Sips equation

Sips [20] considered a combination of the Langmuir and Freundlich equations  

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Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

Adsorption isotherms play a key role in either the design of the adsorption-based process for the disposal of wastes containing VOCs or modeling the catalytic oxidation process. The equilibrium data for mesoporous sorbents are fitted to combined model of Langmuir and Sips equations. This hybrid isotherm model with four isotherm parameters... 

In this section, we present a number of popularly used isotherm equations. We start first with the earliest empirical equation proposed by Freundlich, and then Sips equation which is an extension of the Freundlich equation, modified such that the amount adsorbed in the Sips equation has a finite limit at sufficiently high pressure (or fluid concentration). We then present the two equations which are commonly used to describe well many data of hydrocarbons, carbon oxides on activated carbon and zeolite Toth and Unilan equations. A recent proposed equation by Keller et al. (1996), which has a form similar to that of Toth, is also discussed. Next, we... 

Figure 3.2-7 Fitting of the propane/activated carbon data with the Sips equation (symbol -data line ...

The optimal parameters from the fitting of the Sips equation with the experimental data are tabulated in Table 3.2-3. 

Table 3.2-3 Optimal parameters for the Sips equation in fitting propane data on activated carbon...

For useful description of adsorption equilibrium data at various temperatures, it is important to have the temperature dependence form of an isotherm equation. The temperature dependence of the Sips equation... 

Here b is the adsorption affinity constant at infinite temperature, bo is that at some reference temperature Tq, Uq is the parameter n at the same reference temperature and a is a constant parameter. The temperature dependence of the affinity constant b is taken from the that of the Langmuir equation. Unlike Q in the Langmuir equation, where it is the isosteric heat, invariant with the surface loading, the parameter Q in the Sips equation is only the measure of the adsorption heat. We shall discuss its physical meaning in Section 3.2.2.2.I. The temperature-dependent form of the exponent n is empirical and such form in eq. (3.2-18c) is chosen because of its simplicity. The saturation capacity can be either taken as constant or it can take the following temperature dependence ... 

This temperature dependence form of the Sips equation (3.2-18) can be used to fit adsorption equilibrium data of various temperatures simultaneously to yield the parameter bo, o> Q/RTq, Uq and a. 

ET an le 3.2 2 Fitting of propane/AC data with temperature dependent Sips equation... 

Using the data of propane at three temperatures 283, 303 and 333 K (Table 3.2-1) simultaneously in the fitting of the Sips equation (3.2-18), we get the following optimal parameters ... 

To obtain the isosteric heat for the temperature dependence form of the Sips equation as given in eq. (3.2-18), we use the van t Hoff equation... 

The following table shows the variation of the isosteric heat the amount adsorbed. Table 3.2-5 Isosteric heat as a function of loading using the Sips equation... 

Note the pattern of the isosteric heat with respect to the fractional loading, and the three curves intersect at the same point corresponding to the fractional loading of 0.5 and the isosteric heat of Q. This is the characteristics of the Sips equation. 

The previous two equations have their limitations. The Freundlich equation is not valid at low and high end of the pressure range, and the Sips equation is not valid at the low end as they both do not possess the correct Henry law type behaviour. One of the empirical equations that is popularly used and satisfies the... 

Several hundred sets of data for hydrocarbons on Nuxit-al charcoal obtained by Szepesy and Hies (Valenzuela and Myers, 1989) can be described well by this equation. Because of its simplicity in form and its correct behaviour at low and high pressures, the Toth equation is recommended as the first choice of isotherm equation for fitting data of many adsorbates such as hydrocarbons, carbon oxides, hydrogen sulfide, alcohols on activated carbon as well as zeolites. Sips equation presented in the last section is also recommended but when the behaviour in the Henry law region is needed, the Toth equation is the better choice. 

This measure of heat of adsorption is much higher than that derived from the fitting of the data with the Sips equation earlier, where we have found a value of 28750 J/mol for Q. This large difference should cause no alarm as the parameter Q is only the measure of adsorption heat. For example in the case of the Sips equation, Q is the isosteric heat of adsorption at a fractional loading of 0.5, while the parameter Q in the case of the Toth equation is the isosteric heat of adsorption at zero fractional loading as we shall show in the next section. 

Like the Sips equation, the isosteric heat of adsorption is a function of pressure (or loading), and it takes a value of infinity at zero loading and minus infinity at very high loading, which limits the applicability of the Toth equation in its use in the calculation of isosteric heat at two ends of the loading. The meaning of the parameter Q in the Toth equation is now clear in eq. (3.2-19f). It is equal to the isosteric heat when the fractional loading is zero... 

The isosteric heats calculated by the Toth equation are lower than those calculated by the Sips equation (Table 3.2-5). The above table shows the percentage differences between the values calculated by the Sips and Toth equations. The difference is seen to be significant enough for the isosteric heat to be used as the criterion to better select the isotherm equation. 

Various practical isotherm equations have been presented and they are useful in describing adsorption data of many adsorption systems. Among the many equations presented, the Toth equation is the attractive equation because of its correct behaviour at low and high loading. If the Henry behaviour is not critical then the Sips equation is also popular. For sub-critical vapours, multilayer isotherm equations are also presented in this chapter. Despite the many equations proposed in the literature, the BET equation still remains the most popular equation for the determination of surface area. When condensation occurs in the reduced pressure range of aroimd 0.4 to 0.995, the theory of condensation put forward by Kelvin is useful in the determination of the pore size as well as the pore size distribution. 

The set of equations presented in the previous section (5.3.3) in general can not be solved analytically hence it must be solved numerically. Even if the spreading pressure equation (5.3-21) can be integrated analytically, the inverse of the hypothetical pure component pressure versus spreading pressure is not generally available in analytical form with the exception of the Langmuir, Freundlich and Sips equations (see Table 5.3-3). 

The above conclusion for the Langmuir equation does not readily apply to other isotherms. For example, if the pure component isotherm is described by the Sips equation... 

We let u = (bP) ", then the Sips equation written in terms of this new variable as... 

This equation can also be derived by formally applying the lAST, using the Sips equation, Eq. (47), to describe pure component data. Hence, for adsorption of gas mixtures onto activated carbon, the use of Eq. (50) is generally justified. 

Because the exponential term a 1, the Sips equation does not have a correct low pressure limit, which is contrary to experimental observations. To overcome this problem and extend the range of applicability of isotherms of this type, Staudt et al. [77] introduced the pressure- and temperature-dependent exponents to ensure that Henry s law holds when pressure is sufficiently low. The new generalized form of Eq. (48) is... 

The Sips equation applies to adsorption on non-homogeneous surfaces over the entire pressure range. 

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