Rate expressions for mass transfer

This example also illustrates the use of the three basic concepts on which the analysis of more complex mass transfer problems is based namely, conservation laws, rate expressions, and equilibrium thermodynamics. The conservation of mass principle was implicitly employed to relate a measured rate of accumulation of sugar in the solution or decrease in undissolved sugar to the mass transfer rate ftom the crystals. The dependence of the rate expression for mass transfer on various variables (area, stirring, concentration, etc.) was explored experimentally. Phase-equilibrium thermodynamics was involved in setting limits to the final sugar concentration in solution as well as providing the value of the sugar concentration in solution at the solution-crystal interface. 

The various sections of this chapter develop and distinguish the conservation laws and various rate expressions for mass transfer. The laws of conservation of mass, energy, and momentum, which ate takm as universal principles, are formulated in both macroscopic and difletential forms in Section 2.2. 

Thus the driving force for fuming is approximately equal to that for free evaporation. Using dre experimental data, and the normal expression for mass transfer across a boundary layer, it is concluded that the boundary layer thickness which would account for this rate should be about 2 x 10 cm (Turkdogan et al., 1963). 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

The enhanced rate expressions for regimes 3 and 4 have been presented (48) and can be appHed (49,50) when one phase consists of a pure reactant, for example in the saponification of an ester. However, it should be noted that in the more general case where component C in equation 19 is transferred from one inert solvent (A) to another (B), an enhancement of the mass-transfer coefficient in the B-rich phase has the effect of moving the controlling mass-transfer resistance to the A-rich phase, in accordance with equation 17. Resistance in both Hquid phases is taken into account in a detailed model (51) which is apphcable to the reversible reactions involved in metal extraction. This model, which can accommodate the case of interfacial reaction, has been successfully compared with rate data from the Hterature (51). 

Substituting the rate expression into (El) leads to an inequality for mass transfer coefficient in the liquid phase ... 

The theory is equally applicable when bulk flow occurs. In gas absorption, for example where may be expressed the mass transfer rate in terms of the concentration gradient in the gas phase ... 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

For the case where there is a mass transfer resistance in the fluid external to the particle (mass transfer coefficient hn), express the mass transfer rate in terms of the bulk concentration C , rather than the concentration Cts at the surface of the particle. 

The overall driving force for mass transfer is AT = Pg—Pi, where Pi is the concentration of oxygen in the liquid phase expressed as an equivalent partial pressure. For the experimental conditions, T/ 0 due to the fast, liquid-phase reaction. The oxygen pressure on the gas side varies due to the liquid head. Assume that the pressure at the top of the tank was 1 atm. Then Tg = 0.975 atm since the vapor pressure of water at 20°C should be subtracted. At the bottom of the tank, 1.0635 atm. The logarithmic mean is appropriate AT =1.018 atm. Thus, the transfer rate was... 

The interpretation is straightforward. At reaction conditions the concentration in the film is lowered by reaction, and, as a consequence, the driving force for mass transfer increases. In a homogeneous system this results in high values of Ha. In a slurry reactor this enhancement can occur if the catalyst particles are so small that they accumulate in the film layer. Table 5.4-4 summarizes expressions for the reaction rate or enhancement factor for various regimes. 

Here Jv is the volumetric flow rate of fluid per unit surface area (the volume flux), and Js is the mass flux for a dissolved solute of interest. The driving forces for mass transfer are expressed in terms of the pressure gradient (AP) and the osmotic pressure gradient (All). The osmotic pressure (n) is related to the concentration of dissolved solutes (c) for dilute ideal solutions, this relationship is given by... 

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

Equation 9.1-15 equates the rate of heat transfer by conduction at the surface to the rate of heat transfer by conduction/convection across a thermal boundary layer exterior to the particle (corresponding to the gas film for mass transfer), expressed in terms of a film coefficient, h, and the difference in temperature between bulk gas at Tg and particle surface at Ts ... 

It is more convenient to express the mass transfer coefficient in terms of a humidity difference, so that IcgA(Ps — P, ) kA(M, — M). The rate of drying is thus determined by the values of h, AT and A, and is not influenced by the conditions inside the solid, h depends on the air velocity and the direction of flow of the air, and it has been found that h = CG 0 S where G is the mass rate of flow of air in kg/s m2. For air flowing parallel to plane surfaces, Shepherd et alnv> have given the value of C as 14.5 where the heat transfer coefficient is expressed in W/m2 K. 

Fig. 6. Comparison of the computed numerical results ( ) with the results obtained from the approximate analytical expression ( ), given by eqs (24) and (31)-(34) for mass transfer with first-order chemical reaction, vs reaction rate constant. Parameter values are given in Table 4.

Example 17.4 gives the final rate expression for film mass transfer followed by a second-order rate expression for reaction on a plane surface. Please derive this expression and show that it is correct. 

Now the overall rate expression for the reaction will have to account for the mass transfer resistance (to bring reactants together) and the resistance of the chemical reactions step. Since the relative magnitude of these resistances can vary greatly we have a whole spectrum of possibilities to consider. 

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