Molar Flux Analysis

Now that one has obtained the basic information for the molar density of reactant A within the liquid-phase mass transfer boundary layer, it is necessary to calculate the molar flux of species A normal to the gas-liquid interface at r = l bubbie, and define the mass transfer coefficient via this flux. Since convective mass transfer normal to the interface was not included in the mass transfer equation with liquid-phase chemical reaction, it is not necessary to consider the convective mechanism at this stage of the development. Pick s first law of diffusion is sufficient to calculate the flux of A in the r direction at r = /fbubbie- Hence, 

The appropriate mass transfer coefficient in the boundary layer on the liquid side of spherical interfaces with first-order or pseudo-first-order irreversible chemical reaction predominantly in the liquid phase is 

The mass transfer boundary layer thickness on the liquid side of the interface is very thin at large Damkohler numbers. Hence, 

Illustrative Problem. Consider a spherical solid pellet of pure A, with mass density pa, which dissolves into stagnant liquid B exclusively by concentration diffusion in the radial direction and reacts with B. Since liquid B is present in excess, the homogeneous kinetic rate law which describes the chemical reaction is pseudo-first-order with respect to the molar density of species A in the liquid phase. Use some of the results described in this chapter to predict the time dependence of the radius of this spherical solid pellet, R(t), (a) in the presence of rapid first-order irreversible liquid-phase chemical reaction in the diffusion-limited regime, and (b) when no reaction occurs between species A and B. The molecular weight of species A is MWa. 

SOLUTION. The steady-state molar density profile of reactant A, given by (13-14) in the presence of a first-order irreversible chemical reaction, is employed to calculate the r component of the molar flux of A at the solid-liquid interface [i.e., r = (t)]. Then, one constructs an unsteady-state macroscopic mass balance 

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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... 

Some aspects of the theoretical development which have been presented here follow along the lines of an important paper by Newman and Simon (1980). Their analysis differs from the simplified analysis presented here in two respects. First, the idea that ttd, is a measure of the type of bubble growth which occurs was not incorporated in the Newman-Simon analysis. Second, Newman and Simon used a more realistic expression for the molar flux. ... 

MS can measure the ratio between molar fractions of mass isotopomers. The ratio between two mass isotopomer pools of masses nti and m2 is defined in the present work as intensity ratio Jmi/m2- K identical with a mass spectral intensity ratio. If more than two mass isotopomer pools are assessed, their relative ratios, normalized to the sum, are named mass isotopomer distribution. The mass distribution of a compound can be thus obtained from the analysis of ions, which contain the intact carbon skeleton of the analyte. In the area of me-tabohc flux analysis, mass distributions of various metaboHtes have been assessed by MS. The major method used is GC/MS, whereby the analytes are deriva-tized into forms with desired physico-chemical properties such as increased volatihty, thermal stabiHty and suitable MS properties [62]. The mass of the formed derivate must be sufficiently high (usually above 175 apparent mass units) to avoid background interference [48]. To obtain the mass distribution of a compound, ions with the entire carbon skeleton of the analyte have to be present. For accurate quantification of the mass distribution of such ions, they should occur in high abundance and preferably be unique species, thus being formed by only one fragmentation pathway. 

These equations are written in terms of the (laboratory referenced) velocities V. However, from analysis of the problem it is clear that the molar flux of each species is constant, and this knowledge can be used in formulation of the governing equations. 

ANALYSIS The molar fluxes in the vapor phase will be calculated from... 

Revise the analysis of Example 11.5.3 and show how a method based on the film models of Chapter 8 could be used to compute the rates of mass transfer. Then use the Krishna-Standart method (of Sections 8.3 and 8.8.3) and compute the molar fluxes. Binary pair mass transfer coefficients may be estimated using the Chilton-Colburn analogy. 

Equation (1-77) is one of the most fundamental relations in the analysis of mass transfer phenomena. It was derived by integration assuming that all the molar fluxes were constant, independent of position. Integration under conditions where the fluxes are not constant is also possible. Consider, for example, steady-state radial diffusion from the surface of a solid sphere into a fluid. Equation (1-70) can be applied, but the fluxes are now a function of position owing to the geometry. Most practical problems which deal with such matters, however, are concerned with diffusion under turbulent conditions, and the transfer coefficients which are then used are based upon a flux expressed in terms of some arbitrarily chosen area, such as the surface of the sphere. These matters are discussed in detail in Chapter 2. 

Table 1. Substrate cycling in the anaplerotic reactions of C. glutamicum (expressed as % of the molar glucose uptake rate) undar various cultivation conditions as identified by C labelling-based flux analysis...

Fig. 5. Flux analysis data [12,31,32] on C3-C4 conversions in isogenic strains derived from lysine-producing C. glutamicum MH20-22B in chemostat cultures revealing a strong correlation with the L-lysine production rate. Rates are molar and expressed as % of the glucose uptake rate...

This model has been successfully used by Chin and Sabde [25] for crevice cathodic protection using numerical analysis based on the dilute solution theory and reduction reaction of dissolved oxygen and Aa+, Cl, and OH ions at the crevice surface. Hence, the Nemst-Plank equation, eq. (4.2), can be generalized as a differentiable and continues scalar diffusion molar flux function... 

The ferrous Fe+ ions are not included in the analysis since the crevice cathodic protection is for preventing the formation of this t5 e ion. For a coupled diffusion and migration molar flux under steady-state conditions, dCjIddt = 0, the molar flux becomes the continuity equation for mass transfer under steady state conditions... 

The mass and molar fluxes with respect to the volume average velocity v can be formulated similarly, but their uses are limited, and thus we omit them. Table 4.3 summarizes the various definitions and relations for the mass and molar fluxes of a binary system. The most frequently used definitions are those of molar fluxes N, and J, and mass flux j,. Actually, N, is used in engineering applications, because it offers the advantage of a fixed coordinate system, whereas the fluxes j, and J are the usual measures of diffusion rates. Both definitions will be used in the subsequent analysis of mass transfer. 

This forms the basis of constructing an enthalpy budget in which the total enthalpy flux is compared with the scalar heat flux, 7q(W m-3), obtained from dividing heat flow by size (volume or mass) of the living matter. If account is made of all the reactions and side reactions in metabolism, the ratio of heat flux to enthalpy flux, the so-called energy recovery ( Yq/H = Jq/Jh) will equal 1. If it is more than 1, then the chemical analysis has failed fully to account for heat flux and if it is less than 1, then there are undetected endothermic reactions. Account for all reactions may seem a formidable task, but it should be borne in mind that anabolic processes dissipate insignificant amounts of heat compared with those of catabolism and that ATP production and utilization are balanced in cells at steady-state. Catabolism is generally limited to a relatively few well-known pathways with established overall molar enthalpies. So, as will be seen later, the task is by no means mission impossible. ... 

An analysis of the driving force of evaporation is necessary, as an increase in gas mass flow yields to a lower outlet humidity. An approximation for the evaporated molar mass flux is given by ... 

NH3 and to a lesser extent mono-, di-, and trimethylamines are the only significant gaseous bases in the atmosphere, and there has been considerable interest in whether the oceans are a source or sink of these gases. Early attempt to assess the air-sea flux from concentration measurements are probably suspect because of the ease with which sample contamination can occur during laboratory processing and analysis. It should be noted here that due to its high solubihty (low value of Henry s law constant), the air-water transfer of NH3 (and the methylamines for the same reason) is under gas phase control (see Section 6.03.2.1.1). The first reliable measurements were probably from the North and South Pacific and indicated that the flux of NH3 from sea to air is of a size similar to that for emission of DMS (Quinn et al., 1990, 1988). Indeed, the authors showed that this similarity was mirrored in the molar ratio of (non-sea-salt) sulfate to ammonium (1.3 0.7) in atmospheric aerosol particles collected on the cruise, indicating that for clean marine air remote from terrestrial sources, the emission of DMS and NH3 from the sea appears to control the composition of the aerosol. 

The electric field intensification in the surfactant-mediated separation processes is shown in Table In the feed, surfactant and metal ion concentrations are denoted by Csf and Cmf. respectively, while the corresponding concentrations in the permeate under steady-state conditions are denoted by C p and Cmp. Surfactant and metal ion rejections at a steady state are defined as Rs = (1-Csp/Csf) and Rm = (1-Cmp/Cmf)- The molar ratio of metal ion to surfactant is denoted by Fms- The separation of the electrodes is 3 mm. In Table 2, the initial current, pH, and solution conductivity are also given. It shows that both metal and surfactant are separated effectively under an electric field in which the permeate flux, surfactant, and metal ion rejections are enhanced. Economic analysis of the process indicates that some 20- to 50-fold efficiency increase is achieved compared with the no electric field case. For the process to be economical, low-solubility surfactants that can form multilamellar droplets should be used as carriers. 

In this section, we provide further analysis of qinterdiffusion by combining partial molar Gibbs free energies and partial molar entropies. The interdiffusional flux of thermal energy in a binary mixture was defined in (26-16) as ... 

Analysis of hydrodynamic equations for the flow in the fuel cell channel shows that this flow is incompressible [13]. In other words, the variation of pressure (total molar concentration) along the channel is small. Consider first the case of zero water flux through the membrane. Each oxygen molecule in the cathode channel is replaced with two water molecules. Pressure is proportional to the number of molecules per unit volume. To support constant pressure, the flow velocity in the channel must increase. The growth of velocity provides expansion of elementary fluid volume the expansion keeps pressure in this volume constant. 

First of all, however, in light of the preceding general discussion, let us consider, as an example, the precipitation of cementite e C out of an iron matrix which is supersaturated with carbon [40]. Difficulties arise because of the difference between the molar volume of the precipitate and the volume per mole of the precipitating components in the matrix. This causes a stress field to be formed around the precipitate. As long as no internal voids or creep processes occur, in the steady state case there will be a coupled diffusional flux of carbon towards the precipitate and of iron towards the supersaturated matrix, because the volume of the Fq C precipitate is greater than the volume of 3Fe. This problem was first analyzed by Hillert [41] for the system Fe-C. This analysis, however, has a quite wide and general applicability. 

When a particular component in a mixture is displaced in a given direction, it moves with a certain velocity. This velocity leads to a flux of the species, which is the molar rate of species movement per unit area in any given frame of reference. The nature of the displacements and the forces that cause the displacements leading to species velocity and flux are considered first in this section. Expressions for species velocities and fluxes are then studied to provide the foundations for a quantitative analysis of separation later. 

A quantitative analysis of separation in UF requires first a knowledge of the transport rates of the solvent and the macrosolutes through the UF membrane. When the macrosolute molecular weights are not high (>1000), the membrane pores may have dimensions in the range l-2nm the osmotic pressure of a concentrated solution of such macrosolutes will be significant with respect to the applied pressure difference, AP. The molar solvent flux under ideal conditions will be described by the flux expression (3.4.54) (Vilker etal., 1981) ... 

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