Ind. Eng. Chem. Res. 2000, 39, 4315-4319
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A Hierarchical Optimization Method for Reaction Path Synthesis Mingheng Li, Shanying Hu, Yourun Li,* and Jingzhu Shen Department of Chemical Engineering, Tsinghua University, Beijing, 100084 China
In this paper, a hierarchical optimization procedure for reaction path synthesis is presented in which the overall reaction is not necessarily desirable with respect to the thermodynamics or mechanism of reaction. According to the specified raw materials, desired product, and probable byproducts, the overall reactions are determined through the generation method of simple stoichiometric reactions and evaluated by economic potential and other criteria such as thermodynamic feasibility and kinetic desirability. For an overall reaction that is unachievable thermodynamically or kinetically while potentially desirable with respect to profit, a novel procedure for overall reaction decomposition by adding intermediate chemicals is proposed to realize it by several thermodynamically feasible division reactions. When this method is used, the optimal solution of reaction path synthesis can be either an overall reaction or several division reactions. A case study is described to illustrate the proposed procedure. Introduction Increased awareness of the environmental impacts of chemical processes leads designers to realize pollution prevention and waste minimization with systematic methods. As one of the main areas of process synthesis, reaction path synthesis has become a crucial preliminary tool in identifying the most desirable reaction routes. Because of the large scale and inherent difficulties of this problem, any attempt to take all of its engineering characteristics into consideration is impractical. In this sense, simplification and accentuation is a desirable way to reduce the size of the problem to a manageable one. Stephanopoulos and co-workers put the emphasis of path synthesis on the thermodynamic properties of reactions.1-3 Rotstein et al. revealed several invariable algebraic properties shared by chemical reactions in the ∆G (Gibbs free energy change) vs T (temperature) space, based on which a screening tool is developed to create and identify alternative reaction paths. Later, Fornari et al. extended this procedure to a reactive system having 2 degrees of freedom, and Fornari and Stephanopoulos take into consideration other factors such as economics and processing safety when the reaction path is synthesized. The work of Crabtree and El-Halwagi is conducted with the property of reaction under thermodynamic equilibrium in a CSTR. A novel concept of synthesizing environmentally acceptable reactions (EARs) was introduced following which a systematic approach is deduced to formulate the synthesis problem as a mixedinteger nonlinear optimization program. The thermodynamically feasible, environmentally acceptable overall reaction that can yield the maximum economic potential using the same reactor will be identified by solving this optimization problem.4 Subsequently, Buxton et al. proposed a systematic method to decompose the problem of reaction path synthesis into several steps.5 A guided enumeration procedure was put forward to screen simple stoichiometries between raw materials and the * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: 86 10 62784572. Fax: 86 10 62770304.
desired product. Promising alternative reactions are selected from them using the methodology for environmental impact minimization (MEIM).6 In this paper, a hierarchical method is put forward to decompose the large problem of reaction path synthesis into two levels, each of which can be solved separately as optimization problems. The first level is to identify all the meaningful overall reactions with the generation method of simple stoichiometric reactions (SSRs).7 It has been proved that the number of overall reactions determined by this guided enumeration procedure is very limited. To enlarge the hunting zone of reaction routes, these overall reactions are not necessarily feasible thermodynamically or desirable in terms of reaction kinetics. A novel optimization method for overall reaction decomposition is presented at the second level to replace an unachievable overall reaction with a series of thermodynamically feasible division reactions. From an engineering standpoint, all the reactions should be validated by experiment. Problem Statement The reaction path synthesis problem in this paper can be stated as follows: Given a desired product, determine the raw materials and the corresponding reaction pathways for the production of this product that can yield the maximum economic potential while meeting thermodynamic and other constraints. A Hierarchical Optimization Method The hierarchical optimization procedure for reaction path synthesis, as outlined in the previous passages, can be divided into two levels. The task of the first level is conducted with the identification and evaluation of all the meaningful overall reactions. Although the number of probable reactions is infinite for a reactive system whose reactants and products are all given, it is found that those reactions whose reactive system has only 1 degree of freedom is very limited, and nearly all the reactions used in industry can be categorized into this kind of reaction. After a rich set of prospective raw materials
10.1021/ie9904596 CCC: $19.00 © 2000 American Chemical Society Published on Web 09/28/2000
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and probable byproducts are selected on the basis of the atom and functional group constituents of the desired product, the generation method of SSRs will be executed to screen these overall reactions. Subsequently, all the reactions are ranked according to their economic potential. Thermodynamics and kinetics are also evaluated to divide them into two categoriessfeasible overall reactions, which are achievable, and infeasible ones, which are not compassable but may be accomplished by several feasible division reactions. Among all the feasible overall reactions, the reaction that has the largest profit is selected as the reference reaction. The overall reactions whose economic potential is smaller than that of the reference reaction, whether it is feasible or not, will be spurned. The other overall reactions, which are infeasible while still having larger economic potential, should be decomposed at the second level by introducing intermediate chemicals. If feasible division reactions can be obtained, they may be better than this reference overall reaction. Therefore, the optimal solution for the synthesis problem can be either an overall reaction or several division reactions, thus extending the methods proposed by Crabtree and Buxton. Determination of Overall Reactions The determination of the optimal overall reaction can be formulated as an optimization problem.4 To identify all the other probable meaningful reactions besides the optimal reaction, we use a guided enumeration method in this paper. Let us consider a reaction scheme with s potential species, including candidate raw materials, byproducts, and the desired product, whose rank of the atomic matrix is r. All the simple stoichiometric coefficients of the overall reactions can be founded by enumerating the full rank matrix Er×r with the following expression,
υr×s ) (E1E2...Er)
[
-Er×r-1Er×(s-r) U
]
(1)
where Er×r and Er×(s-r) are two parts of Er×s, which represents the atomic matrix of the reactive system, U a unit matrix, and E1, E2, ..., Er row (or column) translation matrices, and each row vector of υr×s represents the stoichiometric coefficients of all the species in an overall reaction.7 It should be noted that all the reactions whose reactive system has 1 degree of freedom would be generated by this method. Therefore, those reactions whose stoichiometric coefficient of the desired product is zero should be weeded out because there is no production of the desired product. Evaluation of Reactions Once an overall reaction is generated, its evaluation can be made using several criteria such as economic potential, thermodynamics, and mechanism of reaction. Economic Potential. The difference between the value of the products and the cost of the raw materials, or gross profit, is used as the economic objective of this problem. It is invariable once a reaction is determined, which is particularly useful in early economic screening. Thermodynamics. When stoichiometric coefficients of all the chemicals in an overall reaction are determined, its Gibbs free energy change and conversion ratio of raw materials to products at a specified temperature
can be easily determined. An overall reaction whose Gibbs free change is always larger than zero in the given temperature range is considered infeasible. Mechanism. Not all the reactions that satisfy thermodynamic constraints also meet the need for reaction rate and selectivity. But in most cases, the mechanism of a reaction can only be explored by laboratory investigation. A solution is considered desirable only after it has been validated by experiment. Environmental considerations can also be incorporated into the synthesis problem using MEIM6 after stoichiometries are generated. Decomposition of Infeasible Overall Reactions If an overall reaction is not achievable thermodynamically and/or undesirable with respect to the mechanism of reaction, it may be decomposed into several thermodynamically feasible division reactions that are accomplished under different operating conditions. The sum of the division reactions is equivalent to the overall reaction, but intermediate chemicals will appear in these division reactions. Generally, those intermediates are chosen by experience according to the constituents of chemicals involved in the overall reaction. Let m represent the number of constituents in an overall reaction determined at the first level, a the number of additive intermediates, and r the rank of the reactive system after intermediate constituents are added. The freedom of the reaction scheme is (m + a r). Therefore, there are at most total (m + a - r) independent division reactions, whose sum can be equal to the overall reaction. Coefficient matrix µ(m+a)×(m+a-r) is introduced to represent the stoichiometric coefficients of all the (m + a) constituents in the (m + a - r) division reactions. Mass conservation equations will be composed of two sections: (i) The sum of division reactions should equal the overall reaction, or
µ(m+a)×(m+a-r)e ) υm+a
(2)
where e ) (1,1,...,1)T. (ii) And each division reaction should satisfy the atom balance equation, or
Er×(m+a)µ(m+a)×(m+a-r) ) 0
(3)
It is worth pointing out that there are redundant equations in eqs 2 and 3 because the overall reaction has already satisfied the atom balance equation. By deduction we can conclude that eq 2 can be simplified as
µ(m+a-r)×(m+a-r)e ) υm+a-r
(4)
where the matrix µ(m+a-r)×(m+a-r) is composed of the last (m + a - r) row vectors of µ(m+a)×(m+a-r) and υ(m+a-r) the last (m + a - r) elements of υ(m+a). An initialized coefficient matrix, φ(m+a)×(m+a-r), that has already satisfied eq 3 (for specified method, see Appendix 1) is introduced and suppose
µ(m+a)×(m+a-r) ) φ(m+a)×(m+a-r)η(m+a-r)×(m+a-r) (5) where η(m+a-r)×(m+a-r) is a (m + a - r) × (m + a - r) matrix.
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It can be proved that eq 5 is equivalent to eq 3 (see Appendix 2) if the column vector in matrix φ(m+a)×(m+a-r) is linearly independent. In addition to the stoichiometric constraints, the thermodynamic constraints of each division reaction are also considered, which limit the maximum extent of the reactions that can occur between potential reactants. The chemical equilibrium constant of each division reaction can be related to the Gibbs free energy change by
( )
Kj ) exp
-∆Gj RTj
j ) 1, 2, ..., r
(6)
where j is the index of the division reaction and Kj and ∆Gj are the chemical equilibrium constant and Gibbs free energy change of reaction j, respectively. The Gibbs free energy change can be calculated form the Gibbs energy formation of each of the species by
where υi is the stoichiometric coefficient of species i in the overall reaction. Combining eq 8 with eq 11, we can get m+a
niυ ∏ i)1
i
m+a
µij ∏j Vj ∑ i)1
µij∆Gfi(Tj) ∑ i)1
j ) 1, 2, ..., r
(
∑j
)
µij∆Gfi(Tj) ∑ i)1 RTj
(12)
When Vj and Tj are the same for all reactions, eq 12 can be reduced to the overall reaction, and the conversion ratio of the raw materials can be calculated from this equation because it is related to nj. For division reactions, we can also get an equivalent conversion ratio by this equation, which is called the pseudo-conversion ratio. When the approach proposed by El-Hawagi4 is used, products and reactants can be defined as
m+a
∆Gj )
) exp
m+a
-
(7)
(2 × Iij - 1)(µij - ξ) g0
(13)
(1 - 2 × IIij)(µij + ξ) g0
(14)
and where µij is the stoichiometric coefficient of species i in reaction j and Tj the temperature of reaction j. Combining eq 6 with eq 7, we can get
(
)
N
-
∏j Kj ) exp ∑j
µij∆Gfi(Tj) ∑ i)1 RTj
(8)
respectively, where ξ is an arbitrary number. Iij and IIij are binary integer variables. The definition of this kind of binary integer variable can be used to limit the maximum number of reactants, products, and intermediates in each division reaction and in the overall reaction.
On the other hand, the equilibrium constant is defined as the ratio of the product of equilibrium concentrations of products to that of the reactants m+a
Kj )
∏ i)1
j ) 1, 2, ..., r
cijµij,
(9)
where cij is the equilibrium concentration of species i in reaction j. Or m+a
Kj )
nijµ ∏ i)1 m+a
Vj
j ) 1, 2, ..., r
,
(10)
where nij is the equilibrium molecule of species i in reaction j and Vj is the volume of reactor j. To simplify eq 10, we suppose (i) each intermediate chemical only appears in at most two reactions, as a reactant and product, respectively, and (ii) the molar number of intermediate chemicals in each division reaction is the same. After such suppositions, eq 10 yields m+a
∏j Kj )
j ) 1, 2, ..., r
(15)
∑i IIi,j e NR,
j ) 1, 2, ..., r
(16)
∑j Iim,j e 1,
im ∈Intermediates
(17)
∑j IIim,j e 1,
im ∈Intermediates
(18)
ij
µij ∑ i)1
∏j ∏ i)1 (
∑i Ii,j e NP,
m+a
nijµij)
m+a
niυ ∏ i)1
i
)
m+a
µij µij ∏j Vj ∑ ∏j Vj ∑ i)1 i)1
(11)
where NP and NR are the maximum number of products and reactants in each reaction, respectively. Because each division reaction should be feasible thermodynamically, the Gibbs free energy should satisfy
∆Gj(Tj) e 0,
j ) 1, 2, ..., r
(19)
and
TL e Tj e TU,
j ) 1, 2, ..., r
(20)
where TL and TU are the lower and upper bounds of the temperature of each division reaction. Maximize the pseudo-conversion ratio of raw materials to the desired product using constraints (4)∼(5) and (12)∼(20); the problem of overall reaction decomposition can be formulated as a mixed-integer nonlinear optimization program. If there is a solution to the problem listed above, or the maximum pseudo-conversion ratio is larger than the actual maximum conversion ratio of the single overall reaction, we consider that the overall
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Table 1. Overall Reactions Identified by the Generation Method of SSRsa no.
1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0 0 0 0 0 0 0 0 0 0 0 0 -0.5 -0.75 -1 -1 -1
0 0 -1 -1 -2 -1 0 -1.5 -1 0 -1 0 -0.5 0 0 0 0
3
4
-1 2 -1 1.5 0 1 0 0.5 1 0 0 0 -1 0 0.5 0 0 0 -1 0 0 0 -1 0 0 0 -0.25 0 0 0 0 0 0 -0.5
5
6
7
0 -0.5 0 -0.5 0 0 0 -0.5 -0.5 -0.5 -1 -2 -0.5 -0.5 0 -0.5 -0.5
0 0 0 0 0 -0.5 -1 0 -0.25 -0.75 0 0 0 0 0 0.25 0
8
profit 9 (/mol)
-1 0 1 0 0 1 -1 0 1 0 0 1 -1 0 1 -1 1 1 -1 2 1 0 0 1 0 0.5 1 0 1.5 1 1 0 1 3 0 1 0 0 1 0 0 1 -1 0 1 0 -0.5 1 0 0 1
0.251 0.208 0.178 0.135 0.105 0.103 0.100 0.099 0.098 0.095 0.093 0.080 0.074 0.062 0.055 0.051 0.013
a
1, acetylene; 2, ethylene; 3, ethane; 4, hydrogen; 5, chlorine; 6, oxygen; 7, hydrogen chloride; 8, water; 9, vinyl chloride.
reaction is decomposed. Otherwise, it is unachievable with these additive intermediate chemicals. When the overall reaction is decomposed, as many as possible intermediates should be selected to enlarge the hunting zone of optimization. However, the calculation effort will increase significantly with the size of the matrix η(m+a-r)×(m+a-r). From eq 5 it can be concluded that the corresponding column vector in φ(m+a)×(m+a-r) will be zero if one column vector of η(m+a-r)×(m+a-r) is set to zero. Therefore, we can adjust the calculation effort of optimization by setting zero column vectors in η(m+a-r)×(m+a-r). Case Study Take the case of the production of vinyl chloride. Possible constituents chosen include acetylene, ethylene, ethane, hydrogen, chlorine, oxygen, hydrogen chloride, and water. The atomic matrix of this problem is
[
2 2 EE×(s+1) ) 0 0
2 4 0 0
2 6 0 0
0 2 0 0
0 0 0 2
0 0 2 0
0 1 0 1
0 2 1 0
2 3 0 1
]
C H O Cl
Table 2. Data Involved in Overall Reaction Decomposition parameters
value
optimization variables
value
ξ TL TU NR NP
1 × 10-6 250 K 800 K 4 4
∆G1 ∆G2 pseudo reaction ratio
-90.48 kJ/mol -28.16 kJ/mol 0.42
If dichloroethane and hydrogen chloride are introduced as intermediate chemicals, the atom matrix of this scheme can be written as
[
2 4 Em+a ) 0 0
0 0 0 2
0 0 2 0
2 3 0 1
0 2 1 0
2 4 0 2
0 1 0 1
]
C H O Cl
(23)
Using the procedure proposed, we can get a coefficient matrix whose column vectors are linearly independent while satisfying eq 5:
[
-2 -1 -0.5 2 1 0 0 0 0 1 0 φ(m+a)×(m+a-r) ) -1 -1 0 -1 -1 0 1 0 0 1
]
T
(24)
When the data from ref 8 is used and the the proposed method is followed, the optimization problem at stage two is solved by LINGO mathematical programming software in 105 CPU s to yield a local optimum solution:
[ ]
-0.5 -0.5 0 µ(m+a)×(m+a-r) ) 0 0 0.5 0
-0.5 0 -0.25 1 0.5 -0.5 0
0 0 0 0 0 0 0
C2H4 Cl2 O2 C2H3Cl H2O C2H4Cl2 HCl
(25)
Some important data involved in this program are listed in Table 2. The corresponding division reactions are
1 1 1 ethylene + chlorine w dichloroethane (26) 2 2 2 (21)
According to the generation method of SSRs, 17 overall reactions with the production of vinyl chloride are generated at the first level. Their stoichiometries and gross profit are shown in Table 1. On the basis of the thermodynamics and kinetics already known, reaction (11) is selected as the reference reaction. All the reactions whose profit is larger than that of the reference reaction should be decomposed at the second level. Let us cite the example of reaction (9) to illustrate the decomposition method of the overall reaction. Reaction (9) is expressed as
1 1 1 ethylene + oxygen + dichloroethane w 2 4 2 1 vinyl chloride + water (27) 2 The overall reaction decomposition program is executed repeatedly for the other nine reactions selected. Finally, reaction (9) with these two division reactions is selected as the optimum solution. Division reactions should be validated by experiment before they can be used as the final solution. However, this mathematical method proposed can be very useful as a preliminary screening tool to identify potential reaction routes. Conclusion
1 1 ethylene + chlorine + oxygen w 2 4 1 vinyl chloride + water (22) 2
A hierarchical optimization method is put forward in this paper to decompose the large problem of reaction path synthesis into two stages. A prominent feature of this procedure is that overall reactions are synthesized
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4319
before their feasibility is judged, which broadens the search space of reaction routes. Acknowledgment Financial support from the National Science Foundation of China (29836140) is gratefully acknowledged.
[
c11 c η) 12 l c1(m+a-r) Nomenclature
Appendix 1 If r is the rank of the atom matrix EE×(m+a), there must be r row vectors which are linear independent each other. So we can write atom balance equation
EE×(m+a)µm+a ) 0
(A1)
as
[
]
µ )0 [Er×r Er×(m+a-r)] µr m+a-r
(A2)
where Er×r is a full rank matrix. Then,
[ ][
]
µr -Er×r-1Er×(m+a-r) ) µm+a-r µm+a-r U
(A3)
where U is a unit matrix. Finally, we will get a coefficient matrix:
[
-Er×r-1Em+a-r U
φ(m+a)×(m+a-r) )
]
(A4)
As for Em+aµ(m+a)×(m+a-r) ) 0, suppose column vector ζi represents column vector i in matrix µ(m+a)×(m+a-r), we will get
Em+a(ζ1, ζ2, ..., ζ(m+a-r)) ) 0
c21 c22 l c2(m+a-r)
... ... ··· ...
c(m+a-r)1 c(m+a-r)2 l c(m+a-r)(m+a-r)
]
(A8)
(A9)
a ) number of additive chemicals e ) (1, 1, ..., 1)T E ) row (or column) translation matrix im ) index of intermediate chemicals I ) binary integer product flag II ) binary integer reactant flag j ) index of division reaction K ) chemical equilibrium constant m ) number of chemicals in the overall reaction r ) rank of atomic matrix s ) number of species other than the desired product T ) temperature (K) TL ) lower bound of the temperature (K) TU ) upper bound of the temperature (K) U ) Unit matrix V ) volume of the reactor (m3) E ) atomic matrix µ ) stoichiometric coefficient matrix υ ) stoichiometric coefficient vector of overall reaction ξ ) small positive number φ ) initialized stoichiometric coefficient matrix ∆G ) Gibbs energy of reaction (kJ/mol)
Literature Cited
Appendix 2
(A5)
Consider linear equations Em+ax(m+a) ) 0, and let β1, β2, ..., β(m+a-r) be the linearly independent solution vectors, according to the theorem of the structure of the linear system of equations, we can write
ζ1 ) c11β1 + c21β2 + ... + c(m+a-r)1βm+a-r (A6a) ζ2 ) c12β1 + c22β2 + ... + c(m+a-r)2βm+a-r (A6b) l ζm+a-r ) c1(m+a-r)β1 + c2(m+a-r)β2 + ... + c(m+a-r)×(m+a-r)βm+a-r (A6c) where c11, c12, ..., c(m+a-r)×(m+a-r) are constants. Writing eqs A6a-A6c in matrix form, we get
µ(m+a)×(m+a-r) ) φ(m+a)×(m+a-r)η(m+a-r)×(m+a-r) (A7) where
φ(m+a)×(m+a-r) ) (β1, β2, ..., βm+a-r)
(1) Rotstein, E.; Resasco, D.; Stephanopoulos, G. Studies on the Synthesis of Chemical Reaction PathssI. Reaction Characteristics in the (∆G, T) Space and a Primitive Synthesis Procedure. Chem. Eng. Sci. 1982, 37 (9), 1337-1352. (2) Fornari, T.; Rotstein, E.; Stephanopoulos, G. Studies on the Synthesis of Chemical Reaction PathssII. Reaction Schemes with Two Degrees of Freedom. Chem. Eng. Sci. 1989, 44 (7), 15691579. (3) Fornari, T.; Stephanopoulos, G. Synthesis of Chemical Reaction Path Synthesis: Economic and Specification Constraint. Chem. Eng. Commun. 1994, 129, 159-182 (4) Crabtree, E. W.; El-Halwagi, M. M. Synthesis of Environmentally Acceptable Reactions. Pollut. Prevent. Process Product Modifications AIChE Symp. 1994, 117-127. (5) Buxton, A.; Livingston, A. G.; Pistikopoulos, E. N. Reaction Path Synthesis for Environmental Impact Minimization. Comput. Chem. Eng. 1997, 21, 959-964. (6) Pistikopoulos, E. N.; Stefanis, S. K.; Livingston, A. G. A Methodology for Minimum Environmental Impact Analysis. Pollut. Prevent. Process Product Modifications AIChE Symp. 1994, 139150. (7) Mingheng, L.; Shanying, H.; Yourun, L.; Jingzhu, S. Reaction Path Synthesis for Mass Closed-Cycle System. 7th International Conference on Process Systems Engineering, in press. (8) Chemical Engineering Handbook, 2nd ed.; Chemical Industry Press: Beijing, 1996.
Received for review June 25, 1999 Revised manuscript received May 24, 2000 Accepted July 22, 2000 IE9904596
