Diffusion equations, liquid phase chemical

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 membrane and diffusion-media modeling equations apply to the same variables in the same phase in the catalyst layer. The rate of evaporation or condensation, eq 39, relates the water concentration in the gas and liquid phases. For the water content and chemical potential in the membrane, various approaches can be used, as discussed in section 4.2. If liquid water exists, a supersaturated isotherm can be used, or the liquid pressure can be assumed to be either continuous or related through a mass-transfer coefficient. If there is only water vapor, an isotherm is used. To relate the reactant and product concentrations, potentials, and currents in the phases within the catalyst layer, kinetic expressions (eqs 12 and 13) are used along with zero values for the divergence of the total current (eq 27). 

A correlation between surface and volume processes is described in Section 5. The atomic-molecular kinetic theory of surface processes is discussed, including processes that change the solid states at the expense of reactions with atoms and molecules of a gas or liquid phase. The approach reflects the multistage character of the surface and volume processes, each stage of which is described using the theory of chemical kinetics of non-ideal reactive systems. The constructed equations are also described on the atomic level description of diffusion of gases through polymers and topochemical processes. 

As mentioned, from the reaction kinetics viewpoint the behavior of zeolite catalysts shows large variability. In addition, the apparent kinetics can be affected by pore diffusion. The compilation of literature revealed some kinetic equations, but their applicability in a realistic design was questionable. In this section we illustrate an approach that combines purely chemical reaction data with the evaluation of mass-transfer resistances. The source of kinetic data is a paper published by Corma et al. [7] dealing with MCM-22 and beta-zeolites. The alkylation takes place in a down-flow liquid-phase microreactor charged with catalyst diluted with carborundum. The particles are small (0.25-0.40 mm) and as a result there are no diffusion and mass-transfer limitations. 

In the above equations n and N are the atom fractions of nitrogen-15 species in the gas phase and the liquid phase respectively c (moles/cc.), d (cm.2/sec.), b (cm.) for the gas phase are respectively the concentration of oxides of nitrogen, the diffusion coefficient, and the thickness of the boundary layer, while C, D, and B are the same quantities for the liquid phase k (cc./moles-sec.) is a rate constant for the exchange of oxides of nitrogen between the gas and liquid phase. The specific transfer rate kr (moles/sec.-cm. ) when multiplied by the interfacial area a (cm.2/cc.) in a 1 cm. length of column per cm. of cross-sectional area gives an interphase transfer rate fc a (moles/sec.-cc.). If chemical reaction is rate limiting, fc a will be determined by the first term of Equation 25, otherwise it will be determined by the diffusion terms. 

Relaxation and self-diffusion techniques in solution are widely used to study hydrogen-bonded systems. The nuclear quadrupole coupling constant (NQCC) presents a sensitive probe for the strength of hydrogen bonding. Unfortunately in the liquid phase this property cannot be measured in a direct way. Two new indirect methods are now presented for determining NQCC in H-bonded liquids. Ferris and Farrar showed that for the OD deuteron of ethanol the DQCC is related to the chemical shift of the hydroxy proton by a linear equation. Thus the straightforward measurement of the chemical shift... 

The reaction was carried out by dissolving gaseous HCl in a stirred vessel containing the alcohol. The resulting concentration-time data could be correlated with a rate equation half-order in alcohol concentration. However, the rate constant was found to vary with the gas (HCl) flow rate into the reactor, suggesting that the observed rate was influenced by the resistance to diffusion of dissolved HCl in the liquid phase. A method of analysis which took into account the diffusion resistance indicated that the chemical step was probably first order in dissolved HCl and zero order with respect to lauryl alcohol. 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

The following development applies to almost all chemical absorptions. Solute molecules in the gas diffuse to and across the interface, then diffuse in the liquid until meeting a reactant. If the reaction is very fast, the nonreactive mass transfer relations, previously discussed, apply—but very conservatively the effective rate is higher. The flux equation for liquid phase transfer of component i is modified as follows ... 

One-Dimensional Analytical Model With Diffusive Vapor Loss At Upper Boundary. This model was developed by Jury et al. (16) to provide a computational method for classifying organic chemicals for their relative susceptibility to different loss pathways (volatilization, leaching and degradation). Although the basic equation is essentially the same as Equation 2, in contrast to Equation 2 it includes transport in both the vapor and liquid phases. An effective diffusion coefficient, Dg, is defined such that it includes both the vapor component, KjjPq, and liquid component, Dl, in the following manner ... 

Vapor and Liquid Diffusion Coefficients. The values of 247.1 cm2/hr for Dp r and 6 184 x 10" - m /day for D ate were calculated by methoos utilizing chemical properties (3). Other approximations suggested by Jury et al. were used to obtain the effective diffusion coefficient, Dj > the vapor phase diffusion coefficient, Dq, and the liquid phase diffusion coefficient, Dl> for soils. Using the equation DEp = DqKh + Dl for the Maui soils with 59% porosity and 30% water, a value of DEp = 0.195 cm2/hr was obtained (3.). The calculated vapor phase diffusion coefficient, Dq, for these same conditons was 11.5 cm2/hr. Experimental measurements on a similar Hawaii soil by Pringle et al. (26) gave a DBCP liquid-vapor diffusion... 

The polymerisation of PO and EO, initiated by polyfunctional starters, to make short chain polyether polyols is a reaction that is strongly dependent on diffusion. The consumption rate of PO or EO is given by two simultaneous factors the rate of the chemical reaction in the liquid phase and the efficiency of the monomer mass transfer from the gaseous phase to liquid phase (see details in section 4.1.5). The PO (or EO) consumption rate, considering the mass transfer, is described by equation 13.27 [45-50] ... 

In the absence of chemical reactions and at steady-state conditions, since moles are conserved, the rate of diffusion across the gas-phase film must equal the rate across the liquid phase film. Applying (16-8) and (16-9) to this case, and equating diffusion rates on each side of the interface yields the two-film model of Whitman ... 

In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. If the kinetics are not zeroth-order or first-order, then the methodology exists to find the best pseudo-first-order rate constant to match the actual rate law and obtain an approximate analytical solution. The concentration profile of reactant A in the liquid phase must satisfy... 

Design a two-phase gas-liquid CSTR that operates at 55°C to accomplish the liquid-phase chlorination of benzene. Benzene enters as a liquid, possibly diluted by an inert solvent, and chlorine gas is bubbled through the liquid mixture. It is only necessary to consider the first chlorination reaction because the kinetic rate constant for the second reaction is a factor of 8 smaller than the kinetic rate constant for the first reaction at 55°C. Furthermore, the kinetic rate constant for the third reaction is a factor of 243 smaller than the kinetic rate constant for the first reaction at 55°C. The extents of reaction for the second and third chlorination steps ( 2 and 3) are much smaller than the value of for any simulation (i.e., see Section 1-2.2). Chlorine gas must diffuse across the gas-liquid interface before the reaction can occur. The total gas-phase volume within the CSTR depends directly on the inlet flow rate ratio of gaseous chlorine to hquid benzene, and the impeller speed-gas sparger combination produces gas bubbles that are 2 mm in diameter. Hence, interphase mass transfer must be considered via mass transfer coefficients. The chemical reaction occurs predominantly in the liquid phase. In this respect, it is necessary to introduce a chemical reaction enhancement factor to correct liquid-phase mass transfer coefficients, as given by equation (13-18). This is accomplished via the dimensionless correlation for one-dimensional diffusion and pseudo-first-order irreversible chemical reaction ... 

If the rate of chemical reaction is much faster than the rate of mass transfer via diffusion, then A 1 and tanh A -> 1. Hence, the mass transfer enhancement factor Sh -> A in the diffusion-limited regime via equation (24-24) or (24-26). The final form for the liquid-phase mass transfer coefficient of component j in the diffusion-limited regime is... 

In the absence of convective mass transfer and chemical reaction, calculate the steady-state liquid-phase mass transfer coefficient that accounts for curvature in the interfacial region for cylindrical liquid-solid interfaces. An example is cylindrical pellets that dissolve and diffuse into a quiescent liquid that surrounds each solid pellet. The appropriate starting point is provided by equation (B) in Table 18.2-2 on page 559 in Bird et al. (1960). For one-dimensional diffusion radially outward, the mass transfer equation in cylindrical coordinates reduces to... 

This book provides the results of research into the development of scientific fundamentals, the technical design and industrial application of brand new technologies and devices for fast liquid-phase turbulent processes in the chemical, petrochemical, and petroleum industries. The macrokinetic approach, considering processes of diffusion, hydrodynamics, and heat transfer, has been developed and used for the advancement of fundamental knowledge and technologically important equations to enable the calculation of mass and heat transfer processes, which accompany fast chemical reactions and well-developed turbulence. This new family of chemical devices has been proposed for the intensification of fast liquid-phase processes through the creation of optimal hydrodynamic conditions for reacting media flows in a reaction zone. 

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