Heat Transfer by Condensing Vapor on Vertical Tubes C. G. KIRKBRIDE, S t a n d a r d Oil Company (Indiana), Whiting, Ind. E R T I C A L - t u b e conby Equation 1, varied 100 per Data are presented on liquid j l m s flowing densers are not a s cent a t times from theoretical down vertical walls. Film thicknesses f o r isocommon in i n d u s t r i a l results. Unfortunately they did thermal flow are measured and correlated with not present their data in detailed practice as horizontal-tube conthe dimensionless groups 2m3p2g/C2and 4C/p. form, so that a further analysis densers. Thisis due t,othree imData available in the literature on heat transfer is difficult. portant reasons: Tube bundles Badger, Monrad, and cannot be "pulled" for cleaning by condensing vapor on vertical surfaces were Diamond (1) p r e s e n t e d data as easily as horizontal tubes, less taken on condensing diphenyl is known about predicting heat correlated by the coordinates hm(k;2gy/3 and on a vertical tube. I n a paper transfer coefficients for vertical by Monrad and Badger (9) an tubes, and it is believed by many which are suggested by the Nusselt equaattempt was made to correlate that lower coefficients of heat the data with Nusselt's equatransfer are obtained with vertition f o r viscous flow of condensate films. While tion (1). They pointed out that cal tubes than with horizontal the data of certain investigators appear to check the data were a t times 400 per tubes. Where cleaning problems the Nusselt theoretical equation f o r values of cent higher than predicted by do not limit the design, it seems Nusselt's theory. The reason that the vertical-tube condenser 4Cp less than 2500, above this value the data in for this was given as turbulence deserves more serious considerathe literature on condensation indicate lhat it is i n t h e condensate film w h i c h tion, and consequently an accuprobably no longer valid but instead gives results Nusselt had assumed did not rate method of predicting heat that are loo low. transfer coefficients for vertical exist. They observed a general trend of increasing heat transfer tubes becomes of considerable importance. A method which is valid for all rates of condensa- coefficients with increasing temperature differences but were tion has not yet been presented in the literature, but several apparently unable satisfactorily to correlate the data so that excellent papers dealing with this subject have been published. they could be applied generally. Xusselt (10) developed the theoretical equation: Callendar and Nicolson ( 2 ) presented data taken on condensing steam inside of a short vertical tube (2 feet or 61 cm.). Their investigation was made in order to determine the amount h, = 0.943( P L ( L - t w ) of condensation in the cylinders of steam engines, but the data for computing the average heat transfer coefficient, h,, over are applicable to the present subject. Jordan (7) has presented data on a vertical tube one meter the tube length, L, where the condensate flows down the tube in viscous motion. The thermal conductivity, IC, the den- (3.3 feet) in length. Monrad and Badger showed that these sity, p., and the viscosity, p, of the condensate are taken a t data checked Nusselt's equation only qualitatively. the arithmetic average of the tube-wall temperature, tu, and The major portion of data on condensing vapors on vertical of the condensate film surface temperature, t,. The heat ab- tubes in the literature are higher than predicted by Nusselt's sorbed above t,, per unit mass of condensate, is equal to the theory. Two different explanations have been proposed for latent heat of condensation plus the sensible heat of the these results. The first and probably the correct one is turvapor' and is represented by X. The acceleration due to bulence in the condensate film. The second is that drops gravity, g, is 4.18 X los feet per hour per hour (989 cm. per form on the tube surface and, upon becoming large enough, second per second) a t sea level. drain from the tube in streams. Schmidt, Schurig, and Jakob and Erk (6) presented experimental results on con- Sellschopp (11) presented pictures to substantiate this exdensing steam on a short vertical tube (46.6 cm. or 18.35 planation. They pointed out that it was necessary, however, inches). They observed that Nusselt's theory gave only fair to have a highly polished surface or a film would form. agreement. Their results, which were lower than predicted They condensed steam on a highly polished copper surface but noticed that, as the surface became slightly oxidized, film 1 Nusselt considered the special cam of condensing saturated vapor, but, condensation occurred. if superheated vapor is condensed, the superheat must be added.
(Zlb
TABLE I. PROPERTIES OF OILS, DIPHENYL, AND WATER LIQUID
l o O D F.
--
-VISCOSITY
SOo F. Centipoises----
70' F
DENSITY
60' F. Lb./cu. ft.
011 1
11.3
17.6
23.5
54.9
Oil 2
23.6
49.0
75.0
55.8
Oil 3
1.15
1.95
2.60
51.2
Oil 4 Water
0.49
0.56 0.86
0.62 0.96
46.2 62.3
200' F. 1.15
400' F. 55.2
Diphenyl
0.68
600' F. 0.24
400'
F.
0.39
425
103' F.
THERMAL CONDUCTIVITY 70' F.
0.0677 6oo/600 0.0677 sp. gr. 6oo/600 0.0677 k - sp. gr. 6oo/600 0.0677 sp, gr, 6oo/600 0.36
k =
gp. gr,
~
=
0.0035 Cp
(+)I/'
- 0.0003(t
- Wl - 0.0003(t - 32)1 11- 0.0003(t - 3211 [1 - 0.0003(t - 32)1 il
0.34
INDUSTRIAL AND ENGINEERING CHEMISTRY
426
Vol. 26, No. 4
f = - 2gm3p2
C2
and this quantity is a function of Reynolds number, 4CI'p. For viscous flow this function is obtained by rearranging Equation 2 as follows:
-
densation occurred only with steam and at very feeble condensate rates. Since, in industrial p r a c t i c e , low cond e n s a t e rateq are n o t c o m m o n , the possibility of dropforming condensation is n o t f a v o r able (see also citation 4). It was observed that at the t o p of t h e t u b e * clear films existed and a few i n c h e s
FILM
TUBE
-_ 2. _I'
FILMTHICKNESS The t,hickness of a liquid layer on a vertical surface under conditions of viscous motion is given by the theoretical equation: m = (3g)"3
If the amount of liquid flowing should become great enough, it might be expected that the layer would become turbulent, and the criterion for this change should be the Reynolds number which, for flow of liquid layers, can be expressed : Re = 4mpV - = "._c !J
!J
Equation 2 can be rewritten in terms of C / p to assist in contrasting the regions of viscous and turbulent motion as: m
(9)"'
=
(a>
113
4c (--)
(4)
To obtain data to compare with this equation, the phenomenon of condensate flowing down a vertical tube was reproduced artificially with a simple apparatus shown diagraminatically in Figure 1. The liquid under study was passed u p through the vertical tube and down the outer wall. It was caught in a catch basin over a given period of time and weighed to 0.01 pound. The thickness of the liquid film on the outside of the tube was measured with a micrometer. The thickness was taken as that at which permanent contact mas made between the liquid and micrometer. The film thickness measured was therefore the maximum and not the average. Where there is no disturbance on the film surface, the maximum thickness will be equal to the average, but such is not always the case. Four hydrocarbon oils and water were studied under isothermal conditions. Their properties are given in Table I. When taking the data, it was made certain that the film thickness was constant all around the tube, and then all observations were made from one location of the micrometer. Observations were made a t various points for each liquid varying from 1 to 2.5 feet (30 to 76 cm.) from the top of the tube, but the thickness was not found to be a function of the height. The observed data are plotted in Figure 2 as f vs. Re/4 and are compared to Equation 4.2 It is seen that t h e agreement with the theoretical equation is good up to a value of Re = 8 . At this point, apparently, the ripples cause a difference in maximum and average film thicknesses, so that generally higher results were obtained above this value. Were it not for the data of Hopf (6) and those of Cooper, Drew, and McAdams (S), the theoretical equation might, on the basis of the present data, be questionable above a value of Re = 8. It is not possible, moreover, to determine the criticai value of 4C;p from the data. hIore exact data are necessary, and there are a t least two methods which might be used with considerably more success than that employed by the author. The first uses a photoelectric cell. A light is directed on a group of lenses which centralize the light rays into a fine beam. The beam is directed through the condenser tube and condensate film, and finally on the photoelectric cell. The current produced in the photoelectric cell circuit is measured, and by use of a calibration curve the film thickness is obtained. The liquid must have considerable color for this method to be employed with auccess. The second method makes use of an interferometer, one beam of light passing through the air and the other through the tube and liquid film. The band widths are measured, and after calibration of the apparatus the actual thickness can be determined. CORRELATION O F HEATT R a N S F E R DlTa
113
=
0.908
(2a)
To compare the results with flow in pipes, it is desirable to derive from this equation a friction factor in the Fanning equation: (3)
Noting that for a vertical film, Ah = L, and V = C / m p ,
The Nusselt equation for condensation on vertical surfaces was derived by expressing the heat transfer coefficient a t any point as equal to the thermal conductivity of the condensate divided by the condensate layer thickness a t that point. The point condition equation was then integrated over the length of the condenser, assuming a constant temperature difference. The condensate layer was assumed to be in 2
Tables giving the detailed data wlll appear in Trans. A m . Inst. Chem.
Engrs.
I N D U S T R I A L AND E N G I N E E R I N G C H E M I S T R Y
April, 1934
viscous motion with a thickness given by Equation 2 . If the layer should be in turbulent flow, its thickness would be greater than calculated by Equation 2 , but its resistance to the flow of heat would doubtless be less. It is therefore desirable to express the Nusselt equation in terms of the Reynolds number for the condensate layer, so that the region of viscous motion, where condensation data may be expected to check the theoretical equation, will be clearly defined, and beyond that point an empirical correlation of data may be obtained. In the derivation of the Susselt equation, the mean heat transfer coefficient over the length of condenser was defined by the equation for the total heat transferred, q , per unit of periphery: q = h,AiL
=
X
427
-
D A T A OF H O P F
f
XC
Substituting for X,’AtL its equivalent, h,/C, in Equation 1 and rearranging, the following expression is obtained : 2
113
h, (k$g)
=
0.925
(;)c
-113
(5)
This equation is also obtainable by dimensional analysis. Although reliable data for condensation in the viscous range are lacking, it is felt that this equation may be used with a fair degree of confidence for values of 4C/p less than 2000, since FIGURE2. V4RIATION OF FRICTION F4CTOR WITH REYNOLDS NUMBER data for laver thickness show good agreement with the thkoretical equation f i r viscks flow. Furthermore, the fair check of condensation data on hori- Lawrence-Sherwood curves as presented by Kirkbride (8). zontal tubes ( 8 )with the theoretical equation for viscous flow Over the range of the data the curve in Figure 3 has a slope of the condensate is additional support for this region. of roughly 0.4 and can be expressed by dimensionless equations For values of 4 C l p above 2000, the type of flow of con- 6 and 6a: densate is not certain. Nost industrial practice occurs above this value, and according to data in the literature it is doubtful that Nusselt’s equation applies. 113 The coordinates suggested by Equation 5 were used to = 0.0084 $);4 correlate the data available in the literature in the turbulent region. The data of Badger, Rlonrad, and Diamond, of Callendar and Nicolson, and of Jordan were successfully Such a curve indicates that the most economical type of concorrelated (Figure 3). One test was also available on a verti- denser would be one which has rather long tubes. Considercal-tube condenser being used to condense a light hydrocar- ably more data are needed, however, to determine the exact bon in commercial service. No tube temperatures were taken, position of the curve in the turbulent region. so the condensate film coefficient was computed from the Figure 3 is plotted in terms of dimensionless groups, which over-all coefficient and a predicted coefficient on the water must be expressed in consistent units. If the designer wishes side. The coefficient on the water side was calculated from a to use certain engineering units, the following multiplying curve which is a combination of the Morris-Whitman and factors may be used:
(
R 8 NICOLSON
I
2
3
5
7
10
20 30
50 70 V X
Kx)
400
700 1000 2ooo
yxx)
10000
lAL FIGURE3. HEATTRANSFER COEFFICIENT AS FUNCTION OF REYNOLDS NUMBER
Using:
hm h m ($)'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 ( = 4 m V p / 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.
