Derivation of the Mass Transfer Equation

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

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The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

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 ... 

The assumptions inherent in the derivation of the Hertz-Knudsen equation are (1) the vapor phase does not have a net motion (2) the bulk liquid temperature and corresponding vapor pressure determine the absolute rate of vaporization (3) the bulk vapor phase temperature and pressure determine the absolute rate of condensation (4) the gas-liquid interface is stationary and (5) the vapor phase acts as an ideal gas. The first assumption is rigorously valid only at equilibrium. For nonequilibrium conditions there will be a net motion of the vapor phase due to mass transfer across the vapor-liquid interface. The derivation of the expression for the absolute rate of condensation has been modified by Schrage (S2) to account for net motion in the vapor phase. The modified expression is... 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

The derivative bq/bt represents the rate of mass transfer from the fluid phase to the zeolite crystals. The form of the mass-transfer rate equation depends on the nature of the controlling resistance. 

Blaedel and Engstrom [48] noted that for a quasi-reversible process the current could be simply expressed in terms of the rate constant and mass-transport coefficient. Application of a square wave step in the rotation rate of a RDE (i.e., PRV, see Section 10.4.1.3) resulted in modulation of the diffusion-limited current and hence modulation of the mass-transfer coefficient. By solving the appropriate quadratic equation it was possible to derive a value for the heterogeneous rate constant for the electrochemical cathodic, kf, or anodic, kb, process of interest. Values for the standard heterogeneous rate constant and transfer coefficient were subsequently... 

The experimental technique involves batch gas absorption (by surface aeration) in a liquid. The pressure of the enclosed gas phase in the reactor decreases with time because of the absorption. This decrease in pressure with time allows the estimation of the mass-transfer rate and the volumetric mass-transfer coefficient, kLaL. The total pressure decrease until equilibrium is reached gives the equilibrium solubility C. The relevant equations for the calculations of C and kLaL are derived by Albal et al. (1983), Deimling et al. (1985), and Karandikar et al. (1986). These can be expressed as... 

Introduction of r, p = [P]/C , = nF/RT(E - E), and in the Volmer-Butler rate law [Eq. (143)] readily yields Eq. (168). The latter shows that a convenient dimensionless rate of electron transfer is A = k (5/D, since it compares the intrinsic value of the rate constant to that of the mass transfer process. Thus Eq. (168) reformulates as Eq. (169). Let us now examine the time- and space-dependent partial derivative equation of the kind demonstrated by Eq. (158), which describes variations in the concentration profiles in the stagnant layer adjacent to the electrode. For any species S, introducing t, y, and s leads to reformulation of Eq. (158) as in Eq. (170) ... 

The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi-component mixtures are treated first, before the material laws are introduced and the partial differential equations for the velocity, temperature and concentration fields are derived. As typical applications we consider heat and mass transfer in flow over bodies and through channels, in packed and fluidised beds as well as free convection and the superposition of free and forced convection. Finally an introduction to heat transfer in compressible fluids is presented. 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

The above derivatives must be evaluated at the interface composition before use in computing the Jacobian elements. This additional complexity in evaluating the derivatives of the vapor-phase mass transfer rate equations arises because we have used mass fluxes and mole fractions as independent variables. If we had used mass fractions in place of mole factions the derivatives of the rate equations would be simpler, but the derivatives of the equilibrium equations would be more complicated. For simplicity, we have ignored the dependence of the mass transfer coefficients themselves on the mixture composition and on the fluxes. 

Equations 15.9 and 15.10 are empirical with respect to the dehnition of the mass transfer coefficients, but the form of the equations is based on molecular diffusion theory. Applying the theory to a multi-component mixture where each component has a distinct diffusivity is impractically complex and must rely on diffusivity data for all the components in the mixture. To derive usable equations from the diffusion theory, certain simplifying assumptions must be made. The basis for the derivation of Equations 15.9 and 15.10 is to assume that mass transfer takes place either as equimolar counterdiffusion or as unimolar diffusion under dilute conditions. 

This equation is not restricted to liquids with constant density, as one might suspect from the fact that the overall mass density p appears to the left of the substantial derivative of the mass fraction of component i. Since v in the mass transfer equation represents the mass-averaged velocity of the mixture. 

Now, the hierarchy is summarized once again Ca depends on P, P depends on is a function of y and Sc, and Sc depends on t. One shonld not consider that independent variable y depends on r and t, because this was done previously, leading to a convective mass transfer term that contains the radial fluid velocity with respect to the motion of the interface. The appropriate partial derivatives of interest in the mass transfer equation are expressed in terms of P, f, and Sc. Detailed calculations are exactly the same as those provided by equations (11-58) ... 

Combination of variables will be successful if the mass transfer equation can be written exclusively in terms of f. For example, if one substitutes the three previous partial derivatives of the dimensionless molar density profile into the mass transfer equation for species A, then the following equation is obtained after multiplication by S ... 

Comparison with Exact Results. It is not unreasonable to suspect that truncation errors in the numerical approximation of first and second derivatives might accumulate in the computational scheme used to integrate the mass transfer equation. One check for accuracy involves a comparison between numerical results and exact analytical solutions. Of course, only a limited number of analytical solutions are available. For example, the following solutions have been obtained analytically for catalytic duct reactors ... 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

According to Harriott, Equations 6.14 and 6.15 should yield the minimum value of the mass transfer coefficient, (value of atu= Vp). Harriott s (1962a) smdy can be broadly divided into two parts (i) the effect of physical properties, in particular the liquid viscosity, density, and solute diffusivity (components of Schmidt number) along with the particle diameter, was derived from the variation of as calculated from Equation 6.14 or 6.15 for the respective parameters, (ii) The effect of power input (obtained from the impeller diameter, type, and speed of agitation) was derived... 

Correct splitting of a 2D problem in Figure 1.5 requires an accurate accoimt of the mass transfer through the channel/GDL interface. In this section, we derive the general mass conservation equation for the cathode channel of a PEFC or DMFC. 

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