Calculation of Condensation with a Portion of Condensate Layer in Turbulent Motion’ ALLAN P. COLBURN,E. I, du Pont de N e m o u r s & Company, Inc., Wilmington, Del.
T
HE,paper by Kirkbride (5) attacks the problem of film-forming condensation by dimensional analysis, whereas i t is felt that the complications of the case warrant an attempt at a more exact solution. Such a solution requires the determination of an expression for the condensation rate at each point down the length of the vertical condenser, and then an integration of the total condensation for the apparatus. Both steps are complicated for cases where the condensate layer becomes so thick near the lower end of the condenser that a portion of it flows in turbulent motion, but a n attempt wiU be made to carry them out with several simplifying assumptions.
McAdams (4) show that the friction factor for turbulent flow in conduits can be used for flow of liquid layers. For heat transfer, the use of the analogous heat transfer factor is suggested even though such a procedure assumes that there is the same thermal resistance to the flow of all of the heat across the entire layer as there is for flow partially across, as is the case when heat flow results from s cooling of the flowing layer. The justification of this assumption is the fact that the main resistance to heat transfer for turbulent flow is across the viscous film next the surface. The friction factor, f , for flow of liquid layers can be expressed:
POIKT CONDITIONVALUESOF HEATTRANSFER COEFFICIENT (5) AND CONDENSATION RATE For viscous flow of the condensate layer, the heat transfer The heat transfer factor, j, is as follows ( 3 ) : coefficient a t any point is equal to the quotient of the thermal j = conductivity, k, of the fluid divided by the layer thickness, . . . . m. This statement assumes that the amount of heat ab- where G = weight velocity, expressed as weight rate per unit stracted from or added to the condensate layer in passing cross-sectional area c = specific heat of fluid the point is negIigible compared to the heat conducted across the layer resulting from condensation. Nusselt (7) Noting that G = C/m and rearranging Equation 6, the showed that, since for viscous flow the thickness is expressed, following is obtained:
(2)(5)”’”
m
3
($y3
(7)
where C = rate of flow of liquid layer, expressed as weight per unit time and unit periphery p = viscosity of liquid p = density g = acceleration due to gravity
For turbulent flow in conduits it has been shown (3) that j . Substituting this equality in Equation 5 and solving for m gives: $f
I
the heat transfer coefficient for viscous flow, h, a t any given point on a vertical surface is equal to: Substituting Equation 8 in 7 and collecting variables into dimensionless groups : Rearranging and multiplying both sides by p2I3:
3c
By analogy with turbulent flow in conduits,
The differential rate of condensation, dC, over a differential length, dL, can be expressed: where X
dC = hAt d L / h = latent heat of vaporization
At =
j = a
The constants a and n are tentatively taken as 0.027 and 0.2; then,
(3)
temp. drop across condensate layer
4c
(kGg)lla (T) )(; 1f3
h
Substituting the value of h from Equation 2a in Equation 3:
= 0.056
0.2
(11)
The differential rate of condensation for turbulent flow becomes :
(4)
For turbulent flow of the condensate layer the evaluation of the heat transfer coefficient is not so simple. I n the absence of experimental data it is possible to make use of an analogy with the case of heat transfer between a fluid and the walls of a rectangular conduit with a large aspect ratio-i. e., ratio of width to heighth. Cooper, Drew, and Presented as discussion of the three preceding papers a t the session on Principle8 of Chemical Engineering a t the meeting of the American Institute of Chemical Engineera, Roanoke, Va., December 12 t o 14, 1933. 1
4c -* (y)
INTEGRATION OF TOTAL CONDENSATION AND DETERMINATION O F AVERAGEHEATTRANSFER COEFFICIENP An exact general integration of Equations 4 and 12 over their respective ranges is difficult, since the thermal properties and temperature difference usually vary along the length of the condenser. ,4n exact integration of a particular problem
432
April, 1934
INDUSTRIAL AND ENGINEERING CHEMISTRY
The critical Reynolds number, 4Cc/p, a t which the type of Aow changes from viscous to turbulent may vary over quite a range, as for flow in pipes. While the value is not known accurately, it will be tentatively assumed as follows:
can be carried out by a graphical method. Sufficient accuracy for most work can be achieved by taking the thermal properties a t a mean temperature (3) and assuming a constant At, and the general integration mill be carried out assuming them constant. For viscous region, integration of Equation 4 gives:
(18)
over the Substituting Equations 18 and 19 in 17 gives:
and combining Equations 13 and 14 so as t o eliminate L gives : 118 4 c -113 (&g) = 1.47
Combining Equation 20 with 14 so as to eliminate L gives:
(--)
(k&jg)li3
hm
-
-
4Cb/lL [(4Cb/p)0.8- 3641
22
3 - (
Alternatively, combining Equations 13 and 14 so as to eliminate C gives the following form of the Nusselt equation:
+ 12,800
(21)
Alternatively, combining Equation 20 with 14 so as to eliminate C:
q'3]1'4
[( ")( ALL
= 0.945
1600
The value of AtLJxp is obtained from Equation 13 when the above equality for 4 C c / p is substituted, giving:
xc At,L
h, =
= b =
@b/p
(13) Defining the mean heat transfer coefficient, h,, length, L:
433
(16)
k3pZg
For the combined viscous and turbulent regions, integration of Equation 12 from a value of C, at L, to Cb a t Lb gives: Equations 21 and 22 can also be expressed generally by including the constants a and b from Equations 10 and 18, respectively, rather than substituting numerical values, as
0 09 OB 07 08
0 5
A.
IO 00
2
3
4
5
6
7
8
9
low
2
3
4
5
6
7
8
9
O
4c -
W
2
3
4
5
6
7
a
When number and diameter of tubes are assumed
o
oowo
10 09
oa 07
06
05
When length
B.
of tubes is as-
sumed 02
01
loop
FIGURE
1. CONDENS~TION FILXCOEFFICIENT
specific heat o i condensate, P. c. u./lb. c = condensate, lb./(hr.)(ft..perimeter of tubes), a t bottom of condenser 0 - acceleratlon of gravity, 4.18 X lo8 Et /(hr.)(hr.) h = condensing vapor-film heat transfer'coefficient, P. c. u./(hr.)(sq. ft.)(O C.). k - thermal conductivity of condensate, P. e. u./fhr.)(sq. ft.)( C./ft.). c =
L = length of tubes, ft. At = temp. difference between vapor and tube surface, X = latent heat of vaporization, P. c. u./lb.
= viscosity of condensate, lb./(hr.)(ft.). = density of condensate Ib./cu. it. (Any other set of self-consiitent units may also be used.)
P P
C.
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INDUSTRIAL AND ENGIKEERING
Figure 1A shows plots of Equations 15 and 21 and data on condensing diphenyl (1). While the data scatter considerably, they show, in general, fair agreement with the predicted line for the turbulent region for a value of c p / k = 5 . Inasmuch as the value of c p / k of water is equal to 1.8 at 100” C. and 1.0 a t 190’ C., data on condensing steam (under am-forming conditions rather than dropwise) would be expected to lie near the lower curve shown on the figure. This plot is convenient to use where the number and diameter of tubes can be conveniently assumed, thus fixing the value of 4C;p and the length is left variant. Alternatively, Figure 1B shows a plot of Equations 16 and 22 and the same data. This plot is convenient t o use where the condenser length is fixed, and the number and diameter of tubes are left variant. I n the latter case it is necessary to estimate the mean temperature difference across the condensate layer, which may require cut-and-try calculations, based on the
CHEMISTRY
Vol. 26, No. 4
value for the over-all logarithmic mean temperature difference. Since condenser lengths are more often fixed than the number and diameter of the tubes, Figure 1B will be preferred by many engineers. High vapor velocities in a downward direction will decrease the film thickness and increase the heat transfer coefficient thereby; they may also cause a transition from viscous to turbulent flow before the value of 4C/p becomes 1600. It is emphasized also that these equations do not hold under conditions causing drop-forming condensation ( 6 ) , but the evidence available indicates that film-forming condensation predominates with organic materials. It is therefore believed that the figures presented make possible a conservative determination of the hest transfer surface.
LITERATURE CITED Badger, W. L., Monra4, C. C., and Diamond, H. W., IND.ENQ. C H E U . , 22, 700-7 (1930). Colburn, A. P., Ibid., 25, 873-7 (1933). Colburn, A. P., Trans. A m . Inst. Chem. Engrs., 29, 174-239 (1933). Cooper, C. M., Drew, T. B., and McAdams, W. H., IND.ENG. CHEV., 26, 428 (1934). Kirkbride, C. G., Ibtd., 26, 425 (1934). Nagle, W. M., and Drew, T. B., paper presente3 a t Roanoke, Va., meeting, Am. Inst. Chem. Eigrs., Dec. 12 to 14, 1933. Susselt, W., 2.Ver. deut. Ing., 60, 541-6, 559-75 (1916).
RECEIVEDJanuary 20, 1931. This paper is Contribution 142 from the Experimental Station of E. I. du Pont de Nemours & Company, Inc.
The Chemistry of Soft Rubber Vulcanization 111. Comparison of Vulcanized Rubber with Unmilled Crude Rubber Reclaims, and Unvulcanized Stocks Containing Stiffeners or Gas Black B. S. GARVEY,JR., The B. F. Goodrich Company, Akron, Ohio Tough unmilled crude rubber, reclaims, and uncured stocks containing gas black or stiffeners have some of the properties of more or less vulcanized rubber. By suitable tests, however, they can be differentiated clearly f r o m vulcanized rubber. T h e s imilar it ies complicate the measurement of vulcanization and make it desirable to exclude these materials f r o m the compounds used in a study of vulcanization. T h i s exclusion is justified by evidence that the
I
h’ PART I (1) of this series’ it was pointed out that the
measurement of vulcanization is complicated by the use of unmilled rubber (e. g., latex stocks), stiffeners, gas black, and reclaims. Therefore limitations in compounding were accepted and a standard method of processing was adopted. These limitations were accepted in recognition of the fact that compounds containing these materials show some properties akin to those of vulcanized rubber. For example, some crude rubbers have stress-strain characteristics almost identical with those of certain types of vulcanizate. Uncured gas black stocks likewise show some of the characteristics of vulcanized rubber, and the reenforcing action of gas black is sometimes spoken of as a sort of vulcanization ( 2 ) . Such limitations are desirable only if it can be shown that the phenomena excluded are different from those being studied. 1
P a r t I1 appeared in IND.ENG.CHEM., 26, 1292 (1933).
phenomena they produce are different f r o m those resulting f r o m vulcanization. The different combinations of properties suggest that there are at least three distinct types of structure incolved: a crude rubber structure, a pigment structure, and a vulcanized structure. Stiffeners tend either to retain the crude rubber structure or to build up a similar one. I n reclaims the vulcanized structure seems to be only partly broken down. The experiments2 reported in this paper show how the tough, hard, crude rubbers can be differentiated from vulcanized rubber and how the effect of gas black and stiffeners can be differentiated from that of vulcanization. The generally accepted view that reclaims are distinct in character from either vulcanized or unvulcanized rubber is supported by this investigation. The experiments also show why it is desirable, for the present, to exclude these materials from the compounds used to study vulcanization. CRUDERUBBER Moderately milled crude rubbers showed no signs of vulcanization by any of these tests. They gave results similar to those of unvulcanized rubber-sulfur compounds (1). With tough crude rubber which was unmilled, or only slightly 2
The tests used were described in detail in the previous paper ( 1 ) ,
