- Email: [email protected]
Process synthesis by task assignment
Chemical
Engineering
Science,
1974, Vol. 29, pp. 2033-2040.
PROCESS
Pergamon
Press.
Printed
in Great Britain
SYNTHESIS
BY TASK
TOM10
and AK10
UMEDA
ASSIGNMENT
SHINDO
R&D Division, Chiyoda Chemical Engineering and Construction Company, Yokohama, Japan and ATSUNOBU ICHIKAWA Department of Control Engineering, Tokyo Institute of Technology, Tokyo, Japan (Received 4 March 1974; accepted
12 March 1974)
Abstract-An approach by task assignment is presented to determine processing system structures in chemical process design. For given design conditions of major subsystems, which were specified by the results of basic or applied research, a basic flowsheet is determined by task assignments on energy recovery and material flow transportation. An illustrative example shows the usefulness of the present method: INTRODUCTION
A large part of process design work has been carried out with the aid of digital computers and it becomes possible to make the optimal design for a given processing system structure without any particular difficulty. From the viewpoint of process design practice, the determination of a reasonable processing system structure has been made on the comparison of optimal design results for various alternative system structures which have been developed by intuitive and/or more systematic ways. Due to the lack of sufficient theoretical guidance, processing system structures have often been developed from previous experiences and by analogy with other similar processes. In the recent years, much attentions have been paid for the process synthesis-the determination of processing system structures. The first technical symposium on computer aided process synthesis was held at the A.1.Ch.E 71st National Meeting, Dallas, Texas, February 1972, and also the important works on this subject have been reviewed by Hendry et al. [l], one of the authors [2] and possibly by others. “Process synthesis” written by Rudd et al.[3] is the first publication in this area. Two sequential steps in process engineeringprocess development and design-are widely recognized. One of the major tasks in the former stage is to develop basic processing schemes so as to satisfy process objectives, and heuristics, evolutionary strategies and methods of direct search can be effectively utilized. The work in the latter stage is usually more definitive and much degrees of free-
dom are not left for the process synthesis. The work starts with a given basic plan for processing scheme and design conditions for a few major processing units as the suggestions from research results. To complete a basic flowsheet, it is necessary to assign tasks for filling up the gaps of design conditions between two adjacent major processing units. The determination of an energy recovery system and the assignment of material flow transportation belong to the task assignment. It is also required simultaneously to determine optimal design conditions for the processing units including the major units. This paper is intended to present a task assignment concept for process synthesis in the process design stage. That is, the present study limits the process synthesis problems in more definitive stage of process design. After the synthesis problem is stated, a method for solving the problem is presented and a practical example of flowsheet determination is illustrated. STATEMENT
OF PROBLEM
of process design and development works shows some differences in character. The work of process development aims to find feasible design conditions of a few major processing units and it is usual not to have a complete set of design conditions of processing units of which a processing system consists. This is due to practical requirements from the restriction of time and money for process development. On the other hand, the work of process design is to construct a
2033
A careful examination
2034
TOMIOUMEDA,AKIO SHINDOand
more complete processing scheme and optimal design conditions must also be determined. It is usual that the results of process development become an initial condition of process design and no much degrees of freedom are left on the specification of designing the major processing units. From this viewpoint, Rudd’s [4] decomposition principle of design synthesis is useful concept in process development. The synthesis problem at hand is decomposed into subproblems where existing technologies are available to solve them. McGalliard and Westerberg[5], on the other hand, described that chemical processing systems can be characterized as ‘inherently’ and ‘sparsely’ structured. This may be considered that this characteristics is due to the existence of a number of restraints which reduce the degrees of freedom in the process synthesis. This observation seems to the analysis of process design work rather than that of process development work. State transition in chemical processes For the purpose of making a clear presentation of the problem under consideration, mathematical statement of process synthesis problem is given as follows:
ATSUNOBU ICHIKAWA
On the other hand, process synthesis in the process design is to determine paths by connecting given states of major processing units. This can be accomplished by assigning some appropriate task. A set of these major processing units is denoted by a basic processing scheme. Thus this does not mean a basic flowsheet which expresses a completely connected structure of the processing system. In Fig. 1, above mentioned problem is schematically illustrated.
In11
d’/ -
Basic unit
Any states in a chemical process are described by an ordered n-tuple of real numbers, z”r = (Z”‘, z.*, . . . , I”“)=,
n=1,2
,...,
N.
These real numbers are the components of vector znT and those correspond to component-wise material flow rate, enthalpy flow rate, temperature, and pressure by which any states can be expressed. The collection of all such k-dimensional vector is the real k-dimensional vector space which is denoted by Rk. Chemical processes make transition from given initial states to specified final states, and the process synthesis can be defined as to determine an appropriate path in the k-dimensional space which connects initial and final states. The path can be determined by a sequence of transformation from one state to another. This corresponds to a composite mapping from state to state. Process synthesis in the process development step starts with given initial, final states and is to determine states of major processing units between them. There are in general several mappings from a given state to adjacent states which become starting points of further change of states. This character inherents so-called the combinatorial problems.
---
Task
assigned
_
state
tronsltlon
in bosx
units
Fig. 1. Concept of task assignment in process synthesis in state space.
Statement of synthesis problem under consideration In the scope of present problem of process synthesis, a basic processing scheme is assumed to be given beforehand and it is aimed in this study to synthesize a system to meet processing requirements which are satisfied in the basic processing scheme. The present study limits the synthesis problems in more definitive design stage. Task assignment makes a complete interconnection among the major processing units and gives a corresponding basic flowsheet. The task assignment and its result depend on the design conditions of processing units of which the basic processing scheme consists, and it is generally required to find the optimal design conditions of processing units, on which the optimal processing system should be synthesized. Let the basic processing scheme specify more
Process synthesis by task assignment 2035 , precisely to make it clear what is given. The basic gap in temperatures is the problem of synthesizing processing scheme contains the following items: heat exchange systems, and two kinds of problem exist; one is to find an optimal system and the other is to find practically acceptable systems. These two (1) Processing units of which the basic scheme consists. synthesis problems are defined as follows: for given values of temperature, flow rate, and physical (2) Design or operating conditions for these processing units such as inlet and outlet temperaproperties of a set of streams, find an optimal or a tures and pressures for them. practically acceptable system of heat exchange with auxiliary heating and cooling facilities so as (3) A set of streams which shows indirect connections between the processing units. to have minimum or close to minimum value of predetermined objective function. The problem of (4) A set of streams does not initially satisfy temperature and pressure specifications for task assignment for bridging the gap in pressures is the associated processing units. That is, there the problem of assigning materials transport deexist some temperature and pressure differvices and similar definition of problem can be made ences between the indirectly connected profor given values of pressure, flow rate, and physical cessing units. properties of a set of streams. As one of optimal design problems, the present Thus the problem of synthesizing the optimal problem can be defined as follows: system is to assign suitable units, more generally minimize F 4,(x., n., d.,) tasks, to make up the temperature and pressure differences. Figures 2a and 2b show the difference between a + c 9twt, z:, Tf(ijl(di), &7 &jl I basic processing scheme and a feasible processing d,, T,cii,, Sit and Sti system with complete inter-connection. In Fig. 2b by choice of temperature and pressure differences between the z, = f(x., d.), z: = f:(x:, d:) subject to connected units can be made up by providing x: = &Zi, xj = pumps or compressors for pressure differences, and coolers or heaters for temperature differences. or 0 Whether this equipment is necessary or not deand pend on the design conditions for the associated g, (x., zn, d.) s 0, processing units. It is possible to eliminate some of where x., and z. are input and output state vectors them by changing the design conditions. It is the of the n th unit, respectively. x: and z’,are also input purpose of the present study to determine the optiand output state vectors of the t th assigned task, mal design conditions of these processing units and respectively. d, and d’, are decision vector of the to assign automatically the tasks for making up the k th unit and the t th assigned task, respectively. 6~ temperature and pressure differences. (or 6,j) is a structure parameter which expresses the The problem of task assignment for bridging the S,jZyt
Sit,
~_-__----___--___-__~ I
I
;----
&j
=
1
1
m--__&_-_p+___++__&__&_fl Fig. 2a. A basic processing
scheme.
Fig. 2b. A feasible flowsheet generated by task (T,, Th) assignment.
TOM~OUMEDA,
2036
AKIO
SHINDOand ATSUNOBUICHIKAWA
interconnection between the i th unit and the t th task (or the t th task and the j th unit). f,, f: and g, are vector-valued functions. 4” and 4, are objective functions associated with the n th unit and the t th task, respectively.
METHOD
Decomposition
FOR SOLUTION
method for problem solving
The present problem previously stated can be solved by applying a method of decomposition on the level of available technology as described by Rudd[4], since the problem under consideration has been given by a decomposable form. The problems of task assignment and of designing the basic processing units are subproblems of which the present problem consists. The handling of the decomposable problem can be made by multi-level approach. The solution of design problem of the basic processing units is obtained on the second-level, where some values for coordination variables have been given beforehand. Based on these solutions, the task assignment problems are solvable on the first-level. The problem of coordinating these subproblems can be solved on the third-level. This multi-level approach is illustrated in Fig. 3.
1
Coordmotion
problem
Deslqn problems for
bow
3rd-level
2nd-level
units
terized by decomposing a system into sub-systems in such a way that all the constraints are satisfied during the iterative computations for optimization and physically meaningful results are always obtained. Model-coordination approach [8], projection method[9, lo] and information reversal strategy [l l] belong to the present class of feasible decomposition methods. In applying the feasible decomposition method, the basic processing system has been a priori decomposed into individual processing units by choice of inlet and outlet temperatures and pressures at each processing unit. These variables are defined as coordination variables in multi-level approaches. Some appropriate values are assigned for the coordination variables and those become a set of the input and output conditions for each subproblem. Some or all of the coordination variables are treated as decision variables on the upper-level in the multi-level approach. On the lower-level, the differences in a set of the coordination variables between two units, that is, outlet temperature and pressure of the ith unit and inlet temperature and pressure of the j th unit, are made up by providing heat exchangers (coolers or heaters) and materials transporting devices, respectively. To make the feasible decomposition applicable, it is necessary to satisfy the condition that there remain some degrees of freedom for the subproblems in the lower-level [6]. Furthermore, the analysis of the decomposition problem shows that it is better from the viewpoint of effective computations to treat temperature and pressure at the outlet as dependent variables. Along this line, the feasible decomposition method for optimal process synthesis can be mathematically described as follows: Subproblem PD,: the n th processing unit designon the second level
Task
osslqnment
problems
Ist-
level
Fig. 3. Multi-level approach for solving the integrated problems.
To solve the present problem effectively, a feasible decomposition method [6,7] is considered to be applicable. The feasible decomposition is charac-
tin the present study, the relationships of interconnection between subsystems are limited to single input and single output cases for simplicity. Those are easily extended to multiple imputs and outputs cases.
minimize
4” (x,z, z,z, d:, d.))
by choice of
dn2
subject to
2”* = f;(x;, g.‘(x,‘,
d;, d,))
2,2, d:,
where d.’ is determined
d,S) 5 0
on the third-level,
D. xD. xD, = D., (2 A”,,zf,) = (t”2, P.*)EZ,z.
and
Process synthesis by task assignment Subproblem system-on
SH: synthesis of optimal heat exchange the first level
minimize
4. (x )li,z )li,d iii, &Ii,)
by choice of
Shiiand d iii
subject to
h,Cxk, z&
2037
Step 5. Check whether or not the assumed and computed values of d,’ satisfy a stopping criterion. If satisfied, stop the computations. Otherwise return to Step 6. Step 6. Revise the values of di and return to Step 2.
d$ = 0, & = 1 or 0
For solving the subproblems PD, and the coordination problem, some suitable methods for parameter optimization such as gradient, direct search where fhii,and ghii are vector-valued functions and fhii methods can be applied. For carrying out the synthcorresponds to the equation for calculating heat esis in Step 3, it seems to be important how to solve transfer area. xhi and zli are determined on the the so-called combinatorial problems. There seems second-level. to be appropriate to use some specific characteristics of the problem. For solving the subproblem SP, Subproblem SP: assignment of optimal materials there seems to be no combinatorial problems, since transporting units-on the first-level it is generally sufficient to consider a few alternative cases for selecting materials transporting deminimize 4p (xbi, z;i, a;,, &ii) vices. With respect to the subproblem SH, several synthesis strategies for heat exchange network are by choice of S,, and d& applicable. Those strategies have been reviewed by subject to f,ij(Xii, zbi, d&) = 0, S,, = 1 or 0 Hendry et al. [l]. To solve the subproblem SH, it is particularly desired to use a method which requires g,i, (xfi, zbj,dbii)5 0 less computing time. This consideration aims to rewhere f,, and gpij are vector-valued functions and fpii duce too much computing time for solving the integrated optimization problem. corresponds to performance equations for materials transporting devices. xii and zb are determined on Method for solving the subproblem SH the second-level. Since the input and output states of the basic processing scheme are given, several strategies can be Coordination problem on the third level directly applied to solve the subproblem SEI. Aiming at the reduction of the computing time, the apminimize plication of heuristic rules is useful, though the solution does not claim the optimum. In the present study, the concept of optimal assignment approach which was presented by authors[l2] is used with the combination of a heuristic rule. Firstly, each of by choice of d,’ = {XL,,, x;.,] hot and cold streams is subdivided into smaller heat exchange units with approximately equal heat duty. where all the variables excluding de’ are determined The size of a heat exchange unit is taken as an on the first or second level. approximate value for the greatest common divisor According to the above mathematical description, of heat exchange duties of the hot and cold streams. the following problem-solving procedure based on Secondly, use the following heuristic rule: it is the feasible decomposition method can be derived: preferable to make heat exchange of units in such a way that the hot and cold streams are matched in decreasing order of their inlet temperatures. This Step 1. Assume values of dz, n = 1,2,. . . , N gives the optimal solution for minimizing the total Step 2. Solve optimization problems associated heat exchange area in some cases, such as the heat with the subproblems PD., n = 1,2,. . . , N, exchange between one cold (or hot) stream and more by choice of d.2. x.’ and z: are optimally dethan two hot (or cold) streams, or heat exchange termined in this step. problems with satisfying necessary conditions for Step 3. Solve synthesis problems SH and SP by deriving the rule. After determining a practically choice of &ii, d&, Spij and db+ acceptable heat exchange system, all blocks of Step4. Solve the coordination problem by choice exchange units of the same temperature are merged of d,‘. ghij(xbe &, db,) 5 0
TOMIO UMEDA, AKIO SHINDO and
2038
to form new blocks which give more practical results. More detailed description of the above method has been given elsewhere[l2, 131. Task assignment criteria for solving the subproblem SP Where there is a pressure difference between two adjacent units, some type of material transporting unit must be assigned. The type of units can be classified by the following two items: one is the state of fluid to be handled, and the other is the degree of pressure difference with taking the direction of pressure change (negative or positive difference) into consideration. To present the idea, criteria of task assignment, though this is not complete, are given as follows:
State
of fluid
Gas or non-condensable vapour
Liquid
Difference
ATSUNOBU ICHIKAWA
This may often require the increase of system pressure. It is also preferable not to use compressors as possible and to make efforts to replace the transporting units for gas or non-condensable vapor by those for liquid. These strategies for solving the subproblems are applied under the control of three-level approach on the basis of feasible decomposition method as illustrated in Fig. 3. ILLUSTRATIVE EXAMPLE For illustration of the cess synthesis in process ample is taken up. The shown in Fig. 4 has been
proposed method for prodesign stage, a simple exbasic processing scheme given beforehand. Table 1
I I
I I
Type
L___-_______________--_---_-_____L Fig. Positive, Small Positive, Large
Fans, Blowers
Negative Positive Negative
Valves Pumps Valves
4. A basic
processing
scheme
for
illustrative
example.
Compressors
where the pressure difference is specified by (pi2-piI), that is, the difference between output of the i th unit and input of the j th unit. Direct application of these criteria gives us a large number of units in the basic flowsheet. This is not practical and it is necessary to merge the units, if possible. This merging is usually made on heuristic rules given below, which are derived from economic considerations: It is preferable to reduce the number of units as possible by eliminating units in down streams and adding the correspondent pressure difference to the unit in the upper stream.
shows a typical set of information used to make up the gap in temperature differences by the solution of the subproblem SH. Figure 5 is a matrix presentation of the heuristic rule on heat exchange synthesis. After several iterations, a processing system shown in Fig. 6 is generated without applying merging strategies for heat exchange and material transportation. The result of merging gives us a practically acceptable system which is shown in Fig. 7. CONCLUSION A careful examination of process synthesis activities in process design stage shows that the basic flowsheet can be generated by appropriate task assignments for a given basic processing scheme defined in the first section. Based on the characteristics of the basic processing scheme, a feasible de-
Table 1. Initial information on a basic processing scheme M-----R
Unit name
,---__
R,___-_S,---__S,------_s,------
Flow rate 26,006
(kg/W
Heat duty (X IO”kcal/hr) No. of subdivision (-) Flow rate of subdivided Whr)
Heat capacity (kcahkg “C)
26,006
26,006
3.35
0.68
5
1
- 8.45 12
17,211
16,807
10,086
0.67
0
- 0.69
1
0
1
stream 5,201
0.503
26,006
0.531
2,167
0.534
17,211
0.408
0
0.499
10,086
0.431
Process synthesis by task assignment Hot
streams -Stream
number
234567
_I-;---------__-,
2-l I -I
the subproblems SH and SP, heuristic rules are applied to reduce the computing time, though this does not claim the optimal solutions in general cases. Heuristic rules of merging units give us practical solutions for process synthesis.
I I
2
(6i,)
I
3
I
4
I
Acknowledgments-The authors wish to acknowledge Chiyoda Chemical Engineering and Construction Company for the support of the present study and the permission for its publication. They also express their thanks to Hideki Sugiyama in solving examples.
I I I
I I I II :__-------_____+_ I I ,
5 4-I
NOTATION
c
t Streom
2039
number
Fig. 5. Matrix for heat exchange network. composition method is considered to be applicable to solve the integrated synthesis problem. To solve
blower or compressor D set of decision vectors d decision vector function of system f vector-valued g vector-valued constraint A4 mixer
(PI Fig. 6. Preliminary
flowsheet
synthesized
(before
merging).
(PI
Fig. 7. Flow sheet synthesized (after merging).
CES VOL.
29. NO.
lC-C
equation
TOMIO UMEDA, AKIO SHINDO and A~su~oeu
2040
p p
pump
pressure R reactor S separator T task t temperature V valve x input state vector z output state vector Greek symbols
6 4
structure parameter objective function
Superscripts T transpose of vector i the i th level in multi-level
’
approach (i - I,2 and 3) state vector of assigned unit or task
Subscripts h concerned i, j, n
P
with heat exchange number of subsystem or unit concerned with pressure
ICHIKAWA REFERENCES
111HendryJ.E.,RuddD.F.andSeaderJ.D.,A.I.Ch.E.JI 1973 19 1. [21 Ichikawa A. Kagaku Kogaku Chem. Engng, Jupan 1972 36 1053. r31 Rudd D. F., Powers J. G. and Siirola J. J., Process Synthesis, Prentice-Hall Englewood Cliffs, New Jersey 1973. r41 RuddD. F., A.1.Ch.E. JI 1968 14 343. VI McGalliard R. L. and Westerberg A. W., Paper presented at A.I.Ch.E. Meeting, Dallas, Texas, gebriary 1972. [61 Tazaki E., Shindo A. and Umeda, T., Automaticn 1972 8 543. 171 Umeda T., Shindo A. and Tazaki E., Ind. Engng. Chem. Proc. Des. Dev. 1972 11 I. [81 SchoelTler J. D. In Optimization Methods for LargeScale Systems (Edited by D. A. Wismor) p. 1. McGraw-Hill, New York 1971. [91 Geoffrion A. M., Managt. Sci. 1970 16 11. [lOI Geoffrion A. M.. Mana~t. Sci. 1970 16 652. [Ill Westerberg A. w., In Dkomposition of Large Scale Problems (Edited by D. M. Hemmelblau) p. 379. North-Holland Amsterdam 1973. [121 Kobayashi S., Umeda T. and Ichikawa A., Chem. Engng Sci. 1971 26 1367. r131 Nishida N. Kobayashi S. and Ichikawa A., Chem. Engng Sci. 1971 26 1841.
Engineering
Science,
1974, Vol. 29, pp. 2033-2040.
PROCESS
Pergamon
Press.
Printed
in Great Britain
SYNTHESIS
BY TASK
TOM10
and AK10
UMEDA
ASSIGNMENT
SHINDO
R&D Division, Chiyoda Chemical Engineering and Construction Company, Yokohama, Japan and ATSUNOBU ICHIKAWA Department of Control Engineering, Tokyo Institute of Technology, Tokyo, Japan (Received 4 March 1974; accepted
12 March 1974)
Abstract-An approach by task assignment is presented to determine processing system structures in chemical process design. For given design conditions of major subsystems, which were specified by the results of basic or applied research, a basic flowsheet is determined by task assignments on energy recovery and material flow transportation. An illustrative example shows the usefulness of the present method: INTRODUCTION
A large part of process design work has been carried out with the aid of digital computers and it becomes possible to make the optimal design for a given processing system structure without any particular difficulty. From the viewpoint of process design practice, the determination of a reasonable processing system structure has been made on the comparison of optimal design results for various alternative system structures which have been developed by intuitive and/or more systematic ways. Due to the lack of sufficient theoretical guidance, processing system structures have often been developed from previous experiences and by analogy with other similar processes. In the recent years, much attentions have been paid for the process synthesis-the determination of processing system structures. The first technical symposium on computer aided process synthesis was held at the A.1.Ch.E 71st National Meeting, Dallas, Texas, February 1972, and also the important works on this subject have been reviewed by Hendry et al. [l], one of the authors [2] and possibly by others. “Process synthesis” written by Rudd et al.[3] is the first publication in this area. Two sequential steps in process engineeringprocess development and design-are widely recognized. One of the major tasks in the former stage is to develop basic processing schemes so as to satisfy process objectives, and heuristics, evolutionary strategies and methods of direct search can be effectively utilized. The work in the latter stage is usually more definitive and much degrees of free-
dom are not left for the process synthesis. The work starts with a given basic plan for processing scheme and design conditions for a few major processing units as the suggestions from research results. To complete a basic flowsheet, it is necessary to assign tasks for filling up the gaps of design conditions between two adjacent major processing units. The determination of an energy recovery system and the assignment of material flow transportation belong to the task assignment. It is also required simultaneously to determine optimal design conditions for the processing units including the major units. This paper is intended to present a task assignment concept for process synthesis in the process design stage. That is, the present study limits the process synthesis problems in more definitive stage of process design. After the synthesis problem is stated, a method for solving the problem is presented and a practical example of flowsheet determination is illustrated. STATEMENT
OF PROBLEM
of process design and development works shows some differences in character. The work of process development aims to find feasible design conditions of a few major processing units and it is usual not to have a complete set of design conditions of processing units of which a processing system consists. This is due to practical requirements from the restriction of time and money for process development. On the other hand, the work of process design is to construct a
2033
A careful examination
2034
TOMIOUMEDA,AKIO SHINDOand
more complete processing scheme and optimal design conditions must also be determined. It is usual that the results of process development become an initial condition of process design and no much degrees of freedom are left on the specification of designing the major processing units. From this viewpoint, Rudd’s [4] decomposition principle of design synthesis is useful concept in process development. The synthesis problem at hand is decomposed into subproblems where existing technologies are available to solve them. McGalliard and Westerberg[5], on the other hand, described that chemical processing systems can be characterized as ‘inherently’ and ‘sparsely’ structured. This may be considered that this characteristics is due to the existence of a number of restraints which reduce the degrees of freedom in the process synthesis. This observation seems to the analysis of process design work rather than that of process development work. State transition in chemical processes For the purpose of making a clear presentation of the problem under consideration, mathematical statement of process synthesis problem is given as follows:
ATSUNOBU ICHIKAWA
On the other hand, process synthesis in the process design is to determine paths by connecting given states of major processing units. This can be accomplished by assigning some appropriate task. A set of these major processing units is denoted by a basic processing scheme. Thus this does not mean a basic flowsheet which expresses a completely connected structure of the processing system. In Fig. 1, above mentioned problem is schematically illustrated.
In11
d’/ -
Basic unit
Any states in a chemical process are described by an ordered n-tuple of real numbers, z”r = (Z”‘, z.*, . . . , I”“)=,
n=1,2
,...,
N.
These real numbers are the components of vector znT and those correspond to component-wise material flow rate, enthalpy flow rate, temperature, and pressure by which any states can be expressed. The collection of all such k-dimensional vector is the real k-dimensional vector space which is denoted by Rk. Chemical processes make transition from given initial states to specified final states, and the process synthesis can be defined as to determine an appropriate path in the k-dimensional space which connects initial and final states. The path can be determined by a sequence of transformation from one state to another. This corresponds to a composite mapping from state to state. Process synthesis in the process development step starts with given initial, final states and is to determine states of major processing units between them. There are in general several mappings from a given state to adjacent states which become starting points of further change of states. This character inherents so-called the combinatorial problems.
---
Task
assigned
_
state
tronsltlon
in bosx
units
Fig. 1. Concept of task assignment in process synthesis in state space.
Statement of synthesis problem under consideration In the scope of present problem of process synthesis, a basic processing scheme is assumed to be given beforehand and it is aimed in this study to synthesize a system to meet processing requirements which are satisfied in the basic processing scheme. The present study limits the synthesis problems in more definitive design stage. Task assignment makes a complete interconnection among the major processing units and gives a corresponding basic flowsheet. The task assignment and its result depend on the design conditions of processing units of which the basic processing scheme consists, and it is generally required to find the optimal design conditions of processing units, on which the optimal processing system should be synthesized. Let the basic processing scheme specify more
Process synthesis by task assignment 2035 , precisely to make it clear what is given. The basic gap in temperatures is the problem of synthesizing processing scheme contains the following items: heat exchange systems, and two kinds of problem exist; one is to find an optimal system and the other is to find practically acceptable systems. These two (1) Processing units of which the basic scheme consists. synthesis problems are defined as follows: for given values of temperature, flow rate, and physical (2) Design or operating conditions for these processing units such as inlet and outlet temperaproperties of a set of streams, find an optimal or a tures and pressures for them. practically acceptable system of heat exchange with auxiliary heating and cooling facilities so as (3) A set of streams which shows indirect connections between the processing units. to have minimum or close to minimum value of predetermined objective function. The problem of (4) A set of streams does not initially satisfy temperature and pressure specifications for task assignment for bridging the gap in pressures is the associated processing units. That is, there the problem of assigning materials transport deexist some temperature and pressure differvices and similar definition of problem can be made ences between the indirectly connected profor given values of pressure, flow rate, and physical cessing units. properties of a set of streams. As one of optimal design problems, the present Thus the problem of synthesizing the optimal problem can be defined as follows: system is to assign suitable units, more generally minimize F 4,(x., n., d.,) tasks, to make up the temperature and pressure differences. Figures 2a and 2b show the difference between a + c 9twt, z:, Tf(ijl(di), &7 &jl I basic processing scheme and a feasible processing d,, T,cii,, Sit and Sti system with complete inter-connection. In Fig. 2b by choice of temperature and pressure differences between the z, = f(x., d.), z: = f:(x:, d:) subject to connected units can be made up by providing x: = &Zi, xj = pumps or compressors for pressure differences, and coolers or heaters for temperature differences. or 0 Whether this equipment is necessary or not deand pend on the design conditions for the associated g, (x., zn, d.) s 0, processing units. It is possible to eliminate some of where x., and z. are input and output state vectors them by changing the design conditions. It is the of the n th unit, respectively. x: and z’,are also input purpose of the present study to determine the optiand output state vectors of the t th assigned task, mal design conditions of these processing units and respectively. d, and d’, are decision vector of the to assign automatically the tasks for making up the k th unit and the t th assigned task, respectively. 6~ temperature and pressure differences. (or 6,j) is a structure parameter which expresses the The problem of task assignment for bridging the S,jZyt
Sit,
~_-__----___--___-__~ I
I
;----
&j
=
1
1
m--__&_-_p+___++__&__&_fl Fig. 2a. A basic processing
scheme.
Fig. 2b. A feasible flowsheet generated by task (T,, Th) assignment.
TOM~OUMEDA,
2036
AKIO
SHINDOand ATSUNOBUICHIKAWA
interconnection between the i th unit and the t th task (or the t th task and the j th unit). f,, f: and g, are vector-valued functions. 4” and 4, are objective functions associated with the n th unit and the t th task, respectively.
METHOD
Decomposition
FOR SOLUTION
method for problem solving
The present problem previously stated can be solved by applying a method of decomposition on the level of available technology as described by Rudd[4], since the problem under consideration has been given by a decomposable form. The problems of task assignment and of designing the basic processing units are subproblems of which the present problem consists. The handling of the decomposable problem can be made by multi-level approach. The solution of design problem of the basic processing units is obtained on the second-level, where some values for coordination variables have been given beforehand. Based on these solutions, the task assignment problems are solvable on the first-level. The problem of coordinating these subproblems can be solved on the third-level. This multi-level approach is illustrated in Fig. 3.
1
Coordmotion
problem
Deslqn problems for
bow
3rd-level
2nd-level
units
terized by decomposing a system into sub-systems in such a way that all the constraints are satisfied during the iterative computations for optimization and physically meaningful results are always obtained. Model-coordination approach [8], projection method[9, lo] and information reversal strategy [l l] belong to the present class of feasible decomposition methods. In applying the feasible decomposition method, the basic processing system has been a priori decomposed into individual processing units by choice of inlet and outlet temperatures and pressures at each processing unit. These variables are defined as coordination variables in multi-level approaches. Some appropriate values are assigned for the coordination variables and those become a set of the input and output conditions for each subproblem. Some or all of the coordination variables are treated as decision variables on the upper-level in the multi-level approach. On the lower-level, the differences in a set of the coordination variables between two units, that is, outlet temperature and pressure of the ith unit and inlet temperature and pressure of the j th unit, are made up by providing heat exchangers (coolers or heaters) and materials transporting devices, respectively. To make the feasible decomposition applicable, it is necessary to satisfy the condition that there remain some degrees of freedom for the subproblems in the lower-level [6]. Furthermore, the analysis of the decomposition problem shows that it is better from the viewpoint of effective computations to treat temperature and pressure at the outlet as dependent variables. Along this line, the feasible decomposition method for optimal process synthesis can be mathematically described as follows: Subproblem PD,: the n th processing unit designon the second level
Task
osslqnment
problems
Ist-
level
Fig. 3. Multi-level approach for solving the integrated problems.
To solve the present problem effectively, a feasible decomposition method [6,7] is considered to be applicable. The feasible decomposition is charac-
tin the present study, the relationships of interconnection between subsystems are limited to single input and single output cases for simplicity. Those are easily extended to multiple imputs and outputs cases.
minimize
4” (x,z, z,z, d:, d.))
by choice of
dn2
subject to
2”* = f;(x;, g.‘(x,‘,
d;, d,))
2,2, d:,
where d.’ is determined
d,S) 5 0
on the third-level,
D. xD. xD, = D., (2 A”,,zf,) = (t”2, P.*)EZ,z.
and
Process synthesis by task assignment Subproblem system-on
SH: synthesis of optimal heat exchange the first level
minimize
4. (x )li,z )li,d iii, &Ii,)
by choice of
Shiiand d iii
subject to
h,Cxk, z&
2037
Step 5. Check whether or not the assumed and computed values of d,’ satisfy a stopping criterion. If satisfied, stop the computations. Otherwise return to Step 6. Step 6. Revise the values of di and return to Step 2.
d$ = 0, & = 1 or 0
For solving the subproblems PD, and the coordination problem, some suitable methods for parameter optimization such as gradient, direct search where fhii,and ghii are vector-valued functions and fhii methods can be applied. For carrying out the synthcorresponds to the equation for calculating heat esis in Step 3, it seems to be important how to solve transfer area. xhi and zli are determined on the the so-called combinatorial problems. There seems second-level. to be appropriate to use some specific characteristics of the problem. For solving the subproblem SP, Subproblem SP: assignment of optimal materials there seems to be no combinatorial problems, since transporting units-on the first-level it is generally sufficient to consider a few alternative cases for selecting materials transporting deminimize 4p (xbi, z;i, a;,, &ii) vices. With respect to the subproblem SH, several synthesis strategies for heat exchange network are by choice of S,, and d& applicable. Those strategies have been reviewed by subject to f,ij(Xii, zbi, d&) = 0, S,, = 1 or 0 Hendry et al. [l]. To solve the subproblem SH, it is particularly desired to use a method which requires g,i, (xfi, zbj,dbii)5 0 less computing time. This consideration aims to rewhere f,, and gpij are vector-valued functions and fpii duce too much computing time for solving the integrated optimization problem. corresponds to performance equations for materials transporting devices. xii and zb are determined on Method for solving the subproblem SH the second-level. Since the input and output states of the basic processing scheme are given, several strategies can be Coordination problem on the third level directly applied to solve the subproblem SEI. Aiming at the reduction of the computing time, the apminimize plication of heuristic rules is useful, though the solution does not claim the optimum. In the present study, the concept of optimal assignment approach which was presented by authors[l2] is used with the combination of a heuristic rule. Firstly, each of by choice of d,’ = {XL,,, x;.,] hot and cold streams is subdivided into smaller heat exchange units with approximately equal heat duty. where all the variables excluding de’ are determined The size of a heat exchange unit is taken as an on the first or second level. approximate value for the greatest common divisor According to the above mathematical description, of heat exchange duties of the hot and cold streams. the following problem-solving procedure based on Secondly, use the following heuristic rule: it is the feasible decomposition method can be derived: preferable to make heat exchange of units in such a way that the hot and cold streams are matched in decreasing order of their inlet temperatures. This Step 1. Assume values of dz, n = 1,2,. . . , N gives the optimal solution for minimizing the total Step 2. Solve optimization problems associated heat exchange area in some cases, such as the heat with the subproblems PD., n = 1,2,. . . , N, exchange between one cold (or hot) stream and more by choice of d.2. x.’ and z: are optimally dethan two hot (or cold) streams, or heat exchange termined in this step. problems with satisfying necessary conditions for Step 3. Solve synthesis problems SH and SP by deriving the rule. After determining a practically choice of &ii, d&, Spij and db+ acceptable heat exchange system, all blocks of Step4. Solve the coordination problem by choice exchange units of the same temperature are merged of d,‘. ghij(xbe &, db,) 5 0
TOMIO UMEDA, AKIO SHINDO and
2038
to form new blocks which give more practical results. More detailed description of the above method has been given elsewhere[l2, 131. Task assignment criteria for solving the subproblem SP Where there is a pressure difference between two adjacent units, some type of material transporting unit must be assigned. The type of units can be classified by the following two items: one is the state of fluid to be handled, and the other is the degree of pressure difference with taking the direction of pressure change (negative or positive difference) into consideration. To present the idea, criteria of task assignment, though this is not complete, are given as follows:
State
of fluid
Gas or non-condensable vapour
Liquid
Difference
ATSUNOBU ICHIKAWA
This may often require the increase of system pressure. It is also preferable not to use compressors as possible and to make efforts to replace the transporting units for gas or non-condensable vapor by those for liquid. These strategies for solving the subproblems are applied under the control of three-level approach on the basis of feasible decomposition method as illustrated in Fig. 3. ILLUSTRATIVE EXAMPLE For illustration of the cess synthesis in process ample is taken up. The shown in Fig. 4 has been
proposed method for prodesign stage, a simple exbasic processing scheme given beforehand. Table 1
I I
I I
Type
L___-_______________--_---_-_____L Fig. Positive, Small Positive, Large
Fans, Blowers
Negative Positive Negative
Valves Pumps Valves
4. A basic
processing
scheme
for
illustrative
example.
Compressors
where the pressure difference is specified by (pi2-piI), that is, the difference between output of the i th unit and input of the j th unit. Direct application of these criteria gives us a large number of units in the basic flowsheet. This is not practical and it is necessary to merge the units, if possible. This merging is usually made on heuristic rules given below, which are derived from economic considerations: It is preferable to reduce the number of units as possible by eliminating units in down streams and adding the correspondent pressure difference to the unit in the upper stream.
shows a typical set of information used to make up the gap in temperature differences by the solution of the subproblem SH. Figure 5 is a matrix presentation of the heuristic rule on heat exchange synthesis. After several iterations, a processing system shown in Fig. 6 is generated without applying merging strategies for heat exchange and material transportation. The result of merging gives us a practically acceptable system which is shown in Fig. 7. CONCLUSION A careful examination of process synthesis activities in process design stage shows that the basic flowsheet can be generated by appropriate task assignments for a given basic processing scheme defined in the first section. Based on the characteristics of the basic processing scheme, a feasible de-
Table 1. Initial information on a basic processing scheme M-----R
Unit name
,---__
R,___-_S,---__S,------_s,------
Flow rate 26,006
(kg/W
Heat duty (X IO”kcal/hr) No. of subdivision (-) Flow rate of subdivided Whr)
Heat capacity (kcahkg “C)
26,006
26,006
3.35
0.68
5
1
- 8.45 12
17,211
16,807
10,086
0.67
0
- 0.69
1
0
1
stream 5,201
0.503
26,006
0.531
2,167
0.534
17,211
0.408
0
0.499
10,086
0.431
Process synthesis by task assignment Hot
streams -Stream
number
234567
_I-;---------__-,
2-l I -I
the subproblems SH and SP, heuristic rules are applied to reduce the computing time, though this does not claim the optimal solutions in general cases. Heuristic rules of merging units give us practical solutions for process synthesis.
I I
2
(6i,)
I
3
I
4
I
Acknowledgments-The authors wish to acknowledge Chiyoda Chemical Engineering and Construction Company for the support of the present study and the permission for its publication. They also express their thanks to Hideki Sugiyama in solving examples.
I I I
I I I II :__-------_____+_ I I ,
5 4-I
NOTATION
c
t Streom
2039
number
Fig. 5. Matrix for heat exchange network. composition method is considered to be applicable to solve the integrated synthesis problem. To solve
blower or compressor D set of decision vectors d decision vector function of system f vector-valued g vector-valued constraint A4 mixer
(PI Fig. 6. Preliminary
flowsheet
synthesized
(before
merging).
(PI
Fig. 7. Flow sheet synthesized (after merging).
CES VOL.
29. NO.
lC-C
equation
TOMIO UMEDA, AKIO SHINDO and A~su~oeu
2040
p p
pump
pressure R reactor S separator T task t temperature V valve x input state vector z output state vector Greek symbols
6 4
structure parameter objective function
Superscripts T transpose of vector i the i th level in multi-level
’
approach (i - I,2 and 3) state vector of assigned unit or task
Subscripts h concerned i, j, n
P
with heat exchange number of subsystem or unit concerned with pressure
ICHIKAWA REFERENCES
111HendryJ.E.,RuddD.F.andSeaderJ.D.,A.I.Ch.E.JI 1973 19 1. [21 Ichikawa A. Kagaku Kogaku Chem. Engng, Jupan 1972 36 1053. r31 Rudd D. F., Powers J. G. and Siirola J. J., Process Synthesis, Prentice-Hall Englewood Cliffs, New Jersey 1973. r41 RuddD. F., A.1.Ch.E. JI 1968 14 343. VI McGalliard R. L. and Westerberg A. W., Paper presented at A.I.Ch.E. Meeting, Dallas, Texas, gebriary 1972. [61 Tazaki E., Shindo A. and Umeda, T., Automaticn 1972 8 543. 171 Umeda T., Shindo A. and Tazaki E., Ind. Engng. Chem. Proc. Des. Dev. 1972 11 I. [81 SchoelTler J. D. In Optimization Methods for LargeScale Systems (Edited by D. A. Wismor) p. 1. McGraw-Hill, New York 1971. [91 Geoffrion A. M., Managt. Sci. 1970 16 11. [lOI Geoffrion A. M.. Mana~t. Sci. 1970 16 652. [Ill Westerberg A. w., In Dkomposition of Large Scale Problems (Edited by D. M. Hemmelblau) p. 379. North-Holland Amsterdam 1973. [121 Kobayashi S., Umeda T. and Ichikawa A., Chem. Engng Sci. 1971 26 1367. r131 Nishida N. Kobayashi S. and Ichikawa A., Chem. Engng Sci. 1971 26 1841.