Constitutive Coefficients

The beam stiffness coefficients on the principal diagonal of the matrix P associated with extension, Pn, bending, P55, Pee, and warping, P77, depend on the shell stiffnesses associated with the lengthwise extension, An(s), bending, 5ii(s), and the coupling Bn(s)  

The other beam stiffness coefficients on the principal diagonal associated with shear, P22, P33, and torsion, P44, are functions of the shell stiffnesses associated with shear, k33 s), twist, 633(5), and the coupling 633(5)  

Off-Diagonal Mechanical Coefficients without Laminae Level Coupling 

The beam stiffness coefficients, which couple shear in both directions and 

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Note that values for the constitutive coefficients A,s and C-mt are introduced without any experimental validation of the model. However, this choice of... 

When every compound is described as a line vector of constitutional coefficients, a system of N constituents may be defined by a matrix of N x M elements. Clearly the elements of this matrix must be non-negative integers. With regard to the physical sense of the matrix, none of its rows or columns may be composed exclusively of zeros. 

Let us consider a system composed of three elements (C, H, O) and five constituents (CH4, H2O, H2, CO, CO2). The matrix of constitution coefficients will take the form... 

The system is uniquely defined by the two matrixes, the matrix of constitution coefficients and the matrix of stoichiometric coefficients. The matrix of constitution coefficients, however, is of fundamental significance because it describes the qualitative properties of the system and the matrix of stoichiometric coefficients may be derived from it. 

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. 

The rank of the matrix of constitution coefficients enables us to classify all constituents present in the system into two groups, fundamental (primary, basic) and derived (secondary). In every system there exists a certain minimum number of constituents by means of which the system is described in qualitative terms. These constituents are called fundamental ones (or also components) and their number is determined by the rank of the matrix of constitution coefficients. The maximum number of... 

Let us consider the system form example 1. There applies that the rank of the matrix of constitution coefficients is H = 3, since e.g. from the rows 1, 2, 3 a square matrix may be constructed which will have a non-zero determinant... 

Generally, particularly in the case of more complicated systems, N M) the rank of the matrix of constitution coefficients is given by the number of elements, i.e. H — M, In practical calculations, we usually have M 6. When, however, there are P linear relationships between the columns of the matrix of constitution coefficients, the rank of this matrix will be H = M — P. 

The matrix of stoichiometric coefficients may be determined from the matrix of constitution coefficients by means of the equations (2.10). Let us assume, without loss of generality, that the first H constituents are fundamental ones, i.e. 

The set of vectors of an arbitrarily selected number H of rows of the matrix of constitution coefficients is linearly independent, i.e. the set of basic constituents... 

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... 

Matrix of constitution coefficients of basic constituents is singular ... 

In a closed system with N constituents and M elements, in which R reactions take place (R N — H), numbers of moles of the individual constituents are not independent variables, as they are linked by relationships which we call mass balance equations. There are two fundamental ways of expressing the mass balance by means of constitution coefficients or of stoichiometric coefficients. We shall show that the two ways are equivalent under the assumption R =... 

The mass balance equation constructed by means of the constitution coefficients has the form... 

Construct matrixes of constitution coefficients for the following systems, and determine their rank ... 

When, on the other hand, the mass balance equations are expressed in terms of constitution coefficients, i.e. in the form of... 

This procedure does not necessitate a stoichiometric analysis of the system, since knowledge of the matrix of constitution coefficients will suffice for the application of the methods. It will be shown later (see section 5.4), that the problem of finding the minimum of the function (5.2) on a set of points (5.4) can be converted to solving a set of at most (M -1- 1) non-linear equations for (M + 1) unknown variables. It was only the development of methods of this kind that enabled solution of large system (N > M), since even in these systems values of M are rarely greater than six or seven, while the value of P = iV — M is in no wise limited. 

The equations (5.59) express the material balance of the system. Let us assume that the matrix of constitution coefficients A is of rank M. This can always be achieved by eliminating linearly dependent columns in matrix A, thus formally decreasing the number of elements in the system. It should be stated, however, that very few cases are encountered in technical practice in which this procedure would be necessary. Usually there holds N M and thus the probability of finding a linearly dependent column in matrix A is slight. 

It is essential for the further procedure that the matrix of constitution coefficients be rearranged and modified so as to contain only H linearly independent columns the first H rows, representing a qualitative description of the basic constituents, must likewise be linearly independent. An essential condition of the selection of basic constituents is the fact, that they must include all elements. It is then possible to define the two sets of algebraic relationships balance relationships for every basic constituent, using the equations (5J2), and equilibrium relationships (5.34) for individual chemical conversions, expressed as linear combinations of row vectors of the basic constituents by the equations (5.33). 

Rank of the matrix is if = 4, thus the system contains four independent constituents. Considering the required condition that the independent constituents should be represented in the highest concentration, it is advisable to select CO2, H2O and N2. The fourth constituent selected is CO. A determinant formed from the constitution coefficients of these four constituents is ) = 4, providing that the constituents are linearly independent. The remaining six constituents shall be considered to be derived using the procedure described in Example 5, we now construct the matrix of stoichiometric coefficients for six linerlay independent stoichiometric equations in the form of synthesis reactions ... 

Before the White-Johnson-Dantzig method can be applied directly, the rank of the matrix of constitution coefficients must be determined. Let us use the example of methanol synthesis... 

The matrix of constitution coefficients will take the form of... 

The third category includes systems in which one of the columns in the matrix of constitution coefficients is a linear combination of the other columns. In such cases a corresponding number of linearly dependent columns is left out with respect to the following calculation it is advantageous to keep as far as possible those with a maximum number of zeros. This greatly simplifies the process of solving the set (5.121). 

Evidently, if = 2, but M = 3. Since there are only two basic constituents, two balance relationships will suffice to describe the system, so that one column of the matrix of constitution coefficients must be deleted. The resulting set of linear equations according to (5.130) will take the general form of... 

Let us employ the example of 1-hexene isomerisation. The reaction mixture is formed by a total of 17 isomeric hexenes. Clearly the matrix of constitution coefficients will consist of 17 equal rows (6 12). The rank of this matrix will be H = 1 16 linearly independent reactions are possible, i.e. skeleton isomerisation, doublebond shift along the chain and cis-trans conversion. Basing the calculation on 1 mole of 1-hexene, the set (5.121) will be reduced to a single equation of the type... 

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