The Rate Vector

1 Description In the same manner that one might define a mixture concentration compactly as a vector of species concentrations, it is also possible to form a vector, of equal dimension to the concentration vector, associated with the rate of reaction. If benzene and toluene are the components of interest to the problem, the concentration vector is C = [Cb, c-p]. It is natural to express the corresponding rates of reaction for benzene and toluene as a two-component column vector as well. The rate vector, r(C), in Ca-Cp space is hence defined as follows  

rg and rp are the rate expressions belonging to benzene and toluene, respectively. The rate vector r(C) is distinguished from the overall reaction rate for a particular species i, r, by boldface type. Note that r(C), as opposed to q, is a vector containing two elements. The size of r(C) is determined by the number of species rate expressions supplied. This is analogous to the concentration vector C, where the size of C is determined by the number of components in concentration space. Whereas r,- is a scalar valued function of the concentration vector C, r(C) is a vector valued function of the concentration vector C. We have been careful to indicate that the vector r(C) is a function of C. This is important as the rate expressions for each component in the system is a function of the local composition given at C. To see this, observe that 

For the species identified, the rate vector r(C) depends on only benzene, ethylene, and toluene. In general, species rate functions (r,) may be a function of all components in the system, and thus the rate vector is also a function of all components in the system. For an -component system, the rate vector is 

2 Evaluating Rate Vectors Let us draw from some familiar properties of vectors discussed in Chapter 2. Values for benzene, ethylene, and toluene may be substituted into 

Suppose that a mixture is formed with the following concentration vector for the BTX system  

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Vectors, such as x, are denoted by bold lower case font. Matrices, such as N, are denoted by bold upper case fonts. The vector x contains the concentration of all the variable species it represents the state vector of the network. Time is denoted by t. All the parameters are compounded in vector p it consists of kinetic parameters and the concentrations of constant molecular species which are considered buffered by processes in the environment. The matrix N is the stoichiometric matrix, which contains the stoichiometric coefficients of all the molecular species for the reactions that are produced and consumed. The rate vector v contains all the rate equations of the processes in the network. The kinetic model is considered to be in steady state if all mass balances equal zero. A process is in thermodynamic equilibrium if its rate equals zero. Therefore if all rates in the network equal zero then the entire network is in thermodynamic equilibrium. Then the state is no longer dependent on kinetic parameters but solely on equilibrium constants. Equilibrium constants are thermodynamic quantities determined by the standard Gibbs free energies of the reactants in the network and do not depend on the kinetic parameters of the catalysts, enzymes, in the network [49]. 

In steady state, we denote the rate vector v by the flux vector J (this will be the only vector denoted by a capital letter) to obtain ... 

None of the methods so far were able to deal with dynamics of intracellular networks. They were not able to describe the changes in the concentrations of the network intermediates as function of time upon perturbations made to the network, such as the addition of nutrients, growth factors, or drugs. This is what kinetic modelling does. A kinetic model starts from equation (2) by substituting rate equations into the rate vector. Rate equations describe the dependence of a rate of a reaction in the network with respects to its substrates, products, and effectors by the identification of the enzyme mechanism and the parameterisation of its kinetic constants. An example of a rate equation is the following two substrate ( i and 2) and two product (p and p2) reaction with the non-competitive inhibitory effect of x ... 

To linearize the rate vector R(c), we expand it about Co in a Taylor series. The subscript o denotes evaluation at some time t = to- Thus... 

No reaction vector in the AR boundary can point out of the AR. If this were the case, the AR could be extended further by PFR reactors, which have trajectories that are always tangent to the rate vectors. 

No reversed reaction vector in the complement of the AR can point back into the AR. Note that a CSTR can be represented in the AR by a line with ends at the feed and outlet concentrations, with the rate vector at the CSTR outlet collinear with this line. Thus, this condition ensures that the AR cannot be extended further by a CSTR. 

Example 4 Here, we revisit the van de Vusse reaction of Example 1 with altered rate constants. The objective function again is the yield of intermediate species B. The rate vector is given by R X) = [-X X, — 2X, 2X, ... 

Step 2 When the PFR trajectory bounds a convex region, this constitutes a candidate attainable region. When the rate vectors at concentrations outside of the candidate AR do not point bad into it, the current limits are the boundary of the AR and the procedure terminates. In Figure 6.12, the PFR trajectory is not convex, so proceed to the next step. 

If the rate expressions given in Section 4.2 can be used, compute the rate vector r(C). 

Note that an ethylene concentration must still be specified in order to compute the rate vector, even though it is not displayed in Cg-Cj. space. 

Observe that r(C) for the BTX system depends only on the concentrations of benzene, toluene, and ethylene. Had the values of Cx, c, or Cg been different to those supplied, the rate vector would have been the same. 

Observe that each rate vector has a unique direction and magnitude corresponding to the concentration vector of interest. Since we wish to plot the rate vectors in Cg-c space, the components belonging to benzene and toluene are used and values for ethylene are ignored. 

Equilibrium points are points in the rate field where the rate vector is the zero vector. 

Equilibrium is reached when the rate vector is the zero vector. Since both the kinetics and the concentration vectors have been supplied, computing equilibrium points is a simple task of substituting the concentration vectors into the kinetic expressions to determine whether the associated rate vector produces zero for each component. Table 4.3 summarizes the concentration vector and corresponding rate vectors for this system. 

The rate vector r(C2) is the only vector in the list that contains nonzero values for all components. Rate vectors r(Ci), r(C3), and r(C4) are all associated with a select number of components that have zero entries. These entries correspond to rates of reaction for the individual components, but not all components in the system. The final rate vector rfCj) is the only vector that is the zero vector, and thus C5 is an equilibrium point for the system. Note that although the component concentrations of C4 and C5 appear to be similar, the corresponding rate vectors are different. 

The terms on the left-hand side of the equation indicate a standard concentration vector C introduced in Chapter 2 C = [Ca, Cb]. Similarly, the vector on the right-hand side contains elements representing the rate expressions for A and B in the system r(C) = [r, rB]. Observe that this vector is the rate vector evaluated at C. 

The vectors in Equation 4.5 bear special significance. They represent the set of concentrations that are obtained, specifically by a PFR. Although the rate vector is generalized in that it is a function of all components, the particular... 

Geometric Interpretation An elegant relation exists between the PFR trajectory and the rate vector at a point C, on the PFR trajectory. To understand this relation, consider again the vector version of the PFR equation as follows ... 

This result shows that AC/At is an approximation to the gradient of the curve, and that the term dC/dr is the fine gradient of the line C(t). The gradient of the PFR solution trajectory is thus linked closely to the rate field specified by the system kinetics r(C), which is the rate vector evaluated at the same point C. 

For every point C on the PFR trajectory, the gradient of the trajectory is equal to the rate vector r(C) evaluated at C. Rate vectors are thus tangent at all points along the PFR trajectory. 

Here, Z is the mass fraction vector, Z=[z, Zg], and is analogous to a concentration vector for each component in the system. G is the total mass flow rate flowing through the CSTR (G is therefore always constant). It is also assumed that the rate vector is now expressed in terms of mass fractions and catalyst mass W, so that the units of r(Z) are (mass of component reacted/(mass of catalyst X time)). 

For n components in the system, n CSTR equations may be written corresponding to each component. Difficulties in solution arise when nonlinear rate expressions are used, since then multiple solutions may exist for a fixed t and Cf. This is due to the fact that the rate vector is evaluated at the exit eoncentration C and not at the feed concentration Cf. 

Since the concentration vector at the product stream is specified, the rate vector r(C) is also known and thus the feed vector Cf may be calculated directly from the CSTR expression as follows ... 

The rate vector may then be calculated using the values given in C, which gives r(C) = [-0.0245, 0.0245] mol/(L.s). Solving for the feed vector therefore gives... 

This is a somewhat more challenging calculation. To compute the exit concentration of the CSTR, we must again solve the CSTR equation. However this time, the rate vector is no longer known as it relies on the concentration vector C. Writing out the rate CSTR equation for component A gives... 

Figure 4.18 demonstrates the geometric behavior of the CSTR equation. For each point C on the CSTR locus, the rate rector r(C) is collinear with the vector v = C - Cf. This is in contrast to the rate vector for a PFR, which is tangent to every point on the PFR trajectory. 

The geometric interpretation of the CSTR allows for a convenient method for solving the CSTR equation instead of solving a nonlinear system of equations by standard numerical methods (i.e., Newton s method), we can find CSTR solutions by forming the vector v = C-Cf and then testing the rate vector at C for colinearity between r(C) and v. 

Figure 4.18 (a) Geometric interpretation of the CSTR. (b) CSTR solutions are collinear with the rate vector evaluated at that point and the feed. (See color plate section for the color representation of this figure.)... 

Solution via the geometric interpretation follows a somewhat different methodology. We shall use the fact the mixing vector between the CSTR effluent concentration and the feed vector Cf is collinear with the rate vector evaluated at the exit concentration r(C). That is we have... 

CSTR solutions are shown as x s from a hypothetical feed point. Although the region in Figure 4.20(a) is convex, it still cannot be the true AR. This is because CSTR points must be collinear with the rate vector and the feed point, and thus rate vectors evaluated at the CSTR points may point out of the region. 

Geometric interpretation Each point on the trajectory is tangent to the rate vector at that point. 

The tangent to each point on the DSR trajectory lies in a direction somewhere in between the rate vector and the mixing vector (C — C). 

The rate vector associated with point 3 is exactly tangent to the boundary. Point 3 must therefore be a concentration belonging to a PFR trajectory. Portions of the resulting curve therefore either lie on the AR boundary, or inside the region. 

Note that the value of t must be the same for all components participating in the reaction (there is always only one value of r for all components) however. The value of C may be substituted into the rate equation and evaluated to find the rate vector r(C) ... 

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