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Sizing intermediate storage with stochastic equipment failures under general operation conditions
European Symposium on Computer Aided Process Engineering - 13 A. Kraslawski and I. Turunen (Editors) • © 2003 Elsevier Science B.V. All rights reserved.
239
Sizing Intermediate Storage with Stochastic Equipment Failures under General Operation Conditions Eva Orban-Mihalyko^ and Bela. G. Lakatos^ ^Department of Mathematics and Computing , ^Department of Process Engineering University of Veszprem, H-8201 Veszprem, PO Box 158, Hungary email:[email protected], [email protected]
Abstract An algorithm and simulation program have been developed for sizing intermediate storages of batch/semicontinuous systems taking into account stochastic equipment failures under general operation conditions. The method is based on the observation that any process of this system can be built up from a random sequence of failure cycles of finite number. By means of simulation and statistical evaluation of the results of the simulation runs the storage is sized at a given significance level. The computation time is favourable, and it appears to be a linear function of the number of the failure cycles.
1. Introduction The intermediate storage has an important role in improving operating efficiency of batch processing systems. It increases the variability of the system and reduces the process uncertainties. Over a long time horizon, it can also buffer the effects of equipment and batch failures when it is sized adequately. The sizing of an intermediate storage often can be treated as a deterministic problem, but, when the equipment and batch failures are significantly of random nature, then its sizing becomes a stochastic rather than deterministic task. Deterministic variations in the failure frequency and recovery time have been considered by Karimi and Reklaitis (1985) and Lee and Reklaitis (1989), while stochastic variations have been studied by Odi and Karimi (1990,1991), and Mihalyko and Lakatos (1998). In these studies, however, constants filling and withdrawing intensities were assumed. The aim of the present contribution is to study the problem under generalized conditions, i.e. when the intensity of filling and withdrawing of material into and from the intermediate storage may be changed arbitrarily during the operation.
2. Process Model Let us consider a non-continuous processing system with n upstream and m downstream units (usually n?^) with an intermediate storage. We assume that the processing units are operated periodically, and stochastic failures of the units under general conditions may occur. Furthermore, we suppose that the filling and removal rates may vary in time arbitrarily, i.e. they are described by general functions exhibiting at most finite number of jump discontinuities. Let o\^,... co^^ and^2,1 v^2,m denote the operation periods of the upstream and downstream units, while t^^ ,-hn
^^^ h,\ ^"•h,m ^^^ the filling and removal times, respec-
tively. Further, let t^^ ,..• ^i°„ and t^^ .— tl^m denote the corresponding delay times. We
240 suppose that t^^ +1^^^ < co^^ for any unit. Then, the mathematical model of the process is formulated under the following assumptions: 1. The ratio of two arbitrary periods is a rational number. 2. Only one upstream unit may suffer failure at the same time. Let its index be 1. 3. The failed unit does not transfer material into the storage during its repairing. 4. The failure of a unit does not affect the operation of the remaining ones. 5. The differences between the serial numbers of the failure periods, denoted by ^„ are integer random numbers of the same distribution having bounded range. We suppose that 4 /=1,2,... are independent. j
Then the serial number of the f^ period of failure is expressed as ^ ^ / , its initial /=i
moment is ^{^i -1)^1,1 , and the fmal moment is ^ ^ / ^ ^ • As a consequence, the/^ j-i
failure cycle is, by definition, the interval
S^/-^u'S^/-^i,
as it shown in Fig. 1.
The filling and removal rates are described by the following functions showing at most finite number of discontinuities: 0, if
0< CO, 'd,s
\(o.'d,s bef.Ai)-
fciM
CO,
t^ +t
M,s
), if
CO, 'd,s
(1)
0),
\co.
!lilh±<. t \co.'d,s co,^
0, if
where /^^ > 0 is some positive function, while d=l and s=l,....n for the upstream, and d=2 and ^=l,....m for the downstream units. The amount of the material transferred by the s^^ unit during an operation period is t=0
1
'
^i'i
tt•
^^^^'^
•
1 L
(^7+&)^7.7
1
[ ik i L
t
•
^ ^
^ ^ w 2^^ failure cycle w Moment of the 7^' Moment of the 2"' failure failure
V^ failure cycle
Fig.l. Characteristic time intervals of the process with equipment failures. ^
\bef,,{x)dx.
(2)
241 and the total amount of the material transferred by the s^^ unit in the time interval [0,f] may be expressed as
Vj-{t)^\bef,,{x)cbc.
(3)
When a failure of the 7" upstream unit occurs, then the filling intensity is described by the function ^e/i,(0,
7 >
fg
if
^ •«i.i /=1
befli^^(t) =
0, if
7J
re
(4)
j
•^u )
and the total amount of material transferred of the V unit is expressed as f
(5)
Then, the amount of material in the intermediate storage is varied in time according to /o
\
0
"
0
'^
0
(6) 5=2
5=1
where (taf) denotes a vector of dimension (n+m) of the delay times.
3. Sizing of the intermediate storage In order to have material in the intermediate storage always sufficient to operate the system without problem, it is necessary to have an initial amount y(0) in the starting moment of operation as
-minVh^'At).
(7)
Oil
There will no overflow if the volume of the storage is equal to maxVh^'\t)
+ V(.0)
(8)
As a consequence, the volume of the storage sufficient to operate of the system is expressed
0
0
(9)
Since in the present case the failures of the units are stochastic, so is also the function (,o )
Vh ''\ and the goal is to determine the distributions of the maxima and minima of this very function. In order to achieve this we proceed in the following way. We divide the process in the interval [0,0 into sub-processes according to the failure cycles and investigate the variation in time of the amount of material in the storage in
242 these sub-processes consecutively. The method developed is based on the following statements. Consider the function in the/'' failure cycle having the following form Vh^'''Xt)-Vh^''\^^i
'CO,0 for te
S^/-^U'S^i
CO,
. Then we have the
Statement 1. The actual delay times related to the initial moment of the /^ cycle becomes modified compared to the original ones, and are given as a function of the initial moment the cycle 7-1
^1,1
y-1 1=1
'(^ny w...+^".-E^/'^u+^'/.^-^
(0,
rl(Z^rco,0 =
(10) •7-1
" 7-1
S^/-^i.i
/=!
1=1
,+^°.+'.,.-Efi'^u
1=1
_ while the amount of material in the storage is varied in time according to Vh'^^'\t)-Vh^''\Y^^,
•co,,)=:Vh '' -
( / - ^ f , -co,,).
(11)
Statement 2. The number of actual delay times r^^ for all upstream and downstream units is finite. Statement 3. The number of vectors \fd,s} formed from the possible actual delay times 1. TT TT u ^u Pu IS equal to I I ^1 ^ • I I ^2 5 » where —— = '
s=2
5=1
'
CO,s
q,^,
^1,1 ,
P2,s =
(O^^s
Q2,s
. . , , and p^ ^ and q^ ^ are rel-
ative primes. The direct consequence of the Statement 3 is the Statement 4. The number of sub-processes arising in an arbitrary process is finite and is n
m
given by A' • J][^i^ '11^2,^ ' where K is the number of possible values of ^^. Statement 5. The maximum and minimum problems, providing the maximum and minimum values of the variation of amount of material in the intermediate storage in any failure cycle, are reduced to finding maxima and minima of finite number of functions. Statement 6. Let M^'-'^k) and m^^'^k) denote the global maximum and global minimum of problem of Statement 5, where k stands for the possible values of random variable ^1. Then the maximum and minimum value of the amount of material in the intermediate storage in the whole/^ failure cycle can be obtained as a sum of the initial /0
)
(0
\
amount of the cycle, and the global maximum M^'^''(^') and minimum m^'^'^^p, respectively. Subsequently, the maximum and minimum values in the whole interval of the process can be obtained recursively.
243 Statement 7. Any process can be built up from a random sequence of functions related to the failure cycles of finite number. By means of simulation and statistical evaluation of the results of simulation runs the storage can be sized at a given significance level.
4. Example For the sake of illustrating the method developed, consider a batch system with two upstream and two downstream units the operation parameters of which are as follows: ^1,1 = '7 ' ^1,2 = 4, 6^2,1 = '7 '^2,2 = ^ ' ^1,1 = 4, t^2 = 3' ^2,1 = 5, ^2 2 = 2 , t^^ = 2, ^1^2 = ^ ,
^21 = 1, ^2,2 "= i • Let us assume that the filling and removal rates of the units are described by the functions of the form: f^^{t) = 4t, fiji^) = 4t, /2,i(0 = i, fi,!^^) - St. Then, the number of the possible vectors of delay times is equal 12 that are given in Table 1. If the first failure of the unit 1 occurs in the 7^^ 2""^ or 3'"^ period of operation, respectively, then 36 different sub-process variations of the changes of the amount of material in the storage during the first failure cycle are possible. Three such subprocesses, corresponding to the vector of delay times (2,0,1,1), are shown in Fig.2, the Vh'"'"'•"(t)
\
\ \
V
a) ' b) ' c) ' Fig. 2. Variation of the amount of material in the storage in the r^ failure cycle when the r^ failure occurs in the a) V^ period, b) 2" period, c) 3^ period of operation Table 1. The characteristics of the possible vectors of delay times. m(k) and M(k) minimum and maximum for failure in the k^ period of operation; EVh - expected value of the amount of the transferred material in the storage during one cycle Vector of the delay times (2,0,1,1) (2,0,1,0) (2,0,1,-1) (2,1,1,1) (2,1,1,0) (2,1,1,-1) (2,-2,1,1) (2,-2,1,0) (2,-2,1,-1) (2,-1,1,1) (2,-1,1,0) (2,-1,1,-1)
m(l) -18.5 -20 -20.5 -24.5 -24.5 -30.5 -16.5 -18.5 -26.5 -10.5 -16 -22.5
Mil) 3.6 0 0 0 0 0 10 6 10 11.5 2 3.5
m(2) -1 -11 -13 -8 -17 -8 -17 -15 -4 -9 -10 -8
M(2) 23.5 19.5 13.5 19.5 9.5 11.5 19.5 17 17 21.5 21.5 21.5
m(3) 0 -8.5 -10 -8 -14.5 -12 -4 -6.5 -6.5 0 -0.67 -6.5
M(3) 33 31 31 30.5 26.5 21 39 35 27 41 31 33
EVh 3 -13/6 -11/3 -3 -25/3 -29/3 13/3 -1 -113 19/3 1 -1/3
values of minima and maxima obtained for these 36 variations, together with the expected values of the amount of transferred material in the storage during one cycle, are presented in Table 1. The global minima and maxima of these functions were
244 determined numerically by the combination of the grid and modified simplex method. According to Statement 7, each process of the system under investigation can be built
1000
1200
1400
t
a) b) Fig. 3. Variation of the amount of material in the storage through a) 15 and b) 100 randomly selected failure cycles, corresponding to the vector of delay times (2,0,1,1). up from these 36 sub-processes. These processes differ from each other in the sequence of the sub-processes sequenced randomly one after the other. These processes can be generated by computer simulation, selecting one of the possible sub-processes randomly according to a defined probability distribution. Such process is shown in Fig.3 that consists of 15 and 100 failure cycles, respectively, the sequence of which were generated randomly by using the discrete uniform distribution. Carrying out 10,000 simulation runs, consisted of 100 failure cycles under the same operation and simulation conditions, we obtained mean value of minima=-483.08, mean value of maxima=27.9316, and mean value of their differences=509.79. Based on the nu-merical results, it can be concluded that the difference is smaller than 684.1619 with probability 0.95, hence 684.2 is a sufficient size for the storage at significance level 95%. Notice that, in the example, the expectation of the transferred material in the r^ cycle is positive on the basis of the initial delay time but the remaining delay times, as it well seen in Table 1, cause a decreasing tendency in the variation of the process.
5. Conclusions A method and simulation program has been developed for sizing intermediate storages in batch/semi-continuous systems taking into account stochastic equipment failures under general operation conditions. The method is based on the observation that any process of this system can be built up from a random sequence of failure cycles of finite number. With the aid of the analysis of the problem, optimisation on large time intervals of the processes is replaced by optimisation on smaller intervals of the sub-processes of that very process. By means of simulation and statistical evaluation of the results of the simulation runs the storage may be sized at a given significance level. The computation time is favourable, since it is a linear function of the number of the failure cycles.
6. References Karimi, LA. and Reklaitis, G.V., 1985, AIChE Journal, 31,44. Lee, E.S. and Reklaitis, G.V., 1989, Computers chem. Eng., 13, 1235. Mihalyko, E.G. and Lakatos, B.G., 1998, Computers chem. Eng., 22, S797. Odi, T.O. and Karimi, LA., 1990, Chem. Eng. Sci., 45, 3533. Odi, T.O. and Karimi, LA., 1991, Chem.Eng.Sci., 46, 3269.
239
Sizing Intermediate Storage with Stochastic Equipment Failures under General Operation Conditions Eva Orban-Mihalyko^ and Bela. G. Lakatos^ ^Department of Mathematics and Computing , ^Department of Process Engineering University of Veszprem, H-8201 Veszprem, PO Box 158, Hungary email:[email protected], [email protected]
Abstract An algorithm and simulation program have been developed for sizing intermediate storages of batch/semicontinuous systems taking into account stochastic equipment failures under general operation conditions. The method is based on the observation that any process of this system can be built up from a random sequence of failure cycles of finite number. By means of simulation and statistical evaluation of the results of the simulation runs the storage is sized at a given significance level. The computation time is favourable, and it appears to be a linear function of the number of the failure cycles.
1. Introduction The intermediate storage has an important role in improving operating efficiency of batch processing systems. It increases the variability of the system and reduces the process uncertainties. Over a long time horizon, it can also buffer the effects of equipment and batch failures when it is sized adequately. The sizing of an intermediate storage often can be treated as a deterministic problem, but, when the equipment and batch failures are significantly of random nature, then its sizing becomes a stochastic rather than deterministic task. Deterministic variations in the failure frequency and recovery time have been considered by Karimi and Reklaitis (1985) and Lee and Reklaitis (1989), while stochastic variations have been studied by Odi and Karimi (1990,1991), and Mihalyko and Lakatos (1998). In these studies, however, constants filling and withdrawing intensities were assumed. The aim of the present contribution is to study the problem under generalized conditions, i.e. when the intensity of filling and withdrawing of material into and from the intermediate storage may be changed arbitrarily during the operation.
2. Process Model Let us consider a non-continuous processing system with n upstream and m downstream units (usually n?^) with an intermediate storage. We assume that the processing units are operated periodically, and stochastic failures of the units under general conditions may occur. Furthermore, we suppose that the filling and removal rates may vary in time arbitrarily, i.e. they are described by general functions exhibiting at most finite number of jump discontinuities. Let o\^,... co^^ and^2,1 v^2,m denote the operation periods of the upstream and downstream units, while t^^ ,-hn
^^^ h,\ ^"•h,m ^^^ the filling and removal times, respec-
tively. Further, let t^^ ,..• ^i°„ and t^^ .— tl^m denote the corresponding delay times. We
240 suppose that t^^ +1^^^ < co^^ for any unit. Then, the mathematical model of the process is formulated under the following assumptions: 1. The ratio of two arbitrary periods is a rational number. 2. Only one upstream unit may suffer failure at the same time. Let its index be 1. 3. The failed unit does not transfer material into the storage during its repairing. 4. The failure of a unit does not affect the operation of the remaining ones. 5. The differences between the serial numbers of the failure periods, denoted by ^„ are integer random numbers of the same distribution having bounded range. We suppose that 4 /=1,2,... are independent. j
Then the serial number of the f^ period of failure is expressed as ^ ^ / , its initial /=i
moment is ^{^i -1)^1,1 , and the fmal moment is ^ ^ / ^ ^ • As a consequence, the/^ j-i
failure cycle is, by definition, the interval
S^/-^u'S^/-^i,
as it shown in Fig. 1.
The filling and removal rates are described by the following functions showing at most finite number of discontinuities: 0, if
0< CO, 'd,s
\(o.'d,s bef.Ai)-
fciM
CO,
t^ +t
M,s
), if
CO, 'd,s
(1)
0),
\co.
!lilh±<. t \co.'d,s co,^
0, if
where /^^ > 0 is some positive function, while d=l and s=l,....n for the upstream, and d=2 and ^=l,....m for the downstream units. The amount of the material transferred by the s^^ unit during an operation period is t=0
1
'
^i'i
tt•
^^^^'^
•
1 L
(^7+&)^7.7
1
[ ik i L
t
•
^ ^
^ ^ w 2^^ failure cycle w Moment of the 7^' Moment of the 2"' failure failure
V^ failure cycle
Fig.l. Characteristic time intervals of the process with equipment failures. ^
\bef,,{x)dx.
(2)
241 and the total amount of the material transferred by the s^^ unit in the time interval [0,f] may be expressed as
Vj-{t)^\bef,,{x)cbc.
(3)
When a failure of the 7" upstream unit occurs, then the filling intensity is described by the function ^e/i,(0,
7 >
fg
if
^ •«i.i /=1
befli^^(t) =
0, if
7J
re
(4)
j
•^u )
and the total amount of material transferred of the V unit is expressed as f
(5)
Then, the amount of material in the intermediate storage is varied in time according to /o
\
0
"
0
'^
0
(6) 5=2
5=1
where (taf) denotes a vector of dimension (n+m) of the delay times.
3. Sizing of the intermediate storage In order to have material in the intermediate storage always sufficient to operate the system without problem, it is necessary to have an initial amount y(0) in the starting moment of operation as
-minVh^'At).
(7)
Oil
There will no overflow if the volume of the storage is equal to maxVh^'\t)
+ V(.0)
(8)
As a consequence, the volume of the storage sufficient to operate of the system is expressed
0
0
(9)
Since in the present case the failures of the units are stochastic, so is also the function (,o )
Vh ''\ and the goal is to determine the distributions of the maxima and minima of this very function. In order to achieve this we proceed in the following way. We divide the process in the interval [0,0 into sub-processes according to the failure cycles and investigate the variation in time of the amount of material in the storage in
242 these sub-processes consecutively. The method developed is based on the following statements. Consider the function in the/'' failure cycle having the following form Vh^'''Xt)-Vh^''\^^i
'CO,0 for te
S^/-^U'S^i
CO,
. Then we have the
Statement 1. The actual delay times related to the initial moment of the /^ cycle becomes modified compared to the original ones, and are given as a function of the initial moment the cycle 7-1
^1,1
y-1 1=1
'(^ny w...+^".-E^/'^u+^'/.^-^
(0,
rl(Z^rco,0 =
(10) •7-1
" 7-1
S^/-^i.i
/=!
1=1
,+^°.+'.,.-Efi'^u
1=1
_ while the amount of material in the storage is varied in time according to Vh'^^'\t)-Vh^''\Y^^,
•co,,)=:Vh '' -
( / - ^ f , -co,,).
(11)
Statement 2. The number of actual delay times r^^ for all upstream and downstream units is finite. Statement 3. The number of vectors \fd,s} formed from the possible actual delay times 1. TT TT u ^u Pu IS equal to I I ^1 ^ • I I ^2 5 » where —— = '
s=2
5=1
'
CO,s
q,^,
^1,1 ,
P2,s =
(O^^s
Q2,s
. . , , and p^ ^ and q^ ^ are rel-
ative primes. The direct consequence of the Statement 3 is the Statement 4. The number of sub-processes arising in an arbitrary process is finite and is n
m
given by A' • J][^i^ '11^2,^ ' where K is the number of possible values of ^^. Statement 5. The maximum and minimum problems, providing the maximum and minimum values of the variation of amount of material in the intermediate storage in any failure cycle, are reduced to finding maxima and minima of finite number of functions. Statement 6. Let M^'-'^k) and m^^'^k) denote the global maximum and global minimum of problem of Statement 5, where k stands for the possible values of random variable ^1. Then the maximum and minimum value of the amount of material in the intermediate storage in the whole/^ failure cycle can be obtained as a sum of the initial /0
)
(0
\
amount of the cycle, and the global maximum M^'^''(^') and minimum m^'^'^^p, respectively. Subsequently, the maximum and minimum values in the whole interval of the process can be obtained recursively.
243 Statement 7. Any process can be built up from a random sequence of functions related to the failure cycles of finite number. By means of simulation and statistical evaluation of the results of simulation runs the storage can be sized at a given significance level.
4. Example For the sake of illustrating the method developed, consider a batch system with two upstream and two downstream units the operation parameters of which are as follows: ^1,1 = '7 ' ^1,2 = 4, 6^2,1 = '7 '^2,2 = ^ ' ^1,1 = 4, t^2 = 3' ^2,1 = 5, ^2 2 = 2 , t^^ = 2, ^1^2 = ^ ,
^21 = 1, ^2,2 "= i • Let us assume that the filling and removal rates of the units are described by the functions of the form: f^^{t) = 4t, fiji^) = 4t, /2,i(0 = i, fi,!^^) - St. Then, the number of the possible vectors of delay times is equal 12 that are given in Table 1. If the first failure of the unit 1 occurs in the 7^^ 2""^ or 3'"^ period of operation, respectively, then 36 different sub-process variations of the changes of the amount of material in the storage during the first failure cycle are possible. Three such subprocesses, corresponding to the vector of delay times (2,0,1,1), are shown in Fig.2, the Vh'"'"'•"(t)
\
\ \
V
a) ' b) ' c) ' Fig. 2. Variation of the amount of material in the storage in the r^ failure cycle when the r^ failure occurs in the a) V^ period, b) 2" period, c) 3^ period of operation Table 1. The characteristics of the possible vectors of delay times. m(k) and M(k) minimum and maximum for failure in the k^ period of operation; EVh - expected value of the amount of the transferred material in the storage during one cycle Vector of the delay times (2,0,1,1) (2,0,1,0) (2,0,1,-1) (2,1,1,1) (2,1,1,0) (2,1,1,-1) (2,-2,1,1) (2,-2,1,0) (2,-2,1,-1) (2,-1,1,1) (2,-1,1,0) (2,-1,1,-1)
m(l) -18.5 -20 -20.5 -24.5 -24.5 -30.5 -16.5 -18.5 -26.5 -10.5 -16 -22.5
Mil) 3.6 0 0 0 0 0 10 6 10 11.5 2 3.5
m(2) -1 -11 -13 -8 -17 -8 -17 -15 -4 -9 -10 -8
M(2) 23.5 19.5 13.5 19.5 9.5 11.5 19.5 17 17 21.5 21.5 21.5
m(3) 0 -8.5 -10 -8 -14.5 -12 -4 -6.5 -6.5 0 -0.67 -6.5
M(3) 33 31 31 30.5 26.5 21 39 35 27 41 31 33
EVh 3 -13/6 -11/3 -3 -25/3 -29/3 13/3 -1 -113 19/3 1 -1/3
values of minima and maxima obtained for these 36 variations, together with the expected values of the amount of transferred material in the storage during one cycle, are presented in Table 1. The global minima and maxima of these functions were
244 determined numerically by the combination of the grid and modified simplex method. According to Statement 7, each process of the system under investigation can be built
1000
1200
1400
t
a) b) Fig. 3. Variation of the amount of material in the storage through a) 15 and b) 100 randomly selected failure cycles, corresponding to the vector of delay times (2,0,1,1). up from these 36 sub-processes. These processes differ from each other in the sequence of the sub-processes sequenced randomly one after the other. These processes can be generated by computer simulation, selecting one of the possible sub-processes randomly according to a defined probability distribution. Such process is shown in Fig.3 that consists of 15 and 100 failure cycles, respectively, the sequence of which were generated randomly by using the discrete uniform distribution. Carrying out 10,000 simulation runs, consisted of 100 failure cycles under the same operation and simulation conditions, we obtained mean value of minima=-483.08, mean value of maxima=27.9316, and mean value of their differences=509.79. Based on the nu-merical results, it can be concluded that the difference is smaller than 684.1619 with probability 0.95, hence 684.2 is a sufficient size for the storage at significance level 95%. Notice that, in the example, the expectation of the transferred material in the r^ cycle is positive on the basis of the initial delay time but the remaining delay times, as it well seen in Table 1, cause a decreasing tendency in the variation of the process.
5. Conclusions A method and simulation program has been developed for sizing intermediate storages in batch/semi-continuous systems taking into account stochastic equipment failures under general operation conditions. The method is based on the observation that any process of this system can be built up from a random sequence of failure cycles of finite number. With the aid of the analysis of the problem, optimisation on large time intervals of the processes is replaced by optimisation on smaller intervals of the sub-processes of that very process. By means of simulation and statistical evaluation of the results of the simulation runs the storage may be sized at a given significance level. The computation time is favourable, since it is a linear function of the number of the failure cycles.
6. References Karimi, LA. and Reklaitis, G.V., 1985, AIChE Journal, 31,44. Lee, E.S. and Reklaitis, G.V., 1989, Computers chem. Eng., 13, 1235. Mihalyko, E.G. and Lakatos, B.G., 1998, Computers chem. Eng., 22, S797. Odi, T.O. and Karimi, LA., 1990, Chem. Eng. Sci., 45, 3533. Odi, T.O. and Karimi, LA., 1991, Chem.Eng.Sci., 46, 3269.