Mass transfer equation solutions

Fig. 10. Numerical solutions of the forced-convection mass-transfer equation for the case of irreversible first-order chemical reaction [after Johnson et al. (J4)] (Solid lines— rigid spheres dashed lines—circulating gas bubbles).

The sign of the transfer term will depend on the direction of mass transfer. Assuming solute transfer again to proceed in the direction from volume Vl to volume V( the component mass balance equations become for volume Vl... 

The mass transfer equation applicable to the transport-limited extraction of a solute from an aqueous solution to an organic phase (sink conditions), was derived ... 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

The size, shape and charge of the solute, the size and shape of the organism, the position of the organism with respect to other cells (plankton, floes, biofilms), and the nature of the flow regime, are all important factors when describing solute fluxes in the presence of fluid motion. Unfortunately, the resolution of most hydrodynamics problems is extremely involved, and typically bioavailability problems under environmental conditions are in the range of problems for which analytical solutions are not available. For this reason, the mass transfer equation in the presence of fluid motion (equation (17), cf. equation (14)) is often simplified as [48] ... 

An analytical solution of these mass-transfer equations for linear equilibrium was found by Thomas [36] for fixed bed operations. The Thomas solution can be further simplified if one assumes an infinitely small feed pulse (or feed arc in case of annular chromatography), and if the number of transfer units (n = k0azlu) is greater then five. The resulting approximate expression (Sherwood et al. [37]) is... 

In order to estimate resolution among peaks eluted from a chromatography column, those factors that affect N must first be elucidated. By definition, a low value of Hs will result in a large number of theoretical plates for a given column length. As discussed in Chapter 11, Equation 11.20 obtained by the rate model shows the effects of axial mixing of the mobile phase fluid and mass transfer of solutes on Hs. 

FORMULATION OF THE MASS TRANSFER EQUATIONS AND THEIR SOLUTIONS IN LAPLACE FORM... 

The basic mathematical model consists of water and solute mass balances in the concentrating and diluting tanks that are to be coupled with the solute—Eq. 11—and water—Eqs 12 and 13—mass transfer equations and voltage equation—Eq. 18—for the ED loop concerned. 

To illustrate the system behavior, the ternary mixture 1 = iso-propanol, 2 = water, and 3 = air is considered here. In order to obtain an algebraic solution, both the dif-fusivities of iso-propanol in air and iso-propanol in water vapor were assumed to be approximately the same, which is not far from reality. The liquid phase mass transfer resistance was negligibly small, as will be shown below. The phase equilibrium constants K/,c and Kjrs were calculated with activity coefficients from van Laar s equation. Water vapor diffuses 2.7-fold faster in the inert gas air than iso-propanol. The ratio of the respective mass transfer coefficients kj3 equals the ratio of the respective diffusivities to the power of 2/3rd according to standard convective mass transfer equations Sh =J Re, Sc). 

For large values of the Thiele modulus, the differential mass transfer equation has an asymptotic solution ... 

Hixson and Knox employed a more fundamental theoretical approach in their analysis of the effect of agitation on growth rate of single crystals (H7). They started with the basic equation for mass transfer of solute across a plane parallel to an interface and at a fixed distance from it ... 

Helfferich [2,3,30] states that in addition to the mutual interference of substances i and j, characterized by the phenomenological cross coefficients of the type L,j, one should take into account the presence of a coion in the ion exchanger as well. As a result, the simplified solution is inappropriate, even to the problem of ordinary IE. By use of only one diffusion mass-transfer equation, as in this case, account for the presence of co-ion has been neglected. It is, as a consequence, necessary to consider the Nemst-Planck relation for the co-ion also. 

A field-flow fractionation (FFF) channel is normally ribbonlike. The ratio of its breadth b to width w is usually larger than 40. This was the reason to consider the 2D models adequate for the description of hydrodynamic and mass-transfer processes in FFF channels. The longitudinal flow was approximated by the equation for the flow between infinite parallel plates, and the influence of the side walls on mass-transfer of solute was neglected in the most of FFF models, starting with standard theory of Giddings and more complicated models based on the generalized dispersion theory [1]. The authors of Ref. 1 were probably the first to assume that the difference in the experimental peak widths and predictions of the theory may be due to the influence of the side walls. 

The flow rate of both phases, viscosity, density, surface tension, and size and shape of the packing determine the value of a . These same factors affect the value of the mass transfer coefficients Ky and Kx. Therefore, it is expedient to include a in the mass transfer equation and define two new quantities KyU and Kxa. These quantities would then be correlated with the solution parameters as functions of various chemical systems. If A is the absorption tower cross-sectional area, and z the packing height, then Az is the tower packing volume. Defining Ai as the total interfacial area ... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

The model was solved using orthogonal collocation on finite elements (OCFE). Orthogonal collocation on finite elements was developed by Carey and Finlayson (26) for solution of boundary layer problems. Carey and Finlayson used OCFE to solve the simultaneous heat and mass transfer equations describing a catalyst pellet and found the new method to be more efficient than finite difference techniques. They also showed that OCFE was applicable to boundary layer problems that could not be solved by global orthogonal collocation. Jain and Schultz (27)... 

Hern idez, J.A., Pavon, G., and Garcia, M.A. Analytical solution of mass transfer equation considering shrinkage for modeling food-drying kinetics, /. Food Eng., 45,1, 2000. 

To apply the above equations to electrochemical mass transfer, the solution velocity [v in Equation (26.54)] must be known, which requires the solution of a separate set of fluid mechanics equations. Also, the transport parameters, D and M , must be specified, as will be discussed in some detail below. 

S.l. Exact Solutions of Linear Heat and Mass Transfer Equations... 

Polyanin, A. D., Vyazmin, A. V., Zhurov, A. I., and Kazenin, D. A., Handbook on Exact Solutions of Heat and Mass Transfer Equations, Faktorial, Moscow, 1998 [in Russian]. 

The sharp concentration fronts shown in Fig. [4.1-2 and 14.1-3 never occur in practice. The zones are always diluted and broadened by mass transfer and dispersion for both linear and nonlinenr isotherms. The complete solution of (he equilibrium equations, mass balances, and mass transfer equations for nonlinear systems is a formidable task requiring numerical solutions. For I incur systems the task is much easier and very useful solutions have been developed. Even thongh large-scale chromatography often is operated in the u out inear range, the linear analyses are valnable siace they can provide a qualitative feel (quantitative for linear systems) for band broadening effects. 

RATE OF MASS TRANSFER. Equations for mass transfer in fixed-bed adsorption are obtained by making a solute material balance for a section dL of the bed, as shown in Fig. 25.8. The rate of accumulation in the fluid and in the solid is the difference between input and output flows. The. change in superficial velocity is neglected ... 

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