Driving Force LDF Model

The linear driving force (LDF) model can be classified in the group of equilibrium transport dispersive models (Fig. 9.5). For this model it is no longer assumed that the mobile and the stationary phases are permanently in equilibrium state, so that an additional mass-balance equation for the stationary phase is required. Assuming a linear concentration gradient an effective mass-transfer coefficient keff is implemented, where all mass-transfer resistances and the diffusion into the pores of the particle are lumped together. In this model a constant local equilibrium between the solid and the liquid in the pores is assumed. 

Adding a mass-transfer term to Eq. (9.9) it is valid in this model for the mobile phase. Therefore the surface area of the adsorbent interfaced between the mobile and the stationary phase is determined for each differential volume element. 

The volume dVo, which is filled with the free fluid in a differential volume element of the column, can be calculated from  

The volume dVp, which is occupied by the stationary phase in the volume element, can be calculated from  

From the particle radius rp the number of particles per volume element p is determined. 

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The linear driving force (LDF) model with a constant diffusivities dqi 60Di. 

For a single-zone equivalent TMB model, an analytical solution is available for a linear isotherm, considering both axial mixing and a finite rate of mass transfer which is accounted for with the linear driving force (LDF) model model 2a) [18]. 

The most rigorous formulation to describe adsorbate transport inside the adsorbent particle is the chemical potential driving force model. A special case of this model for an isothermal adsorption system is the Fickian diffusion (FD), model which is frequently used to estimate an effective diffusivity for adsorption of component i (D,) from experimental uptake data for pure gases.The FD model, however, is not generally used for process design because of mathematical complexity. A simpler analytical model called linear driving force (LDF) model is often used. ° According to this model, the rate of adsorption of component i of a gas mixture... 

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 ... 

The solution of the model presented in Table 9.5-1 for a thin layer in a fixed bed, a pellet, and a microcrystal is very time consuming. Therefore, the linear driving force (LDF) model has been extended to take into accoimt further mass transfer resistances. The following extended LDF model takes into account the resistance provided by the concentration boundary layer around a pellet with volume-based outer surface a ... 

As a first approach, the linear driving force (LDF) model is adopted with negligible temperature effect. Mass conservation of adsorbate is given in dimensionless form as... 

Profile of q, z) at the end of the other step must be employed as an initial condition for each step. In Eqs. (11-32) and (11-36) an LDF model with partial pressure difference as a driving force is used, uho and Ulo are molar flow rates of the inert component in the adsorption and desorption steps, where total pressures are P and Pt. N,h and N,x represent the rates of mass transfer of component i between particle and fluid at the adsorption step and at the desorption step expressed in terms of linear driving force (LDF) model by taking the partial pressure difference or difference in amount adsorbed as the driving force of mass transfer. 

The linear driving force (LDF) model may be considered to be one reliable model which can be used for this purpose. In the LDF model, overall mass transfer coefficient ks<7v is the only rate parameter, which is usually related to the intraparticle diffusion coefficient. A, as... 

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. 

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 ... 

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. 

One of the most widely used adsorption rate model is the so-called Linear Driving Force (LDF) approximation [Carta Lewus, 2000]. It is not surprising that the models are applicable to the situation where adsorption/desorption can take place, i.e., the active zone where both adsorbents and desorbents are present. In contrast, the conventional rate models are normally not applicable to the inactive zones where adsorption/desorption cannot occur due to lack of a limiting-component. Such a rate... 

Fig. 9. Uptake curves for N2 in two samples of carbon molecular sieve showing conformity with diffusion model (eq. 24) for sample 1 (A), and with surface resistance model (eq. 26) for example 2 (0)j LDF = linear driving force. Data from ref. 18.

The solution for (Eq. 9.9) requires two boundary conditions on c, one on v an initial condition on c and similarly one initial condition on q. Finally we must prescribe the sink/source term for the adsorption. This can be done in the most general case by writing another pde to describe adsorption, which is the transport of the adsorbing species into the crystal structure of the formed adsorbent. This model must be sufficiently broad to allow us to calculate the uptake at any location in the packed bed and at any time during the process. In many cases it wiU be found expedient and quite satisfactory to prescribe the uptake term as some kind of linear driving force model (LDF). 

For adsorption rate, LeVan considered four models axial dispersion (this is not really a rate model but rather a flow model), external mass transfer, linear driving force approximation (LDF) and reaction kinetics. The purpose of this development was to restore these very compact equations with the variables of Wheeler equation for comparison. 

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... 

The most important mass transfer resistance is pore diffusion in the adsorberrt pellets. This depends on the diflusivity of the component / in the pores of the stationary phase (s) and the particle diameter = 2R. The LDF (linear driving force) model resirlts in the Gliickairf eqiration (see previous section) ... 

The linear driving force model (LDF) applies to both the crystal and the pellet. 

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