Comparison Between Sequential and Simultaneous Approaches in Process Simulation Tomio Umeda' and Masatoshi Nishio Systems Development Team, Process Engineering Group, Chiyoda Chemical Engineering & Construction Co., I-okohama, J a p a n
Two basic approaches in process simulation-sequential and simultaneous-are investigated. The method of successive substitutions i s used to make a comparison of the basic characters of those two approaches. Sufficient conditions for the simultaneous approach to give better convergence are presented, and two numerical examples with widely different characters are studied for the demonstration of utilizing the present results.
T h e steady-state simulation of chemical processes plays a n important role in process design and operation. I n particular, optimal design by any search method includes a simulation routine within a n optimization routine, and thus a n overall computational efficiency is greatly affected by the effectiveness of the simulation method used. I n performing process simulation, there are two basic approaches -sequential and simultaneous. While the choice of a numerical method for solving nonlinear algebraic equations is a n important matter, the order of sequence in computation or network analysis is a dominant consideration for the sequential approach, as well as the linearization of nonlinear models for the simultaneous approach. In the past decade, many research results and reviews were reported. Although this is not a place for reviewing, there are several papers that have particular relevance worth mentioning. With respect to the numerical methods for solution, Rosen (1962), Kesler and Griffith (1963), and Shannon e t al. (1966) have applied the method of successive substitutions or its modification. Rubin (1962), Saphtali (1964), and Ravicz and Norman (1964) have applied NewtonRaphson's method. Cavett (1963) has reviewed various methods for numerical solution. Kliesch (1967), Orback and Crowe (1971) have made the extensive use of Wegstein's method. On the network analysis, Rubin (1962), Sargent and Resterberg (1964), Lee e t al. (1966), Lee and Rudd (1966), and Hiraizumi e t al. (1969) have presented various algorithms. Christensen and Rudd (1969) have made a n expansion of those ideas to reduce the difficulties of iteration by the choice of suitable parameters. For the simultaneous approach, Kagiev (1957) developed a mathematical theory of recycle processes. Rosen (1962), Ravicz and Korman (1964), Naphtali (1964), Kishimura et al. (1967), and Maejima (1970) have used the linearized models with renewal of parameters on every iteration. Though the sequential approach x i t h accelerated convergence methods for solution is commonly used in the steadystate simulation, not many studies on t'he simultaneous approach and the comparison of these two approaches as to their basic characters have been made. I n this paper, these two approaches are investigated by using the method of successive substitutions which has been mathematically well investigated and also has been widely used in practice. The two numerical examples presented are To whom correspondence should be addressed
for the purpose of illustrating how to make the choice between the sequential and simult'aiieous approaches which most appropriately meet the character of the system in question. Representation of System Characteristics
The characteristics of a total processing system are obtained by using those of the individual processing units and the system structure xhich expresses how these processing units are interconnected. I n the steady-state conditions, the general form of each unit is represented by: zn =
Y,
=
T n ( x n , dn)
S,(X,,d,)
n
=
(1)
1,2,
,S
. .
(2)
where T , and S, show, respectively, t'ransformation from input vect'ors to output vectors. d, is decision vector. x , , y,, and z, are state vectors of the input, system-output, and output, respectively. The following set of relalions shows how the individual units are interconnected :
.ir
x,
=
Cmnz, f o r k
=
n=l
1,2,
.
,
.,K
(3)
where C,, is a matrix, whose elements are zeros and unities, expressing the interrelations b e h e e n inputs and out'puts. The steady-state behavior of the total processing system is obtained by performing the process simulation using Equations 1-3. d, must be given beforehand, and the calculations start from specified system-inputs. These system Equations 1 and 2 are generally nonlinear, and it'erative computations are required to handle complex system structures with recycles. I n addition bo these equations, there are various inequality constraints which must be satisfied to have feasible solutions. Inequality constraints encountered in process design are those such as stable operating region of an exothermic reactor, maximum allowable temperature for material of construction of equipment, and so on. The mathematical expression of those inequality constraints is given by g f ( X n , Y n , z n , dn)
2. 0 i
=
n
=
1,2, . ,I 1,2, . . , N
(4
where g i is a linear or nonlinear function expressed in explicit or implicit form. Ind. Eng. Chern. Process Des. Develop., Vol. 11, No, 2, 1972
153
r
-7
a basic recycle process consisting of a single process with a recycle h o p is taken. Input-output relation a t the junction point where feed and recycle streams are connected gives the following form of iteration:
Sequential aPProoch,
x,,,’+’ = Gseq(x’) = XO
I
-
aPPrOaCh
xSimk+1= GLllrn . (xk)
Figure 1. Comparison of sequential and simultaneous epproaches
I n simulating processing systems by the sequential approach, Equations 1 and 2 are directly used, and the iterative computations are carried out for state variables of streams. I n simulating the processing systems by the simultaneous approach, the system equations of the subsystems and the interrelations among them must be linear or linearized. Since the interrelations expressed by Equation 3 are linear, only the subsystems equations expressed by Equations 1 and 2 need to be linearized. By introducing linearizing parameters, these two equations may be given by the following forms (Nishimura et al., 1967) : zn =
Ln(Bn)* xn
(5)
Jn
L’n(%)
(6)
xn
where Ln(Bn)and L’%(Bn)are matrices whose elements are linearizing parameter vector 0,. The linearizing parameter vector is not generally constant, but is an implicit function of the state and decision vectors as expressed by the following relations :
For a given set of the linearizing parameters, Equations 5 and 6 can be solved for the state vectors by matrix manipulation. The linearizing parameters can be obtained iteratively by applying various methods such as successive substitution, Newton-Raphson, Wegstein, and a weighting factor (Rosen, 1962), and by using the state vectors to solve for the parameters in each processing unit. These procedures are repeated until the convergence criterion is satisfied.
=
G(x) or xk+l
=
G(xk)
(8)
where x, xk, and G are vectors of variables and functions, respectively. The superscript k denotes an iteration number. To demonstrate the difference in the basic characters of the two approaches by the method of successive substitutions, 154 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 2, 1972
M(Xsimk)
*
[I - M(x,imk)l-’ (10)
/Ix*
limxk
=
x*
k+m
Under this sufficient condition for the existence of convergent sequence, it is attempted to compare the rate of convergence between the sequential and the simultaneous approaches by taking a norm for a distance between x* and x n . Since it may be assumed that the computing time required for the computation of unit operations is about the same between these approaches and the additional time for the simultaneous approach is required for obtaining linearized parameters and for manipulation of the matrices, the comparison of the rate of convergence can be made by the number of iterations, if the additionally required time is short enough compared with the computing time for unit operations. If the following inequality is satisfied, the simultaneous approach gives a better convergence character:
JIx*- xo
+ x,eq*. M(xse,*)II >
I~X*-
xo
+ xo
~ ( ~ a i m k )
[I - ~(xsimk)I-1// (1.1)
By taking notice of the relationship,
x*
- xo = fi
M(x*)
‘
[I - M(x*)l-’,
the following relationships can be derived as sufficient conditions :
< M(xk) < M(X*)
(12)
< XO. [I - M ( x k ) ] - 1
(1%
M(9)
Mathematical descriptions an the convergence by the method of successive substitutions are made under the following form:
*
where Z denotes the identity matrix. To have xSimk positive, < I, must be satisfied. Equation 10 an inequality, !iM(x8imk)li shows that a constant matrix gives the solution without iterations. Theorems on contraction mapping give conditions on a convergent sequence to a unique fixed point. A sufficient condition for the convergence of the method of successive substitutions is described as follows (Russell, 1970) : Let G:R + R be continuous differentiable in the neighborhood of point x* such that x* = G ( x * ) .If /IaG/ax// < 1, there - x0ll < e and xlc is generated by is an e > 0 such that if x k f l = G(xk)k = 0,1,2,. , ., then
Two Approaches in Solving Basic Recycle Problem
x
=
+ XO
XO
+
(9)
where XO is a constant vector corresponding to the systeminput, and M(xaeqk)is a matrix whose elements are linearizing parameters. These are not generally constant but a function of xk. Equation 9 expresses the recurrence relation for the sequential approach. The simultaneous approach carries out the iterative computations in the following manner :
0 Simultaneous
+ xaeqn. M(x.,,’)
and Xk
for
XO
< xk < X*
k = 1,2,. . .
Furthermore, it can be derived from these relationships that M(xk) increases monotonically with xk within the region 0 < M(x~< ) 1. Thus, in short, the relationship IlaG/axlI < 1 in
Figure 2. Signal flow diagram for muitirecycle system
cation to basic recycle problems, but they can be extensively applied t o more complex recycle problems. The following are possible cases of extended application. If multirecycle loops involved in those loops comare joined at one point, M(xk)s prise a composite matrix-that is,
where M c ( x k )and Ml(xk) denote the composite and the lth loop matrices, respectively. If more than one recycle loop is involved and they are not joined a t one point, a set of the vector-valued equations in the form of Equation 8 is to be solved. For this case, Equations 9 and 10 are replaced b y : Xlk+l = X&lk
+ XZ*.M(XLk)
(15)
and xp+1 =
XZ2
+ XJ
*
M(XLki)
. [I - M ( x , ” ] - ’ k
Figure 3. Four flash drums system
the existence theory, Equations 12 and 13 comprise the sufficient conditions for the simultaneous approach to give a better convergence than the sequential approach. I n the case of a single variable, the above sufficient conditions are represented by : XO
-[I
Xk
+ M(xO)l- 1 < M ( x k ) and 1 - [l - M ( x O ) ]XOXk < M ( x k )
from ((aG/axll < 1, M ( z k ) < M(z*) and xk
-
