Mole fluxes mass transfer

The difference in mole fractions is most significant in the case of S02 where this difference is 15% of the bulk phase level. This result indicates that external mass transfer limitations are indeed significant, and that this difference should be taken into account in the analysis of kinetic data from this system. Note that there is a difference in nitrogen concentration between the bulk fluid and the external surface because there is a change in the number of moles on reaction, and there is a net molar flux toward... 

JA = flux rate of component A [moles (total moles)-1 s-1 m-2] k1A = gas phase mass transfer coefficient (s-1 m-2) k2A = water phase mass transfer coefficient (s-1 m-2)... 

Gas phase mass transfer fluxes (Stefan flux in the gas phase is negligible as long as the vapor phase mole fractions are below say 20 %, which means either moderate temperatures and/or high sweep gas flow rates) are ... 

Negligible and medium interaction regimes. Experiments were carried out with an aqueous 2.0 M DIPA solution at 25 °C in a stirred-cell reactor (see ref. [1]) and a 0.010 m diameter wetted wall column (used only in negligible interaction regime, see ref. [4,5]). Gas and liquid were continuously fed to the reactors mass transfer rates were obtained from gas-phase analyses except for CO2 in the wetted wall column where due to low C02 gas-phase conversion, a liquid-phase analysis had to be used [5]. In the negligible interaction regime some 27 experiments were carried out in both reactors. The selectivity factors were calculated from the measured H2S and CO2 mole fluxes and are plotted versus k... 

Two experimental runs were performed. The H2S- and CO2 mole fluxes were obtained from the measured concentration curves by numerical differentiation and are plotted in figure 8a,b together with penetration and film model calculations. It is evident that forced desorption can be realized under practical conditions and can be predicted by the model. In general, measured H2S mole fluxes are between the values predicted by the models, whereas the CO2 forced desorption flux is larger than calculated by the models. The CO2 absorption flux, on the other hand, can correctly be calculated by the models. This probably implies that the rate of the reverse reaction, incorporated in equation (5), is underestimated. Moreover, it should be kept in mind that especially the results of the calculations in the forced desorption range are very sensitive to indirectly obtained parameters (diffusion, equilibrium constants and mass transfer coefficients) and the numerical differentiation technique applied. 

The following symbols are used in the definitions of the dimensionless quantities mass (m), time (t), volume (V area (A density (p), speed (u), length (/), viscosity (rj), pressure (p), acceleration of free fall (p), cubic expansion coefficient (a), temperature (T surface tension (y), speed of sound (c), mean free path (X), frequency (/), thermal diffusivity (a), coefficient of heat transfer (/i), thermal conductivity (/c), specific heat capacity at constant pressure (cp), diffusion coefficient (D), mole fraction (x), mass transfer coefficient (fcd), permeability (p), electric conductivity (k and magnetic flux density ( B) ... 

In these equations cj and cf are the molar densities of the superscripted phases, yj is the mole fraction in the bulk vapor phase, xf is the mole fraction in the bulk liquid phase, and xj and yj are the mole fractions of species i at the phase interface. Also is the total molar flux in phase p, and kj and are the mass-transfer coefficients for... 

Comparison of equations 3 and 10 shows the essential difference between the stationary states of closed and continuous, open systems. For the closed system, equilibrium is the time-invariant condition. The total of each independently variable constituent and the equilibrium constant (a function of temperature, pressure, and composition) for each independent reaction (ATab in the example) are required to define the equilibrium composition Ca- For the continuous, open system, the steady state is the time-invariant condition. The mass transfer rate constant, the inflow mole number of each independently variable constituent, and the rate constants (functions of temperature, pressure, and composition) for each independent reaction are requir to define the steady-state composition Ca- It is clear that open-system models of natural waters require more information than closed-system models to define time-invariant compositions. An equilibrium model can be expected to describe a natural water system well when fluxes are small, that is, when flow time scales are long and chemical reaction time scales are short. 

Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution... 

The computation of the fluxes from either of Eqs. 8.3.24 necessarily involves an iterative procedure (except for the special cases discussed above), partly because the themselves are needed for the evaluation of the matrix of correction factors and also because an explicit relation for the matrix [0] cannot be derived as a generalization of Eq. 8.2.16 for binary mass transfer there is no requirement in matrix algebra for the matrices [FFq] be equal to each other even though the fluxes calculated from both parts of these equations must be equal. Indeed, these two matrices will be equal only in the case of vanishingly small mole fraction differences (yg Tg) and vanishingly small mass transfer rates. In almost all cases of interest these two matrices are quite different. An explicit solution was possible for binary systems only because all matrices reduce to scalar quantities. 

In problems where the flux ratios are known (e.g., condensation and heterogeneous reacting systems where the reaction rate is controlled by diffusion) the mole fractions at the interface are not known in advance and it is necessary to solve the mass transfer rate equations simultaneously with additional equations (these may be phase equilibrium and/or reaction rate equations). For these cases it is possible to embed Algorithms 8.1 or 8.2 within another iterative procedure that solves the additional equations (as was done in Example 8.3.2). However, we suggest that a better procedure is to solve the mass transfer rate equations simultaneously with the additional equations using Newton s method. This approach will be developed below for cases where the mole fractions at both ends of the film are known. Later we will extend the method to allow straightforward solution of more complicated problems (see Examples 9.4.1, 11.5.2, 11.5.3, and others). 

Given the binary mass transfer coefficients and the mole fractions and there are three unknown quantities in these equations the molar fluxes, N-, N2, However, there are only two independent mass transfer rate equations. Thus, one more equation is needed this will be the bootstrap relation ... 

Given the bulk fluid conditions (mole fractions and temperatures), the system pressures, the low flux mass and heat transfer coefficients (or methods to evaluate them), and appropriate equilibrium models, what is the most effective means of obtaining the rates of mass and heat transfer ... 

When taking these partial derivatives it must be remembered that, in general, the molar densities, the mass transfer coefficients, and thermodynamic properties are functions of temperature, pressure, and composition. In addition, H is a function of the molar fluxes. We have ignored most of these dependencies in deriving the expressions given above. The important exception is the dependence of the K values on temperature and composition that cannot be ignored. The derivatives of the K values with respect to the vapor mole fractions are zero in this case since the model used to evaluate the K values is independent of the vapor composition. 

The above derivatives must be evaluated at the interface composition before use in computing the Jacobian elements. This additional complexity in evaluating the derivatives of the vapor-phase mass transfer rate equations arises because we have used mass fluxes and mole fractions as independent variables. If we had used mass fractions in place of mole factions the derivatives of the rate equations would be simpler, but the derivatives of the equilibrium equations would be more complicated. For simplicity, we have ignored the dependence of the mass transfer coefficients themselves on the mixture composition and on the fluxes. 

The total mass transferred between phases may be determined by summing the contributions from the bubble formation zone and each bubble population. The total number of moles transferred in the /cth bubble population is the product of the average molar flux given by Eq. 12.1.69 and the total interfacial area in that population ( //, The... 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. 

The species transfer can thus be expressed as a mass flux Ni k)Ai kg/sm ) in line with (3.180), simply by multiplying the conventional mole flux by the molecular weight ... 

This is the solution for an instantaneous flux rate at the interface, since we are considering dilute solutions any diffusion-induced convection can be neglected. This means that the total mole flux is equal to the diffusion flux, and that we can write the instantaneous mass transfer rate directly in the form derived for the diffusion flux ... 

The species mole (mass) flux at the interface can be defined both in terms of Tick s first law and the mass transfer coefficient concepts ... 

The fundamental quantity describing the irreversible process is the flux vector Ji for species i. In the case of mass transfer, it describes a flow in a given direction in space in units of moles or grams crossing unit area per second. It can also be described as the product of the local concentration c,- times the velocity v,- at which molecules or ions are moving. Thus,... 

The units on 4> a are moles per square meter per second. This is the convective flux. The student of mass transfer will recognize that a diffusion term like — Adaldz is usually included in the flux. This term is the diffusive flux and is zero for piston flow. The design equation for a variable-density, variable-cross-section PER is... 

Use the film theory approach to predict the effect of a simultaneous mass transfer flax on the heat transfer flux as in the coedensation of a binary vapor on a cold surface. The bulk gas condition are temperature T2 and mole fraction yAa2, while the coedilions at ihe liquid surface are T, and yAil. 

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