Runge semi-implicit

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

M72 Solution of stiff differential equations semi-implicit Runge-Kutta method with backsteps Rosenbrock-Gottwa1d-Wanner 7200 7416... 

The basic formula of the semi-implicit Runge-Kutta methods is similar to... 

One-step algorithms have been developed and used by Prothero et al. [146, 166], Cdme et al. [156, 168], Pratt [177], Villadsen et al. [178] and Layokun and Slater [179]. Embedded semi-implicit Runge—Kutta algorithms have been discussed by Lapidus and co-workers [180]. 

IIIN) Prokopakis, G. J., Seider, W. D. Adaptive Semi-implicit Runge-Kutta Method for Solu-... 

An extension of semi-implicit Runge-Kutta methods is the Rosenbrock-Wanner method. In this method eq. (10.63) is replaced by the expression... 

To illustrate the principle this device is based on, let us consider the following two-terms semi-implicit Runge-Kutta method ... 

Michelsen s third order semi-implicit Runge-Kutta method is a modified version of the method originally proposed by Caillaud and Padmanabhan (1971). This third-order semi-implicit method is an improvement over the original version of semi-implicit methods proposed in 1963 by Rosenbrock. 

Caillaud, J.B., and L. Padmanabhan, An Improved Semi-Implicit Runge-Kutta Method for Stiff Systems, Chem. Eng. J. 2, 22 -2i2 (1971). 

Weimer, A.W., and D.E. Clough, A Critical Evaluation of the Semi-Implicit Runge-Kutta Methods for Stiff Systems, AIChEJ. 25, 730-732 (1979). 

The computational cost of the LU-SSOR scheme is comparable to that of the two-step explicit scheme. The damping properties of the error of the LU-SSOR method tend to be a bit worse when compared to explicit multistep methods, such as the simplified Runge-Kutta method. However, implicit or semi-implicit methods are preferred to solve stiff systems of equations. 

Caillaud JB, Padmanabhan L (1971) An improved semi-implicit Runge-Kutta method for stiff systems. Chem Eng J 2 227. 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

For stiff differential equations, an explicit method cannot be used to obtain a stable solution. To solve stiff systems, an unrealistically short step size h is required. On the other hand, the use of an implicit method requires an iterative solution of a nonlinear algebraic equation system, that is, a solution of k, from Equation A2.4. By a Taylor series development of y + /=i il i truncation after the first term, a semi-implicit Runge-Kutta method is obtained. The term k, can be calculated from [1]... 

When using the semi-implicit Runge-Kutta method, the calculation of the Jacobian matrix can be critical. The Jacobian, J, influences directly the values of the parameters k, and, thereby, the whole solution of y. An analytical expression of the Jacobian is always preferable over a numerical approximation. If the differentiation of the function is cumbersome, an approximation can be obtained with forward differences... 

TABLE A2.1 Coefficients in a Few Semi-implicit Runge-Kutta Methods... 

After calculating k, according to Equation A2.4, is easily obtained from Equation A2.3. The coefficients an and bi for some semi-implicit Runge-Kutta methods are summarized in Table A2.1 [1-3]. 

Systems 3.1 and 3.2 contain N number of equations. The algebraic equation system can be solved with the Newton-Raphson method [1] and the ordinary differential equations with the semi-implicit Runge-Kutta method, Michelen s semi-implicit Runge-Kutta method [2], or, alternatively, with the Rosenbrock-Wanner semi-implicit Runge-Kutta method [2],... 

When using the Newton-Raphson method, the subroutine NONLIN [3] is used. The subroutine NONLIN solves the equation system 3.1 with respect to y. Furthermore, x is considered as a continuity parameter. The solution to equation system 3.1 is thus obtained as a function of the parameter x [3]. When using the semi-implicit Runge-Kutta methods, the subroutines SIRKM [4] and ROW4B [5] are used. These are used to solve equation system 3.2. 

A similar equation can be set up for the pressure drop. The combined model now contains K+1 coupled parabolic partial differential equations and one ordinary first order differential equation. They are solved by discretization in the radial direction by use of the orthogonal collocation method, and integration of the resulting set of coupled first order differential equations by use of a semi-implicit Runge-Kutta method. With this model and the used solution method, one can now concentrate on the effective transport properties given in PeH, Pcm and the wall heat transfer coefficient, with the latter being the most important parameter for design. 

The semi-implicit scheme is chosen to solve the mass and momentum conservation equations where only the pressure is implicitly solved. The Runge-Kutta... 

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