wall of the condenser tube and about three times more than the condensing film itself. This shows that the thickness of the Teflon film should be reduced to improve the over-all performance of the Teflon-coated condenser tube. These films can be made to l , ’ 5 as thick as the one used in this program; however, the production of such a thin film is most difficult. T h e estimated performance of a n aluminum tube with a Teflon coating 0.00005 inch thick is also shown in Figure 5. Comparison of the top and bottom curves shows that heat fluxes 1007‘ greater than with uncoated tubes can be obtained with Teflon-coated tubes. Conclusions
Filmwise steam condensation heat transfer in horizontal condenser tubes can be adequately predicted by Peck and Reddie’s (73) equation. Dropwise steam condensation heat transfer can be adequately predicted by Fatica and Katz’s (4) equation, although practical difficulties in its use are involved. Teflon is a. good promoter for dropwise condensation, and it can offer heat transfer rates increased u p to 1 0 0 ~ o when the film is made 0.00005 inch thick. Nomenclature
sq. ft. F. filmwise condensation heat transfer coefficient predicted from Nusselt’s theory, B.t.u./hr. sq. ft. ’F. thermal conductivity, B.t.u./hr. ft. F. temperature of condensing surface, ’ F. saturated vapor temperature, ’ F. over-all heat transfer coefficient, B.t.u./hr. sq. ft. O F: average cooling water velocity, ft./sec. heat of vaporization, B.t.u./lb. viscosity, lb./ft. sec.
h
= heat transfer coefficient, B.t.u./hr.
hxu
=
k
= =
T,
T,, = Go = V = X p
= =
Literature Cited (1) Baer, E., McKelvey, J. M., Delaware Science Symposium,
,4CS, University of Delaware, Newark, Del., February 1958. (2) Davies, J. T., Rideal, E. K., “Interfacial Phenomena,” p. 7, Academic Press, New York, 1961. (3) Drew, T . B., Nagle, W. M., Smith, W. G., Trans. A m . Znst. Chem. Engrs. 31, 605 (1935). (4) Fatica, N., Katz, D. L., Chem. Eng. Progr. 45, 66’. (1949). (5) Groher, H., Erk, S., Grigull, J., “Fundamentals of Heat Transfer,” p. 353, McGraw-Hill, New York, 1961. (6) Hampson, H., Engineering 172, 221 (1951). (7) Hampson, H., “International Developments in Heat Transfer, Part 11,’‘p. 310, Am. SOC.Mech. Engrs., New York, 1961. (8) Hodgman, C. D., ed., “Handbook of Chemistry and Physics,” 36th ed., Chemical Rubber Publishing Co., Cleveland, Ohio, 1954. (9) McAdams, W. H., “Heat Transmission,” 3rd ed., p. 345, McGraw-Hill, New York, 1960. (10) McCormick, J. L., Baer, E., J . Colloid Sci. 18, 208 (1963). (11) McCormick, J. L., Baer, E., 8th Midwestern Mechanics Conference, Case Institute of Technology, Cleveland, Ohio, April 1963. (12) Nusselt, W., 2. Ver. Deut. Ingr. 60, 541 (1916). (13) Peck, R. E., Reddie, W. A , , Ind. Eng. Chem. 43, 2926 (1951). (14) Pierce, 0. R., Intern. Sci. Technol. No. 11, 31 (1962). (15) Schmidt, E., Schurig, W., Sellschopp, W., Tech. Mech. u . Thermodynam. 1, 53 (1930). (16) Shea, F. L., Krause, N. W., Trans. A m . Znst. Chem. Engrs. 36, 463 (1940). (17) Sugawara, S., Michiyoski, N., Proc. 2nd Japan. Natl. Congr. Appl. Mech., Part 111, p. 289, 1952. (18) Topper, L., Baer, E. J., J . ColloidSci. IO, 225 (1955). (19) Watson, R. G., Brunt, J. J., Birt, D. C., “International Developments in Heat Transfer, Part 11,” Am. SOC. Mech. Engrs., p. 296, 1961. (20) Welch, J. F., Ph.D. dissertation, University of Illinois, Urbana, Ill., 1960. (21) Westwater, J. W., Welch, J. F., “International Developments in Heat Transfer, Part 11,” p. 302, Am. SOC.Mech. Engrs., 1961.
RECEIVED for review September 23, 1963 ACCEPTED December 10, 1963
ANALOG COMPUTER METHOD FOR DESIGNING A COOLER-CONDENSER WITH FOG FORMATION D.
R. COUGHANOWR AND E. 0. STENSHOLT,
School of Chemical Engineering, Purdue University, Lafayette, Ind. An analog computer method is given for designing a cooler-condenser for the separation of a condensable vapor from an inert gas accompanied b y fog formation. The design equations which describe the process are applied to an example (air-benzene) in which fog i s likely to form. The unique feature is the method b y which the heat balance is modified when fog is present and the computer implementation which decides when the heat balance should b e modified. A comparator monitors the difference between the gas temperature and the saturation temperature. Whenever the two temperatures become equal, a relay operated by the comparator changes the circuit to modify the heat balance. HE basic design equations for the condensation of mixed Tvapors were presented by Colburn and Drew (2). Colburn and Hougen (4)developed design equations for removal of condensable from an inert gas. O’Brien and Franks ( 6 ) have recently discussed the design of a cooler-condenser by means of a n analog computer for the H20-CO2 system. They mention that fog may be formed, but d o not solve this case; however, they refer to the work of Schuler and Abell ( 8 ) ,who account 1 Present Sorway.
address,
Institut
for Kjemiteknikk,
Trondheim,
for the formation of fog in a n experimental study of the condensation of Tic14 from N2. T h e purpose of the present work is to show what modifications are needed in the usual design equations to account for fog formation, a n d how the equations can be solved by analog computation. Design Equations
T h e following design equations are developed for a doublepipe condenser in which the gas flows through the center tube and cooling water flows cocurrently to the gas outside the tube. VOL. 3
NO. 4
OCTOBER 1964
369
Since the equations are well known, and have appeared frequently in the literature (2, 4, they are listed without derivation. Material Balance for Gas. The rate of change of condensable flow rate, VT, with distance z may be obtained by writing a material balance over a differential segment of the condenser tube. The result is
Mass and Neat Transfer Coefficients. I n this analysis, the Chilton-Colburn analogies ( 7 ) for heat and mass transfer coefficients in tubes are used :
For convenience, the mass velocity, G, in Equations 6 may be written G = ( V , Vt)iti,,/ut. Substituting this expression for G in Equations 6 and replacing c v Mayby C, gives
+
iC0 h, = - (V,
Heat Balance for Gas. The rate of change of gas temperature, T,, with distance is obtained by writing a heat balance over a differential length of the exchanger. The result is
T h e factor C$ accounts for the effect of mass transfer on the cooling of the gas. As the condensation flux, N , approaches zero, C$ + 1-Le., for very low rates of condensation, Equation 2 [given by Colburn and Drew‘(2)l takes the usual form for the case of heat transfer without mass transfer. The heat balance expressed by Equation 2 is written for the case of no fog formation. The modification needed to account for the heat released by fog formation is described later. Condensate Heat Balance. The condensate loses heat by conduction through the wall, scale, and coolant film. I t receives latent heat from the condensing vapor plus sensible heat from the bulk gas. A heat balance a t the interface gives
The over-all coefficient, U, accounts for the resistance of the coolant film, scale, metal in wall, and condensate film. This coefficient will generally vary along the tube because of changes in physical properties of the coolant and in thickness of the condensate film. However, U is taken as a constant in this analysis. O’Brien. Franks, and Munson (7) discuss a method to account for the variation in resistance of the condensate film as it increases in thickness down the wall. The Ackermann coefficient, Ac, which is discussed by Colburn and Drew ( Z ) , accounts for the effect of mass transfer on heat transfer. For N -+ 0-Le., no mass transfer-Ac -+ 1. For many condensing systems, the term including Ac is very small compared with the latent heat term, NX, and consequently Ac may be taken as 1.O without introducing appreciable error. In fact, the entire term may be neglected under some circumstances. Coolant Heat Balance. The flow of cooling water in condensers is usually low enough to give a significant temperature rise along the tube. T h e coolant temperature, T,, may be determined by a heat balance. For cocurrent flow the result is
dT, ~- - UrDt(Tt - T,) dz wctc
(4)
For countercurrent flow, Equation 4 holds if the right side is multiplied by - 1. Vapor-Liquid Equilibrium. The mole fraction condensable in a saturated gas mixture a t constant total pressure is a function of temperature only : Y
=f(T)
(5)
It is assumed that the gas and condensate at the interface separating rhe two phases are always in equilibrium. Furthermore, wheii fog is present, the assumption is made that the gas h saturated and follows this equilibrium relation. 370
l&EC P R O C E S S DESIGN A N D DEVELOPMENT
a1
+ Vi)(Pr)-2’3
(7)
Fog Formation. The mass balance (Equation 1) and the heat balance (Equation 2) can be used to determine the composition and temperature of the gas at any location z. However, these equations may give a temperature which is lower than the saturation temperature corresponding to the gas composition. This will mean that the gas mixture is subcooled, which is unlikely under most industrial conditions where particles will be present in the gas to serve as nuclei for the formation of fog droplets. In this analysis, it is assumed that the gas never becomes subcooled, and that fog will form by an amount necessary to maintain the gas saturated. If one assumes that there is no resistance to mass transfer to the fog droplets, the gas temperature and composition will lie on the equilibrium curve whenever fog is present. The tendency for a gas mixture to form fog in a coolercondenser depends on the ratio of sensible heat transfer to mass transfer. The higher this ratio, the greater the tendency to form fog. The convective transfer coefficients are related to each other through analogies, such as the Chilton-Colburn analogy just presented. An inspection of these equations shows that the ratio of the heat transfer coefficient to the mass transfer coefficient is proportional to (Le)2i3,where Le is the Lewis number (Sc/Pr). From this result, one concludes that fogging is likely to occur for systems having large Lewis numbers. This has been verified experimentally by Colburn and Edison ( 3 ) . The tendency for a gas mixture to fog will also depend on the inlet conditions. For example, if a gas mixture is superheated a t the inlet of a condenser, fog may not start to form until some distance from the entrance, if it forms at all. To account for the fog formation, a term must be added to the heat balance (Equation 2) to account for the latent heat released during the transformation from vapor to liquid. One method which accounts for fog formation in the heat balance was successfully programmed for the analog computer ( 9 ) . If the gas mixture is saturated (with or without fog), the gas temperature can always be determined from the gas composition (known from material balance) through the equilibrium relation of Equation 5 . For the case where the gas is saturated with fog present, Equation 2 can be written in the following modified form :
-(VT
+ VdCg dTof
=
horDt(Toe
-
dz
(9)
The term T o ein Equation 9 is the saturation temperature of the gas (an equilibrium value) corresponding to the gas composition, y g ; consequently, the right side of Equation 9 gives the rate of sensible heat transfer when the vapor is saturated. The left side is the product of the gas flow rate, the average heat capacity of the gas mixture, and the change of gas temperature one would have if no fog were formed-Le., if the gas remains subcooled. Temperature T,f is therefore a
fictitious variable. since it has been assumed that the gas will actually remain saturated if fog is present. The difference: 7'ge - Tsf. is proportional to the amount of fog formed according to the equation
This equation states that the heat released by the formation of fog. AFT,% just equals the sensible heat needed to bring the temperature of the gas from T V lback to the saturation temperature, T g e . In summary, if fog forms, Equation 2 is replaced by Equations 9 and 10. Prevention of Fog b y Addition of Heat. Colburn and Edison (S) showed that fog may be prevented by adding sensible heat to the gas stream by means of an auxiliary heat source such as a heated Jvire running along the center of the tube. The rate of heat released per unit length. q: from the wire may be included in these design equations by adding a term to Equations 2 and 9 to obtain the following modified equations:
tions the range of variables and physical properties were determined as shown in Table I . From this table, it can be seen which variables may be taken as an average of the inlet and outlet values. Summary of Equations. To avoid an excessive amount of computer equipment the following quantities were taken as constant throughout the condenser: 4 = 1, C, = 13.2 B.t.u.j(lb. mole)(' F.), (Pr)-213 = 1.18, ( S c ) - * I 3 = 1.16, j = 0.0035, and X = 14,000 B.t.u./(lb. mole). T h e average values of C,. Prp213,S C - ~ /and ~ . j are the averages of the corresponding quantities in Table I. Other constant quantities are P = 1 atm., u t = 0.00412 sq. foot. T o improve the accuracy of the function generator representation of the equilibrium relationship. new temperature deviation variables, defined as T' = T - 50, are introduced. The following list of equations was obtained after substituting all constants into Equations 1, 5 to 8, 10, and 11 :
If T o > T o e ,use - -0.228 h g ( T g ' -
dT,' __ (no fog present)
~
c,
dz
(Ila)
- T,')
vu + vi
+
9 13.2(VU
+ V,) (Bl)
(no fog present)
and
if T o < To,, use dT,,' _ _ _ _ _-0.228 h,(To,' (fog present)
(1 1b)
Summary of Design Equations. T h e system of design equations to be used will depend on whether or not fog is present. If no fog is present [ T o> Tge.where T,,= f ( y o ) ] , Equations 1, 3 to 8. IO, and l l a apply. If fog is present ( T o < T o e ) Equation . l l a should be replaced by Equation l l b . To illustrate the use of these equations and to see how they are solved on an analog computer, consider the following example.
-
dz
c,
v,
- Ti')
+ Vf
ho = CdVU
T o e f= fi(yo)
q +
+
13.2 ( V u Vi) (fog present)
(B2)
+ vo
(C)
(equilibrium relation)
V, = 0.00094 (Vn
(D)
+ Vi)(Tge' - Toy')
(E)
Additional equations needed to solve the problem are : The mole fraction of condensable vapor in the gas is given by
vu
Example
yo =
Statement. An equimolal mixture of air and benzene vapor at 150' F. and 1-arm. total pressure (dew point = 135' F.) enters a cooler-condenser at a rate of 0.65 pound mole of mixture per hour. T h e inside diameter of the tube through which the vapor flows is 0.8'7 inch. .4ssume that the interface temperature, Ti. remains at a constant value, say 100" F., throughout the exchanger. Devise an analog computer circuit which can be used to determine the conditions of the gas (T,. J , ) and the quantity of fog present (if any) as a function of condenser 1eng;th. 'The assumption of constant interface conditions (7-,.? & )was used to reduce the amount of analog computer equipment by eliminating the need for Equations 3 and 4. Ll'ith this assumption. it is still possible to illustrate hoiv the equations are implemented on the computer when fog forms. Physical Properties. T o solve the design equations for the specific example under consideration. the physical properties of the gas mixture must be known; however, these will vary Fvith the temperature and composition. To account for the variation in all the physical properties would require a very complex computer program. Therefore, to simplify the problem, some of the properties will be taken as constant at some average value. If the exchanger \vere of infinite length, the gas would approach the surface temperature of 100' F.as z -* m , and if one assumed the gas to be saturated a t z -P a the composition of the gas and the amount of condensate formed would also be knoivn from a material balance. From these terminal condi~
Table 1.
V,,
Range of Physical Properties and Variables
Vulue r=O 150 100
Variable
T,. O F. T;; F. y o , mole fraction y,, mole fraction V , , Ib. mole/hr. V , , Ib. mole/hr. M," P m , atma p , Ib./cu. ft. C,, B.t.u./(lb. mole)(lo F.) c y , B.t.u./(lb.)(' F.) k , B.t.u./(hr.)(ft.)(80 F.) p , lb./(ft.)(hr.) D,,sq. ft./hr. Pr2I3 Sc2'3 Le
0.50 0.22 0,325 0.325 53.5 0.64 0.12 15.5 0.29 0.012
0.034 0.40 0.88 0.80 0.86
ke
18,000
Rr0.2 ---
7 1
j h,, B.t.u./(hr.)(sq.f t . ) ( " F.) k,, Ib. mole/(hr.)(ss. ft.Matrn.) ,g,Ib. mole/(hr.)(sq. f t . )
0'6032 9.1 0.80 0 30
U
0 52
b
0 76 a
z =
m
100 100
0 0 0 0
22 22 325 09
40 0 0 78 0 10 10 8 0 27 0 014 0 039
0.37 0 82 1 02 1 4 7000 5 9
6 0039 5 2 0 48 0 0 1 0
Based on arithmetic mean.
VOL. 3
NO. 4
OCTOBER 1 9 6 4
371
The logarithmic mean partial pressure of inert in the gas film ( p m f ) , which is needed in the Chilton-Colburn analogy for determining k,. may be approximated by the arithmetic mean if the partial pressure of condensable is not too great. Since the total pressure is 1 atm.?pB.,f is numerically equal to This approximation takes the form yB.tf. ,YBM
= 1 -
0 . 5~ ~0.5,~c
_,A”
I
(G)
A simple material balance gives
p’c. =
L’T
-
(H1
VL
Equation B1 is to be used when the gas does not contain fog and B2 when the gas contains fog. In obtaining Equation L7J has been replaced B2 from Equation 11B the term (7, L’,), since the difference between these terms lyill be by ( V c slight if only a small quantity of fog is formed. The computer results showed this simplification to be justified. Computer Circuit. Equations .4 to H were programmed for solution by an analog computer (Applied Dynamics, Inc., analog computer, Model AD-2-32PB). and a scaled circuit diagram is shown in Figure 1. The procedure used for magnitude scaling is to divide the maximum value of a variable into 100 volts (the maximum voltage produced by the particular amplifier used). For example, the maximum value of y G (which occurs at the entrance) is 0.5. The scale factor is therefore 100 ’0.5 = 200 and the output of the amplifier which corresponds to y e is labeled [200 y o ] . This scaling method, kvhich is described more fully by Jackson (5. Chap. 3): \vas applied to the system of design equations to produce the scale factors shown in Figure 1. No time scaling was necessary. for it was convenient to let 1 second of computer time be equivalent to 1 foot of condenser length. T h e symbols used in the computer diagram are standard and can be found in texts on analog computers (5). Multiplication and division (represented by function blocks on the computer diagram) wrre performed by circuits which used a quarter-square multiplier card in conjunction with a high-gain amplifier. A pair of relay switches. operated by a comparator. was used to determine whether Equation B1 or B2 was to be solved. - T g ’ ) ; if this quantity is Amplifier 1 generates -5(T,,’ positive (7.Q’ > T Q e ’ )the , relay coil is energized and the sivitches are in position 1 as shown in Figure I , with the result that Equation B1 governs the solution. LVhen the output of amplifier 1 goes negative ( 7 0 ’ < Toe’).the relay is no longer energized, the switches are in position 2. and Equation B2 governs the solution. The operation of the comparator ir described by Jackson (5. p. 198). Amplifier 3 produces T S f ‘ or T’o> depending on whether or not the gas is saturated. The equilibrium relation of Equation D was produced by a diode function generator (DFG) using five line segments. The function generator \\-as set u p to give [ T g e ’L’S. ] [200 y o ] . The auxiliary heat term is the second on the right side of Equations B and will be called Q for convenience; thus
+
+
9 = 13.2(V,
+ Vi)
In a practical situation, q might be generated by an electrically heated wire or a steam-heated tube running along the condenser. In such a case, q would vary along the exchanger, the specific relationship depending on the convective heat transfer coefficiect between the heat transfer surface and the flowing gas, To avoid the need for consideration of such a detailed analysis. and to illustrate the effect of the auxiliary heat source on prevention of fog with the simplest computer implementation, the computer was programmed to maintain Q constant, 372
l&EC PROCESS DESIGN AND DEVELOPMENT
Figure 1.
Analog computer circuit
100
80
60
40 >
20 0
-0
2
4
6
8
10
12
14
16
18
2. f t
Figure 2.
- rg =
--
-._.
Profiles in condenser 1500 F.,
=
o
T, = 140’ F., q = 0 .T , = 150° F., auxiliary heat a d d e d
Table II. Potentiometer Settings for Figure 1 Pot. h$. Value Pot. .Yo. Value A1 0.224 B1 1.5T,’:100 A2 0 650 B2 2YI .% 3 0.0376 B3 0.25 A4 0 325 B4 1-0. 5yl A5 0.500 B5 0’. 625 A6 0.8Qs100 B6 0 15 A7 0 01 B, 0.50 .48 0 122 B8 A9 0 8T,’:100 A10 0.188 * Adjusted to g i w output soltag? f r o m ampl;S;~r2 of about 80 uolts *hen l o i w r diode is conducting. This aoltage iras suficient to e n r r t i r p the r e l q (Potitr and Rrun$eld. S o . R C P 1 4 . 70 kziohms).
by simply introducing a constant voltage into the input of amplifier 3. The potentiometer settings for Figure 1 are given in Table 11. Results
The circuit sho\\-n in Figure 1 was operated for several different inlet gas temperatures, To. and tube surface conditions. 7‘vasmade at this tube surface temperature. I n Figure 2. TG‘. Toe’..yG. VT. and VI, are plotted (in terms
of the corresponding computer variables) against condenser length. The solid curves represent the case for T , = 150’ F. and no auxiliary heat generation iq = 0 ) . T h e effect of inlet gas temperature on the formation of fog is shown in Figure 2 by plotting the fog rate. V L . against t for a different inlet gas temperature, 140’ F., all other variables remaining the same ,as for the solid curves. This curve is shown by the dashed line. As one might expect, the position at which fog forms moves further domm the tube as the inlet gas temperature increases. T h e effect of adding auxiliary heat, q , is also shown in Figure 2 by plotting V L against z. Two typical curves for this case are sho\vn dotted in Figure 2 ; with the exception of auxiliary heat, all other variables remain the same as for the solid curves. Fog was completely prevented when T, = 150’ F. and Q = V$,this corre2.5, O n the basis of the inlet value of V, sponds to
+
Nomenclature =
at
= cross-sectional area of tube, sq. ft. = Ackermann coefficient = a / ( l - e-.)
Ac C, cg
C,
D, D, G h,
j k k,
Le q = (2.5)(13.2)(0.65) = 22 B.t.u./’(ft.)(hr.)
M,,. 1%’
P T h e fraction of the total condensable which is fog at any location in the condenser can be calculated from the curves of Figure 2. For example, at t = 8 feet for Q = O? [5.3 X 1 0 3 V ~ ] = 30 volts and [200 V ,] = 17 volts. T h e per cent fog is therefore
yGfog
=
VL VT
1 00
=
30/(5.3 X lo3) 17/200
,bB.v
Pr Q
Q
Re sc
100 = 6.7
This calculation shows that the amount of fog is a very small fraction of the total condensable present in the gas stream. T h e curves representing V L in Figure 2 appear to be segmented rather than smooth. This is probably caused by the diode function generator and quarter-square multipliers, for in both of these devices, straight-line segments are used to produce the functional relations.
Vi V, VL VT W
YO Discussion
Yi
Some of the assumptions which have been used in constructing the set of design equations, or model of the process, are open to question. In particular, assumptions difficult to justify are that the gas never becomes subcooled, and that there is no resistance to mass transfer to the drops. Very little experimental evidence is available which can settle such questions. I n an actual system, there is undoubtedly some transfer of fog droplets from the gas to the condensate, however, in this analysis, it was assumed that no such transfer occurs. A number of assumptions and approximations were made in the computer solution to save computing equipment. With a sufficiently large computer and additional effort, refinements can be made in the computer circuit to account for the variation of parameters which were taken as constant. For example, @ could be continuously generated by using appropriate nonlinear computer circuits. T h e example assumed that the tube surface temperature remains constant. If this is not the case, Equations 3 and 4 must be introduced into the computer circuit, with the result that an integrator would be needed to solve Equation 4: and a tunction generator would be needed to relate Tt to y t , which will vary along the tube.
AVC,/hg.dimensionless
a
z
X p
p
6
average heat capacity of gas mixture, B.t.u./(lb. mole) (’ F.) = average heat capacity of gas mixture, B.t.u./(lb.)(”F.) = heat capacity of coolant, B.t.u. /(lb. mole) (” F.) = inside diameter of tube, ft. = diffusivity of condensable vapor in inert gas. sq. ft./hr. = mass velocity of gas mixture, Ib./ (hr.)(sq. ft.) = convective heat transfer coefficient for flow of gas through tube, B. t.u./(hr.)(sq. ft.) (” F.) = j factor used in Chilton-Colburn analogies, dimensionless = average thermal conductivity of gas mixture, B.t.u.1 (hr.)(ft.)(O F.) = convective mass transfer coefficient for flow of gas through tube, lb. moles/(hr.) (sq. ft.) (atm. driving force) = Lewis number = Sc/Pr, dimensionless = average molecular weight of gas mixture, lb.,’lb. mole = condensation flux, Ib. mole/(hr.)(sq. ft.) = total pressure. atm. = log mean partial pressure of inert gas (mean of partial pressure in bulk gas and a t interface), atm. Prandtl number of gas mixture, = cnu,’k. dimensionless rate of auxiliary hgat generated pe;’ foo’t of condenser tube, B.t.u./(hr.)(ft.) q / ( V u VJC,, ” F./ft. Reynolds number = D i G / p , dimensionless Schmidt number of gas mixture = p / p D , , dimensionless temperature at interface, O F. temperature of bulk gas, O F. temperature of coolant, O F. saturated temperature of gas, ” F. fictitious temperature used in Equation 9, ” F. over-all heat transfer coefficient between bulk water temperature and interface temperature, B.t.u. / (hr.)(sq. ft.)(” F.) flow of inert gas, Ib. mole/hr. flow of condensable vapor, lb. mole/hr. flow of condensable in form of fog, Ib. mole/hr. total flow of condensable (vapor plus fog), Ib. mole/hr. flow of coolant, Ib. mole/hr. mole fraction condensable in mixture of condensable vapor and inert gas mole fraction condensable in mixture of condensable vapor and inert, a t interface = distance measured along tube from entrance. ft. = heat of condensation of condensable, B.t.u.Ab. mole = average viscosity of gas mixture, lb./(ft,)(hr,) = density of gas mixture, lb./cu. ft. = a / ( e a - l ) , dimensionless
=
+
SUPERSCRIPT Prime refers to deviation temperature; thus T‘
=
T - 50, ’ F.
Literature Cited
(1) Chilton, T. H., Colburn, A. P., Ind. Eng. Chem. 2 6 , 1183 ( 19 34) . (2) Colburn, A. P., Drew, T. B., Trans. A.I.Ch.E. 33, 197 (1937). (3) Colburn, A. P., Edison, A. C.: Znd. Eng. Chem. 33, 475 (1941). (4) Colburn. A. P., Hougen, D. A,, Ibid..2 6 , 1178 (1934). (5) Jackson, A. S., “Analog Computation,” McGraw-Hill, New York, 1960. (6) O’Brien, N. G., Franks, R. G., Chem. Eng. Progr. Symp. .Cer. 31, 37 (1960). (7) O’Brien, N. G., Franks, R. G.: IMunson, J. K.: Ibzd., 29, 177 (1959). (8) Schuler, R. W., Abell. J. B., Ibid.,5 2 , No. 18, 51 (1956). (9) Stensholt, E. O., “Application of the Analog Computer to Chemical Engineering Problems,” M.S. thesis. Purdue University, Lafayette, Ind., January 1963. RECEIVED for review November 26, 1963 ACCEPTED ilpril 13. 1964
VOL. 3
NO. 4
OCTOBER
1964
373
