Exact Solution for the Kinetic Equations of First Order Reversible

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Exact Solution for the kinetic equations of first order reversible reaction systems through flow graph theory approach Nurul Amira Syakilla Binti Hasan, and Periyasamy Balasubramanian Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie303501t • Publication Date (Web): 09 Jul 2013 Downloaded from http://pubs.acs.org on July 9, 2013

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Industrial & Engineering Chemistry Research

Exact Solution for the Kinetic Equations of First Order Reversible Reaction Systems through Flow Graph Theory Approach Nurul Amira Syakilla Binti Hasan, Periyasamy Balasubramanian Chemical Engineering Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Perak Darul Ridzuan, Malaysia.

Abstract In a first order monomolecular reversible reaction system, the time evolution of molar concentration of the reacting species in a batch reactor is governed by linear ordinary differential equations. In this work, a flow graph theory approach is considered to derive the analytical solution for the kinetic equations of two and three species reacting systems. The flow graph is based on the image of reaction stoichiometry and utilizes Cramer’s method of determinants to find an analytical solution for the chemically reacting system. The exact solutions derived for the reversible reaction systems through flow graph theory approach are consistent with the reported analytical solutions obtained through Laplace transforms. Keywords: Kinetics, reversible reactions, flow graph theory, batch reactor, determinants Introduction Kinetic modeling is indispensable in optimization and control of activity of the feedstock, and selectivity towards the desired product in a typical chemical process. It is also a concept of mathematical description of reactions which occur in a chemical reactor and are based on description of transport processes, and the reaction mechanisms involved. The time evolution of concentrations of the chemically reacting species in a batch reactor is governed by ordinary differential equations. The kinetic model for the first order monomolecular reversible reactions is

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governed by linear ordinary differential equations. The governing model equations may be solved by using either analytical or numerical methods. Analytical solution for the linear model equations can be obtained using the techniques such as Laplace transforms, eigenvalue method, classical integration, and so on. Analytical solution is exact by definition and this is applicable for the solution of linear system of differential equations. Numerical methods are always considered for the solution of complex engineering problems. For example: numerical solution of nonlinear differential equations which arise in modeling of catalytic reactors. Nowadays, the system of mass and energy balance equations for a batch reactor or an ideal plug flow reactor are easily solved using the mathematical software available without any constraint. This involves the numerical solution of the governing ordinary differential equations using Runge-Kutta algorithms. However, the numerical methods are always an approximation and it may cause error during the estimation of derivatives while performing parameter optimization. In the literature, researchers had developed explicit mathematical expressions for the first order reversible reactions’ using various methods such as Laplace transforms, eigenvalue methods, and so on. In 1962, Wei and Prater1 proposed a general solution structure for the first order monomolecular reactions and determined the values of the kinetic constants using characteristic directions. Later, the analytical solutions for the first order three species reversible reacting system were derived using various methods such as classical integration2, eigenvalue method3, approximation method4, and Laplace transforms5,6.

Chu proposed a general solution

methodology for a first order reversible reaction system using eigenvalue method7. Recently, a flow graph theory approach was proposed to derive an analytical solution for the pharmokinetic model by Socal and Bâldea8, 9. In chemical kinetics, the flow graph represents the image of reaction stoichiometry. The reactor performance of the first order monomolecular


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reversible reactions is governed by linear ordinary differential equations. The analytical solution for the linear differential equations can be represented as the sum of constants multiplied with the time dependent exponential function1,10,11. In flow graph theory approach, the constants are the ratio of determinants of formation and consumption flow graphs8,9, and are determined by using Cramer’s method. Bhusare and Balasubramanian12 derived a general analytical solution for the first order monomolecular irreversible reactions using flow graph approach. However, this method is not applied to derive an exact solution for the first order monomolecular reversible reaction system in the literature. Therefore, in the present work, the analytical solutions for the two and three species first order monomolecular reversible reactions are derived using the flow graph theory approach. Flow Graph Theory A flow graph is drawn based on the reaction mechanisms involved in a chemical process. This approach can be used to represent the evolution of a physical system and to obtain the relationships between the system variables. Cramer’s method with determinants can be applied to determine the analytical solution of the first order monomolecular reaction system. In signals system, a flow graph represents a network in which nodes are connected by directed edges. Each node in a flow graph represents a system variable, and each edge connecting two nodes acts as a signal multiplier. The direction of signal flow is indicated by placing an arrow on the edge. The transmittance is shown along the edge. The signal flow graph shows the flow of signals from one system to another and gives the relationships between the signals. The same principles can be applied to derive an analytical solution for the chemically reacting systems. In kinetics, the node represents a component undergoing a chemical transformation.



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Edge is a directed line segment joining two nodes. Weighting of an edge is a real gain between nodes and represents the kinetic constant for a reaction. The input and output nodes have only outgoing and incoming edges, respectively. In kinetics, the input and output nodes are the reactant and product, respectively. The mixed node is a node which has both outgoing and incoming edges. This can be the intermediate products in chemical kinetics. The definition of flow graph is illustrated in Figure 1. Here, the label A is the reactant and is an input node. The labels p1 and p2 represent the output nodes. Also, the labels B and C are the mixed nodes. The kinetic constants for the three species reactions are kBA, kCB, and kBC. The further details for the flow graph theory approach are available elsewhere8,9. Model Equations for Reversible Reaction System A general stoichiometry of first order reversible reaction system13 can be represented as k

j ,i → si ← sj

ki , j

(1)

where, i and j vary from 1 to Ns, and Ns represents the number of species involved in a reaction system. A matrix form of kinetic constants (h-1) is given by

 0  k  2,1  k 3,1  K =  k 4,1  M  k N s −1,1 k  N s ,1

k1, 2 0

k1,3 k 2 ,3

k1, 4 k 2, 4

L L

k1, N s −1 k 2, N s −1

k 3, 2 k 4,2

0 k 4 ,3

k 3, 4 0

L L

k 3, N s −1 k 4, N s −1

M

M

M

k N s −1, 2

k N s −1,3

k N s −1, 4

M L

M 0

k N s ,2

k N s ,3

k Ns ,4

L k N s , N s −1

k1, N s  k 2, N s  k 3, N s   k 4, N s  M   k N s −1, N s  0 

(2)

In matrix K, the subdiagonal and superdiagonal elements represent the kinetic constants for the forward and reverse reactions, respectively. Each column represents the kinetic constants for all


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possible parallel reactions from the reactant i. Unsteady state mole balance in a batch reactor is identical to that of steady state mole balance in an ideal plug flow reactor when the volume change due to chemical reaction is neglected. Therefore, it is assumed that the reaction occurring in a batch reactor is similar to an ideal plug flow reactor for the liquid phase reactions. This assumption is not applicable for variable volume reactor system. Also, it is assumed that the reactor is operated under isothermal condition. The vector form of kinetic equations for the first order reversible reactions can be represented as

c& s = Rk c s

(3)

where, cs is a vector of molar concentration of species si (i varies from 1 to Ns) in mol·m-3, and Rk is a matrix form of coefficients of the kinetic equations1 and is given by  Ns − ∑ k j ,1  j =2   k 2,1    k 3,1   Rk =   k 4,1   M   k N s −1,1     k N s ,1 

k1,3

k1, 4

L

k1, N s −1

− ∑ k j,2

k 2,3

k 2, 4

L

k 2, N s −1

k 3, 2

− ∑ k j ,3

k 3, 4

L

k 3, N s −1

− ∑ k j ,4 L

k 4, N s −1

k1, 2 Ns

j =1 j ≠2

Ns

j =1 j ≠3

Ns

k 4,2

k 4,3

j =1 j≠4

M

M

M

M

k N s −1, 2

k N s −1,3

k N s −1, 4

L −

k Ns ,2

k N s ,3

k Ns ,4

L

M Ns

∑k

j =1 j ≠ N s −1

j , N s −1

k N s , N s −1

    k 2, N s    k 3, N s     k 4, N s    M  k N s −1, N s    N s −1  − ∑ k j ,Ns  j =1  k1, N s

(4)

The properties of matrix Rk are: (i) sum of all kinetic constants for the disappearance of reactant i is represented by the diagonal elements, (ii) subdiagonal and superdiagonal elements represent



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the kinetic constants for the forward and reverse reactions, respectively, and (iii) sum of elements in each column of the matrix is zero. It is cumbersome to derive a general exact solution for eq. (3) using flow graph theory approach and therefore, it is decided to demonstrate the applicability of this approach for the simple reaction schemes such as two and three species reacting systems. Illustrative Examples In the following, the derivation of analytical solutions for the two and three species reacting system using flow graph theory approach is presented. Two Species Reacting System: The stoichiometry of two species reacting system6 which occurs in a batch reactor is given by k2,1

s1

s2

(5)

k1,2

It is assumed that the first order reversible reaction occurs in a constant-volume isothermal batch reactor. The rate of reactions (mol·m-3·h-1) for the reactant s1 and product s2 can be represented as rs1 = − k 2,1c s1 + k 1, 2 c s2 , and

(6a)

rs2 = k 2 ,1c s1 −k 1, 2 c s2

(6b)

The molar concentration balance equations for the reactant and product are dc si dt

= rsi for i = 1 and 2

(7)

The analytical solution for the linear system of differential equations can be represented as the sum of constants multiplied with the time dependent exponential functions12,13. Thus, the analytical solution for eq. (7) can be written as


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Ns

c si (t ) = ∑ Ai , j exp(− γ j t )

(8)

j =1

where, Ai , j =

∆f si (γ j ) ∆C (γ j )

and i = 1, 2,…, Ns.

In eq. (8), the constants Ai,j represent the ratio of formation and consumption determinants8,9. These determinants are formulated from the formation and consumption flow graphs. The factors γj (j = 1, 2, …, Ns) are calculated from the determinant of the consumption flow graph. Also, these factors are function of kinetic constants involved in the reversible reaction system. Consumption Flow Graph: The consumption determinant of the two species reacting system can be calculated based on the consumption flow graph and is shown in Figure 2. In Figure 2, the labels p1 and p2 represent the formation of final products from the species s1 and s2 respectively, with the zero kinetic constants8,9 in the two species reacting system. It is

assumed that every reacting species undergo chemical reactions to produce a final product with a constant reaction rate, even if it is zero as depicted in Figure 2. Thus, the consumption determinant for the aforementioned flow graph is given by

s1 ∆C =

s1 k 2,1 − γ s 2 − k 2,1

s2 − k1, 2

(9)

k1, 2 − γ

In eq. (9), the diagonal elements represent the disappearance of reactant si to the products in all possible ways with the positive sign in front of it. This is because all the transmittances of edges which are outgoing from the node si. Therefore, the diagonal elements have positive sign in the consumption determinant. The second element in the first column represents the formation of product s2 from the reactant s1 with the kinetic constant k2,1. Here, this element has negative sign



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as a result of the transmittance of edge outgoing from s1 and incoming to s2. Similarly, the kinetic constant k1,2 has negative sign as a result of the transmittance of edge outgoing from s2 and incoming to s1. The values of γ are determined by making ∆C = 0 and the resulting expression can be represented as

∆C = γ 2 − γ (k2,1 + k1, 2 ) = (γ 1 − γ )(γ 2 − γ ) = 0

(10)

The roots of eq.(10) are

γ 1 = 0, and γ 2 = k 2,1 + k1, 2 or γ 1 = k 2,1 + k1, 2 , and γ 2 = 0

(11)

The expression for the consumption determinant in eq. (10) is the second order polynomial in γ. For Ns species reaction system, the consumption determinant expression is nth order polynomial in γ. Also, the expression for the consumption determinant can be conveniently represented as the products of the differences of the factors γi (i = 1, 2, …, Ns). Thus, the general formula for finding the determinant of the consumption flow graph8,9 can be written as Ns

∆C (γ i ) = ∏ (γ j − γ i ) ≠ 0

(12)

j =1 i≠ j

Formation Flow Graph: The formation flow graph for the two species reaction system is deduced from the consumption flow graph with the consideration that the interest reacting species being a target one and by adding a new source input8,9. The formation flow graph for the reactant s1 is depicted in Figure 3. Here, f1 and f2 represent the source terms for the reactant s1 and product s2, respectively. The initial concentrations c s1, 0 and c s2, 0 represent the reactant and


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product, respectively. Thus, the determinant of the formation flow graph for the reactant s1 can be written as f

s2 − k1, 2 k1, 2 − γ

s1 c s1, 0 s 2 c s2 , 0

∆f s1 =

(13)

In eq. (13), the elements of first column represent the initial concentrations of the reacting species. This formation determinant is deduced from eq. (9) by replacing the first column with the source terms according to Cramer’s method. Eq. (13) gives,

∆f s1 (γ j ) = c s1, 0 (k1, 2 − γ j ) + c s2, 0 k1, 2 ,

for j = 1, and 2

(14)

The constants Ai,j for the reactant s1 are

A1,1 =

∆f s1 (γ 1 )

A1, 2 =

∆C (γ 1 )

∆f s1 (γ 2 ) ∆C (γ 2 )

=

=

c s1, 0 (k1, 2 − γ 1 ) + c s2 , 0 k1, 2

γ 2 −γ1

, and

c s1, 0 (k1, 2 − γ 2 ) + c s2 , 0 k1, 2

γ1 − γ 2

(15a)

(15b)

The formation flow graph for the product s2 is shown in Figure 4. The determinant for the formation flow graph of the product s2 is obtained by replacing the second column of eq.(9) with the source terms. The determinant is given by s1 ∆f s 2 =

s1 k 2,1 − γ s 2 − k 2,1

f c s1, 0 c s2 , 0

(16)

Eq. (16) results,



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∆f s2 (γ j ) = c s2, 0 (k 2,1 − γ j ) + c s1, 0 k 2,1 , for j = 1, and 2

(17)

The constants Ai,j for the product s2 are

A 2 ,1 =

A 2, 2 =

∆f s2 (γ 1 ) ∆C (γ 1 )

∆f s2 (γ 2 ) ∆C (γ 2 )

=

=

c s1, 0 k 2 ,1 + c s2 , 0 (k 2,1 − γ 1 )

γ 2 − γ1

, and

(18a)

c s1, 0 k 2 ,1 + c s2 , 0 (k 2 ,1 − γ 2 )

(18b)

γ1 − γ 2

The exact solutions for the reactant s1 and product s2 are obtained by substituting eqs.(11, 15, and 18) in eq. (8). Thus, the explicit mathematical expressions for the two species monomolecular reversible reaction system are

c s1 (t ) =

c s2 (t ) =

c s1, 0 k 2,1 + k1, 2

k 2,1c s1, 0 k 2,1 + k1, 2

{k

1, 2

+ k 2,1 exp[− (k 2,1 + k1, 2 )t ]} +

{1 − exp[− (k

2 ,1

+ k1, 2 )t ]} +

k1, 2 c s2 , 0 k 2,1 + k1, 2

c s2 , 0 k 2,1 + k1, 2

{k

2 ,1

{1 − exp[− (k

2 ,1

+ k1, 2 )t ]} , and

+ k1, 2 exp[− (k 2,1 + k1, 2 )t ]}

(19)

(20)

The derived analytical solutions for the two species reversible reaction system using flow graph theory approach are consistent with the reported expressions obtained using Laplace transforms6. The simulated time evolution of molar concentrations of the reactant and product obtained using eqs.(19 and 20) are shown in Figure 5. Eqs. (19 and 20) are solved using MATLAB software. The initial concentration of pure reactant s1 considered here is 100 mol/m3. The assumed values for the kinetic constants k2,1 and k1,2 are 1.24 h-1 and 0.35 h-1, respectively. Three Species Reacting System: The stoichiometry for the three species reacting system1 can be represented as


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k2,1 s2

s1

k1,2 k3,1

k2,3

k1,3

k3,2

s3

It is assumed that the above-mentioned first order reversible reactions occur in a constantvolume isothermal batch reactor. The rate of reactions (mol·m-3·h-1) for the reactant s1, and products s2 and s3 are rs1 = −(k 2 ,1 + k 3,1 )c s1 + k1, 2 c s2 + k1,3 c s3 rs2 = k 2,1c s1 − (k1, 2 + k 3, 2 )c s2 + k 2 ,3 c s3

(21) , and

rs3 = k 3,1c s1 + k 3, 2 c s2 − (k1,3 + k 2,3 )c s3

(22) (23)

The molar concentration balance equations for the species s1, s2, and s3 are given by dc si dt

= rsi ,

for i = 1, 2, and 3

(24)

In the following, the derivation of analytical solution for eq. (24) using flow graph approach is presented. Consumption Flow Graph: The consumption determinant for the three species reacting system is calculated from consumption flow graph and is shown in Figure 6. Thus, the consumption determinant can be represented as



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s1 s1 ∆C = s 2 s3

k 2 ,1 + k 3,1 − γ − k 2,1 − k 3,1

s2 k1, 2

− k1, 2 + k 3, 2 − γ − k 3, 2

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s3

k1,3

− k1,3 − k 2,3 + k 2,3 − γ

(25)

In eq. (25), the diagonal elements represent the disappearance of reactant si in all possible ways with positive sign in front of it. The off-diagonal elements have negative sign in front of it as a result of the transmittances of edges outgoing from si and incoming to sj with i ≠ j . The values of γ are determined by making ∆C = 0 and the resulting expression can be represented as ∆C = −γ 3 + γ 2 (k 2,1 + k1, 2 + k 3,1 + k1,3 + k 3, 2 + k 2,3 ) − γ (k 2,1 k1,3 + k1, 2 k1,3 + k 3, 2 k1,3 + k 2 ,1 k 2 ,3 + k 3,1 k 2 ,3 + k1, 2 k 2 ,3 + k 2 ,1 k 3, 2 + k 3,1 k1, 2 + k 3,1 k 3, 2 ) = 0

(26)

The roots of this cubic equation (26) are

γ 1 = 0, γ 2 =

a + a 2 − 4b a − a 2 − 4b , and γ 3 = 2 2

(27)

where, a = k 2,1 + k1, 2 + k 3,1 + k1,3 + k 3, 2 + k 2 ,3 = γ 2 + γ 3 , and b = (k 2 ,1 k1,3 + k1, 2 k1,3 + k 3, 2 k1,3 + k 2 ,1 k 2 ,3 + k 3,1 k 2 ,3 + k1, 2 k 2 ,3 + k 2 ,1 k 3, 2 + k 3,1 k1, 2 + k 3,1 k 3, 2 ) = γ 2 γ 3

Formation Flow Graph: The formation flow graph for the reactant s1 is depicted in Figure 7. The formation determinant for the reactant s1 is deduced from eq. (25) by replacing the first column with the source terms. Therefore, the formation determinant is given by


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f

s2

s3

s1

c s1, 0

− k1, 2

∆f s1 = s 2 s3

c s2.0 c s3 , 0

k1, 2 + k 3, 2 − γ − k 3, 2

− k1,3 k1,3

− k 2,3 + k 2,3 − γ

(28)

The following expression is obtained after expanding the determinant.

(

)

∆f s1 (γ j ) = c s1, 0 γ 2j − αγ j + β + c s2 , 0 (β − k1, 2 γ j ) + c s3, 0 (β − k1,3γ j ) , for i = 1, 2, and 3

(29)

where, α = k1, 2 + k1,3 + k 3, 2 + k 2,3 , and β = k1, 2 k1,3 + k 3, 2 k1,3 + k1, 2 k 2,3

Similarly, the formation flow graph for the product s2 is depicted in Figure 8 and the formation determinant for the product s2 is given by s1 ∆f s 2

s1 = s2 s3

f

s3

k 2 ,1 + k 3,1 − γ − k 2,1

c s1, 0 c s2 , 0

− k1,3 − k 2,3

− k 3,1

c s3 , 0

k1,3 + k 2,3 − γ

(30)

It results,

(

)

∆f s2 (γ j ) = c s1, 0 (ε − k 2,1γ j ) + c s2 , 0 γ 2j − ϕγ j + ε + c s3, 0 (ε − k 2 ,3γ j ) , for i = 1, 2, and 3

(31)

where, ε = k 2,1 k1,3 + k 2,1 k 2,3 + k 3,1 k 2 ,3 , and ϕ = k 2,1 + k 3,1 + k1,3 + k 2,3 .

Furthermore, the formation flow graph for the product s3 is depicted in Figure 9, and the formation determinant for the product s3 is



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s1 s1 ∆f s3 = s 2 s3

k 2,1 + k 3,1 − γ − k 2,1

s2 k1, 2

− k 3,1

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f

− k1, 2 + k 3, 2 − γ

c s1, 0 c s2 , 0

− k 3, 2

c s3 , 0

(32)

After simplifications, we get

(

)

∆f s3 (γ j ) = c s1, 0 (δ − k 3,1γ j ) + c s2 , 0 (δ − k 3, 2 γ j ) + c s3, 0 γ 2j − θγ j + δ , for i = 1, 2, and 3

(33)

where, δ = k 2 ,1 k 3, 2 + k 3,1 k1, 2 + k 3,1 k 3, 2 , and θ = k 2,1 + k 3,1 + k1, 2 + k 3, 2 .

Finally, the analytical solutions for the reacting species s1, s2, and s3 are obtained by substituting Eqs. (12, 27, 29, 31, and 33) in eq. (8). After making rearrangement and simplifications, the analytical expressions can be written as

 β  γ 22 − αγ 2 + β γ 32 − αγ 3 + β − exp(− γ 2 t ) + exp(− γ 3t ) + c s1 (t ) = c s1, 0  γ 2 (γ 3 − γ 2 ) γ 3 (γ 3−γ 2 )  γ 2γ 3  k1, 2γ 2 − β k1, 2γ 3 − β  β  c s2 , 0  + exp(− γ 2 t ) − exp(− γ 3t ) + γ 3 (γ 3 − γ 2 )  γ 2γ 3 γ 2 (γ 3 − γ 2 ) 

(34)

k1,3γ 2 − β k1,3γ 3 − β  β  c s3.0  + exp(− γ 2 t ) − exp(− γ 3t ) γ 3 (γ 3 − γ 2 )  γ 2γ 3 γ 2 (γ 3 − γ 2 ) 

k γ −ε k γ −ε  ε  c s2 (t ) = c s1, 0  + 2,1 2 exp(− γ 2 t ) − 2,1 3 exp(− γ 3t ) + γ 3 (γ 3 − γ 2 )  γ 2 γ 3 γ 2 (γ 3 − γ 2 )  2 2  ε  γ − ϕγ 3 + ε γ − ϕγ 2 + ε c s2 , 0  − 2 exp(− γ 2 t ) + 3 exp(− γ 3 t ) + γ 3 (γ 3−γ 2 )  γ 2 γ 3 γ 2 (γ 3 − γ 2 )  k 2, 3γ 2 − ε k 2, 3γ 3 − ε  ε  c s3.0  + exp(− γ 2 t ) − exp(− γ 3t ) γ 3 (γ 3 − γ 2 )  γ 2 γ 3 γ 2 (γ 3 − γ 2 ) 


(35)

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k γ −δ k γ −δ  δ  c s3 (t ) = c s1, 0  + 3,1 2 exp(− γ 2 t ) − 3,1 3 exp(− γ 3t ) + γ 3 (γ 3 − γ 2 )  γ 2γ 3 γ 2 (γ 3 − γ 2 )  k γ −δ k γ −δ  δ  c s2 , 0  + 3, 2 2 exp(− γ 2 t ) − 3, 2 3 exp(− γ 3t ) + γ 3 (γ 3 − γ 2 )  γ 2γ 3 γ 2 (γ 3 − γ 2 )  2 2  δ  γ − θγ 3 + δ γ − θγ 2 + δ c s3.0  − 2 exp(− γ 2 t ) + 3 exp(− γ 3t ) γ 3 (γ 3−γ 2 )  γ 2 γ 3 γ 2 (γ 3 − γ 2 ) 

(36)

The time evolution of molar concentration of the three species reacting system is illustrated with the following example. Consider the isomerization of butenes over alumina catalyst at a temperature of 230oC in a glass flow reactor1,10. The stoichiometry of butene isomerization1 given by Lago and Haag is 0.5327 1-butene (s1)

cis-2-butene (s2) 0.2381 0.1736

0.0515

0.1918

0.2892

trans-2-butene (s3) The unit of kinetic constants included in the above-mentioned stoichiometry is h-1. The molar concentrations of butenes in a reactor are calculated using eqs. (34-36) with the following two initial conditions: (i) pure cis-2-butene, and (ii) a mixture of 1-butene and cis-2-butene. The simulated molar concentration profiles obtained for isomerization of pure cis-2-butene, and a mixture of 1-butene and cis-2-butenes using eqs.(34-36) are depicted in Figures 10 and 11, respectively.



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Page 16 of 24

The time dependent behavior of the first order monomolecular reaction system in a batch reactor is governed by the linear differential equations. In this work, the flow graph theory approach8,9 is used to derive the analytical solution for simple first order monomolecular reaction systems. This approach utilizes Cramer’s rule with determinants for finding the solution for the linear system of differential equations. The analytical solution for the linear differential equations can be conveniently represented as the sum of constants multiplied with the time dependent exponential functions. In flow graph theory approach, the constants are determined as the ratio of the formation and consumption determinants8,9. The consumption determinant is calculated from the consumption flow graph and is the image of the reaction stoichiometry. The formation determinant is calculated from the formation flow graph. This flow graph is deduced from the consumption flow graph by adding the initial feed concentrations appropriately. The constants included within the exponential functions are calculated from the consumption determinant. The flow graph theory approach eliminates the usage of classical integration, eigenvalue method and Laplace transforms for determining the exact solutions of the linear system of differential equations which arise in chemical kinetics. This approach is not applicable for solution of nonlinear differential equations. However, the flow graph approach can be applied to derive the analytical solution of the complex reaction systems such as methane pyrolysis, thermal cracking of ethane and so on. This will be the future scope of the present work. Conclusion In this paper, the analytical solutions for the first order reversible reaction systems are derived using flow graph theory approach. This method is demonstrated for the two and three species reacting systems. The flow graph theory approach eliminates the usage of Laplace transforms


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and eigenvalue method for determining the analytical solution of simple first order reversible reaction systems. The derived analytical solutions for the two and three species reacting system are consistent with the solution obtained through Laplace transforms5,6. Author Information Corresponding Author: E-mail: [email protected], Tel.: +6053687548, Fax: +6053656176 Notes The authors declare no competing financial interest. Literature Cited (1) Wei, J.; Prater, C. D. The structure and analysis of complex reaction systems. Adv. Catal., 1962, 13, 203-392. (2) Himmelblau, D. M.; Jones, C. R.; Bischoff, B. C. Determination of rate constants for complex kinetics models. I & EC Fundamentals, 1967, 6 (4), 539-534. (3) Pogliani, L.; Santos, M. N. B.; Martinho, J. M. G. Matrix and convolution method in chemical kinetics. J. Math. Chemistry, 1996, 20, 193-210. (4) Chrastil, J. Determination of the first-order consecutive reversible reaction kinetics. Comp. Chem., 1993, 17(1), 103-106.

(5) Korobov, V. I.; Ochkov, V. F. Chemical kinetics with Mathcad and Maple. Springer-Verlag. 2011, ISBN: 978-3-7091-0530-6, (Pages. 35-72).



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(6) Berberan-Santos, M. N.; Pogliani, L.; Martinho, J.M.G. A convolution approach to the kinetics of chemical and photochemical reactions. React. Kinet.Catal. Lett., 1995, 54 (2), 287-292. (7) Chu C. A generalized stochastic model for systems of first-order reactions-probability generating function approach. Chem. Eng. Sci., 1971, 26, 1651-1658. (8) Socol, M.; Bâldea, I. New method of finding the analytical solutions directly on the base on the reaction mechanism. J. Math. Chem., 2009, 45, 478-487. (9) Socol, M.; Baldea, I. A new approach of flow graph theory applied in physical chemistry. J. Chinese Chem. Soc., 2006, 53, 773-781.

(10)

Li, G. A new approach to determination of the rate constants of complex first-order

chemical reaction systems. Chem. Eng. Sci., 1985, 40 (6), 939-949. (11)

Shenvi, N.; Geremia, J. M.; Rabitz, H. Nonlinear kinetic parameter identification through

map inversion. J. Phys. Chem. A, 2002, 106, 12315-12323. (12)

Bhusare, V. H.; Balasubramanian, P. A New paradigm in kinetic modeling of complex

reaction systems. International Review on Modelling and Simulations 2010, 3 (5), 11451152. (13)

Bhusare, V. H.; Balasubramanian, P. A general kinetic modeling methodology for first

order irreversible reaction systems. Research Bulletin of Australian Institute of High Energetic Materials. 2011, 1, 85-105, ISBN: 978-0-9806811-9-2.


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kCB

kBA

A

C

B γ

kBC

γ

Figure 1. A flow graph for a reaction system.

,

,

γ

γ

Figure 2. The consumption flow graph for two species reacting system.

,

,

,

,

γ

Figure 3. The formation flow graph for the reactant s1 in the two species reacting system.



,

,

γ

,

,

Figure 4. The formation flow graph for the product s2 in the two species reacting system.

100 s1 s2

90 80 70 60 3

csi (mol/m )

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Page 20 of 24

50 40 30 20 10 0

0

1

2

3 t (h)

4

5

6

Figure 5. The concentration versus time plot for the two species reacting system.


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,

,

γ

γ ,

, ,

,

γ

Figure 6. The consumption flow graph for the three species reacting system.

,

,

,

γ

, , ,

,

,

γ

,

Figure 7. The formation flow graph for the reactant s1 in the three species reacting system.



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Page 22 of 24

,

,

,

, ,

,

,

,

γ

γ

,

Figure 8. The formation flow graph for the product s2 in the three species reacting system. ,

,

,

,

γ , γ

,

, ,

,

Figure 9. The formation flow graph for the product s3 in the three species reacting system.


Page 23 of 24

110 1-butene 100

cis-2-butene trans-2-butene

90 80 70 csi mol/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 50 40 30 20 10 0

0

2

4

6

8

10 t (h)

12

14

16

18

20

Figure 10. Isomerization of pure cis-2-butene over alumina at 230oC.



110 1-butene 100

cis-2-butene trans-2-butene

90 80 70 csi (mol/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

60 50 40 30 20 10 0

0

2

4

6

8

10 t (h)

12

14

16

18

Figure 11. Isomerization of 1-butenes over alumina at 230oC.


20

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Exact Solution for the Kinetic Equations of First Order Reversible

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