Equations parabolic diffusion

Figure 6.26. Kinetics of Mn transformations (described by the parabolic diffusion equation) among three main labile solid-phase fractions of Israeli arid soils. Soils were incubated at the saturated paste regime (after Han and Banin, 1996. Reprinted from Soil Sci Soc Am J, 60, Han F.X., Banin A., Solid-phase manganese fractionation changes in saturated arid-zone soils Pathways and kinetics, p 1079, Copyright (1996), with permission from Soil Sci Soc Am)...

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

Equation (57) is empirical, except for the case where v = 0.5, then Eq. (57) is similar to the parabolic diffusion model. Equation (57) and various modified forms have been used by a number of researchers to describe the kinetics of solid phase sorption/desorption and chemical transformation processes [25, 121-122]. 

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

The parabolic diffusion law or equation can be used to determine whether diffusion-controlled phenomena are rate-limiting. This equation... 

Chien and Clayton (1980) compared several equations for describing P04 release from soils and found that the Elovich equation [Eq. (2.49)] was best based on the highest values of the simple correlation coefficient (r2) and the lowest SE. The two-constant rate equation also described the data satisfactorily. The parabolic diffusion equation was judged unsatisfactory due to low r2 and high SE values. 

However, when data from many of the kinetics studies on pesticide-soil interactions were plotted according to the parabolic diffusion equation, initial nonlinearity resulted (Fig. 6.4). This suggested that only at longer times did the reaction process conform to PD. The rate-limiting step for this reaction is diffusion into or out of micropores. 

A number of additional equations are often used to describe reaction kinetics in soil-water systems. These include the Elovich equation, the parabolic diffusion equation, and the fractional power equation. The Elovich equation was originally developed to describe the kinetics of gases on solid surfaces (Sparks, 1989, 1995 and references therein). More recently, the Elovich equation has been used to describe the kinetics of sorption and desorption of various inorganic materials in soils. According to Chien and Clayton (1980), the Elovich equation is given by... 

The parabolic diffusion equation is used to describe or indicate diffusion control processes in soils. It is given by (Sparks, 1995 and references therein)... 

When diffusion is considered as the rate-determining step, the parabolic diffusion equation can be applied (Zimens 1945 Boyd et al. 1947 Crank 1956 Chute and Quirk 1967 Wollast 1967 Jardine and Sparks 1984 Sparks 1999). 

Several workers have attempted to develop dissolution rate equations to model apatite dissolution (Olsen 1975, Smith et al. 1977, Christoffersen et al. 1978, Fox et al. 1978, Chien et al. 1980, Onken and Metheson 1982, Hull and Lerman 1985, Hull and Hull 1987, Chin and Nancollas 1991). Rate equations from these models include zero order, first order, parabolic diffusion, mixed order, and other forms. The most current model (Hull and Hull 1987) focuses on surface dissolution geometry, which the authors argue fit the experimental results better than previous dissolution models. These experiments and the dissolution rate equations derived from them are missing the experimental conditions that replicate the natural dissolution processes and agents in soils, as they do not include the range of apatite mineralogies likely to be naturally weathering in soils. 

Fig. 6. Plotting of Cd desorption by chloride or citrate from Fe oxides formed at initial citrate/Fe(II) MRs of (a) 0 and (b) 0.1 based on overall parabolic diffusion equation.

For the parabolic equation, the diffusion of ions or migration of electrons through the scale is controlling, and the rate, therefore, is inversely proportional to scale thickness. 

Example 6.2 Solution of Parabolic Partial Differential Equation for Diffusion. 

Phenomena of propagation such as vibrations are characterized by equations of hyperbolic type which are essentially different in their properties from other classes such as those which describe equihb-rium (elhptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows ... 

Parabolic The heat equation 3T/3t = 3 T/3t -i- 3 T/3y represents noneqmlibrium or unsteady states of heat conduction and diffusion. 

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