Heat Transfer, Mass Transfer, and Fluid Friction RELATIONSHIPS IN TURBULENT FLOW THOMAS K, SHERWOOD Massachusetts Institute of Technotogy, Cambridge, Mass. The present status of the theory and experimental data relating the three interphase transfer processes of heat transfer, mas11 transfer, and friction is outlined. The nature of flow of fluids in pipes and the concepts of moleoular difision, turbulence, and eddy diffusion are reviewed, and the theories of the over-all process of transfer from a pipe wall to a turbulent fluid stream are summarized. Repreaentative data on the three processes are compared in order to illustrate their basic similarity and the simplicity of the relationships between them.
W
K. LEWIS once listed the chemical engineers' three principal tools as stoichiometry, equilibria data, and the so-called rste equations. Certainly, the laws of conservation of energy and of combining weights are. fundamental to every calculation, design, or investigation which the chemical engineer makw. Equilibria data tell him how far he may expect aprocess to go, and the rate equations tell how fast it will go. Chemical kinetics leads to equations for the rates of chemical reactions. Of at leaat equal importance in chemical engineering are the rates of transfer of fluids through conduits, of heat between fluids and solids, and of diffusion of material from one phaw to another. Fluid friction and heat transfer are important in almost every type of chemical engineering equipment. Diffusion or maa~transfer is often the controlling factor in dryers, gas absorption equipment, cooling towers, humidifiers, and many other devices of engineering importance. Fluid friction involves the transfer of momentum, and can be considered as a rate process along with heat transfer and mass transfer. The important application of these three rate processes is the transfer between a moving fluid and a conduit or other conking wall. The purpose of this paper 19to outline the basic similarities of these three transfer p r o c e m and the relationships connecting them. The subject outlined is broad and complicated, and can be covered only in general outline. For this purpose the discuesion will be limited to transfer between a pipe wall and a fluid passing in turbulent flow through the pipe. Furthermore, the complicated effects observed near the pipe inlet will be ignored and the turbulent nature of the flow assumed to be fully developed. NATURE OF FLUID FLOW IN PIPES
Before proceeding with an outline of the nature of the three transfer processes, it is necessary to review the elements of present knowledge of fluid flow in pipes. If the path of each element of fluid in its passage through the tube were known, it would be possible to calculate rates of heat and mass transfer between the pipe wall and the fluid; inoomplete understanding of the transfer proceases is due in large part to incomplete knowledge of fluid dynamics. In order to express the shear stress at the tube wall under conditions of turbulent flow, the familiar Fanning equation for fluid friction may be written
The empirical friction factor, f, hrae been related to the Reynolds number, Re, by the well known correlation of Stanton and Pannell, who obtained a single curve off us. Re for the flow of air, water, and oils in turbulent motion in smooth pipes. In connection with a discussion of the transfer procesaes, i t is the velocity gradient resulting from a shear stress that is of epecial significance, inasmuch m the rolling eddies that cause the mixing of adjacent layers of fluid are the result of velocity gradients. The fluid velocity, U,is zero at the wall and reaches a maximum, Urn,a t the center of the pipe; the velocity gradient, dU/dr, 19 zero a t the center and a maximum at the wall. In smooth is about 80% of Urn, but in general pipes the average velocity, Uo, both the ratio, Uo/U,, and the velocity gradient, dU/dT, depend on both Re end the relative roughness of the pipe wall. The shear stress, To, at the wall is greater for a rough than for a smooth wall a t the same average flow rate, so dU/dr at the wall must be larger to transfer the greater shear strew. Consequently, Uo/Umis smaller in the case of the rough pipe. If To is held oonstant by varying Uo in experiments with pipes having varying degrees of wall roughness, it is found (1) that the different velocity distributions obtained are represented by an equation of the form
u a utu f f ( T )
(3)
where f ( r ) is the same in each case, but the constant U, has a different value for each wall roughnw. In other words, the velocity deficiency defined as Um U,is the fame function of the radius, r, for all degrees of wall roughness, providing To is constant. The semitheoretical quantitative relation is
-
(4)
which represents the data on water in pipes very well over a wide range of wall roughness. This equation does not give the complete velocity distribution unleea Umis known, but does provide the following general relation for the velocity gradient: (5)
Ae suggested above, ratio UoIUrn depends on Re and the wall roughness, and is found to be a unique function off, BO Urn may be eliminated from Equation 4 to give
2077
2078
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
25
I
I I
LAMINAR
1
I Ill1
I
BUFFER
j
1
I 1 IIll
I
TURBULENT --
Vol. 42, No. 10
eddies which result iu rapid radial mixing and transfer; this process is rapid in comparison with that due to molecular motion. In the intermediate or transition region both procegses contribute to radial transfer.
I I j
I
eo
MOLECULAR DIFFUSION
Although the laminar region near the wall is thin, it may present a relatively large resistance to transfer from fluid to pipe wall, because the fluid in this region is unmixed radially, and transfer is by the slow proceas of molecular motion. In the case of heat transfer the thermal flux, &, per unit area is proportional to the temperature gradient:
IS
4+ IO
5
0
I
/O
YS
IO
/OJ
The proportionality constant, k, is the thermal conductivity, which is small for most tluids. For mass transfer the rate is likewise proportional to a potential gradient. In the case of diffusion in a liquid, for example,
Figure 1. Generalized Velocity Distribution for Turbulent Flow of Water in Pipes After Mnrtiaelli
For simplicity, this is written u + = 5.5
+ 2.5 In y +
(7)
where
and
Here u + is a dimensionless velocity group and y + is a Reynolds number based on u+ and the distance, y, from the wall, Equation 7 is an excellent representation of the veIocit,y distribution across the diameter of a pipe for conditions of turbulent flow in both rough and smooth pipes, but breaks down in the regions very near the wall and near the center line of the pipe. The latter is not important, but good representation of the velocity near the wall is essential if the information is to be used in connection with theories of heat or mass transfer. Explorations of the velocity gradient within a few hundredths of an inch of the pipe wall are very difficult, and only a few such measurements have been made. Figure 1 [after Martinelli (II)] shows the results of Nikuradse and of Reichardt and coworkers. The curved branch at the left is the laminar region near the wall in which the velocity, U,is proportional to the distance from the wall. The straight-line branch at the right is the region of fully developed turbulence, in which Equation 7 applies. The intermediate region is sometimes called the “transition” or “buffer” layer; the velocity here is represented by the empirical equation indicated on the figure. Turning now to the transfer processes-heat transfer, mass transfer, and momentum transfer-it is evident that quite different mechanism may be expected to apply at different distances from the pipe wall. Immediately adjacent to the wall the flow is laminar, with no flow or mixing in a direction normal to the wall. Whatever transfer may take place in this laminar region will be the result of molecular motion in a fluid that may be regarded as stagnant. At the other extreme, the fluid in the main turbulent stream (y+>30) is made up of swirling
.+‘a =
-
n , dc dtl
(9)
where N A is the rate of diffusion per unit area, c is the concentration of diffusing solute, and DL is the molecular diffusivity. Because of the manner in which the diffusivity, D,,in gases is defined, the wrresponding equation for mass transfer by molecular diffusion in gases is slightly more complicated. In streamline or laminar flow momentuin is transferred by the process of viscous ahear, as expressed by the definition of visrasity:
where T is the shear stress on a plane normal to the velocity gradient. This may also be looked upon aa a rate equation, analogous to Equations 8 and 9, expressing the rate of transfer of momentum as proportional to a driving force, where the driving force is momentum per cubic foot. The lefbhand side, T.gd,has the dimensions T’gc
=(
(foot lb./hour) (hour)(sq. foot)
--
momentum (hour)(sq. foot)
(Ill
The potential, pU, is expressed m (foot Ib./hour)/(cu. foot), 01 momentum per cubic foot. Thus the kinematic viscosity, p / p . is a coefficient of momentum transfer in the same way that k and DL are heat and mass transfer coefficients. The rate of momentum transfer to the wall, T o ,is given by Equation 1. TURBULENCE
At high fluid velocities the thickness of the combined laminttr arid buffer layers is small and turbulence is fully developed in the main portion of the stream. By turbulence is meant a condition of irregular motion evidenced by fluctuations in the fluid velocitj a t a point. Thus the instantaneous velocity at a point is U * IA. where U is the time-average velocity and u is the “deviating velocity.” A small Pitot tube may indicate the velocity, U of air in the center of the pipe to be 3000 feet per hour; actually the velocity may be 3035, 2990, 3070, 3018, 2945, etc., feet p e r hour a t intervals of a few milliseconds of time. At any point in a fluid stream the components of the timeaverage velocity are U ,V , and W ; and u, v , and w represent the
-
m
INDUSTRIAL A N D ENGINEERING CHEMISTRY
October 1950
components of the deviating velocity. Inasmuch aa the latter may be positive or negative, it is convenient to employ root mean square values and which for brevity will be represented by u’, u’, and w‘. When these are equal the turbulence is said tombeisotropic. Values of u’, u’, and w’ may be measured by special hotcwire anemometers with fine wires of very low heat capacity. The component ur is often taken as a measure of the intensity of turbulence, and LOOu‘/U is referred to as the “per cent turbulence.” I n addition to information regarding the intensity of turbulence, it is necessary to know something of the “scale” of the turbulence-Le., of the statistical size of the eddies. There are various approaches to this equation, none of them completely satisfactory.
4 2 , 43,
G,
1.0
@
ck Y
8
0.5
2
0
i= P
G
E 0
0.2
0
0.4
0.6
PROBC SEPARATION. /N.) Perhaps the simplest and best known is the “mixin Figure 2. Correlation of Velocities at Two Points in Air Stream length” conce t of Prandtl. For this it is w u m e d that the lengtf, L, which a particle of fluid, or eddy, Separated by Distance y Normal to Flow travels before losing its identity corresponds to the distance between two layers moving parallel to one anAU,where AU = u’. In other other with velocities U and U NA,may be taken aa proportional to the concentration gradient, words, an eddy s u r p ahead at a velocity u‘ greater than providing the average velocity, will blend in with the stream of velocity, U AU,and lose its identity. If AU is small, then
+
+
11’-
dU L-&
The mixing length, L has been described as analogous to the mean free path of moiecules in a gas. In round pipes L varies fr?m zero at the wall to about one seventh of the pipe radius at the
where the ro ortionality constant, E is defined as the ‘‘eddy diffusivity? iecause d y is defined aa the mixing length, L,
E = 2 V‘L
(18)
am.
Another characterization of the scale of turbulence is based QU the idea that there should be some correlation between the velocities at two points within a single eddy. The length, Ll,is defined by P m
where RY is the correlation between two values of U at two points separated by a distance y along a line normal to the direotion of flow. Within an eddy the correlation is high, approaching unity, but, if y is greater than the eddy size, the two velocities %rein random relation to each other and R, is aero. A graph of R , us. y, therefore, is a curve falling from R, 1 at y 0 to R, = 0 at some finite value of y. Figure 2 represents meaaupementa of RY in air flowing in the center of a Cinch pipe, well downstream from any bend or other disturbance. In this c88e the scale is found to be 0.22 inch at several air velocities.
- -
The prooeseee of technical importance are those of transfer between the main fluid stream and the pipe wall. Before considering these over-all processes, however, it is pertinent to consider transfer of momentum, ma&9, and heat by the single mechanismof turbulent mixing, or eddy diEusion. At a point in a turbulent stream the v e p t y is U and the deviating velocity normal to the pipe wall is u The concentration of one com onent of the fluid mixture is e pound moles er cubic foot. TEen the velocity of an edd across a plane paratel to the direction of flow is v’ durin a smalfperiod of time, de, and durin a second instant is - V I . n! the first interval, ds, the eddy travel a ditance, L,defined by Equation 12, and lose its identity, so that the eddy returning in the second interval, de, will have a composition c de characteristic of the fluid at the plane g dy. The net flow oithe component diffubing will be
.
v’cde
wd +
in time 2de.
- u’(e +
=
- u’dcd8
where E R ~ C is , the proportionality constant between thermal flux and temperature gradient. The correa onding relation for the eddy diffusion of momentum relates &e shear stress to u‘, Consider two ad’acent layere or lamina of fluid, one moving at velocity U and b e second at velocity U u’, as illustrated by Figure 3. $?hen there will be a
+
0 fSUEAR f $ -STRESS ”+&
EDDY b1FFUSION
+
The process is obviously analogous to the molecular ditrusion of a gas through itself (self-diffusion), in which case the molecular diffusivity, D., is equal to one third the product of the mean free path and the average @peedof the molecules. Exactly the same reasoning applied to the transfer of heat across a turbulent stream a t constant pressure leads to the relation
(14)
The rate of transfer per unit ares, represented by
@ j
3-U
Figure 3. Momentum Transfer between Two Parallei Layers in Turbulent Stream Aftsr Bakhmeteff
lateral motion carryin particles at a velocity u‘ from the second into the first. On the%asis of a unit area of the lane separating the two lamina the rate of mass transfer from tEe second to the first is pu’, and if the particles so transferred become part of the first stream at velocity U,then there will have been a transfer of momentum from the second to the first at the rate of pll’u‘ foot ounds per (hour)* per unit area of the plane separating the two. %he resulting shear stress is equal to the rate of momentum transfer : r g . = -pu’u‘
(18)
(A more careful reasoning leads to -pi% on the right-hand side; ?Erepreaents the time-average of the product u X u . )
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 42, No. 10
roughly 60% greater than cp-i.e., the coefficient of mam transfer is some 60% greater than the coefficient of momentum transfer. Put another way, the turbulent Schmidt group is 0.63 and is essentially independent of Re in Woerts’ tests. Forstall and Shapiro ( 4 ) tabulate values of the turbulent Prandtl and Schmidt groups from several sources; the first ranges from 0.70 to 0.77, and the latter from 0.63 to 0.74. Both can be said to be constant at 0.70, and, unlike the corresponding groups for molecular motion, to be independent of what diffuses. OVER-ALL PROCESS OF TRANSFER FROM STREAM TO PIPE WALL
From the foregoing it is evident that any theory of an over-ail transfer process from a turbulent stream to the pipe wall must allow for the varying contributions of molecular and eddy diffusion across the diameter. Although the Reynolds analogy does not do this, it represents a milestone in the search for better understanding of the transfer processes. Originally presented as an analogy between heat and momentum transfer, it may also be developed for mass transfer. The basic equations for mass and momentum transfer have been shown to be
REYNOLDS NUMBER
Figure 4.
Eddy Diffusion in Turbulent Gas Streams Data of Woertz
n‘* = and
Eliminating u‘ by Equation 12, this becomes
where the “eddy viscosity,” e, replaces v‘L/2. The similarity of this relation to those for the eddy diffusion of mass and of heat is evident. In the general case the transfer coefficients, cp and p, may be added, and the shear or rate of momentum transfer due to the combined action of molecular and eddy motion becomes
The viscosity under isothermal conditions may be assumed constant across the conduit, but e varies from a small value at the wall to a relatively large value in the main stream. Referring to Figure 1, the three regions correspond to conditions of I/. large compared with ep (laminar layer); p and ep of the same order of magnitude (buffer layer); and e p large compared with p (turbulent region). The following relation, obtained from Equations 1, 5 , and 19, gives the approximate variation of across the pipe diameter:
If the variation of the shear stress, T,and the mass flux, N A , be assumed similar to the extent that T I N A is constant, and if p and D are assumed negligible in comparison with ~p and E, then the last two equations may be integrated to give (24)
where CL is the average solute concentration in the main fluid stream and Ci is the concentration at the wall. If the turbulent Schmidt group be assumed constant at 0.70, Equation 24 reduces to
7.0
6.0
I-
I t may be shown that e would be constant across the pipe if U, U were proportional to r2, which is approximately the case.
- (D + E ) dcdY
-
Data on mass transfer by eddy diffusion have been obtained by Towle ($0)and Woerta ($4) in the central turbulent portions of gm streams in round and rectangular ducts. Towle’s data were for the spread of hydrogen and of carbon dioxide in air; Woertz studied the transfer of water vapor in helium, air, and carbon dioxide. Figure 4 shows the data of Woertz. Towle’s results on carbon dioxide in air by a different technique are given as a dashed line, to show the agreement with Woertz’ data on water vapor in air. Figure 6 shows that Woertz’s results with different gases are correlated by plotting Ep DB. Re. The well known Prandtl and Schmidt groups, Cpp/k and p / p D , for molecular motion have counterparts in turbulent flow known as the turbulent Prandtl number, t / E x , and the turbulent Schmidt group, € / E . These are the ratios of the coefficient of momentum transfer, E , to the coefficients of eddy diffusion, EH and E. The dotted line of Figure 5 represents values of ep obtained by Woerta from careful velocity traverses in the apparatus he used to measure E. It is evident that Ep is
5.0 T
k
$ P
4.0
2
3.0
s \ 1
\ 2.0
1.0
0
0
20000 40000 60000 80000 /110000
Rf YNOLDS NUMBER
Figure 5. Correlation of Eddy Diffusivities and Comparison with Eddy Viscosity Data of Woertz
INDUSTRIAL AND ENGINEERING CHEMISTRY
Oatober 1950
+
where is the ratio of the Schmidt group to the turbulent Schmidt group, rEltpD. The constants 5 and 8 originate in empirical equations shown on Figure 1 used to repreaent the relation between u+ and g+. The heat transfer equations of Reichardt (18) and of Boelter, Martinelli, and Jonessen (9)have been transformed into mass transfer equations by Pigford with the following results:
(la,
Reichardt 6~ =
L PDY
Figure 6. Vaporization of Liquids into a Turbulent Air
Stream
CompPriwn of thebraticalequations with date
ks
-
f Us ~4
Here R, is the ratio of the mean concentration difference to the maximum concentration difference wall to center line; Ry is the ratio of average velocity to maximum velocity; and a is a small correction for the variation in the ratio N A / T from wall to center line. If the turbulent Schmidt group is aesumed constant, Equation 26 may be written
If the same derivation is followed with cp and E assumed small in comparison with @ and D, then the result is similar, except that the [Johmidt group, p / p D , replaces the turbulent Schmidt group, 0. The results obtained with two limiting aasumptions suggest the relation
where +D is Borne function of p / p D . The Von K 4 d theoretical equation is of this form. In place of a theoretical derivation of +D, Chilton and Colburn (3)deduced empirically that +D could be expressed simply as (p/pD)’f 8, whence
Because the indications are that e / E is essentially constant, it should be possible to relate the left-hand side to some function of p / p D only for turbulent flow, The corresponding derivation The correspondingrelation for heat transfer is for heat transfer suggests a relation between the heat transfer coefficient, the friction factor, and the Prandtl group. In recent yeam there have been various extensions (IS) of the Reynolds analogy to allow for the varying importance of E and D near the IO’&? pipe wall, and, in the cam of heat transfer, for the variation of physical properties of the fluid with temperature. Perhaps the most instructive of these is that developed by 10-J Von I(&rm& (81) for heat transfer; the corresponding equation for maw transfer has ala0 been obtained (19). Von K h & n ’ s procedure is to employ the generalized velocity distribution shown 10-4 in Figure 1 w a basis for the oalculation of velocity gradients from which p t p is obtained from Equation 23 from the wall to g + 80 by using To for T. I n 10-5 the main stream (g+>30) the I IO IO IO I O ‘ turbulent Schmidt number Y is taken constant at B, and PDL the maas transfer rate Equation 22 integrated. The reFigure 7. Unton’r Data on Rate of Solution Compared with Theories for Mass Transfer in Pipes sult is
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
2082 0.10
1
I
Vol. 42, No. 10
0 o/ 0.008
so the tube diameter did not change appreciably during a test. For the saltwater combinations studied the Schmidt group was in the range lo00 to 3000, and the data make it possible to extend the comparison of Figure 6 over a much wider range. This is done on Figure 7, where the four points shown represent four series of Linton’s tests dissolving cinnamic acid and 2naphthol in water a t a Reynolds number of 1000. It is evident that the empirical Chilton-Colburn equation fib these data more closely than the theoretical equations or the empirical equation of Gilliland. The latter was based on data for which the Schmidt group varied over the narrow range 0.6 to 2.4, and should not be extrapolated so far.
ODob
COMPARISON O F DATA ON THREE TRANSFER PROCESSES
Q OR
I
WETTED-WALL COLUMNS I
I
I
}
I I I I I
0.02 ----JOHNSTON€ AND P/GFORD B/NARY RECT/f/CAT/ON
$0
a004
0 002
o.m/
io00
2
4
6
a/0000
2
4
REYNOLDS NUMBER
Figure 8. Heat, Mass, and Momentum Transfer for Turbulent Flow in Pipes
where h is the heat transfer coefficient and C,p/k is the Prandtl group. In the absence of a complete theory for the over-all transfer process, this approach has proved to be of enormous value tn the correlation of mass and heat transfer data, and in relating both to friction. COMPARISON OF THEORY AND EXPERIMENTAL DATA
From the foregoing, it is evident that the most convenient method of comparing data on heat transfer, mass transfer, and friction is to compare j,, jD, and f/2 assthese quantities vary with Re. This is done for turbulent flow in pipes in Figure 8, which shows points representing several sets of data on absorption and vaporization for gases in turbulent flow in pipes. These points fall on either side of the line for f/2 and 0 to 60% above McAdams’ line (9)forjH, representing a large amount of data on heat transfer. It is at least a fair approximation to conclude that ja = jH = f/2. Linton’s data at the very high Schmidt numbers fall slightly higher than the vaporization data, as suggested by Figure 7. Figure 9 shows points representing data on vaporization from flat plates plotted as jD us. Rez, where Re, is based on the downwind length of the wet plate (8, 12, 14, 16, 17, 29). The solid line is the line for f/2 as given by Goldstein (8). Here again j ~ jH, , and f / 2 agree well. Actually, the situation is more complicated in the case of flat plates than is suggested by Figure 9. A laminar boundary layer extends part way down the plate from the leading edge, and it is important whether all, part, or none of the wet or heated surface is in this zone. A complete correlation of data on flat plates would include, as an additional variable, the length of the dry or unheated zone upstream from the wet or heated zone (10).
The comparison of the several theories with the data on heat transfer is complicated by the fact that the process is not isothermal, and the physical properties of the fluids vary so widely with temperature. For the present purpose it will suffice to compare the theory 0.06 with two sets of mass transfer data: one for mass transfer in gases, the other for xruw transfer in water. 0.04 Figure 6 shows representative data of Gilliland (6) on the evaporation of liquids into a turbulent air stream in a wetted-wall column. Several liquids were 0.02 employed, fed at the top of the vertical pipe so as to form a falling film of liquid. The data shown are for a JO Reynolds number of 10,000 for air flow in 0.01 a 1-inch tube, with f taken as measured a t 0.0081. The turbulent Schmidt group, p, 0 006 is assumed to be 0.63, as indicated by 0.006 Figure 5. The empirical equation derived by Gilliland passes through the points, as it should, and the theoretical 0.004 Von K & r m h relation is also in excellent agreement. This is considered remarkable in view of the fact that the only data employed in placing the Van K6rm6n 0 002 line are the velocity gradients in water from Figure 1, the friction factor, and the value of 0.63 for 8. Linton (7) has recently reported a study 0 001 of the rates of solution of cast tubes of several salts into water flowing in turbulent motion through the tubes. The salts were ~ i i l yslightly soluble in water, Figure 9. Comparison of Data for Flow over Flat Plates
INDUSTRIAL A N D ENGINEERING CHEMISTRY
October 1 9 s 0.10
f
0.08
go
0.06
0
0
0.04
POWELL ,EVAPORATfW OF W T E R -- LORISCH ,EVAPORATION OF WATER
h jD
ja
2083
= friction factor dimensionleas = 4.17 X 10s (ib. mass) X (feet)/ (hour)l(lb. force) = heat transfer coefficient, B.t.u.1 (hour)(sq, foot)(' F.) k
-
6 h
(y)2'a
conductivity, B.r .u.1 (hour)(s , foot)(' F./fo&) JO ko masa trans7er coefficient, lb. mole#/ (hour)(sq. foot) (Ib. moh/cu. 0.Of foot) 0.008 L = Prandtl mixing length feet L1 = scale of turbulence, ket, defiued 0.m by Equation 13 N A = mass transfer rate, Ib. moles/ (hour)(sq. foot) 0.004 P = pressure, Ib. force/sq. foot. Q = t h e r m a l flux, B.t.u./(hour)(sq. foot) 0.002 r = radial distance from pipe axis, feet ro = pipe radius, feet R, = ratio of m e a n c o n c e n t r a t i o n difference to maximum concen100 200 400 600 f00 2000 40006400 1OooO 200rx) 40oaO tration difference, wall to center REYNOLDS NUMBER line Ru = Uo/Um Figure 10. Comparison of Data for Flow Past Single Cylinders R , = rorrelation coefficient b e t w e e n values of U a t two points y feet apart along line normal to flow Re, = Reynolds nuqber based on downstream length Figure 10 shows points represeutiug data on evclporatioii aiid absorption of water from air for single cylindeld placed transverw t = temperature, F. shear stress, on plane parallel to flow, Ib. (force)/sq. foot 2'' to a turbulent air stream. The dotted line represents the' exstress at pipe lb. (force)/sq. foot To = tensive data of Maisel (IO) for evaporation of water into air, = deviating in 2, T / 1 feet/hour "I carbon dioxide, and helium, and of benzene and carbon tetrau', v', w' = root mean square deviating velocities, feet/hour chloride into air from single cylinders. The comparison with jH and f/2 shows even better agreement than in the preceding u+ = cases, and the three may be assumed equal within the'precisioii U = time-average velocity at a point, feet/hour of the data. In this case f/2 represents the skin friction only, I fluid velocity at pipe axis, feet/hour obtained by subtracting the form drag from the total drag as u, velocity of fluid stream, feet/hour given by Goldstein. At high values of Re the skin friction drops in Equatioll 8) u, = velocity near pipe wall to less than 5% of the total drag, and the precision is poor. z = distance downstream, feet In the case of single spheres it is not possible to compare j~ y = distance from pipe wall, feet and jo with f/2, because the skin friction has not been segregated (To TW~P g+ from the total drag, of which it is a very small part. Heat and P mass transfer may he compared on this basis, however, as shown fluid density, Ib,/cu. foot in Figure 11. The points are Maisel's data on vaporization of = viecosity, Ib./(hour)(foot) water from single spheres into turbulent air streams. Maisel's e = time, hours data on benzene, not shown, fall somewhat lower. The upper line t = eddy viscosity, (feet)Z/hour represents the excellent correlation obtained by Williams (%3), @ = turbulent Schmidt number, ~ / p h ' ,dimensionlw based mainly on evaporation from small drops. From these comparisons for several shapes it appears that ju, j D , and f/2 agree closely when the last is based on skin friction only; in oases 90 where skin friction is not known j x and j~ agree fairly closely. As the theory developb further, especially for cases other than flow in tubes, it may be anticipated that the once relatively distinct subjects of heat transfer, masa transfer, Q 10 and fluid friction will become special applicllr % b tions of the broad subject of fluid mechanics. 9
k
0.02
$
= dermal
4;
4;
-
*
6
NOMENCLATURE
C
= concentration of
-
moles/cu. foot
diffusing solute,
4
Ib.
Ci = concentration at wall, Ib. moles/cu. foot
CL
4
= =
E = En =
average concentration of main stream, Ib. moles/cu. specific heat,foot B.t.u./(lb.)( O F.) molecular diffusivity, (feet)*/hour, DL in liquids, D,in gasea masa eddy diffusivity, (feet)*/hour heat eddy diffusivity, (feet)s/hour
2 2
4
6
8 /04
2
4
6
8 10"
RKVNOLDS NUMBER
Figure 11. Comparison of Heat and Mass Transfer Data for Single Spheres
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
2084
ratio of Schmidt group to turbulent PDP D ~ P Schmidt group = Schmidt group, dimensionless c
p/pD
uo do = f2kc e / E = turbulent Schmidt group, dimensionless LITERATURE CITED
(4) Bakhmeteff, B. A., “Mechanics of Turbulent Flow,” Princeton,
N. J., Princeton University Press, 1936. (2) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Trans. Am. Soc. Mech. Engrs., 63,447 (1941). (3) Chilton, T. H., and Colburn, A. P., IND.ENG.CHEM.,26, 1183 (1Q.74). \ _ _ _ _ ,
(4) Forstall, W., Jr., and Shapiro, A. H., Mass. Inst. Tech., Meteor Rept. 39 (July 1949). ( 5 ) Gilliland, E. R., and Sherwood, T. K., IND.END.CHEM.,26, 516 (1934). (6) Goldstein, S., “Modern Developments in Fluid Mechanics,” London, Oxford University Press, 1938. (7) Linton, W. H., Jr., Massachusetts Instituto of Technology, Sc.D. thesis in chemical engineering, 1949. (8)Lurie, M., and Michailoff, M., IND.ENG.CSEM.,28,345 (1936).
Vol. 42, No. 10
(9) McAdams, W.H.,“Heat Transmission,” 2nd ed., New York McGraw-Hill Book Co., 1942. (10) Maisel, D. S., Massachusetts Institute of Technology, Sc.D. thesis in chemical engineering, 1949. ill) Martinelli, R.C., Trans. A m . SOC.Mech. Engrs., 69,947 (1947). (12) Millar, F.G.,Can. Meteor. Mem., 1, No.2 (1937). ENG.CHEM.,24,726(1932). (13) Murphree, E.V., IND. (14) Pasquill, F.,Proc. Roy. SOC.(London),182A,75 (1943). (15) Pigford, R.L., private communication, 1948. (16) Powell, R.W., Trans. Inst. Chem. Eng. (London),18,26 (1940). (17) Powell, R.W., and Griffiths, E., Trans. Inst. Chem. Engrs. (Lond o n ) , 13, 175 (1935). (18) Reichardt, H.,Angeu. Math. Mech. Forschung, 20, 6, 21 (December 1940); Natl. Advisory Comm. Aeronaut., Tech. Mem. 1047 (September 1943). (19) Sherwood, T. K.,Trans. A m . Inst. Chem. Engrs., 36,817 (1940). (20)Towle, W.L.,and Sherwood, T. K., IND.ENG.CHEM.,31,457 (1939). (21) Ton K4rm4n, Th., Trans. Am. SOC. Mech. Engrs., 61, 705 (1939). (22) Wade, S. H., Trans.Inat. Chem. Engrs. (London),20,1 (1942). (23) Williams, G. C., Massachusetts Institute of Technology, 9c.D. thesis in chemical engineering, 1942. (24) Woertz, 13. B.,and Sherwood, T. K., Trans. Am. Inst. Chew. Engrs., 35, 517 (1939).
RECEIVED March 30, 1950.
A Small Industrial Research Laboratory DESIGN AND CONSTRUCTION K. G . CHESLEY Crossett Lumber Company, Crossett, Ark.
T
HE many new labora T h e details of design and construction of a small laboplant designandconstruction ( l e , 16, 18). None of these tories which have been ratory for industrial research on lumber, pulp, paper, and built within the past few wood chemicals are given. The problem of combining publications, however, haa touched on the specific probspaces for offices, laboratories, and large scale or pilot years have directed a conBiderable amount of attenplant equipment into a single, attractive building is dislems of designing and constructing a small industrial cussed. The use of wood to reduce costs and give a versation to the subjects of laboratory design and constructile, attractive interior is high-lighted. research laboratory. The tion. Several papers have value of research and development by smaller indusbeen published recently on the design of educational (1, %, 6, 7, 9,11, 14, 16, 28) and industrial organizations and the desirability of providing the proper facilities are recognized. The advantages of designing and contrial laboratories (9,6, 8, 18, 20-22). In addition to this, many structing separate buildings for chemistry laboratories, pilot papers have appeared on the specialized problems of design for plants, offices, and special functions are obvious. This type of radiochemistry laboratories (10, 19, 17,19, 23-26) and on pilot
Figure 1. Building Floor Plan
Figure 2.
Floor Plan for Expansion
