INDUSTRIAL AND ENGINEERING CHEMISTRY
478
The thinness of water film necessary to give such coefficients as these is surprising. For the over-all coefficient we may write K =
I
1 d
1
x+x+Tc
where K I and Kz are the conductances of the films on either side of the tube, d is the thickness, and X the conductivity of the metal. I n this case X is 327 calories per square meter per meter per hour, and d is 0.0016 meter. For the cases where K1 is 10,000 and 15,000, we have, respectively, 1
E
+
= 0.000095 and 0.000070
If we assume the film resistance on the steam side to be twice that of the water side, KZ (the water film conductance) would be about 30,000 and 50,000, respectively. Using 0.42 as-the thermal conductivity of water at 100” C. in metric unitszthe film thickness on the water side would be 0.014 and 0.009 mm., respectively. This indicates what an extremely slight deposit would be necessary to double the water film thickness, or even to cut the over-all coefficient in half. . ACKNOWLEDGMENTS The authors wish to acknowledge their indebtedness to many students who, at one time or another, have helped in the work. Mention should be made of the work of D. k t e a g a , F. C. Calhooh, and J. T. Garber, who helped on Series AZ. Especial mention should be made of the work of W. A. Myers, who assisted in the work of Series A Z in the spring of 1923, and who was entirely responsible for reassembling the machine in the new laboratory, and for carrying out the work on Series A Z in 1924.
where S is the thickness of the scale (for thin deposits only), Q is the heat which has passed through the heating surface, and a is the proportionality constant. The Newton equation for heat flow may be written in the form
d_Q de
= H A At
(2)
when dQ is the heat flowing through A 30 square feet of heating s u r f a c e during the time de with the over- 2.0 all temperature difference At and the over-all coefficient of 1.0 heat transmission H . According to the well-known “resisI I tance concept,” H is ‘o I 2 3 Dags Since Cleaninq the reciprocal of the FIG. 1 sum of the resistances of the several films and layers of solids through which the heat flows in series. Therefore, let
, , ,
where a is the combined resistances of the heating surface when it is free from scale, and PQ is the resistance which is proportional to the thickness of the layer of scale. I n Equation 2, A is substantially constant (for thin scales) and At is constant, while H is a variable. In order to integrate the equation it is necessary to express H as a function of either Q or 8.8 This is conveniently done by differentiating Equation 3 as follows:
T
HE serious effect of scale formation on the over-all coefficient of heat transfer in evaporators has long been appreciated by all who are connected with the evaporator industry, and it may be safely said that scale formation is often the most important factor in evaporator performance. In spite of this, the writers have been unable to find in the literature any quantitative analysis of the rate a t which scale deposits, although engineers make the qualitative statement that the scale increases as time goes on and the heat transfer decreases in proportion. It is believed that the following analysis of the rate of scale formation is wholly new. It is based upon an assumption as to the mechanism of scale formation which the writers feel is well justified by the subsequent experimental data. It is assumed that under constant operating conditions with respect to liquid velocities and temperature difference, and where the feed enters the evaporator a t about boiling temperature, the amount of scale deposited on the heating surface is directly proportional t o the amount of water evaporated. Since the water evaporated is proportional to the heat flow through the heating surface, the following relation may be written: S = aQ
(1)
Received January 18, 1024. 2 Contribution from the Department of Chemical Engineering, Massachusetts Institute of Technology. 1
a
+ BQ = 1g
Q E - -l-
or
MASSACHUSETTS ~ N S T I T U T EOF TECHNOLOGY, CAMBRIDGE, MASS.
,
,
(3)
Evaporator Scale Formation’’* By W. L. McCabe and Clark S.Robinson
Vol. 16, No. 5
a ! P dH
H
Whence
d@ =
be s u b s t i t u t e d in Equation 2, giving
I 1
- PH2 j
(4)
j , i
,1
which, when integrated in ‘the usual manner, gives stant (of integration) (5)
Since 2 PA At is a constant, call i t c , giving 1 = ce + constant H*
This is the desired relation between the coefficient of heat transmission and the time. I n order to test its validity, the writers obtained data from three well-known evaporator experts.
* Hitchcock and Robinson, “Differential Equations in Applled Chemirtry.”
John Wilep & Sons, Inc., New York, 1988.
May, 1924
INDUSTRIAL A N D ENGINEERING CHEMISTRY
DATAAND REFERENCES
-First
H
I n Bulletin 149 of the Louisiana State University, Professor Kerr gives the following test on a quadruple-effect Kestner evaporator, evaporating cane juice at Guanica:
464 324 264 256 230 216 200 190 186 176 164 162 151 140
Heat Transfer Coefficient in Fourth Body 244 209 193
Days since Cleaning 0 2 3
&
These data have been plotted in Fig. 1 with plotted against 0 (days). It will be seen that this gives substantially a straight line, as required by Equation 6. The second set of data was given to the writers by R. M. Torrey, of E. B. Badger & Sons Company. These data were obtained on an experimental evaporator, evaporating concentrated brine. , Coefficient (H) 1500 1090 950 800 960
Time ( 0 ) in Hours 0 1 2 3 4 5 6 7 8 9 10
700 700 680 660 630 610
e
c S e c o n d Run-
H
e
(Minutee) 22 44 66 91 116 140 164 189 214 240
-Third
H
Run-
e
(Minutes) 16 32 54
79 103 129
CONCLUSION I n conclusion, it may be pointed out that this method of plotting may be used to predict the change in the over-all coefficient with respect to time, provided that two points on the H vs. 0 curve are known. Extrapolation within reasonable limits should be reasonably safe, provided conditions remain constant and no scale cracks off the heating surface.
The Film Concept of Heat Transmission Applied to a Commercial VVater Heater ’
&
These data have been plotted in Fig. 2 with plotted against 0. It will be seen that the first four points make a straight line and the last five points make another, but that the two intermediate points are irregular. The writers attribute this to a sudden change in evaporator conditions, with possibly some of the scale dropping off about the fourth hour. I
The last set of data was obtained through the kindness of
W. L. Badger, of the University of Michigan. These were data from an experimental basket-type evaporator, evaporating sodium sulfate solution (see following table). These data have been plotted in Fig. 3 in the same way as in the 1,wo previous cases, and it will be seen that the agreement with the straight lines is very satisfactory. The writers hope that this analysis of rate of scale formation will be of considerable assistance to the evaporator industry, and that it will stimulate further investigation on this important subject
.
Run-
(Minutes) 25 50 75 100 125 150 175 200 225 255 290 325 360 295
479
By D. K. Dean2 ALBERGER PUMP& CONDENSER Co., BOSTON, MASS.
HE objects of this paper are, first, to derive a method for calculating the amount of heating surface required for a steam heater using the “film concept” as the basis of the theory of heat transmission; and second, to apply to an actual heater the values of the film transferences so far determined by the investigators of the Chemical Engineering Department of the Massachusetts Institute of Te~hnolpgy.~.~J,6 I n the design of steam heaters, as well as other heat transfer apparatus, it has been the common practice to make use of an over-all rate of heat transmission from the heating fluid to the fluid to be heated, assuming a constant rate of heat transmission throughout the apparatus for the given conditions. Such a method of calculation has proved quite satisfactory for the determination of a certain set of conditions, but necessarily it is inadequate to predict complete characteristics of a heat transfer apparatus. The “film concept” of heat transmission of fluids is not new. As early as 1874 Reynolds’ derived a theoretical equation for the coefficient of heat transfer between a fluid and a metal wall, and other investigators since that time have studied the problem from this standpoint. Generally speaking, however, this method has not been followed in practice, probably because the values determined were in the nature of instantaneous rates for elementary portions of the heating surface; and since these rates are expected to vary throughout an apparatus, the complications of the application of this method presented difficulties too great for everyday use. It may be mentioned that most of the test results on record have been made with single tubes in order that complications
T
Received January 21, 1924. District Sales Manager. a McAdams and Frost, THIS JOURNAL, 14, 13 (1922). Ibid., 14, 1101 (1922). 8 Lewis, McAdams, and Frost, J . Am. SOC. Heatzng Ventilating Eng., 28 (1922). 8 McAdams and Frost, “Heat Transfer of Water Flowing inside Pipes,” paper delivered before American Society of Refrigerating Engineers, December, 1923. 7 Proceedings Manchester Lilcrary and Philosophical Society, 1874. 1
1
(I
