on heat transfer. A few methods have been proposed, but they have each been tested only for particular types of roughness, and their applicability in predicting or correlating results for other types of wall roughness has not been thoroughly assessed. A successful method - one which applies for a l l roughness geometries and distributions - would be useful not only for general industrial design, but also for the design of special heat exchange apparatus for nuclear reactors. In the operation of these reactors high temperatures and unusually large heat fluxes are encountered and i t is essential to achieve high heat transfer rates. To this end, several recent studies (3, k, 5, 6, 7) have been devoted to improving heat transfer rates by using artificially-roughened surfaces. In addition to the practical problem of predicting heat transfer in rough pipes, there is an academic Interest as well. The success of a method would verify the theory on which i t was based, or at least strengthen some of the concepts involved In the theory; its failure would perhaps point out errors in the theory and modifications necessary to describe the phenomena involved correctly. In either case, the mechanisms of the turbulent transfer of heat and momentum would be better understood. '"^yi Thus, i t was the object of this work to evaluate methods for predicting heat transfer in rough pipes. Particular emphasis has been given to a proposed method which uses the velocity profile equations of von Karman (8) and Rouse ( 9 ) in the fundamental equation for the Nusselt number derived by Lyon (lO). method was chosen for several reasons. This In the f i r s t place, Smith and Epstein ( l l ) found quite good agreement between their experimental results and results-, predicted by substituting rough-pipe friction factors into the smooth-tube equation of Martinelli (12). It is known that Icon's and Martinelli's equations yield similar results for smooth tubes; the small difference between predicted values (generally less than ten per cent) can be attributed to the fact that Lyon's equation is slightly more rigorous than Martinelli's. Moreover, In the derivation of his equation, Martinelli used smooth-tube velocity profile equations which therefore, s t r i c t l y speaking, restrict the use of his equation accordingly. Lyon's equation, on the other hand, is not so restricted and the use of rough-pipe velocity profile data is perfectly permissible. Other prediction methods, nearly a l l of which are based on the analogy between turbulent heat and momentum transfer, are available. Several of these, as pointed out by Smith and Epstein ( l l ) , provide l i t t l e hope of being generally successful. They do not show any dependence of J H on the Reynolds number after the friction factor has become independent of Re, whereas experimental evidence suggests that there is such a dependence. Several methods, however, namely those of Martinelli (12), Nuteer (13), and Mattioli (l*0, do conform to this requirement and therefore merit testing with a l l the available data. In addition to these, an empirical method due to Epstein (15) for correlating rough-pipe heat transfer results has shown promise. Calculations have been done using ther.above methods. The results have been compared with one another and with the available experimental results in order to determine which, i f any, are successful. The separate but intimately related topic of temperature profiles in rough pipes has been given some attention in an appendix to this thesis. Equations for the prediction of such profiles have been derived using Nunner's (13) model of turbulent flow in rough pipes, and comparison of some predicted profiles with profiles measured by Nunner has been made. k. DESCRIPTION OF THE PROPOSED METHOD FOR PREDICTING HEAT TRANSFER IN ROUGH PIPES 1. Development o f the Method From t h e o r e t i c a l considerations of the problem o f t r a n s f e r r i n g heat t o a f l u i d flowing i n a pipe, Lyon ( 1 0 ) derived an equation f o r the Nusselt number which i s v a l i d f o r a l l flow regimes and w a l l conditions. The assumptions made i n the d e r i v a t i o n are: (a.) Heat t r a n s f e r r e d by the process o f thermal r a d i a t i o n i s n e g l i g i b l e . (b) Macroscopic steady state e x i s t s . (c) F l u i d p r o p e r t i e s ore constant, independent o f temperature, pressure, and hence o f p o s i t i o n i n the pipe. (d) At any c r o s s - s e c t i o n in. the pipe, q u a n t i t i e s such as temperature, pressure, and v e l o c i t y are i n r a d i a l symmetry. (e) There are no end e f f e c t s , i . e . v e l o c i t y and normalised temperature p r o f i l e s are fully-developed and do not change along the length o f the (f) pipe. The heat f l u x a t the w a l l i s constant along the length of the pipe. " d ^ (f^*) A consequence o f these assumptions i s that = = constant. The d e r i v a t i o n begins w i t h the d e f i n i n g equation f o r the heat t r a n s f e r coefficient: % - hA (t -t ) w w b ( An enthalpy balance on a d i f f e r e n t i a l annulus of f l u i d of radius r 1 ) _ yields an expression f o r the d r i v i n g force ( t - t ) which i n v o l v e s the temperature w d i s t r i b u t i o n ( t - t ) across the pipe. w D By considering the r a d i a l heat f l u x i n t o a c y l i n d r i c a l element o f f l u i d of radius r<j_ (r^ < r ) , an* independent w equation f o r (tw-t) i s obtained. L a s t l y , an o v e r a l l enthalpy balance r e l a t e s <j_ t o the o v e r a l l temperature r i s e o f the f l u i d . w When these equations are s u b s t i t u t e d i n t o equation ( l ) , an expression i s obtained 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 o f the radius, the time-average p o i n t v e l o c i t y and Tliis expression i s . rendered dimensionless a n d the t o t a l r a d i a l conductivity. simplified to provide N the equation _ U 2 To evaluate the dimensionless in smooth Rouse's equation the region velocity that for data + i s necessary c a n be determined /r between exist the laminar U/ * + u the edge f o r rough results functions the velocity from an expression of Nikuradse profiles due t o (lb, Rouse Yj). (3) 3.75 + core i n the absence o f v o n Karman + (8) and the pipe of such data wall, no i t was assumed applied: + . , y 00 <5 leyer, i / o - 3 . 0 5 + 5.olny Tlie t e r m K / k c a n b e r e l a t e d the t o t a l radial and the r a d i a l + , 5<y <30 eddy t o the velocity _ I + En/f C | _ € H k/ C ? K = P I + i s p _ I € tyf> + M/f « M ln.Ea.Pr € H M/f k/fCp the follows. By molecular M p + as Thus, k/j»c | profile t h e sum o f conductivity. K _ k EH " k + (5) + conductivity k or ) sublayer f o r the buffer Furthermore ( K / k ) and V as core, of the turbulent pipes; equations + conductivity t o know For the turbulent 2.5ln(y J = the smooth-tube definition, 2 ^ on the experimental u =y and ( i s u For radius. and rough pipes i s based i t number: A R.(K/k) equation(2) of (6) w h i c h 1 \ f o r the Nusselt _ | ? -f- £H €H f € /i M Pr (6) 6. For one-dimensional turbulent flow, the eddy viscosity, E^j, is defined by (7) ^7 But according to the law of linear stress distribution, T . T w f e ) . T R (a) w Substituting equation (8) into equation (7) and this result into equation (6) gives (9) Mdu - d y Equation (9) can be put into a more useful form by making use of the definition of the friction velocity, u* : U = (10) It is shown in many standard texts on fluid flow, for example," Knudsen and Katz ( l ) , that this definition is equivalent to U1 u* _ (ll) 2 Combining equations ( 1 0 ) and ( l l ) , solving for ^w<jc J 8 X 1 ( 1 substituting the result into equation (9) yields -K k = I + 1 I* Pr <M U (f/2)(?)(R) _ a (12) 4y The ratio of the eddy diffusivities of heat and momentum, a separate problem. €fl. , poses In analyses of turbulent heat transfer i t is commonly assumed that this ratio is unity, although there is experimental evidence to show that this assumption is somewhat in error. Azer and Chao ( l 8 ) , in reviewing the experimental work, indicate that values of and less than unity are reported. It appears that JLH. = £JL greater than and one would expect that wall roughness should also be included in this relationship. Nevertheless, not enough experimental data are available to determine the form of the function, so that for the purpose of the present work, the value of has again been taken as unity. To evaluate equation (2) for various conditions, i t is necessary to have equations for K/k for the particular regions of the velocity profile as defined by equations (3) to (5), and also to express the velocity profile V = V(R). equations in the form u _ u V _u_ y £-r Noting that From equation ( l l ) , , so that = _ I -R equation (3) becomes V = 2.5 f In(l-R) + 3.75 (13) In the turbulent core, then, jRVdR = ^75j?+ljykdR + 2.5 £jRln(l-R)dR This expression integrates by standard procedures to yRWR = |?(375|I+-lj + ^.5jTJ^I-Rj|(l-R)ln(l-R) _ Ind-R) ,(I-R) x I To evaluate K/k in the turbulent core, .du is required. From equation (3), u + = _si 2.5 . U In v x UfT t J _ 3.75 UfT Therefore u _ 2.5 U NJ 2 X ll 4. 3-75 f u 2 constant (ih ) 8. and du _ 2.5UJT/2 In - I Substituting this result and into equation (12): 6- K k (i-XFV) U C f / 2 ) C ? ) ( R ) a _ | Pr k (ReNJra(i-RKf« _ " (15) By noting that (16A) expressions for V, RVdR, and K/k are similarly obtained for the buffer The equations are summarized in Table 1. layer and laminar sublayer. It w i l l be noted that in deriving these equations, no assumptions involving roughness geometry and distribution have been made; the sole measure of roughness is the overall value of f at a given Re • The values of R for which the equations outlined in Table 1 are applicable must now be determined. Solving equation (l6A) for y_ , substituting the result into equation (3), and making use of equation ( l l ) gives u _ + 2.5 In y + - In 2.5 + 3.75 (22) This equation can be written as u* = 5.5 + 2-5 In y + - 2.5ln^Re f j - 1.75 But U + = 5.5 + 2.5 In y + is the equation of Nikuradse (l6) smooth pipes. (23) for the turbulent core velocity profile for 9-. Table 1. SUMMARY OF EQUATIONS REQUIRED FOR CALCULATING VALUES OF Nu BY EQUATION (2) Turbulent Core V = 2.5J£ In(l-R) /RvaR | ( .75g l = K 3 V = + 1 + ) Pr = k Buffer r + + t. 3.7SJ? (,sjf)(.-R) (13) Q-RMnfl-R) - U l - R ) - (itR) + I (R.)(JTS)(1-RVR^ .+. costant n . 04) 05) Layer 5.oJIIn(l-R) + j f [ * RWR _ f | 5. ( & | | ) -3.05J o l n + 0 , n (^J|) - 3 0 (16) 5 ^.oj£J(l-R)"(l-RlWl-R> - In(l-R) - (l-W (ReY^Xl-RUR) _ "I consitant 07) Tt (18) 10 Laminar S u b l a y e r V = RVdR = Kk fReCl-R) 4 R*(3-2R) f Re IO (19) (20) constant t T T (20 Limits of application of these equations are explained i n the text, beginning on page 8 These equations are quadratic i n R and i t i s therefore possible to obtain K/k = 0 i n the region to 0 ^ R < 1.0. In order that values of K/k correspond the p h y s i c a l l y r e a l s i t u a t i o n , i t i s necessary to impose the condition K/k > 1 . 0 . The actual equation derived Is K/k = 1 + Pr(R-l). In t h i s region, however, f l u i d motion i s by d e f i n i t i o n laminar (hence E^ = 0 ) ; therefore K/k = 1 . 0 . ib. (u \ )^ + Therefore = W u pipC ~ 2 moothf v 5 , n ( % Jr ) r - '- 75 Thus, on a plot of u versus (log) y" (see Figure l ) , the turbulent core line + 4 for rough pipes has the same slope as that for smooth pipes but an intercept whose value is a function of both Re and f. 'For a fixed Re, J= < ^ 24 22 1 I and I I I I l I 2.5ln(&iJS) > 2.5ln(J^|E) I I I I l I I I I I I I II - 20 eqn.(22)or{23) 16 - U 10 eqn(5) F i g u r e 1. S k e t c h o f V a r i o u s 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 i3> the rough-pipe line must f a l l below the smooth-pipe line. Depending on the magnitude of Re and/or f, three separate cases arise. Case (l) Intersection of the curve for the turbulent core with that for the buffer layer. Solving equations (5) and (22) simultaneously gives 2.72 y (25) Solving equation (l6A) for R, R = (26) Rejf/i" and using the value of y obtained. + calculated from equation (25), a value R=R, is The boundary between the buffer layer and the laminar sublayer is defined to be at y + = 5, and by inserting this value into equation (26), the corresponding value of R, designated as Rg, is obtained. Thus, for this particular case, the turbulent core extends from R=0 to R=R-j_ , the buffer layer from R=R- to' R=R , and the laminar sublayer from R=R to £=1.0. L Case (2) 2 2 Intersection of the curve for the turbulent core with that for the laminar sublayer. Solving equations (k) and (22) simultaneously gives 2.5 In f - f = 2.5 In/Be I\ - - 3.75 (27) In this case, no buffer layer exists) the turbulent core extends from R=0 to R=R]_ (calculated now from equations (27) and (26)) and the laminar sublayer from R=R to R=1.0. 1 Case (3) u + =0. Intersection of the curve for the turbulent core with the curve In this case, y + is calculated from equation (22) with u zero, that is from ext _1.5 + set equal to In order that u =0, u must be zero since U and are finite. + The turbulent core again extends from R=0 to R=R}_, with R-j_ calculated from equations ( 2 8 ) and ( 2 6 ) , and from R=R]_ to R=1.0 the fluid is stagnant. The possibility of this condition is also mentioned by Hinze ( l 9 ) > p.U8U. To facilitate calculation, i t would be desirable to be able to determine which case applies for given values of Re and f. For case ( 2 ) to apply, values of Re and f must be such that y* < 5- The lower limit is found by determining the value of y + at which the turbulent core curve is tangent to the laminar sublayer curve. _ 'turbulent cor» y Therefore, y + Thus, from equations (22) and (k), 2.5 _ (du + .0 \d V / laminar / Sublayer = 2.5 If case ( 2 ) is to apply, then 2.5 < y + ^ 5- Substituting these limite:; into equation ( 2 7 ) : 2.5 In(2.5) -2.5 +3.75 > i.sln/feelf^ - _ L Letting Y = 2 5 > 2.5ln(5) -5.0 + 3.75 , r i m (29) - then for case ( 2 ) to apply, 2-77 < Y < 3-54. and i f Y > 3-54, case ( 3 ) applies. If Y<2-77, case (l) applies, A convenient visual aid is obtained by plotting Y = 2.77 and Y = 3.5U on f-versus-Re coordinates. Figure 2 shows such a plot along with the friction-^factor curve for smooth tubes determined by Hikuradse ( l 6 ) . 2. Computational Details The calculation of the boundary values of y or Y > 3.54 t i f c j quite straightforward. Wewton-Raphson method (20) was used. for the cases where Y < 2.77 For 2-77 < Y < 3.5U, however, equation (27) cannot be rendered explicit in y . + + In order to compute values of y , the + The general iteration expression is and for the equation under consideration, the particular iteration expression is + 7n = + 3.75 + 7n- - 25—3] t (30) Although there w i l l generally be two roots to this equation, only one is of interest. Uniform convergence to the appropriate root was obtained by setting < + y]_ =5i iteration was stopped when + io" 7 Since the equations involved in calculating values of Nu do not lend themselves to analytical integration, computations were done by numerical integration performed on a modified Alwac III-E digital computer. The integral t for Nu was split up into appropriate parts which were evaluated separately by successive applications of Simpson's rule. A flowsheet showing the logic of the program is given in Figure 3For the case where the flow regime is composed of the turbulent core and a stagnant layer of fluid adjacent to the pipe wall (Y > 3-5O> the contribution i of the stagnant layer to the value of Nu can be obtained analytically. Although in this region V=0, the term / RVdR is constant and equal to R, ° / RVdR; furthermore, because the fluid is stagnant, K/k = 1 . 0 . -V R, I R RVdR] , j (jRVdR) D Ro.J< 2NJu R, / R t K J R f [JRVdRj D " / Thus, /7'_ R J D \% RaK ^For example, i f Y < 2.77, case (l) applies; thus, from equation (2), _ ! _ 2Nu = / ( ( / R V a R ) R, K AR9 ^Wtwlent cor*) = * J + / ( W f e r layer) J . /(laminar sublayer) J ' v Input Data <» I Bring Pr, Set I j=j Bring Re^, f 8J I Set n=n i S.J n=n-| if > 0 _ N if < 0 Calc. Y Y - 2.77 if < 0 Calc.R. (turb. core) /(lam. sub.) Calc- R ~ J(buff. layer) • J(la a R, T if > 0 Y-354 if < 0 sublayer) /(turb. core) ~ * /(lam. ^ Calc. R a 0 R, Add if > 0 X 2 Calc R. /(turb. core) -( JrfvdrJ.(lnR) Reciprocal = N u Output Nu n/o. n-l = n if n = o ]7o j-l = j if j 3 o , i-l Figure 3. Flowsheet Shoving Logic for Computer Program for Calculating Values of Nu According to the Proposed Method. if i = o To establish the characteristics of the proposed method of prediction, i t was decided to calculate values of Nu for the following range of the variables: Pr = 0.001 to 1,000 Re = if, 000 to 10,000,000 f = f s T to 0.020 . After selecting representative combinations of the above variables, the convergence of the numerical integration was tested by doubling the number of increments in each region (turbulent core, etc.) u n t i l the values of Nu became constant. These preliminary calculations indicated that convergence became more d i f f i c u l t with increasing Pr and that convergence was influenced by values of Re and f. (These problems are taken up in the "Discussion of Results"). Values of Nu were considered acceptable i f the second significant figure remained unchanged when the number of increments in any region was increased. To check the accuracy of the calculation, the data for two worked examples on smooth pipes provided by Knudsen and Katz (l) were used. Thus, for Example 15-2, p.M+7, in which Pr = 7-74, Re = 30,000 and f = O.OO585, Nu is worked out and reported to be 185; and for Example l o - l , p.460, in which Pr = O.OO38, Re = 30,000 and f = O.OO585, Nu is found by the authors to be 8.80. calculated by the computer were 201 and 7*9^, respectively. calculation, values of the integrand To check this, ^J^RVdRJ^R^K were determined on a desk calculator and plotted. Values f o r Pr = 7.7U Three, independent graphical integrations were performed and the results are summarized in Table 2. Values of fg were determined using Nikuradse's equation 0.40 In calculating values of Nu, for a particular Re, f > f g 17- Table 2. Check on Correctness of Computer Calculation of Nu by Equation (2) Method of Calculation Conditions Pr = 7.7U numerical integration (computer) Re = 30,000 f=fg =0.00585 Nu Reported by 201 This work . graphical integration (trapezoidal rule) 217 same as previous method but using a larger scale for plotting n ti 196 11 11 graphical integration (counting squares) 208 it 11 graphical integration (K?) I85 Knudsen and Katz (l) p.U51 The differences in results obtained by graphical integraions are due to the rapidly changing curvature of the curve near the wall (R = 1.0). Other values of Nu for smooth-tube conditions are reported by Knudsen and Katz ( l ) , p.^36, for Pr = 10" , IO" , and Re = lo\ 10^, 10^; the difference between computed 1 2 and reported values was found to be less than seven per cent. It was concluded that the values calculated by the computer were correct. 18. DESCRIPTION OF OTHER METHODS FOR PREDICTING HEAT TRANSFER IN ROUGH PIPES 1. Equation of Martinelli (12) It was pointed out previously that Martinelli's equation for Nu applies s t r i c t l y only to turbulent flow in smooth tubes. Since i t predicted the experimental rough-pipe results of Smith and Epstein ( l l ) quite well, i t was decided to see how well i t predicted other available results for rough pipes. A l l the assumptions made in the deriviation of the Lyon equation also apply in Martinelli's derivation. In addition, Martinelli made the following assumptions: (g) For y + > 30, p, « Ej , while for y C 5, both % and E + 4 H are zero. < 30, To T (h) For y (i) For y > 30, u = U; i.e. in the turbulent core, the velocity profile is flat. This assumption was used by Martinelli because, and only insofar as, i t leads to the consequence that + W and 4. A = \ A /r=r w + ft ) /A r (1/ )r.r Is linear with r. A w Using equations (k), (5), and (23), Martinelli derived Nu = (31) i a Pr + ln(l + 5 i i - P r j + 0.5 F In Re IT For the purposes of calculation in this work, values of la. , t-wtc , and F t -t, were taken as unity. values of t ~tc tw-U w Although Martinelli provides methods for calculating and F, these methods do not apply to rough pipes. In contrast, the equations developed for the proposed prediction method account for these factors. 2. Equation of Nunner (13)On the basis of his experimental results (particularly with regard to the effect of the type of roughness on the heat transfer results and the effect of roughness on temperature profiles) Nunner postulated a model for the flow process and transfer mechanism In rough tubes from which he derived an equation T for the Nusselt number. equation of Prandtl roughness. His equation is actually the smooth-tube analogy (2l) modified in such a way to include the effect of Assumptions (a) to (f), page k , apply in the derivation and i t is further assumed that (*•) ( X ) (l) The viscous wall layer (laminar sublayer) has the same thickness in both smooth and rough pipes. (m) In the turbulent core, u << and k << E^ whereas in the viscous wall layer, E^ and Eg are both zero. (n) In the viscous wall layer, T = %mot \\ i p j p e s - ( X pou^h pipftt and J L /_k] A \A/r.-r = Nurtaer's equation is Na = ^RePr ** , , I + 1.5 Re"* I V (Pr± (32) e r r 1 . |j where f„ is the smooth-tube friction factor corresponding tbcthe particular Re. 3. Equation of Mattioli (lk) Mattioli's theory of turbulent heat transfer is based on his hydrodynamic theory of turbulence, in which he postulates that both momentum transfer and vorticity transfer must be considered in an analysis of transfer phenomena ( i . e . , the angular momentum as well as the linear momentum of eddies must be considered). The assumptions involved in his derivation of the expression A description of Nunner's model is given in the Appendix. 20. for Nu are those listed on page k as well as assumptions (j) and (n), above. In addition i t is assumed that in the turbulent core u = U and that i f the tube is completely rough, i . e . i f Re - or better s t i l l , Re e a - is greater 8 than 60, the kinematic viscosity, tf, and the thermal diffusivity, c< may be f neglected across the entire tube. Mattioli's equation is 4 ReFV Nu = ' Re„ > 60 The two empirical constants, A R (33) and n, must be determined from experimental results; i t is estimated from the theory that | - < n < l . The factor A is a universal dimensionless quantity for which a value of 7-7 is recommended. If the above Equation Is valid a plot of the experimental results according to 7.7 + I-st 2 st versus il (til R should yield a straight line from which the ceonstants. o A r and n can be obtained. h. Empirical Method of Epstein (15). In searching for a method for correlating rough-pipe heat transfer results, Epstein found that the coordinates that were most successful In correlating his and Smith's results ( l l ) were S t versus Re y eg possible coordinate systems is shown in Figure h. . The improvement over other It was thought worth-while, therefore, to see i f this empirical method applied as well to the experimental results ^bf other investigatiSrsi? .006 St T-n— i—I—r 1—I—rTT •004 0 •002 2 xio' J <b 0 00 i I L 8 i I0 Z i i i 8 10 e.U T 08 r 1 1 1 I 1 1 r St. o o o © o .05 J 03 2x|0 6 o l i i i I 6 8 I0 J L 6 V 8 10 Re. 08 -i 1 i—i—r-r 1—i—i—r—r Si •05 0 00 * •03 2-0 i Legend: Figure h. i 4 I I 6 I L 8 J_ 10 • /e Std- Steel Pipe 0 j/V S*d- Steel Pipe ft A J/2 Karbafe Pipe 3 I I 6 Re, I II 8 10 O /8 Galvanized Pipe O Galvanized Pipe 3 O V s Galvanized Pipe Epstein's Empirical Plots of the Results of Smith and Epstein 22. RESULTS 1. Basic Results Obtained from the Proposed Method. Values of Nu were calculated for the following range of variables: Pr = 0.001, 0 . 0 1 , . . . , 1,000. Re = (k, 7, 10, 20) x 103, f = f , , 10? 0.0025/0.004, 0.006, s , 0.020 Plots of Nu as a function of Re for several values of f and constant Er were made; some of these are shown in Figures 5a to 5e. In order to show the effect of Pr, a plot of Nu versus Re was made in which f was held constant at 0.010 and Pr varied. Figure 6 shows this plot. Figure 7, a plot of Nu as a function of f for Pr = 0.10 and a l l values of Re used, shows some further characteristics of the proposed method, namely the occurrence of maxima in such plots. 2. (a) Comparison of Experimental Results with Results Predicted from the Proposed Method and from the Equations of Martinelli and Nunner Comparison with the Results of Smith and Epstein ( l l ) Fluid friction and heat transfer measurements were made by Smith and Epstein for air flowing through a smooth copper pipe and six commercial pipes. The range of Re investigated was Re = 10,000 to 80,000. Heating was achieved by surrounding the experimental pipe with a steam jacket so that the wall temperature was essentially constant along the length of the pipe. The heat transfer results for each pipe were correlated by equations of the form JH = a R e / (35) For the purposes of this work, predicted values of Nu were calculated for the experimental range of Re using the aveasage film Prandtl number of 0.69 and values of f determined by substituting the reported average values of D/e in s the Colebrook (22) equation, io 3 io 4 io io 5 6 io 7 Re Figure 5d. Basic Results Calculated from the Proposed Prediction Method; Nu as a Function of Re Figure ^e. Basic Results Calculated from the Proposed Prediction Method: Hu as a Functionof Re 29. Figure 7. Some Further Character!sties o f the Proposed Prediction Method: Occurrence of Maxima with Increasing f and Re 30. Figure 8a. Comparison of Predicted ReBUlts with Experimental Results of Smith and Epstein 32. 33. 3h. rf + = 4K (|-J o 2.28 - 4loo 10 I+ 4.67 D/es (36) CompariBon w i t h t h e smoothed e x p e r i m e n t a l r e s u l t s was made b y p l o t t i n g Nu as a f u n c t i o n o f Re^ ; some o f t h e s e p l o t s a r e shown b y F i g u r e s 8a t o 8d. (b) Comparison w i t h t h e R e s u l t s o f Sama (23,24). Sams' f i r s t i n v e s t i g a t i o n was conducted w i t h a i r f l o w i n g t h r o u g h e l e c t r i c a l l y - h e a t e d I n c o n e l t u b e s a r t i f i c i a l l y roughened b y c u t t i n g v a r i o u s t y p e s o f square t h r e a d s i n them. studied. B u l k R e y n o l d s numbers up t o 350,000 were Thread p i t c h , h e i g h t , and w i d t h were v a r i e d and c o r r e l a t i o n o f non- i s o t h e r m a l f r i c t i o n r e s u l t s was o b t a i n e d f o r t h e r e g i o n o f complete t u r b u l e n c e by an e q u a t i o n w h i c h d i d n o t i n c l u d e t h e c o n v e n t i o n a l roughness r a t i o : f = f 0.00T2 ( s f 8 ° (- l ) 1 7 , 0 R e f > 20,000 (37) Heat t r a n s f e r r e s u l t s were c o r r e l a t e d b y t h e e q u a t i o n Nu f = 0.040 ^Re f °(P )° r f 4 > 600 < Re j?£ f < 20,000 w i t h a maximum s c a t t e r o f l e s s t h a n + 15 p e r c e n t f o r a l l t h e r e s u l t s . (38) In c a l c u l a t i n g p r e d i c t e d v a l u e s o f Nu^ , an average P r o f O.67 v a s u s e d and f v a l u e s o f t h e f r i c t i o n f a c t o r were d e t e r m i n e d f r o m Sams' c o r r e l a t i n g e q u a t i o n . Comparison o f t h e p r e d i c t e d and e x p e r i m e n t a l r e s u l t s i s shown i n F i g u r e 9* In h i s second i n v e s t i g a t i o n , Sams employed a s i m i l a r e x p e r i m e n t a l set-up, b u t used w i r e h e l i c e s sprung t i g h t l y a g a i n s t t h e t u b e w a l l f o r t h e roughness elements. The p a r t i c u l a r emphasis i n t h i s work was on t h e i n f l u e n c e o f w i r e d i a m e t e r and c o i l p i t c h on t h e h e a t t r a n s f e r and f r i c t i o n c o e f f i c i e n t s . The e x p e r i m e n t a l r e s u l t s were g i v e n i n t h e f o r m o f graphs; no c o r r e l a t i n g e q u a t i o n s were r e p o r t e d . F o r t h e c a l c u l a t i o n s i n t h i s work, an average P r a n d t l number o f O.67 was u s e d and f r i c t i o n d a t a were o b t a i n e d d i r e c t l y f r o m Sams' p u b l i s h e d graphs. F i g u r e s 10a and 10b compare t h e p r e d i c t e d r e s u l t s 10a. Comparison 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 R e s u l t s o f Sams, W i r e - C o i l - T y p e -Roughness 38. with the experimental r e s u l t s . (c) Comparison with the Results of Edwards and S h e r i f f (25) In Figure 11 i s shown a comparison of predicted r e s u l t s with the experimental r e s u l t s obtained by Edwards and S h e r i f f f o r the case of a i r flowing over a p a r t i c u l a r rough surface. Their apparatus consisted of an open c i r c u i t wind tunnel of uniform rectangular cross-section. Wires were attached to the bottom of the tunnel, at r i g h t angles to the f l u i d flow; t h i s surface was heated e l e c t r i c a l l y to produce a uniform heat f l u x . Bulk and surface temperatures were not p a r t i c u l a r l y high, so that an average value of Pr = 0.70 was used i n the c a l c u l a t i o n s . F r i c t i o n data were obtained from a p l o t of f versus Re reported by the authors. There i s a s l i g h t v a r i a t i o n of Re i n the r e s u l t s shown i n Figure 11; actual experimental values were used i n the c a l c u l a t i o n s . (d) Comparison with the Results of Lancet (26) Lancet obtained heat transfer and f l u i d f r i c t i o n data f o r a i r flowing through very small smooth and rough rectangular ducts. over the range Re = 3,000 to 27,000, Measurements were made and experimental conditions were such that the heat f l u x at the w a l l was e s s e n t i a l l y constant along the length ,of the ; tube. Roughness elements were composed of c u b i c a l protrusions, i n t e g r a l with the wall* The f r i c t i o n data indicated that the "smooth" duct was i n f a c t quite rough, with the f r i c t i o n f a c t o r e s s e n t i a l l y constant f o r Re > 7>000. Nevertheless, the.heat transfer r e s u l t s were correlated by an equation generally v a l i d f o r smooth tubes: N".«.« = 0.023 Re Pr (39) For the rough duct, c o r r e l a t i o n of the heat .tsiansfer r e s u l t s was achieved by the equation _ 0-8 Nu m e a n = 0.042 Re 1/3 Pr (ko) 39. 24 P i t c h ,in- R e x i o i 2-22 2 2-37 4 206 6 2.16 12 1-93 22 5 3-9 3.2 21 1 73 1.35 20 -Nunner 8 mean Proposed Method 16 1-4 1.2 -Edwards 1-0 0 2 _L 4 J 6 8 I 8 Sheriff I 10 L 12 14 Pitch, inches Figure 11. Comparison of Edwards of Predicted and Sheriff Results with Experimental Results ko. Nu F i g u r e 12. Comparison 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 of Lancet Results Ul. In the above equations, fluid properties were evaluated at the film temperature. No experimental values of Pr were given so It was assumed for calculation purposes that the average value was 0.69- The reported values of the f r i c t i o n factor were used in calculating values of Nu. The experimental and predicted results are compared ln Figure 12. (e) Comparison with the Results of Nunner (13)Nunner conducted an extensive experimental Investigation for air flowing through rough pipes for Reynolds numbers up to 80,000. Both naturally-rough and artifically-rough pipeB were used; the latter were produced by placing split rings in a smooth brass pipe. The shape of the cross-section, the height, and the separation of the rings was varied to produce different roughness geometries. steam jacket. Heating was achieved by surrounding the pipes with a No direct comparison of Nunner's experimental data has been made with results predicted by the proposed method and by Martinelli's equation. Since Nunner's data cover the same range of Pr, Re, and f as those for the above investigators, and since his theoretical expression for Nu (equation 32) predicts his results within 15 to 20 per cent, predicted and experimental results have in effect been compared indirectly In Figures 8 to 12. (f) Comparison with the Results of Cope (27). Cope was one of the very few investigators who used water, rather than air, as the working fluid in his experiments. Data were obtained for three rough pipes whose inner surfaces were milled in such a way to produce a series of pyramids geometrically similar in shape but varying in absolute size from pipe to pipe. The test section was a single-pass double-pipe heat exchanger orienteSvertically, with shell-side cooling water flowing parallel to the working fluid. The range of Re studied was 2,000 to 6py000. Actual values of Pr, Re, and f reported by.Cope were used to calculate predicted values of Nu. For low values of Re, some experimental friction k2. factors f e l l below the commonly accepted smooth-tube values and were therefore not used i n the c a l c u l a t i o n s . Comparison of the experimental and predicted r e s u l t s i s made i n Figures 13a to 13c. Although Cope d i d not draw curves through h i s experimental points, dashed curves have been drawn through them here to indicate more c l e a r l y the trend of h i s r e s u l t s . 3. Test of the Effectiveness of M a t t i o l i ' s Method f o r Correlating Rough-Pipe Heat Transfer ResultsA p l o t of the available experimental data according to the coordinates suggested by M a t t i o l i ' s theory Is shown i n Figure lk. p l o t t e d s a t i s f y the requirement that Re Figure lh of those that do. e Not a l l the points > 60; an i n d i c a t i o n i s given i n In c a l c u l a t i n g values of the coordinates of the points, the values of Pr and f used were those obtained as outlined previously. Values of D/e s needed i n c a l c u l a t i n g values of the abscissa were determined using the reported f r i c t i o n data and equation (36) which, i n c i d e n t a l l y , reduces to Nikuradse's ^ = 4I.J.D (17) equation + 2.28 f o r s u f f i c i e n t l y high values of Re and f, i . e . when f becomes independent Re. Points f o r the r e s u l t s of Nunner (13) were calculated using equation i n which Pr was taken as O.69 of (32), and values of f were obtained from Nunner's f r i c t i o n - f a c t o r curves. k. Test of the Effectiveness of Epstein's Empirical Method f o r Correlating Rough-Pipe Heat Transfer Results. • • Figure 15 versus R e es . here as w e l l . shows a p l o t of the available experimental r e s u l t s as S t The above comments regarding values of Pr, f , and D/e $ r apply 4=4=- Figure 13P- Comparison of Predicted Results with Experimental Results of Cope 1—I TT o 10 • o - ° » - f/2-St o o ° o o O a „ a _a a 9 Sams (Square Threads) A + =8 Q>o° 9 « e I I I II O Pipe A Pipe B- o„ . * a ° o Aa ^ 'ipe C 10 4 A *" • A • • A B a • a I •0 10 I I I I I I I I 10 • • ° o° • a 'Re^>60 (air data) 7 2 10 2 Nunner Sams (Wire Coils) 0 Lancet O Cope (Pipe A ) 9 Cope (Pipe B ) O Cope (Pipe C ) I III 4 A a •Re^>60 (water data) I 2 ° * 4 7 10 Pe. 3 2 I I I I I I 7 I 10 4 4=0 N Figure lk. Plot of Rough-Pipe Heat Transfer Results on Mattioli's Coordinates o r — i — i 1 i i i 11 I 1—I I I II 1—i—i l l I I | l O Smith i i i 'a • i I II ij 8 Epstein ^ Nunner • Lancet a Sams (Wire Coils) • Cope (Pipe A) • Cope (Pipe B ) • Cope (Pipe C ) -i 10 * 1 St Q U —* o 6 —ts&r££&-&x 5*^^^s; H o oo c rr" a 0 o a o„ n a - S a m s (Square atrarfJ Threads) -2 10 i 2*10 2x|0 J I i i i i 1.0 I I 10 ' ' ' ' J 10 Re, Figure lg. Plot of Rough-Pipe Heat Transfer Results on Epstein's Coordinates 10 ' I I I I I I 10 DISCUSSION OF RESULTS 1. Values of Nu Calculated by the Proposed Method The accuracy of computed values of the Nusselt number using the proposed method was limited severely by poor convergence of the numerical integration. If convergence had not been such a problem, calculated values would^have been accurate to at, least four significant figures, instead of two. The convergence became more d i f f i c u l t as the Prandtl number was increased. The influence of this variable in these calculations is limited to the turbulent core and buffer layer regions where i t appears in the expressions for K/k (equations 15 and 1 8 ) . For large values of-Pr, the curvature of the plot of versus R« increases greatly near the wall ( R « — » - 1 . 0 ) . Since the area under the curve is calculated by successive applications of Simpson's rule which is exact only for a third degree polynomial, a large number of increments must be taken where the curvature changes greatly in order to obtain an accurate estimate of the area. The turbulent core zone was therefore sub-divided into three separate regions, in which the number of increments could be varied independently. By much t r i a l and error, the positions of these intermediate boundaries and the number.soe>f increments were found which gave convergent values of Nu for particular values of Pr,' Re, and f. Not only did the magnitude of the Prandtl number influence the convergence of the integration, but also the values of Re and f. Convergence was especially difficult i f the points were slightly to the right of the curve Y = 3.5U (see Figure 2 ) , but relatively easy - even for large Pr - i f the points were remote from this curve. The reasons for this behaviour were rather obscure, but a close examination revealed two effects which provided explanations. In the f i r s t place i t was found that for f = constant, the value of R, calculated from equations (28) and (26) was independent of the Reynolds number. Thus, the thickness of the stagnant layer, (l-R, ), remains constant for a fixed value of f. greater than 3 ' 5 ^ > vas reached. For Secondly, i t vas revealed that for Y only slightly K/k "became unity before the end of the turbulent core (R,) This behaviour is a consequence of the form of equation ( 1 5 ) . the purpose of analysis, let R be that value of R in the turbulent core K T at which K/k just becomes unity and f for which R i R, . K . It was desired to determine values of Re Using equations ( 2 8 ) and ( 2 6 ) , values of R, were calculated for various values of f. was solved explicitly for Re. After setting K/k = 1 . 0 , equation (15) Using the values of f and R, in this equation, the corresponding values of Re were calculated. These particular (Re,f) coordinates were then plotted to give the curve designated as R = R, in K Figure 2 . Thus, between the curves Y = 3 - 5 ^ and R = R, , K/k in the turbulent K core reaches unity before R = R, ; for values of (Re,f) to the right of the curve R = R, , K/k is greater than unity at R = R K ( . Convergence was not nearly so difficult for Y< 2 . 7 7 or for 2 . 7 7 Calculations indicated that R > R for both of these cases. K ( £ Y <3«5U. Since equation ( l 8 ) has exactly the same form as equation ( 1 5 ) . > i t is quite possible that in the buffer layer K/k becomes unity before R 2 is reached. No extremely difficult convergence problems were encountered in this region, so a detailed 1 analysis was not pursued. The consequences of these characteristics of the proposed method on the convergence of the numerical integration can be seen most clearly in the sketches of the integrand plotted against R^ in Figure ( l 6 ) . As the Reynolds number increases, the area bounded by R and R, decreases . K R = 0 and R = R of K also decreases, but i t becomes a more significant fraction the total area. Two The area between Since the major portion of this fraction isunear the values, corresponding to the two roots of the quadratic w i l l be obtained; however only the root nearer 1 . 0 case. is of any significance in this •Increasing Fixed Figure l 6 . Pr and f Sketch (not to scale) of several Influence of Re Re - RVdR R„ K -versUs-R^ Purves Showing the 51. boundary R K where the curvature of the curve is rapidly increasing, a very large number of increments must be taken in this region in order to obtain an accurate estimate of the total area. For sufficiently high Re, R > R, (as K in the right-hand sketch of Figure l 6 ) and the turbulent core area becomes small relative to that of the stagnant layer, especially for higher values of Pr. Since the area due to the stagnant layer is essentially independent of Re the flattenlng-out of the Nu-versus-Re curves (in Figures 5b to 5c) is •t&us explained. The effects of the proposed method's characteristics on these curves would have been more evident had Nusselt numbers been evaluated at smaller intervals of Re. Figure 17 shows the curve for Pr = 100 and f = 0.010; a sufficient number of values of Re were used to carefully define the curve in each region. For lower values of Pr, the points of inflection are less abrupt, as indicated by the curves in Figure 6. From the comparisons of the predicted and experimental results i t can be seen that the proposed method is not satisfactory. In Figures 8 to 12 the predicted values are only f a i r l y good when the pipes are not too rough; the faults in the method are not too well exposed due to the combined effects of rather low Pr and values of f not far removed from smooth-tube values. When, however, the fraiction factor is very large as in Figure 10a, the absolute magnitude of the predicted values n.dis very poor and the points show the wrong trend with increasing Re. In Figures 13a 4sbdl3c the faults in the proposed method are more obvious because of the larger values of Pr ( 6 to 8 ) . Cope used water as the working fluid. heat transfer coefficients pipes. Other investigators besides Pohl ( 2 8 ) , for example, reported that for rough pipes were lower than those for smooth His data are considered unreliable due to his experimental method of determining the average temperature driving force and consequently were not used in the calculations. Grass (k) reported his results graphically as ro Figure 17. Results of More Detailed Calculations Based on the Proposed Prediction Method: of Points of Inflection pressure drop and heat transfer coefficient versus average fluid velocity. He gave no indication of the operating temperatures so that values of Re and Pr required for the calculations could not be estimated. The work of Savage (29) has just recently become available. In addition to the experimental work in which air and water were used as the working fluids, some work (30, 31* 32) has been done in which liquid metals were used. The test pipes were naturally-rough, although there was no intention to study the effects of roughness on the heat transfer to these fluids. The experimental results are either f a i r l y well predicted by Lyon's (lO) simplified equation for smooth pipes or f a l l well below the predicted values ( 3 3 ) ' If the fact that the pipes were slightly rough had been taken into consideration by using £he proposed method, predicted values would have bean even higher in relation to the experimental results. Ln any event, there is some question (33) s to the r e l i a b i l i t y of the data due to a the possibility of surface fouling and to the methods of measuring temperatures. Furthermore, wettability may be an additional uncontrolled variable in some of these experiments ( l ) . Apart from the defects in the proposed method which are mainly the result of the particular velocity profiles assumed, predicted values cannot be expected to correspond exactly to the experimental results because the experimental conditions do not s t r i c t l y satisfy a l l the subsidiary assumptions used in deriving the theoretical equation. For example, in ijone of the experimental work referred to were temperature profiles fully-developed. Also, some of the experimental results correspond to the condition of constant wall temperature rather than constant heat flux at the wall. Seban and Shimazaki (3*0 show that the former condition yields values of flu that are lover than those for the latter condition; the differences are small, however, for Pr > 0.7, and diminish for increasing Pr. These authors provide a rather Involved iterative method for converting values of Nu (and temperature profiles, also) for any wall temperature condition to tne case of constant wall terroerature. Ine assumption that fluid properties are independent of temperature is also not rigorously satisfied. Evaluating fluid properties at some "film" temperature largely - but not completely - eliminates the influence on fluid properties of radial temperature Gradients, and therefore ic of some assistance- in sinking i t possible to conpare predicted and experimental on the sane basis. results Generally, the conditions of the experimental work cited f u l f i l l the other assumptions involved in the theory. Thus, the: fact that a l l the assumptions are not exactly satisfied does not seriously invalidate a comparison between predicted and experimental results. One of the most objectionable features of the proposed method is that for Y > 3o^- i t predicts a stagnant layer of fluid adjacent to the pipe wall. This layer, i t was found, is responsible for most of the resistance to heat transfer, especially for higher values of Fr. Moreover, i f the f r i c t i o n factor is constant, this resistance remains almost constant as Re increases. It is not unreasonable to suppose that i f this resistance were removed, the predicted values of ITu might correspond more closely to experimental values. A rationale for this modification can be offered. The region (f-R, ) is considered to be occupied partly by stagnant fluid and partly by roughness elements which also protude into the turbulent core. Calculations show that the thickness (l-R, ) of this region is only about J _ the height' of the equivalent sand roughness for 0.004 < f < 0.020. The fluid in this region is therefore in contact with a very large fin-like wall area and can be supposed as equilibrating i t s e l f to the temperature of the pipe v a i l To test this hypothesis, the rough-pipe data for Figure 10a was used to recalculate values of Nu on the assumption of no thermal resistance in the zone (J'-R, ); the results are compared with the previous results in the ssne figure. Although the modified method apparently overcorrects for the effect 55. of the stagnant layer, i t does have the v i r t u e that the trend of the experimental r e s u l t s i s c o r r e c t l y predicted. Furthermore, recognition to the f i n action of the roughness elements. i t gives some Calculations f o r the other experimental data were not done; l e t I t be noted, though, that i f Y > 3«5^> a^d i f the. predicted curve l i e s above the experimental curve (as i n Figure 1 0 b ) , then removing the resistance of the stagnant layer w i l l only raise the predicted curve further. 2. (a) Comparison of Other Method of Prediction with the Experimental Results Equation of M a r t i n e l l i ( 1 2 ) Although M a r t i n e l l i ' s equation i s s t r i c t l y applicable only to heat transfer i n smooth pipes, i t gives good r e s u l t s f o r rough-pipe data - exen i n i t s s i m p l i f i e d form. The maximum error i n predicted values f o r a l l the a i r data U3ed was approximately + 3 0 per cent. For the l i m i t e d amount of water data used, predicted values of Nu were not nearly as good; a maximum error of about lkO per cent was found. In general, however, the errors i n predicted values were s i g n i f i c a n t l y less than these figures. The assumption made by M a r t i n e l l i i n h i s derivation that the turbulent core v e l o c i t y p r o f i l e i s f l a t (and hence "the r a d i a l heat f l u x i s l i n e a r l y distributed) i s not nearly so v a l i d f o r flow i n rough pipes as i t i s f o r flow i n smooth pipes. I t i s a well-known experimental f a c t that as the roughness (as measured by the value of f ) increases, rough-pipe become less f l a t andivmore pointed. profiles In view of ithe good p r e d i c t i o n of iheat transfer r e s u l t s , i t must be concluded that the shape of the v e l o c i t y p r o f i l e i n the turbulent core does not have a decisive influence on the heat transfer c o e f f i c i e n t , at least f o r values o f Pr i n the v i c i n i t y of 0 . 7 In addition, the evidence suggests i n d i r e c t l y that a laminar sublayer exists for flow i n rough pipes° M a r t i n e l l i reached the same conclusion a f t e r analysing Cope's r e s u l t s , and the comparisons with other r e s u l t s made i n t h i s work seem to substantiate i t . For constant P r / Martinelli's simplified equation reduces to the form Nu = p (Re/T). In order to correlate their results for a l l degrees of roughness, Sams (23) and Cope (27) empirically replaced the average velocity by the friction velocity in dimensionless groups involving the velocity of the fluid. Since the variation in the Prandtl number vas f a i r l y small in both cases, the coordinates they used - those for Figure 9 and Figures 13a to 13c, respectively - Imply that Nu = Y ^ e ^ T ) ; moreover Sams determined 1.0 the form of Y to be Nu = b(Re Jf)'.'.. Comparison of the experimental and predicted results suggest that the forms offtand Y correspond most closely at lower values of Re/T . The effect of simplifying the Martinelli equation by assuming F and ^ ~^ w c to be unity Is shown in Figure 8a. Using correct values of these tw ~tb factors raises predicted Nusselt numbers for smooth pipes approximately 20 per cent - at least for Pr O.69 and Re = 10^ to 10^. If methods were available for estimating these factors for rough-pipe conditions, corrections fifould also be applied to a l l the other predicted values. For the data of Edwards and Sheriff (25), values of h calculated using the simplified s Martinelli equation are about lk per cent lower than those that would be obtained using the more rigorous form. for calculating values of h simplified equation. r , the ratio Since corrections are not available _jic was determined using the In view of the close correspondence between predicted and experimental values (Figure l l ) , i t appears that the effects of these approximations are largely self-cancelling, (b) Equation of Nunner (13). The equation derived by Nunner appears to be f a i r l y successful, though predicted values are often somewhat higher than experimental results. Maximum errors observed were (approximately) + kO per cent and + 75 per cent for air and water data, respectively, but were generally much lower. Since Nunner himself worked with air and nearly a l l experimental data used in the present study are for a i r , Nunner's equation remains relatively untested for fluids whose Prandtl number is significantly different from 0.7' For very low values of Pr, the equation can predict negative or infinite values of Nu, especially at lower Reynolds numbers.^" Nevertheless, because of the good prediction for Pr = 0.7, Nunner's model of the flow process and transfer mechanism in rough pipes appears to be verified, at least in a gross sense. The results, moreover, strengthen his proposition that the magnitude of f alone, at a given Re, is a sufficient measure of the roughness in its effect on the heat transfer coefficient; the type of roughness ( i . e . , the geometry and distribution of roughness elements) need not be considered. It should be noted that i f f = f s , Nunner's equation reduces to one form of Prandtl's (21) equation for smooth pipes. More than one form is available due to several methods of describing the ratio of the velocity at the edge of the laminar sublayer to the average velocity. the empirical expression, 1.5 R e / ^ Nunner used P r / ^ , of Hofmann (35); Knudsen and -1 - 1 Katz ( l ) , on the other hand, employed equation (h) to obtain value of this ratio for use in their form of Prandtl's equation, (c) Equation of Mattioli (lh). The results shown In Figure (lh) indicate that the coordinates•suggested by Mattioli are not successful in correlating rough-pipe heat transfer results. In the f i r s t place, they do not generally eliminate the effect of roughness; a plot (not shown) using the actual roughness heights rather than the equivalent sand roughness heights produced no improvement. results of Cope (27) for which Re e > Only those 60 plotted as straight lines having a slope near the predicted range 0.5 to 1.0, but here also the effect of roughness has "not been correctly accounted for. Secondly, the fact that the results for a i r and for water are significantly separated indicates that This behaviour is due to the assumption that, in the turbulent core, k < < E H 58. variations in Pr have not been eliminated. Since other methods more adequately represented the experimental results, the reasons for the failure of Mattioli's method were not sought, (d) (15). Method of Epstein Epstein's empirical method of plotting St encouraging results. T against Re es produced quite With the exception of Cope's results, the effect of roughness appears to be largely accounted for by using the equivalent sand roughness height as the characteristic dimension and the friction velocity as the characteristic fluid v e l o c i t y / using friction data and equation (36), Since values of _D_ are determined the actual geometry and distribution of the roughness elements does not enter the correlation. . Although the experimental data were obtained for many different types of roughness the use of this particular dimension, rather than-the pipe diameter, does seem t o J be justified from the results of Figure 15It w i l l be noted that the method, as presented here, does not attempt to account for variations in Pr. The separation of the air and water data' in Figure 15 could be eliminated to some extent by applying an empirical correction for these variations. St Pr r m Thus, assuming that = a Rea, an estimate could be obtained for the value of m which would minimize the separation of the points. The value obtained, however, would have to be considered tentative u n t i l more data for water (and, for that matter, for fluids other than air and water) were available. ^Similar empirical plots (not shown) of Sams'(2*0 results were made in which various geometrical factors involving wire diameter and c o i l pitch ...were -.used in the shear Reynolds number, Re , instead of the pipe diameter. Of those factors tried, the one giving the best correlation - virtually as good as the equivalent sand roughness height used in Figure 26 - was T (wire diameter)^ c o i l pitch This method of correlation can he c r i t i c i z e d on fundamental grounds in that i t does not allow for the case of smooth pipes- A smooth pipe is only a special case of a rough pipe so that the correlation should provide for the limiting conditions of e valid only for e s s = 0; equation ( 3 6 ) , however, is also > 0. It would be interesting to see i f this method could be justified theoretically. Specifically, a theoretical investigation should provide answers to the following questions: (i) Is there a straight-line relation between (log) St T and (log) Re i f so, what slope does this line have and over what range of Re does the relation apply? (ii) Can the method be modified to allow for the case of smooth pipes (e, (iii) = 0)? Is the method of correlation restricted to a particular wall temperature condition? (iv) Can variations in Pr be accounted for? e 60. CONCLUSIONS AND RECOMMENDATIONS 1. The proposed method for predicting heat transfer in rough pipes contains serious faults which render i t not useful. Its failure can be attributed mainly to a" lack: of knbwled^eyofveloeity' conditions near the walls of rough pipes 2. Both Nunner'B equation and Martinelli's simplified equation give quite good prediction of experimental results in the range of variables investigated. With air as the working f l u i d , maximum errors of approximately +k0 per cent and +30 per cent, respectively, can be expected; generally, however, the errors w i l l be significantly lower. These equations are recommended for use at the present time, but with the reservation that they be limited to fluids for which 0.5 < Pr < 1.0. 3. From comparison of predicted and experimental results there is the suggestion that the fluid adjacent to a rough wall is not stagnant but is in motion, probably laminar. k. Epstein's empirical method for correlating heat transfer results for rough pipes shows promise and merits closer investigation, both theoretically and experimentally. 5. Much more experimental data ishneeded before the roughness problem can be completely solved. From the point of view of testing theoretical refinements to equations for predicting heat transfer in rough pipes, i t would be desirable to have (a) More values of Nu for Pr = 0.001 to 0.1 and 10 to 1000, Re = 2,000 to 1,000,000, and f = f s to 0.04, for both heating and cooling conditions. (b) Velocity profile data In the wall region of rough pipes. (c) Temperature profile data across the entire pipe cross-section. (d) Values of £*_ as functions of Pr, Re, f, and R. 61. NOMENCLATURE a - constant, dimensionless A - area, f t b - constant, dimensionless Cp - s p e c i f i c heat at constant pressure, D - inside diameter of pipe, f t . e - average height of roughness elements, f t . 2 Btu/(11a)( F) 0 e g - equivalent sand-roughness height of roughness elements, f t . E H - r a d i a l eddy conductivity, E M - eddy v i s c o s i t y , l b / ( h r ) ( f t ) - Fanning f r i c t i o n f a c t o r f (Btu)(ft)/(hr)(ft )(°F) 2 M = 9 c ^ ,. (- Afliislisa \ TfU*\ L , dimensionless J f r - rough-pipe Fanning f r i c t i o n factor , dimensionless f s - smooth-pipe Fanning f r i c t i o n f a c t o r , dimensionless F - f a c t o r i n equation (31) g - denotes function , dimensionless g' - denotes f i r s t derivative of function, gj. - conversion f a c t o r h - heat transfer c o e f f i c i e n t , = = %/(Eg + ) , dimensionless dimensionless U.17 x 10 ( l b ) ( f t ) / ( l b ) ( h r ) 8 2 M F Btu/(hr)(ft )(°F) 2 h r - rough-pipe heat transfer c o e f f i c i e n t , Btu/(hr)(ft )(°F) h g - smooth-pipe heat transfer c o e f f i c i e n t , 2 Btu/(hr)(ft )(°F) 2 2/3 «5H " " J - f a c t o r " f o r heat transfer k - Molecular thermal conductivity K - t o t a l r a d i a l conductivity L - length of t e s t section of pipe, f t . Ma - Mach number n - exponent of Pe dimensionless = U/U e s = = St(Pr) ' , dimensionless , (Btu)(ft)/(hr)(ft )(°F) 2 k + % , (Btu)(ft)/(hr)(ft )(°F) 2 , dimensionless i n equation (33) and of Re^ i n equation (35 )> Nusselt-number = hD/k , dimensionless absolute pressure, l b p / f t roughness Peclet number Prandtl number = = (Pr)(Re ef> ), dimensionless , dimensionless heat transfer rate, Btu/hr. point radius, f t . point radius, f t . point radius, f t . radius of pipe, f t . dimensionless radius = r/ry dimensionless radius = dimensionless radius = £q/r r w t/ w r value of R i n the turbulent core at which K/k = 1.0, dimensionles boundary radius defined i n text, dimensionless boundary radius defined i n text, dimensionless Reynolds number = _PUf_ dimensionless f roughness Reynolds number based on average height of roughness elements =* ell? IT , dimensionless roughness Reynolds number based on equivalent sand roughness height of roughness elements shear Reynolds number = = e ^ = sU° , dimensionless > dimensionless distance between threads, f t . Stanton number = = — N u _ U^Cp shear Stanton number temperature temperature = Stj^ at radius r , °F bulk temperature dimensionless Re Pr , °F at r = 0 , °F > dimensionless 63v - temperature a t a d i s t a n c e y , °F u point velocity max maximum f l u i d v e l o c i t y U (time-average) i n Z - d i r e c t i o n , ( v e l o c i t y a t r = 0), ft/hr. ft/hr. (shear v e l o c i t y ) = U* - f r i c t i o n velocity U* - dimensionlesB v e l o c i t y > U u/u* average v e l o c i t y , f t / h r . s U = " sonic v e l o c i t y , ft/hr. V dimensionless v e l o c i t y = u/u V dimensionless v e l o c i t y = u/U w width of t h r e a d s , f t . w mass f l o w r a t e , m a x lbj^/hr. y - d i s t a n c e from w a l l o f p i p e , y-+ - dimensionless d i s t a n c e from w a l l of p i p e Y - v a r i a b l e d e f i n e d "by e q u a t i o n Z - coordinate i n d i r e c t i o n of p i p e a x i s or d i r e c t i o n of flow f l u i d , dimensionless Greek ft. = .yJL (29) of Letters cx - thermal d l f f u s i v i t y S, - thickness of laminar sublayer Sj - d i s t a n c e between t h e p i p e w a l l and t h e edge o f t h e t u r b u l e n t c o r e , A - constant i n e q u a t i o n (33)> dimensionless R - c o n s t a n t i n e q u a t i o n (33)* dimensionless € H - eddy d l f f u s i v i t y o f h e a t ( M - eddy d l f f u s i v i t y o f momentum - dimensionless temperature A 9 = - J i - = , , ft /hr. 2 ft. Eft/yCp = = , J^L t -t w y tw-tc jx - molecular v i s c o s i t y , lbj^/(ft )(hr) ^ - kinematic v i s c o s i t y , ft /hr. 2 ft /hr. 2 , ft /hr. 2 ft. ? - density, l b j ^ f t 3 T - shear stress, l b / f t 2 0 - denotes function or functional relationship, dimensionless V - denotes function or functional relationship, dimensionless F Subscripts f - denotes fluid properties evaluated at the film temperature = 3 ( t g + t g. ) ; w>av t jau w - denotes conditions at the inner wall of the pipe or, for dimensionless numbers, denotes that fluid properties evaluated at the average inside wall temperature S - denotes condition at y = %, - denotes condition at y = S ( £ 2 a . LITERATURE CITED Knudsen, J . G . , and Katz, D . L . , "Fluid Dynamics and Heat Transfer", McGraw-Hill Book Company, Inc., 1958. McAdams, W.H., "Heat Transmission", 3rd ed., McGraw-Hill Book Company, Inc., 195^Durant, W.S., and Mirshak, S., "Roughening of Heat Transfer Surfaces as a Method of Increasing the Heat Flux at Burnout", DP-38O Progress Report No.l, July, 1959' (As found in Nuclear Science Abstracts, l4, 2520) Grass, Gunther, "Verbesserung der Warmeubertragong an Wasser durcb, kunstliche Aufrauhung der Oberflachen in Reetktoren oder Warmeaustauschern", Atomkern-Energie, 3_, 328-31, Aug.-Sept., 1958. Draycott, A . , and Lawther, K.R., "Improvement of Fuel Element Heat Transfer by Use of Roughened Surfaces and the Application to a 7-Rod Cluster". Paper presented at the 1961 International Heat Transfer Conference, U n i v e r s i t y of Colorado, Boulder, Colorado. Preprint published by the American Society of Mechanical Engineers, New York, N.Y., Part III, Section A, p.5^3 (l96l). Hall, W.B., "Heat Transfer in Channels Composed of Rough and Smooth Surfaces", U.K.A.E.A Industrial Group, IGR-TN/W-832, 1958. Brauer, H . , "Flow Resistance and Heat Transfer in Annuli with Rough Inner Tubes", Kerntecknik, 3_, 387-391 (1961). Karman, T. von., J . Aeronautical S c i . , 1, 1 (193*0 - as found in Knudsen and Katz (l). Rouse, H . , "Elementary Mechanics of Fluids", John Wiley & Sons, Ine*, New York, 19U8. Lyon, R.N., "Liquid Metal Heat-Transfer Coefficients", Chem. Eng. Progr., kl, (2), 75 (1951). Smith, J.W., and Epstein, Norman, "Effect of Wall Roughness on Convective Heat Transfer in Commercial Pipes", A.I.Ch.E. Journal, 3, (2), 2k2 (1957). Martinelli, R . C . , "Heat Transfer to Molten Metals", Trans. A.S.M.E., 6_9_, 9^7 (19^9). Nunner, W., "Warmeubergang and Druckabfall in rauhen Rohren", VDIForschungsheft-. 455 (1956). Also A.E.R.E. Lib/Trans. 786. Mattioli, G.D., "Theorie der Warmeubertragung in flatten und rauhen Rohren", Forschung auf dem Gebiete des Ingenieurwesens, 11, (k), 1^9-158, (19^0). Translated by LVM.K. Boelter, NACA Technical Memorandum N0.IO37, 19*+2. 66. Epstein, N . , private communication (1961). Nikuradse, J . , "Gesetzmaszigkeiten der turbulenten Stromung in glatten Roliren",ViD^I.Forschungsheft 356 (1932). Nikuradse, J . , "Stromungsgesetze in rauhen Rohren", Vi.'E.I. Forschungsheft 361 (1933). Translation available: The Petroleum Engineer, XI; No.6, l6k (Mar. 19^0); No.8, 75 (May, 19^0); No.9, 12U (June, 19UO); No.11, 38 (July, I9U0); No.12, 83 (Aug., 19^0). Azer, N . Z . , and Chao, B . T . , "A Mechanism of Turbulent Heat Transfer in Liquid Metals", Int. J . Heat and Mass Transfer, 1, 121-138 (I960). Hinze, J . O . , "Turbulence", McGraw-Hill Book Company, Inc., 1959Pipes, Louis A . , "Applies Mathematics for Engineers and. Physicists", 1st ed., McGraw-Hill Book Company, Inc., 19^6. Prandtl , L . , "A Correlation between Heat Exchange and Flow Resistance of a Fluid", Phys. Z . , 11, IO72-IO78 (1910). Colebrook, C F . , "Turbulent Flow in Pipes, with Particular Reference to the Transition between the Smooth and Rough Pipe Law", J . Inst. C i v i l Engrs., 11, 133 (1938-39)Sams, E.W., "Experimental Investigation of Average Heat-Transfer and Friction. Coefficients for Air Flowing in Circular Tubes Having Square-Thread-Type Roughness", NACA RM E52 D17 (1952). Sams, E.W., "Heat Transfer and Pressure Drop Characteristics of WireCoil-Type Turbulence Promoters", United States Atomic Energy Commission Technical Information Service Extension, Oak Ridge, Term. TID-7529 (Pt'.l) Book 2, Physics and Mathematics, 390-415 (1957) Edwards, F . J . , and Sheriff, N . , "The Heat Transfer and Friction Characteristics for Forced Convection Air Flow over a Particular Type of Rough Surface", Paper presented at the 1961 International Heat Transfer Conference, University of Colorado, Boulder, Colorado. Preprint published by the American Society of Mechanical Engineers, New York, N.Y., Part II, Section B, 1+15 (1961). ' Lancet, R.T.,'"The Effect of Surface Roughness on the Convection Heat Transfer Coefficient for Fully-Developed Turbulent Flow in Ducts with Uniform Heat Flux", Trans. ASME, Series C , 8l, (2), 168-17^(1959). Cope, W.F., "The Friction and Heat Transmission Coefficients of Rough Pipes", Proc. Inst. Mech. Engrs (London), l4_5, 99-105 (19^1)• Pohl, W., "EInflujS, der Wandrauhigkeit auf den Warmeubergang an Wasser", Forsch. Ing. Wes., h, 230 (1933)- 67. (2$) Savage, David W i l l i a m , " E f f e c t o f Surface Roughness on Heat and Momentum Transfer", Ph.D. Thesis, Purdue U n i v e r s i t y , Lafayette, Indiana, 196l. (30) Johnson, H.A., Hartnett, J.P., and Clabaugh, W.J., "Heat Transfer to Molten Fh-Bi Eutectic i n Turbulent Pipe Flow", Trans. ASME, 7_5, 1191 (1953)- (31) Johnson, H.A., Hartnett, J.P., and Clabaugh, W.J., "Heat Transfer t o Meicu^y i n Turbulent Pipe Flow", Trans. ASME., 7_6, 505 (195*+). (32) S t y r i k o v i c h , M.A., and Semenovker, J . Tech. Phys., 10, 132^ (I9U0); as reported by Kutateladze, S.S., e t a l , "Liquid-Metal Heat Transfer Media", (Translated from the Russian), Consultants Bureau, Inc., New York, 1959« (33) Doody, T.C, and Younger, A.H., "Heat Transfer C o e f f i c i e n t s f o r L i q u i d Mercury and D i l u t e Solutions of Sodium i n Mercury i n Forced Convection", Chem. Eng. Progr. Syrap. S e r i e s , 33 (1953). (Note the remarks i n the "Discussion" and e s p e c i a l l y the communication o f I s a k o f f and Drew). (3h) Seban, R.A., and Shimazaki, T.T., "Temperature D i s t r i b u t i o n s f o r A i r Flowing Turbulently i n a Smooth Heated Pipe", Trans. ASME, 7_3_, 803 (1951). (35) Hofmann, E., "Der Warmeubergang b e i der Stromung im Rohr", Z.ges. Kalte-Ind., Uk, 99-107 (1937)- 68. APPENDIX TEMPERATURE PROFILES IN ROUGH PIPES It can "be readily demonstrated that the Nusselt number is directly related to the dimensionless temperature profile. The rate of flow of heat from the pipe wall to the fluid can be written as <J = h A ( t - t ) W w w (1) b or alternatively as = -UA ftt (h2) w y/y = o Combining equations (l) and (k-2) and solving for h gives 0*3) Now ht Ty >(t -t) ^y = w Substituting equation (hk) into equation (U3) and multiplying through by 2fw gives k h (zr\ k Therefore ~o(t -t) k Mu = 2 Nu = 2 Alternatively, - t•k|_ tL-t w k w j y•=0 ( 1 * 5 ) o ( Y / r J jy=o ftw-t>\ Vtw-tJ 0*6) /jy= In the derivation of Lyon's (10) equation for the Nusselt number, an expression for the temperature profile, expressed as ty-t, is developed. By integrating this expression across the tube radius an equation for t^-t Is c obtained which, when combined with the previous equation, yields the expression for the normalized dimensionless temperature profile: Thus, temperature profiles in rough pipes can be predicted using equation (hf) and the equations in Table 1. Martinelli (12) developed equations for the normalized dimensionless temperature profile applicable for turbulent heat transfer in smooth pipes. It was previously found that by using rough-pipe friction factors in the simplified equation for the Nusselt number, quite good prediction of experimental values was obtained for the range of experimental data tested. It is not necessary, however, that a similar degree of success would be obtained in using his equations to predict the corresponding rough-pipe temperature profiles. In the light of equation (h6) only the values of the slopes of the profiles at the wall could be expected to agree with experimental values; it does not follow from equation (h6) that the overall shapes of the predicted and experimental profiles would be the same. Although Nunner (13) derived an equation for the Nusselt number based on his model of the flow process in rough tubes, he did not derive expressions for the temperature profile. It is a l l the more surprising that he did not do so since he is the only one to report experimental temperature profiles in rough pipes. These equations are derived below. Before proceeding with the derivations, i t is necessary to give a brief description of Nunner's model for the flow conditions in rough pipes. From his Sheat transfer measurements, Nunner found that there was no separate effect on -the Nusselt number due to the type of roughness: at a given Re (and Pr) the value of the Nusselt number depended only on the overall value of the f r i c t i o n factor. He considered that the total frictional resistance of a rough wall was- <St&e,'t'0!.tHe :«^liitfdC«ffeeti9 o f t v o separate processes - skin friction of a hypothetical smooth wall and shape resistance of the roughness elements. Physically, the model consists of a viscous wall layer (or laminar sublayer) caused by the resistance of the hypothetical smooth wall, a roughness zone where separate roughness particles areppresent yielding the form d^ag of the actual roughness elements, and the turbulent core. Figure (1$) shows a sketch of this model and the resulting shear stress distribution. On the basis that the experimental temperature profiles are "scarcely affected by roughness" Nunner assumed, finally, that the thickness of the viscous wall layer was the same in both smooth and rough pipes. The dferfvatdL-oitsare as follows. In the laminar sublayer where molecular transfer of momentum and heat prevails and where the velocity and temperature profiles can be assumed to be linear, = 2k*. =juM ±L = M ^ M ) T y d =-kA 4!l and = w dy , w y (HQ) ' «i kA (twt ) y y * «, = kA (h9) w s, Combining equations ( 1 + 8 ) and ( U 9 ) , kA Tw^ y w c and t w Let _ t j (51) = be the distance between the pipe wall and the edge of the turbulent core; thus, in terms of Nunnerfts model, (iS^ - i ] ) Is the thickness of the "roughness zone" in which the form drag of the roughness elements is added to the skin friction of the pipe wall. In the roughness zone, let the velocity and temperature profiles be flat due to the large eddy diffusivlty caused by turbulent wakes, let u = i u i.e. (52) c 1 71and (53) Furthermore, since and S.« r bloy#r and w a rou^Knfftt Zone urbulfn^ core (5*0 and even more so is (55) Is* In the turbulent core, apply Reynolds analogy according to which w(u-0 WC (U-t ) = p (56) b -TURBULENT CORE PIPE WALL ^-LAMINAR SUBLAYER ROUGHNESS Figure 18- Solving equation ZONE Sketch of Nunner*s Model of the Flow Conditions in Rough Pipes (56) for (t - t ) and making use of equation S) b A w C T ^ P S i (55)> (57) c Reynolds analogy also postulates U -us, Solving equation ( 5 8 ) for ( t i( - t ) and then making use of equation (57) gives y Aw C- \g P (59) t 72. Adding equations (5l) and (57), (60) Adding equations (51) and (59)/ t.-t,- (t.-t„) (t ,t ) + l - ) + ^ (61) ) Dividing equation (6l) by equation (60) and multiplying numerator and C %^ u denominator by t -t w P _ Y } kr cr w P Hr Sl = \uArJ u u (62 ) At the same Re, however, Is* 1 w _I r_ T (63) s Substituting equation (63) into equation (62) yields the expression for the dimensionless temperature profile in the turbulent core: t -1 > w u*H") + ' The corresponding expression for the laminar sublayer is obtained by dividing equation (50) by equation (60) and substituting equation (63) into the result. tw (65) -U In order to compare profiles predicted by equations (6k) and (65) with Nunner's experimental temperature profiles, i t is necessary to have velocity profile data. Since Nunner's measured velocity profiles in the turbulent core agreed with those of Nikuradse (l6,17), the Rouse (9) equation ft - JI( - -£) 2 5ln + 375 ff + 1 <> 6 73which is based on Nikuradse's data, can be used for this region. Also, since Nunner postulated smooth-tube conditions in the laminar sublayer, equation (k) can be used for this region. Lastly, the edge of the laminar sublayer can be determined by considering equation (k) and the expression used by Nunner for U li_ , v i z •, U Noting that equation (k) can be written as or ^ Re r 4 U . (69) w Equating (67) and (69), there results rw (70) Re Pr' f< 9/8 /6 Thus,substituting equations (66) and (67) into equation (6k) gives for the turbulent core t -t w t w - i.5Re" Pr"(Prfr/f -Q + 2 - 5 ^ In y/r g y = s b t w + 3-75>fc/£ + I l . 5 R e " * r V ± ( r K / f , -l) + I Equation (71) is valid for 81. (by equation 70) < y r \, . w < 1.0 For the laminar sublayer, substituting equations (67) and (68) into equation (65) gives t w "^ l w t b = , O < + < 4L (by eon.ro) P r R e f r (72) 15 Re"* Pr" (Pr f /f - l) + I 7 r & It w i l l be noted that the equations for the temperature profile are consistent with Nunner's expression for the Nusselt number as required by equation (k$). 74. Differentiating equation (72) with respect to y / r w and doubling the answer yields equation (32). tw -t Nunner's experimental profiles are reported as so that Y tw-t, before a comparison of these profiles with those computed from equations (71) and (72) can be made, calculated values of tw-t must f i r s t be multiplied y tw-U by the normalizing factor can be obtained by setting reciprocal of the result. tw-t. t w _ t t -U t w ~ tc . w y- Since t c at y = r w this factor Thus, the desired factor is 1.5 Re"* Pr"* (ft t = t = 1.0 in equation (71) and taking the i-sRe^FV^R-^-l) = y f /f r s + I ( 7 3 ) - l ) + 3.75jf^¥ + I |~An analagous procedure can be used to predict the position at which the point temperature is equal to the bulk temperature. By setting t y = t in b equation (71); i t is found that t y = t b at = e"' = 0.223 1 5 It is predicted^ therefore, that the location of this position is independent of the values of Pr, Re, and f. u r = U at Similarly, from equation (66), JC. - 0.223 J Temperature and velocity profiles have been calculated using the above equations for the cases reported by Nunner; no profiles have been calculated using the equations of Lyon or Martinelli. A representative selection of these profiles is shown in Figures 19a to 19e along with the corresponding experimental profiles. With f held constant and Re varied, as in Figures 19a to 19c, the experimental temperature profiles become more rounded as Re increases. predicted profiles exhibit a similar, though less pronounced, trend. The This behaviour could be expected from equation (7l)> which indicates that the effect of Re on the shape of the profiles diminishes as Re increases. Figure 1 9 a (top) and 19b (bottom): Comparison of Predicted and Experimental Temperature and Velocity Profiles in Rough Pipes 76. 0 or V Re = 6 x i o f = 00575 o- 0 Nunner 4 ol i I 2 0 or L .4 6 •8 10 V Re = 6 x i o f = 002175 4 ® j Nunner _l_ •4 J 6 y / r F l g u t e 19c •8 L 10 » ( t o p ) and F i g u r e 19d ( b o t t o m ) : Comparison o f P r e d i c t e d and E x p e r i m e n t a l Temperature and V e l o c i t y P r o f f c i e s i n Rough P i p e s 77- Profiles are- also-shown for Re held constant and f varied. The predicted curves do not follow the same trend as the experimental curves shown; in fact for f = f s and f = 0.02175* the predicted curves are almost identical. The reason for this behaviour can be found be examining the equation for the turbulent core profile. = 9 ^ t y Combining equations (71) and (73) gives l . 5 R e ~ M ( P r Wf - l ) +2.5jl^lnv/r + 3.75^ + I z = w l.5Re"*Pr"*(R-fr/f -0 +3.75^ s For fixed values of Pr, Re, and y / r of f (or equally well, r Figure 20. fr. fj +1 ( ? u ) " , equation (7*0 is a quadratic in terms w ), and displays the characteristics shewn in An examination of a l l of Nunner's experimental profiles indicates that they do not have these characteristics; moreover they do not 6eem to follow any particular trend as f increases. r It may Taell be that geometry and distribution of the roughness elements influence the experimental profiles The correspondence between the experimental and predicted velocity profiles in the turbulent core is good, as expected, since Nunner claimed that his experimental results agreed with those predicted from equation (66).^" . Only the predicted profiles are shown in Figures 1:9a and ;l9b; no experimental profiles were reported for these cases. In calculating the velocity profiles i t was found that ( ^ Y * i ) b y touaTion (68) = ^ (^ *')by a eijuatie >((><>) and \ *7 by equation ( 6 8 ) 7 \ " b y e^uatio ^Equation (66) gives values of plotted in Figure \$. itx U =u y , whereas values of V = ifv max at y =r w U = = u^fiL ) = ±L(ZISK + \) are , the normalizing factor is calculated from equation (66) by setting Uma« V Since (66) 3— ^ = 1.0; hence (75) f o r y s u b s t a n t i a l l y greater than S, except f o r f r = f s where the reverse was true; the same behaviour was found f o r c a l c u l a t e d values of t - ty w tw-tc The p r e d i c t e d curves shown, however, were l i m i t e d to the unambiguous p o r t i o n of the turbulent core, as were the experimental measurements. The o b j e c t i o n to the use of Nunner's equation f o r the Nusselt number applies equally w e l l to the equations f o r temperature p r o f i l e s . values of t -t U-t w Unrealistic can r e s u l t f o r very low values of Pr. y e In conclusion, then,by c o n s i d e r i n g Nunner's model, equations f o r temperature p r o f i l e s i n rough pipes have been derived. In most cases, the p r e d i c t e d p r o f i l e s show f a i r l y good absolute agreement with the experimental 0 0 1 0 Figure 20. f r A Sketch of C h a r a c t e r i s t i c s E x h i b i t e d by Equation (~{k) p r o f i l e s , though the e f f e c t s of the v a r i a b l e s f and Re ( p a r t i c u l a r l y f ) on the shapes of the p r o f i l e s are not too w e l l accounted f o r . l ne r e s u l t s r substantiate Nunner's conclusion that, i n general, temperature and v e l o c i t y p r o f i l e s i n rough pipes are not s i m i l a r .
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The prediction of heat transfer in rough pipes McAndrew, Murray Alexander 1962
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Title | The prediction of heat transfer in rough pipes |
Creator |
McAndrew, Murray Alexander |
Publisher | University of British Columbia |
Date Issued | 1962 |
Description | An evaluation of methods for predicting turbulent heat transfer in rough pipes has been made with the intention of obtaining a better understanding of the transfer processes involved and of providing a general design equation, valid for all types of roughness shapes and distributions. The equations of Martinelli, Nunner, and Mattioli, along with an empirical method suggested by Epstein, have been tested using the available experimental data. In addition, particular attention has been given to a proposed method which makes use of the velocity profile equations of Rouse and von Karman in Lyon's fundamental equation for the Nusselt number. The results indicate that the proposed method is not successful, largely because of ignorance of velocity conditions near the walls of rough pipes. Mattioli's equation also does not give a satisfactory correlation of experimental results. Epstein's empirical method, which, in the pertinent dimensionless groups, uses friction velocity and equivalent sand-roughness height of the roughness elements in place of the average fluid velocity and the pipe diameter, respectively, shows promise but requires further investigation. Nunner's equation and Martinelli's (simplified) equation give good prediction of the experimental results and are recommended for use at present, providing 0.5 < Pr < 1.0. The success of these latter equations gives support to the hypothesis that the fluid adjacent to a rough wall is probably in laminar motion. Using Nunner's model of the flow conditions in rough pipes, equations have been derived for predicting temperature profiles from velocity profile data. Generally, the absolute agreement between predicted profiles and Nunner’s experimental profiles is good, but the influences of Re and especially f are not too well accounted for. Nunner's conclusion that temperature and velocity profiles in rough pipes are not similar is substantiated by the results. |
Subject |
Heat -- Conduction |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0059149 |
URI | http://hdl.handle.net/2429/39614 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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