Numerical solutions general values

There are four unknowns ag = ag)out which is independent of z and a g,a, and ai, which will generally vary in the z-direction. Equations (11.6) and (11.7) can be used to calculate the interfacial concentrations, and nj, if Ug and ai are known. A numerical solution for the general case begins with a guess for Ug. This allows Equation( 11.31) to be integrated so that nj, and are all calculated as functions of z. The results for are substituted into Equation (11.32) to check the assumed value for Ug. Analytical solutions are possible for a few special cases. 

Numerical solution. Numerical methods of solution do not produce the general solution given by Equations (/) and (g) but require that specific numerical values be provided for the parameters and give specific results. Suppose that px = 100 kPa and p4 = 1000 kPa. Let the gas be air so that A = 1.4. Then (k — 1 )/k = 0.286. Application of the BFGS algorithm to minimize W in Equation (c) as a function of p2 and p3 starting with p2— p3 = 500 yields... 

The control strategies for determining the feed policies were decided on the basis of a numerical solution of the terpolymerisations described by equations 1 - 3 using a microcomputer and a general purpose simulation package, BEEBSOC (10). Where necessary, these data were acquired in the course of this study, otherwise literature values were used. The apparent first order rate constants in terpolymerisations have been shown to be composition dependent. The variation in rate constants with... 

The resultant equations are non-linear and in this general case numerical solution techniques must be used. However, there exists a special case where an analytical solution may be obtained. If the increase in biomass concentration during flow through the reactor is small then an average value for the biomass concentration, independent of the distance Z along the fermenter, may be used. The material balance for the substrate over the reactor element may then be written ... 

In polymetallic systems, the larger the number of coupled ions, the larger the spreading of the S levels. As a consequence, even relatively small / values give rise to large separations of the S levels and therefore to depopulation of the highest levels. The general theory is still the same. Analytical solutions, as in dimeric systems, are seldom possible. Often, numerical solutions are possible. 

In general, the numerical solution of PDEs is much more difficult to automate than the solution of initial-value ODEs. The best method to be used is very dependent on the problem being solved. 

Differences in the Responses of the Different Types of Models. The basic differences that exist in the heat and mass balances for the different types of models determine deviations of the responses of types I and II with respect to type III. In a previous work (1) a method was developed to predict these deviations but for conditions of no increase in the radial mean temperature of the reactor (T0 >> Tw). In this work,the method is generalized for any values of T0 and Tw and for any kinetic equation. The proposed method allows the estimation of the error in the radial mean conversions of models I and II with respect to models III. Its validity is verified by comparing the predicted deviations with those calculated from the numerical solution of the two-dimensional models. A similar comparison could have been made with the numerical solution of the one-dimensional models. 

No such closed-form solution exists for the more general case v / 0. The general form of Eq. 18 can be solved for small values of the deposition modulus, [3, though as we will see later, such solutions are not applicable to the problem of interest." Fortunately, very accurate numerical solutions to the boundary value problem posed by Eqs. 8 and 18 are readily obtained using a numerical shooting technique. 

A numerical solution is given below for a feed volume of 150 cm and a 2 1 progression in time, values are given in Table (8.5) and these are inserted into the general equation to give the F values presented in Table 8.9. The dimensions of the eentrifuge bowl are such that y,. 364. 

The two equations of motion we discussed so far can be found in classical mechanics text books (the differential equations of motion as a function of the arch-length can be found in [5]). Amusingly, the usual derivation of the initial value equations starts from boundary value formulation while a numerical solution by the initial value approach is much more common. Shouldn t we try to solve the boundary value formulation first As discussed below the numerical solution for the boundary value representation is significantly more expensive, which explains the general preference to initial value solvers. Nevertheless, there is a subset of problems for which the boundary value formulation is more appropriate. For example boundary value formulation is likely to be efficient when we probe paths connecting two known end points. 

It is necessary to note that (44) is an approximation, because the value of y is lower than unity. This approximation is widely used in qualitative discussions, because it permits the simple mathematical treatment of electrochemical processes with relatively small errors and with clear physical meaning. If y 1 is included in the derivation of the general polarization curve equation, simple analytical solutions are not available and numerical solutions are required. 

A consequence of the complex interplay of the dielectric and thermal properties with the imposed microwave field is that both Maxwell s equations and the Fourier heat equation are mathematically nonlinear (i.e., they are in general nonlinear partial differential equations). Although analytical solutions have been proposed under particular assumptions, most often microwave heating is modeled numerically via methods such as finite difference time domain (FDTD) techniques. Both the analytical and the numerical solutions presume that the numerical values of the dielectric constants and the thermal conductivity are known over the temperature, microstructural, and chemical composition range of interest, but it is rare in practice to have such complete databases on the pertinent material properties. 

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