2962
INDUSTRIAL AND ENGINEERING CHEMISTRY
cause of the greater availability in a high molecular weight molecule of hydride ions far removed from the positively charged carbon atom. The occurrence of a molecular \wight maximum in an intermediate temperature range can also be explained. This texnperature represents an optimum condition of balance betryeen the proton transfer reaction n hich tends to lower molecular weight and the hydride transfer reaction which spreads the molecular weight distribution yielding high molecular \\-eight fractions. These reflections would indicate that the activation energy for proton transfer is higher than that for hydride transfer, which in turn is higher than the 10.3 kcal. previously indicated for propagation ( 4 ) . These considerations would also indicate the existence of an optimuin in weight average molerulnr weight but not in number average molecular weight. It is interesting to consider in more detail the hydride transfer process, In view of the differences observed between aluminum chloride and aluminum bromide in the polymerization of 1-olefins (6),it seems probable that the ease of hydride transfer depends not only upon the nature of the carbonium ion and the hydride ion source but also upon the nature of the anionic portion of the active complex. If it is assumed that the hydride ion enters the sphere of influence of the carbonium ion from the qide opposite the position of the anion, then the nature of the carbonium ion-anion bond would be expected to influence the activation energy for hydride transfer. With increasing covalency or decreasing dipole moment in this bond, carbonium ion carbon would become more tetrahedral and result in an increase in the
ErigFn7ring
Vol. 44, No. 12
activation energy for hydride transfer. Many of the observed differences among Friedel-Crafts catalysts and between FriedelCrafts and acid type catalysts may possibly be explained on this basis. The unique catalytic activity of promoted aluminum bromide in the polymerization of the monoalkylethylenes appears to be associated with its high activity for hydride ion transfer resulting from its large size and a high degree of ionic charnoter in the anion-carbonium ion bond. literature Cited (1)
Bartlett, P. D., Condon, F. E., and Sohnieder, A., J. Am. Chem.
Soc., 66, 1531 (1944). (2) Denbigh, K. G., Tmns. Faiadag Soc.. 40,352 (1944). ( 3 ) Donnell, C. K., and Kennedy, R. R I . , paper presented before Division of Fuel, Gas, and Petroleum Chemistry, XIIth International Congress of Pure and Applied Chemistry, Sew York, September 1951. ( 4 ) Fontana, C. &I,, and Kidder, G. A,, J . Am. Chem. Soc., 70, 3745 (1948). (5) Fontana, C. JI., Kidder, G. A , , and Herold, R. J., IND.ENG. CHEM.,44, 1688 (1952). (6) Georgi, C. W.,“Motor Oils and Engine Lubrication,” p. 196, Xev York, Reinhold Publishing Corp., 1950. (7) MacDonald, R. W., and Piret, E. L., Chem. Eng. Progress, 47,363 (1951). (8) Meier, R. L., J . Chem. Soc., 1950,3656. (9) Schmerling, L., paper presented before Division of Petroleum Chemistry, a t the 119th Meeting of the AYEHICAN CHEMICAL SOCIETY, Cleveland, Ohio. RECEIVED for review March 7, 1952.
.kCCEPTED
July 1, 1052.
Heat Transfer in Condensation
Bocess development
Effect of Temperature Variation around a Horizontal Condenser Tube I
LEROY A. BROMLEY, ROBERT S. BRODKEY’,
AND
N O R M A N FISHMAN2
Department o f Chemisfry and Chemical Engineering, Univerrify of California, Berkeley, Calif.
UMEROUS investiyatois (1, 2, 8-10, 14) have reported that there is a large variation in temperature around a horizontal condenser tube. Kusselt ( l a ) assumed in his derivation 4or the heat transfer coefficient that the temperature of the tube is constant, and although this is a fair approximation for a small, heavy-walled copper tube, it is a poor one for a large diameter, thin, stainless steel or Monel tube. It is the purpose of this paper to investigate both theoretically and experimentally the error introduced by the assumption of a constant tube temperature. Recently, Peck and Reddie (14) have made an entirely different derivation and concluded, as vi11 be done here, that conduction of heat around a tube has a negligible effect on the over-all heat transfer coefficient,
Here At is the temperature drop across the liquid film at 4 and
r is the amount of condensate flowing past this point per unit time per unit length of tube. Writing a heat balance on the condenser tube for that section located between the top and (O using the average gradient around the tube a t any point in order to simplify the equation,
is obtained. As long as Atm is not too large in relation to Atw, approximately hwr,dpAt, =
k,r,dplt, (To
Derivation
The Nusselt equation ( I @ , before Susselt makes the assumption of a constant-tube wall surface temperature, may be written
1 Present address, Chemioal Engineering Department, University of Wisoonsia, Madison, Wis. f Present address, Western Regional Research Laboratory, $lbany, Calif.
- rz)
(3)
may be written. Conibining Equations 1, 2, and 3, reniembering that the sum of the individual temperature drops adds up to the total, A&, there results the approximate equation (6):
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1952
Effect of Number of Increments Taken on
Table 1.
Calculation of
Afav
At,
Atav. Att 0.169 0.1683 0.16818 0.0178 0.01630 0.016222 ~
B 6.463
No. Increments 15 50 175 15 50 175
86.176
where /3 and a are dimensionless groups defined by
(5)
and
Numerical Solution for Limiting Case of 01 = 0. Equations 10 and 11 may be solved by successive approximations and integration to give the local temperature drop a t q as well as the average temperature drop across the film. It was found necessary, to obtain any accuracy, to use a very large number of increments in the solution of the equation. Two points were calculated with 50 intervals on a desk calculator. All the points were calculated on an IBM 602A calculating punch, using about 175 increments spaced 3' apart near 0 and 0.01" near 180. Simpson's rule was successively applied to obtain the average value of At. The effect of the number of increments may be seen from Table I. It appears certain, therefore, that the fourth place after the decimal point in the IBM calculation must be correct, although the fifth place may be somewhat uncertain. The heat transferred through the tube is proportional to
A comparison of values of
(7)
2963
Atz s v .
att for numerous values of
Nusselt's Equation 9 ( a = ( a = 0) is given in Table 11.
m)
6 for
and this paper's Equation 10
The latter is a measure of the tendency of the tube to conduct heat circumferentially rather than radially. Equation 4 has not been solved explicitly; however, the two limiting cases have been. Tube of Infinite Circumferential Thermal Conductivity. For this case At is exactly constant and, of course, there is Nusselt's solution. Nusselt's constant depends on the value of the integralt J r
sin'/a
qdp,
which he reported as 3.428. This in-
tegral has been re-evaluated by means of gamma functions as given by Pierce (15) using the tables by Dwight (7). T h e correct value is 3.4495 f 0.0002, and Nusselt's equation for a horizontal tube should read (11) h = 0.7280
DpAt
-0 7
0'4L 03
Here At = Atav., the average temperature drop across the condensate film, and is assumed constant. The equation may be rewritten in terms of p ( a = a)as follows:
0.2b 01-
0
Tube of Zero Circumferential Thermal Conductivity. The other limiting case is obtained when 01 approaches zero, for in this case there is no circumferential heat flow. With this assumption Equation 4 reduces to (6) At _ Att
(
(?)l'*
and Atav. =
?r
1
)''a
At, At, ev. o - (*
CY
80 I
100 I
120 I
140 I
160 I
x"
Eo t tom
~-
Afa sv.
Comparison of -for Various p Values Atz
Att
Af d q
B m
-
0) are sufficiently close together, i t is anticipated that either equation could be used with satisfactory accuracy for design purposes.
180
Figure 1. Calculated Temperature Variation around a Horizontal Condenser Tube When There Is Negligible Heat Conduction around the Tube
ev. -
= a, Hence, if Nusselt's solution and t h e one derived here
(a =
60 I
Angle 4
To D
Table II.
Tube of Finite Conductivity. When a is finite it seems reasonAtav will lie between the values obtained for (Y 0 and Atc
40 I
(10)
+
where
able that -'
8.291
20 I
86.176 65.700 32.316 25.100 15.081 11.650 6.4633 4.9800 3.8779 2.1544 1.0772 1.0607 0.53860 0.25500 0.21000 0.15081 0.0
Nusselt's Eq.9 1.0 0.98484 0.98023 0.96086 0.95035 0.92077 0.90045 0.83716 0.80080 0.76102 0.65083 0.50373 0.50040 0.36172 0.23622 0.20952 0.16958
0.0
At,
-
Atz B V . At, Author's Eq. 10
Deviation,
%
0.000 0.108 0.134 0.237 0.284 0.412 0.483 0.642 0.700 0.75 0.79 0.72 0.72 0.58 0.43 0.37 0.23
...
2964
INDUSTRIAL AND ENGINEERING CHEMISTRY
The dserenoe between the Nusselt values and the author's values is negligible for design purposes. However, the individual heat transfer coefficient across the condensate film a t p = 86 may be in error 6.5%. The Nusselt equation gives a higher Coefficient.
-
Tube Thermocouple Reading, Millivolts (1 Division 0.02 Mv.)
At for various values of p as calculated from equation At; 10 are plotted in Figure 1. There is a large variation in tem-
Curves of
perature when there is negligible heat conduction around the tube (Le., a = 0), and At approaches At; as 4 goes to B O 0 , because the condensate film becomes infinite in length. In order to better approximate conditions at the bottom of the tube, one must substitute a finite drop in place of the infinite drop postulated in Equation 10. Equations accounting for this finite drop (4) show that it is possible to predict the general shape of experimental curves in the vicinity of 4 = 180°, b u t it has not been possible to determine the curve exactly. The general solution to Equation 4 for all CY has not been undertaken as it seemed that little would be gained. It remains to show experimentally that for real tubes of relatively poor conductors that the theoretical Equation 10 predicts approximately the correct temperature distribution except a t the bottom of the tube where the entire theory breaks down because of drop formation.
Figure 3. Typical Tube Thermocouple Reading versus Tube Rotation for Condensation of n-Butyl Alcohol Time, one division = 1 5 seconds or approximately 6.43' rotation
Apparatus Much of the data reported in this paper for temperature variation with angular rotation of the tube were taken with the equipment shown diagrammatically in Figure 2. The rest of the data were taken on a similar tube a t Illinoia Institute of Technology (3,6). Vapor from the stainless steel boiler passes over the stainless ~ t e e condenser l tube and also through the annular space between the 3- and 4-inch glass pipe. Thus, it is possible to observe continuously the nature of the condensation with minimum heat loss from the tube. The condensate from the condenser tube is returned to the boiler after passing through a cooler and rotameter. Excess vapor is condensed in the auxiliary glass condenfer. Water from a constant head tank is passed through a n orifice meter to measure the flow, a thermocouple to give the inlet temperature, and then through the condenser tube. After the water leaves the condenser it passes through a mixer and then over the outlet thermocouple.
FROM CONSTANT LEVEL TANK
Pl
HORIZONTAL
CONDENSER
AUXILA ORIFICE METER
n k
THERMOCOUPLE
TO BOILER
Figure 2.
Vol. 44, No. 12
Diagram of Experimental Horizontal Condenser
tC
INDUSTRIAL A N D ENGINEERING CHEMISTRY
December 1952 1.0
--Q-
Throretlcal
I
(a.0 )
- Erprlmentd
2965
couple 3 for run 4 as the tube was rotated continuously through 360'. There was a random fluctuation of the temperature of about f 1O F . There is no noticeable dip in the temperature a t the top of the tube. This is in contrast to the first data taken ( 6 ) . The explanation is that the temperature at the top of the tube is extremely sensitive to the curvature of the solder and thermocouple a t this point. If there is a slight flat (as there was in the older tube), then a minimum in temperature will be observed at the top of the tube. If there were a bump, then a rather sharp maximum wvpuld be expected. This is predicted theoretically from Equation 10 as at the top of the tube AL is indeterminate and, hence, a small change in curvature will sin rp violently affect -as rp 0, and hence, At a t (p = 0. rp
~
Table IV.
I top
I
I
I
I
I
I
I
20
40
60
80
100
120
140
Angle
I
1
160 bottom
0
Figure 4. Comparison of Typical Experimental Temperature Distribution (a = Finite) with Calculated Ones (CY = 0 )
~~
~~~~
Tube Temperatures and Operating Conditions
(Run No. 4, n-butyl alcohol: average water rate, 4428 lb./hr: average Condensate rate, 143 6 lb./hr ; avera e vapor temperature, i43.9' F.; average inlet water temperature. 67.8' %,: and average outlet water temperature, 75.Q0F.) Thermocouple Readinrrs, O F. Tube Position, Degrees 3 4 5 6 109.2 109.9 0 112.7 118.4 108.2 109.3 112.6 114.4 45 105.0 106.6 108.9 111.6 90 101.0 102.5 135 99.3 106 9 92.8 95.0 96.8 169 91.5 88.2 89.9 180 85.9 93.0 90.1 92.3 92.8 -169 99.4 -135 100.2 100.1 102.3 105.8 105.9 106 6 109.8 112.5 90 -45 108.1 106.2 112.7 115.6 109.2 109 2 112.7 115.6 0
-
The condenser tube is free to rotate and during part of the rum was connected to a gear reducer and turned continuously a t a rate of 1 revolution in 15 minutes. This rate of rotation is slow enough, as i t was observed that thermal equilibrium is established in about 5 seconds after vapor completely fills the condenser, Physical data on the condenser tubes used a t both schools are given in Table 111. < . The four copper-advance (constantan) tube thermocouples were very carefully installed so as to read as nearly a8 possible tube surface temperature. Great care was taken in the installation to have the curvature over the thermocouples the same as that of the tube, because it was thought that a dip in the temperature a t the top of the tube observed previously by one of the authors (6) was due to a slight flat over the thermocouple. The junction mas a t the surface to 0.005 inch below and, thus, all temperature readings are uncertain in this amount or a maximum of about 3" F. The electromotive force was measured by a Leeds and Northrup precision potentiometer, Type 8662. The tube thermocouple electromotive force was recorded by a General Electric portable photoelectric potentiometer recorder, model 8CE5CL19. The multirange instrument was normally used on the 1-mv. full scale range and the necessary back electromotive force was applied. More details of the apparatus and the complete experimental procedure are described by Brodkey (4). Data and Results
Table IV represents typical experimental data taken on the University of California tube. The data are for run4 withn-butyl alcohol. All other data are recorded (4, 6). Figure 3 is a photograph of the trace made by the photoelectric recorder for thermoTable 111.
D a t a on Horizontal Condenser Tubes Matedal of Tube Type KA2 Monel stainless steela metalb
Therms1 conductivity, B t u./hr.-ft.-' F. 14 Len& exposed topapor, in. 79 Outside diameter, in. 0.877 Inside diameter, in. 0.766 Data from University of California. b Data from Illinois Institute of Technology.
16
63 1.313 1.049
Figure 4 is a typical comparison of the calculated curve for 0 with several experimental curves for finite values of a. It is apparent that with this stainless steel tube to assume a negligible heat flow around the tube is certainly nearer to the truth than to assume a constant tube temperature. The rather large deviations occurring at the bottom of the tube are caused by two effects. Since a # 0, there is a rounding of the temperature in this region caused by circumferential conduction. Also, the rather wide departure in the case of water is due to the very large drops formed which cover and, therefore, cause to be produced increased cooling of the tube near the bottom. Thus, one would expect this latter effect to be smaller as the liquid surface tension is reduced. This is found to be the case. a =
Experimental values of
At,
Atw or Ptl Aft &I.
-'
versus 6 are shown in
Figure 5. The curve is that given by Equations 10 and 11; it is almost identical to that given by Equation 9 and, hence, is a check on the Nusselt Equation 8 which is in more common use. The values of a range from 0.066 to 0.175 for the University of California tube and from 0.10 and 0.59 for the Illinois Institute of Technology tube. Qualitative Observations on the Physical Nature of Condensation. Except for acetone and water the liquid film formed covered the entire tube and was extremely smooth except, of course, for the drops a t the bottom of the tube. In the condensation of water vapor it was observed that when the vapor first entered the condenser, and for about 10 minutes thereafter, a smooth film similar t o that of organic vapors formed. Shortly thereafter a disturbance occurred which can be described as follows: The condensate film formed in ridges, which moved along the tube in the direction of vapor flow. Their velocity of travel was estimated to be between 0.3 and 0.4 foot per second (probabl caused by va or flow). The ridges were about I/, inch in widtg and extendecf completely around the tube; there were about three ridges in 1 inch of tube length. The boundaries of the ridges were very well defined although somewhat irregular. They would often run into each other. The valleys between the ridges appeared to be a very thin film of condensate, which in places showed disturbances because of the flow of t u = -
INDUSTRIAL AND ENGINEERING CHEMISTRY
2966 a98
Design Application
0.02
-
01
0.9
ptaw Att
0 .I 0.1
I
42
09 0.6 .e 10 .
2
I 6I 8l l
4
3
IO
I
20
0.9 40
60 80 100
P Figure 5.
Vol. 44, No. 12
The design procedure of Peck and Bromley ( I S ) is, of course, substantiated by the above data. An alternate procedure, which will require some trial and error, is as follows: Figure 5 may be used to give a solution to condenser design for a horizontal condenser. If the average temperature, rate of flow of the cooling water, and tube diameter are fixed for a proposed condenser, then from Equations 5 and 6 p may becalculated directly. Remembering that for n tubes directly above each other, p for one tube is divided by n1/3to obtain the p for n tubes. From Figure 5 Aiz _av.- is knoxn and from q = -A- d t z a v Att
Variation of Average Temperature Drop from Tube Outer Surface to Cooling Fluid Abav, with Parameter, p
B defined by Equation 5
Solid lines calculated from Equations 10 throush 15
ridges. At times the disturbances would almost completely disappear; the valleys in this case appeared somewhat like holes in the film surface. At other times the disturbance would completely disappear, and One have the smooth film condensation for a short period. When either the cooling water or vapor flow was cut off, the condensate formed a smooth film on the condenser tube surface. This fact would indicate that the surface was clean and that the disturbance was not caused by any drop promoter. When a small amount of air n-as introduced into the system during operation a smooth film was immediately formed, but after some time the irregularities would again appear. Somewhat similar behavior was observed with acetone (one of thematerials which was used to clean the equipment) except that the ripples were more regular although less pronounced. I n the assembly of the equipment only graphite and water were used in the joints. No oil or other possible drop promoter was used anywhere in the system.
RZ
( I 6 ) the heat flow Or the mea may be calculated. If drsired, one may, of course, use the Xusselt Equation 8, remembering that the At in the equation is the mean value. One may also use the correlation of Peck and Reddie (14). Conclusions
Although the Nusselt Equation 8 is derived on the very poor assumption that the temperature of a condenser tube is constant, it has been shown both theoretically and experimentally that the error in the calculated heat flox- through a condenser tube from this cause alone is negligible. The temperature distribution derived, assuming no heat conduction around a condenser tube, approximately represents that observed for a 7/8-inch stainless steel tube except near the bottom of the tube. Acknowledgment
One of the authors (6) wishes t o acknowledge the generous help of Ralph E. Peck of the Illinois Institute of Technology who first introduced him to the subject and under whose direction some of the work described herein was performed. The authors also wish to thank Paul L. Morton of the Computer Laboratory, University of California, for his assistance in the use of the IBM system.
(HEAT TRANSFER IN CONDENSATION)
Effect of Heat Capacity of Condensate LEROY A. BROMLEY
F
OR the condensation of vapors a t moderate pressure the latent heat of vaporization is large compared to the possible contribution of the sensible heat to the heat flow. However, for organic vapors a t high pressures and large temperature differences between the saturated vapor and the tube temperature, the contribution of the sensible heat term may significantly affect the heat transfer. Nusselt (18) has derived an equation for this effect but because of a fundamental error in writing his enthalpy balance his answer is incorrect, even as regards to sign. Thus Nusselt's equation predicts that the effect of the condensate film having heat capacity is to decrease the heat transfer coefficient. That this is incorrect may be seen from the folloning argument. Consider two identical condenser tubes, numbered 1 and 2, - + the same temperature and in vapors whose physical proper-
ties are alike except that the heat capacity of the condensate is zero on tube 1 and finite on tube 2. As a first approximation, assume that at a given time a film is established on tube 2 which is of the same thickness as that present on tube 1 a t dynamic equilibrium (see Figure 6). All of the condensate on the tubes except that immediately adjacent to the tube is undergoing cooling. Thus the liquid is deposited at the boiling point but drops to a lower temperature as more liquid is added on the outer surface and a linear temperature gradient tends to be established. Because of this cooling, heat will be removed from the condensate on tube 2. There will be, therefore, for tube 2 a steeper temperature gradient at the tube wall (hence a higher coefficient) and a flatter gradient in the liquid a t the vapor boundary. This flatter gradient must result in less condensation and hence a thinning of the film which will result in a further increase
