Semianalytical Methods

Semianalytical methods attanpt to solve analytically Equation 11.31 by introducing diverse approximations this methodology has attracted several researchers (Gamble, Underdown, and Langford 1980 Gamble and Langford 1988 Buffle et al. 1990 Nederlof, Van Riemsdijk, and Koopal 1992, 1994), but currently numerical techniques are preferred. The main approaches are the differential equilibrium function (DEE) and the local isotherm approximations (LIA). 

The DEE approach starts with the application of the mass action law to the overall ligand system. In this case, the metal binding to the substrate is represented as (dropping charges for simplicity) 

A plot of log Ko (differential equilibrium function) has been accepted as characterization of heterogeneity by some authors (Gamble and Langford 1988), but later an estimation of the true cumulative distribution function was proposed as (Buffle et al. 1990) follows  

It can be easily shown (Nederlof, Van Riemsdijk, and Koopal 1992) that the DEF distribution implies a stepwise sorption-desorption process that is, the site types fill or empty sequentially. 

In the LIA approach, different approximations are introduced as the integral kernel (local isotherm) function in Equation 11.30. The simplest form is the condensation approximation (Harris 1968), C A, where a step function is introduced as the local isotherm  

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The solution of Eq. (2) can also be obtained by a numerical analysis similar to the calculus of finite differences. However, an analytical or semianalytical method based on Eq. (2) is not suitable for discussing the time-dependent distribution function because the calculation is lengthy. 

Bell (1960, 1963) developed a semianalytical method based on work done in the cooperative research program on shell and tube exchangers at the University of Delaware. His method accounts for the major bypass and leakage streams and is suitable for a manual calculation. Bell s method is outlined in Section 12.9.4 and illustrated in Example 12.3. 

Semianalytical Method for Parabolic Partial Differential Equations (PDEs)... 

Sometimes it is important to know how the temperature or concentration at the center of the cylinder changes with time. Analytical solutions for cylinders involve Bessel functions and an infinite series. With our semianalytical method, we can find how the temperature varies analytically with time. The time dependent variable at the center of the cylinder varies with time as ... 

Semianalytical Method for Parabolic Partial Differentia] Equations (PDEs) 387 = a8331161339e-l 5 09JU 0.013334P7307 + 0.0008469294252 ... 

Example 5.7. Semianalytical Method for PDEs with Known Initial Profiles... 

The semianalytical method developed earlier can be used to solve partial differential equations in composite domains also. Mass or heat transfer in composite domains involves two different diffusion coefficients or thermal conductivities in the two layers of the composite material.[6] In addition, even in case of solids with a single domain and constant physical properties, the reaction may take place mainly near the surface. This leads to the formation of boundary layer near one of the boundaries. In this section, the semianalytical method developed earlier is extended to composite domains. 

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