Systems of non-linear algebraic equations

Systems of non-linear algebraic equations are directly encountered in the calculation involving a CFSTR and indirectly in the calculations involving batch and plug flow reactors. 

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In order to apply the concepts of modern control theory to this problem it is necessary to linearize Equations 1-9 about some steady state. This steady state is found by setting the time derivatives to zero and solving the resulting system of non-linear algebraic equations, given a set of inputs Q, I., and Min In the vicinity of the chosen steady state, the solution thus obtained is unique. No attempts have been made to determine possible state multiplicities at other operating conditions. Table II lists inputs, state variables, and outputs at steady state. This particular steady state was actually observed by fialsetia (8). 

Application of computer analytical methods. Extensive use of computer analytic methods are thought to intensify theoretical analysis drastically. They will be applied, in particular, to study kinetic models of complex reactions that can be represented by systems of non-linear algebraic equations, for the detailed bifurcation analysis, etc. 

The instantaneous copolymer composition X generally doesn t coincide with the monomer feed composition x from which the copolymer was produced. Such a coincidence X = x can occur only under some special values of monomer feed composition x, called azeotropic . According to definition these values can be calculated in the case of the terminal model (2.8) from a system of non-linear algebraic equations ... 

The basic element in a modular simulator is the unit operation model. A simulation model is obtained by means of conservation equations for mass, energy and momentum. These lead finally to a system of non-linear algebraic equations as ... 

M. J. D. Powell, A Fortran subroutine for solving systems of non-linear algebraic equations, AERE-R 5947, AERE, Harwell, 1968. 

This converts the system of ODEs into a system of non-linear algebraic equations that can be solved with standard root-finding methods. The steady-state assumption is not strictly necessary when solving a microkinetic model numerically, but the transient start-up and shut-down behavior is typically short in comparison to steady-state operation. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

The behaviour of the system is described by equations 7.26-7.30. The non-linear coupled two-point boundary value differential equations are difficult to solve as a part of the maximization procedure due to the excessive computational effort involved. The solid phase equations will therefore be recast into an equivalent set of non-linear algebraic equations using the orthogonal collocation method (Villad-sen and Michelsen, 1978). The application of this method to this problem is explained in Appendix C. 

The resulting model of raulticonponent enulsion pjolymerization systems is consituted by the Pffil 17, an integro-differential equation, a set of ordinary differential equations (equation 18 and 25 and the equations for pjoiymer conposltlon) and the system of the remaining non linear algebraic equations. As expected the conputatlonal effor t is concentrated on the solution of the PBE therefore, let us examine this aspect with some detail. 

Both methods are implicit by their nature because fn usually is a non-linear function of y . Eqns (10.66) and (10.67) imply a solution of a non-linear algebraic equation system of the type... 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

As discussed in Section 1.6.1 the microkinetic model may be solved as a system of ODEs or non-linear algebraic equations using the steady-state assumption. It turns out that, regardless of which approach you want to use, the function that must be passed to an ODE solver or numerical root-finding method is the same Here, the more general case of the ODE system is chosen. Note that we named the previously defined function get ratesQ. 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

We will get a non-trivial solution for these, if and only if the determinant of the associated matrix of the linear algebraic system given by above equations is zero i.e.,... 

The degrees of freedom analysis DOF) allows the user to determine the variables needed to be specified to execute a simulation. In steady state simulation the degrees of freedom are the number of variables that must be assigned to solve the non-linear algebraic system describing the operational unit, Here we adopt the approach called variable-minus-equations, in which DOF is equal with the number of variables minus the number of independent equations ... 

The systems discussed above belong to the class of non-linear dynamical systems. In such cases, the non-linear equation represents evolution of a solution with time or some variable like time. Such non-linear equations may be (i) algebraic, (ii) functional, (iii) ordinary partial differential equations and (iv) integral equations or a combination of these. Non-equilibrium systems can be defined by the type of equations as defined above involving the bifurcation parameter. The solution may change depending on the particular values of parameter. The solutions change at bifurcation points. Such situations do occur in the form of bistability and oscillations. 

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