Mole fluxes

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

The mole flux of species at the bounding surfaces of the gas channel is governed by Faraday s law ... 

Equations (9.7) and (9.36) can be used to solve for the spatial distribution of mole fluxes and mole fractions for all J components. 

The temporal change of the molar quantity within the differential balance area (Fig. 16.1a) is affected by inflow and outflow as well as by transferred mole flux into the volume element... 

The separation efficiency is determined by the difference in gas concentration between the inlet and outlet of the fluidized bed. A dimensionless quantity defined by Hill [40] - the so-called absorption factor - can be used in reaction technology at a constant volume flow rate if mole fluxes are replaced by concentrations... 

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. 

Figure fla. Measured and calculated mole fluxes during extreme interaction experiments. Key , / O, /co, ---------, penetration theory and-------> film theory. 

Figure 8b. Measured and calculated mole fluxes during extreme interaction experiments. Key is the same as in Figure 8a.

In these equations JA is the mole flux of A in moles of A per second per square metre flowing through surface area of the catalyst pores. This is not the same as the mole flux in moles of A per second per square metre flowing through surface area of the catalyst pellet. This is elaborated in Appendix E. The term DAP is the Maxwell diffusion coefficient of A in a binary mixture with P and DM is the Knudsen diffusion coefficient of A inside the catalyst pores. ka is the mole fraction of A. J, Dpdp Dn, kp etc. are similarly defined. VA is a factor that accounts for viscous flow inside the pores. If VA is much smaller than one, viscous flow can be neglected. We will neglect viscous flow for all components and substitute... 

By defining as the mole flux of A in moles of A per second per square meter flowing through area of the catalyst pellet (Figure A.4), then... 

Figure A.4 Schematic drawing which illustrates the difference between the mole flux through the cross-sectional area of the pellet JA and the mole flux through the cross-sectional area of the catalyst pores/,. ...

For a completely mixed flow the needed material balance information consists f the ste y mole fluxes per unit volume of A and B to the system, given by tCa and rCe, where r = g/K is the fluid transfer rate constat for the system (the rate of flow Q divided by the volume V) and Q and Cb are the inflow concentrations of A and B, respectively. For the steady state, dCJdt = dC ldt = 0, and assuming inflow and outflow rates to be the same, we have... 

For the open system we must consider the material balance to the system of steady mole fluxes of A and B per unit volume, V. These fluxes can be designated and where r = Q/V, and Q is again the overall volume rate of flux in units such as liters per second. In this context (A) and (B)o are the inflow concentrations. For steady state, by definition d A)/dt = d B)ldt = 0, and inflow and outflow rates must be equal. Skipping several steps we find that for steady-stale conditions... 

In particular cases it is desired to work with an explicit expression for the mole flux of a single species type s, J, avoiding the matrix form given above. Such an explicit model can be derived manipulating the original Maxwell-Stefan model (2.303), with the approximate driving force (2.301), assuming that the mass fluxes for all the other species are known ... 

The classical models are generally expressed in terms of diffusive mole fluxes. 

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 mole flux is commonly used in research of electrochemical processes. 

Figure 3 shows the influence of the CO2 gas-phase concentration on H2S and CO2 mole fluxes, h2 > respec-... 

This forced desorption originates from the comparatively high amine consumption by the H2S absorption reaction and the induced shift in the net rate of reaction (3). At a CO2 concentration of approximately 0.80 mole/m the positive driving force balances the high local CO2 concentration in the penetration zone and results in a net zero mole flux. Above this point the CO2 mole flux increases steadily with the CO2 gas-phase concentration. 

The H2S mole flux is hardly affected by CO2 up to gas-phase concentrations of about 4 moles/m. Calculated free DIPA concentration profiles in the penetration zone correspondingly show no depletion. At higher [C02]q the amine becomes depleted and leads to a decrease of the H2S flux at a constant H2S driving force. CO2 concentrations exceeding 40 moles/m yield forced desorption of H2S according to the mechanism mentioned earlier. 

The effects of mass-transfer coefficients and CO2 gas-phase concentrations are briefly indicated in figure 5. Doubling of the gas-phase mass-transfer coefficient kg hardly affects the largely liquid-phase controlled CO2 absorption but increases the substantially gas-phase limited H2S mole flux. In the forced desorption concentration region however, the doubled kg causes a higher desorption flux due to a facilitated H2S transport to the gas phase. 

The experimental set-up is given in figure 6. A closed reactor-detector system was used to enable detection of small mole fluxes. The stirred cell reactor is 0.10 m in diameter and was filled before each experiment with 120 ml of charged 2.0 M DIPA solution. The gas phase in the system was circulated by means of a flexible tube pump over a flow-through cell in a Perkin Elmer model 257 Infrared Grating Spectrophotometer for CO2 detection. Although spectrophotometers are not exceptionally well-suited for quantitative measurements, we preferred this type of analysis compared to gas chromatography for example because it does not influence the gas phase. 

Four experiments were caried out at different stirrer speeds under conditions as given in tad le 2, Mole fluxes were obtained from measured concentration curves and plotted in figure 8 together with calculated fluxes for both penetration and film theory models. The latter model was derived from our numerical penetration model by zeroing time derivatives and equalizing the penetration depth to the film thickness. 

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