Mass-transfer coefficients, analogy with

In the absence of any experimental data at all, overall mass transfer coefficients for nonchemically reacting systems can be predicted from values of individual film mass transfer coefficients coupled with (16-23) or its HTU analog... 

The preceding method has an inherent problem—namely, the use of the interfacial compositions. This can be overcome by using overall mass transfer coefficients (analogous to the overall heat transfer coefficients, U, of Chapters 6 and 8). In order to do this we use values of X (liquid mole firaction in equilibrium with the vapor) and Y (vapor mole firaction in equilibrium with the vapor). 

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. 

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

The mass-transfer coefficient with a reactive solvent can be represented by multiplying the purely physical mass-transfer coefficient by an enhancement factor E that depends on a parameter called the Hatta number (analogous to the Thiele modulus in porous catalyst particles). 

When liquid-liquid contactors are used as reactors, values of their mass-transfer coefficients may be enhanced by reaction, analogously to those of gas-liquid processes. Reactions can occur in either or both phases or near the interface. Nitration of aromatics with HN03-H2S04 occurs in the aqueous phase [Albright and Hanson (eds.), Industrial and Laboratory Nitrations, ACS Symposium Series 22... 

As is shown in Figure 2, in the two-phase model the fluid bed reactor is assumed to be divided into two phases with mass transfer across the phase boundary. The mass transfer between the two phases and the subsequent reaction in the suspension phase are described in analogy to gas/liquid reactors, i.e. as an absorption of the reactants from the bubble phase with pseudo-homogeneous reaction in the suspension phase. Mass transfer from the bubble surface into the bulk of the suspension phase is described by the film theory with 6 being the thickness of the film. D is the diffusion coefficient of the gas and a denotes the mass transfer coefficient based on unit of transfer area between the two phases. 6 is given by 6 = D/a. 

Just as with the gas holdup, gas-liquid interfacial area should also be divided into two parts. The literature, however, gives a unified correlation. The same is true for volumetric gas-liquid mass transfer coefficients and mixing parameters for both gas and liquid phases. The fundamental r.echanism for inter-phase mass transfer and mixing for large bubbles is expected to be different from the one for small bubbles. Future work should develop a two phase model for the bubble column analogous to the two phase model for fluidized beds. 

The treatment is divided into four sections. Section II deals with estimation of coefficients of heat transfer and of mass transfer. Because most, or all, of the latent heat of evaporation of the moisture is normally derived from the sensible heat of the carrier gas, our knowledge of the pertinent coefficients of heat transfer from the gas to the surface of the drying solid is summarized. A summary of the analogous mass-transfer coefficients records in condensed form gives our current knowledge of the means of estimating the rate of transport from the solid to the gas of the vapor evolved. 

Consider a circular pipe of inner diameter D = 0.015 m whose inner surface is I covered with a layer of liquid water as a result of condensation (Fig. 14-49). In I order to dry the pipe, air at 300 K and 1 atm is forced to flov/ through if with an average velocity of 1.2 m/s. Using the analogy between heat and mass transfer, determine the mass transfer coefficient inside the pipe for fully developed flov/. 

Air flows in a 4-cm-diameter wet pipe at 20 C and 1 atm with an average velocity of 4 m/s in order to dry the surface. The Nussell number in this case can be determined from Nu = 0.023Re Pi"- where Ke = 10,550 and Pr = 0.731. Also, the diffusion coefficient of water vapor in air is 2.42 X 10 mVs. Using the analogy between heal and mass transfer, the mass transfer coefficient inside the pipe for fully developed flow becomes... 

The powerful analogy that exists among momentum, heat, and mass transport permits useful values of convective mass transfer coefficients to be calculated from known values of convective heat transfer coefficients. For a particular drying system with a specific geometry and flow characteristics, the following relationship is recommended. " ... 

In this the Nusselt number Nu = aL/X and the Sherwood number Sh = f3L/D are formed with the local heat and mass transfer coefficients. The ratio Nu/Re and Sh/Re is independent of the characteristic length as this is also contained in the Reynolds number Re = wmL/v. The equation (3.130) is known as the Reynolds analogy. The heat and mass transfer coefficients can be calculated with this as long as the friction factor is known. 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

Thus, for dilute solutions, the design equation for an absorber is identical in form to that for heat exchangers with the overall mass transfer coefficient Ky analogous to the overall heat transfer coefficient U and the log-mean of y-y at the two ends of the absorber analogous to the log-mean AT. 

In Figure 10.8 we have plotted the variation of the ratios of mass transfer coefficients 12/ 11 k i/k22 for an acetone-benzene-helium system considered in Example 11.5.3. The Chilton-Colburn analogy predicts that these ratios would be independent of Re, as shown by the horizontal lines in Figure 10.8. The von Karman turbulent model, on the other hand, predicts that the influence of coupling should decrease with increase in Re. The latter trend is in accord with our physical intuition. Depending on the driving forces for mass transfer, the Chilton-Colburn and the von Karman turbulent models could predict different directions of transfer of acetone (see, e.g., Krishna, 1982). 

A comparison of the interactive film models that use the Chilton-Colburn analogy to obtain the heat and mass transfer coefficients with the turbulent eddy diffusivity models. 

For heat transfer the film theory is the most commonly used model, and the physical picture of a laminar film in which the whole temperature difference is situated leads to a result analogous to the mass transfer coefficient model [5]. After integrating Fourier s law over the film, a comparison with the heat transfer coefficient model (5.126) yields ... 

At high enough qualities and mass fluxes, however, it would be expected that the nucleate boiling would be suppressed and the heat transfer would be by forced convection, analogous to that for the evaporation for pure fluids. Shock [282] considered heat and mass transfer in annular flow evaporation of ethanol water mixtures in a vertical tube. He obtained numerical solutions of the turbulent transport equations and carried out calculations with mass transfer resistance calculated in both phases and with mass transfer resistance omitted in one or both phases. The results for interfacial concentration as a function of distance are illustrated in Fig. 15.112. These results show that the liquid phase mass transfer resistance is likely to be small and that the main resistance is in the vapor phase. A similar conclusion was reached in recent work by Zhang et al. [283] these latter authors show that mass transfer effects would not have a large effect on forced convective evaporation, particularly if account is taken of the enhancement of the gas mass transfer coefficient as a result of interfacial waves. 

Explain the concept of a mass-transfer coefficient for turbulent diffusion by analogy with molecular diffusion. 

Most heat-transfer data are based on situations involving no mass transfer. Use of the analogy would then produce mass-transfer coefficients corresponding to no net mass transfer, in turn corresponding most closely to k G, k c, or k (= F). Sherwood numbers are commonly written in terms of any of the coefficients, but when derived by replacement of Nusselt numbers for use where the net mass transfer is not zero, they should be taken as Sh = FllcDAB, and the F used with equation (2-1). 

Replacing Nuov with ShQV and Pr with Sc in the given heat-transfer correlation, the analogous expression for the mass-transfer coefficient is... 

This relation between mass and momentum transfer is termed Reynolds analogy, although the terminology more frequently is used to denote a correspondence between heat and momentum transfer. It is evident that the same argument used above could have been applied to heat transfer for a constant wall temperature, and with v = az relation identical to Eq. (4.5.9) could be obtained with a dimensionless heat transfer coefficient in place of the mass transfer coefficient. 

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