Gradient profiles kinetic equations

The VERSE method was extended to describe the consequences of protein de-naturation on breakthrough curves in frontal analysis and on elution band profiles in nonlinear isocratic and gradient elution chromatography [45]. Its authors assumed that a unimolecular and irreversible reaction taking place in the adsorbed phase accormts properly for the denaturation and that the rate of adsorption/desorption is relatively small compared with the rates of the mass transfer kinetics and of the reaction. Thus, the assumption of local equilibrium is no longer valid. Consequently, the solid phase concentration must then be related to the adsorption and the desorption rates, via a kinetic equation. A second-order kinetics very similar to the one in Eq. 15.42 is used. 

Helfferich [90] had given a mathematical description of the diffusion of ions in ion-exchange beads based on the Nernst-Planck and the Nernst-Einstein equations, Hornby et al [82] combined a case of this treatment with the Michaelis-Menten kinetic equation (Table IX) in order to calculate diffusion reactions with linear profiles in an idealized quasi-Nernst layer of thickness x limited on one side by a bulk solution of concentration Sq and on the other one by a charged enzyme-layer the rate of reaction in the steady state equals the sum of and Jj the fluxes of the charged substrate due respectively to the electrical and concentration gradients ... 

The energy balance equation was solved for the most exothermic case (Run 5), (Tables I and II) together with the mass balance equation (1). Thus, the r. were deduced from a well-fitted but with respect to the kinetic expression still arbitrary description of the experimental concentration profile along the reactor. Since the AHj are known, it remains to choose hw and Xeff so that the experimentally measured temperature gradient is correctly described. For this,... 

Assumption 6 In the general case, we shall assume that there are no thermal effects and neglect the influence of the heat of adsorption on the band profile. In principle, heat or enthalpy balances for the mobile and stationary phases should be included in the system of equations, as discussed in Section 2.1.4. In practice, however, the temperature excursion associated with the migration of a band seems to be small and no detectable effect has been demonstrated [31]. Accordingly, we ignore the thermal effect in the rest of this book, except in Section 2.1.5. We assiune that the column is operated isothermally. A temperature gradient along the column would require appropriate adjustments in Eq. 2.2 to account for the variations of the isotherm and the kinetic parameters, of u, Di i, and, improbably, F with T, hence, in this case, z. 

The mentioned thermodynamic prerequisite that the formal potential of the substrate redox system must be more positive than the formal potential of the catalyst redox system means that, in principle, reduction of S is easier compared to Cat , but that kinetic constraints essentially hinder this process at potentials where the catalyst is oxidized. Then, the direct reduction of S does not proceed electrochemically at potentials where Cat is reduced (or maybe even at no accessible potential at all) but only via homogeneons redox reaction (Equation (3.2)) with CaC". In this context, the regeneration of the catalyst leads to much steeper concentration profiles of the catalyst in the diffusion reaction layer that is, to a steeper concentration gradient that (see Chapter 1) means larger current. 

Mass flux of reactant A into the catalyst across its external surface is employed to develop analytical expressions for the effectiveness factor in terms of the intrapellet Damkohler nnmber. Reactant molar density profiles for diffusion and first-order irreversible reaction have been developed in three coordinate systems, and these profiles in Chapter 17 represent the starting point to calculate the dimensionless concentration gradient on the external surface of the catalyst. In each case, the reader should verify these effectiveness factor results by volumetri-cally averaging the dimensionless molar density profile throughout the pellet via equations (20-47) with n = 1, realizing that it is not necessary to introduce a critical dimensionless spatial coordinate when the kinetics are first-order. 

Answer Two. The thermal energy balance is not required when the enthalpy change for each chemical reaction is negligible, which causes the thermal energy generation parameters to tend toward zero. Hence, one calculates the molar density profile for reactant A within the catalyst via the mass transfer equation, which includes one-dimensional diffnsion and multiple chemical reactions. Stoichiometry is not required because the kinetic rate law for each reaction depends only on Ca. Since the microscopic mass balance is a second-order ordinary differential eqnation, it can be rewritten as two coupled first-order ODEs with split boundary conditions for Ca and its radial gradient. 

In the first case, if Po > 1 (sq > K), substrate gradient is very high and reaction is limited only by enzyme kinetics, so substrate profile within the stagnant film will be negligible and ss = Sq. In that case the rate equation reduces to a simple Michaelis-Menten equation that can be linearized as already described in Chapter 3. Using the double reciprocal plot ... 

With this equation, the temporal profile of porosity during the SPS of A1 powder can be derived numerically, with the available experimentally measured temporal profile of temperature, together with the applied voltage gradient and constant pressure. It is found that the model satisfactorily predicts the shrinkage kinetics of the powder compacts. 

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