The Method of Lines

Divide the tube length into a number of equally sized increments, = LjJ, where J is an integer. A finite difference approximation for the partial derivative of concentration in the axial direction is 

This approximation is called a forward difference since it involves the forward point, z + Az, as well as the central point, z. (See Appendix 8.2 for a discussion of finite difference approximations.) Equation (8.16) is the simplest finite difference approximation for a first derivative. 

The approximations for the radial derivatives are substituted into the governing PDE, Equation (8.12), to give 

In this formulation, the concentrations have been discretized and are now given by a set of DDEs—a typical member of the set being Equation (8.21), which 

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Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

The Grank-Nicholson implicit method and the method of lines for numerical solution of these equations do not restrict the racial and axial increments as Eq. (P) does. They are more involved procedures, but the burden is placed on the computer in all cases. 

Example 8.5 Use the method of lines combined with Euler s method to determine the mixing-cup average outlet for the reactor of Example 8.4. ... 

The zero slope boundary condition at = 0 assumes S5anmetry with respect to the centerline. The mathematics are then entirely analogous to those for the tubular geometries considered previously. Applying the method of lines gives... 

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

The boundary conditions are unchanged. The method of lines solution continues to use a second-order approximation for dajdr and merely adds a Vr term to the coefficients for the points at r Ar. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz. ... 

The overall solution is based on the method of lines discussed in Chapter 8. The resulting DDEs can then be solved by any convenient method. Appendix 13.2 gives an Excel macro that solves the DDEs using Euler s method. Figure 13.9 shows the behavior of the streamlines. 

For both the finite difference and weighted residual methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear s method are popular. 

Apply the method of lines to the solution of the unsteady state dispersion reaction equation with closed end boundary conditions for which the partial differential equation for a second order reaction is,... 

Formulate the solution by the method of lines for a steady state reaction in a vessel where dispersion occurs radially and axially. 

The method of lines reduces a partial differential equation to a system of ordinary differential equations which can be solved by readily available software. It is applicable to PDEs that have only the first derivative of one of the variables, for example,... 

The method of lines replaces a partial differential equation with a set of ordinary differential equations. In an equation like that of P8.01.01, for instance,... 

Apply the method of lines to the heat and material balances of P8.01.04. The differential equations that apply except at the center and the wall are,... 

W. Pascher and R. Pregla, Vectorial analysis of bends in optical strip waveguides by the method of lines, Radio Science 28, 1229-1233 (1993). 

R. Pregla, The method of lines for the analysis of dielectric waveguide bends, J. Lightwave Technol. 14, 634-639 (1996). 

W. Pascher, Analysis of Waveguide Bends and Circuits by the Method of Lines and the Generalized Multipole Technique, Assoc. Prof, thesis, (FemUniversitat Hagen, Germany, 1998). 

R. Pregla, and W. Pascher, The method of lines, in T. Itoh (ed.). Numerical Technics for Microwave and Millimeter Wave Passive Structures, Wiley, New York, 381-446 (1989). 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

The method of lines can handle size-dependent growth rates, fines removal and product classification and is not restricted in the choice of the elements of the output vector y (t). The population densities at the grid points are system states, thus moments, L, CV, population densities at the grid points and the number or mass of crystals in a size range can be elements of y (t). 

The use of system identification and the method of lines will now be illustrated in an example. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

Figure 3 Response of L q to a step of +.4 1/s on Q obtained with the nonlinear model, i stem identification and the method of lines (201 grid points).

The method of lines and system identification are not restricted in their applicability. System identification is preferred because the order of the resulting state space model is significantly lower. Another advantage of system Identification is that it can directly be applied on experimental data without complicated analysis to determine the kinetic parameters. Furthermore, no model assumptions are required with respect to the form of the kinetic expressions, attrition, agglomeration, the occurence of growth rate dispersion, etc. 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

This implementation is second-order accurate with respect to the time and the size step. The scheme is general applicable and as shown in the next section, this scheme is also sensitive for discontinuities in Gn as caused by the R-Z model for fines removal. The oscillations are however less severe than for the method of lines. Also for this method a first-order scheme was Implemented. Here the so-called Lax scheme wcus chosen (8) ... 

This removal function gives rise to a discontinuity in the population density at the cutsize of the fines. The nucleation parameters are given in equation 19 In Figure 3 the responses are shown of the population density at 120 pm and of the growth rate after a step in the heat input to the crystallizer from 120 to I70 kW for three simulation edgorithms. The cut-size of the fines was 100 pm, a size dependent growth rate was used as described by Equation 4 with a= -250 and the number of grid points was kOO. When the simulation was performed with the method of lines, severe oscillations are present in the response of the population density at 120 pm, which dampen out rather slowly. Also the response of the Lax-Wendroff method shows these oscillations to a lesser extend. 

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