- Email: [email protected]
Global approach to the computation of separations by approximate methods
Uwmicol Engineering Wencc Vol. 31. No. 3, pp, 425-432, 1982 Printed in GW.I Britain.
CQl9-2S09182/0M(25-08SO3-o85o3.oo10 @ I!%82Pergaman Press Ltd.
GLOBAL APPROACH TO THE COMPUTATION OF SEPARATIONS BY APPROXIMATE METHODS EGON ECKERT Department of Chemical Engineering, Prague Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia (Received 15 June 1981; accepted 20 July 1981) Abstract-A global approach to the computation of simple separation columns for non-ideal mixtures by approximate methods is described. The procedure based on a modification of the Smith-Brinkley method enables us to compute practicallyany type.of problem formulation, i.e. both simulation and design problems. The technique can also be used for formulationof semirigorousproceduresfor generalarrangementsof more complex types of equipment. 1. INTRODUCTION
It holds further
An earlier paper[l] presented an effective modification of the Smith-Brinkley methodi for the simulation of separation processes in cases where non-ideal behaviour has to be considered for calculations of vapor-liquid equilibria. As realistic assumptions were used in the derivation of the method, very satisfactory results are usually obtained in the applications. Here we shall present a generalization of the above method used to design computations, where a number of trays, the feed location etc., have to be determined. For such computations the application of rigorous methods is usually cumbersome and requires repeated calculations. The well-known methods of computation[3-6], together with some recently suggested techniques[7,8], are thus used in this area of computer-aided design[9]. The parameters, which have to be determined in the computations (e.g. number of trays, feed location, etc.) can not be usually expressed in an explicit way. Therefore a global approach to the solution of the equations describing separation equipment is appropriate for the use of the approximate methods for design computations.
2. METHOD OF COMPUTATION
Let us illustrate the approach on the case of a simple distillation column schematically shown in Fig. I. The component mass balance relations are in the form Fxfi-&i-&~i=O
i=
1,2,...,n
(1)
where fi fi, s are flow rates of feed, top product (vapor or liquid) and bottom product, respectively; and xF, x, x, are compositions of these streams. The ratios of the components in the feed stream which appear in the
7
XDi -
1= 0
(3)
+,-l=O.
(4)
For the reboiler
z$Lxs,
- 1= 0
(5)
for the total condenser
or for the partial condenser (6b) where K,, KD are equilibrium constants evaluated for composition, pressure and temperature in both the reboiler and condenser, respectively. The total number of eqvs (l)-(6) is 2n t4. In a simulation problem, xP, P and D (or f,,,,), the reflux ratio R, location of the feed tray M and number trays N ar,e usually known and we calculate X, x,, B and fD (or D and f,,j (i = I, 2, . . . n; i# h)), the temperature in condenser TD and the temperature in reboiler T,, i.e. all together there are 3n t 3 unknown variables. We still need therefore, II - 1 equations. According to Smith-Brinkley[Z], we can use the relations
I
1- S:-” + R(1- S”,) 1-h-l-S~-“tR(1-S.,)+H,S~-M(l-s~:,”)(1-S.i)i(l-S~i)=o’ i=l2
distillate fD can be expressed in the form dx,, - A,jm = 0,
i=1,2 ,,.., n.
) ,*..,K
(7)
To the vector of unknown variables we add the temperature TF, which is necessary for evaluation of “average equilibrium distribution coefficients”
(2) 425
E. ECKERT
426
D
can approximate only the changes of y in the column by the relations (13) and (14), [ 11. The relationships (Q-0-j) represent now the system of 3n t 4 nonlinear equations with binding relatio?s (8)-(14) for the unknown variables xD, xB, fD (or D and f~i (i = 1, 2, * . . I n ; i# h)), TD, TB, TF, B. In a design problem xF, R R/R,i,, fDh, fm (or b,) are usually known, and we calculate xh xB, R, 8, T’ r,, fDi (i= 1,2,. . . ) n:i#h,1)(ori)andf,(i=1,2 ,..., n; i# h)), N, M, R,,.; i.e. all together there are 3n t 5 unknown variables. In this case we have to add to the base system of eqns (l)-(6) n t 1 equations. In petrochemical problems, the Fenske-UnderwoodGilliland method is very often used and we can add n - 2 equations
,hLN+l
"N
-
RD
N
I+-_i
N-l
M+2 M*l M
xJaD~ - a N”;“xBi/xBh= 0 i=l,2 ,..., n
(16)
ifh,!
where the relative volatilities a can be estimated by Fig. 1. Simple distillationcolumn.
a; = X@GXK~BKD,&~))
I’&= exp (Ai t
i=l,2,...,n
(17)
__. The Gilliland correlation [6] gives another equation; with the mathematical form
BJ(Ci + OS(T,, t T,))), i=1,2,...,n
(8)
KBi= exp (Ai t Bi/(Ci t0.5(TB t Tp))), i=l,2,...,n
(9)
N-N,,;, ~-
l+
N+l
by means of which we can determine S”,= &v/L,
i=l,2,...,n
smi=KBiVl~,
i=1,2 ,..*, Il.
(10) (l,)
(1+54@&))(*-1) exp
i
where Nmi, = In
(~~~x&~x~~))/ln ~1.
(19)
i=I , 2,..., n.
(12) The evaluation of coefficients A and B was described earlier [I] Ai = In KBi- In (I&/&&(
(18)
(II t 1,7.2&+))(W) i
According to the feed vapor fraction I$ we calculate the values of H as Hi = L(d t (1- ~)Iz&?~i)lL’,
=.
TBt Ci)/(?‘D+ Ci) - I), (13)
For the estimation of the minimum reflux ratio R,, the equations of Underwood [ 121are used 7 wdai
~ 0) t Rmint 1 = 0
(20)
Ta&/(ai-8)tq-l=O
(21)
i=1,2,...,n Bi = In (KDilKBi)/(lI(TDt CJ - l/(TB + CM
(14)
i=l,2,...,n
For location of the feed, we can use [9] the relation 0’ - M)/ N -In (.r~~~J(x~,x,))/( &,,cr~) = 0
where the values of C are chosen. For low-pressure vapor-liquid equlibria we can often write Ki = $‘?I P = YiKF
i=l2 , ,.-.t n
Since the temperature dependence
of K”
is known,
WI we
(22a) or Kirkbride’sequation[4] (N-M - 1)lM - ((iri~,)(xnJxn)(xs,/~,h~~)~)~~*~ = 0, (22b) The relations (l)-(6), (16), (18), (2OH22) represent the
42~
Global approach to lhe computationof separationsby approximate methods
system of 3n t 6 nonlinear equations for 3n t 6 unknown variables (# is a new unknown!). We can also use a combined method, i.e. the eqns (l)-(7), (20)-(22) for calculation of x,,, .+, R B, T,, TB, T, fDi (for i = 1,2, . . . , n; i#h), . 1 n; ifh, I)(or Dand jm for i=l,2,. Rmln, 0, M, N. When the Fenske-Underwood-Gilliland method for simulation problems is used, the number of equations and variables is only 3n t 5. We know +, @,d (or jDh), R, M, N and we can calculate X&xB,B, T,,, TB, 0, R/R,,+,, fD (or fi and jDt, i = I, 2, , . . , n; i# h). In this case we solve the system of eqns (l)-(6), (16), (lS), (20t(21). But the Fenske-Underwood-Gilliland method cannot be applied for equilibrium stage distillation processes where strongly non-ideal behaviour of the mixture in the column has to be considered in the calculation of vapor-liquid equlibria. For design problems where the equilibrium behaviour is strongly non-ideal, we can use the modified Smith-Brinkley method; i.e. eqns (l)-(7) and (Zlb) (there are no relative volatilities) to calculate xD, xs, ti, TD, Ta, r,, N, M, f,,; (i= 1, 2, , n; i# h); i.e. all together there are 3n t 5 unknown variables. To sohe the system of nonlinear equations we can use, for instance, Newton-Raphson’s method, the corresponding Jacobian being evaluated either numerically by means of difference approximation or analytically. Combined procedure can also be used.
3. EXAMPLES 0F THE CALCULATION The proposed ilobal approach to the solution of distillation problems has been tested on a number of systems and proved to be effective. The convergence properties of the Newton-Raphson method were satisfactory in the solved problems because most of the equations have a linear character.
Example 1. A butane-pentane splitter[l] Statement of the problem Feed. Mixture of vapor and liquid, temperature
358.4K, vapor fraction 0.0932, feed rate 100 mole/unit time. Feed composition. i Component
IFi
1 propane 2 isobutane 3 n-butane 4 isopentane 5 n-pentane
0.05 0.15 0.25 0.20 0.35
Type of condenser. Total. Pressure. 8.274~ lo5 Pa (the pressure drop in the column is not considered). Thermodynamic specification. The equilibrium distribution coefficients are calculated from the ChaoSeader correlation. Constants for evaluation “average equilibrium distribution coefficients” are C = - 43.15K. Reference results for comparison purposes in this example were obtained through using the CHESS, version 3 (January 1978),particularly the modules DISC and MSEQ[9]. In the MSEQ subroutine, a multistage equilibrium process of a specified number of theoretical stages is treated as a series of adiabatic flash calculations which are recycled until convergence is obtained. The DISC routine uses Fenske’s total retlux equation for feed tray location and for splitting, Underwood’s equations for minimum reflux, Kirkbride’s equation to predict the optimum feed location, and Gilliland’s correlation for stage and reflux ratio relations to solve the distillation process.
Tablel(a). Results of examDIela and lb oesqn
Simulation Variable
'Dl %2 xD3 *lJ4 %5 ‘Bl %2 ‘B3 *B4 XB5 N
MSB+
DISC opt. =2
48.86
50.33
46.80
48.98
49.05
51.14
49.61
53.12
51.02
50.95
336.7
336.9
336.2
336.6
336.7
383.1
384.4
381.4
383.3
383.2
0.1023
0.0993
0.1064
0.1020
0.1019
0.3002
0.2950
0.3050
0.2997
0.2985
0.4734
0.4757
0.4642
0.4721
0.4715
0.0712
0.0786
0.0691
0.0733
0.0754
0.0530
0.0514
0.0553
0.0528
0.0528
0.0001
0. pooo
0.0002
0.0001
a.0001
0.0065
0.0031
0.0132
0.0063
o.ca7i
0.0366
0.0213
0.0610
0.0367
0.0368
0.3231
0.3230
a.3155
0.3216
0.3200
0.6337
0.6526
0.6101
0.6353
0.6361
8.83
11.77
1
4.43++
M
+MSB
:
**DISC,
MSB+
DISC opt.=1
MSEQ
Modified
opt.=3,
Smith-Brink+& Mc4.81
method
5.58
E. ECKEXT
428
(a) Simulation In addition to the above given values of parameters, the values M = 5, N = IO, R = 2.579, IDS= 0.074 were also chosen. The results of rigorous and approximate calculations are presented in Table la.
Example 2. Example 4 from Hanson’s book[l3] Statement of the problem Feed. Boiling liquid, 1.0mole/unit time. Feed composition.
Component i xFi 1 0.239 2 0.045 3 0.716
(b) Design In addition to the above given values of parameters, the values R = 2.5991,jDs = 0.074,jDs = 0.9251were also chosen. For the feed location in the modified SmithBrinkley method (MSB) we have used the Kirkbride
relation equation (22b). The results from the DISC routine and MSB method are also presented in Table la. In both cases the modified Smith-Brinkley method gives very good results. It is interesting, that in the case of the design computation, the number of theoretical stages (and thus also the location of the feed tray) computed by the modified Smith-Brinkley method is less “optimistic” when compared with the results obtained by the standard methods for petrochemical problems.
Type of condenser. Total. Thermodynamic specification.
The equilibrium coefficients are calculated from eqn (15) using the following dependence K”= K”(T) In Kin = Cli t C2j(C3i t T),
three suffix Margules equation has been adopted[13]
>I
Table 2(b). Numericalvalues of the binary interactioncoefficientsA,
,
I
!
Table2(c).Resultsof examples _ 2(a)and 2(b) variable
Simulation Rigorous FUG
MSB
Design FUG
MSB
B
0.7
0.7
0.7
0.7
0.7
TD
327.2
327.2
327.3
327.2
327.7
TB
337.9
337.4
335.7
338.2
336.1
XD1
0.6056
0.5970
0.5635
0.6208
0.5195
xD2
0.1100
0.1057
0.0847
0.1101
0.1101
XD3
0.2843
0.2973
0.3518
0.2692
0.3705
XB?
0.0819
0.0856
0.0999
0.0754
0.1188
'B2
0.0171
0.0190
0.0280
0.0171
0.0171
XB3
0.9010
0.8954
0.8721
0.9075
0.8641
0.9467
-
1.121
%in M N
_
(23)
Numerical values of the constants are presented in Table Za. To describe the non-ideality in the liquid phase, the
Table Z(a).Numericalvaluesof the constantsC&
1,
i=l,2,3.
3.37
1,ia
8.60
2.H
Global approach to the computationof separations by approximatemethods
429
and set R = 1.0. Both results are presented in Table 2c. As the deviation of the vapor-liquid equlibria from the ideal behaviour is only small, the agreement between the method and the exact trayFenske-UnderwoodGilliland by-tray solution is excellent. The results obtained by the modified Smith-Brinkley method are also very good. Example 3. A strongly non-ideal mixture[l5]
(24 Statement of the problem Feed composition.
Following Wohl[ 141the htik term is defined as follows ATjk=O.S(hij “tAji f A* t Ati + hjk + A,).
i component
(25)
The values of binary interaction coefficients are shown in Table 2b. The constants for the evaluation “average equilibrium distribution coefficients” are C = - 43.15 K.
I n-hexane 2ethano,
XFi
Normal boiling point K
0.3 0.1
341.9 351.5
0.3 0.3
345.0 353.3
3 methylcyclo-
(a) Simulation In addition to the above given values of parameters, the values R = 1.0, d = 0.3 mole/unit time, M = 5, N =
pentane 4 benzene
11 were chosen. The results of rigorous and approximate calculations using the Fenske-Underwood-Gilliland method (FUG) and the modified Smith-Brinkley method (MSB) are presented in Table 2c.
Type of condenser. Total. Feed. Boiling liquid, 100 mole/unit time. Pressure. 1.034~ l@ Pa, the pressure drop in the column is not considered. specification. The equilibrium Thermodynamic
(b) Design
coefficients are calculated from eqn (15) using the following dependence
In addition to the above given values of parameters, the values d = 0.3 mole/unit time and fDZ= 0.7338were chosen. For the design calculation using the FenskeUnderwood-Gilliland method the value R/R,, = 1.3 was chosen. In the modified Smith-Brinkley method we have used the Kirkbride eqn (22b) for the feed tray location
1nPt = Cl, t CJ(C,it T)t C.+iTtCsi In 7’. (26) Numerical values of constants are presented in Table 3a. To describe the non-ideality in the liquid phase Wilson’s
Table 3(a).Numericalvalues of the constants C,-C,
~~
Table 3(b).Numericalvalues of the binary interaction coefficientsA,-A,
4
57.525
66.158
81.24C
0.0
Table 3(c).Numericalvalues of the coefficientsal-a,
E. ECKERT
430
Table 3(d). Results of examples 3(a)and 3(t1) Simulation
Design
Variable
Rigorous
MSB
B
50.0
50.0
50.0
334.4
339.0
339.1
TD TB xD1 xD2 %3 “D4 xB1 XB2 XB3 xB4 M
MSB
345.2
345.2
345.0
0.3504
0.3679
0.3585
0.2000
0.2000
0.1998
0.2117
0.2910
0.2828
0.1779
0.1411
0.1589
0.2496
0.2321
0.2415
0.0000
0.0000
0 - 0002
0.3283
0.3090
0.3172
0.4221
0.4589
0.4411 3.87
N
equation has been adopted
5.51
5. DISCUSSIONAND
CONCLUSIONS
The use of the global approach to the solution of the set of equations describing simple separation equipment (27) In Yi= 1 -In C ljA;j -q (&A$ C xjAkiJ enables us to use the modified Smith-Brinkley method i I for non-ideal mixtures for practically any type of the where problem. The set of equations is not large (maximum 3n t 7 equations) and the convergence properties are hi; z Vi exp ( -(A, - Aii)/ T)/Uj. (28) good, when the Newton-Raphson method is used (5-10 iterations for 0.1% accuracy). Therefore, the present The values of the binary interaction coefficients are method has an advantage in cases where only small shown in Table 3b. Table 3c presents numerical values of computers are available. The method will be helpful for the temperature dependence of molar volumes the rapid location of the feed tray and for evaluating the number of stages. This avoids the trial and error calV* = Uji t UziTt UsiT (2% culations with more complex and time-consuming, rigorous algorithms. The constants for the evaluation “average equilibrium The proposed global approach has proved advancoefficients” are C = - 43.15K. tageous and reliable in two cases: (i) in computation of hydrocarbon fractionators, where Chao-Seader correlation is used for description of vapor-liquid equlibrium (a) Simulation data; (ii) in the case of nonideal mixtures, where activity In addition to the above given values of parameters, the values R = 2.0, d - 50 mole/unit time, M = 3, N = 9 were chosen. As the selection of heavy and light keys in this example is very difficult, the Fenske-UnderwoodGillilandmethod either diverges or gives incorrect results and cannot be applied. The application of the modified Smith-Brinkley method, however, presents no problems and the results are very good (cf. Table 3d).
In addition to the above given values of parameters, the values R = 2.0, d = 50 mole/unit time and fn2 = 0.999 were chosen. For the optimum location of the,feed tray the eqn (22b) was used. The calcuIated results are also given in Table 3d. It is e\iident from all the sets of presented results that the described modified Smith-Brinkley method gives satisfactory results for estimation of the composition and temperature of the outlet streams. This is not only the case in petrochemical problems but also in cases where strongly non-ideal behaviour of the distilled mixture has to be considered.
M12 M+l
t
t
F’
B
Fig. 2. Generalizedseparationequipment.
Globalapproach to the computationof separationsby approximatemethods
431
eA
4
-43 4
4 4
4 * 4
9
ibl
ICI
Ml
Fig. 3. Non-standard columnarrangement:,(i) distillationcolumn with side cut; (b) conventional tray-by-tray model; (c) semi-tray-by-traymodel(Ohmuraand Kasahara);(d)globalshort-cutmodel: A, classicalstage; B, partly Lumpedcomputationalblock (Edmister’stype); C, lumpedcomputational block (Smith-Brinkley’s type). coefficients (evaluated e.g. from the equation of Wilson, Margules or others) are used for characterization of non-ideal behaviour. The method is unfortunately not appropriate for computation of azeotropic distillations. The described approximate method of computation can evidently be applied also to the problem of computation of a generalized type of a separation equipment[2] (see Fig. 2). Hence we can solve, in an analogous way, not only distillation, but also absorption (absorption with reboiler), extraction, stripping etc. of nonideal mixtures. Moreover, the same approach can be used for the approximate (semi-rigorous)computation of a non-standard column arrangement with multiple feed streams or side withdrawals, similar to the description of Ohmura and Kasahara[lO]. These authors have used, for approximation of certain parts of a column,_an absorption section computed by means of the Edmister method[I I] (see Fii. 3). It is evident that the application of our method substantially decreases the necessary number of sections and thus also decreases the dimension of the set of equations which has to be solved. Acknowledgement-Part of this work was supported by Alexander von HumboldtFoundationduringthe stay of the author at the Institute for Systemdynamicsand Control, University Stuttgart, West Germany. NOTATION
A, B,C constants in temperature dependence of equilibrium coefficient constants in temperature dependence of molar volume molar flow rate of bottom product molar flow rate of top product molar feed flow rate
separation factor in column parameter in the Smith-Brinkley method equilibrium distribution coefficient liquid flow rate in column number of trays below feed stage number of trays in column number of components vapor pressure, pressure in the column thermal condition of a feed stream reflux ratio stripping factors in rectifying and stripping part of column, respectively temperature molar concentration vapor flow rate in column molar volume Greek symbols
(Y y C$ A, A B
relative volatility activity coefficient feed vapor fraction interaction coefficient variable in the Underwood equations
Subscripts B bottom product
D distillate product F feed h heavy key i component i I light key min minimum value 2 average value X vector x” ideal behaviout X’ stripping part of column
E. ECKERT
432 REFERENCES
[I] Eckert E. and Hlavacek V., C/rem. Engng Sci 197833
77. [Z] Smith B. D., Design of Equilibrium Stage Processes. McGraw-Hill,New York 1963. 131Hengstebeck R. I., An Improved Shortcut for Calculating Diflcult Multicomponent Distillations, Chemical Engineering. 1969, 13, is.
[4] Kirkbride C. G.. Per. ReJ 194423 32. [5] Fenske M. R., Ind. Engng Chem. 193224 482. [6] GillilandE. R., Ind. Engng Chem. 1940 32 1101, [I] Scheller Wm. A. and Akhave S. R., Proc. Symp. Cornput. Design Erection Chemical Plants, p. 147. Karlovy Vary 1975.
[S] Scheller Wm. A. and Bhate S. M., Pmt. 5th Symp. Compvt. Chemical Engng, The High Tatras, p. 189. 1977.
191 Motard R.
L. and Lee H. M.. CHESS. User’s guide. Uni-
_ . versityof Houston 1971.
IlO] Ohmura S. and Kasahara S., 1 Chem. Engng lap. 197811 185. 111) Edmister W. C., Ind. Engng Chem. 194335 837. [12] Underwood A. J., J. Inst. Petrol. 194632 614. [13] Hanson D. N., Duffin .I. H. and Somerville G. F., Computation of Multistnge Seporntion Processes. Reinhold,New York 1962. [14] Wohl K.. Trans. A.1.Ch.E. 1946 42 215: Chem. Engng Prog. 195349 218. 1151Boston I. F., Ph.D. Thesis. Tulane University 1970.
CQl9-2S09182/0M(25-08SO3-o85o3.oo10 @ I!%82Pergaman Press Ltd.
GLOBAL APPROACH TO THE COMPUTATION OF SEPARATIONS BY APPROXIMATE METHODS EGON ECKERT Department of Chemical Engineering, Prague Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia (Received 15 June 1981; accepted 20 July 1981) Abstract-A global approach to the computation of simple separation columns for non-ideal mixtures by approximate methods is described. The procedure based on a modification of the Smith-Brinkley method enables us to compute practicallyany type.of problem formulation, i.e. both simulation and design problems. The technique can also be used for formulationof semirigorousproceduresfor generalarrangementsof more complex types of equipment. 1. INTRODUCTION
It holds further
An earlier paper[l] presented an effective modification of the Smith-Brinkley methodi for the simulation of separation processes in cases where non-ideal behaviour has to be considered for calculations of vapor-liquid equilibria. As realistic assumptions were used in the derivation of the method, very satisfactory results are usually obtained in the applications. Here we shall present a generalization of the above method used to design computations, where a number of trays, the feed location etc., have to be determined. For such computations the application of rigorous methods is usually cumbersome and requires repeated calculations. The well-known methods of computation[3-6], together with some recently suggested techniques[7,8], are thus used in this area of computer-aided design[9]. The parameters, which have to be determined in the computations (e.g. number of trays, feed location, etc.) can not be usually expressed in an explicit way. Therefore a global approach to the solution of the equations describing separation equipment is appropriate for the use of the approximate methods for design computations.
2. METHOD OF COMPUTATION
Let us illustrate the approach on the case of a simple distillation column schematically shown in Fig. I. The component mass balance relations are in the form Fxfi-&i-&~i=O
i=
1,2,...,n
(1)
where fi fi, s are flow rates of feed, top product (vapor or liquid) and bottom product, respectively; and xF, x, x, are compositions of these streams. The ratios of the components in the feed stream which appear in the
7
XDi -
1= 0
(3)
+,-l=O.
(4)
For the reboiler
z$Lxs,
- 1= 0
(5)
for the total condenser
or for the partial condenser (6b) where K,, KD are equilibrium constants evaluated for composition, pressure and temperature in both the reboiler and condenser, respectively. The total number of eqvs (l)-(6) is 2n t4. In a simulation problem, xP, P and D (or f,,,,), the reflux ratio R, location of the feed tray M and number trays N ar,e usually known and we calculate X, x,, B and fD (or D and f,,j (i = I, 2, . . . n; i# h)), the temperature in condenser TD and the temperature in reboiler T,, i.e. all together there are 3n t 3 unknown variables. We still need therefore, II - 1 equations. According to Smith-Brinkley[Z], we can use the relations
I
1- S:-” + R(1- S”,) 1-h-l-S~-“tR(1-S.,)+H,S~-M(l-s~:,”)(1-S.i)i(l-S~i)=o’ i=l2
distillate fD can be expressed in the form dx,, - A,jm = 0,
i=1,2 ,,.., n.
) ,*..,K
(7)
To the vector of unknown variables we add the temperature TF, which is necessary for evaluation of “average equilibrium distribution coefficients”
(2) 425
E. ECKERT
426
D
can approximate only the changes of y in the column by the relations (13) and (14), [ 11. The relationships (Q-0-j) represent now the system of 3n t 4 nonlinear equations with binding relatio?s (8)-(14) for the unknown variables xD, xB, fD (or D and f~i (i = 1, 2, * . . I n ; i# h)), TD, TB, TF, B. In a design problem xF, R R/R,i,, fDh, fm (or b,) are usually known, and we calculate xh xB, R, 8, T’ r,, fDi (i= 1,2,. . . ) n:i#h,1)(ori)andf,(i=1,2 ,..., n; i# h)), N, M, R,,.; i.e. all together there are 3n t 5 unknown variables. In this case we have to add to the base system of eqns (l)-(6) n t 1 equations. In petrochemical problems, the Fenske-UnderwoodGilliland method is very often used and we can add n - 2 equations
,hLN+l
"N
-
RD
N
I+-_i
N-l
M+2 M*l M
xJaD~ - a N”;“xBi/xBh= 0 i=l,2 ,..., n
(16)
ifh,!
where the relative volatilities a can be estimated by Fig. 1. Simple distillationcolumn.
a; = X@GXK~BKD,&~))
I’&= exp (Ai t
i=l,2,...,n
(17)
__. The Gilliland correlation [6] gives another equation; with the mathematical form
BJ(Ci + OS(T,, t T,))), i=1,2,...,n
(8)
KBi= exp (Ai t Bi/(Ci t0.5(TB t Tp))), i=l,2,...,n
(9)
N-N,,;, ~-
l+
N+l
by means of which we can determine S”,= &v/L,
i=l,2,...,n
smi=KBiVl~,
i=1,2 ,..*, Il.
(10) (l,)
(1+54@&))(*-1) exp
i
where Nmi, = In
(~~~x&~x~~))/ln ~1.
(19)
i=I , 2,..., n.
(12) The evaluation of coefficients A and B was described earlier [I] Ai = In KBi- In (I&/&&(
(18)
(II t 1,7.2&+))(W) i
According to the feed vapor fraction I$ we calculate the values of H as Hi = L(d t (1- ~)Iz&?~i)lL’,
=.
TBt Ci)/(?‘D+ Ci) - I), (13)
For the estimation of the minimum reflux ratio R,, the equations of Underwood [ 121are used 7 wdai
~ 0) t Rmint 1 = 0
(20)
Ta&/(ai-8)tq-l=O
(21)
i=1,2,...,n Bi = In (KDilKBi)/(lI(TDt CJ - l/(TB + CM
(14)
i=l,2,...,n
For location of the feed, we can use [9] the relation 0’ - M)/ N -In (.r~~~J(x~,x,))/( &,,cr~) = 0
where the values of C are chosen. For low-pressure vapor-liquid equlibria we can often write Ki = $‘?I P = YiKF
i=l2 , ,.-.t n
Since the temperature dependence
of K”
is known,
WI we
(22a) or Kirkbride’sequation[4] (N-M - 1)lM - ((iri~,)(xnJxn)(xs,/~,h~~)~)~~*~ = 0, (22b) The relations (l)-(6), (16), (18), (2OH22) represent the
42~
Global approach to lhe computationof separationsby approximate methods
system of 3n t 6 nonlinear equations for 3n t 6 unknown variables (# is a new unknown!). We can also use a combined method, i.e. the eqns (l)-(7), (20)-(22) for calculation of x,,, .+, R B, T,, TB, T, fDi (for i = 1,2, . . . , n; i#h), . 1 n; ifh, I)(or Dand jm for i=l,2,. Rmln, 0, M, N. When the Fenske-Underwood-Gilliland method for simulation problems is used, the number of equations and variables is only 3n t 5. We know +, @,d (or jDh), R, M, N and we can calculate X&xB,B, T,,, TB, 0, R/R,,+,, fD (or fi and jDt, i = I, 2, , . . , n; i# h). In this case we solve the system of eqns (l)-(6), (16), (lS), (20t(21). But the Fenske-Underwood-Gilliland method cannot be applied for equilibrium stage distillation processes where strongly non-ideal behaviour of the mixture in the column has to be considered in the calculation of vapor-liquid equlibria. For design problems where the equilibrium behaviour is strongly non-ideal, we can use the modified Smith-Brinkley method; i.e. eqns (l)-(7) and (Zlb) (there are no relative volatilities) to calculate xD, xs, ti, TD, Ta, r,, N, M, f,,; (i= 1, 2, , n; i# h); i.e. all together there are 3n t 5 unknown variables. To sohe the system of nonlinear equations we can use, for instance, Newton-Raphson’s method, the corresponding Jacobian being evaluated either numerically by means of difference approximation or analytically. Combined procedure can also be used.
3. EXAMPLES 0F THE CALCULATION The proposed ilobal approach to the solution of distillation problems has been tested on a number of systems and proved to be effective. The convergence properties of the Newton-Raphson method were satisfactory in the solved problems because most of the equations have a linear character.
Example 1. A butane-pentane splitter[l] Statement of the problem Feed. Mixture of vapor and liquid, temperature
358.4K, vapor fraction 0.0932, feed rate 100 mole/unit time. Feed composition. i Component
IFi
1 propane 2 isobutane 3 n-butane 4 isopentane 5 n-pentane
0.05 0.15 0.25 0.20 0.35
Type of condenser. Total. Pressure. 8.274~ lo5 Pa (the pressure drop in the column is not considered). Thermodynamic specification. The equilibrium distribution coefficients are calculated from the ChaoSeader correlation. Constants for evaluation “average equilibrium distribution coefficients” are C = - 43.15K. Reference results for comparison purposes in this example were obtained through using the CHESS, version 3 (January 1978),particularly the modules DISC and MSEQ[9]. In the MSEQ subroutine, a multistage equilibrium process of a specified number of theoretical stages is treated as a series of adiabatic flash calculations which are recycled until convergence is obtained. The DISC routine uses Fenske’s total retlux equation for feed tray location and for splitting, Underwood’s equations for minimum reflux, Kirkbride’s equation to predict the optimum feed location, and Gilliland’s correlation for stage and reflux ratio relations to solve the distillation process.
Tablel(a). Results of examDIela and lb oesqn
Simulation Variable
'Dl %2 xD3 *lJ4 %5 ‘Bl %2 ‘B3 *B4 XB5 N
MSB+
DISC opt. =2
48.86
50.33
46.80
48.98
49.05
51.14
49.61
53.12
51.02
50.95
336.7
336.9
336.2
336.6
336.7
383.1
384.4
381.4
383.3
383.2
0.1023
0.0993
0.1064
0.1020
0.1019
0.3002
0.2950
0.3050
0.2997
0.2985
0.4734
0.4757
0.4642
0.4721
0.4715
0.0712
0.0786
0.0691
0.0733
0.0754
0.0530
0.0514
0.0553
0.0528
0.0528
0.0001
0. pooo
0.0002
0.0001
a.0001
0.0065
0.0031
0.0132
0.0063
o.ca7i
0.0366
0.0213
0.0610
0.0367
0.0368
0.3231
0.3230
a.3155
0.3216
0.3200
0.6337
0.6526
0.6101
0.6353
0.6361
8.83
11.77
1
4.43++
M
+MSB
:
**DISC,
MSB+
DISC opt.=1
MSEQ
Modified
opt.=3,
Smith-Brink+& Mc4.81
method
5.58
E. ECKEXT
428
(a) Simulation In addition to the above given values of parameters, the values M = 5, N = IO, R = 2.579, IDS= 0.074 were also chosen. The results of rigorous and approximate calculations are presented in Table la.
Example 2. Example 4 from Hanson’s book[l3] Statement of the problem Feed. Boiling liquid, 1.0mole/unit time. Feed composition.
Component i xFi 1 0.239 2 0.045 3 0.716
(b) Design In addition to the above given values of parameters, the values R = 2.5991,jDs = 0.074,jDs = 0.9251were also chosen. For the feed location in the modified SmithBrinkley method (MSB) we have used the Kirkbride
relation equation (22b). The results from the DISC routine and MSB method are also presented in Table la. In both cases the modified Smith-Brinkley method gives very good results. It is interesting, that in the case of the design computation, the number of theoretical stages (and thus also the location of the feed tray) computed by the modified Smith-Brinkley method is less “optimistic” when compared with the results obtained by the standard methods for petrochemical problems.
Type of condenser. Total. Thermodynamic specification.
The equilibrium coefficients are calculated from eqn (15) using the following dependence K”= K”(T) In Kin = Cli t C2j(C3i t T),
three suffix Margules equation has been adopted[13]
>I
Table 2(b). Numericalvalues of the binary interactioncoefficientsA,
,
I
!
Table2(c).Resultsof examples _ 2(a)and 2(b) variable
Simulation Rigorous FUG
MSB
Design FUG
MSB
B
0.7
0.7
0.7
0.7
0.7
TD
327.2
327.2
327.3
327.2
327.7
TB
337.9
337.4
335.7
338.2
336.1
XD1
0.6056
0.5970
0.5635
0.6208
0.5195
xD2
0.1100
0.1057
0.0847
0.1101
0.1101
XD3
0.2843
0.2973
0.3518
0.2692
0.3705
XB?
0.0819
0.0856
0.0999
0.0754
0.1188
'B2
0.0171
0.0190
0.0280
0.0171
0.0171
XB3
0.9010
0.8954
0.8721
0.9075
0.8641
0.9467
-
1.121
%in M N
_
(23)
Numerical values of the constants are presented in Table Za. To describe the non-ideality in the liquid phase, the
Table Z(a).Numericalvaluesof the constantsC&
1,
i=l,2,3.
3.37
1,ia
8.60
2.H
Global approach to the computationof separations by approximatemethods
429
and set R = 1.0. Both results are presented in Table 2c. As the deviation of the vapor-liquid equlibria from the ideal behaviour is only small, the agreement between the method and the exact trayFenske-UnderwoodGilliland by-tray solution is excellent. The results obtained by the modified Smith-Brinkley method are also very good. Example 3. A strongly non-ideal mixture[l5]
(24 Statement of the problem Feed composition.
Following Wohl[ 141the htik term is defined as follows ATjk=O.S(hij “tAji f A* t Ati + hjk + A,).
i component
(25)
The values of binary interaction coefficients are shown in Table 2b. The constants for the evaluation “average equilibrium distribution coefficients” are C = - 43.15 K.
I n-hexane 2ethano,
XFi
Normal boiling point K
0.3 0.1
341.9 351.5
0.3 0.3
345.0 353.3
3 methylcyclo-
(a) Simulation In addition to the above given values of parameters, the values R = 1.0, d = 0.3 mole/unit time, M = 5, N =
pentane 4 benzene
11 were chosen. The results of rigorous and approximate calculations using the Fenske-Underwood-Gilliland method (FUG) and the modified Smith-Brinkley method (MSB) are presented in Table 2c.
Type of condenser. Total. Feed. Boiling liquid, 100 mole/unit time. Pressure. 1.034~ l@ Pa, the pressure drop in the column is not considered. specification. The equilibrium Thermodynamic
(b) Design
coefficients are calculated from eqn (15) using the following dependence
In addition to the above given values of parameters, the values d = 0.3 mole/unit time and fDZ= 0.7338were chosen. For the design calculation using the FenskeUnderwood-Gilliland method the value R/R,, = 1.3 was chosen. In the modified Smith-Brinkley method we have used the Kirkbride eqn (22b) for the feed tray location
1nPt = Cl, t CJ(C,it T)t C.+iTtCsi In 7’. (26) Numerical values of constants are presented in Table 3a. To describe the non-ideality in the liquid phase Wilson’s
Table 3(a).Numericalvalues of the constants C,-C,
~~
Table 3(b).Numericalvalues of the binary interaction coefficientsA,-A,
4
57.525
66.158
81.24C
0.0
Table 3(c).Numericalvalues of the coefficientsal-a,
E. ECKERT
430
Table 3(d). Results of examples 3(a)and 3(t1) Simulation
Design
Variable
Rigorous
MSB
B
50.0
50.0
50.0
334.4
339.0
339.1
TD TB xD1 xD2 %3 “D4 xB1 XB2 XB3 xB4 M
MSB
345.2
345.2
345.0
0.3504
0.3679
0.3585
0.2000
0.2000
0.1998
0.2117
0.2910
0.2828
0.1779
0.1411
0.1589
0.2496
0.2321
0.2415
0.0000
0.0000
0 - 0002
0.3283
0.3090
0.3172
0.4221
0.4589
0.4411 3.87
N
equation has been adopted
5.51
5. DISCUSSIONAND
CONCLUSIONS
The use of the global approach to the solution of the set of equations describing simple separation equipment (27) In Yi= 1 -In C ljA;j -q (&A$ C xjAkiJ enables us to use the modified Smith-Brinkley method i I for non-ideal mixtures for practically any type of the where problem. The set of equations is not large (maximum 3n t 7 equations) and the convergence properties are hi; z Vi exp ( -(A, - Aii)/ T)/Uj. (28) good, when the Newton-Raphson method is used (5-10 iterations for 0.1% accuracy). Therefore, the present The values of the binary interaction coefficients are method has an advantage in cases where only small shown in Table 3b. Table 3c presents numerical values of computers are available. The method will be helpful for the temperature dependence of molar volumes the rapid location of the feed tray and for evaluating the number of stages. This avoids the trial and error calV* = Uji t UziTt UsiT (2% culations with more complex and time-consuming, rigorous algorithms. The constants for the evaluation “average equilibrium The proposed global approach has proved advancoefficients” are C = - 43.15K. tageous and reliable in two cases: (i) in computation of hydrocarbon fractionators, where Chao-Seader correlation is used for description of vapor-liquid equlibrium (a) Simulation data; (ii) in the case of nonideal mixtures, where activity In addition to the above given values of parameters, the values R = 2.0, d - 50 mole/unit time, M = 3, N = 9 were chosen. As the selection of heavy and light keys in this example is very difficult, the Fenske-UnderwoodGillilandmethod either diverges or gives incorrect results and cannot be applied. The application of the modified Smith-Brinkley method, however, presents no problems and the results are very good (cf. Table 3d).
In addition to the above given values of parameters, the values R = 2.0, d = 50 mole/unit time and fn2 = 0.999 were chosen. For the optimum location of the,feed tray the eqn (22b) was used. The calcuIated results are also given in Table 3d. It is e\iident from all the sets of presented results that the described modified Smith-Brinkley method gives satisfactory results for estimation of the composition and temperature of the outlet streams. This is not only the case in petrochemical problems but also in cases where strongly non-ideal behaviour of the distilled mixture has to be considered.
M12 M+l
t
t
F’
B
Fig. 2. Generalizedseparationequipment.
Globalapproach to the computationof separationsby approximatemethods
431
eA
4
-43 4
4 4
4 * 4
9
ibl
ICI
Ml
Fig. 3. Non-standard columnarrangement:,(i) distillationcolumn with side cut; (b) conventional tray-by-tray model; (c) semi-tray-by-traymodel(Ohmuraand Kasahara);(d)globalshort-cutmodel: A, classicalstage; B, partly Lumpedcomputationalblock (Edmister’stype); C, lumpedcomputational block (Smith-Brinkley’s type). coefficients (evaluated e.g. from the equation of Wilson, Margules or others) are used for characterization of non-ideal behaviour. The method is unfortunately not appropriate for computation of azeotropic distillations. The described approximate method of computation can evidently be applied also to the problem of computation of a generalized type of a separation equipment[2] (see Fig. 2). Hence we can solve, in an analogous way, not only distillation, but also absorption (absorption with reboiler), extraction, stripping etc. of nonideal mixtures. Moreover, the same approach can be used for the approximate (semi-rigorous)computation of a non-standard column arrangement with multiple feed streams or side withdrawals, similar to the description of Ohmura and Kasahara[lO]. These authors have used, for approximation of certain parts of a column,_an absorption section computed by means of the Edmister method[I I] (see Fii. 3). It is evident that the application of our method substantially decreases the necessary number of sections and thus also decreases the dimension of the set of equations which has to be solved. Acknowledgement-Part of this work was supported by Alexander von HumboldtFoundationduringthe stay of the author at the Institute for Systemdynamicsand Control, University Stuttgart, West Germany. NOTATION
A, B,C constants in temperature dependence of equilibrium coefficient constants in temperature dependence of molar volume molar flow rate of bottom product molar flow rate of top product molar feed flow rate
separation factor in column parameter in the Smith-Brinkley method equilibrium distribution coefficient liquid flow rate in column number of trays below feed stage number of trays in column number of components vapor pressure, pressure in the column thermal condition of a feed stream reflux ratio stripping factors in rectifying and stripping part of column, respectively temperature molar concentration vapor flow rate in column molar volume Greek symbols
(Y y C$ A, A B
relative volatility activity coefficient feed vapor fraction interaction coefficient variable in the Underwood equations
Subscripts B bottom product
D distillate product F feed h heavy key i component i I light key min minimum value 2 average value X vector x” ideal behaviout X’ stripping part of column
E. ECKERT
432 REFERENCES
[I] Eckert E. and Hlavacek V., C/rem. Engng Sci 197833
77. [Z] Smith B. D., Design of Equilibrium Stage Processes. McGraw-Hill,New York 1963. 131Hengstebeck R. I., An Improved Shortcut for Calculating Diflcult Multicomponent Distillations, Chemical Engineering. 1969, 13, is.
[4] Kirkbride C. G.. Per. ReJ 194423 32. [5] Fenske M. R., Ind. Engng Chem. 193224 482. [6] GillilandE. R., Ind. Engng Chem. 1940 32 1101, [I] Scheller Wm. A. and Akhave S. R., Proc. Symp. Cornput. Design Erection Chemical Plants, p. 147. Karlovy Vary 1975.
[S] Scheller Wm. A. and Bhate S. M., Pmt. 5th Symp. Compvt. Chemical Engng, The High Tatras, p. 189. 1977.
191 Motard R.
L. and Lee H. M.. CHESS. User’s guide. Uni-
_ . versityof Houston 1971.
IlO] Ohmura S. and Kasahara S., 1 Chem. Engng lap. 197811 185. 111) Edmister W. C., Ind. Engng Chem. 194335 837. [12] Underwood A. J., J. Inst. Petrol. 194632 614. [13] Hanson D. N., Duffin .I. H. and Somerville G. F., Computation of Multistnge Seporntion Processes. Reinhold,New York 1962. [14] Wohl K.. Trans. A.1.Ch.E. 1946 42 215: Chem. Engng Prog. 195349 218. 1151Boston I. F., Ph.D. Thesis. Tulane University 1970.