Linear driving force assumption

In this report, a kinetic model based on the solid film linear driving force assumption is used. Unlike the equilibrium-dispersive model, which lumps all transfer and kinetic effects into an effective dispersion term, the kinetic model is effective when the column efficiency is low and the effects of column kinetics are significant. 

There are a number of other solutions primarily bttsed on various assumed mechanisms of mass transfer. One such approach replaces the mass balance equation for diffusion within the particle by simplifying assumptions. In what is known as the linear driving force assumption, Glueckauf (1955b) su ested that, for (Dipt/r ) >0.1,... 

Note that since there are only two phases, i = (l- 2). Consider steady state the terms on the left-hand side disappear. We now employ the linear driving force assumption (7.1.5b) along with linear equilibrium relation (7.1.18a), C,i/Ca=Ka and the relation v z - z/t) relating the solid-phase velocity to time and axial coordinates ... 

The following equations constitute the approximate solutions of the fixed-bed model under the constant pattern and plug-flow assumption for the favorable Langmuir isotherm and linear driving forces (Perry and Green, 1999) ... 

To represent the adsorption dynamics in column, the linear driving force (LDF) approximation model for overall mass transfer coefficient was applied. The LDF model for gas adsorption dynamics is frequently and successfully used for analysis of column dynamics because it is simple, analytic, and physically consistent [7J. We assumed that the velocity of the gas in column is constant, and radial temperature, concentration and velocity gradients within the bed are negligible in this model. With the ideal gas-law assumption, the set of equation for this work is as follow ... 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

A brief comment should be made concerning the use of the Nernst-Planck equations for ion transport across the liquid film (e.g., Copeland and Marchello [1969], Kataoka et al. [1987]). This is a nonlinear, three-ion problem because of the presence of at least one coion at comparable concentration. The Nernst film model relies on the assumption of a linear concentration gradient in the liquid film. The film has no physical reality, and the calculation of nonlinear concentration profiles in it overburdens the model and offers little improvement over the much simpler linear driving force approximation. For higher accuracy, more refined and complex hydrodynamic models would have to be used (Van Brocklin and David, 1975). 

Pb removal from water by means of a Na-exchanged phillipsite-rich tuff was studied, using a diffusional model. The model is based on the linear driving force (LDF) approximation and takes into account both fluid-particle and intra-particle resistances to diffusion, making no specific assumption on their relative magnitude or on the form of the ion exchange isotherm. 

The model presented in Table 9.5-1 is complex and calculatiorrs are time consuming. Therefore, many simplifications of the model are known and experimental breakthrongh curves have been compared with results obtained from simplified models. Most known is the linear driving force (LDF) model. As a rule the following assumptions are made ... 

Despite the wide choice of mass transfer models which are available, the simplest and most popular adsorption rate expression is the linear driving force model because it represents actual processes reasonably well and reduces the computational effort required. An example of how this and various other simplifications and empirical correlations can be incorporated into the design and analysis of pressure swing adsorption processes is provided by White and Barkley (1989). The example used is the drying of air. Examples of how simplifying assumptions can aid the modelling of PSA air separation processes is provided by Knaebel and Hill (1985) and by Kayser and Knaebel (1989). Further information on cycle models can be found in Ruthven (1984), Yang (1987), Ruthven (1990) and Ruthven et al. (1994). 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

We examine next the cyclic voltammetric responses expected with nonlinear activation-driving force laws, such as the quasi-quadratic law deriving from the MHL model, and address the following issues (1) under which conditions linearization can lead to an acceptable approximation, and (2) how the cyclic voltammograms can be analyzed so as to derive the activation-driving force law and to evidence its nonlinear character, with no a priori assumptions about the form of the law. 

The linearity or proportionality between fluxes and conjugate driving force is also valid for the contributions to the flux of one species from the forces on the other species. Hence, with this assumption, one can write Eq. (4.265) in the form... 

The first postulate of irreversible thermodynamics is that the fluxes (or dependent variables) are directly proportional to the driving forces (or independent variables). [Actually, it may be shown that the assumption of local equilibrium follows from the assumption of a linear relation between the fluxes and driving forces (Truesdell, 1969).] If we take the di as dependent variables and the (m, — Wy) as independent variables we may, therefore, write... 

In Chap. 5 it was noted that for many physical phenomena that entail transport — whether it is charge, mass, or momentum — the assumption is usually made that the flux J is linearly proportional to the driving force F, or... 

For the particular case of the system air-water at near-ambient conditions, we are dealing with very dilute solutions of water vapor in air. For this reason, it is justified to assume that the total gas flow rate will remain relatively constant at its inlet value and that the gas phase will behave as if it were dry air. If we assume that the temperature and pressure will remain constant, it follows that the dimensionless numbers Re and Sc will not change along the column, justifying the assumption of a constant value for the mass-transfer coefficient. Dilute gas solutions, and NB - 0, also justify the use of a k-type coefficient coupled to a linear concentration driving force. These assumptions simplify considerably the analysis of the problem. 

We motivate (4.521), (4.522) here by plausible additional constitutive assumptions according to Truesdell [188], [13, Lect7] and Muller [18, Sect. 6.6] Let us consider a non-reacting three-constituent linear fluids mixture (n = 3 generalization on more constituents is possible [188]). To prove (4.521) it suffices to consider the special case with g = o, hj, = o (in driving force (4.512) gradyt = o, see below (4.511)) because vpg does not depend on them. Then by (4.137), (4.24)... 

To obtain the moisture profile in the machine direction, Nissan et al. [90] have made several assumptions about the drying process, principally that the drying was a first-order process (linear falling rate period) and that the pressing felt in phase 2 of the drying cycle reduced the evaporation rate to one tenth that in the sheet s free traverse between cylinders under similar temperature driving forces. Over each of the periods when the sheet touches the cylinder, it is assumed that the sheet temperature is constant (or linearly varying about an arithmetic mean), both in the plane of the sheet and normal to it. These concepts lead to a relatively simple relationship for the sheet temperature after a time interval t ... 

Assuming that the reaction (atom attachment and detachment) at the grain interface is linearly proportional to its driving force, Wagner derived a kinetic equation of interface-reaction-controlled grain growth and a stationary size distribution of grains. This assumption is questionable (see Section 15.2.3) and... 

---