Computers and Chemical Engineering 25 (2001) 1199–1217 www.elsevier.com/locate/compchemeng

A systematic method for reaction invariants and mole balances for complex chemistries

Sagar B. Gadewar, Michael F. Doherty *, Michael F. Malone

Department of Chemical Engineering, Uni6ersity of Massachusetts, Amherst, MA 01003-3110, USA

Received 8 May 2000; received in revised form 16 February 2001; accepted 19 February 2001

Abstract

Reaction invariants are quantities that take the same values before, during and after a reaction. We identify a set of reaction invariants that are linear transformations of the species mole numbers. The material balances for chemically reacting mixtures correspond exactly to equating these reaction invariants before and after reaction has taken place. We present a systematic method for determining these reaction invariants from any postulated set of chemical reactions. The strategy presented not only helps in checking the consistency of experimental data, and the reaction chemistry but also greatly simpliﬁes the task of writing material balances for complex reaction chemistries. For examples where the reaction chemistry is not known, we employ Aris and Mah’s (Ind. Eng. Chem. Fundam. 2 (1963) 90) classic method to determine a candidate set of chemical equations. Application of reaction invariants in validating proposed reaction chemistry is discussed. One of the important applications of this method is the automation of mole balances in the conceptual design of chemical processes. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Conceptual design; Reaction invariants; Mole balances; Complex reactions

Nomenclature

A atomic matrix A* echelon form of the atomic matrix obtained by row operations on A B nonsquare matrix of dimension (R, c) c total number of reacting and inert components D square matrix of dimension (R, R) e number of elements in the reaction species F 0 column vector of inlet molar ﬂow rates of components

EP2 economic potential for Level 2 decisions 0 F i inlet molar ﬂow rate of component i 0 F lr inlet molar ﬂow rate of the limiting reactant T F lr total molar ﬂow rate of limiting reactant at the reactor inlet F T column vector of total molar ﬂow rates at the reactor inlet 0 F Ref column vector of the R reference inlet molar ﬂow rates g total number of components in the gaseous purge G molar ﬂow rate of the gaseous purge stream I identity matrix of dimension (R, R) l total number of components in the liquid purge

* Corresponding author. Present address: 3323 Engineering II, University of California, Santa Barbara, CA 93106, USA. Tel.: +1-805-8935309; fax: +1-805-8934731. E-mail address: [email protected] (M.F. Doherty).

0098-1354/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0098-1354(01)00695-0 1200 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217

L molar ﬂow rate of the liquid purge stream

Mi molar ratio of component i w.r.t. the limiting reactant M column vector of the molar feed ratios n column vector of c outlet mole numbers n 0 column vector of the c inlet mole numbers nRef column vector of the outlet mole numbers for the R reference components 0 n Ref column vector of the inlet mole numbers for the R reference components ni number of moles of component i at the outlet 0 n i number of moles of component i at the inlet Ni mole number transform of component i 0 N i mole number transform of component i based on inlet mole numbers P column vector of outlet molar ﬂow rates of components O zero matrix

Pi outlet molar ﬂow rate of component i

PRef column vector of the R reference outlet molar ﬂow rates R number of independent reactions R rank of the atomic matrix Rg column vector of ﬂow rates in the gas recycle

Rl column vector of ﬂow rates in the liquid recycle RG molar ﬂow rate of gas recycle RL molar ﬂow rate of liquid recycle Rmax maximum number of independent reactions

Si selectivity to component i V nonsquare matrix of dimension (c, R) of stoichiometric coefﬁcients for the c components in the R reactions

VRef square matrix of dimension (R, R) of the stoichiometric coefﬁcients for the R reference compo- nents in the R reactions VI, VII stoichiometric coefﬁcient matrix x column vector of c mole fractions in the liquid purge stream xR column vector of c mole fractions in the liquid recycle stream xi mole fraction of component i in the liquid purge stream y column vector of c mole fractions in the gaseous purge stream yR column vector of c mole fractions in the gaseous recycle stream yi mole fraction of component i in the gaseous purge stream 0 column vector of zeroes

Greek letters h r dependence constant for linear combinations column vector of the R molar extents of reaction m r molar extent of reaction r w i,r stoichiometric coefﬁcient of component i in reaction r § T i row vector of the stoichiometric coefﬁcients for component i in each reaction

Subscripts and superscripts 0 initial −1 inverse of matrix i components r reactions Ref reference components T total S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1201

1. Introduction For a continuous reactor, the number of moles of component i in the outlet can be represented in terms Three important areas of investigation of a reaction of the inlet moles as system are, stoichiometry, kinetics and mechanism. 0 § Tm Frequently, it is assumed that the stoichiometry of a ni =n i + i , i=1, … , c, (1) reaction system is known and, therefore, the main § T emphasis is laid on determining the mechanism and where i is a row vector of the stoichiometric coefﬁ- cients of component i in R reactions and is a kinetics. However, getting the stoichiometry correct is column vector of the extents of reaction. For simple equally important and can also be difﬁcult. Smith reaction chemistries where each reaction contains at and Missen (1979) deﬁne chemical stoichiometry as least one species that is unique only to that reaction, the constraints placed on the composition of a closed it is easy to write each extent in terms of a single chemical system by the necessity of conserving the component. For example, amount of each elemental or atomic species in any physiochemical change of state occurring within the 0 w m w m w m ni =n i + i1 1 + i2 2 +···+ iR R, (2) system. Alternatively, Gibbs’ rule of stoichiometry must be satisﬁed by a chemical system (Aris & Mah, w and if component i occurs only in reaction 1, i2 = 1963). A classic approach for determining the reac- w w m i3 =···= iR =0. Therefore, 1 can be expressed solely tion chemistry for a reaction system on the basis of in terms of moles of i. Similarly, the extent of each limited experimental observations was published by reaction can be found, and mole balances can be Aris and Mah (1963). This approach also helps in written by expressing these extents in terms of other determining the stoichiometric degrees of freedom. components, and eliminating the extents. However, no The information needed to implement the method is systematic methods are available for writing the mate- an accurate knowledge of all the chemical species rial balances for reaction systems of arbitrary com- present in the reaction system. Some of the earliest plexity where the extent of each reaction cannot be work in this subject was published by Jouguet (1921) expressed in terms of mole numbers of a single com- and Defay (1931) (implications of the presence of iso- ponent. Standard textbooks provide a good in- mers on the method of Aris and Mah were troduction for developing intuition and skills for writ- discussed by Whitwell & Dartt, 1973). This treat- ing mole balances (Nauman, 1987; Himmelblau, ment was extended by Happel (1986) to incorporate 1996). Rosen (1962) published an iterative pro- transient species which may not be observed necessar- cedure for solving material balances over a reactor, ily in the inlet or outlet of the reaction system. A however, the technique had problems with conver- reaction mechanism involves detailed reactions gence. Sood and Reklaitis (1979), Sood, Reklaitis, between reactants, intermediates that may or may not and Woods (1979) proposed a procedure which re- be transient, and the ﬁnal products in a reaction mix- quired no iterations for solving material balances for ture. The reaction chemistry is determined from a ﬂowsheets. Their procedure, however, needed speciﬁ- proposed mechanism by eliminating the transient cation of the extents of reaction for solving balances around a reactor. Schneider and Reklaitis species in the overall scheme. The dependent reactions (1975) were among the earliest to consider the rela- in the reaction chemistry generated are then elimi- tionship between mole balances and element nated. The mechanistic approach is used widely and balances for steady-state chemically reacting systems. is frequently successful. Sometimes, the reaction For simple reaction stoichiometries (e.g. involving chemistry generated from a reaction mechanism one, two or three reactions) it is often possible to might contain less than the maximum number of in- write the material balances based on intuition and dependent reactions. However, Aris and Mah’s experience. This task, however, becomes much more method always generates a set containing the maxi- difﬁcult for complex reaction chemistries with many mum number of independent reactions. Aris and reactions. It is, therefore, useful to devise a systematic Mah’s approach, therefore, should be considered method to determine the material balances for reac- complimentary to the mechanistic approach. Bonvin tion systems, supposing that the reaction chemistry is and Rippin (1990), and Amrhein, Srinivasan, and known. Bonvin (1999), used a method called ‘Target Factor To facilitate the numerical treatment of complex Analysis’ for determination and validation of reaction reaction systems, many authors have proposed linear stoichiometry based on experimentally measured data. and nonlinear transformation of variables (e.g. Waller Their method is also applicable to systems where the &Ma¨kila¨, 1981). In their review, Waller and Ma¨kila¨ molecular formulae of some of the species is not describe various ways to decompose a state vector known. (consisting of all variables needed to describe the sys- 1202 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217

w w w X tem) into variants and invariants. Denn and Shinnar 1,rA1 + 2,rA2 +···+ c,rAc 0, r=1,2,…, R, (3) (1987) used invariants for checking mass balances for where A are the reacting species and w is the stoichio- coal gasiﬁcation reactors. These invariants depend on i i,r metric coefﬁcient of component i in reaction r.The the molar feed ratios to the reactor, but are indepen- convention used is w \0 if component i is a product, dent of the type of gasiﬁer used. Reaction invariants are i,r w B0 if it is a reactant and w =0 if component i is an variables that take the same values before, during and i,r i,r inert. after the reaction. They are independent of the extent of In the absence of knowledge about the reactions reaction, although they may change with ﬂow rate and occurring in the reaction system, it is possible to use other parameters. Asbjørnsen and co-workers Aris and Mah’s (1963) method to determine feasible (Asbjørnsen & Fjeld, 1970; Asbjørnsen, 1972; Fjeld, candidates for the reaction chemistry based on a knowl- Asbjørnsen, & A,stro¨m, 1974) demonstrated the use of edge of the species present in the reaction system. This reaction invariants to reduce the dimensionality of the method always gives the maximum number of indepen- differential equations describing the process dynamics dent chemical equations for the system. Also, based on of continuous stirred tank reactors. Srinivasan, Am- the starting point we get a different set of chemical rhein, and Bonvin (1998) extended this methodology to equations. We note, however, that each set of chemical include ﬂow invariants for such systems. Most of the equations can be obtained from any other by linear literature dealing with reaction invariants is related to combinations. The material balances derived from all the process control aspects of a system (Waller & sets containing the maximum number of independent Ma¨kila¨, 1981). Our aim is to employ them for process chemical equations are equivalent (i.e., give the same design. We use linear transformations to determine the answers), in spite of the fact that some sets of chemical reaction invariants which are easy to understand and equations do not correspond to reasonable chemical straightforward to implement. This transformation can pathways! be used effectively to go from limited experimental The reaction invariants for the system are given by observations to setting up molar balances for a reactor the transformed mole numbers (Appendix A). Trans- system at steady state. Also, this methodology is useful formed mole numbers take the same value before, in planning experiments as it provides an estimate of during, and after reaction. These are given as degrees of freedom necessary for checking data consis- tency. The use of this systematic methodology in pro- 0 0 7 T −1 0 N i =n i − i (VRef) n Ref, i=1,…, c−R, (4) cess design applications is demonstrated after the basic 7 T −1 method is developed. Ni =ni − i (VRef) nRef, i=1, … , c−R, (5)

0 where n i is the number of moles of component i at the inlet, ni is the number of moles of component i at the 2. Reaction invariants § T outlet, i is the row vector of dimension c of the stoichiometric coefﬁcients of component i in all of the Consider a reaction system consisting of c compo- 0 R reactions. N i are the reaction invariants based on the nents undergoing R independent chemical reactions. A inlet molar ﬂow rates and Ni are the reaction invariants block diagram for such a system is shown in Fig. 1. based on the outlet molar ﬂow rates. When we equate This process block can contain any complex combina- these reaction invariants, the mole balances for the tion of unit operations. The inlet to the process is reacting system are simply represented by a c-dimensional column vector of 0 0 inlet molar ﬂow rate of species, n ; the outlet of the N i =Ni, i=1, … , c−R. (6) process is represented by a vector of outlet ﬂow rate of We note that the number of mole balances for R species, n.TheR independent chemical reactions are independent reactions is c R. written as − The transformed mole numbers represent a linear transformation of the number of moles of species to give conservation relationships which depend only on the reaction chemistry and these linear transformations correspond exactly to the material balances.

3. Properties of mole balances

Property 1. Mole balances for all sets, each containing the maximum number of independent reactions, are Fig. 1. Block diagram of a reaction system. equi6alent. S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1203

R−1 % w m h m We consider a reaction chemistry consisting of inde- = i, r( r + r R). (8) pendent reactions. Using a given set of independent r=1 reactions, we can generate a new set by linear combina- m h m m% Redeﬁning ( r + r R)= r, to be the extent of rth tions. The maximum number of independent reactions reaction, we have expressed the composition is c−R, where R is the rank of the atomic matrix changes in terms of the ﬁrst (R−1) reactions (Smith & Missen, 1979). The property states that the alone. The procedure in Appendix A eliminates the mole balances for every set with the maximum number extents of the reaction, so the mole balances can be of independent chemical equations are equivalent, i.e., determined from the ﬁrst R−1 reactions alone and they can be obtained from mole balances for any other therefore, give mole balances equivalent to the ones for set with the maximum chemical equations by linear a set containing only the independent reactions. combinations. A proof for this property is given in More details on this can be found in Aris (1965, pp. Appendix B. It also follows from this proof that the 14–25). mole balances for two sets of chemical equations, each having the same number (but less than the maximum Property 2. The mole balances for the maximum number number) of independent reactions, are equivalent if one of independent reactions are equi6alent to the element set of chemical equations can be obtained from the balances. other by linear combinations. This means that writing the mole balances for a Corollary 1. Mole balances for two sets of chemical reaction chemistry using the maximum number of inde- 6 equations are not equi alent if one set cannot be obtained pendent reactions is equivalent to writing the balances from the other by linear combinations. for conservation of individual elements which constitute the species (e.g. C, H etc.). This property of mole If two sets of chemical equations have the same balances and its mathematical proof was published by reaction species and the same number of independent Schneider and Reklaitis (1975). reactions (less than or equal to the maximum number), but if one set cannot be arrived at from the other by Corollary. Element balances form a subset of the mole linear combinations, the mole balances are not balances for less than the maximum number of indepen- equivalent. dent reactions. 6 Corollary 2. Mole balances are equi alent in the presence If the reaction chemistry is represented by less than of dependent chemical equations. the maximum number of independent reactions, then the element balances are incomplete and form a subset This means that the mole balances for a given set of of the mole balances. In such cases, the element bal- independent reactions (a set with maximum or less than ances have more degrees of freedom than the mole maximum number of independent reactions) are equiv- balances. A proof of this corollary is given in Appendix alent to those for the same set with additional depen- C. dent reactions. For example, consider a set of R reactions, out of which R−1 are independent. There- fore, the Rth reaction is dependent on the ﬁrst (R−1) m 4. Categories of mole balances reactions. If r is the extent of the rth reaction, it w m contributes i,r r moles to the total change in the num- ber of moles of the ith species. Therefore, the change in Case 1. R=0. number of moles of the species is given as For a nonreactive system with c components, the R mole balances are given as 0 % w m ni =n i + i, r r. r=1 0 N i =Ni, i=1, … , c. (9) Since the Rth reaction is dependent, we can write, From the deﬁnitions of the mole number transforms, R−1 w % h w the mole balances given in Eq. (9) are equivalent to i, R = r i, r, r=1 n 0 =n , i=1, … , c. (10) h i i where r are the dependence constants used for linear combinations. Therefore, we have, Eq. (10) are the regular mass balances on c components for nonreactive systems. R−1 0 % w m w m ni −n i = i, r r + i, R R (7) r=1 Case 2. R=Rmax. 1204 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217

The mole balances for a system with the maximum These three independent mole balances have ten number of independent reactions is given as variables. The species in the reaction system are numbered as 0 N i =Ni, i=1, … , R, (11) follows: toluene (1), hydrogen (2), benzene (3), methane where R is the rank of the atomic matrix (Gadewar, (4) and diphenyl (5). Using the method of Aris and Mah (1963), we determine a set containing the maxi- 2001) and Rmax =c−R is the maximum number of independent reactions. Usually, the rank of atomic mum number of independent reactions. The atomic matrix is same as the number of elements in the react- matrix for this reaction system is given as: ing species. Therefore, the number of mole balances is equal to the number of elements in such cases. These mole balances are equivalent to the element balances. (17) When the rank of the atomic matrix is less than the number of elements, the element balances are not inde- pendent. This occurs when every molecule in the system Here, the columns correspond to the species, and the obeys the same ﬁxed stoichiometric relationship(s) be- rows represent the elements. Performing elementary tween its constituent elements (Appendix D). row operations, we reduce the matrix A to its echelon B form given as: Case 3. R Rmax.

In the case when the number of reactions is less than the maximum number of independent reactions, the (18) mole balances can be written as

0 N i =Ni, i=1, … , R+r, (12) where r=R −R. Therefore, there are more mole max The rank of atomic matrix A is, 2. Therefore, balances than independent element balances. For such R= the maximum number of independent chemical equa- chemistries, element balances give an incomplete view tions is, c 3. These chemical equations are deter- of stoichiometric constraints imposed due to reactions. −R= mined from matrix A*. Each column in A* after the unit matrix represents a chemical equation. The chemi- Example 1. Hydrodealkylation process. cal equations corresponding to matrix A* can be deter- mined as follows (Smith & Missen, 1979): Let us consider the hydrodealkylation (HDA) of Consider the third column in A*, the coefﬁcient in toluene to benzene to elucidate our procedure and the ﬁrst row of this column is multiplied by the species exemplify the properties of mole balances. The species designated to the ﬁrst column of the unit matrix. Add present in the reaction system are toluene (C H ), hy- 7 8 this to the coefﬁcient in the second row of the third drogen (H ), benzene (C H ), methane (CH )and 2 6 6 4 column multiplied by the species designated to the diphenyl (C H ). Although there are many more spe- 12 10 second column of the unit matrix. This summation is cies present in the process, we only consider the above equated to the species designated to the third column species in the simpliﬁed scheme of reactions given as giving a chemical equation. Similarly, other chemical (Douglas, 1988, p. 126): equations are written by considering the fourth column C7H8 +H2 C6H6 +CH4 (13a) and onwards. These chemical equations are given as follows: X 2C6H6 C12H10 +H2 (13b) X 6 3 We determine the reaction invariants for deﬁning the C6H6 C7H8 − H2 (19a) 7 7 mole balances by choosing methane and diphenyl as the reference components. The mole balances determined 1 10 CH X C H + H (19b) using the method outlined in Appendix A are given as: 4 7 7 8 7 2

0 0 (nC H −n C H )+(nCH −n CH )=0, (14) 12 13 7 8 7 8 4 4 X C12H10 C7H8 − H2 (19c) (n −n 0 )+(n −n 0 )−(n −n 0 )=0, 7 7 H2 H2 CH4 CH4 C12H10 C12H10 (15) The set of reactions given by Eqs. (19a)–(19c) can be (n −n 0 )−(n −n 0 )+2(n −n 0 ) rearranged by linear combinations to give C6H6 C6H6 CH4 CH4 C12H10 C12H10 =0. (16) C7H8 +H2 C6H6 +CH4 (20a) S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1205

X 2C6H6 C12H10 +H2 (20b) 5. Applications X 6CH4 C6H6 +9H2 (20c) Reactions 20a and 20b are identical to reactions 13a 5.1. Validating experimental data and 13b, while reaction 20c is known to occur on MoO3 supported on oxides at temperatures above 923 K Experimental data for reacting mixtures are often (Solymosi, Erdohelyi, & Szoke, 1995). Solymosi et al. described with concentration and ﬂow rate measure- ments. It is useful to check the measured data for also found that products like CO2,H2O and CO are formed due to the presence of oxygen during the reac- consistency by formulating material balances. The sys- tion. The process of HDA of toluene (usually given by tematic method developed in this paper simpliﬁes and Eqs. (20a) and (20b)) is conducted catalytically at tem- automates the task of setting-up material balances for an experimental reaction system for checking the con- peratures of 773–923 K at high pressures over Cr2O3, sistency of measured data. Mo2O3 on Al2O3 supports (Weissermel & Arpe, 1993, p. 327). Reactions 20a and 20b are also known to be carried out homogeneously at temperatures of 895–978 5.2. Checking proposed reaction chemistry K (Douglas, 1985). Using our procedure, we determine the mole number transforms for this reaction chemistry. This is a complement of the previous application. If We choose components (3), (4) and (5) as the reference there is reasonable conﬁdence in the measured data, components. Rearranging the mole balances deter- mole number transforms can be used to determine the mined using the procedure given in Appendix A, we get molar balances for a proposed reaction chemistry. The 0 0 0 agreement of the predictions with the measured data 7(nC7H −n C7H )+6(nC6H −n C6H )+(nCH −n CH ) 8 8 6 6 4 4 provides validation of the proposed reaction chemistry. 0 +12(nC H −n C H )=0, (21) 12 10 12 10 We will exemplify this application by considering real 7(n −n 0 )−3(n −n 0 )+10(n −n 0 ) examples. H2 H2 C6H6 C6H6 CH4 CH4 −13(n −n 0 )=0. (22) C12H10 C12H10 5.3. Complementing experimental data We can write element balances for this system as follows: The degrees of freedom for solving the mole balances Hydrogen balance determines the minimum number of variables to be 0 0 0 measured in order to close the balances. For a reaction 4(nC H −n C H )+(nH −n H )+3(nC H −n C H ) 7 8 7 8 2 2 6 6 6 6 system consisting of c components undergoing +2(n −n 0 )+5(n −n 0 )=0, (23) CH4 CH4 C12H10 C12H10 R reactions, there are c−R independent mole balances Carbon balance and 2c variables. Therefore, these balances can be 0 0 0 solved by specifying c+R variables. Sometimes it is 7(nC H −n C H )+6(nC H −n C H )+(nCH −n CH ) 7 8 7 8 6 6 6 6 4 4 difﬁcult to measure some species in a reaction +12(n −n 0 )=0. (24) system due to adsorption on catalyst, gaseous state, C12H10 C12H10 etc., therefore, mole balances help in predicting these Eqs. (23) and (24) can be obtained from Eqs. (21) and values if we can measure c+R variables to close the (22) by linear combinations. Therefore, the mole bal- balances. ances for the maximum number of independent chemi- cal equations are equivalent to the element balances as expected from Property 2. Also, we see that Eqs. (23) 5.4. Conceptual design and (24) can be obtained from Eqs. (14)–(16) by linear combinations. Note that there are three independent Heuristic and hierarchical methods are widely used in mole balance equations for less than the maximum the chemical process industries (Tyreus & Luyben, number of chemical equations (given by Eqs. (14)–(16)) 2000). A hierarchical decision procedure for process and two independent element balances. The element synthesis was published by Douglas (1985) in which balances form a subset of the mole balances and, the decisions are decomposed into levels with increasing therefore, element balances are incomplete for complexity. We must set-up material balances chemistries that are represented by less than the maxi- for evaluating economic tradeoffs, which are used as mum number of independent chemical equations (Corol- objectives while proceeding through these decision lev- lary to Property 2). This means, for any postulated els. One of the foremost uses of our mole balance reaction chemistry with less than the maximum number methodology is automating the task of formulating of independent reactions, element balances need more mole balances in Douglas’s hierarchical procedure. We speciﬁcations than mole balances in the estimation of will also show this application of our method later in process ﬂow rates usually needed in process design. the paper. 1206 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217

X Example 2. Dehydrocyclization of propane. C3H6 C3H8 −H2 (27d)

X 4 1 The process of dehydrocyclization of propane is used C4H10 C3H8 − H2 (27e) for upgrading lower alkanes to aromatics which have a 3 3 higher octane number. Price, Kanazirev, Dooley, and 4 4 C H X C H − H (27f) Hart (1998) published an experimental study of this 4 8 3 3 8 3 2 reaction over a cation-containing, proton-poor MFI X 5 2 zeolite catalyst. The gas phase reaction is carried out at C5H12 C3H8 − H2 (27g) 475°C. Measurements for a batch recirculating reactor 3 3 system give the product distribution at different conver- X C6H6 2C3H8 −5H2 (27h) sions of propane. However, there is no information in the paper about the reaction chemistry. We therefore X 7 16 C7H8 C3H8 − H2 (27i) use the method of Aris and Mah (1963) to postulate a 3 3 reaction chemistry and then apply our systematic pro- Rearranging the above equations, we get the following cedure of mole number transforms to determine the set of chemical equations: mole balances. Our aim is to setup the mole balances X and cross-check the solutions from these mole balances C3H8 +2H2 3CH4 (28a) with the experimental observations to determine the 2C H X 3C H +2H (28b) consistency of our postulated reaction chemistry. 3 8 2 4 2 X The reaction species reported in Price et al. (1998) are 2C3H8 +H2 3C2H6 (28c) propane (1), hydrogen (2), methane (3), ethene (4), C H X C H +H (28d) ethane (5), propene (6), butane (7), butene (8), pentane 3 8 3 6 2 X (9), benzene (10) and toluene (11). Trace amounts of 4C3H8 3C4H10 +H2 (28e) other aromatics are also reported, but we will neglect X them in the subsequent analysis. We apply the method 4C3H8 3C4H8 +4H2 (28f) of Aris and Mah (1963) to determine a candidate X 5C3H8 3C5H12 +2H2 (28g) reaction chemistry for this system.The atomic matrix, X A, is given as: 2C3H8 C6H6 +5H2 (28h) X 7C3H8 3C7H8 +16H2 (28i) (25) This set of chemical equations is one among 55 sets that can be determined using the method of Aris and Mah (1963). We know that if we choose any one of these 55 Here, the columns correspond to the species, and the sets, we will get equivalent mole balances (Property 1), rows represent the elements present. Performing ele- so we choose the above set and proceed. Using the mentary row operations, we reduce the matrix A to its procedure outlined in Appendix A, we determine the echelon form given as: mole balances for this reaction chemistry. We choose components (1) and (4–11) as the reference components. The mole balances determined using the (26) method outlined in Appendix A and after rearrange- ments are: 3(n −n 0 )+(n −n 0 )+2(n −n 0 ) C3H8 C3H8 CH4 CH4 C2H4 C2H4 The rank of atomic matrix A is, 2. Therefore, the +2(n −n 0 )+3(n −n 0 ) R= C2H6 C2H6 C3H6 C3H6 maximum number of independent chemical equations 0 0 +4(nC H −n C H )+4(nC H −n C H ) is, c−R=9. These chemical equations are determined 4 10 4 10 4 8 4 8 from matrix A*, each column of A* from the third +5(n −n 0 )+6(n −n 0 ) C25H12 C25H12 C6H6 C6H6 column onwards yields a chemical equation. The chem- 0 +7(nC H −n C H )=0, (29) ical equations corresponding to matrix A* are given as: 7 8 7 8 0 0 0 2(nC H −n C H )−(nH −n H )+2(nC H −n C H ) 1 2 3 8 3 8 2 2 2 4 2 4 CH X C H + H (27a) 4 3 8 2 +(n −n 0 )+3(n −n 0 ) 3 3 C2H6 C2H6 C3H6 C3H6 0 0 2 2 +3(nC H −n C H )+4(nC H −n C H ) C H X C H − H (27b) 4 10 4 10 4 8 4 8 2 4 3 3 8 3 2 +4(n −n 0 )+9(n −n 0 ) C5H12 C5H12 C6H6 C6H6 2 1 X 0 C2H6 C3H8 + H2 (27c) +10(nC H −n C H )=0. (30) 3 3 7 8 7 8 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1207

First, we choose the outlet molar ﬂow rates of methane and hydrogen as unknown variables in the mole balances. Since the outlet ﬂow rate of hydrogen is not measured, we can only cross-check the predicted values of outlet ﬂow rate of methane with the experimentally measured values from Price et al. (1998, Table 1). Fig. 2 shows a parity plot of the predicted vs measured values of methane ﬂow rate at the outlet. Perfect agreement between the predicted values and measured values means that the points lie on the 45-degree line. In Fig. 2, there is a good agreement of our predicted values with the measured values, and the proposed reaction chemistry is consistent with the experimental observations. Now, if we solve the molar balances by choosing outlet ﬂow rates of ethene and hydrogen as unknowns, we can compare the predicted values for ethene with the experimentally measured values from Price et al., Table 1. Fig. 3 again validates the applicability of the proposed reaction Fig. 2. Parity plot of predicted vs measured production rate of chemistry in predicting the dependent molar ﬂow rates. methane. Example 3. Oxidation of methanol and formaldehyde.

Oxidation of methanol is used industrially to produce formaldehyde which is used primarily as an intermediate in the production of various other chemicals (e.g., Weissermel & Arpe, 1993, p. 35). Cheng (1996) published a study of oxidation of methanol and formaldehyde to study the effect of oxidation of desired product, formal- dehyde, into undesired products. The gas phase reaction is carried out at 300°C in the presence of molybdenum oxide catalyst. However, there is no information in the paper about the reaction chemistry. We therefore use the method of Aris and Mah (1963) to propose a set of chemical equations and then apply the systematic proce- dure of mole balances. Like the last example, our aim is to set up the mole balances and cross-check the solutions from these mole balances with the experimental observa- tions to determine the validity of the proposed reaction chemistry. In this example, however, we will also inves- tigate the effect of using a reaction chemistry with less Fig. 3. Parity plot of predicted vs measured production rate of ethene. than the maximum number of chemical equations. The species present in the reaction system as reported Our procedure allows these mole balances to be deter- by Cheng (1996) are formaldehyde, methanol, oxygen, mined in a completely automated fashion. nitrogen, water, carbon monoxide, carbon dioxide, dimethyl ether (DME), methyl formate (MF) and methy- 5.4.1. Degrees of freedom lal (MYL). Nitrogen is an inert in this reaction mixture The number of variables in the above two equations and we choose to neglect it from subsequent analysis. The is 22 (2c), and there are 20 (c+R) degrees of freedom. system consists of 9 species, comprised of 3 elements. The experimental observations from Price et al. (1998, Since the rank of the atomic matrix is 3, the maximum Table 1) are used to specify these 20 degrees of freedom number of independent chemical equations is 6. One such so that we can solve Eqs. (29) and (30). The inlet stream set is given below. consists only of propane, while the outlet consists of all components. All the outlet ﬂows except hydrogen are 1 CH OH+ O X HCHO+H O (31a) measured. Therefore, we have 21 measurements, and 3 2 2 2 hence, one extra measurement to cross-check our predic- X tions. CH3OH+CO 2HCHO (31b) 1208 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217

1 chemistry might not be the most realistic reaction 2HCHO+ O X CH OH+CO (31c) 2 2 3 2 chemistry for this system, but irrespective of which set we choose, we will arrive at equivalent mole balances 1 X (Property 1). Using the formulation developed in Ap- CH3OCH3 + O2 HCHO+CH3OH (31d) 2 pendix A, we get the following mole balances for the X reaction chemistry given by Eqs. (31a)–(31f) 2HCHO HCOOCH3 (31e) 0.5(n −n 0 )−(n −n 0 )−0.5(n −n 0 ) CH3OH CH3OH O2 O2 CO CO 1 X C3H8O2 + O2 2HCHO+CH3OH (31f) 0 0 0 −(nCO −n CO )+(nDME −n DME)+(nMYL −n MYL)=0, 2 2 2 This is one among the 84 possible sets of chemical (32) equations that can be generated using the method of 0 0 0 (nCH OH −n CH OH)+(nH O −n H O)−(nCO −n CO) Aris and Mah (1963). The above candidate reaction 3 3 2 2 −(n −n 0 )+(n −n 0 )+(n −n 0 )=0, CO2 CO2 DME DME MYL MYL (33) 0.5(n −n 0 )+0.5(n −n 0 ) HCHO HCHO CH3OH CH3OH +(n −n 0 )+0.5(n −n 0 )+0.5(n −n 0 ) MF MF CO CO CO2 CO2 0 0 +(nDME −n DME)+1.5(nMYL −n MYL)=0. (34) We chose formaldehyde, methanol, carbon monoxide, carbon dioxide, DME and MYL as the reference com- ponents in getting Eqs. (32)–(34).

5.4.2. Degrees of freedom The number of variables in Eqs. (32)–(34) is 18 (2c), therefore, there are 15 (c+R) degrees of freedom. The experimental observations from Cheng (1996, Table 2) are used to specify these 15 degrees of freedom for solving the above mole balances. The inlet consists only of formaldehyde and oxygen while the outlet consists of all the components. There are 16 measurements leaving one variable to be cross-checked with the predictions. Fig. 4. Parity plot of predicted vs measured production rate of MF. We choose the outlet ﬂow rate of MF as one of the unknowns in the mole balances so we can compare its prediction with the measured values. Fig. 4 shows the parity plot of predicted vs measured production rate of MF. There is a good agreement of the predicted values (determined using the mole balances) with the measured values (Cheng, 1996). Therefore, the proposed reaction chemistry is consistent with the experimental observa- tions. The proposed reaction chemistry consists of six independent reactions, which is the maximum number for the reaction system. Neglecting any one or more of these reactions may affect the predictions. The mole balances must be deter- mined again for the new set of reactions, which we have done using our procedure. Fig. 5 shows the parity plot of predicted vs measured production rate of MF using mole balances determined after neglecting reaction 31d from the proposed reaction chemistry. There is a signiﬁ- cant effect on the accuracy of predictions. Fig. 6 shows the parity plot for the case when reaction 31f is ne- glected from the set of maximum number of indepen- Fig. 5. Parity plot of predicted vs measured production rate of MF dent reactions. In this case there is little effect on the neglecting reaction 31d. accuracy of predictions. It should however be noted S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1209

C5H8N2 (DHMP)+2H2O+2H2 (35a)

C2H8N2 (ED)+C3H8O2 (PG) C5H6N2 (MP)+2H2O+3H2 (35b)

C2H8N2 (ED)+C3H8O2 (PG) C4H4N2 (P)+H2O+3H2 +CH3OH (M) (35c) C3H8O2 (PG) C3H6O(A)+H2O (35d)

C2H8N2 (ED)+C3H8O2 (PG) 0.5C6H8N2 (DMP)+0.5C4H4N2 (P)+2H2O+3H2 (35e) 2C2H8N2 (ED) C4H4N2 (P)+2NH3 +3H2 (35f) The set of reactions 35a–35f has less than the maxi- mum number of independent reactions. The mole bal- ances for the chemical equations 35a–35f are

0 0 0 (nED −n ED)−0.5(nA −n A)+0.5(nM −n M)

0 0 +0.5(nH O −n H O)+(nNH −n NH )=0, (36) Fig. 6. Parity plot of predicted vs measured production rate of MF 2 2 3 3 0 0 0 neglecting reaction 31f. (nPG −n PG)+0.5(nA −n A)+0.5(nM −n M)

0 +0.5(nH O −n H O)=0, (37) that the amount of MYL formed in the experiments as 2 2 0 0 0 reported by Cheng (1996) is not signiﬁcant. Therefore, (nMP −n MP)−(nA −n A)+2(nDMP −n DMP) it is not detrimental to neglect reaction 31f in the 0 0 0 +2(nM −n M)+(nH O −n H O)+1.5(nNH −n NH ) analysis of the reported experimental data, but reaction 2 2 3 3 −(n −n 0 )=0, (38) 31d cannot be neglected. The methodology developed is H2 H2 not only useful in validating the experimental data, but 0 0 0 (nDHMP −n DHMP)+1.5(nA −n A)−1.5(nM −n M) also in determining the importance of individual chemi- −1.5(n −n 0 )−1.5(n −n 0 )+(n −n 0 ) cal reactions in the overall mole balances. Also, if a H2O H2O NH3 NH3 H2 H2 reaction chemistry is proposed based on intuition and =0, (39) experience (which is done routinely), one can determine 0 0 0 if the reaction chemistry is consistent with the experi- (nP −n P)−(nDMP −n DMP)−(nM −n M) mental observations. −0.5(n −n 0 )=0. (40) NH3 NH3 Example 4. Synthesis of 2-methylpyrazine. We chose acetone, dimethylpyrazine, methanol, water, ammonia and hydrogen as the reference components in Many processes for the production of pharmaceuti- the procedure for setting the mole balances. cals and pesticides are characterized by complex reac- 5.4.3. Degrees of freedom tion chemistries. Also, biological systems are often The number of variables in Eqs. (36)–(40) is 22 (2c), represented by complex reaction chemistries. Here, we therefore, there are 17 (c R) degrees of freedom. If apply our methodology to a process for the synthesis of + the inlet ﬂow rates for the reaction system are known, 2-methylpyrazine, a pharmaceutical intermediate. Forni the new degrees of freedom are 6 (R). In an experimen- and Miglio (1993) performed experiments to study the tal protocol, therefore, six product ﬂow rates have to be kinetics of cyclization of ethylenediamine and propyl- measured to close the mole balances. More than six ene glycol to 2-methylpyrazine over a Zn–Cr–O/Pd product ﬂow rates must be measured for data consis- catalyst. The species present in the reaction mixture are tency checks. Therefore, the mole balance methodology reported as ethylenediamine (ED), propylene glycol provides vital input for planning experiments. (PG), 2-methylpyrazine (MP), dihydro-2-methylpyra- zine (DHMP), acetone (A), pyrazine (P), dimethyl- pyrazine (DMP), methanol (M), water, ammonia and 6. Application to conceptual design hydrogen (c=11). They suggest a mechanism for the reactions in the system and the overall reaction chem- The aim of conceptual design is to ﬁnd the best few istry based on the mechanism is candidate process ﬂowsheet(s) and to estimate optimum

C2H8N2 (ED)+C3H8O2 (PG) operating conditions for a given reaction chemistry and 1210 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 production rate(s). Douglas (1985), Douglas (1988, p. P. The components in the process may be removed in 117) proposed a hierarchical approach for conceptual the purge streams; G, denoting the gaseous purge designs. This procedure needs the speciﬁcation of the stream, and L representing the ﬂow rate of liquid purge streams entering and leaving the process units. The stream. The overall material balances at Level 2 can be concept of reaction invariants can be applied to sim- written using the transformed mole numbers as plify the task of identifying the degrees of freedom and § T −1 the molar balances needed in the hierarchical decision Pi +Gyi +Lxi − i (VRef) (PRef +GyRef +LxRef) procedure for process synthesis of systems with multiple 0 § T −1 0 =F i − i (VRef) F Ref, i=1, … , c−R. (41) chemical reactions. The hierarchy of decisions is given as (Douglas, 1985, 1988, 1995) Here, xi and yi are mole fractions of component i in the Level 0: input data gas and liquid purge stream, respectively. Accumulation Level 1: number of plants of trace components in a recycle loop can make a Level 2: input/output structure and plant process inoperable, purge streams can be used to avoid connections this accumulation. Joshi and Douglas (1992) published Level 3: recycle structure of the ﬂow sheet and a systematic procedure for identifying exit points in a reactor considerations ﬂowsheet for the removal of trace components. Our Level 4: general structure of the separation system- analysis relies on the assumption that all the species in phase splits the reaction system are known. If some compounds are Level 4a: vapor recovery system formed in small amounts by reactions not accounted Level 4b: solid recovery system for, their accumulation may cause problems in oper- Level 4c: liquid recovery system ability. Therefore, provisions for their removal in the Level 4d: combine separation system for multiple purge streams must be made. The molar outlet ﬂow of plants each component is given as the sum of its amount in Level 5: energy integration the product and purge streams. The dimension of each Level 6: evaluate process alternatives reference vector is R, which is the number of indepen- Level 7: control system synthesis dent reactions. Level 8: hazops analysis 7.1. Degrees of freedom

7. Level 2: input–output structure of the ﬂow sheet If the number of components in the gaseous and liquid purge is g and l, respectively, the total number of A block ﬂow sheet for a typical chemical process is 0 unknowns for a Level 2 balance is 2c+l+g. These shown in Fig. 7. The process inlet, given by F ,isa consist of 2c variables for fresh feed input streams, and vector of fresh feed ﬂow rates into the process while the product streams, (l−1) independent mole fractions in vector of product and byproduct ﬂow rates is given by the liquid purge, (g−1) independent mole fractions in the gas purge, liquid purge ﬂow rate and gas purge ﬂow rate. There are c−R molar balances among these variables. Therefore,

DOF=c+g+l+R. (42)

In order to solve the Level 2 balances, it is necessary to specify c+g+l+R variables.

Fig. 7. Block diagram for Level 2 of hierarchical procedure for conceptual design. 8. Level 2 balances for Example 3

We consider the oxidation of methanol and formal- dehyde discussed in Example 3 to demonstrate the application of our method for evaluation of the eco- nomic potential at Level 2 (Douglas, 1985). Methanol is the limiting reactant in the process of manufacturing formaldehyde. We assume that there are no liquid or gas purges (i.e., there is a complete recovery of all the Fig. 8. Block ﬂow sheet for production of formaldehyde from reactants). Fig. 8 shows the Level 2 block ﬂow sheet for methanol. the process. We use the product distribution vs conver- 1211 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 material balances on the process block shown in Fig. 8. At Level 2, we specify that there are no reactants in the outlet streams and no products in the inlet streams. These correspond to c speciﬁcations. The re- maining R degrees of freedom are normally ﬁxed by specifying the production rate of the primary product together with R−1 selectivities. For this example, these correspond to P , S , S , S , S and HCHO HCHO CO CO2 MYL S . Eqs. (43)–(45) are then solved for F 0 , P and MF O2 DME P . H2O Experiments need to be carried out to determine the product distribution (i.e., the R−1 selectivities) at dif- ferent conversions of the limiting reactant at speciﬁed values of the experimental parameters (e.g., reactor conﬁguration, molar feed ratios, pressure, temperature etc.). These are usually decided by the chemist, perhaps in collaboration with an engineer. The selectivities to products and byproducts can then be established either as discrete data points or a selectivity relationship as a Fig. 9. Economic potential at Level 2 vs conversion of methanol. function of the experimental parameters. One useful source for getting such information is the ‘‘Attainable Region’’, which is a collection of all the product and byproduct compositions (or selectivities) that can be achieved by using reaction and mixing in all possible combinations (Glasser, Hildebrandt, & Crowe, 1987; Feinberg, 2000). Many processes for making commodity monomers, specialty polymers and specialty chemicals, pesticides etc., need large number of reaction steps and sometimes separations are required between reactors. Similar char- acteristics are observed in integrated plant complexes for producing petrochemicals. Douglas (1990) extended the hierarchical procedure for conceptual design to Fig. 10. Block diagram for Level 3 of hierarchical procedure for conceptual design. plant complexes. Our methodology for Level 2 balances can also be used for plant complexes. sion data reported in Cheng (1996, Fig. 4). The product 8.1. Economic potential at Le6el 2 distribution is reported in terms of the selectivity to the main product, formaldehyde, and the byproducts. We Single plants for the manufacture of formaldehyde reformulate the mole balances given by Eqs. (32)–(34) have a capacity of up to 200000 t year−1 in terms of selectivities. The Level 2 balances are given (Weissermel & Arpe, 1993). We assume that the plant capacity, as follows: −1 PHCHO =100000 t year for estimating the economic SHCHO SHCHO potential. The remaining degrees of freedom are spe- F 0 −0.5S −S +P +S O2 CO CO2 DME MYL PHCHO PHCHO ciﬁed using the product distribution given in Cheng (1996, Fig. 4). Eqs. (43)–(45) are solved in conjunction =0, (43) with the deﬁnitions of selectivities in terms of compo- SHCHO SHCHO nent ﬂows enabling the estimation of all the unknown P −S −S +P +S =1, H2O CO CO2 DME MYL PHCHO PHCHO ﬂow rates. The economic potential at Level 2, EP2,is (44) deﬁned as

SHCHO EP2 = (Value of formaldehyde) PDME +0.5SCO +0.5SCO +0.5SHCHO +SMF P 2 HCHO −(Cost of methanol and oxygen). (46)

+1.5SMYL =0.5, (45) The plot of EP2 vs conversion of methanol is shown in where Si represents the selectivity deﬁned as the moles Fig. 9. Here, we have assumed that there is no value of of component i produced per mole of limiting reactant the byproducts and we also do not consider the cost of consumed. There are c+R degrees of freedom for the waste treatment. The graph shows that the economic 1212 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 potential increases with conversion before decreasing formulation, we may no longer have control over some slightly at very high conversions (\0.95). variables we speciﬁed at Level 2 (like purge liquid and gas compositions), which can then be calculated once the recycle balances are solved. This will occur if the 9. Level 3: recycle structure of the ﬂow sheet condition given in Eq. (47) is satisﬁed. The recycle ﬂow rate of the limiting reactant can be A simpliﬁed process ﬂow sheet is shown in Fig. 10. written in terms of the feed ﬂow rates by a material The ﬂow sheet consists of a reaction system and a balance around the mixing point separation system. The determination of recycle ﬂows is T 0 critical in evaluating the economic potential for the Rlr =F lr −F lr, (48) process. Also, it is necessary for reactor design since the where F T is the total molar ﬂow rate of the limiting unreacted feed is recycled back to the reactor. Compar- lr reactant at the reactor inlet and F 0 is the fresh molar ing Fig. 7 and Fig. 10, we ﬁnd that the overall molar lr feed rate of the limiting reactant. The recycle ﬂow rates balances for Level 3 are the same as for Level 2 since for all the components can be written as the inlet and outlet streams are identical in both cases. However, at this stage, based on process requirements 0 F lr 0 (often determined by reaction constraints or optimiza- Rl +Rg = ·M−F . (49) Xlr tion), some new design variables are often imposed on the process which may include specifying values for Here, Rl and Rg are the column vectors of the recycle molar feed ratios at reactor inlet, equality of composi- ﬂow rates of c components in the liquid and gas recycle, tion of the recycle stream and purge stream for gaseous respectively, Xlr is the conversion of the limiting reac- components in the absence of a gas recovery system in tant, M is a column vector of the molar feed ratio of the separation block of the ﬂow sheet, etc. This will components to the limiting reactant at the reactor inlet, T T lead to speciﬁcation of new variables which were not where Mi =F i /F lr is the molar feed ratio of component speciﬁed at the Level 2 balances. Depending on the i with respect to the limiting reactant, and F 0 is the overall degrees of freedom at Level 3, this may result in column vector of the fresh molar feed rates of c compo- cancellation of some of the speciﬁcations made at Level nents. The molar feed ratios at the reactor inlet are 2. This happens when usually kept constant during operation to avoid distur- bances in the process conditions. Since there are l (Number of restrictions at Level 3) components in the liquid recycle, c−l values in the

\(DOF at Level 3−DOF at Level 2). (47) column vector Rl are known to be zero. Similarly, c−g values in the column vector R are known to be zero. For example, this occurs when the selectivities (used at g Therefore, we have introduced c+l+g+1 new vari- Level 2) depend on the molar feed ratios at the reactor ables which include c molar ratios, l+g recycle ﬂow inlet. rates, and conversion of the limiting reactant. We have c+1 additional equations at Level 3, c equations given 9.1. Degrees of freedom T 0 in Eq. 49 and F lrXlr =F lr. The variables that we spe- 9.1.1. Case 1 ciﬁed earlier but need not be speciﬁed in the new If the mole fractions of the components in the purge formulation at Level 3 are l−1 mole fractions in the streams are the same as the corresponding recycle liquid purge and g−1 mole fractions in the gaseous purge. stream, xR is the same as xpurge and yR is the same as ypurge in Fig. 10. Therefore, there are two new variables introduced at this level compared to Level 2, which are 9.1.2. Case 2 the recycle ﬂow rates of the liquid and gaseous streams, If a separation system is used to separate the purge respectively. Therefore, stream from the recycle stream, the component mole fractions in the recycle stream will be different from the DOF=c+l+g+R+2. corresponding purge stream. Therefore, xR is different

However, since c+g+l+R variables are already spe- from xpurge and yR is different from ypurge in Fig. 10. If ciﬁed at Level 2, we must specify two additional degrees the number of components in the gas and liquid purge of freedom to solve the Level 3 balances. Usually, streams is g and l, respectively, there are l+g new molar feed ratios at the reactor inlet are chosen. Also, variables at Level 3 as compared to Level 2. These l+g conversion of the limiting reactant is an important new variables consist of (l−1) independent mole frac- variable which is known before solving these balances. tions in the liquid recycle stream, (g−1) independent In order to incorporate this information, we formulate mole fractions in the gas recycle stream, recycle ﬂow the recycle balances in terms of molar ratios. However, rate of liquid and recycle ﬂow rate of gas, given as RL since the degrees of freedom are unchanged by this new and RG, respectively, in Fig. 10. Therefore, S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1213

DOF=c+2l+2g+R. The R independent chemical reactions can be written as: However, since c+g+l+R variables are already spe- w w w X ciﬁed at Level 2, we must specify l+g additional 1, rA1 + 2, rA2 +···+ c, rAc 0, r=1, 2, … , R. degrees of freedom to solve the Level 3 balances. The (A.1) balances given in Eq. (49) along with the Level 2 w balances can be solved in this case if the molar feed Here Ai is the reacting species and i,r is the stoichio- ratios of components at the reactor inlet are known. metric coefﬁcient of component i in reaction r.The w \ convention used is i,r 0 if component i is a product, w B w i,r 0 if component i is a reactant and i,r =0if 10. Conclusions component i is an inert. Intuitively, one might think that c molar balances We have developed a systematic treatment of input- can be written for a reaction system with c components. output mole balances for complex chemistries by using However, we should take into consideration the con- the concept of reaction invariants to determine mole straints imposed by R independent reactions. These balances for a complex reaction system. If the reaction constraints can be deployed by using ‘extent of reac- chemistry is not known, we employ a systematic tion’ in relating the molar quantities of components. An method by Aris and Mah (1963) to determine a consis- extent of reaction for reaction r can be deﬁned as: tent set of chemical equations. We demonstrate the 0 m (ni −n i )r applicability of this method in data reconciliation for r = w , (A.2) two examples, viz., propane dehydrogenation, and oxi- i,r m 0 dation of formaldehyde and methanol. We prove that where r is the extent of reaction r,(ni −n i )r is the the mole balances for the maximum number of inde- number of moles of component i reacted in reaction r w pendent reactions are in fact the element balances and and i,r is the stoichiometric coefﬁcient of component i these balances are equivalent for any set of maximum in reaction r. number of independent chemical equations. This Since a component can be present in more than one methodology also gives the degrees of freedom for reaction, we express the overall consumption (produc- experimental analysis of a reaction system and thus tion) of reactants (products) in terms of the ‘extents of provides insight for planning experiments. One of the reaction’ as foremost applications of reaction invariants is in the 0 § T automation of the hierarchical decision procedure for ni =n i + i , i=1, … , c, (A.3) process synthesis published by Douglas (1985, 1988). where n 0 is the initial number of moles of component i, This systematic methodology greatly simpliﬁes the task i ni is the number of moles of component i at any given of setting mole balances for complex chemistries at § T time t, i is the row vector of dimension R of the Levels 2 and 3 of Douglas’s hierarchical procedure. stoichiometric coefﬁcients of component i in each of the R reactions:

Acknowledgements § T w w i =( i,1, … , i, R) (A.4) We are grateful to the sponsors of the Process Design and is the column vector of the R extents of reaction and Control Center, University of Massachusetts, for each of the R reactions: Amherst. Sagar B. Gadewar would like to thank GE =(m , … , m )T. (A.5) Plastics for the award of a fellowship. We are also 1 R thankful to Dr. Jean Heuschen, Brian Watson and Eq. (A.3) can be written as Alberto Nisoli of GE Plastics for their help and encour- agement. We are indebted to Prof. J. M. Douglas for n=n 0 +V (A.6) his guidance. We acknowledge Drs. Ernesto Vera and with, Raymond Rooks of the Union Carbide Corporation, for useful discussions. This research was partially sup- Á6 ··· 6 Â ported by the National Science Foundation (Grant no. Ã 1,1 1,RÃ = Ã 6 Ã CTS-9613489). V Ã i,r Ã 6 6 Ä c,1 ··· c,R Å

Appendix A. Mole number transforms is a nonsquare matrix of dimension (c, R) of the stoichiometric coefﬁcients for the c components in the T Consider a reaction system with a total of c compo- R reactions, and n=(n1,…,nc) is the column vector of 0 0 0 T nents undergoing R independent chemical reactions. dimension c of mole numbers and n =(n 1,…,n c) is 1214 S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 the column vector of dimension c of the initial mole the maximum number of independent reactions, are numbers. equi6alent. We can eliminate the R extents of reaction from the c equations given by A.6. This can be accomplished by Proof 1. Consider a stoichiometric coefﬁcient matrix choosing a subsystem of R equations from among the c VI which represents a set containing the maximum Eq. (A.6). These are the reference components and the number of independent reactions. Let VII represent reference equations are given as another set with the same number of independent reac- tions. We can get VII from VI by elementary column 0 nRef =n Ref +VRef . (A.7) operations since each reaction is represented by a Here, column in the stoichiometric coefﬁcient matrix. There- fore, Á6 ··· 6 Â Ã (c−R+1),i (c−R+1),RÃ (VII)T =D(VI)T, (B.1) = Ã 6 Ã (A.8) VRef Ã i,r Ã where D is a nonsingular permutation matrix (Strang, 6 6 Ä c,1 ··· c,R Å 1988, p. 215) of dimension (R, R) that produces the column operations. T and nRef =(n(c−R+1),…,nc) is the column vector of From Eqs. (A.6) and (A.9), we can write the mole dimension R of mole numbers for reference compo- balances for the two reaction chemistries as 0 0 0 T nents, n Ref =(n (c−R+1),…,n c) is the column vector of 0 I I I −1 0 I dimension R of the initial mole numbers for the refer- (n −n) =V (VRef) (n Ref −nRef) , (B.2) ence components. For convenience, the components are (n 0 −n)II = II( II )−1(n 0 −n )II. (B.3) numbered such that the reference components are at V VRef Ref Ref the end of the column vector of mole numbers. The To prove that both reaction chemistries give equivalent reference components should be chosen such that the mole balances, we must show that if (n 0)I is the same as 0 II I II square matrix VRef is invertible. (n ) ,then(n) is the same as (n) . We accomplish this Using Eqs. (A.7) and (A.8), the extents of reaction by showing that the RHS of Eqs. (B.2) and (B.3) are can be expressed as the same. Step 1: Assuming we choose the same reference =( )−1(n −n 0 ). (A.9) VRef Ref Ref components for evaluating the mole balances for both Substituting relation A.9 in Eq. (A.3), we get, the reaction chemistries, then we can always arrange that 0 § T −1 0 ni =n i + i (VRef) (nRef −n Ref), i=1, … , c−R. 0 I 0 II (A.10) (n Ref −nRef) =(n Ref −nRef) . (B.4)

We now deﬁne the ‘transformed mole numbers’ as: The degrees of freedom for mole balances are c +R for each set of mole balances given in Eqs. (B.2) and (B.3), § T −1 Ni =ni − i (VRef) nRef, respectively. We set them as follows:

0 0 § T −1 0 0 I 0 II N i =n i − i (VRef) n Ref. (n i ) =(n i ) , i=1, … , c,

I II These quantities allow us to write the mole balances in (ni,Ref) =(ni,Ref) , i=(c−R+1), … , c. Eq. (A.10) in the compact form The reaction species are arranged in the same order 0 Ni =N i , i=1, … , c−R. (A.11) for both the reaction chemistries. Therefore, Eq. (B.4) is satisﬁed by specifying the c+R degrees of We can deﬁne c R transforms and there are c R − − freedom. mole balances for a system with R independent reac- Step 2: A matrix of stoichiometric coefﬁcients for the tions. Eq. (A.11) says ‘‘transformed number of moles reference components is a submatrix of the full stoi- in’’ ‘‘transformed number of moles out’’, like a mole = chiometric coefﬁcient matrix and, therefore, can be balance for nonreactive mixtures. These transformed represented as mole numbers also form a basis for transformed com- I I positions used in applying the lever rule for reacting VRef =BV , (B.5) systems (Ung & Doherty, 1995a,b). II II VRef =BV , (B.6) where B is a nonsquare matrix of dimension (R, c) Appendix B. Proof of Property 1 given as

Property 1. Mole balances for all sets, each containing B=(O,I). S.B. Gadewar et al. / Computers and Chemical Engineering 25 (2001) 1199–1217 1215

Here, O is a zero matrix of dimension (R, c−R)andI where O is a zero matrix of dimension (e, R). Now we is an identity matrix of dimension (R, R). check whether (n−n 0)*, which solves C.1, also satisﬁes Step 3: From Eqs. (B.3) and (B.6), we get, C.2. We check this by direct substitution of (n−n 0)* from C.1 into C.2, giving 0 II II II −1 0 II (n −n) =V (BV ) (n Ref −nRef) . (B.7) A(n−n 0)=AV. (C.4) Taking the transpose on both sides of Eq. (B.1), we have, Using Eq. (C.3),

II I T V =V D . (B.8) AV=0. (C.5) Substituting Eq. (B.8) in Eq. (B.7), and using Eqs. (B.4) Therefore, (n−n 0)* satisﬁes both Eq. (C.1) (mole bal- and (B.5), we get, ances) and Eq. (C.2) (element balances). 0 II I T I T −1 0 II Eq. (C.1) corresponds to (c−R) independent mole (n −n) =V D (BV D ) (n Ref −nRef) , (B.9) balances (Appendix A), and, Eq. (C.2) corresponds to I I −1 0 II \ =V (BV ) (n Ref −nRef) , (B.10) (c−Rmax) independent element balances. Since, Rmax R, the number of mole balances is greater than I I −1 0 I =V (VRef) (n Ref −nRef) , (B.11) the number of element balances. Therefore, the =(n 0 −n)I. (B.12) element balances form a subset of the mole balances for less than the maximum number of independent reac- Therefore, mole balances are equivalent for every set tions. containing the maximum number of independent chem- ical equations. Appendix D. Independence of element balances

Appendix C. Proof of Corollary to Property 2 Consider a reaction system consisting of ethylene

oxide (C2H4O), water (H2O), ethylene glycol (C2H6O2) Corollary. Element balances form a subset of the mole and diethylene glycol (C4H10O3). The species are num- balances for less than the maximum number of indepen- bered as follows: ethylene oxide (1), water (2), ethylene dent reactions. glycol (3) and diethylene glycol (4). The atomic matrix for the system is given as Proof. Let Rmax be the maximum number of indepen- dent reactions and R be the number of reactions (as- sume each is independent) for a set containing less than the maximum number of independent reactions, there- (D.1) B fore, R Rmax. The change in number of moles of c components while R reactions are proceeding is given by Eq. (A.6) as Each row of the atomic matrix represents an element V=(n−n 0), (C.1) balance given by the summation of the product of the coefﬁcient in the row and the corresponding species at where V is a stoichiometric coefﬁcient matrix for a the top of the column. Performing elementary row chemistry with R chemical equations. For given extents operations and reducing the matrix A to its echelon 0 of reaction, *, we can determine (n−n )* as the form, we get solution to the above equation. For the case of the maximum number of independent chemical equations,

Rmax, the element balances are (Schneider & Reklaitis, 1975) (D.2) A(n−n 0)=0, (C.2) where A is the atomic matrix of dimension (e, c) and 0 The rank of matrix A is R=2. There are c−R=2 is a zero vector of dimension e. There are (c−R ) max independent chemical reactions, independent element balances in this case, where c− R 0e. Since the R chemical equations given by X max V C2H4O+H2O C2H6O2 (D.3a) are stoichiometrically balanced, conservation of ele- X mental species leads to, C2H4O+C2H6O2 C4H10O3 (D.3b)

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