1318 (19) (20) (21) (22) (23)
I N D U S T R I A L A N D E N G I N E E R I N G C H E 11.1: I S T R Y
Krafft and Vorster, Ibid., 26, 2813 (1893). M a u t h n e r , Ibid., 39,3593 (1906). Oddo, Gatr. chim. itnl., 41 ( l ) , 15 (1911). R o s e n m u n d a n d H a r m s , Ber., 53, 2232 (1920). Stenhouse, Proc. Roy. Soe. (London), 14, 351 (1865); Ann., 140, 284 (1866).
Vol. 24, No. 11
(24) Wuyts and Cosyns, Bull. SOC. ehim.,[3] 29, 686 (1903); Wuyts, Ibid., [4] 5, 405 (1909). (25) Ziegler, Be?., 23, 2469 (1890). RECEIVEDJuly 23, 1932.
A Note on the Prandtl-Taylor Equation A. E. LAWRENCE, E. I. du Pont d e Nemours & co., - ~ N DJ. J. HOGAN,Massachusetts I n s t i t u t e of Technology, Cambridge, Mass.
0
SBORNE REYNOLDS ( I S ) was apparently the first to use an a n a l o g y between heat transfer and friction to d e r i v e a f o r m u l a for heat transfer between a moving f l u i d a n d a solid in c o n t a c t with it. He assumed that the r a t i o of heat a c t u a l l y transferred to that amount of heat which would have been transf e r r e d had the fluid r e a c h e d the t e m p e r a t u r e of the wall w a s e q u a l to t h e ratio of the energy lost in fluid friction to the e n e r g y which would have been lost had the fluid reached the velocity of the wall-that is, if the fluid had come to rest. From this c o n s i d e r a t i o n , the following simple equation was obtained: h = fcG/2
The
Prandtl-Taylor
h
=
is to the precise experimental data of Stender on heat transfer to water flowing in turbulent flow in a aertical pipe. The calculations on these data are made in a manner in f u l l accordance with the deriflation of the equation. It is found that the value of p is a complicated function of inlet water aelocify, inlet water temperature, and average temperature difference between pipe wall and water. It is also found that, ezen under the most favorable conditions, the Prandtl-Taylor equation does not correlate the data as well as the equation of the dimensionless form:
The value of such equations lies in the p o s s i b i l i t y of the prediction of heat transfer or diffusion rates for a given condition of t e m p e r a t u r e , pressure, velocity, and material, in an a p p a r a t u s for w h i c h the f r i c t i o n is known under the s a m e conditions. Ordinarily, friction factors can be obtained e x p e r i m e n t a l l y more easily and accurately than can coefficients of heat transfer. For this reason and because of the wide i n t e r e s t in the PrandtlTaylor equation in this country and in Europe, the f o l l o w i n g study was made of the applicability of the relation.
k
in which the physical properties of the jluid are taken at main-body temperature.
(1)
As stated by McAdams (9) this equation agrees well with the experimental data on fluids of low viscosity flowing inside a horizontal pipe; however, with fluids of high viscosity the values of h predicted by this equation are much higher than those actually obtained. A fluid in turbulent flow within a pipe was later conceived to be divided into two parts; one, next the wall, a thin layer in which the particles move in a straight-line flow parallel to the wall; the other, the main body of the fluid wherein the particles move in all directions. The first section has been designated as the laminar layer, and the second as the turbulent core. Prandtl (IO) in 1910, and Taylor (16) in 1916 stated that, to be in accordance with this concept, the momentum loss-heat transfer analogy should be applied only to the turbulent core. On this basis, they derived, independently, the following equation cfcGl2) (2) 1 T[(CP/k) - 11 where t = ratio of fluid velocity at boundary b e h e e n turbulent core and laminar layer t o average velocity of fluid. h=
equation,
few2 1 + r [ ( c p / k )- 11,
+
Equation 2 is known as the Prandtl Or Prandtl-Taylor equation. By a similar consideration of the analogy between momentum loss and mass transfer (diffusion), together with considerations introduced by a previous study of diffusion, Colburn (dl derived an equation for ,.he rate of diffusion of a soluble gas from an inert carrier gas to 8. Stationary absorbent. I n this equation also the ratio r occurs.
PREVIOUS REPORTS
S e v e r a1 investigators h a v e reported a v e r a g e values of r which they found best to correlate certain experimental data on heat transfer to liquids flowing in pipes. Stanton (14) gave values of 0.29 and 0.34 for r for water, and ten Bosch (1-3), 0.35. Shemood and Petrie ( I d ) obtained individual values of T for each run and gave arithmetic averages of 0.59 for water, 0.48 for acetone, 0.43 for benzene, 0.31 for kerosene, and 0.38 for n-butyl alcohol; they also showed that the individual values of r varied from 50 to 200 per cent of the averages. It is therefore reasonable to assume that the value of r would vary with the nature of the flow and with the fluid considered. Prandtl (11), from an empirical equation for the velocity distribution in the core, the assumption of undisturbed laminar motion in the boundary layer, and the Blasius equation for pressure drop, derived the equation: r = 1.75 (dG/p)-lIa
(3)
However, Prandtl(I.2) stated that the data require that the value of the proportionality coefficient should range from 1 1 to 1.2 instead of being 1.75; he attributes the discrepancy to the lack of a sharp boundary between the turbulent core and the laminar layer. Also, according to ten Bosch ( I ) , the data of Stender (16) indicate that the coefficient ranges from 1.1 to 1.5, and furthermore that the use of a mean value of 1 Colburn (4) gave this equation as r = l.g(dG/p) -1'9. Prandtl ( 1 ) gave T = l.G(Re)-'/s, where Re is the Reynolds number containing the radius rather than the diameter. T o substitute diameter for radius, the coefficient must be divided b y 2-1'8 which gives 1.75. Using the equation f o . o ~ ( ~ G- I// ~~, )and the incorrect value of 1.6, Colburn obtained r = the correct expression would be r = 5 9 f i . 53 E
November, 1932
INDUSTRIAL AND ENGINEERING
STENDER'S DATAOK COOLING WATERFLOWI~V: Do\\> INSIDE VERTICAL BRASSPIPE0.669 INCHI. D . = 1.954 ft./sec. ti = 104' F.
VI
AI = 3.3-3.7' F. where u = initial water velocity t = water inlet temp.
nz = 5.07 ft./sec. Ir = 158OF. 12.2'F.
At
A = av. temp. difference between wall and waxer
1.2 correlates the data less well than does a constant value of
r of 0.35. To present the results of their experimental study of the heating of water fluid in turbulent flow in a pipe, Eagle and Ferguson (6) gave an equation of the same type as Equation 2, but with the addition of one term in the denominator. According to ten Bosch ( I ) , their data might just as well have been correlated by the simpler equation.
CHEMISTRY
1319
The value of r from each set of data is determined by trial and error in the following way: The value of h is found from the data. Then, by assuming a film temperature, the values of p and k are determined, and r is calculated from Equation 2 . Inserting this value of r in Equation 4, the boundary temperature is calculated and a check obtained on the original assumption of film temperature. These calculations have been carried out on two sets of Stender's data. One set was obtained on heating water flowing downwards in a 0.669-inch i. d. brass pipe; the other, on the cooling of water under the same conditions. The effects of m t e r velocity, water temperature, and temperature difference between wall and water were separately investigated. To avoid confusion in plotting, selected representative points from these calculations are shown in Figures 1 and 2, where the full lines are for variable velocity, the broken lines for variable inlet temperature, and the dotted lines for variable temperature difference; in each case the other two variables are constant. From these figures it is seen that r decreases with increasing velocity, and increases with increasing temperature and, usually, with increasing temperature difference. This variation of r with temperature and temperature difference indicates that it is not a function of the Reynolds number alone. Since c ( p / k ) / is dependent on both temperature and temperature difference, the experimental values of r were plotted vs. ( c p / k ) ~ . Figure 3 shows a selected representative group of points from the data on heating water. The desired correlation of the separate effects of temperature and temperature difference is not obtained by this method. It was found that the value of r was sensitive to small changes in the measured quantities. Because of this and be-
CALCULATIOSS WITH STENDER'S DATA To obtain precise information of the variation of r with temperature, a study was made of the accurate data of Stender (15). The appropriate values of the physical properties mere determined from the following considerations: Because of the manner of their use in the derivation of the equation, c must be taken a t the temperature of the main body of the fluid; p and k , however, are to be taken a t the film temperature. The correct mean film temperature has been shown by ten Bosch (3) to be the arithmetic average of the wall and boundary temperatures. The boundary temperature is determined by an equation given by Gibson ( 7 ) :
The friction factor is that for conditions of heat transfer. Ileevil and McAdams ( 8 ) have investigated the effect of heat transfer on pressure drop, and they found, for a given material in a given apparatus, that the friction factor is affected by the temperature difference between fluid and wall. The effect is more marked in laminar flow than in turbulent. The use of a friction factor determined a t a Reynolds number taken a t an average film temperature gives results in reasonably good accordance with the experimental data, and in this work was so determined from Lees' equation for the data of Stanton and Pannell.
%,th&
STENDER'S D.4T.h UI
ON
= 1.954 ft./sec.
HEATING W A T E R FLOWING DOWNINSIDE BRASSPIPE 0.669 INCHI. D. na = 3.28 ft./sec.
VI =
VERTICAL
5.07 ft./seo.
ti = 54' F. Iz = 104OF. t i = 158'F. AI = 3.76-4.27' F. Az = 12.4-13.2' F. As = 34.4-37.2' F . where u. t , and A have aame significance as in Figure 1 .
I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T I< Y
1320
Vol. 24, No. 11
coordinates according to the equation proposed by Dittus and Boelter ( 5 ) for heating2 fluids in turbulent flow in horizontal pipe: = (0.024)
(d;)'.8 -
(Y)"'~
(5)
wherein all the physical properties are taken a t the mainbody temperature of the fluid. The data on the heating of water flowing downward in the 0.669-inch i. d. brass pipe are shown in Figure 8. The correlation of the data is better than appears from the figure, for it was impossible to show all the check points. The line representing Equation 6 is somewhat too low over most of the range of values covered by the data; Table I gives the values of a and n in the equation
--
STENDER'S DATAON HEATING WATERFLOWING DOWNINSIDE VERTICALBRASS PIPE 0.669 INCH I. D. AT WATER-INLETTEMPERATURE OF 53.6" F.
dC/p from 8,200 t o 10.400 0 = dG/p from 19,600 to 21,700 dG/p from 21,300 to 26,200 0 = dG/p from 16,700t o 20,800 at-water-inlet temp. of 66.Z°F. A dG/p from 25,400 to 29,900
X
A
-
wherein all the physical properties are taken a t the main-body temperature. Table I also describes the conditions of the experiments and the range of values covered by each set of data. The average value of a is 0.035 and of n is 0.825, which indicate that Equation 5 will, in general, give values lower than were obtained in these experiments.- However, since Equation 5 represents with reasonable accuracy the data on the heat transfer to fluids in turbulent flow in horizontal pipes, it is recommended for vertical pipes also.
cause the film coefficient of heat transfer F, is not directly proportional to r in the Prandtl-Taylor equation, a fair test of this relation demanded a comparison of the values of h calculated from this equation with the values determined experimentally. This comparison is shown in Figure 3c30 4 where the calculated values of h are plotted as ordinates against the experimental values as abscissas. zM0 The arithmetic average value of T of 0.480 for the heating runs was used for calculating this set of data. loo This value of T is probably lacking in general significance since it is affected by the relative number of runs a t various conditions; however, it should give the best correlation possible of the data from which it was obtained. Because there were a number of points close together, it was impossible to plot them all, and the distribution above and below the line representing agreement between the two values seems (dCbj(10-? uneven. While the percentage spread of lalues in SrENDER'S DATAO U H E 4 T I U G WATER FLOWING DOWNINSIDE Figure 4 is not as great as in Figures 1 and 2 , thus VERTICALBRASSPIPE 0.669 INCH I. D. qhowing that r is disproportionately sensitive to test conditions, the correlation is poor. Since the calculations mere made under conditions particular1Y to =and Boelter also gave an equation for the cooling of fluids i n the equation, it would appear that the Prandtl-Taylor equa- turhulent flow in 3 horizontal pipe' tion is not generally applicable. hd = (00265) (54 Since there is no good correlation of data on heat transfer k between a fluid in turbulent flow in a vertical plpe and the I t ua8 found that the single aet of data obtained b y Stender on the cooling pipe, these data were plotted on logarithmic paper with of uater was correlated equally well by elther equation
(";'>"'(;)>"'
TABLE
Bras8
INSIOE PIPE DIAM. Inches 0.669
Braas
0.669
PIPE
DIRECTIOX OF WATER FLOW
OF I. COXDITIOSS
~ T E S D E R ' . : ESI'ERIMESTS
DIRECTION OF HEATFLOW
a
n
Down
Wall to water
0.0146
0.85
Down
Water to wall
0.0151
0.75
F. 0.85
104-158
4-30
10,000
55-158
4-60
55-158
6-55
Brass
0 669
UP
Wall t o water
0.669
Down
Wall to water
0.0448
Wall t o water
0.0294
0.79 0.85
Down
Brass
1.101
UP
Wall t o water
0.0183
Steel
1,101
Down
Wall t o water
0.0231
0.82
Rusted steel
0.669
Down
Wall to water
0.0191
0 82
65,000 22,000
68,000
55-158
Av. At 0 P. 440
0.85
Steel
1,101
RANGEOF DATA--Entrance temp.
10,000 65,000 14,000 65,000
0.0157
Braas
dG/p
55-158
2-70
10,000
55-104
8-60
25,000 10,000 20,000 13,000
55- 87
42-70
73- 77
30-60
10,000
40,000
30.000
November, 1932
INDUSTRIAL AND ENGINEERING CHEMISTRY (At),
ACKNOWLEDGMENT The authors wish to acknowledge t h e valuable suggestions and encouragement of W. H. McAdams a n d T. K. Sherwood of the Department of Chemical Engineering of the Massachusetts Institute of Technology. NOMENCLATURE All values in consistent units; examples in foot-pound-second system. a = coefficient c = sp. heat of fluid. B. t .u./(lb.) (” F.) d = inside pipe diameter, ft.
f =
friction factor in Fanning equation, A p
=
2 f l p v1 -__ gd
9 = gravit ntional acceleration, ft./ (sec. ) 2
h = coefficient of heat transfer, B. t. u./(sec.) (ft.)2 (” F.) k = thermal conductivity of fluid, B. t. u./(sec.) (ft.)’
(” F./ft.) I = length, ft.. n = exponent t = ratio of linear velocity a t boundary between laminar layer to av. linear velocit G = mass velocity of fluid, lb./S;ft.)z (sec.) v = linear velocity, ft./sec. Af = difference between wall temp. and av. temp. of fluid, ” F.
=
p =
1321
difference between wall temp. and temp. at boundary between laminar layer and turbulent core viscosity of fluid, Ib./(sec.) (ft.) LITER.4TURE CITED
(1)
Bosch, ten, 2. Ver. deut. Z n g . , 70, 911 (1926). Bosch, ten, Schweiz. Baur., 96, 325 (1930). Bosch, ten, 2. Ver. deut. I n g . , 75, 4 0 (1931). Colburn, IKD.ENG.CHEM.,22, 967 (1930).
(2) (3) (4) Pub. 2, 443 (1930). ( 5 ) Dittus and Boelter, UniV, (6) Eagle and Ferguson, Engineerzng 130, 692 (1930); Proc Inst. N e c h . Engrs. (London), 1930, 985, 1074. (7) Gibson, “Mechanical Properties of Fluids,” p. 178, Van Nostrand, 1925. (8) Keevil and McAdams, Chem. 62 Met. Eng., 36, 464 (1929). (9) McArIams, Trans. 2nd World Power Conf , Berlin, 18, 48 (1930). (10) Prandtl, Physik. Z . , 11, 1072 (1910). (11) Prandtl, Ibid., 29, 487 (1028). (12) Sherwood and Petrie, IKD.EXG.CHEM.,24, 736 (1932). (13) Stanton, Trans. Roy. Soc. (London), A190, 67 (1897); this (14)
paper gives derivation of Reynolds’ analogy and attributes derivation to Reynolds. Stanton, Dictionary of Applied Physics, Vol. I, p. 401, MacmiUan, 1922.
(15) (16)
Gtender, Wiss. Verofentlich. Siemens-Konzern, 9, 88 (1930). Taylor, Tech. Rept. Brit. Advisory Comm. for Akronautics, Rept. and M e m . 272, Vol. 11,p. 423 (1916-17).
RECEIVEDJune 2, 1932.
BOO-K REVIEWS INTERSBTIONAL HAND-BOOK OF T H E BY-PRODUCT COKE INDUSTRY. By W . Gluud. American edition (based on revised German edition), by D. L. Jacobson, The Koppers Company, Pittsburgh, Pa. 879 pages. The Chemical Catalog Company, Inc., 419 Fourth Ave., New York, and W. Knapp, Halle (Saale), Germany, 1932. Price, $15.00. THISbook was first published in Germany (Volume I in 1927 and Volume I1 in 1928). After revision of the German text it was rendered into English by A. Thau of Berlin, and this was then expanded and revised for the American edition by the staff of The Koppers Company, Pittsburgh, D. 1,. Jacobson in charge. An English edition also, revising and supplementing the American, has been prepared by E. M. Myers of Dorman, Long and Company, Ltd. Gluud is director of the Gesellschaft fur Kohlentechnik at Dortmund-Eving. A preface to the American edition, by F. W. Sperr, Jr., formerly Director of Research and now consultant to The Koppers Company, says: “The primary objective in presenting this material has been to draw a complete picture of the international by-product coke industry.” The author has accomplished very well indeed the difficult task of placing on an international basis a description of the varied methods and apparatus used in this industrial field by different countries. The book is divided into a Scientific Section, 8 chapters, 154 pages, and a Technical Section, 14 chapters, 682 pages. The reader will have difficulty in segregating always the parts written by the German author and those added by the American firm revising the work. There is no distinct separation in most instances. Arrangement of the material is not always well done by captions of the chapters and subchapter headings. In the Scientific Section, for example, is given a 14-page appendix of Statistics of Coke Production throughout the world. Also a most excellent treatment of the ultimate and proximate chemical make-up of coal, and the nature of nitrogen and sulfur combinations is given the heading “Ultimate Analysis, Moisture Content, and Ash in Coal.” In the Scientific Section is included a rather brief discussion of the processes that go on in a coking mass of coal and the path of travel of the gases in an oven. The question of direction of this travel is left undecided. Very little is given in this connection as to processes of decomposition of the primary gases and tars, and the mechanism of formation of final products. It is stated that the gases leave the oven chamber usually a t a temperature reaching 850” C. a t the end of the period. and sometimes with fast coking as high as 1100” C.
In the Technical Section is found an excellent and full description of coke-oven machinery and accessoiies, both American and European. On the subject of coke-oven types a considerable variety of European types is described, and under American practice the old type Koppers oven and the newer Becker oven, now built by Koppers, are fully covered by the use of papers (previously published) of C. J. Ramsburg and D. L. Jacobson. Preparation of coal by washing is treated fully, several European processes being described. Under American practice, an excellent description of the Koppers Rheolaveur process is presented in detail. By-product recovery apparatus is fully described and illustrated. Recovery of sulfur and of hydrocyanic acid by the Koppers Seaboard wet processes is given 50 pages. In an appendiu, the new European Koppers recirculation oven is described It will be noticed by coke-oven engineers and operating men that this book gives them very little on the extremely important and practical subject of coke-oven heating. Temperatures within the oven and in the flues, and heat transmission and losses are rather meagerly discussed. The book will be found exceedingly valuable by those desiring a comprehensive descriptive survey of plant practice in this field in various countries. HORACE C. PORTER CHEMICAL ENCYCLOPEDIA. AN EPTTOMTZED DIGEST OF CHEMISTRY AND ITS INDUSTRIAL APPLICATIONS. Bv C. 2‘. Kinazett. 5th edition, 1014 pages. D. Van Nostrand Company. Inc.. 250 Fourth Ave., Newkork. 1932. Price, $10.00. THIS book contains about 6000 headings with texts varying in length from 2 words, or a reference, to 13 pages. The subjects treated comprise pure and applied chemistry; descriptions of the elements and their chief compounds; natural and industrial products and their uses; chemical processes; chemical and physical terms and theories; trade names of chemical products; and an occasional nondescript subject, such as Import Duties Act, Legal Matters, and Copyright of Scientific Publications. In an endeavor to increase its value, numerous references are given to recent publicatirms which usually amplify the text. Where, however, the seeker after information is referred immediately to another book or a paper, with no information on the subject being given directly, it can lead only to annoyance. There is entirely too much loose writing encountered throughout the volume to justify a recommendation. Although most of
