Matrix stoichiometric reaction coefficients

So matrix of stoichiometric coefficients for key components Sr matrix of Btoichiometric coefficients for key reactions T absolute temperature (T)... 

Ccn Ce, and Cl, are the concentrations in the chips, the entrapped liquor and the free liquor 7c and 77 are the chips and the liquor temperatures and are the chips and liquor velocities a is the dispersion coefficient (Peclet number) A and An are the heat and mass transfer coefficients is the heat of reaction coefficient y stoichiometric coefficient matrix rj is the volume fr action of the chips e is the chip porosity E is the activation energy Da is Damkdhler number and Cci°° is the concentration in which the component does not react. The definition of the dimensionless groups and the parameter values are presented in Wisnewski et al. (1997) and Funkquist (1997). 

For every system containing a greater number of constituents, a set of stoichiometric equations may easily be constructed by means of a suitable combination of row vectors of the matrix of constitution coefficients. Only a certain number of these equations will, however, be mutually linearly independent. The other equations may then be expressed by a linear combination of the preceding reactions. It is essential for the following considerations to determine the maximum number of linearly independent stoichiometric equations, and to select out of all possible combinations those which describe a given system in the simplest possible manner. 

In this case every chemical reaction, and thus also every row of the matrix of stoichiometric coefficients must contain at least H + I non-zero terms. The reason is, that if e.g. a chemical reaction should involve only H constituents, then the rows of the matrix of constitution coefficients of these constituents would have to be linearly dependent, which disagrees with our assumption. Let us chose in relation (2.18) r,H+i = (Kronecker s delta), where 5 i = 0 r + i and 5,. i = 1 for r = /, i.e. the r-th coordinate of the vector > r,iv) is equal to one and the rest... 

Consider a system with N chemical components undergoing a set of M reactions. Obviously, N > M. Define the A x M matrix of stoichiometric coefficients as... 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

The columns of the orthonormal matrix Vp are linear combinations of reaction invariants . In fact, the only invariants for the batch reaction being analyzed can be stoichiometric coefficients. Hence the matrix Vp may be interpreted as containing the stoichiometric information (Waller and Makila (1981)) and its rank Nr can be considered to be equal to the number of independent... 

The task now is to select the linear combinations that will most probably correspond to independent parts of the reaction network with easily interpretable stoichiometry. A simplification of the data in the matrix can be achieved by such a rotation that the axes go through the points in Fig. A-2 (this is equivalent to some zero-stoichiometric coefficients) and that the points of Fig. A-3 are in the first quadrant (this corresponds to positive reaction extents) if possible. Rotations of the abscissa through 220° and the ordinate through 240° lead to attaining both objectives. The associated rotation matrix is ... 

To test for independence, form the matrix of the stoichiometric coefficients of the above reactions with vki in the kth row and the ith column. 

For elementary reactions (Hill 1977), the values of the stoichiometric coefficients are constrained by the fact that all chemical elements must be conserved in (5.1). Mathematically, this can be expressed in terms of an E x K element matrix A where E is the total number of chemical elements present in the reacting flow. Each column of A thus corresponds to a particular chemical species, and each row to a particular chemical element. As an example, consider a system containing E = 2 elements O and H, and K = 3 species H2, O2 and H20. The 2x3 element matrix for this system is... 

An eigencomponent routine confirms that the matrix ATA has one eigenvalue equal to zero with corresponding eigenvector [0.485, —0.485, +0.2425, —0.485, -0.485]1. It is common practice to use integers as stoichiometric coefficients. This can be achieved by dividing each component by the component of smallest modulus (0.2425), which produces the vector [2, - 2,1, - 2, - 2] corresponding to the mineral reaction... 

Non-negative integers a,y, are stoichiometric coefficients of component / in reaction i. Matrix F = (y,y) is a stoichiometric matrix of some rank (rk), with elements jij — fiij—ocij. 

Up to the scaling, they are co-factors Ag of elements of any column of stoichiometric matrix F (see, for instance, Bykov et ah, 1998 Lazman and Yablonskii, 1991). We can always assign the directions of elementary reactions so that all stoichiometric coefficients are non-negative and this will be assumed later. 

The elementary reactions in Eqs. (1) are not necessarily linearly independent, and, accordingly, let Q denote the maximum number of them in a linearly independent subset. This means that the set of all linear combinations of them defines a 0-dimensional vector space, called the reaction space. In matrix language 0 is the rank of the S x A matrix (2) of stoichiometric coefficients which appear in the elementary reactions (1) ... 

This displays the convention, tacitly assumed later, that the positive direction of a step corresponds to the advancement from left to right of the stated chemical equation. The matrix of stoichiometric coefficients for these reactions is shown in Table II. The diagonalization of the matrix in Table II gives the matrix in Table III, from which the steady-state mechanism is S + 2s2 + 2s3 + 2s4. In Horiuti s terminology the stoichiometric numbers are 1 for Sj and 2 for s2, s3, and s4. 

The diagonalization of the matrix of stoichiometric coefficients is simplified in this case if the rows are not ordered as in steps (35). The result of diagonalizing is given in Table XI. Then, using the methods of Section IV,B, we find all the direct mechanisms of Table XII for the overall reaction... 

Diagonalization of the matrix of stoichiometric coefficients gives Table XVI, from which we read off the general steady-state mechanism (37) and its overall reaction (38), where p, a, and (j> are unrestricted ... 

The vki are integer stoichiometric coefficients and Xk is the chemical name of the fcth species. Normally an elementary reaction involves only three or four species (reactants plus products). Hence, as in the gas phase, the matrix is very sparse for a large set of reactions. 

A stoichiometric matrix is one whose elements are the stoichiometric coefficients of the reacting substances. Its rows correspond to the reactions and its columns are the reacting substances. 

What requirements must the stoichiometric matrix T fit Chemists choose stoichiometric coefficients such that, in each reaction, the number of atoms on the left-hand and right-hand sides are the same for every element. Hence the law of constant mass for atoms of a given type must hold over the reaction steps. In matrix representation, this requirement is of the form... 

Here vT is the transposed matrix of stoichiometric numbers and Tint is the matrix of stoichiometric coefficients for intermediates. Elements of the latter are taken to be negative if substance is consumed in a given reaction step, positive if it is formed, and zero if substance is not involved in the reaction step. Multiplication of matrix vT (P-by-s) by matrix Tmt (s-by-/tot) gives the matrix vTrint whose size is (P-by-/tot) (s is the number of steps). 

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]. 

Brinkley (4 postulated C species at equilibrium, p species, referred to as "components," were selected to have linearly independent formula vectors, where p is the rank of the atom matrix, (mjk), and Yj is the formula vector for the jth species, [mj, mj2f...mjE]. Given the choice of components, the stoichiometric coefficients for an independent set of chemical reactions are computed ... 

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