Mathematical model for mass transfer

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

Huang CJ and Huo CH. Mathematical models for mass transfer accompanied by reversible chemical reaction. AICHE J. 1965 11(5) 901 910. 

Mathematical Models for Mass Transfer with Reaction... 

II. MATHEMATICAL MODELS FOR MASS TRANSFER WITH REACTION IN LIQUID-LIQUID DISPERSIONS... 

Despite its frequent occurrence in chemical engineering systems, a theoretical model for mass transfer with a reversible reaction has not been completely analyzed, mainly because of the mathematical difficulties. 

Chalermchat, Y, Fincan, M., and Dejmek, P, Pulsed electric field treatment for solid-hquid extraction of red beetroot pigment mathematical modelling of mass transfer, J. Food Eng., 64, 229, 2004. 

A Perspective on the Skovborg-Rasmussen Model. With its simplification, the Skovborg-Rasmussen model represents perhaps the most approachable mathematical treatment for mass transfer controlled growth. As with all models, it should be studied carefully and used with caution. There are several limitations to the model ... 

The form of equation (21) is interesting. It shows that the uptake curve for a system controlled by heat transfer within the adsorbent mass has an equivalent mathematical form to that of the isothermal uptake by the Fickian diffusion model for mass transfer [26]. The isothermal model hag mass diffusivity (D/R ) instead of thermal diffusivity (a/R ) in the exponential terms of equation (21). According to equation (21), uptake will be proportional to at the early stages of the process which is usually accepted as evidence of intraparticle diffusion [27]. This study shows that such behavior may also be caused by heat transfer resistance inside the adsorbent mass. Equation (22) shows that the surface temperature of the adsorbent particle will remain at T at all t and the maximum temperature rise of the adsorbent is T at the center of the particle at t = 0. The magnitude of T depends on (n -n ), q, c and (3, and can be very small in a differential test. 

Refined models for mass transfer to a disk electrode are presented in Section 11.6. The equivalent circuit presented in Figure 20.12 was regressed to the impedance data. The mathematical formulation for the model is given as equation (17.1). Four models were considered for the convective-diffusion term Zd (/) ... 

Significant successes are reached achieved in the field of mathematical modeling process mass transfer in vortical devices. The received dependences enable theoretically to define determine influence of various factors on efficiency of process and to generalize experimental data. At the same time kinetics mass transfer at separate stages, for example at dispersion liquids, impact of drops, it is studied investigated insufficiently full. 

Mao and White developed a mathematical model for discharge of an Li / TiS2 cell [39]. Their model predicts that increasing the thickness of the separator from 25 to 100 pm decreases discharge capacity from 95 percent to about 90 percent further increasing separator thickness to 200 pm reduced discharge capacity to 75 percent. These theoretical results indicate that conventional separators (25-37 pm thick) do not significantly limit mass transfer of lithium. 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

The reaction is reversible and therefore the products should be removed from the reaction zone to improve conversion. The process was catalyzed by a commercially available poly(styrene-divinyl benzene) support, which played the dual role of catalyst and selective sorbent. The affinity of this resin was the highest for water, followed by ethanol, acetic acid, and finally ethyl acetate. The mathematical analysis was based on an equilibrium dispersive model where mass transfer resistances were neglected. Although many experiments were performed at different fed compositions, we will focus here on the one exhibiting the most complex behavior see Fig. 5. 

In order to solve the mathematical model for the emulsion hquid membrane, the model parameters, i. e., external mass transfer coefficient (Km), effective diffu-sivity (D ff), and rate constant of the forward reaction (kj) can be estimated by well known procedures reported in the Hterature [72 - 74]. The external phase mass transfer coefficient can be calculated by the correlation of Calderback and Moo-Young [72] with reasonable accuracy. The value of the solute diffusivity (Da) required in the correlation can be calculated by the well-known Wilke-Chang correlation [73]. The value of the diffusivity of the complex involved in the procedure can also be estimated by Wilke-Chang correlation [73] and the internal phase mass transfer co-efficient (surfactant resistance) by the method developed by Gu et al. [75]. 

Recently, Pacheco et al developed and validated a pseudo-homogeneous mathematical model for ATR of i-Cg and the subsequent WGS reaction, based on the reaction kinetics and intraparticle mass transfer resistance. They regressed kinetic expressions from the literature for POX and SR to determine the kinetic parameters from their i-Cg ATR experimental data using Pt on ceria. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

Blasco et al. [12] proposed two-dimensional mathematical model for the drying process of dense phase pneumatic conveying. However, heat and mass transfer were not considered and therefore their model may be used for dense phase pneumatic transport only. In their paper, both experimental and numerical predictions for axial and radial profiles for gas and solid velocity, axial profiles for solid concentration and pressure drop were presented. 

During the extraction an unsteady process prevails. The present paper presents an unsteady state mathematical model for a fixed bed extractor (model 1). The overall mass transfer coefficients were calculated by matching the calculated and experimental values of oil loading in CO2. The results are compared with those obtained by the model developed by Catchpole et al, 1994 (model II). Good agreement between both models results and our experimental measurements were obtained, although the model II allows the best fit over the entire extraction curve. 

The method can be applied to investigate the bidisperse pore structures, which consist of small microporous particles formed into macroporous pellets with a clay binder. In such a structure there are three distinct resistances to mass transfer, associated with diffusion through the external fluid film, the pellet macropores, and the micropores. Haynes and Sarma [24] developed a suitable mathematical model for such a system. 

Although under certain experimental condition, step 2 or step 3 may be ignored due to its non-conspicuous influence to the flnal pyrolysis product distribution, however, a good mathematical model for biomass pyrolysis should be versatile applicable to other pyrolysis conditions, and thus it should be involved the above-mentioned three steps of process, of course heat and mass transfer equations should be included also. This paper presents this kind of mathematical model. Although the model is constructed based on sawdust pyrolysis, it is quite straightforward to apply the same approach to other cases such as straw and municipal solid waste pyrolysis even if to biomass or coal gasification or metal ore reduction. 

The mathematical models for the convective-diffusion impedance associated with convective diffusion to a disk electrode are developed here in the context of a generalized framework in which a normalized expression accoxmts for the influence of mass transfer. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

Further study of the mass and heat transfer, the kinetics of substrate consumption, cell growth and product formation, estabhshment of a workable mathematical model for process scale up and optimization for sohd-state fermentation. 

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