Diffusion mixture model

In this section we derive the diffusion mixture model equations starting from the time averaged multi-fluid model expressed in terms of phase- and mass weighted variables [112]. The relative movement of the individual phases is given in terms of diffusion velocities. 

Mixture models (such as those of Scheffe) are still useful, especially when there are three or more such excipients with fairly large ranges of variation. In solid formulations, this is often the case for diluents (or fillers) and also for the polymers or waxes incorporated into controlled-release tablets to form a matrix through which the drug diffuses slowly out when immersed in aqueous fluid, i.e., in the gastrointestinal tract. 

Today two models are available for description of combined (diffusion and permeation) transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[21,22] and the Dusty Gas Model (DGM)[23,24]. Both models enable in future to connect multicomponent process simultaneously with process as catalytic reaction, gas-solid reaction or adsorption to porous medium. These models are based on the modified Stefan-Maxwell description of multicomponent diffusion in pores and on Darcy (DGM) or Weber (MTPM) equation for permeation. For mass transport due to composition differences (i.e. pure diffusion) both models are represented by an identical set of differential equation with two parameters (transport parameters) which characterise the pore structure. Because both models drastically simplify the real pore structure the transport parameters have to be determined experimentally. 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

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... 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

To determine the dispersed phase velocities as occurring in the phasic continuity equations in both formulations, the momentum equation of the dispersed phases are usually approximated by algebraic equations. Depending on the concept used to relate the phase k velocity to the mixture velocity the extended mixture model formulations are referred to as the algebraic slip-, diffusion- or drift flux models. 

When the diffusion velocity is eliminated, the main difference between the particular diffusion- and algebraic-slip mixture models presented in this... 

It must be emphasized, however, that the treatment presented in Eqs. (68)-(80) is too restrictive since it considers a simple diffusive relaxation only such a treatment applies to solid binary mixtures ( model B in the Hohenberg-Halperin [144] classification), while in a fluid binary mixture it is necessary to include the long range order parameter fluctuations that are transmitted by velocity fluctuations [145]. The resulting model model H in the Hohenberg-Halperin classification leads to a renormalization of the Onsager coefficient A(q) due to mode-coupling effects [146]. Equation [77] remains valid but A(q) is replaced by... 

Rull (2002) recently provided Raman spectroscopic evidence supporting the mixture model, a major fraction consisting of domains with linear HBs in a tetrahedral like configuration, the other of interstitial molecules, with either bifurcated or else weak or no HBs. Soper (2010) commented on the two-state model that the different domains must be very short lived, in view of the rapid diffusion of the water molecules, one of them moving over 150 molecular diameters away in 1 ms. 

The selection of a proper sorbent for a given separation is a complex problem. The predominant scientific basis for sorbent selection is the equilibrium isotherm. Diffusion rate is generally secondary in importance. The equilibrium isotherms of all constituents in the gas mixture, in the pressure and temperature range of operation, must be considered. As a first and oversimplified approximation, the pure-gas isotherms may be considered additive to yield the adsorption from a mixture. Models and theories for calculating mixed gas adsorption (Yang, 1987) should be used to provide better estimates for equilibrium adsorption. Based on the isotherms, the following factors that are important to the design of the separation process can be estimated ... 

Fainerman and Miller [35] found that displacement of an initially adsorbed surfactant by a second, more surface-active species allowed measurement of the desorption rate of the former. For example, competitive adsorption of sodium decyl sulfate and the nonionic Triton X-165 gave a desorption rate constant for the former of 40 s". Mul-queen and coworkers [36] recently developed a diffusion-based model to describe the kinetics of surface adsorption in multicomponent systems, based upon the Ward-Tor-dai equation. Experimental work with a binary mixture of two nonionic alkyl ethoxy-late surfectants [37] showed good agreement with the model, demonstrating a similar temporal adsorption profile to that found by Diamant and Andehnan [34],... 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

Thermal property evaluation took place with respect to thermal expansion, diffusivity, specific heat, and ultimately thermal conductivity. Instantaneous CTE data at 20°C is presented in Figure III. The thermal expansion of these materials is an important consideration to take account of, as many applications require matching CTE s to help reduce thermal mismatch stresses during cycling. Results are plotted with a rule of mixture model as well as the Turner and Kemer models for CTE. These are shown in Equations 5, 6, and 7 respectively. These predictions were also based on the SiCrSi system, with the property inputs provided in Table II. 

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