Collocation balance equations

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

The CNMMR model with laminar flow liquid stream in the annular region consists of three ordinary differential equations for the gas in the tube core and two partial differential equations for the liquid in the annular region. These equations are coupled through the diffusion-reaction equations inside the membrane and boundary conditions. The model can be solved by first discretizing the liquid-phase mass balance equations in the radial direction by the orthogonal collocation technique. The resulting equations are then solved by a semi-implicit integration procedure [Harold etal., 1989]. 

Using this technique and after some manipulation, the bulk phase mass and heat balance equations can be written in terms of the variables at the N intraparticle collocation points as follows ... 

For these simulations, the discretization of the mass balance equations used first order backward finite differences over a uniform grid of 100 intervals in the axial direction and a third order orthogonal collocation over 50 finite elements in the radial direction [16]. 

Since the mass balance equation (Eq. 8.120) is valid at any point inside the domain [0,1], we evaluate it at the ith interior collocation point as follows (note, as a reminder, the residual is zero at the collocation points)... 

The mass balance equation is discretized at the ith interior collocation point as before, and we have... 

We saw in the last example for the elliptic PDE that the orthogonal collocation was applied on two spatial domains (sometime called double collocation). Here, we wish to apply it to a parabolic PDE. The heat or mass balance equation used in Example 11.3 (Eq. 11.55) is used to demonstrate the technique. The difference between the treatment of parabolic and elliptic equations is significant. The collocation analysis of parabolic equations leads to coupled ODEs, in contrast to the algebraic result for the elliptic equations. 

To further demonstrate the simplicity and the straightforward nature of the orthogonal collocation method, we consider the adsorption problem dealt with in Section 12.2 where the singular perturbation approach was used. The nondimensional mass balance equations are... 

We are now ready to apply the orthogonal collocation to each subdomain. Let us start with the first subdomain. Let N be the number of interior collocation points hence, the total number of collocation points will be N + 2, including the points c, = 0 (center of particle) and y, = 1 Qunction point between the two subdomains). The mass balance equation (Eq. 12.237) will be valid only for the interior collocation points, that is. 

The mass balance equation (eqs. 9.5-23) and its boundary and initial conditions (eqs. 9.5-24) are cast into the non-dimensional form for the subsequent collocation analysis. We define the following non-dimensional variables ... 

Substituting eqs. (9.6-28) and (9.6-29) into (the heat and mass balance equations (eqs. 9.6-24), we can solve them numerically using the combination of the orthogonal collocation method and the Runge-Kutta method. Solving these equations we obtain the partial pressures of all components at N interior collocation points and temperature as a function of time. The adsorbed concentrations at these N collocation points are then calculated from eq. (9.6-27a). Hence the volumetric average partial pressures and adsorbed concentrations are obtained from the following quadrature formula ... 

Evaluating the mass balance equation in Section 1 (eq. A8.3-6) at the j-th interior collocation point, we have ... 

Applying the similar collocation analysis to the mass balance equation of Section 2 (eq. A8.3-7), we obtain ... 

The mass balance equation (A9.1-4) is valid at any point within the u domain. Thus, evaluating that equation at the i-th interior collocation point we get ... 

Written in terms of the collocation variables, the mass and heat balance equations are ... 

Similarly applying the transformation of eq. (A9.7-2), the heat balance equation (A9.7-4) can be written in terms of collocation variables as ... 

Next, we evaluate the mass balance equation along the pellet co-ordinate (A 10.4-6) at the i-th collocation point ... 

The interpolation points are chosen with N interior collocation points and the point at the boundary (u=l). Evaluating the mass balance equation (eq. A10.6-6a) at the interior point), we get ... 

We have completed the collocation analysis of the mass balance equation, now we turn to the heat balance equation ... 

This effort is already at an advanced stage steps one to four are, in essence, complete for the case of sandpacks and can be extended with little extra effort to apply to consolidated porous media. A collocation method for the integration of the population balance equations is also nearing completion. 

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

Similar to other collocation methods, the coordinates should be normalized within the interval [0, 1]. For the 1-D population balance equation given by Eq. 3.11, this can be done by introducing the linear transformation x = v/ Vmax, where x is the dimensionless particle volume and v ax is the maximum particle size in the system. The original integral intervals [0, v] and [0, oo] are transformed to [0, x] and [0, 1], respectively. Consequently, Eq. 3.11 becomes ... 

Three population balance equations with different kernel models have been successfully solved by using the wavelet collocation method (28). These kernel models were (1) size-independent kernel ) = = constant, (2) linear size-dependent... 

Website on the wavelet collocation methods for the computations of population balance equations. The operational matrices M and Mi, matrices A and B, together with the integral functions H and Q at various resolution levels are available at http //www.cheque.uq.edu.au/psdc. 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

Figure 7.22 illustrates the numerical solution of concentrations in the liquid phase of a tank reactor. The simulation also gives the concentration profiles in the liquid film, as shown in Figure 7.22b. The algebraic equation system describing the gas- and liquid-phase mass balances is solved by the Newton-Raphson method, whereas the differential equation system that describes the liquid film mass balances is solved using orthogonal collocation. To guarantee a reliable solution of the mass balances, the mass balance equations have been solved as a function of the reactor volume. The solution of the mass balances for the reactor volume, Vr, has been used as an initial estimate for the solution for the reactor volume, Vr -F A Vr. The simulations show an interesting phenomenon at a certain reactor volume, the concentration of the intermediate product, monochloro-p-cresol, passes a maximum. When the reactor volume—or the residence time— is increased, more and more of the final product, dichloro-p-cresol, is formed (Figure 7.22a). This shows that mixed reactions in gas-liquid systems behave in a manner similar to mixed reactions in homogeneous reactions (Section 3.8) [11,12].

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

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