Breakthrough curve analytical solution

To calculate the breakthrough curve, analytical solutions of the system of Equations 6.43a and b can be obtained using the Laplace transform method [99], It is then possible to express the solution of Equations 6.43a and b [99] as follows ... 

When the equilibrium relationship is nonlinear it is generally not possible to determine a general analytic solution for the breakthrough curve. Such solutions have been obtained, however, for a number of special cases of which the irreversible or rectangular isotherm is the simplest. The irreversible isotherm, sketched in Figure 8.14, may be considered as the extreme limit of a favorable type 1 isotherm for which /8 0 and, as such, it represents an important limiting case. 

Breakthrough curves such as those illustrated in Fig. 6 are typically analyzed using the following analytical solution for Eq. (81) [143,154] ... 

When no analyhcal soluhon can describe the process satisfactorily it may be possible, working from Eq. (9.18) (which describes the length of the wave) and either Eq. (9.11) or (9.13) (the expression for the velocity of the adsorption wave), to assemble a simple wave mechanics solution that approximates the length and movement of the mass transfer front in the bed. As with analytical solutions this method can deliver useful results that may approximate the wave shape inside the bed and thus can be used to describe the shape and duration of the breakthrough curve that occurs as the wave intercepts and crosses the end of the bed. Such methods are generally only applicable for one or at most two adsorbable components. 

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. 

Figure 4.27 Examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5).

In Figure 4.27, some examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5) are presented. As is clear, the curve corresponds to the case of equal and combined solid and liquid-film diffusion resistances ([ = 1) which is between the two extremes, i.e. solid diffusion control (l = 10,000) and liquid-film diffusion control ( = 0.0001). 

The analytical solution of this set of differential equations [Eqs. (5) and (6)] was first given by Thomas [38] for the frontal breakthrough curve. The concentration profile is... 

This model has been used by Thomas [83], Goldstein [84], and Wade ei al. [80]. Thomas has derived an analytical solution for a step function boundary condition (i.e., a breakthrough curve or frontal analysis problem) [83]. Goldstein [84] and Wade et al. [80] have derived analytical solutions for pulse boimdary conditions the overloaded elution problem). 

As an example of the experimental validity of the solution derived by Lapidus and Amundson, Figme 6.3 shows the breakthrough curve measured by Rixey and King [26] (symbols) for succinic acid on pre-wet Porapak Q (Waters-Millipore). The experimental results are in excellent agreement with the analytical solution (solid line) derived by Lapidus and Amtmdson [3]. The Peclet number and the external mass transfer coefficient were estimated using conventional relationships taken from the literature [26]. 

From a theoretical viewpoint, frontal analysis and displacement chromatography are important and interesting problems because there are as5onptotic solutions for the breakthrough curves of frontal analysis and for the band profiles in the isotachic train in displacement chromatography. An asymptotic solution is an analytical solution obtained after an infinite migration distance. The existence... 

An analytical solution can be foimd for the breakthrough curve provided the following two assumptions are accepted ... 

The asymptotic solutions given in Eqs. 14.6 and 14.7 were derived assuming that axial dispersion is negligible. Acrivos [15] has discussed the influence on the shape of the constant pattern breakthrough curve of the combination of axial dispersion and mass transfer resistance. An exact analytical solution can be derived only in the case of an irreversible adsorption isotherm (Req = 1/(1 - - bCo) = 0, or b infinite), and assuming a liquid film linear driving force model [15]. 

For this model (Eqs. 14.2 and 14.50), Thomas [23] derived an analytical solution in the case of a step function input, i.e., of a breakthrough curve in frontal analysis. This solution can be written in dimensionless form using the following transformation suggested by Hiester and Vermeulen [9,23]. This transformation uses the parameters defined in Eqs. 14.8a to 14.8 d x, Td, K, Req, Ngy, N/y) and... 

Experimental results [7,8] obtained in the case of the breakthrough curves of binary mixtiues imder constant pattern condition have been compared with the anal5ttical solution. Figure 16.1 compares the experimental breakthrough curves obtained in the case of the vapor phase adsorption of benzene and toluene carried by nitrogen through a bed of activated carbon [8] with the analytical solution calculated from the binary adsorption data and imder the assumption of constant pattern behavior [1,3]. The agreement achieved is excellent. 

An analytical expression for the breakthrough curve can be obtained by solving the equations describing continuity of a sorbate species in a fixed bed, the equilibrium relation between the solute and the sorbate, and the rate of adsorption and mass transfer, with the appropriate initial and boundary conditions. The exact solution of the complete set of equations is often impossible, but affinity chromatography lends itself to several convenient simplifications, with the result that analytical solutions are available. The notation used here is that of Vermeulen (4). 

Affinity chromatography is a particularly simple example of fixed-bed adsorption very tight binding of the solute during the adsorption step means that the shape of the breakthrough curve depends only on the rate-limiting mass transfer (or reaction) mechanism. Analytical expressions are available for a number of cases four that can be useful in the scale-up of affinity chromatography have been presented here. 

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. 

Isotherms can also be obtained by frontal chromatography [171-173]. The MIP is then packed into a column and used as the stationary phase in liquid chromatography. A solution of known concentration of analyte is applied continuously to the column until a breakthrough curve is obtained. After washing the column, the procedure is repeated with increasing concentrations of analyte. The amount analyte bound for each concentration is calculated from the breakthrough curves. 

The apparent diffusion coefficient, D, was determined for the particular leaching conditions of each of the thirteen experiments. This was accomplished using the measured chloride breakthrough (effluent concentration) curve and the analytical solution to Equation 7 with Kd==0. Examples of the observed and calculated chloride concentrations (determined by adjustment of D until a best fit was obtained) are presented for three different experiments (Experiments 7, 8, 11) in Figures 2-4. Values of D and the pore water velocity (v) determined for each experiment are presented in Table III. The value of D increased for cases with large v, and was different between soils for any particular v. This is consistent with the basic relationship be-... 

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). 

Figure 6.10 Comparison with analytical solution of Tang et al. (1981). Breakthrough curves for fracture at x = 0.76 m, for values of D in the range of 10 to lO m s . ...

Figure 2 A theoretical breakthrough curve obtained by percolating an idealized MIP cartridge with a dilute analyte solution.

If the equilibrium isotherm is linear, analytic expressions for the concentration front and the breakthrough curve may, in principle, be derived, however complex the kinetic model, but except when the boundary conditions are simple, the solutions may not be obtainable in closed form. With the widespread availability of fast digital computers the advantages of an analytic solution are less marked than they once were. Nevertheless, analytic solutions generally provide greater insight into the behavior of the system and have played a key role in the development of our understanding of the dynamics of adsorption columns. 

TABLE 8.1. Summary of Analytic Solutions for Breakthrough Curve for Linear, Isothermal Trace Component Systems... 

The solutions given in Table 8.1 were all obtained by Laplace transformation. To obtain the solution of the model equations in the Laplace domain is straightforward but inversion of the transform to obtain an analytic expression for the breakthrough curve or pulse response is difficult. Simple analytic expressions for the moments of the pulse response may, however, be derived rather easily directly from the solution in Laplace form by the application of van der Laan s theorem... 

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. ... 

Cen, P.L., and Yang, R.T., Analytic solution for adsorber breakthrough curves with bidisperse sorbents (zeolites), AIChEJ., 32(10), 1635-1641 (1986). 

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