Fluid surfaces, mass-transfer coefficients turbulent flow

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be. given the following physical properties ... 

The mass transfer coefficient (3C with SI units of m/s or m3/(sm2) is defined using these equations. It is a measure of the volumetric flow transferred per area. The concentration difference Aca defines the mass transfer coefficient. A useful choice of the decisive concentration difference for mass transfer has to be made. A good example of this is for mass transfer in a liquid film, see Fig. 1.41 where the concentration difference cA0 — cM between the wall and the surface of the film would be a a suitable choice. The mass transfer coefficient is generally dependent on the type of flow, whether it is laminar or turbulent, the physical properties of the material, the geometry of the system and also fairly often the concentration difference Aca. When a fluid flows over a quiescent surface, with which a substance will be exchanged, a thin layer develops close to the surface. In this layer the flow velocity is small and drops to zero at the surface. Therefore close to the surface the convective part of mass transfer is very low and the diffusive part, which is often decisive in mass transfer, dominates. 

In Eq. (67), a is an experimental constant and c usually has a value of 1/3 [128-130]. The value of b depends on the type of equipment and system, and most of the theories predict a one-half power on the Reynolds number [131]. The mass transfer from bulk solution to the surface of the membrane is mainly controlled by the turbulence of the fluid motion created by stirring. The characteristic velocity is defined in terms of the stirring speed (u = nd). The values of a and b were determined from the intercept and slope of the line of Sh/Sc against Re for the specified mass transfer coefficients of k, k, and Kg. These parameters are different and are dependent on the system geometry and flow pattern. However, it can be concluded that the exponent value on Re varied from 0.2 to 1.0, depending on the design of the membrane permeation system. 

The time averaged equations of mass, morrrentum, energy, and species conservation can be written in dimensionless form for a fluid in turbulent flow past a surface. If (1) radiant energy and chemical reaction are not present, (2) viscous dissipation is negligible, (3) physical properties are independent of temperature and composition, (4) the effect of mass transfer on velocity profiles is neglected, and (5) the boundary conditions are compatible, then dimensionless local heat and mass transfer coefficients can be shown to be described by equations of the form ... 

Boundary-layer theory. The boundary-layer theory has been discussed in detail in Section 7.9 and is useful in predicting and correlating data for fluids flowing past solid surfaces. For laminar flow and turbulent flow the mass-transfer coefficient fc oc D g. This has been experimentally verified for many cases. 

This is the oldest and most obvious picture of the meaning of the mass-transfer coefficient, borrowed from a similar concept used for convective heat transfer. When a fluid flows turbulently past a solid surface, with mass transfer occurring from the surface rn the fluid, the concentration-distance relation is as shown by the full curve of Fig. 3.6, the. shape of which is controlled by the... 

It is possible to predict theoretically the mass transfer rate (or flux AT,) across any surface located in a fluid having laminar flow in many situations by solving the differential equation (or equations) for mass balance (Bird et al, 1960, 2002 SkeUand, 1974 Sherwood et al, 1975). Our capacity to predict the mass transfer rates a priori in turbulent flow from first principles is, however, virtually nil. In practice, we follow the form of the integrated flux expressions in molecular diffusion. Thus, the flux of species i is expressed as the product of a mass-transfer coefficient in phase y and a concentration difference in the forms shown below ... 

If the surface over which the fluid is flowing contains a series of relatively large projections, turbulence may arise at a very low Reynolds number. Under these conditions, the frictional force will be increased but so will the coefficients for heat transfer and mass transfer, and therefore turbulence is often purposely induced by this method. 

It appears possible to make the following two important generalizations concerning the relative rates of mass transfer to the catalyst pellet and diffusion into the pellet (a) Mass transfer to the external catalyst surface is always faster than diffusion into the internal catalyst surface. This is because turbulence in the fluid stream enhances the effective diffusion coeflacient in the flowing fluid to much larger values than those possible inside a catalyst pellet. Even in the absence of turbulence, the presence of small pores in catalysts depresses the diffusion coefficient to (Knudsen) values lower than the bulk values in the flowing stream, (b) Hence, whenever mass transfer to the external catalyst surface is influencing reaction rate, then the internal surface area can be only partly available to the reaction. We thus get the elementary theorem that whenever a catalyst is sufficiently active so that the reaction rate is influenced by mass transfer (diffusion) to the catalyst pellet, then the internal surface area of that catalyst can be only partially available to the reaction. 

Another analogous relationship is that of mass transfer, represented by Pick s law of diffusion for mass flux, J, of a dilute component, I, into a second fluid, 2, which is proportional to the gradient of its mass concentration, mi. Thus we have, J = p Du Vmt, where the constant Z)/2 is the binary diffusion coefficient and p is density. By using similar solutions we can find generalized descriptions of diffusion of electrons, homogeneous illumination, laminar flow of a liquid along a spherical body (assuming a low-viscosity, non-compressible and turbulent-lree fluid) or even viscous flow applied to the surface tension of a plane membrane. 

---