CONVECTIVE MASS-TRANSFER CORRELATIONS

Your objectives in studying this section are to be able to  

Estimate convective mass-transfer coefficients for the following situations (a) flow paralell to a flat surface, (b) flow past a single sphere, (c) flow normal to a single cylinder, (d) turbulent flow in circular pipes, (e) flow through packed and fluidized beds, and (f) flow through the shell side of a hollow-fiber membrane module. 

Use the corresponding coefficients to solve typical mass-transfer problems. 

Thus far, we have considered mass-transfer correlations developed from analogies with heat transfer. In this section, we present a few of the correlations developed directly from experimental mass-transfer data in the literature. Others more appropriate to particular types of mass-transfer equipment will be introduced as needed. 

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For example, the convective mass transfer correlation for a gas flow channel with square cross-sectional area is given as... 

The blood flow patterns within the fiber interstitial space lead to a complex convection-diffusion process that determines the mass transfer coefficient in the blood phase. Given the complexity of the mass transfer processes in blood, convective mass transfer correlations are often used for design purposes or to analyze experimental data. A convective mass transfer correlation takes the general form ... 

The mass-transfer correlation obtained by Bohm et al. (B9), Eq. (33), in Table VII, is conspicuous for its remarkably high exponent (0.85) on the GrSc product. Since the current is almost independent of diffusivity, this must mean that the reacting ion is depleted at the downstream end of the narrow slit between the cathode and diaphragm. The total current then is determined largely by the convective transport of reactant into the slit, which, in turn, depends on the density difference but not on diffusivity. 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

Here we review some of the correlations of convective mass transfer. We will find that many reactors are controlled by mass transfer processes so this topic is essential in describing many chemical reactors. This discussion will necessarily be very brief and qualitative, and we win summarize material that most students have encountered in previous courses in mass transfer. Our goal is to write down some of the simple correlations so we can work examples. The assumptions in and validity of particular expressions should of course be checked if one is interested in serious estimations for particular reactor problems. We will only consider here the mass transfer correlations for gases because for liquids the correlations are more comphcated and cannot be easily generalized. 

Solution of the shrinking core model at zero time (t=0) depends only on two parameters the solubility of solute in SC CO2 and the external particle to fluid mass transfer coefficient Kq. Hence, knowing the solubility, measurements of the initial extraction rates allow to determine the values of K(j. Detailed discussion on the evaluated mass transfer coefficients are given in [7].These authors found that the overall mass transfer from particles to fluid depends upon both free and forced convection mechanism. Figure 2 illustrates a parity plot of die experimental values of Sh number (evaluated by zero-time solution of the shrinking core model) and the calculated Sh number (using an appropriate mass transfer correlation). 

Here /iq is the convective mass transfer coefficient for an unspecified geometry. For a given geometry, ho would contain the appropriate boundary layer thickness, or it would have to be determined by independent measurements giving correlations that permit /jq to be found from other parameters of the system. More interestingly, Eq. (22) should be compared to Eq. (179) in Chapter 6, which can be written as... 

Mass Transfer Correlation. The development of a correlation for the mass transfer coefficient is based upon the mass transfer coefficients which were obtained through the use of the cell model proposed by Kramers and Alberda ( [). Buoyant effects become important under supercritical conditions because of the small kinematic viscosities which are a consequence of the high densities and low viscosities. Consequently it is necessary to consider both forced and natural convection when attempting to correlate mass transfer coefficients under supercritical conditions. 

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. 

In Equations (26.78a,b), the subscripts F and N refer to forced and natural convection stirring. In Equations (26.79a-d), L is a characteristic electtode dimension, D is the diffusion coefficient, v is velocity, g is the gravity acceleration constant, p is viscosity, is bulk solution density, and Ap is the difference in density between the bulk solution and the solution at the electrode surface. Once a Sh or expression has been found/generated, it is combined with Equation (26.70) (or the analogous equation for a supporting electrolyte system) to obtain an / relationship. Examples of mass transfer correlations follow. 

Explain the concept and importance of dimensional analysis in correlating experimental data on convective mass-transfer coefficients. 

For both forced and natural convection, relations have been obtained by dimensional analysis which suggest that a correlation of experimental data may be in terms of three variables instead of the original six. This reduction in variables has aided investigators who have developed correlations for estimating convective mass-transfer coefficients in a variety of situations. 

Estimate convective mass-transfer coefficients for multicomponent problems based on binary correlations. 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

Local Interfacial Molar Flux. Results for P(0 and Sc(t) via (11-199) and (11-202), respectively, represent basic information from which interphase mass transfer correlations can be developed. Gas-liquid mass transfer of mobile component A occurs because it is soluble in the liquid phase, and there is a nonzero radial component of the total molar flux of A, evaluated at r = R(t). Even though motion of the interface induces convective mass transfer in the radial direction, there is no relative velocity of the fluid with respect to the interface at r = R t). It should be emphasized that a convective contribution to interphase mass transfer in the radial direction occurs only when motion of the interface differs from Vr of the liquid at r = R. Hence, Pick s first law of diffusion is sufficient to calculate the molar flux of species A normal to the interface at r = R t) when f > 0 ... 

Generating the Design Correlation between the Average Residence Time for Convective Mass Transfer and the Intrapellet Damkohler Number... 

The rotating electrode cell employs a working electrode which is rotated about its axis. Because the electrolyte is stirred in the vicinity of the electrode-electrolyte interface, convective mass transfer occurs, for which the following correlation can be employed ... 

In most industrial applications, the jet temperatures are such that the radiative heat transfer contribution is very small (a small percentage) of the convective. Thus, a design based purely on convective heat or mass transfer is conservative. This chapter will therefore consider only convective heat or mass transfer correlations for design. The reader is referred to the literature for full details. 

The drop radius is determined by the mass rate of change due to evaporation or condensation. Taking the convective mass transfer into account, these phenomena can be described by the Frbssling correlation [16],... 

Introduction. For many years mass-transfer coefficients, which were based primarily on empirical correlations, have been used in the design of process equipment. A better understanding of the mechanisms of turbulence is needed before we can give a theoretical explanation of convective-mass-transfer coefficients. Some theories of convective mass transfer, such as the eddy diffusivity theory, have been presented in this chapter. In the following sections we present briefly some of these theories and also discuss how they can be used to extend empirical correlations. 

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