Using:
hm h m ($)'laj$s)lla
ka
to obtain:
multiply by:
(kgg)l'a
hm (kA)l'a hm aP% 0.00241
W
4c IJ
6.31
where z = viscosity, centipoises and D' = tube diam., inches, and other symbols are as defined in the table of nomenclature, in English units.
ACKNOWLEDGMENT The author is indebted to W. H. McAdams of the Massachusetts Institute of Technology, Allan P. Colburn of the du Pont Experimental Station, and 0. A. Hougen of the University of Wisconsin for their many helpful suggestions and criticisms. Thanks are also due R. L. Geddes of the-Research Laboratory of the Standard Oil Company (Indiana). NOMENCLATURE (IN CONSISTENT BRITISHUNITS) A
C
c
D
f g
Vol. 26, No. 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
428
= = = = = =
G h k L
= = = =
m
=
Q
=
M =
area of condensing surface, sq. ft. condensate rate/unit perimeter = W / H D lb./(ft.)(hr.) , heat capacity, B. t. u./(lb.)("F.) diameter of tube, ft. friction factor in Fanning equation acceleration due to gravity, ft./hr./hr. mass velocity = Vp, Ib./(sq. ft.)(hr.) condensing film coefficient, B. t. u./(sq. ft.) ( O F . ) (hr.) thermal conductivity, B. t. u./(hr.)(sq. ft.)("F./ft.) tube length, ft. molecular weight, dimensionless hydraulic radius = layer thickness, ft. duty, B. t. u./hr.
cross-sectional area of flow,. sa.- it. temp., O F , v = mean linear velocity, ft./hr. weight rate of condensation (per tube), lb./(hr.) (tube) Ah = head loss, ft. At = temp. difference through film, a F. A = latent heat of condensation plus sensible heat of vapor, S
=
t
=
w =
B. t. u./lb.
= density, Ib./cu. ft. p = viscosity, lb./(hr.)(ft.) Subscript c = condensate b = bottom of tube m = average w = wall p
LITERATURE CITED (1) Badger, Monrad, and Diamond, IND. ENQ. CHEM., 22,700(1930). (2) Callendar and Nicolson, Engineering, 64, 481 (1897). (3) Cooper, Drew, and McAdams, IND. ENQ.CHEN.,26,428(1934). (4) Drew and Nagel, paper presented before meeting of American Institute of Chemical Engineers, Roanoke, Va., December 12 to 14, 1933. ( 5 ) Hopf, Ann. Phusik, 32, 777 (1910). (6) Jakob and Erk, Porschungsarbeiten, Ver. deut. Ing., 310. (7) Jordan, Engineering, 87, 541 (1909). (8) Kirkbride, IND.ENQ.CHEM.,25, 1324 (1933). (9) Monrad and Badger, Ibid., 22,1103 (1930). (IO) Nusselt, 2. Ver. deut. Ing., 60, 541 (1916). (11) Schmidt, Sohurig, and Sellschopp, Tec. Mech. Thermodynamik, 1, 53 (1930). (12) Wulfinghoff, Mech. Eng., 55, 410 (1933). RECEIYEDNovember 27, 1933. Presented before the session on Principlee of Chemical Engineering a t the meeting of the American Institute of Chemical Engineers. Roanoke, Va., December 12 t o 14, 1933; abridged.
Isothermal Flow of Liquid Layers C. M. COOPER, T. B. DREW,AND W. H. MCADAXS,Massachusetts Institute of Technology, Cambridge, Mass.
T
From the data of six observers dealing with s o l v e d f o r t h e m e a n linear liquids flowing down the wetted smooth walls of velocity, Here densers and of wettedwall towers in connection vertical towers or over flat plates, it is concluded h = drop in head Over length L with such processes as absorption, rectification, and humidithat the theoretical equations f o r steady isothermal Im = friction factor = hydraulicdepth' fication lends importance to the Stream-line flow apply for values of the Reynolds g = acceleration due to gravity number, Re (=4mVp/p), below 2100. For relationships which describe the Re greater than 2100, scattered data indicate that It is necessary to determine motion of thin layers of liquid flowing under t h e a c t i o n o f the Fanning equation, togetherwith the friction experimentally either f or the gravity over wetted surfaces. c o r r e s p o n d i n g C h e z y coeffactor curve for smooth circular pipes, may be The radii of curvature of these ficient d3x For each crosssurfacesareusuallysogreat relaused as an approximation. Data are not avails e c t i o n a l s h a p e and kind of tive to the thickness of the fluid able f o r determining possible effects of gas wetted surface, f is a function of layer that the Problem is essenvelocity upon the flow of a contiguous thin layer the Reynolds number: tially that of a broad, shallow, of open stream on a flat plate. This Re = 4mVp M case of flow in open channels may be found in any treatise on hydraulics, but the velocity range where p = viscosity for which the formulas are customarily used is far above that p = density of fluid met in the circumstances of interest here. That either turbulent or viscous-laminar motion will occur Usually, in turbulent flow is the Same function in a shallow stream, according as the velocity is high or low, of Re whatever the cross-sectionalshape. An exception might has been definitely proved by the classical color-band method, arise in the present if the contiguous gas phase should as in the work Of Schoklitsch (8)* The genera' use by hy- exert appreciable traction at the gas-liquid interface. I n draulic engineers of the Chezy formula and its modifications the viscous range the desired formulas are contained in Lamb,s laws Equations 4 and 10 (7) which are easily derivable from the shows that the in OPen-channe1 Of flow in pipes to flow in 'pen definition of viscosity. Since the pressure gradient as used by channels. The Chezy formula is merely the Fanning equation HE use of v e r t i c a l con-
v.
(1)
1 m P 0.25 diameter for 8 full circular pipe; m = actual depth for a ahallow stream; i n general m ia the ratio of the cros8 eection of the stream to ths. wetted perimeter.
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1934
FIGURE1. FANNING FRICTION-FACTOR f FOR LIQUIDL.4YERS
US.
429
REYNOLDS NUMBER
Theoretically, for stream-line flow, f = 2 4 / R e on flat plates, and f = 16/Rc for circular pipes; the solid curve a t the right is for smooth circular pipes.
Lamb in these equations is to be understood as the sum of the static pressure gradient and the effective gravitational force per unit volume (7n), a simple change in variable makes his Equation 4 become: q = -C= P
mapg sin a
(2)
3P
where q
= volumetric discharge per unit breadth
fi
= absolute viscosity = angle between the plate and the horizontal
= weight rate of flow per unit breadth 7n = depth of stream
C a
P
m3pg sin a
rgm2
3P
2P
$) w
COMPARISON O F
Equation 2 applies when there is no traction a t the gas-liquid interface; if a uniform tractive force, 7,per unit area acts on the exposed surface, the appropriate theoretical formula is: q = -C =
The theoretical velocity distribution corresponding to Equation 3 is: u = - p g sin a (my (4) - /J J ! where u = local linear velocity at height y above the bottom of the channel The result corresponding to Equation 2 is obtained by setting 7 = 0 in Equation 4.
(3)
7 is positive when it acts to oppose gravity. Both for turbulent and for viscous flow a suitable correction for change in kinetic energy must be introduced if the variation in depth is appreciable, and in short channels allowance for end effects may be needed. These refinements are generally not necessary in the case of thin films.
THEORY WITH DATA
For the study of the experimental data it is convenient to present the results in the form of a plot off vg. Re, so that a the known curves for pipes may be made. By the substitution mV = q , which is the volumetric rate of flow per unit breadth, the formula used in computing f from q and the measured depth, m, is easily obtained from Equation 1: 2gm3 sin a (5) f = *a where sin Ct has replaced h/L* By the Same substitution, Re is Seen to be 4 q P / k which does not involve m. Equation 2 , witha free surface, is equivalent to: for viscous f = 24/Re
(6)
INDUSTRIAL AND ENGINEERING CHEMISTRY
430
MICROMETEFS
A
B
PUMP
FIGURE2. APPARATUS OF CHWANG Equation 6 is plotted in Figure 1, together with the friction factor curve for smooth round pipes as given by Drew, Koo, and McAdams (4). Yarious hydraulic engineers have reported measurements on the flow of shallow rivers, but apparently it is difficult or impossible to obtain results of accuracy under other than laboratory conditions (compare Gibson, 6). I n 1910 Hopf ( 6 ) published the earliest results found of use in the present study. He investigated the flow of water and of sugar solutions along a slightly inclined, polished brass trough, 5.2 cm. wide and about 40 cm. long. The depth was measured by a micronieter. The slopes ranged from 0.008:l to 0.059:l. Except for the smallest depths, the narrowness of Hopf's trough renders the data unsuited for testing the present theories. These theoretical equations are not applicable when the depth is appreciable relative to the breadth of the stream. Moreover, in high-velocity runs, ripples originating at the edges made the measurement of layer thickness uncertain. Kevertheless, a few of the data have been plotted in Figure 1 and are in excellent agreement with Equation 6 at low Reynolds numbers. Claassen (2) in 1918, desiring to test the assumptions of certain theories of heat transmission, studied the flow of water, of molasses, and of 25 per cent sodium chloride solution down the outer vertical walls of a polished, an unpolished but bright, and a rough steel pipe. The outside diameter of the first pipe was 5.0 cm. and its wetted length 135.0 cm.; for the other pipes the diameter was 4.0 em. and the wetted length, 196.0 em. The holdup, from which the average layer thickness could be computed, was determined by stopping the flow, weighing the drainage, and adding to the weight thus obtained the weight of the liquid adhering to the surface. The latter quantity was found by wiping the surface with a tared towel and reweighing the towel. Unfortunately, in more than half of his runs, Claassen either used molasses of unrecorded viscosity or operated a t such a high inlet temperature that cooling by evaporation vitiated the results. Hence it is permissible to show but few of his data in Figure 1. They are not very consistent with themselves, but six of the nine points are close to the theoretical curve. I n 1920 Schoklitsch (8), in the course of an extensive series of experiments on the flow of both clear and muddy water in shallow streams, repeated and extended the experiments of Hopf. He used a trough 25.5 em. wide with a plate glass floor and varnished wood sides. The data are reported only in the form of a small graph in which the depth is plotted against the mean velocity. The slopes ranged from 0.0042:l to 0.0340:l. A number of points have been read from the graph and transferred. after the necessary computations, to Figure 1. I n judging these data, possible errors due to the method of transcription must be taken into account. I n 1928 Chwang (I), with an apparatus (Figure 2) similar
Vol. 26, No. 4
to that of Hopf, measured layer thicknesses and rates of flow over a flat glass plate for slopes between horizontal and 0.23:l. Both water and a viscous hydrocarbon oil were used. The plate was about 25 em. wide. To avoid edge effects, the measurements were made only for a narrow strip near the center line of the plate. The data at low velocities with zero angle of plate inclination are plotted in Figure 1. End effects make the runs a t high velocities incomparable with the other data under FOR MEMURING consideration. Willey (S), in collaboration with one of the present authors, studied the flow of dilute s u l f u r i c acid down the inside wall of a verticalglass tube 1.134 cm. in diameter and 63.5 cm. high (Figure 3). The holdup was measured by stopping the flow, draining, washing out the adhering liquid, and determining the acid in the combined drainage and washings by titration. These data are plotted in Figure 1. They cover the range of Reynolds numbers from 1.53 to 192 and are in rough agreement with Equation 6. Warden (Q), in 1930, employed an apparatus (Figure 4) similar to that of Cooper and Willey. Water was the liquid studied, and the holdup was determined by a modification of the method of Claassen ( 2 ) . The viscosity of the water was varied by controlling its temperature, which for the data presented here ranged from 20" to 50" C. The Reynolds number ranged from 2 i 8 to 7330. Warden used a glass tube with a diameter of 6.35 em. and a length of 73.2 em., and a brass tube with a diameter of 6.18 cm. and a length of 76.4 em.* Examination of Figure 1 shows that the data of the various observers, taken as a whole, support the theoretical Equation 6 for Re less than about 1500; and this limit may be raised to
* The detailed data of these three investlgatlons ai11 appear i n Trans. Am. Inst Chem. Engrs
OF COOPER AND WILLEY FIGURE 3. APPARATUS
I
I\' D
u S T R I A I,
A N D I< N G I N I.: II i N G
(1 t i ii M I S.r II Y
431
iogetlier wit11 the iiioal frictioii factor curves slioold b' w e resiilts of sufficient pr ioti for rriatiy purposes. There is no intlieatioii froin tie data i~vailaldet,liat gas traction at the intcrfaoe l>et,~
