Heat Transfer Involving Turbulent Fluids

and Whitman adopted the less logical scheme of correlating the heat transfer coefficients on conditions for the main body of fluid. It was found that ...
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Heat Transfer Involving Turbulent Fluids Correlation of Heating and Cooling Data W. A. KAYEAND C . C. FURNAS,Yale University, New REVIOUS writers h a v e generally conceded that

Haven, Conn.

A theory is advanced to explain the diference between the film coejicients of heat transfer on

whereas, when the fluid is being cooled, it is the hottest Part. This is illustrated in Figure 1, in

fer between a turbulent fluid and a solid wall are different when contact with solid surfaces- The relation besolid walls and the dotted lines tween the two is a simple function of the mainfilm boundariesfor the two cases. the fluid is being heated from stream viscositv. Qualitatively, t h e n , t h e film when it is being cooled. Morris w o u l d be s w e p t a w a y t o a and Whitman c y ) , in Presenting The theory* gives a satisfactory correlation greater extent in the case of coolthe most p u blished between the heating and cooling coejicients for data on heat transfer involving ing liquid or heatinga gas, raisoils and water flowing through conduits and for ing the coefficients in these two oils, a t first attempted to correairflowing through beds of broken solids. cases, since a liquid has a lower late t h e i r heating and cooling viscosity a t higher temperatures, runs on the basis-of mean film against D V / Z (see table while the variation of the viscosity of a gas with temperature properties, plotting (hD/k)/(~Z/k)~,3~ of nomenclature) with the various properties of the fluid taken is just the reverse. at the mean film temperature. This is surely the most logical It should be realized that correlation on mean film propermethod of correlation in view of the accepted theory of con- ties involves merely an inversion of the film for a heating duction through a stationary film to the turbulent body of case as compared with a cooling case having the same value fluid, for the film is the part of the fluid concerned in heat of D V / Z and the same AT across the film. Figure 1 shows transfer. They found for the same value of D V / Z that the such an inversion, for the two cases there illustrated have the cooling coefficient was very much greater than the heating same mean film temperature. I n the case of a liquid, for excoefficient, the discrepancy being 270 per cent in one case. ample, when the liquid is being heated, as in the case on the The same type of correlation on other data leads to divergen- left in Figure 1, the cold side of the film is in contact with the cies between the coefficients in this same direction. In order m a i n stream', i t s to handle the unexplained divergency in their data, Morris viscosity is high so and Whitman adopted the less logical scheme of correlating that it does not tend the heat transfer coefficients on conditions for the main body to be swept away to of fluid. It was found that the cooling coefficients were some- as great an extent as what less than the heating coefficients on this new basis, but when the liquid is they considered this correlation more satisfactory even if less being c o o l e d . I n logical, for the variation between heating and cooling was the latter case, on less. the right in Figure Furnas (4) has found, in working on heat transfer between 1, t h e h o t s i d e of gases and beds of broken solids, that, other conditions being t h e film is toward equal, higher coefficients are obtained when the solid is being the main stream, its cooled, or, to be consistent with the basis of classification for viscosity is s w e p t oils, when the fluid is being heated. To the authors' knowl- away to a greater edge this is the only case in which coefficients for heating gases e x t e n t , t h e film and for cooling gases have been found by the same investigator thickness t e n d s to Heating a liquid Cooling a ltquld with the same apparatus (I, 6, 8). Attempts to correlate re- be cut down, and a sults by different investigators lead to no consistent results, higher heat transfer Identical meon llqutd - film conditions so great are differences in apparatus and technic. The coefficient r e s u l t s . FIGURE 1. TEMPERATURE GRADIENTS IN HEATING AND COOLING OF TURheating-cooling discrepancy for gases, then, is in the opposite In the case of a gas BULENT LIQUIDSTREAM the effect is just the direction from that for liquids. Any explanation of the discrepancies between heating and reverse, higher cocooling coefficients must account for this opposite effect in efhients being obtained for heating the gas because the gases from that in liquids. Such an explanation may be viscosity of a gas increases with temperature. formulated if it is hypothesized that the heat transfer takes In attempting to use such a concept of the exposed surface place through a stationary film with definite boundaries and of a film tending to be swept away with the main turbulent that the main stream is perfectly turbulent-that is, free stream, it should be recognized that doubtless no sharp film from radial temperature gradients, and if it is recognized that boundary, in the form of an exact surface between stagnant the surface of the film in contact with the turbulent main fluid and perfectly turbulent fluid, exists. For the purposes stream has a certain tendency to be swept away with the main of heat transfer considerations, the film ends a t that plane stream and that this tendency is some inverse relation of the parallel to the solid wall a t which the temperature has atviscosity a t the film boundary. Thus the less the viscosity tained main-stream temperature, or simply where the temin the main stream, and hence a t the film boundary, the less perature gradient ends, and the stagnant film concept furwill be the thickness of the film and the greater will be the nishes a satisfactory working hypothesis. At a given plane heat transfer coefficient. When the body of the fluid is being parallel to the solid wall the tendency for the fluid to be swept heated, this surface of the film is the coldest part of the film, into the turbulent main stream, where there is no longer any 783

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temperature gradient, will be greater the further out from the solid wall is the plane under consideration. However, this tendency will also be larger if the viscosity is smaller, so that such a tendency is inversely proportional to the viscosity. d general relationship for the tendency of a liquid in a plane, z, to be swept away may then be set up as a function, F,, on the hypothesis that,

where a, m, n = positive constants x = distance of plane out from solid wall 2, = viscosity a t plane z

CHEMISTRY

Vol. 26, No. 7

But by the hypothesis these critical values are equal: F, = F h

(4)

Then from Equations 2, 3, and 4,

X , and X h are the film thicknesses for cooling and heating the fluid, respectively, since the critical values of F , are a t the film boundary and there is no temperature gradient in the main stream. Likewise, 2, and Zh are main stream viscosities for the same reason. Since heat transmission through a film is by conduction and the temperature drop and mean film temperature are same in the heating and cooling cases under consideration, the trallsfer coefficients are inversely proportional to the film thicknesses:

-

The use of a single term is roughly justified by the consideration that the tendency to be swept away be 0 when z is 0 (at the plane of the solid wall), or when Z is infinity. F , will have different values for different planes parallel to the solid wall. The position of the plane a t which the film must be considered to terminate, beyond which no heat is transmitted by conduction (the dotted lines in Figure l), will be characterized by a certain value of F, a t that plane. All smaller values of F, will characterize actual planes within the film, while any larger value of F, represents a tendency to be swept away so great that the material actually is swept away and is no longer a part of the film. The f l o w of h e a t through the film by c o n d u c t i o n must then be s u c h t h a t the whole temperature change occurs within the film, the temperature a t this terminating p l a n e being equal to the m a i n stream temperature, so that the rate of heat transfer is inversely proportional to film thickness. The value of F, a t such a terminating plane may be called the "critical FIGURE 2. CORREL.4TION OF HEATING value,, of F , for the DATAOF MORRISAND WHITMANON STRAWOIL ON THE BASIS OF MEAN Particular film and FILMTHERMAL CONDITIONS, AND PRE- conditions of flow. DICTED COOLING POINTS Considering a pair of experiments identical except that the film is inverted in one as compared with the other, as in Figure 1, each of the two films would terminate a t a plane at which F, has a critical value, but, since all other conditions of flow are the same, the critical values of F , are the same, for in each case the main stream will cut down the film until the boundary is a plane which has this value of F,. Such a pair of cases is simply a heating and a cooling case under otherwise identical mean film conditions and conditions of flow. For consideration of the relative values of the heat transfer coefficients for heating and cooling, the numerical value of F, need not be known, but only the fact that these critical values of F, are the same in the two cases. Using the subscript c for cooling and h for heating:

(3)

thd

From Equations 5 and 6:

I\

here r

=

n / m , a constant

Equation 7 gives a relationship between the heat transfer coefficients for two cases which are identical as concerns mean film properties, differing only in that one case is a case of cooling while the other is that of heating the fluid. It remains to evaluate the constant, r, by trial. Once it is found for a given fluid, i t should remain a constant for that fluid.

DATAON LIQUIDS Equation 7 has been applied to the data of Morris and Khitman for straw oil, since this is the only set of their data in which heating and cooling data over the same range of mean film conditions are presented for the same oil. Such an application involves the recorrelation of the data on the basis of mean film conditions, which, as has been explained, was not the basis used by Morris and Whitman in their paper. Fortunately sufficient data were published so that ( h D / k ) / (cZ,/k)"3 i and D V / Z may be computed on the basis of mean film conditions for each run. These values for the heating data are plotted as the lower points of Figure 2, through which a curve has been drawn as shown. Each of the points is then projected upward by the use of Equation 7 to an ordinate that would have been found if the case had been one of cooling rather than heating. Since the cooling case would mean a n inversion of the film, such a prediction involves merely the multiplication of the ordinate by the ratio of the viscosity a t the main stream film surface to the viscosity a t the solid wall surface, the ratio being raised to a power, r, which was found empirically to be best taken as 0.5. Each of the lower points in Figure 2 is thus projected upward to give a point in the upper group a t the same value of D V / Z for mean film conditions. A curve is then drawn through these upper points. This is the predicted cooling curve for straw oil, as obtained from heating data alone, with the use of Equation 7 a t r = 0.5, the viscosity data being taken from Morris and Whitman's article. This predicted curve is then traced onto Figure 3. The experimental cooling points correlated on mean film properties are then superimposed on Figure 3. They follow the predicted curve quite closely. It should be emphasized that the curve of Figure 3 is not drawn through the points of Figure 3 but is copied from Figure 2. Its fit to the points in Figure 3 is a measure of the accuracy of the prediction which is quite satisfactory. Dittus and Boelter ( 3 ) have correlated heating and cooling

I N D U S T R I A L .4N D E N G I X E E R I N G C H E M I S T R Y

July, 1934

data for liquids on the basis of main stream propert,ies by the equation :

where a

= 0.0243 L = 0.4 a = 0'0265 L = 0.3

'r

for heating

'> for cooling, McAdams units ,

(6)

Something was done towards including gases in this general equation, but the correlation is rather indefinitt. and may be neglected, especially as it i n c l u d e s n o heating and cooling I500 data by the qame investigator. The IC02 equation holds well eoc 700 for oils and water 600 ( 6 ) . The equation m may he used to test B , a 0 the ability of Equa:d tion 7 to p r e d i c t y 3 0 0 either h e a t i n g o r Pa) cooling coefficients f rom the other. 150 IS EO 30 40 5060 m IM 1% Since its properties DV are obtained f r o m Z FIGCRE 3. PREDICTED COOLINGCURVE s t a n d a r d tabu1aAUD ATORRIS 4 U D \\HITVAN D A T A FOR tion5 ( l o ) ,nater is COOLIIG S T R 4 W O I L CORREL4TED ON chosen as the fluid M E 4iY FILMT H E R W 4 L CONDITIOUS f o r ,,onsideration. The method of testing used is to assume a set of main stream conditions for heating, for it is upon main stream conditions that the equation of Dittus and Boelter is based, and to assume a wall temperature. These assumptions comprise columns 1 , 2 , and 3 of Table I. The next step is to find the main stream conditions in the cooling case, which would give the same mean film conditions. This involves simply an inversion of the film since each film boundary is a t a temperature which is a main stream temperature in one of the cases, and comprises columns 4, 5 , and 6 of Table I. Then, using the equation of Dittus and Boelter, first for heating and then for cooling, the coefficients may be found, and their ratio is recorded in column 7 of Table I. These ratios are for a heating and a cooling case, each pair having the same mean film temperature, and with DT7 the same.

783

Dittus and Boelter equation. Columns 1 and 2 are obtained from Table I for the same three pairs of assumed cases. Column 3 is the mean film temperature, which holds for both the heating and the cooling case which are being compared. Column 4 is the ratio of the main stream viscosities to the same power as was used previously-namely, 0.5. Column 5 is the inverted ratio of the heat transfer coefficients, equal to column 4 by Equation 7 . Column 7 of Table I and column 5 of Table I1 show good correlation. The measurements of Keevil and Chang ( 5 ) on the heating and cooling of Rabbeth Spindle oil, which were correlated by them on a basis of main stream conditions, have been recalculated on the basis of mean film properties arid correlated by Equation 7 ; the results are shown in Figure 4. The experimental points on the lon-er curve are for heating; those on the upper are for cooling. The solid points represent experimental data; the open points predicted values. For each experiment a value of the coefficient for the equivalent inverse case has been obtained by the use of Equation 7 and plotted. That is, from each actual heating experiment a value of the coefficient for a hypothetical cooling experiment based on the same film conditions simply reverqed has been calculated and plotted, while from each cooling experiment a coefficient for a hypothetical heating experiment has been calculated and plotted. It so happens that the values of Keevil and Chang for DGIZ recalculated on mean film conditions do not overlap in heating and cooling, but the good correlation between predicted and actual cases may be seen. Each curve will be seen by comparison with Figures 2 and 3 to agree closely with the corresponding Morris and Xhitman curve of Figures 2 and 3. The value of r used is 0.5 as before. E x a c t l v t h e same p r o c e d u r e h a s been 1039 followed with the data 830 of C l a p p a n d F i t 2 600 Simons on Velocite B 500 ( 2 ) , and the c o r r e l a 400 tion is shown in Figure 3.

DATAOK GASES I n correlating t h e : data of Furnas (4) on ';Jx gases, only the experi&co rnents on Danube ore, Corsica ore, and liineFxperimentol onfa s t o n e w e r e used beo Predicted pafnts cause these provide the 20 30 40 W W 80 100 f u l l e s t sets of data. T.4BLE I. RATIOOF COOLING TO HEATING COEFFICIENT E BY DITTCSAND BOELTER EQVATIOX These data were obz tained from the FIGURE 4. CORRELATIOK OF D 4 T A 1 2 3 4 5 6 7 T perature history curms OF K E E \ I L OY R ~ B B E T OHI L ou THE of b e d s o f b r o k e n B ~ S IOF S &TEAT F I L M T H E R V 4 L C O N D V/Z, ST RE.^ heating heating \Tall Main 11.41~ DITIONS, AND PREDICTED P O I h T S bolids, by the method H E A T I X G (as(asroolstream STHEAM 2 C A S E (ASSCMED)eumed) sumed) InR cooling COOLING hh of S c h u m a n n ( 9 ) . 1 10,000 70 150 70 150 21,600 1.42 Average values of y for the Schumann curves were taken 2 50,000 60 140 60 140 117,000 1.50 3 100,000 50 110 50 110 217,000 1.44 from the tabulation of data, and an average value of the ratio of the temperatures a t the two surfaces of the film thus obTABLE11. RATIOOF COOLING TO HEATING COEFFICIENTS tained for the mass of each group of data using the Schumann 1N C A S E S O F TABLE 1 B Y EQUATION 7 curves to obtain solid and gas temperatures a t points spaced 1 2 3 4 5 through a bed. Since viscosity of air may be assumed t o be directly proportional to absolute temperature ( I O ) , the ratio Main stream Wall Mean hc of the absolute temperatures is equal to the ratio of the CASE heating heating film hh viscosities. The value of r was found by trial to be best 1 70 150 110 1.47 1.47 2 eo 140 100 1.53 1.53 1 for air, so this ratio of temperatures is the predicted taken as 3 50 110 so 1.47 1.47 value of the ratio of the coefficients, by Equation 7 . These predicted ratios form column 1 of Table 111. The actual Table I1 involves the calculation of the same ratios as in values of the ratios of the coefficients found by Furnas are column 7 of Table I by the use of Equation 7 instead of the recorded in column 2 of Table 111. The correlation is seen 7

INDUSTRIAL AND ENGINEERING CHEMISTRY

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t o be well within the experimental error. This correlation is likewise @nthe basis of mean film conditions.

Val. 26, No. 7

ILLUSTRATIVE EXAMPLE

Assume that there is a certain oil for which coefficients have been obtained for heating in steam-jacketed pipes. TABLE111. RATIOOF COOLING TO HEATINQ COEFFICIENT Suppose that these data are represented by the oil curve on IN DATAOF FURNAS page 141 of Walker, Lewis, and McAdams (IO)which is corre1 2 lated on mean film conditions. Now suppose that i t is dehc/hh sired to cool this same oil in water-jacketed pipes, under ORm Predicted Experimental Corsica 0.89 0.87 conditions as follows: Danube 0.87 0.90 Limestone 0.87 0.89 c = 0.5 B. t. u./lb./" F. k = 0.4 B. t. u.)sq.'ft./" F./ft. V = 250 lb./sq ft./sec. The tendency of the data of Furnas to show higher coD = 3 in. efficients when heating gases was checked qualitatively by 240 = 63 centipoises ZOS = 20 centiDoises placing a thermometer in a stream of turbulent gas a t a temZm = 4.8 centipoises p e r a t u r e varying from Toil = 150" F. t h e temperature of t h e TWdi= 40' F. thermometer b y a conWe have D V / Z = 37.5. s t a n t a m o u n t . It was found that, for the same From the figure cited, letting T = 100 as was done in the mean film value of D V / Z original correlation, between the gas and the t h e r m o m e t e r bulb and hD = 390 m k for the same temperature difference, the thermome- Since Walker, Lewis, and McAdams had cZ/k = 12, a correct e r a r r i v e d at a steady tion factor should be applied, in the light of later data which temperature more quickly shows (cZ/L)O.3' to come into the correlation: for cooling than for heat5 x 20 = 25 Our cz - = 0L ing the thermometer. k 0.4 CONCLUSION Correction factor = = 1.31 nv In Equation 7 , r is 0.5 -. hD 510 X 1.5 X 0.4 = 102 = 1.31 X 390 = 510; h = for liquids and 1.O for gases Then 1.5 k 3 FIGURE 5. CORRELATION OF DATA so far as present data go. for the equivalent heating case. OF CLAPP AND FITZ SIMONS ON VELOCITEB ON THE BASIS OF Aside from suggesting an Applying Equation 7 , MEAN FILM THERMALCONDI- explanation for a hitherto t TIONS, AND PREDICTED POINTS unexplained phenomenon, = 102 = 370 B. t. u./sq. ft./hr./" F. t h e p r i n c i p a l u s e of h, = hh 48 Equation 7 is probably the obtaining of a heat transfer coIt may be noted that the greater the temperature gradient efficient for design purposes in any particular case of heating or cooling when data only for the opposite case are known. across the film, the greater the ratio of cooling coefficient to Its use requires correlation of data on mean film properties. heating coefficient. The principal objection,that has been raised to such correlaLITERATURE CITED tion is that it places the heating and cooling curves farther Badger, W. L., "Heat Transfer and Evaporation," Chemical apart than does main stream correlation, but if a method of Catalog Co., N. Y . ,1926. exact correlation of the heating and cooling coefficients on Clapp, M. H., and Fits Simons, O., Thesis, Mass. Inst. Tech., mean film properties can be found, whereas the two are com1928. Dittus and Boelter, Uniu. Calif.Pub. Eng., 2, 443 (1930). pletely discordant when correlated on the basis of main stream Furnas, C. C., Bur. Mines, Bull. 361 (1932). conditions, the use of mean film correlation has a decided Keevil, C. S., Thesis, Mass. Inst. Tech., 1930. advantage. Such a means of correlation of the two separate McAdams, W. H., "Heat Transmission," McGraw-Hill Book curves is offered by Equation 7 . Co., N. Y . ,1933.

(,)""

z-

(2)"'"

(~)'"

Morris and Whitman, IND. ENG.CHEM.,20, 234 (1928).

NOMENCLATURE c

E

ED.

heat

D = &&;-if pipe, in. F = function defined by Equation 1 h = film coefficient of heat transfer, B. t. u./hr./sq. ft./' F. k = thermal conductivity. _ .B. t. u./hr./sq. ft./' F./ft. of thickness r = a constant, found by trial to be 0.5 for liquids and 1 for gaseso T = temp., F. V = mass velocity, lb./sq. ft./sec. x = distance from pipe wall toward center of pipe, any units Z = viscosity, centipoises

Royds, R., "Heat Transmission by Radiation, Conduction, and Convection," Constable & Co., London, 1921. Schumann, J. Franklin Insl., 208, 305 (1929). Walker, Lewis, and McAdams, "Principles of Chemical Engineering," McGraw-Hill Book Co.. N. Y., 1927.

RECEIVED February

19, 1934.

NORWEGIAN CHEMICAL INDUSTRY.The consumption of most industrial chemicals in Norway increased in 1933, as compared with 1932, but was still generally below predepression years. Imports of sulfur, phosphate fertilizers, and soda ash declined as a result of increased domestic production; imports of potassium fertilizers dropped further with the continued difficult position of farm populations. Exports of sulfur, calcium nitrate, The above units have been chosen because of their generally calcium carbide, and ammonium nitrate were considerably accepted use in engineering calculations. If it is desired t o ob- larger than in 1932- but shipments of cyanamide, nitric acid, tain the dimensionless groups D V / Z , hD/k, and cZ/k of Figures and sodium nitrate declined. Conditions during the first quarter 2 t o 5 in any set of consistent units, the value of D V / Z obtained of 1934 generally have been uiet, but with a noticeable increase from the units above must be multiplied by 124,hD/k by 0.0833, in imports of potassium fertiyizers and a sharp drop in soda ash. and cZ/k by 2.42. This correspondsto multiplying the abscissas Exports of calcium nitrate reached the record figure of 145,049 tons for the quarter. of Figures 2 to 5 by 124 and the ordinates by 0.0601.