Mass balance interfacial

The expression for the Stefan velocity is easily obtained from the interfacial mass balance (Eq. 11.123) by summing over all Kg species and noting that the mass fractions must sum to one,... 

As we have shown, the surface forces at an interface depend upon the surface tension gradients there. If adsorbed surface-active materials are distributed at an interface, then this distribution must be known to determine the surface forces, since the surface tension gradients depend on the local surface concentration of adsorbed material. The surface mass concentration of the adsorbed substance follows from an interfacial mass balance. 

The interfacial mass balance, interrelating the change of the surfactant or counterion adsorptions Tj with time and the respective electro-diffusion influx from the bulk, reads... 

Section 4.1.4 shows how the Faraday law can be demonstrated by writing interfacial mass balances. 

In particular, when the geometry is not unidirectional, the surface concentration f) is not necessarily identical at all points on the surface and becomes a function of several variables (time and space). The term needed in the interfacial mass balance is therefore a partial derivative dr /dt, instead ofdrj/dtin unidirectional geometry. 

V The interfacial mass balances can be illustrated for instance by a system with a phase transfer reaction between an aqueous solution containing potassium chloride and a solution containing crown-ether (ligand L) in an organic solvent (figure 4.5). 

The same reasoning can be applied to a species adsorbed at the interface. This type of species is not mobile in either of the two phases (the adsorbed species may possibly be mobile along the interface). The fluxes normal to the surfaces are therefore zero, which in turn simplifies the general equation outlined above. The local interfacial mass balance for a species i adsorbed at the interface is ... 

Faraday s law, which has already been applied in this work in its integrated form (see section 2.1.2.2), can be demonstrated based on the interfacial mass balances. 

Remember that this equation stands in algebraic terms as long as the sign convention chosen is fully respected. The interfacial mass balance is written with the normal oriented from the metal to the electrolyte. As already outlined, this boils down to considering the currents through the anode to be positive and those through the cathode to be negative. Moreover, this equation only applies to species that are mobile in the electrolyte. 

The following results are given by the interfacial mass balance for each species in the system (applying the same reasoning as previously in this section) ... 

The system is also defined in terms of the interfacial mass balance. In this case, NO3" does not react at the anode, yet silver is oxidized. Therefore, at this interface the mass balance is written with the anodic current taken as being positive, as usual (see Faraday s law in section 4.1.4) ... 

As the extent of S is arbitrary, the integrand itself must vanish, and we obtain the differential interfacial mass balance ... 

The resulting differential interfacial mass balance is given by... 

Initially c has its bulk value c for all z > 0. Also, at all times, c - c far from the interface (i.e., as z - oo). Were the solution concentration c(0j) at the interface known, as a function of time. Equation 6.39 could be solved. Instead, it is known from the interfacial mass balance that... 

Starting with Equation 6.1, derive the differential interfacial mass balance of Equation 6.2. 

This equation is obtained on neglecting terms in T in the interfacial mass balance equation (Equation 5.32). The two terms are equal to the flux (mass transfer) through the interface, in the absence of which Equation 7.14 reduces to... 

Combining the general hydrodynamic equations for films of partially mobile surfaces, the interfacial mass balance, and the boundary conditions for the surface stresses, one can... 

In the case of an input of component i to the system by interfacial mass transfer, the balance equation now becomes ... 

Although the Lewis cell was introduced over 50 years ago, and has several drawbacks, it is still used widely to study liquid-liquid interfacial kinetics, due to its simplicity and the adaptable nature of the experimental setup. For example, it was used recently to study the hydrolysis kinetics of -butyl acetate in the presence of a phase transfer catalyst [21]. Modeling of the system involved solving mass balance equations for coupled mass transfer and reactions for all of the species involved. Further recent applications of modified Lewis cells have focused on stripping-extraction kinetics [22-24], uncatalyzed hydrolysis [25,26], and partitioning kinetics [27]. 

For the PFTR the mass transfer area is simply the total interfacial area between the phase in question and the other fluid, while for the PFTR, where Cj is a function of position, [areaWolume] is the area per unit volume of that phase at position Z- The steady-state mass balance is therefore... 

This mass balance concerns the liquid phase, since oxygen must be dissolved in order to be used by the cells. Due to the difficulty in measuring the interfacial area (a), especially when oxygenation is carried out by bubble aeration, it is common to use the product of kL times a (kLa), known as the volumetric oxygen transfer coefficient, as the relevant parameter. 

A more complete procedure to compute the average thicknesses of the water and oil layers will be presented below. It is based on eq 3c, mass balances of components, and phase equilibrium equations. The calculations indicated that the interfacial tension of lamellar liquid crystals is very low, of the order of 10 5 N/m. It will be shown that y = 0 is always an excellent approximation of eq 3 c. 

The above equations assume that the liquid-phase reactant C, the product of the reaction, and the solvent are nonvolatile. The effective interfacial area for mass transfer (nL) and the fractional gas holdup (ii0o) arc independent of the position of the column. The Peclet number takes into account any variations of concentration and velocity in the radial direction. We assume that Peclet numbers for both species A and C in the liquid phase are equal. For constant, 4 , Eq. (4-73) assumes that the gas-phase concentration of species A remains essentially constant throughout the reactor. This assumption is reasonable in many instances. If the gas-phase concentration does vary, a mass balance for species A in the gas phase is needed. If the gas phase is assumed to move in plug flow, a relevant equation would be... 

Eor a complete description of the separation process, it is necessary to include the mass balances in the emulsion reservoirs as well as the interfacial equilibrium expression at the feed-membrane side. 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

If the adsorption rate is high, or the additive concentration low, substantial depletion of the additive concentration occurs next to the electrode. In a supported electrolyte, the interfacial concentration of the additive, Q, is determined by mass balance between adsorption, desorption and the available diffusional flux ... 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

The function is the mass transfer rate equation for the vapor phase is the interfacial energy balance F3 is valid if the liquid phase may be assumed unmixed with... 

In practical applications one normally neglects the terms describing the interfacial effects, thus the jump mass balance may be written as... 

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