Convective Mass-Transfer Coefficients

In the previous sections of this chapter and Chapter 6 we have emphasized molecular diffusion in stagnant fluids or fluids in laminar flow. In many cases the rate of diffusion is slow, and more rapid transfer is desired. To do this, the fluid velocity is increased until turbulent mass transfer occurs. 

To have a fluid in convective flow usually requires the fluid to be flowing by another immiscible fluid or by a solid surface. An example is a fluid flowing in a pipe, where part 

7 Principles of Unsteady-Slate and Convective Mass Transfer 

When a solute A is dissolving from a solid surface there is a high concentration of this solute in the fluid at the surface, and its concentration, in general, decreases as the distance from the wall increases. However, minute samples of fluid adjacent to each other do not always have concentrations close to each other. This occurs because eddies having solute in them move rapidly from one part of the fluid to another, transferring relatively large amounts of solute. This turbulent diffusion or eddy transfer is quite fast in comparison to molecular transfer. 

Three regions of mass transfer can be visualized. In the first, which is adjacent to the surface, a thin viscous sublayer film is present. Most of the mass transfer occurs by molecular diffusion, since few or no eddies are present. A large concentration drop occurs across this film as a result of the slow diffusion rate. 

---

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. 

A spherical benzoic acid particle with a diameter of 2 mm is dropped into a quiescent air duct at ambient conditions (25°C). During the fall, sublimation of the particle occurs. Taking the molecular diffusivity of benzoic acid in air as 4 x 10-6 m2/s, calculate the convective mass transfer coefficient when the particle is falling at its terminal velocity condition. 

C = oxygen concentration at the reaction surface, kg = convective mass transfer coefficient, ks = intrinsic reaction rate constant,... 

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... 

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. " ... 

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. 

The thickness of the fictitious film in the film theory can never be measured. The film theory predicts that the convective mass transfer coefficient k is directly proportional to the diffusivity whereas experimental data from various studies show that k is proportional to the two-third exponent of the diffusivity. In addition, the concept of a stagnant film is unrealistic for a fluid-fluid interface that tends to be unstable. Therefore, the penetration theory was proposed by Higbie to better describe the mass transfer in the liquid phase... 

Convective mass transfer coefficient, kgmol s var mol frac Molar mass of air, kgmol ... 

Here ha is the convection mass transfer coefficient and it has a dimension of Lit. 

Therefore conditions that influence the surface concentration gradient (3Ci/3y)l,o will also influence the convection mass transfer coefficient and the rate of species transfer across the fluid layer near the wall. 

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 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. 

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

KG overall convective mass-transfer coefficient for diffusion of A through stagnant B in dilute solutions with driving force in terms of partial pressures mol/m2-s-Pa. 

L convective mass-transfer coefficient for equimolar counterdiffiision in... 

Convection and boundary conditions at the surface. In Fig. 7.1-2 there was no convective resistance at the surface. However, in many cases when a fluid is outside the solid, convective mass transfer is occurring at the surface. A convective mass-transfer coefficient, similar to a convective heat-transfer coefficient, is defined as follows ... 

The flux J is based on the surface area A, since the cross-sectional area may vary. The value of — z, the distance of the path, is often not known. Hence, Eq. (7.2-2) is simplified and is written using a convective mass-transfer coefficient k. ... 

Again all the mass-transfer coefficients can be related to each other and are given in Sec. 7.2 Convective Mass-Transfer Coefficients 435... 

The diffusivity = 1.0 x 10" mVs- Suddenly, the top surface is exposed to a fluid having a constant concentrationc = 6 x 10 kg moM/m. The distribution coefficient K = cjc = 1.50. The rear surface is insulated and unsteady-state diffusion is occurring only in the x direction. Calculate the concentration profile after 2500 s. The convective mass-transfer coefficient can be assumed as infinite. Use Ax = 0.001 m and M = 2.0. 

As a result, the equation for the local convective mass-transfer coefficient is... 

---