Using PDE solver

Access to powerful computers and to commercial partial-differential-equation (PDE) solvers has facilitated modeling of the impedance response of electrodes exhibiting distributions of reactivity. Use of these tools, coupled with development of localized impedance measurements, has introduced a renewed emphasis on the study of heterogenous surfaces. This coupling provides a nice example for the integration of experiment, modeling, and error analysis described in Chapter 23. 

The perturbed state variables are complex and the coefficients use the steady state solution as functions of x. Instead of a time solution, the PDE solver now has a series of steady state solutions with co as a varying parameter. 

The boundary condition implementations play a very critical role in the accuracy of the numerical simulations. The hydrodynamic boundary conditions for the LBM have been studied extensively. The conventional bounce-back rule is the most popular method used to treat the velocity boundary condition at the solid-fluid interface due to its easy implementation, where momentum from an incoming fluid particle is bounced back in the opposite direction as it hits the wall [20]. However, the conventional bounce-back rule has two main disadvantages. First, it requires the dimensionless relaxation time to be strictly within the range (0.5,2), otherwise the prediction will deviate from the correct result. Second, the nonslip boundary implemented by the conventional bounce-back rule is not located exactly on the boundary nodes, as mentioned before, which will lead to inconsistence when coupling with other partial differential equation (PDE) solvers on a same grid set [17]. 

Solving ID Heat Equation Using the MATLAB PDE Solver... 

Figure 10.30 shows a 3D plot for u (i.e., 0) as a function of x and f, using 200 mesh points for each dimension. Notice that the general behavior or topology of the surface is very similar to that of the anal5d ical solution shown earlier in Fig. 10.24. Keep in mind that the solution given by the MATLAB PDE solver is approximate.

The initial conditions and the spatial array of x points are defined on lines 19 and 20 of the listing. The call to the PDE solver on line 22 then returns a set of solutions from 0 to tmax (0.01 in this example) with 5 time solutions returned and with 5 sub time intervals used in solving the equations. Finally line 23 plots three of the solutions and line 24 saves all the ealeulated results. In order to monitor the progress of the solution set, the nprint = 1 parameters are set on tine 5 for both the odefd() and pdeivbv() functions. This produces printed output such as that shown in the selected output listing. It can be noted in the selected output that the solutions at each time increment are requiring 2 iterations of the basic Newton method used to solve the sets of equations. It is noted that the corrections on the second iteration are always within the machine aceuracy limits. This is because the equa-... 

A few other bookkeeping sections in the code convert the input solution array from an x-y labeled two index matrix into a single column array if needed on lines 454 through 459 and a final loop converts the single column solution array into a j,i two index matrix of solution values on lines 486 through 488. The type of solution desired may be specified as a table of values in order to pass additional parameters to the PDE solver. If a table is specified for the typsola parameter, two additional parameters, rx and ry, can be extracted from the input list as seen on line 444. For a direct sparse matrix solution with the SPM parameter these are not used. For the COE and SOR methods, this feature can be used to specify a spectral radius value with only the rx parameter used. For the ADI method these two values can be used to specify wx and wy parameters. In all cases, default values of these parameters are available, if values are not input to the various functions. 

An orthogonal co-location method can be used to convert the above partial differential equahons (PDE) into the ordinary differential equations (ODEs). An ODE solver, EPISODE can be used to solve the (ODEs) (25). In the model, the diffusivity is obtained from a batch kinetic study while the external mass transfer coefficient can be calculated from empirical equations or is available in the literature. The longitudinal dispersion coefficient (D ) is determined by matching the model output with the experimental data. Readers may like to refer to Chen and Wang s work (9) for detailed information. 

The balance equations described in the previous sections include both space and time derivatives. Apart from a few simple cases, the resulting set of coupled partial differential equations (PDE) cannot be solved analytically. The solution (the concentration profiles) must be obtained numerically, either using self-developed programs or commercially available dynamic process simulation tools. The latter can be distinguished in general equation solvers, where the model has to be implemented by the user, or special software dedicated to chromatography. Some providers are given in Tab. 6.3. 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

Stiff nonlinear PDEs cannot be solved using the Runge-Kutta subroutine (see chapter 2.2.5). Maple s stiff solver can he used to solve stiff nonlinear PDEs efficiently. 

In section 5.2.4, a stiff nonlinear PDE was solved using numerical method of lines. This stiff problem was handled by calling Maple s stiff solver. The temperature explodes after a certain time. The numerical method of lines (NMOL) technique was then extended to coupled nonlinear parabolic PDEs in section 5.2.5. By comparing with the analytical solution, we observed that NMOL predicts the behavior accurately. 

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