Interphase mass and energy transfer

There are scores of expressions available in the literature for estimating heat transfer coefficients we mention only one here, the well-known Chilton-Colburn analogy 

The heat transfer coefficients estimated from correlations or analogies are the low flux coefficients and, therefore, need to be corrected for the effects of finite transfer rates before use in design calculations. We recommend the film theory correction factor given by Eq. 11.4.12. 

We will often be faced with the problem of determining the rates of mass and energy transfer across a phase boundary. It is these fluxes that appear in the equations that model processes, such as distillation, gas absorption, condensation, and so on. Here we present a summary of the relevant equations and suggest a procedure for determining the required fluxes. 

At the vapor-liquid interface we have continuity of the component molar fluxes 

Using the definitions of the mass transfer coefficients we may write, for the diffusion fluxes 

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DA 7/1 A good deal of physical property data is needed for interphase mass and energy-transfer calculations. The data and methods used for this example are summarized below. 

In the rate-based models, the mass and energy balances around each equilibrium stage are each replaced by separate balances for each phase around a stage, which can be a tray, a collection of trays, or a segment of a packed section. Rate-based models use the same m-value and enthalpy correlations as the equilibrium-based models. However, the m-values apply only at the equilibrium interphase between the vapor and liquid phases. The accuracy of enthalpies and, particularly, m-values is crucial to equilibrium-based models. For rate-based models, accurate predictions of heat-transfer rates and, particularly, mass-transfer rates are also required. These rates depend upon transport coefficients, interfacial area, and driving forces. It... 

The operations considered in this chapter are concerned with the interphase transfer of mass and energy which result when a gas is brought into contact with a pure liquid in which it is essentially insoluble. The matter transferred between phases in such cases is the substance constituting the liquid phase, which either vaporizes or condenses. These operations are somewhat simpler—from the point of view of mass transfer—than absorption and stripping, for when the liquid contains only one component, there are no concentration gradients and no resistance to mass transfer in the liquid phase. On the other hand, both heat transfer and gas-phase mass transfer are important and must be considered simultaneously since they influence each other. 

Reformulation of the problems previously explained to account for interphase gradients essentially requires only the change of the surface boundary conditions. Assuming that we may see traditional mass- and heat-transfer coefficients as the rate constants characteristic of interphase transport, the boundary conditions for mass and energy conservation equations become... 

Models considering detailed flow patterns in multi-phase reactors are similar to those presented in Section 13.7 for fluidized bed reactors. The models are based on the Navier-Stokes equations for each of the moving phases. The different phases are assumed to be fully penetrating each other. Interphase mass, momentum, and energy transfer is accounted for. The methods accounting for the fluctuations in the flow field discussed in Chapter 12 for single phase flow can be extended to multi-phase flow. Application to the simulation of a bubble column reactor is illustrated in Example 14.3.6.A. 

Interfacial area measurement. Knowledge of the interfacial area is indispensable in modeling two-phase flow (Dejesus and Kawaji, 1990), which determines the interphase transfer of mass, momentum, and energy in steady and transient flow. Ultrasonic techniques are used for such measurements. Since there is no direct relationship between the measurement of ultrasonic transmission and the volumetric interfacial area in bubbly flow, some estimate of the average bubble size is necessary to permit access to the volumetric interfacial area (Delhaye, 1986). In bubbly flows with bubbles several millimeters in diameter and with high void fractions, Stravs and von Stocker (1985) were apparently the first, in 1981, to propose the use of pulsed, 1- to 10-MHz ultrasound for measuring interfacial area. Independently, Amblard et al. (1983) used the same technique but at frequencies lower than 1 MHz. The volumetric interfacial area, T, is defined by (Delhaye, 1986)... 

In this case, as shown in Figure 4, the subsystems are stoichiometry, material balance, energy balance, chemical kinetics, and interphase mass transfer. The mass transfer phenomena can be subdivided into (1) phase equilibrium which defines the driving force and (2) the transport model. In a general problem, chemical kinetics may be subdivided into (1) the rate process and (2) the chemical equilibrium. The next step is to develop models to describe the subsystems. Except for chemical kinetics, generally applicable mathematical equations based on fundamental principles of physics and chemistry are available for describing the subsystems. 

During interphase mass transfer, concentration gradients will be set up across the interface. The concentration variations in the bulk phases x and y will be described by differential equations whereas at the interface /, we will have jump conditions or boundary conditions. Standart (1964) and Slattery (1981) give detailed discussions of these relations for the transport of mass, momentum, energy, and entropy. It will not be possible to give here the complete derivations and the reader is, therefore, referred to these sources. A masterly treatment of this subject is also available in the article by Truesdell and Toupin (1960), which must be compulsory reading for a serious researcher in transport phenomena. 

Clearly, from these results, it is a species partial molar Gibbs energy difference between phases, rather than a concentration difference, that is the driving force for interphase mass transfer in the approach to equilibrium. See, for example, of J. W. Tester and M. Modell, Thermodynamics and Its Applications, 3rd ed., Prentice Hall, Englewood Cliffs. N.J. (1997) Chapter 7. 

Further temperature increase will diminish the reactant concentration on the outer pellet surface as the influence of external mass transfer becomes important. Finally, interphase mass transfer will be the rate controlling step and the surface concentration drops to zero. Under those conditions, the apparent activation energy corresponds to Ep. 

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