I n Figure 8 the predicted yield decline with catalyst age is shown for two additional cases: k 2 is three times greater than the value used for Figure 3, and El is one-half as great as the value for Figure 3. Xpparently, the increase in k 2 serves primarily to decrease the absolute level of jet yield; a catalyst possessing high selec,tivity for cracking lower-molecularweight compounds could produce this effect. The decrease in E l , b y contrast, primarily influences selectivity changes during fouling. Such a decrease in E l could result either from diffusional limitations on the large feed molecules (which are absent for the smaller jet molecules) or from a refractory feed inherently exhibiting a lower El relative to E2. In any event, the selective site deactivation concept need not be invoked. Acknowledgment
The authors thank Chevron Research Co. for permission to publish this viork. Nomenclature
CJ
= total yield to feed of a narrow boiling range jet fuel,
CJa
=
@,,
=
E
=
E i= F
=
FI = Fo = I.'t = FoL,=
pi= G
=
J
=
wt fraction yield t'o feed of synthetic jet fuel produced by cracking higher-molecular-weight feed, wt fraction predicted value of total yield t'o feed of narrow boiling range jet fuel from X o d e l 3 , wt fraction activation energy of a reaction with rate constant k , kcal!g-mol activation energy of a reaction with rate constant ki, kcal/g-mol symbolic designation of gas oil feed or molar flow rate of this component, mol/hr flow rate of fresh feed to the recycle loop, cc/hr flow rate of unconverted total feed to reactor, mol/hr flow rate of total feed to the reactor, cc/hr flow rate of unconverted total feed to the reactor, cc/hr predicted value of the yield of unconverted feed from Model 3, wt fraction symbolic designation of gasoline product symbolic designat'ion of jet fuel product or mole flow rate of this component, mol/hr
flow rate of jet fuel in the total feed to the reactor, mol/hr J o ~ flow rate of jet fuel in the total feed to the reactor, cc/hr k rate constant for disappearance of gas oil feed, hr-1 ki rate coilstant for cracking in any individual reaction i , hr-* L symbolic designation of butane and lower-molecularweight products average molecular weight of unconverted fresh feed, gm/gm mol average molecular weight of jet fuel product, gm/gm mol number of experimental observations N R gas constant, kcal/g-mol/OR s, liquid hourly space velocity (cc total feed/hr)/(cc of reactor) T average catalyst temperature, OR residence time, hr t X apparent conversion of feed to products, cc fresh feed/ cc total feed XO jet fuel in fresh feed, LV % Jo
GRI:I.:K
LETTERS
a = preexponential factor for reaction with rate constant
k , hr-I a, = preexponential factor for reaction with rate constant
k , . hr-I A = determinant criterion for multiresponse parameter
estimation v i = stoichiometric coefficient for reaction i pF = PJ =
density of fresh feed, g/cc density of jet product, g/cc
Literature Cited
Froment, G. F., Bischoff, K. B., Chem. Eng. S a . , 16, 189 (1961). Froment, G. F., Bischoff, K. B., ibid., 17, 10.5 (1962). Kittrell, J. It., Advan. Chem. Eng., 8, 98 (1970). Szepe, S.,Levenspiel, O., European Symposium on Chem. Engr. ITr, Brussels, Belgium, September 9-11, 1968. Weekman, V. W., Ind. Eng. Chem. Process Des. Develop., 8, 385 (1969). RECEIVEU for review September 11, 1970 ACCEPTED July 23, 1971
Analytical Solutions for Predicting Vapor-Gas Behavior in Cooler-Condensers James 1. Schrodt Chemical Engineering Department, Cnivarsity of Kentucky, Lexington, Ky. 40506
I n a cooler-condenser, vapor and sensible heat are transferred from the vapor-rioncoiideiisahle gas mixture to the cooled surface a t rates equal to the respective products of the transfer coefficients and driving forces. In the classic design procedure developed by Colburn and Hougen (1934), it is assumed the mixture remains saturated throughout the process, implying that the transfer rates are relatively equal. This situation may be true for some mixtures-e.g., water vapor in air-but most others will teiid either to superheat or subcool and form fog. 20
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
Some researchers have theorized that t'hese tendencies stem from the basic transport characteristics of the mixtures. Colburn and Edison (1941) discussed the various systems which can be expected to superheat or subcool, emphasizing that heavy, sloaly diffusing vapors, such as benzene, butyl alcohol, and toluene, in air are likely to form fog in coolercondensers, whereas light,er, faster diffusing systems like water-carbon dioxide and water-helium should tend to superheat. The C"M was extended by Colburn and Drew (1937)
Two general analytical solutions have been obtained for linear models of the cooling-condensation problem: The first applies to unsaturated mixtures of vapor and noncondensable gases; the second applies to mixtures which are saturated or subcooled and may form fog in a condenser. Dynamic characteristics were predicted for mixtures of benzene-air, water-air, and water-carbon dioxide in a hypothetical condenser and were shown to b e a function of the absolute and relative rates of vapor and sensible heat transfer. The proved to b e a useful parameter for predicting the superheating and subcooling tendencies of these mixtures.
to handle superheated mixtures. They proposed a means of following the gas-phase temperature partial pressure relationship by using the following equation at points in a condenser: - 1 dP, o - P, e'__- - P1 __- _ P, . P_.__. dT, Le2/3PB.w T o - T , c
(1)
Recently, Stern and Votta (1968) condensed water in the presence of air, carbon dioxide, and helium in a vertical tube condenser. Results for superheated mixtures compared favorably with those predicted by Equation 1. The complete set of heat aiid material balance equations was solved by Coughanow and Stensholt (1964) aiid Ivanov (1962) for coiideiisers where mixtures pass from a superheated to a saturated state aiid form fog. The former authors solved their equatioiis 011 a n aiialog computer and reported results on the benzene-air system for which 6.70/, of the coiideiisate obtained was fog. Ivaiiov obtaiiied an analytical solution and reported results for a fog-forming mixture of NH3-K:2-H2. The objective of this study is t8hepresentation of general analytical solutioiis of linearized models of tlie processes that can be used to design and dynamically analyze the performance of cooler-condensers. Solutions are obtained for three mixtures condensing in a vertical tube: one that superheats, another whicli teiids to become saturated, and a third which forms fog. For these, temperatures, partial pressures, heat and mass transfer rat,es, aiid heat loads are calculated as a function of the condenser's area. These results are discussed in terms of the Lewis iiumber of each system and several generalizations are concluded.
dV _ dA
-
- P,) - dl" -
-lc,(P,
d.4
The Ackermann (1937) correction factor, e / ( l - e - ' ) , appearing in Equat'ions 2 and 2a accounts for the sensible heat transferred by the diffusing vapor. In the absence of condensation this factor equals 1.0, but increases posit'ively as the rate of condensation increases. The last term in Equation 2a is positive because the heat released from gas-phase condensatioii increases the sensible heat of the mixture. Vapor loss from the mixture by both surface and core stream condensation can be related to the decrease in the vapor's partial pressure by NP,d[P,/(P, - Po)]. This may be substituted for dV in Equatioiis 3 and 3a. The temperature of the coolant increases as sensible and latent heats are conducted through the coiideiised film, tube wall, and a fictitious coolant film. This rate is expressed as
dTw d.4
W C~ =
-ho'(T, - T,)
For cocurreiit flow conditions, the right-hand side of this equation should be positive. The gas-condensate, interface partial pressure, P,,is the equilibrium saturation pressure corresponding to the interface temperature, T,. I n most cases, P, can be approximated by a linear function in T , over the entire surface. Thus,
P,
= a
+ bT,
(5)
When the mixture is saturated, the following relationship is more useful:
Po - P , = b(To - T,)
Mathematical Models
Consider a superheated, single vapor-iioncoiidensable gas mixture flowing downward in a vertical tube condenser with a countercurrent flow of coolant in an aniiulus. The temperature of tlie coolant is everywhere low enough to provide a surface below t,lie saturation temperature of the mixture, a condition which develops concentration and temperature gradients essential for the transfer of heat and mass. If conditions and the condeiiser's length permit, the mixture after a point may become saturated and/or subcooled and thus fog. 'l'he variable thermodynamic properties and their profiles a t a point iii a superheated region of the condenser are depicted in Figure 1 . Over a differential area of t>hecoudeiiser, the core stream seiisil)le heat and mass transfer rates for t'he superheated and saturated regions are given, respectivel~;by
(3)
(4)
(54
If Equation 3 is multiplied b y i and combiiied with Equations 2 and 4, a heat balance can be obtained a t the interface. Hence, N+V
Distance
and Figure 1 . Schematic diagram of cooler-condenser and profiles a t an arbitrary cross section Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
21
Table 1. i
Complexes in Model and Solution Equations
aYj
- - .h,
1
GC,
Pj
E
e'
-1
2
3
4 5
1 2
- PZ'Y1 - P a * T,' P4 - P3
Pl.Y2
64' T,'
3
ho'(T, - T,)
=
+
h,(T, - T,) E k,i (PQ - Pd 1 - e-'
(6)
It follows that:
cesses in the subcooled and saturated parts of the condenser. Heat transfer from radiation effects and condensate subcooling have not been included in the models because these are usually negligible. Exact analytical solutions to the above sets of equations exist for the case of constant coefficients. Analytical Solutions
These same equations apply for the case of fogging, because the latent heat released from the gas-phase condensation is absorbed by the mixture in the form of sensible heat. If the mixture's temperature, T,, falls below the saturation temperature corresponding to P, by a few degrees, AT, this means that a metastable subcooled state has been reached which is conducive to the formation of fog, particularly when any nuclei are present. As this type of condensation decreases the partial pressure of the vapor, the sensible heat of the mixture will increase by a n amount equivalent to the latent heat released, and the mixture will return to a stable saturated state. The temperature increase of the mixture given by
can be used in the equation (9)
to calculate the rate of incipient fog formation, FO. Equations 2-7 define the simultaneous heat and mass transfer processes that take place in the superheated region of the condenser, and Equations 2a, 3a, 4 , 5a, 6-9 define the pro22
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
For the superheated case, Equations 2-4 may be written in the form:
dTw - - - 01z(T,- T,) d -4
(11)
(12) and treated as a system of linear differential equations in the three dependent variables, T,, T,, and P,. The 01 coefficients following directly from the original equations are assumed constants as given in Table I. When Equations 5 and 6 are solved simultaneously, T , may be obtained in terms of T,,P,, and T,.
T,
=
Tg
+
~
1
- a) + __ asT, + + bag (
p
g
(13)
015
Direct integration of Equation 7 subject to the boundary conditions
If Equations 22 and 23 are multiplied by C Y ~ / ( C Y ~ L Yand ~ ) b/b, respectively, and the results added, the common term a3/kg. d F / d S can be eliminated giving dTg - =
gives
d-4
alW(ba4 bai0.4
+ 1) (To - T3)
+
(25)
a3
Substitution of Equation 5a into Equation 6 gives, after rearrangement, With substitut’ionsfrom Equations 5 , 13, and 14, the Equations 10 and 12 are reduced to a simple set of linear differential equations.
dT, dil
dP, __ dd
=
PITO
=
PzT,
+
yip0
+ 61
+ y2Pg +
(15)
P,
=
=
--K1yIeXIA
-
K~(PI
Xl)eXIA
- KZy1eX2A -
+
K*(P~
-
This expression may be used with Equations 24 and 25 to produce two differential equations with two dependent variables:
(16)
62
Since the matrix of the coefficients in t’hesetwo equations is nonsingular and its inverse does exist, solutions were easily obtained. The author used the method advanced by Amundson (1966) to obtain these solutions:
T,
(26)
X2)eXZA
The solutions to these are
(17)
-
$2
(18)
Substitution of these two equations into Equation 14 gives a n expression for the coolant t,emperature. Likewise, t,he interface properties, T , and P,, may be obtained as a function of A from Equations 13 and 5, respectively. The local rate of heat transfer is given b y
and since the milture is saturated
and the heat load up to a given ,4 by
For the case nhere the milture enterq the condenser saturated and remains in thi. state, these equations apply throughout with values of Tu’ and T,’ taken a t .1 = 0. When the mixture enters superheated aiid then becomes ubcooled, T,’ aiid T,’ are values taken after the mivture has ieturiied to the stable saturated state. The local rate of gas-phase conden~atiun1. given by
Q
=
-we,
LA2
dd
Other equations characterizing the dynamics of the process may be developed from the basic solutions. For example, the superheat of the mixture, the difference between the mixture’s temperature and the temperature correbpondiiig to the partial pressure of the vapor, is given b y
P, =
a
+ bT,
(31)
and the amount of fog formed as a function of .L, by
(T,’ - T,’)(eX3A- 1) +PO As the mixture cools and vapor condenses, AT increases, decreases, or remains unchanged, depending on the relative rates of heat and mass transfer. A negative value of AT indicates a subcooled state \+heregas-phase condensat~onis likely to occur. The rate of fog formation and the increaae in the mixture’s temperature to the dew point can be calculated from Equations 9 and 8. At this point the mlvture must be considered as having a potential for both surface and gas-phase condensation. For the saturated case, Equations 2a, 3a, and 4 may be written in the form d--T , = ff1(Tg- T,) - -(YICY? . dF dA k, dil
dP, -_ dd
=
a d P o - P,)
dA
=
+ k,
03 -
.
dF dA
(33)
Method of Application
The general solutions were applied to saturated and superheated mixtures of benzene-air and water-air-carbon dioxide. A 1.0-in. i d . vertical tube condenser of unspecified length was selected for the theoretical studies. Each mixture was iiitro~ presduced into the top of the tube a t a k n o n ~temperature, sure, composition, and flow rate, and the outlet temperature was specified. Cooling water entered the annuliia of the coiidenser a t the bottom a t a knoivn flow rate and its outlet temperature was specified. Cnknown terminal temperature. and concentrations were est,imated from overall material and enthalpy balance. The lat’ter were based on the assumption that each mixture would exit the condenser saturated. By using averaged propert,ien, partial pressures, and flow rates, gas-phase heat and mass transfer Coefficients were calculated from the follo~vingcorrelation (Treybal, 1968) :
a2(T, - Tm)
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
23
~~~~~
Table II.
Terminal and Averaged Property Values
Benzene-air
L , ft N , lb mol/hr V , Ib mol/hr W , lb/hr OF
Tsj "F T,, O F P,, a t m P,, a t m Btu
q1
m Btu
&, hr lb mol
c, hr lb mol F J
hr
PBM,
Water-air
Water-carbon dioxide
0 0,949 0.511 390.0 130.0 97.6 85.0 0.35 0.22
9.20 0.949 0.185 390.0 85.0 74.9 70.4(70.0) 0 . 1 6 (0.16) 0.11
1.256 0.202 250.0 130.0 97.9 85.0 0.14 0.76
6.27 1.256 0,056 250,O 85.0 75.6 71.6(70.0) 0 , 0 4 3 (0.057) 0.029
0 1.24 0.22 275.0 175,O 96.3 85.0 0.15 0.073
9.08 1.24 0.054 275.0 85.0 72.6 7 0 . 1 (70.0) 0,042 (0.057) 0.025
3769.0
1304.0
3881.0
904.0
3398.0
711.0
0.0
5608,O
0
3350.0
0
4098.0
0
0.326
0
0.146
0
0.0054
0
0
atm Btu
hul W
~
F
E
1- ecC Ib mol "' hr ft2a t m Le2/3
0
0.165
0
0
0.79(0.75)
0.93(0.90)
0 . 9 3 (0.90)
15.16
16.76
10.45
1 . 1 7 (1.1)
1 . 0 2 (1.0)
1 . 0 1 (1.0)
1.64
2.78
1.73
1.26
0,760
0.946
Applicat'ions of the j-factor analogy and correlations t o problems involving simultaneous heat and mass transfer have been validated for both parallel (Stern and Votta, 1968) and cross-flow geometries (Schrodt and Gerhard, 1965, 1968). The combined condensate film, tube wall, and coolant film heat transfer coefficient, ho', was taken as a constant' 300 Btu/hr-Et 2-oFfor all the studies. Equations for physical properties as a function of temperature and composition, the solution equations, and the correlation expression were programmed for digital computation, and a series of results were computed for saturated and superheated inlet mixtures a t 1.0 atm.
parentheses correspond to the preliminary estimated values. The predicted Po-T, curves are plotted in relation to the saturation equilibrium lines of benzene and water in Figures 2 and 4. Changes in coolant, interface, and mixture temperatures with condenser length are given in Figure 3 for the benzene-air example, and changes in q and Q with L in Figure 5 for the water-air example. The benzene-air mixture entered the condenser superheated, became subcooled, and, finally, saturated. Fog was
Results and Discussion
Terminal values of T,,P,, q , and so forth and average values of several other quantities, h,, Le2I3,and so forth are given in Table I1 for a n example from each series. Xumbers in
70
80
90
too Tq,
110
Figure 2. Cooling-condensation path mixture in relation to saturation line 24
I20
130
O F
of
benzene-air
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
2.o
4 .O
6.0
8.0
92
L ,ft
Figure 3. Computed temperature variations expressed as a function of the condenser length for benzene-air system
o'2'
001
were indicated b y mixtures with > 1.0. The P,-T, curves of highly superheated mixtures approached the saturation curves because ( P o - P,) b for T , > 101.5'F and dP,/dT, < b for T , < 101.5"F. This agrees with the results in Figure 4. Stern aiid Votta (1968) reported that their superheated mixt'ures of HzO aiid CO, tended to become saturated. No results for initially saturat'ed mixtures were given. Clearly, design procedures based 011 the assumption of continuous core stream saturation could lead to considerable error in the dynamic analysis of tem for which the < 2 > 1.0. Results for both saturated aiid unsaturated water-air mixtures were similar to those obtained b y other iiivestigatorsLe., saturated mixture:: remained saturated, aiid unsaturated mixtures tended to become saturated. S o gas-phase coiideiisatioii was predicted in these cases. Figure 4 shows the ?,-To curve for a n unsaturated water-air mixt,ure. The dotted lines coiinect local P,-T, values in the coiideiiser with correspoiidiiig P,-T, interface values. Curves in Figure 5 show the local heat transfer rate aiid flux for this run as a furictioii of L. The curves were determined b y using Equations 19 aiid 20. Interestingly, the heat transfer rates in this and all the other studies decreased rapidly as the mixture passed through the condenser, indicating clearly that coiideiisatioii of tlie final traces of vapor from a mixture becomes increasingly difficult as the concentration of iioncoiideiisable gas iiicreases, and the mass transfer driving force decreases. The general analytical solutions used to solve the cooliiigcoiideiisatioii problems in this article appear to have a decided advantage over iterative methods; however, in special cases, breaking the problem into matching segments aiid solving each with more properly averaged coefficients may prove desirable. Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
25
Conclusions
T
=
The foregoing general analytical solutions to the linearized cooler-condenser problems have been shown to be useful for designing condensers and analyzing their performance. Equations for calculating vapor-gas mixture and coolant properties as a function of condenser length were given. From these, equations for predicting local heat transfer, core stream and surface condensation rates, and overall thermal loads were developed. Three vapor-noncondensable gas mixtures, typical of those which might be encountered in condenser design problems, were analyzed, and results verify the general applicability of the solutions. The Lewis number has been shown to be a n important criterion of the relative rates of sensible heat and mass transfer in the process.
B
= vapor flow rate, lb mol/hr
w
=
temperature, OF coolant flow rate, lb/hr
e ~- hckermann correction factor 1 - e-c
Le Pr Re Sc
= = = =
Lewis number Prandtl number Reynoldsnumber Schmidt number
SUBSCRIPTS B.11 = logarithmic mean g
s
=
= t = w =
vapor interface total coolant
Nomenclature
A
= condenser area, ft2 a = constant, a t m b = constant, atm/OF C = total rate of condensation, lb mol/hr C, = heat capacity of mixture, Btu/lb mol-”F c, = heat capacity of coolant, Utu/lb-”F F = rate of fog formation, lb mol/hr G = mixture flow rate, lb mol/hr h, = gas-phase heat transfer coefficient, Btu/hr-ft2-”F ho’ = combined heat transfer coefficient, Btu/hr-ftZ-OF i = latent heat of vaporization, Btu/lb mol ,i = factor 12, = mass transfer coefficient, lb mol/hr-ft2-atm L = condenser length, ft Jf = molecular weight, Ib/lb mol N = gas flow rate, lb mol/hr P = pressure, a t m Q = thermal load, Btu/hr p = heat transfer rate, Btu/hr-ft2
literature Cited
Ackermami, G., Ver Deutsch Ing. Forschungsh., 382, 1 (1937). Amundson, N. R .,“hlatheniatical hlethods in Chemical Engineering,” pp 111-127, Prentice-Hall, Englewood Cliffs, N.J., 1966. Colburn, A. P., Drew, T. R., Trans. Amer. Inst. Chem. Engrs., 33, 197 (1937). Colburn, A. P., Edison, A. G., I n d . Eng. Chem., 33, 437 (1941). Colburn, A. P., Hougen, 0. A., ibzd., 26, 1178 (1934). Coughanowr, D. I
