Diffusion, mass Dilution rate

Concentration of A Arrhenius constants Arrhenius constant Constant in equation 5.82 Surface area per unit volume Parameter in equation 5.218 Cross-sectional area Concentration of B Stoichiometric constants Parameter in equation 5.218 Concentration of gas in liquid phase Saturation concentration of gas in liquid Concentration of G-mass Concentration of D-mass Dilution rate DamkOhler number Critical dilution rate for wash-out Effective diffusion coefficient Dilution rate for maximum biomass production Dilution rate for CSTF 1 Dilution rate for CSTF 2 Activation energy Enzyme concentration Concentration of active enzyme Active enzyme concentration at time t Initial active enzyme concentration Concentration of inactive enzyme Total enzyme concentration Concentration of enzyme-substrate complex with substance A... 

The species boundary condition at the stagnation surface follows from the fact that the diffusive mass flux in the fluid is balanced by a heterogeneous chemical reaction rate on the surface. In general, this can involve multiple and complex surface reactions and complex descriptions of the molecular diffusion. Here, however, we restrict attention to a single species that is dilute in a carrier gas and a single first-order surface reaction. Under these circumstances the surface reaction rate (mass of Y consumed per unit surface area) is given... 

Dm denotes the molecular diffusion coefficient F denotes the interphase mass exchange rate between the dense and the dilute phases and Fc = — F, which can be directly calculated with EMMS/matrix model parameters if the reaction source term, Sk, is negligible compared to the bulk gas conservation. For vaporization of A, the source term reads... 

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 ... 

In facilitated transport, unlike solvent extraction and other equihbrium stage wise processes, the overall mass transfer rate is not governed by the usual equihbrium considerations alone. Instead, the solute transport process is controlled by a combination of the diffusion rate and the complexation reaction rate and in case of coupled transport the solute can be transported against its concentration gradient thus opening up the possibilities of separation from even very dilute solute solutions. 

Smoke point of a turbulent flame is defined as the critical fuel mass flow rate (CFMFR) beyond which the flame does not smoke. Goh [88] studied the effects of nitrogen dilution on the smoke point characteristics of propylene diffusion flames in crossflow. Figure 29.20 shows the variation of diluent mass flow rate with fuel... 

The proportionality D is a constant, which is known as the diffusion coefficient or diffusivity, with a unit of m s (SI) or cm s. The diffusion coefficient is a property of materials, which is the most useful parameter to characterize the rate of diffusive mass transport, showing a strong dependence on temperature. Although it is also a function of composition, if the diffusing species are significantly diluted, it can be assumed to be independent on the composition. 

The constant of proportionality D is called the diffusion coefficient or diffusivity and has SI units of m /s (but is more commonly expressed in cm /s). The diffusion coefficient is a material property and is the most useful parameter for characterizing the rate of diffusive mass transport. It is usually a strong function of temperature and is also a function of composition, although in certain limiting cases, such as a dilute concentration of the diffusing species, it can be taken to be independent of the composition. 

If there is no chemical reaction but mass exchange between phases, such as evaporation, subfimation, and condensation, we can replace the reaction rate with certain mass exchange rate tm, and similarly define the heterogeneity index for structure-dependent mass transfer. For example, assume that the concentration of transferred species at particle surface is saturated further, the diffusion rate from the particles to the gas in the dense phase is equal to that from the gas in the dense phase to the gas in the dilute phase and we get... 

In the case of both gas and liquid, assuming that the mass-transfer coefficient follows an Arrhenius-type equation, compute the energy of activation" of mass transfer. Are these high or low in comparison with the energy of activation of typical chemical reactions Note that, for dilute solutions, the identity of the diffusing solute need not have been specified in order to obtain the energy of activation" of mass transfer. What other method might be used to determine whether reaction rate or mass-transfer rate controls ... 

For dilute mixture and low mass transfer rates, mass transfer is quite analogous to the heat transfer rate with Le 1, and mass transfer correlations are derived from that of heat transfer correlations by simply replacing the Nusselt number, Nu, with the Sherwood number, Sh, and replacing the Prandtl number with the Schmidt number. However, one important difference will be the case where only one side of the channel is exposed to the gas diffusion layer and permeable and the rest of the three surfaces are impermeable to species flux. 

Second and more important, diffusion in dilute solutions is easier to understand in physical terms. A diffusion flux is the rate per unit area at which mass moves. A concentration profile is simply the variation of the concentration versus time and position. These ideas are much more easily grasped than concepts Hke momentum flux, which is the momentum per area per time. This seems particularly true for those whose backgrounds are not in engineering, those who need to know about diffusion but not about other transport phenomena. 

The mass of the sedimenting particle could be deduced from its rate of sedimentation at high dilution in a given field, i.e., from its sedimentation constant, if the frictional coefficient / could be determined independently. Hates of diffusion may be utilized to secure this necessary supplementary information, since the diffusion constant D depends also on the frictional coefficient. Thus ... 

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

The activity calculated from (7) comprises both film and pore diffusion resistance, but also the positive effect of increased temperature of the catalyst particle due to the exothermic reaction. From the observed reaction rates and mass- and heat transfer coefficients, it is found that the effect of external transport restrictions on the reaction rate is less than 5% in both laboratory and industrial plants. Thus, Table 2 shows that smaller catalyst particles are more active due to less diffusion restriction in the porous particle. For the dilute S02 gas, this effect can be analyzed by an approximate model assuming 1st order reversible and isothermal reaction. In this case, the surface effectiveness factor is calculated from... 

Information on particle size may be obtained from the sedimentation of particles in dilute suspensions. The use of pipette techniques can be rather tedious and care is required to ensure that measurements are sufficiently precise. Instruments such as X-ray or photo-sedimentometers serve to automate this method in a non-intrusive manner. The attenuation of a narrow collimated beam of radiation passing horizontally through a sample of suspension is related to the mass of solid material in the path of the beam. This attenuation can be monitored at a fixed height in the suspension, or can be monitored as the beam is raised at a known rate. This latter procedure serves to reduce the time required to obtain sufficient data from which the particle size distribution may be calculated. This technique is limited to the analysis of particles whose settling behaviour follows Stokes law, as discussed in Section 3.3.4, and to conditions where any diffusive motion of particles is negligible. 

The rate of sedimentation is defined by the sedimentation constant 5, which is directly proportional to the polymer mass m, solution density p, and specific volume of the polymer V, and inversely proportional to the square of the angular velocity of rotation o), the distance from the center of rotation to the point of observation in the cell r, and the fractional coefficient /, which is inversely related to the diffusion coefficient D extrapolated to infinite dilution. These relationships are shown in the following equations in which (1 — Vp) is called the buoyancy factor since it determines the direction of macromolecular transport in the cell. 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

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