The heat-transfer data for kerosine up to 2OO0C., obtained a t Reynolds numbers in the range 2000 to 10,000 and a t moderate differential temperatures (15" to 30" C.), correlated reasonably well with the empirical relationship of Sieder and Tate; however, analysis of the results indicated that the simple empirical relationship:
Nu = 0.0204 Re0.'*' afforded significantly better agreement. The results were obtained under conditions which were steady for a prolonged period. I n the supersonic transport, the fuel system will be subjected to vibration, and the fuel flow and temperatures will cycle. All these conditions may tend to decrease the amount of deposit laid down or lead to its periodic removal from the heat-transfer surfaces. Acknowledgment
m = mass flow, lb./hr. = Nusselt number hD/k, dimensionless = Prandtl number c , l / k , dimensionless = Reynolds number DG/p, dimensionless = heat flux ( q / A ) ,CHU/hr. sq. ft.
Nu Pr Re Q
q = heat input to a system, product of
m, C,
and
At, CHU/hr. t = temperature, C. (unless stated otherwise) T = time, hr. YL
= distance from tube wall defining laminar sublayer,
inches
y+ = Prandtl dimensionless distance, dimensionless At, = logarithmic mean temperature difference, C. At = temperature difference, C. ,u = absolute viscosity a t mean bulk fuel temperature,
lb./ ft. hr. wiL = absolute viscosity a t mean surface temperature, lb. / ft. hr. Literature Cited
Assistance given by Lucas Gas Turbine Equipment, Ltd., Serck Radiators, Ltd., and the Ministry of Technology under whose sponsorship the work was done, is gratefully acknowledged. Nomenclature
A = area, sq. ft. bo, bi = multiple regression analysis constants, dimensionless C, = specific heat, CHU (centigrade heat unit, the amount of heat required to raise temperature of 1 lb. of water, 1"C.) D = diameter, ft. (unless stated otherwise) f / 2 = friction factor, dimensionless G = mass velocity, lb./hr. sq. ft. h = heat transfer coefficient h, a t time T , h , a t time T = 0, CHU/hr. sq. ft. o C . k = thermal conductivity, CHU ft./hr. sq. ft. 'C. L = length, ft. (unless stated otherwise)
Boelter, L. M. K., Martinelli, R . C., Finn, Jonassen, Trans. A S M E 63, 447 (1941). Dukek, W. G., J . Inst. Petrol. 5 , 273 (1964). Nicholson, J. T., Trans. Inst. Eng. Shipbuilders Scotland 54, 64 (1910). Sheriff, N., Gumley, P., France, J., Chem. Process Eng. 45, 624 (1964). Sieder, E. N., Tate, G. E., Ind. Eng. Chem. 28, 1429 (1936). Smith, J. D., Aircraft Eng. 34, No. 4, 19 (1967). Ziebland, H., Dupree, M. T., J . A m . Rocket SOC.1961, 845.
RECEIVED for review March 7, 1968 ACCEPTED February 3, 1969
OPTIMAL DESIGN OF AN ABSORBER-STRIPPER SYSTEM Application of the Complex Method TOM10
U M E D A
Chiyoda Chemical Engineering & Construction Co., Ltd., Tokyo, Japan The absorber-stripper system, which has a complicated structure with recycle of absorbent as well as recovery of heat, has been successfully optimized by the Complex method developed by Box. The system contains five independent design variables which have to be decided so as to minimize some objective function, like production cost, under various constraints. Like other optimal seeking methods, the Complex method requires several trials by varying initial conditions for checking that the result is the global minimum, rather than a local one, but this does not lessen its applicability to the optimal design of process systems.
THEabsorber-stripper
system is a common one in the process industries, both as a means of purifying a process stream and from the increasingly emphasized viewpoint of pollution abatement. An optimal design of this type of system is therefore important in both reducing the cost of gas treatment and ensuring efficiency of purification. Such a system (Figure 1) follows a well308
I & E C PROCESS D E S I G N A N D DEVELOPMENT
tried and refined arrangement and optimal design within this arrangement will prove worthwhile. The process has a complicated structure with recycle of absorbent as well as recovery of heat, and a hill-climbing method would seem appropriate. The Complex method developed by Box (1965) has been used, since it requires only the iterative computation of the design calculations
Start
Give s p e c i f i e d v a l u e s for independent variables
Figure 1 . Schematic flow sheet of absorberstripper system
variables
Evaluate objective function
I
1
I
and i t is not necessary to find any gradients. This method is widely applicable to the optimal design of process systems. Statement of Problem
The basic flow pattern for the system is shown in Figure 1. The feed gas containing a material to be removed is passed upward through an absorber and comes in contact with an absorbent flowing countercurrent to the gas stream. The rich solution from the bottom of the absorber is heated by a heat exchanger with the absorbent from the bottom of the stripper and then fed to the stripper to separate the absorbent from the rich solution. I n the stripper, the material in the feed gas is obtained as a product from the top of the column. When the purity of the top product is specified, i t is often necessary to have the rectifying section as in ordinary distillation. The absorbent, after being cooled in the heat exchanger and a cooler, is fed to the absorber to complete the cycle. The recoveries of the absorbent and of heat are characteristic of this system, which has been well refined from the viewpoint of an economical process arrangement. Like other process design, the work starts with the setting up of mass and heat balances, followed by sizing the equipment. The optimal design procedure using a direct search method is shown in Figure 2. The optimal design is a constrained optimization problem, in which an objective function has to be minimized or maximized, subject to the various kinds of constraints. The equality constraints correspond to the relationship of mass and heat balances. There are also inequality constraints in the process design, which are required to specify the admissible regions to the design variables. Mass and Heat Balances
The mass balance for the absorber-stripper system would be written as follows:
Fyl + Lxz = Gyz + ( L + D )xi ( L + D)Xi = L x +~ D x ~ F=G+D
for t h e absorber for t h e stripper
for t h e whole s y s t e m
Here we have three independent equations with nine variables so that by specifying six variables, the other three variables can be calculated. I n many cases F , yl, and x 3 have already been specified; therefore we have
Figure 2. Flow diagram of optimal design procedure to choose three variables as decision variables. G, L , and xp are chosen as the decision variables and the other three variables, x i , y2, and D, can be obtained by the following relationship:
The heat balance for the system would be written as follows:
( L + D)Cpzt2-+
Q r
= D~p3t3+ L~p4t4+
Qc
around the stripper, where
for the heat exchanger and Qcooler
= L(cpd5 - C
~ I ~ I )
for the cooler. tr can be calculated by knowing xz’, since it is the bubbling point corresponding to xz and can be obtained by specifying a “q-value,” which is the thermal condition of the feed to the stripper-that is,
tz = tl - ( 4 - l)h,/cpz Ir, these equations, c,, and h, refer to heat capacity and latent heat, respectively. For simplicity, the absorber is assumed to be operated isothermally-i.e., no heat of absorption-and the operating temperature is specified in advance. I n the relationships above, R and t j are chosen to be independent variables and the rest of the variables can be calculated by the use of these equations. Equipment Sizing
Packing Height of an Absorber. The height of a packed bed can be obtained by the equation (Perry, 1963) VOL. 8 N O . 3 J U L Y 1 9 6 9
309
where
= HOP[ (%)log
1 -yz + JY’ Y2
d yy * ) (y-
1
where Hog is the height of a transfer unit and N , is the number of transfer units. When the operating line and the equilibrium line are straight, which is the case for a dilute solution, obeying Henry’s law (yl/xl = y2/ xp = m ) and in the absence of heat effect, it is possible to evaluate the integral explicitly. The first term of the right-hand side of the equation is usually small enough to be neglected in many practical examples. Then N , is approximated by
M = R(R+1) c = 1 + (U - l ) k , x,’ = ~3 - k , x:, = X, - k, X“
/
for rectifying section
\
= intersection of the operating line with q-line
k, = intersection of the operating line with the equilibrium curve
M =
L+qF V - (1- q ) F
c = I+(a-l)k, x: = x 0 - k, x i = xz - 12, x =
1 i
for the stripping section
intersection of the operating line with q-line
k = intersection of the operating line with the equilibrium curve
where m is the slope of the equilibrium line. The height of a transfer unit can be estimated by
Hog= HE+ ( m G / L ) H i where H, and H i are gas and liquid phase transfer units, respectively. Although there are many empirical relationships based on experiments to predict HB and Hi, these do not always give a satisfactory prediction. These simplifying assumptions for the calculation of N , and Hog are made in the numerical example. By ,these simplifications the computing time is shortened but there is no loss of generality in applying the Complex method to the optimal design of an absorber-stripper system. Diameter of an Absorber. The diameter of an absorber can be calculated on the basis of flooding velocity or obtained by the graph given by Lobo et ul. (1945). The following equation given by Sawistowski and Smith (1963) is also useful for computation of the flooding velocity.
Log Gi (a/t3)( 1 / B c P i P s ) ( p w / / + ) o . 2 =
1.74(L ’/ G’) ” * ( p i / p g ) where G/ is the following velocity. The diameter of a packed bed is computed on the basis of about 75% of the flooding velocity. Number of Stages for a Stripper
There are two methods in calculating the number of stages: the “stage to stage” calculation and the shortcut method using Fenske’s (1932) and Underwood’s (1948) equations combined with Gilliland’s (1940) correlation. For a binary system, Smoker’s (1938) method seems to be useful which corresponds to the graphical computation of McCabe-Thiele method on a digital computer; this last method is used for the numerical example. By this method, the number of stages can be calculated by the following equation:
N = log
1
x;{ 1 -
310
M C ( a - l)xL a - MC2
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
The actual number of stages is obtained by dividing the number of theoretical stages by a plate- efficiency which might be found from a manual (American Institute of Chemical Engineers, 1958) or from experimental results for a particular system. Diameter of a Stripper
The diameter of a stripper may be estimated on the basis of a reasonable approach to flooding, which is called Souders and Brown’s (1934) method. The vapor load obtained by V = Q,/h,V,,, is divided by the following linear velocity to find the sectional area of the stripper,
U=K(-)P i - P g ‘
2
PP
where K is a specified constant for a given tray spacing and the height of liquid seal on a tray. Heat Transfer Areas
Heat transfer areas can be calculated by the equation
A, = Qi/ UhL( L M T D ) , where ( L M T D ) , is a logarithmic mean temperature difference and Uhgis an over-all heat transfer coefficient. If phase change occurs during the heat transfer, it is not easy, in general, to find a film coefficient by the calculation, but otherwise the film coefficient can be calculated. For simplicity, the heat transfer coefficients are given as constants in the numerical example. I n the calculation of LMTD’s for a condenser and a reboiler, it is necessary to compute a dew point and a bubbling point in advance. The application of Raoult’s law to a binary system gives the equilibrium constants, K1 and K 2 for components 1 and 2, respectively, in the form
where the vapor pressure for each component has been expressed by Antoine’s equation. The dew point and bubbling point of the binary system can be calculated by solving the following equations with respect to temperature:
Y1/K 1 + y2/ K Z= 1.000 for dew point K1xl+ K2x2= 1.000 for bubbling point The substitution for K1 and Kz gives
f ( T )= p y l ~ 1 0 ~ ~ - ~ l ~ ( ~++ 2 3 0 )
Finally, we have the following inequality constraints
F>G>F
for dew point calculation, and
f ( T )= ( X 1 / P ) 1 @ '
-W(T-230)
+
( x 2 / p ) 1 O A ' - B ' / ( T - 2 3 0 ) - 1.000 =
0
for bubbling point calculation. These equations are highly nonlinear and the NewtonRaphson method must be used to obtain the solutions. The approximate temperature, T , + can be obtained by
x3
- Yl
x3 - mxz
L,,,
> L > mG
R,,,
> R > R,,
T,+1=T , - f ( T n ) / f ' ( T n ) where f(T,) is the relationship for dew point or bubbling point calculation given above. The iterative computation would be terminated upon satisfaction of a specified criterion-for instance, 1 T , - 1 - T,I 5 0.5" C.
R,,
Inequality Constraints
I n the preceding section, the independent variables for this system, G, L , XZ, q, and R , were found. T o have feasible results for the design calculation, these independent variables must satisfy various constraints. In the absorber, the following relationships must be satisfied.
1 - ( m G / L ) > 0 , yl - mxz > 0 , and y 2 - mx2
>0
Since
y2 = x3
+ ( F /G )(yi - ~
3
)
m and
G > F ( x~ y l ) / (x3 - mx2) I n the stripper, the reflux ratio must be larger than the minimum reflux ratio-Le., R > Rmn. The restriction for the q-value may be determined from the heat balance for the heat exchanger. Since Cpe (ti - t 2 )
hi
, t z > tl
we have
I n the heat exchanger, we have
L (cpet4- c&) = ( L + D ) ( c p h- cpltl) and
are used for convenience.
can be evaluated by R,,
=
x3
- yc
yc - x c
where
yc =
axc 1+ ( a - l ) x ,
Point (xc, yc) is the intersection of the q-line with the equilibrium curve in an x-y diagram. Objective Function
We have
q = 1.00 +
where F , Lmax,and R,,
t5
>
tl
The substitution of these relations into the definition of the q-value gives
I t is first necessary to form an appropriate objective function to be minimized or maximized in the optimal design problem. Investment and operating costs have to be taken into account in process design, and, in principle, the flow arrangement should also be done optimally. The flow arrangement problem would not be solved for this particular system, because this process arrangement is thought to be well established. Annual production cost, net profit, or venture profit has been widely adopted as an objective function of an optimal design. Production cost consists of various items (Table I ) , but operating labor and supervision, maintenance, etc., in the direct production costs, plant overhead costs, and general expenses are considered to be independently specified without regard to the process design. Therefore when an objective function is formed for the optimal design, the following items should be considered: raw materials, steam, electricity, fuel, refrigeration, water in the direct production costs, and fixed charges (Peters, 1958). Fixed charges may be estimated by the concept of cost factors, which is based on the fabricating costs of process equipment. The installed or fabricating costs of equipment can be obtained by the following relation, an approximate expression (Chilton, 1960):
Y=cu.x@ VOL. 8 N O . 3 JULY 1 9 6 9
311
Table I. Product Cost Raw materials Operating labor Operating supervision Maintenance and repairs Operating supplies
I
Steam Electricity Fuel power and utilities Refrigeration Water Royalties (if not on lump-sum basis)
J
Rent Insurance Taxes (property) Depreciation
7
Medical Safety and protection General plant overhead Payroll overhead Packaging Restaurant Recreation Salvage Control laboratories Plant superintendence Storage facilities Executive salaries Clerical wages Office supplies Communications Sales offices Salesmen expenses Shippeg. Advertismg Research and development
Direct production costs
- Factory, manufacturing costs
i'
Fixed charges
+ Total product, cost
- Plant overhead costs
7
i
Administrative expenses
*
General expenses
Distribution and expenses
Financing (interest) (often considered a fixed charge)
where x is a parameter-for instance, tower diameter, heat transfer area-and y is an installed or fabricating cost. a and p are constants. For the optimal design problem to be concerned, the following objective function would be defined:
+ C,,(W, + W z )+ C,W, + C d ( H P 1+ HP,) + (investment cost) F,
Objective function = C,F
where
Investment cost = C,ZoPz + C,,NDP + C,(APn+
C, C,, C,, and C, are unit costs of utilities. C,, C,, C, and C, correspond to a , and P,, P,, Pa, and P, correspond to p above. According to the procedure outlined in Figure 2, the optimal design would be carried out by minimizing the objective function defined above under the constraints summarized above. Method of Constrained Optimization
There are many methods of optimization, broadly classified as search methods, mathematical programming, and variational methods. Search methods include gradient 312
I & E C PROCESS D E S I G N A N D DEVELOPMENT
method, pattern search method, ridge analysis, and the generalized IVewton-Raphson method. Mathematical programming consists of linear programming, dynamic programming, and various methods of nonlinear programming. Variational methods are related to Pontryagin's maximum principle and complex chemical processes may be optimized by the extended methods of several workers. Box's Complex method is a search method developed from the Simplex method of Spendley et al. (1962). The Simplex method was applicable only to unconstrained optimization problems, and evaluated the objective function a t a set of points forming a simplex in the space of independent variables. After comparing the values of the objective function t o find the worst value and its position, a new simplex was formed by replacing the worst vortex by its reflection in the centroid of the remaining points. This procedure was continued until a stopping criterion was satisfied. Box extended this idea (Nielder and Mead, 1965) to solve constrained optimization problems-that is, to minimize or maximize a function (xlx p . . . x n ) of n independent variables x l , x2, . . .x, subject to m constraints of the form gk 2 xk 2 hk, k = 1, 2, . . ., m. Here x n f l r . . .x, are functions of xl, x 2 , . . .x, and the lower and upper constraints gk and hk are either constants or functions of x , , xz, . . ., x,. I t is assumed that an initial point, xl, x 2 , . . .x, which satisfies all the m constraints, is available. In this method, K 2 n + 1 points are used, one of which is the given initial point
and the other K - 1 points required to set up the initial configuration are obtained one a t a time by using pseudorandom numbers and the range of each independent variable, x,. Thus n, is given by
x*
I
!I
x , = g, + ( h , - gJrt where rLis a pseudorandom number deviate rectangularly distributed over the interval (0, 1). Each point generated in this way must satisfy explicit constraints, but need not satisfy all the implicit constraints. I t will be convenient to modify some of the implicit constraints to approximate explicit constraints for the generation of an initial configuration of points. After generating all points, the objective function is evaluated a t each vertex of the simplex. The following procedure is repeated; after finding the worst point, it is replaced by a point y times as far from the centroid of the remaining points as the reflection of the worst point in the centroid, the new point being colinear with the rejected point and the centroid of the retained vertices. If this new trial point is also the worst or does not satisfy some constraint, it is moved halfway toward the centroid of the remaining points. Thus as long as a simplex has not collapsed into the centroid, progress will continue. For the twodimensional case, the procedure is shown schematically in Figure 3. The usual method for checking that the global rather than a local maximum or minimum has been found is to restart the above procedure from different initial points. If the calculations all converge to the same result, this is considered to be the global optimum.
co 1
Figure 3. Complex method Two-dimensional
[
1
Start
I Generate initial
points
No
F i n d new point b y m o v i n g h a l f way towards centroid o f remaining p o i n t s
x xi-
--I(
I
w
P ( I - 1 ) -
xw
2
Computation f o r deaign calculation I
Numerical Example of Optimal Design
By applying the Complex method to the absorberstripper system, we might hope to achieve the optimal design of the system. Accordingly, the flow diagram shown in Figure 4 was developed into a program for the CDC-1604. A few of the numerical results, based on the specified values shown in Table 11, are given in Tables 111, IV, and V. The case studies have been carried out five times (cases 1, 2 , 3, 4, and 5 in Tables 111, IV, and V) by varying the initial points. The results would be considered to be the global optimum, since almost the same values of variables and objective function could be obtained (Lmeda, 1966). I t is important to get some feel for the effects of the following parameters on the results of computation and the rate of convergence: initial configuration, an expansion factor and a stopping criterion, and the number of vertices. Some further mathematical experiments were therefore carried out, involving the variation of an expansion factor: 1, 2, 4, and 6; of a stopping criterion: 0.01 to 0.0001; and of the number of vertices: 6 and 8. By these studies, the following information was obtained.
I
Evaluate objective function
1 Reject t h e worst p o i n t and replaced by
x
i
--*
w
1 -
S a t ia f y c o n s t r a i nt s Xi = g, + e l Or
x. = h I
- e l
F i n d new p o i n t b y m o v i n g h a l f way towards c e n t r o i d o f remaining points
l X
1
=
-
xw
2
0
xw
-2
-
The initial configuration seems to have no large effect on the results of computation and the rate of convergence. Although the use of various expansion factors is needed to improve the objective functions, the results do not indicate how to use this to speed the convergence. The stopping criterion was expressed as a relative error, (fm,,, - fmi, )/fm,,,, and must be less than 0.001.A value 0.0001 as the stopping criterion, though suitable, demanded an excessive number of iterations. (The computing time for each iterative computation was about !., second on CDC-1604.) VOL. 8 NO. 3 JULY 1969
313
Table V. Optimal Results
Table II. Optimal Design Basis Specification
E.' = 60 (kg.-moles hr.) j
i
1,
A. k = 8. y = 1, EPSJ = 0.0001
= 0.400 = 0.99
Physical properties =
1500, p2 = 5(kg.;cu. m.)
m, = 58, m! = 154(kg./kg.-mole) ('p, = 0.2. Cr>, = 1.0 (kcal.ikg.'C.)
H, = 50. h. = 500 (kcal..'kg.) Ln PI = 32.9 - 14,300; (1 + 503). PI,P , (mm. Hg) Ln P. = 30.4 - 13,800: ( t + 503)
:ace
L
G
D
nl
X?
y?
4
R
1 2 3 4 5
30.14 30.46 30.36 30.46 30.39
36.32 36.29 36.32 36.33 36.28
23.68 23.72 23.68 23.67 23.72
0.440 0.438 0.439 0.439 0.439
0.0078 0.0088 0.0094 0.0103 0.0080
0.0153 0.0144 0.0154 0.0157 0.0142
1.021 1.003 1.012 0.997 0.996
2.35 2.35 2.35 2.36 2.36
ti
t,
tr
t;
A!
A?
AI
A,
49 53 51 55 55
42 42 42 42 42
77 77 77 76 77
47 40 43 37 37
185.5 186.1 185.5 186.1 186.9
22.7 22.4 22.4 22.2 22.4
28.8 51.2 38.7 65.8 66.3
11.9 8.6 10.3 6.8 6.8
Design data and conditions
P = 760 mm. Hg I1 = 30. t , , = 20; i , ,= 25, t. = 120 ('C.) m = 0.8. a = 2.0
1 2 3 4 5
c,
ci = 500, = 100 = 200(kcal.:sq. m. hr. C.)
= 300.
LJl
c
Constraints = 300 (kg.-moles hr.) = 20
'ma, 'ma,
Cost data
c, c., c, cer c, C" c, c,
1 2 3 4 5
= 50 (S/yr:hr., kg.) = 0.0635 ( $ / y r . .hr.1 kg.) = 35.2 ($, yr.. hr. kg.) = 89.7 ($, yr. 1 hp.) = 350 (S). P , = 0.556 = 363 ($j. P , = 1.085 = 1000 ($1. P,= 0.30 = 600 (Sj. P , = 1
k
EPSJ
DA
S
DS
Wl
Wr
12.9 13.3 13.1 13.3 13.0
0.51 0.51 0.51 0.51 0.51
59 60 59 59 60
0.87 0.86 0.86 0.86 0.86
93730 94015 93754 94041 94442
3140 1811 2437 1300 1283
o p . Cost
Constants for complex method y
Z
1 2 3 4 5
= 1, 2, 4, 6 = 6,8 = 0.0001 (stopping criterion)
Table 111. Generated and Specified Initial Values
L
G
X?
A. Randomly Generated Values 1.0442 56.817 0.14898 233.217 0.41280 1.0213 52.485 54.148 0.17663 1.0133 125.483 47.556 0.44046 1.0049 206.650 58,036 0.20429 1.1257 56.634 282.154 1.0789 0.46811 103.993 57.628 0.23194 1.0928 48.616 176.256
11.096 3.338 16.301 10.512 6.442 17.483 12.522
B. Given Values for First Point Case 1 2 3 4 5
100.00 100.00 150.00 50.00 50.000
40.00 45.00 45.00 45.00 45.00
0.0100 0.0100 0,0100 0.0100 0.0500
1.000 1.000 1.000 1.000 1.000
6.825 8.592 11.932 4.000 4.418
Table IV. Starting Values of Objective Function Case
Point 1 2
314
1 x IO'
2 x IOi
3 x IO4
4 x IO'
5 x 10'
-13.1880 -16.6262 -13.2100 -20.0104 -16.6201 -17.1698 -16.5974 -18.0551
-15.4170 -16.6262 -13.2100 -20.0104 -16.6201 - 17.1698 -16.5974 -18.0561
-18.5252 -16.6262 -13.2100 -20.0104 -16.6201 -17.1698 -16.5974 -18.0551
-12.0112 -16.6262 -13.2100 -20.0104 -16.6201 - 17.1698 -16.5974 -18.0551
-11.2953 -16.6262 -13.2121 -20.0104 -16.6201 -17.1698 -16.5974 -18.0551
1 8 E C PROCESS D E S I G N A N D DEVELOPMENT
X
10'
-6.6092 -6.6002 -6.6010 -6.6052 -6.6048
X
10'
182 566 879 211 303
B. k = 8, y = 2, EPSJ = 0.0001
R
Q
2.1776 2.2314 2.1948 2.2293 2.2455
No. of Trials
Obj. Fn.
Inc Cost
4.4317 x 10' 4.3688 4.4062 4.3759 4.3592
S. 982 972 975 967 971
L
G
D
x
~X2
R
30.76 30.38 40.28 31.35 30.75
36.64 36.32 42.55 36.74 36.38
23.36 23.68 17.45 23.26 23.62
0.438 0.439 0.437 0.433 0.437
0,0185 0,0087 0.197 0.0193 0.0125
yi 0.0238 0.0152 0.158 0.0266 0.0170
4
1 2 3 4 5
1.094 1.010 1.096 1.096 1.041
2.26 2.34 2.26 2.51 2.33
t,
ti
t?
1 2 3 4 5
30 51 30 30 44
42 42 42 42 42
76 77 63 76 76
ti 75 42 63 76 54
A, 178.2 185.5 133.0 191.0 184.2
A1 22.5 22.4 13.5 24.0 22.6
Ai 0.2 41.1 0. 0. 16.8
Ad 19.8 9.9 22.6 20.2 14.8
Z
DA
S
DS
W1
Wz
Si
1 2 3 4 5
11.6 12.7 15.1 9.1 12.8
0.51 0.51 0.56 0.51 0.51
58 60 44 47 58
0.87 0.86 0.75 0.90 0.87
90001 93632 67222 96505 93071
8571 2299 8309 8801 4629
999 973 765 1066 991
o p . Cost 1 2 3 4 5
4.6910 x 10' 4.3912 7.2148 5.0271 4.5030
Jnu. Cost 2.0279 x 10' 2.2098 1.6453 1.8114 2.1293
Ob]. Fn. -6.7189 x 10' -6.6010 -8.8601 -6.8385 -6.6323
N o . of Trials 418 808 405 1078 317
c \
-u0 y1
E
r'.
l
-
I 0)
c z m n W
0
m m a U.
0 W
+ a n
3 0 iL A
I
0
I
0
50
I
I I
I
100
1 150
200
NUMBER
OF
I
I
I
250
I
300
N U M B E R OF
TRIALS
TRIALS
Figure 5 . Variation of flow rate of absorbent
Figure
7. Variation of
mole fraction of absorbent
r \
-0
0
0 I
I
v)
0
c
VI 3
4 I X W
U
I
0
w
c a n
I
I
50
100
I
3 A
I
U
I
I
I
I
I 0 50h
NUMBER
150
I 200
I 250
300
OF T R I A L S NUMBER
Figure 6. Variation of flow rate of exhaust gas
of convergence of objective function is given. (Final results are those for case 5 in Table V, A.) Conclusions
The Complex method is applicable to optimal design of the absorber-stripper system. I t does not require much
OF T R I A L S
Figure 8. Variation of q-value
storage capacity for the digital computation and the computing time seems not to be large. Although the method requires several trials by varying initial conditions for checking that the result is the global minimum, rather than a local one, this does not lessen its applicability to optimal design of process systems. VOL. 8
NO. 3 JULY
1969
315
Acknowledgment
The author acknowledges the invaluable guidance of
R. Aris, whose contribution helped to make this possible. Also he is indebted to the numerical Analysis Center of the University of Minnesota for the use of the facilities during a portion of this study. Finally the author thanks Chiyoda Chemical Engineering & Construction Co. for support of the study. Nomenclature
A , = heat transfer area or a constant in Antoine’s
-0
c
a
LT X
3 A iL W
a
NUMBER
Figure
OF
TRIALS
9. Variation of reflux ratio
X
z
0 L-
o 3 z LI.
-
-
10
504 I
I I
OO
50
100
150
NUMBER
I
I
I
200
250
300
OF T R I A L S
Figure 10. Convergence of objective function 316
equation Ai = heat transfer area of a condenser or a constant in Antoine’s equation heat transfer area of a reboiler or a constant in Antoine’s equation heat transfer area of a heat exchanger heat transfer area of a cooler relative volatility constants in Antoine’s equation a constant in the equation of the installed or fabricating cost of heat exchanger, condenser, and reboiler C C L l = unit cost of cooling water Cd = unit cost of electricity c, = unit price of feed gas C” = a constant in the equation of the installed or fabricating cost of a stripper c, = a constant in the equation of the installed cost of pumps and motors c, unit cost of steam a constant in the equation of the installed or C, fabricating cost of an absorber 1 + (a- 1) k,, 1 + (a- 1) k , C heat capacity i = 1, 2 , . . ., 5 , c,, corresponds CVl tot, top product from a stripper D constant to be reset inside a constraint e, F feed rate to an absorber cost factor/life of a plant F, exhaust gas rate from an absorber, molar or G, G’ weight flooding velocity of a packed bed (weight) conversion factor (between mass and force) lower constraint for x gas phase transfer unit liquid phase transfer unit H.T.U. horsepower of a pump for feed horsepower of a pump for a stripper latent heat of feed to a stripper upper constraint for x,, latent heat upper constraint for xh a constant for tray design equilibrium constant for component 1 equilibrium constant for component 2 number of points to form configuration see section “number of stages for stripper” flow rate of an absorbent upper limit for L (flow rate of an absorbent) a constant in the operating line of a stripper Henry’s constant m number of plates N N.T.U. N, total pressure of stripper P a constant in the equation of the installed or Pa fabricating cost of heat exchanger, condenser, or reboiler a constant in the equation of the installed or P, fabricating cost of a stripper
I & E C PROCESS D E S I G N A N D DEVELOPMENT
P, = a constant in the equation of the installed or P, =
Q,
Qc
or
Qcoal r
q qmx qm, Rmn Rmx
r,
= = = = = = = = = =
s= S I tl
= =
t* = t3 = t4
t j ,
=
TS = tu, =
t,,, = t/ = t, =
u= = u1 = u,=
Uh,
u 3
=
u 4
=
v= v,,= w1 = w2 = XI, x / = 7.2
=
= xi = x, = yl = x3
fabricating cost of pumps and motors a constant in the equation of the installed or fabricating cost of an absorber heat load heat load in a condenser heat load in a cooler heat load in a reboiler q-value maximum q value minimum q value reflux ratio of a stripper upper constraint for R random number number of stages steam consumption in a reboiler operating temperature of an absorber feed temperature to a stripper dew point of top product bubbling point of bottom product from a stripper inlet temperature to a cooler inlet temperature of cooling water outlet temperature of cooling water temperature a t the feed plate in a stripper saturated temperature of steam in a reboiler linear velocity in a stripper over-all heat transfer coefficient over-all heat transfer coefficient in a condenser over-all heat transfer coefficient in a reboiler over-all heat transfer coefficient in a heat exchanger over-all heat transfer coefficient in a cooler vapor load in a stripper specific molar volume cooling water consumption in a condenser cooling water consumption in a cooler feed composition to a stripper composition of an absorbent fed to an absorber (bottom composition in a stripper) composition or purity of top product independent variable see section “inequality constraints” feed composition to an absorber
ye = exhaust gas composition from an absorber y( = see section “inequality constraints” 2 = height of an absorber
GREEKLETTERS a,
P = constants in cost data equation Y = reflection factor in Complex method c = void in a packed bed (fraction) ILE = viscosity of gas or vapor kl = viscosity of liquid = viscosity of water Pa = density of gas or vapor pi = density of liquid
fill
Literature Cited
American Institute of Chemical Engineers, Xew York, “Bubble Tray Design Manual,” 1958. Box, M. J., Computer J . 8, No. 1, 42 (1965). Chilton, C. H., Ed., “Cost Engineering in the Process Industries,” McGraw-Hill, New York, 1960 Fenske, M. R., Ind. Eng. Chem. 24, 482 (1932). Gilliland, E. R., Ind. Eng. Chem. 32, 1220 (1940). Lobo, W. E . , et al., Trans. Am. Inst. Chem. Engrs. 45, 693 (1945). Nielder, J. A., Mead, R., Computer J . 7, Xo. 4, 308 (1965). Perry, J. H., Ed., “Chemical Engineers’ Handbook,” 4th ed., pp. 14-29, McGraw-Hill, New York, 1963. Peters, M. S., “Plant Design and Economics for Chemical Engineers,” McGraw-Hill, Xew York, 1958. Sawistowski, H . H., Smith, W., “Mass Transfer Process Calculations,” Interscience, Kew York, 1963. Smoker, E. H., Trans. A m . Inst. Chem. Engrs. 34, 165 (1938). Souders, M., Brown, G. G., Ind. Eng. Chem 26, 98 (1934). Spendley, W., Hext, G. R., Himsworth, F. R., Technometrics 4, Xo. 4, 441 (1962). Umeda, T., M.S. thesis in chemical engineering, University of Minnesota, 1966. Underwood, A. J. V., Chem. Eng. Progr. 44, 603 (1948). RECEIVED for review May 29, 1968 ACCEPTED January 16, 1969
SORPTION OF SULFUR DIOXIDE BY ION EXCHANGE RESINS LAWRENCE
L A Y T O N ’ A N D
G .
R .
Y O U N G Q U I S T
Department of Chemical Engineering, Clarkson College of Technology, Potsdam, AV.Y . 13676 Rate and equilibrium data are presented for sorption of sulfur dioxide on a weak base, macroreticular ion exchange resin. Rates are limited by intraparticle diffusion. Equilibrium loadings a t 25’ C. compare favorably with other adsorbents.
THEsorption of gases and vapors by ion exchange resins has been given little attention. Cole and Shulman (1960) showed that the equilibrium uptake of sulfur dioxide by several types of dry ion exchange resins compared favorably with that of commercial adsorbents such as silica gel, activated charcoal, and molecular sieves. Although no definitive rate measurements were made, qualitatively the rates of approach to equilibrium were so low as to
‘Present address, E. I. du Pont de Kernours and Co., Inc., Parlin, N. J.
make industrial applications doubtful. The resins used were of the microreticular type, generally characterized by low surface areas (less than 1 sq. meter per gram) and low porosity (less than lei). Pores in the resin consist of spaces between polymer chains, are of molecular size, and may be very small or nonexistent unless the polymer matrix is swollen by the presence of a solvent such as water. The sorption process for a microreticular resin is somewhat akin to absorption and the low sorption rates observed by Cole and Shulman may perhaps be attributed to the slow diffusion of sulfur dioxide through the unswollen polymer matrix of the dry resins. VOL. 8 NO. 3 JULY 1969
317
