Breakthrough curve numerical solution

The breakthrough curve for different values of the Courant number is given in Figure E7.3.1. A lower Courant number, less than 1, adds more numerical diffusion to the solution. If the Courant number is greater than 1, the solution is unstable. This Cou > 1 solution is not shown in Figure E7.3.1 because it dwarfs the actual solution. Thus, for a purely convective problem, the Courant number needs to be close to 1, but not greater than 1, for an accurate solution. In addition to the value of the Courant number, the amount of numerical diffusion depends on the value of the term UAz, which is the topic discussed next. 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Tphe breakthrough curve for a fixed-bed adsorption column may be pre-dieted theoretically from the solution of the appropriate mass-transfer rate equation subject to the boundary conditions imposed by the differential fluid phase mass balance for an element of the column. For molecular sieve adsorbents this problem is complicated by the nonlinearity of the equilibrium isotherm which leads to nonlinearities both in the differential equations and in the boundary conditions. This paper summarizes the principal conclusions reached from a recent numerical solution of this problem (1). The approximations involved in the analysis are realistic for many practical systems, and the validity of the theory is confirmed by comparison with experiment. 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

Numerical Solution of the Breakthrough Curve under Constant Pattern Behavior 657... 

Numerical Solutions of the Kinetic Model for a Breakthrough Curve.671... 

Garg and Ruthven [16] also discussed the influence of axial dispersion on the shape of the constant pattern breakthrough curves. They used a numerical solution of the liquid film linear driving force model. They concluded that the linear addition rule is approximately valid for nonlinear isotherms. The deviation from the result of an addition of the two contributions becomes important only when... 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

Using numerical solution, they showed that, in frontal analysis, denaturation results in multiple, unsymmetrical waves in the breakthrough curves, waves that can be mistakenly attributed to the presence of impurities. However, curves recorded at increasing flow rates permit the differentiation between the effects due to impurities and those due to denaturation. In isocratic elution, peaks of the native and the denatured forms cannot be fully separated because of the denaturation reaction. 

Equation [6] has analytical solutions, one of which is presented in the Appendix. It can be also solved numerically using finite differences (Benson, 1998). An example of the application of the analytical solutions of the FADE to describe solute transport is given in Fig. 2-4 that contains data on chloride transport in unsaturated sand reported by Toride et al. (1995). Measured and breakthrough curves and their fit with the ADE and the FADE at 11-, 17-, and 23-cm depths are shown. Parameters of the ADE and the FADE were optimized using a modified Marquardt-Levenberg algorithm2. Values of parameters and their standard errors are given in Table 2-3. 

Finite-difference techniques were used to compute numerical solutions as column-breakthrough curves because of the nonlinear Freundlich isotherm in each transport model. Along the column, 100 nodes were used, and 10 nodes were used in the side-pore direction for the profile model. A predictor-corrector calculation was used at each time step to account for nonlinearity. An iterative solver was used for the profile model whereas, a direct solution was used for the mixed side-pore and the rate-controlled sorption models. 

FIGURE 7. Simulated resident concentration, C, depth profiles (a,b, and c) and flux concentration, (J, breakthrough curves at 2 m depth (d,e, and f) for different dispersivities, (Black lines are analytical benchmarks, coloured lines are different numerical solutions ). Source Vanderborght J. et al. (2004). 

The set of equations formed by fheseequations is then solved numerically. Such models have been used extensively to describe breakthrough curves onto activated carbon of mono-component solutions of metal ions, micro-organic compounds or dyes [20-22], Some studies have demonstrated that they could be used to model binary namic adsorption [23] but they may not be applied in the case of complex multi-solute solutions. In addition, they do not take into account the pore characteristics of activated carbon materials, which are known to influence strongly the adsorption of micro-orgaiucs. In these cases, statistical tools like neural networks may be used in order to introduce such parameters as explicative variables. 

Modelling Reactive Transport of Organic Solutes in Groundwater 7.2.3 Numerical Evaluation of Breakthrough Curves... 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

The constant-pattern extension of equilibrium theory analysis is applicable only when the transition is, according to equilibrium theory, of shock form. When this condition is not met the only feasible approach to the prediction of the profile is the numerical solution of the coupled differential mass balance equations for the system. This approach has been followed by several authors and brief details of some of these studies are given in Table 9.1. As a representative example, the mathematical model used by Santacesaria et al. is summarized in Table 9.2, while examples of some of the experimental breakthrough curves, together with the model predictions, are shown in Figure 9.8. 

In the calculation of the predicted response curves the axial dispersion coefficient and the external mass transfer coefficient were estimated from standard correlations and the effective pore diffusivily was determined from batch uptake rate measurements with the same adsorbent particles. The model equations were solved by orthogonal collocation and the computation time required for the collocation solution ( 20 s) was shown to be substantially shorter than the time required to obtain solutions of comparable accuracy by various other standard numerical methods. It is evident that the fit of the experimental breakthrough curves is good. Since all parameters were determined independently this provides good evidence that the model is essentially correct and demonstrates the feasibility of modeling the behavior of fairly complex multicomponent dynamic systems. 

Except for the limiting case of the irreversible isotherm discussed above the prediction of the temperature and concentration profiles requires the simultaneous solution of the coupled differential heat and mass balance equations which describe the system. The earliest general numerical solutions for a nonisothermal adsorption column appear to have been given almost simultaneously by Carter and by Meyer and Weber. These studies all deal with binary adiabatic or near adiabatic systems with a small concentration of an adsorbable species in an inert carrier. Except for a difference in the form of the equilibrium relationship and the inclusion of intraparticle heat conduction and finite heat loss from the column wall in the work of Meyer and Weber, the mathematical models are similar. In both studies the predictive value of the mathematical model was confirmed by comparing experimental nonisothermal temperature and concentration breakthrough curves with the theoretical curves calculated from the model using the experimental equilibrium... 

Chapter 5, Yang, 1987). Mass and heat transfer are considered dispersive forces, which have a dispersive or smearing effect on the concentration and temperature fronts. The detailed bed profiles and breakthrough curves can be calculated only by the numerical solution of the mass and heat balance equations (Yang, 1987), coupled with the equations for equilibrium adsorption from mixture. 

Due to both common knowledge and related model calculations, at a technical scale that is characterized by L/2Rp > 200, axial dispersion as compared with convection contributes little to overall mass and heat transfer. Neglecting both steps in the calculation of breakthrough curves causes deviations within the order of magnitude of numerical errors for the solution of the resulting equations for convection. Further simplification is possible by lumping parameters. A comparison of mass transport resistances in a pellet leads to the following overall resistance [101] ... 

There are a number of other analytical solutions for Ci2(z,t) available in the literature for linear isotherms. These take into account axial dispersion, model particles using macropores and micropores, etc. they have been summarized by Ruthven (1984) in his Table 8.1. Widespread use of powerful computers and sophisticated numerical methods have reduced the importance of such analytical solutions for breakthrough curves. 

Various researchers, including Thomas (1944), Lapidus and Amundson (1952), Levenspiel and Bischoff (1963) and Rosen (1954) have produced analytical solutions to the coupled differential equations describing flow of adsorbate through a bed of adsorbent in which mass transfer and diffusion processes occur. Their solutions differ in detail but numerical representation of the breakthrough curves (see Chapters 5 and 6) of adsorbate from the adsorbent bed produces very similar results. Glueckauf and Coates (1947) and Glueckauf (1955) introduced a linear driving force expression for the rate of adsorption... 

Beyond these relatively simple systems and for all other non-linear isotherms, it is necessary to obtain solutions for the breakthrough curves by applying numerical approximation techniques to the model equations. Standard finite difference or collocation methods are commonly used. Table 6.3 provides a brief source list to solutions for plug flow and axially dispersed models with Langmuir, Freundlich or more general isotherms. 

Table 6.3 Summary of available numerical solutions for breakthrough curves in isothermal systems with plug flow or axially dispersed flow and Freundlich, Langmuir or other non-linear isotherms...

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