Parametric continuation

we have focused upon simulating dynamic systems Mx = /(jc 0) however, we can use these techniques to study how the solution Xs(0) of an algebraic system f x 0) = 0 depends upon 0. Let the parameters vary along some path 0(A,), 0 k 1, such as 

O and Param.l are 0q and 0i. The initial guess of the solution for 0 = /(x 0o) is xO. ArcLenParam is an optional structure that varies the performance of the algorithm see the help utihty for its description. x c and Param c are arrays in which each column vector contains the state Xs(A.) or parameter vector 0(A) for a point along the solution curve 0 = f(xs(k) 0(A)). Iambda c contains the value of A for each point, f norm.c contains the function norms for each point (should be near zero). stab c stores a value of 1 if the point is stable or critically stable and a value of 0 if the point is unstable. This routine traces only a single curve. For a more complex routine that can handle branching, see the AUTO package (indy.cs.concordia.ca/auto/). 

Example. Multiple steady states in a nonisothermal CSTR 

We demonstrate the use of arclength.continuation.m for the study ofmultiple steady states in a nonisothermal CSTR. Consider a perfectly-mixed stirred-tank reactor with A reacting to form B 

For constant volume CSTR with no volume change due to reaction, the mole balances for a volumetric feed rate v are 

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In the category of physical continuation methods are the thermodynamic homotopies of Vickery and Taylor [AIChE J., 32, 547 (1986)] and a related method due to Frantz and Van Brunt (AIChE National Meeting, Miami Beach, 1986). Thermodynamic continuation has also been used to find azeotropes in multicomponent systems by Fid-kowski et al. [Comput. Chem. Engng., 17, 1141 (1993)]. Parametric continuation methods may be considered to be physical continuation methods. The reflux ratio or bottoms flow rate has been used in parametric solutions of the MESH equations [Jelinek et al., Chem. Eng. Sd.,28, 1555(1973)]. 

One important advantage of the parametric continuation method is that intermediate problems have a physical implication. Consequently, each intermediate problem has a solution where the variables take up reasonable values. 

When we use a parametric continuation method such as the one described in the previous section. 

The program must include special parametric continuation methods to force convergence under certain hard conditions and to smooth out the nonideality of the problem, as discussed in Section 7.19. 

Consider, for instance, the optimal design of a unit operation or a chemical plant. Constraint equations constitute the most significant part of the overall problem. If we want to optimize the unit design, it is useful to adopt an outer optimizer that manages the small number of parameters dedicated to optimization, by solving the constraint system using a parametric continuation method. 

Chapter 14 introduces parametric continuation in optimization and process control, which is of significant practical interest in chemical and process engineering. This topic is broached in Vol. 4 - Buzzi-Ferraris and Manenti (in press). 

M is a matrix, in general itself a function of x and t, and singular, as it contains a row of all zeros for each algebraic equation. Finally, we present a robust method, based upon IVP solvers, to study how the solution to a set of nonlinear algebraic equations depends upon its parameters, parametric continuation. 

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