Mass Relationship for Liquid and Gas

For a liquid or gas buffer, shown in Fig. 2.3, the mass balance holds. This equation relates the rate of change in mass m to the difference between inlet mass flow and outlet mass flow  

In case there are N inlet flows and M outlet flows, this equation may be written as  

Often the volumetric flow is used in stead of the mass flow  

This is only useful if the density / , and volumetric flow f v.ime measured variables and if the densities of the flows are similar. Usnally, the outlet flows all have the same composition, similar to the composition inside the bnffer. The mass balance, Eqa (2.14), can then be written as  

The density p in Eqn. (2.16) is defined as the mass per unit volume at a certain pressure, temperature and composition  

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Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

For a conventional mechanically agitated biological reactor, the information provided for aqueous gas-liquid and gas-liquid-solid systems in Sections II, III, and VII is applicable here. For power consumption, the most noteworthy works are those by Hughmark (1980) (see Eqs. (6.15) and (6.16)) and Schiigerl (1981). For gas-liquid mass transfer, the relationship kLaL = (P/V, ug) is applicable for biological systems. The relationships (6.19) and (6.20) are also valuable, and their use is recommended. The most generalized relation for kLaL is provided by Eq. (6.18). The intrinsic gas-liquid mass transfer coefficient is best estimated by Eq. (6.23). For liquid-solid mass transfer, the use of the study by Calderbank and Moo-Young (1961) (Eqs. (6.24)-(6.26)) is recommended. For viscous fluids, Eq. (6.27) should be used. 

Surface mass changes can result from sorptive interactions (i.e., adsorption or absorption) or chemical reactions between analyte and coating, and can be used for sensing applications in bodi liquid and gas phases. Although the absolute mass sensitivity of the uncoated sensor depends on the nature of the piezoelectric substrate, device dimensions, frequoicy of operation, and the acoustic mode that is utilized, a linear dependence is predicted in all cases. This allows a very general description of the working relationship between mass-loading and frequency shift, A/ , for AW devices to be written ... 

As shown, [X ], [yJ is on the equilibrium curve. The equilibrium curve is the relationship between the concentration [x] in the liquid phase and the concentration [y] in the gas phase when there is no net mass transfer between the phases. For any given pair of values [x] and [y] in the liquid and gas phase, respectively, the point ([-tj. [yJ) represents the distance that ([x], [y]) will have to move to attain their equilibrium values concurrently at the equilibrium curve. This distance, represented by the line segment ([x], [y] —> [xj, [yJ), is the actual driving force for mass transfer ... 

The densify of a gas is directly proportional to its molar mass. No simple relationship exists between density and molar mass for liquids and solids. 

Mass Transfer Relationships for calculating rates of mass transfer between gas and liquid in packed absorbers, strippers, and distillation columns may be found in Sec. 5 and are summarized in Table, 5-28. The two-resistance approach is used, with rates expressed as transfer units ... 

The mass transfer coefficients, Kg and Ky, are overall coefficients analogous to an overall heat transfer coefficient, but the analogy between heat and mass transfer breaks down for mass transfer across a phase boundary. Temperature has a common measure, so that thermal equilibrium is reached when the two phases have the same temperature. Compositional equilibrium is achieved at different values for the phase compositions. The equilibrium concentrations are related, not by equality, as for temperature, but by proportionality through an equilibrium relationship. This proportionality constant can be the Henry s law constant Kh, but there is no guarantee that Henry s law will apply over the necessary concentration range. More generally, Kyy is a function of composition and temperature that serves as a (local) proportionality constant between the gas- and liquid-phase concentrations. 

In two-phase flow, most investigations are carried out in one dimension in the steady state with constant flow rates. The system may or may not be isothermal, and heat and mass may be transferred either from liquid to gas, or vice versa. The assumption is commonly made that the pressure is constant at a given cross section of the pipe. Momentum and energy balances can then be written separately for each phase, and with the constraint that the static pressure drop, dP, is identical for both phases over the same increment of flow length dz, these balances can be added to give over-all expressions. However, it will be seen that the resulting over-all balances do not have the simple relationships to each other that exist for single-phase flow. 

In the above terms, the quantities (AP/AZ)i or aP/aZ)g are calculated from conventional single-phase correlations on the basis that the liquid or gas is flowing in the pipe alone at the same individual mass-flow rate as in the two-phase case. Lockhart and Martinelli (L6) have given the appropriate expressions for X, and the relationships between tpa, flow regimes. The relationships are shown graphically in Fig. 7. 

The overall coefficients of liquid-liquid mass transfer are important in the calculations for extraction equipment, and can be defined in the same way as the overall coefficients of gas-liquid mass transfer. In liquid-liquid mass transfer, one component dissolved in one liquid phase (phase 1) will diffuse into another liquid phase (phase 2). We can define the film coefficients /C i (nr h" ) and k (m h ) for phases 1 and 2, respectively, and whichever of the overall coefficients A (m h ), defined with respect to phase 1, or Al2 (mh ) based on phase 2, is convenient can be used. Relationships between the two film coefficients and two overall coefficients are analogous to those for gas-liquid mass transfer that is,... 

Psychrometry can be defined as the study of the relationships between the material and energy balances of water vapor/air mixtures. If a system other than air and water is involved, then psychrometry is concerned with the mass and energy balances for the particular liquid(s) and gas(es) at hand. The air/water vapor system is the most common system encountered in the drying of pharmaceutical granulations, but the air/ethanol or air/ethanol-water sy.stems are also frequently encountered. 

The valnes of the mass-transfer coefficient are measured at various impeller speeds using a model gas-liquid system such as CO2-H2O. Using these results of vs. N, a similar relationship is obtained for the given gas-liqnid system using the square root diffusivity relationship. Thus, at any impeller speed, the specific rate of absorption under mass transfer-controlled conditions (regime 2) can be obtained and is eqnal to ki[A]. ... 

As indicated in the first three rows of Table 16.3, (16-8) and (16-9) can be written in terms of component partial pressure (p — p,) or concentration Q — C) driving forces, the corresponding mass transfer coefficients being designated kg and kc, respectively, for the gas, and ki for the liquid, in terms of a concentration driving force. The relationships between k and k in the different mass transfer expressions are given at the bottom of Table 16.3. Here Pbm is the log mean partial pressure of B, p is the gas density, and M is the average molecular... 

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