Mechanism of Heat Transmission

August, 1931 (9) (10) (11) (12) (13) (14) (15) (16)

INDUSTRIAL A N D ENGINEERING CHEMISTRY

Holden, Thesis, Mass. Inst. Tech., 1927. International Critical Tables, V, McGraw-Hill, 1927. Josse, Z. V u . deut. Ing., 63, 322 (1909). Keevil and McAdams, Chem. Met. Eng., 36, 464 (1929). Kiley and Mangsen, Thesis, Worcester Polytech. Inst., 1931. Landolt-Bornstein, Phys. Chem. Tabellen, Springer, 1912. Lawrence and Sherwood, IND.ENG.CHEM.,23, 301 (1931). Leveque, A n n . mines, [12] 13, 201, 305, 381 (1928).

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(17) McAdarns, M e c h . E n g . , 62, 690 (1930). (18) Morris and Whitrnan, I N D . ENG.CHEM.,20, 234 (1928). (19) Nash, Thesis, Penn. State College, 1927. ( 2 0 ) Nusselt, Z . Ver. deut. Ing., 64, 1154 (1910); Forschungsarbeiten, 89, 1 (1910). (21) Poisson, "ThCorie Mathematique de la Chaleur, 11," Bachelier, 1835. (22) Smith, IND. ENG.CHEM.,12, 1246 (1930). (23) White, Thesis, Mass. Inst. Tech., 1927.

Mechanism of Heat Transmission I-Distribution of Heat Flow about the Circumference of a Pipe in a Stream of Fluid' T. B. Drew and W. P. Ryan DEPARIMENT OF CHEMICAL ENGINEERING. MASSACHUSETTS INSTITUTE

OF

TECHNOLOGY, CAXBRIDGE, >f

ASS.

This paper is the first of a series that will describe heat flux and the natures of 0 P R E V E K T miswork completed and in progress on the mechanism of the fluid motion and temunderstanding, it must heating and cooling fluids by "conduction and convecperature d i s t r i b u t i o n are be stated that radiant tion." The following first report is concerned pricompletely determined when heat is not treated. This marily with a preliminary study of the peripheral the external conditions-i. e., entire discussion deals only variation of the rate of heat flow from the surface of kind of fluid, initial fluid temwith those cases of heat exa single, vertical, round pipe placed transversely in a perature, weight rate of flow, change between solids and stream of fluid. That the heat-flow distribution curve surface temperatures, etc.fluids in which radiation plays has maxima at the front and back of the cylinder and are fixed. Hence, in general, a minor part and, at most, minima at the sides, as predicted by Lohrisch from isothe phenomena mentioned can be regarded as a small thermal absorption data, is confirmed by an experimay be expected to be related additive correction. Furthermental method which is self-reproducible within 4 per to each other. Especially more, although some of the cent. is this true of the velocity facts adduced and concluFor a vertical, steam-heated, brass tube 3.19 inches and temperature dis t r i b usions drawn may be equally (8.10 cm.) in outside diameter, and at a temperature of tions, on account of the depertinent, those cases are 212" F. (100" C.) in an air stream with a velocity of 24.8 pendence of the v i s c o s i t y omitted in which t,here is a feet per second (7.56 meters per second) and a temperaand the density upon the change of phase at the heating temperature of the fluid. A ture of 75" F. (23.9' C . ) , the mean effective coefficient surface. These limitations knowledge of the mechanism from pipe wall to air is 8.53 B. t. u./sq. ft./' F./hr. granted, it is self-evident that of conduction and convec( * 2 per cent). This is in agreement with Rieher and the rate of heat-flow from a tion exists when the interLohrisch, but is much higher than the value predicthot surface into a fluid2 is relations of the velocity and able from the results of Hughes on which much used controlled ( a ) by the normal temperature fields, a n d of temperature gradient within current formulas are largely based. the heat flow from various the fluid at the heated surface, and (b) by the thermal conductivity of the fluid at the same parts of the heating surface are understood. Such inplace. Only if the change considered alters the value at the formation, in as precise form as possible, is of vital imporheating surface of the product k(6t/6n), wherein n is measured tance in order that improvements in the design of equipment along the normal to the surface, can the heat flux from a may be facilitated, and safe estimates of performance under particular part of a surface be affected by the alteration of a new conditions may be obtained by extrapolation from existvariable such as the speed of the gas or liquid flowing. Of ing test results. Although the well-known film theory (33) gives a convencourse it is an experimental fact that, other conditions being equal, an increase in the mass-velocity of the fluid passing ient qualitative explanation of the phenomena involved in through an apparatus increases the rate of heat flow from the processes of heat exchange under consideration, the the heating surface as a whole. Nevertheless, it does not details are actually very imperfectly known. For example, follow that the effect is uniform for all parts of the heated until quite recently (Y),the only existing published data on walls. I n fact changes in mass-velocity may reverse the rela- the velocity distribution in a moving fluid when near a solid tive heat-transfer efficiencies of certain parts, and at least con- wall, except for the somewhat incomplete results of Pannell ceivably, may decrease the absolute value of the heat flux (20), had been obtained under isothermal conditions. As a at some places. Such results are easily explained. For ex- consequence of the considerable deficiency of factual material, ample, a change that increases the effectiveness of the up- this laboratory has for some time been conducting a program stream parts of a heating surface may deliver the fluid to of research which it is hoped will result in the elucidation of downstream points a t a relatively higher, and perhaps ab- the mechanism of heat transmission. solutely higher, temperature than that formerly obtaining One of the first problems to be undertaken was that of the there. Analogous statements may be made for other vari- surface variation of the heat flux from a steam-heated pipe ables. placed vertically in a horizontal air duct. The object was Obviously, in the case of a particular piece of equipment, the to extend and complete the work of Reiher (23),who, in 1924, in addition to giving data on mean effective heat-transfer 1 Received June 8, 1931. Presented before the meeting of the American coefficients for pipes perpendicular to a stream, published Institute of Chemical Engineers, Swampscott, Mass , June 10 to 12, 1931. some measurements of the surface temperature distributions In general, discussion and conclusions herein are independent of direcin such cases. An experimental method and preliminary tion of heat flow,save in that an obvious reversal of direction of certain induced effects may he required to be consistent with reversed heat current. results, together with an outline of pertinent available data,

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are discussed below. The final measurements and conclusions will be the subject of a later paper. Other problems now in progress in this field include studies of the velocity distribution about a heated pipe placed perpendicular to a stream, and of the velocity and temperature distributions within a pipe carrying liquid in non-isothermal stream-line flow.

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enough to the wall to give a number of points in the laminar layer sufficient for satisfactory extrapolation. Nevertheless, there still appears to be no reason to suppose that slip occurs. VELOCITY DISTRIBUTION FOR ISOTHERMAL FLOW NEAR A WALL-The isothermal behavior of a moving fluid near the surface of an obstruction in the channel may be most clearly understood if there is first considered van der Hegge Zijnen’s (9) work in 1924 on the velocity distribution near the u p Review of Relevant Data stream edge of a flat plate. This experimenter so placed a KON-EXISTENCE OF SLIPAT WALL-It is generally believed sharp-edged flat plate &-I a wind tunnel- that the surface paralthat in ordinary circumstances, isothermal or not, there is no leled the ai1* stream and, by using a hot-wire anemometer, mapped out the velocity field near the plate. Figure 1 shows, for various velocities of the main stream, what is found a t a distance of 150 em. (59 inches) from the upstream edge. As the distance, n, from the wall increases, the points representing the local linear velocity lie quite closely on 45-degree lines up to a value of n which varies with the speed of the main stream from less than 0.02 em. to about 0.10 cm. After a short transition zone, the points, on the whole, fall on lines with a slope of l/, until the v e l o c i t y of t h e m a i n s t r e a m is reached. This slope is in a c c o r d a n c e w i t h the theory of von Kttrmttn (12). A 45-degree line on a logarithmic plot becomes a straight l i n e t h r o u g h t h e origin when drawn with ordinary uniform r e c t a n g u l a r c o o r d i n a t e s . Hence, these data show the existence near the plate of a thin film of air in laminar flow covered by an outer turbulent film of relatively conFigure 1-Isothermal Velocit Distributions near Surface of Flat Plate a t Point 150 siderable t h i c k n e s s . The interface between c m . f r o m Upstream E&e, for Main-Stream Velocities Shown on Curves these films is naturally not sharply _ - defined, and slip between a moving fluid and a contiguous solid surface. fluctuates somewhat in position. Some authors articularly abroad, apply the terms “film,” I n isothermal laminar flow, the fact that Poiseuille’s experimental law is theoretically derivable on that basis substan- “boundary laydry’ or “Grenzschicht” to the laminar and tiates the conclusion. For turbulent flow, the proof of the hrbulent films combined. Some restrict the terms to mean statement has been accomplished for isothermal conditions the laminar film. I n this discussion, the laminar film only by measuring the velocity distribution in the vicinity of the will be meant by the word “film” when used alone. “Boundboundary with minute Pitot tubes (SO) and with hot-wire ary layer” will denote both films together, as i t includes the anemometers (9). By either of these experimental methods, velocity measurements can be carried out within a very few thousandths of an inch from the and if to the velocities as recorded there are applied corrections for the abnormal behavior of the instruments when near solids, an extrapolation to the wall indicates that the velocity there is zero. At least in the case of the Pitot tube method, the results of Miss Barker (1) appear to justify the method of correction used. The corrections applied in the g anemometer method seem to be entirely legitimate, although conclusive proof of that contention, independent of the results with the Pitot tube, is lacking. d For non-isothermal conditions, the existing I l l published data on velocity distribution are n.DISTANCE FROM WALL (CM) meager, and no measurements have been Dl 42 D 3 DO 01 a2 a3 a4 ID 21) the to be conclu- Figure &Isothermal Velocity Distributions near Surface of Flat Plate a t Various Dismad’ sive. Pannell (20) rerJorts no measurement tances f r o m Uoatream Edee. _ .for Main-Stream Velocity of 39.3 Feet per Second much closer to thewali than 0.01 inch (0.0254 cm.). filiSs ( 7 ) gives points as close as 0.005 inch (0.012 em.), whole layer wherein the velocity differs from that of the main but his main-stream velocity was too high for this to be near stream. If, however, similar data for points near the upstream edge With hot-wire anemometer (P), distance of closest approach was i t is found that the boundanr laver is are

3

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0.0012 inch (0.003 cm.). Theoretically there is no limit in case of Pitot

--+;--,-. ,

Augurt, 1831

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A, and 7.s At quite low values of dG/p (d = diameter of the

E'ieure 3--lsofhermsl Plow of Air a b ~ u (:ydillirr l * I Low Yeiodrie8 ( v = 0.8 m. R~c.,d = 0.05 m.1 [Lohrisch (f6ii

plate for a particular wind velocity (39.3 feet pcr second). The laminar character of the boundary layer at the front of the plate is clearly evidenced by the fact that all except tlic last few points oii the curves for 1: = 2.0 em., 1: = 5 cm., arid z = 20 cm. lie on a 45-dcgee line. In total thickness, m, the houndary layer varies roughly as the square root of the distance z traversed along the plate, the proportionality constant depending upon the air velocity, l., in the wind t,unnel. Wheri the thickness ?a has increased sufficiently to $$%,e the lteyiiolds nrrmber, Re = mVp/p, a value of ahout 3000,' the laminar layer becomes unst.able and collapses into violent eddies made manifest ti) the experimenter by rapid fluctnatioiis in the color of the anemometer wire. The point on the plate d i e r e the collapse occurs naturally varies with the main-stream velocity. Farther downstream the velocity distribut,ion always has the characteristics showu in I%m 1. The nature of isothermal flow a b o u t the front of a cylinder can be described in an analogous m a n n e r . When the surface is first met by the fluid, a retarded boundaty layer develops, wliicli the approaching 8tre;un presses against ..

p h en om e 11a occur. In the case of the plate, no eddy could grow to any great size without being immediately swept away by the air current. With a cylinder, however, aftcr passing around a quarter circumference, the surface drops away from the direction followed by the main stream and, in a sense, the small eddies that have been rolled U D alone the sides are nrovided with a

cylinder, G = mass-velocity of the main stream) i t is possible to.6nd two more or less permanent eddies behind the cylinder, as in Figure 3. Generally, however, such a condition is unstable even at quite moderate velocities, for, as is well shown in F i y r e 4, the eddies tend to accumulate fluid from the main stream, soon become too large to be protected by the cylinder, arid are swept downstream. I n the steady state it is found that eddies are shed alternately, first from one side of the cylinder and then from the other. A study of any of the photographs will show that the direction of rotation of the eddies is such as to give the fluid in contact with rear qiiadrarits of the cylinder a motion opposed to that of the stream. Tliis is most clearly seen in Figures 3 and 4. In consequence, there is a district at the sides of the cylinder where the fluid near the walls is relatively stagnant. Fage, Relf, and their eo-workers have contributed a con&erahie amount of evidence that confirms and completes t.ho picture of isothermal Row around cylinders. FRge (81, in lYZ!). auiiears to Imue demonstrat,&' that the motion in the early part of the boundary layer is laminar in character. IIe found that Pitot t u b e meitsurenients made near the u-all of a cylinder agreed with the theoretical computations of Thorn (31) for the laminar ease. He roughly identified the point a t which the eddies are shed from the cylindrical surface nith the point on the flat p l a t e w h e r e the thick laminar film collapses, and showed that on the cylinder this critical point is moved downstream as the v e l o c i t y increases. Moroover, it was found that the location of the critical point was indicated on a plot of the peripheral variation of normal pressure. Relf and Simmons ($6), in 1925, completed some earlier work (1921) by Relf ($4) on the frequency of the eddies generated behind a cylinder. If .\i is the number of eddies per second shed from the cylinder, it was found that pNd/G was a function of the Reynolds number, Re = dG/p. If consistent units are used, pNd/G is about 0.17 to 0.19 between IZe = 500-and Re = 106; helow Re = 100, pNd/G drops rapidly to zero; above Re = 105, p N d / G rises very rapidly. Relf and Simmons used air and water as the test fluids, and employcd scvcral different methods of experimentation, the results of which appear to agree in so far as the ranges of applicahility overlap. In 1813 Morris ( I ? ) explored, with a hot-wire a,nemonreter, the vicinity of a cylinder in an air stream. If a graph of his readings for points near the surface is made on polar coordinate plotting paper, a somewliat butterfly-shaped figure is obtained. The indicated velocities are low a t tlic front, sides, and back of the cylinder, and high a t intermediate points. The curve is symmetrical ahout a line drawn through the

INU USTRIAL A N D ENGINEERING CHEMISTRY

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data are coricemed, it is apparent that precise infor~nation concerning the iron-isothermal flow ahout cylinders is wanting. ‘rEXPERATURE D I s T R ~ ~ I ~ TIIN~ F NL U I w T h e only experiments known to the authors that involved an investigation of the temperature field around a cylinder during forced convection are those of Ray (#$) in 1920 and 1921, and of I’ramanik ($1) who extended Ray’s work in 1922. The method WRS a pliotographic procedure hy which the heated portions of t,he air surrounding a hot wire were caused to appear lighter in color than the cooler air. A typical photograph is shown in Figure 8. Evidently tirere is a relatively cool region behind the cylinder. Temperature distributions for non-isothermal flow of air in other circornstances have been obtained by I’annell (#ff), Jiirges (IZ), and &is (7) by the making of traverses with ther~nocouples. These sets of data confirm the general belief that there is ordinarily no appreciable discontinuity of temperature between the heated Eurfaee and the fluid, though in F i w r e 7 - 4 s o t h r r m a l Flow of A i l ahout Square Rod ( 1 m.Jsec.1 [Lohrisch (35)l

axis p a r a l l e l t o bhe main strenni, hut n s g n i m e t r i c a l about a sin~ilarlydrawn line perpendicular to the stream. When it is realized that a hot,-wire anemometer cannot show the direction of flow, the agreement between tlre obvervatioiis of Morris and those jnst discwsed bemines apparent. VELOCITY DISTRIHUTION FOR XON-ISOTHEXMAL FLOWTJnder non-isothermal conditiinm, velocity-distribiition data have been obtained by Pannell ($0) in 1916 and Eli&s(7) in 1929. The former studied the flow OS air through a heated pipe, and the latter examined t.he flow of air near a lmt, flat plate. I n neither case did the results differ markedly from those of the isothermal cme. As it was pointed out aboye, t.he experimenters did not approach the wall sufficiently to detect differences from the isothermal that might have existed in the film. &:lidsdid find evidence for the existence of a thick laminar layer near the upstream edge of the plate, thus agroeiiig with Zijnen’s isothermal results. No published data have been found for the caae of a heated cylinder in a stream of fluid, hut rough color hand experiments6 carried out under FIBure 6 -Isothermal Flow Alr -Mus Stream-Lined Pipe (@.am.*e