The Rate and Performance Equations

On the other hand, consider series deactivation. In the regime of strong pore resistance the concentration of product R is higher within the pellet than at the exterior. Since R is the source of the poison, the latter deposits in higher concentration within the pellet interior. Hence we can have poisoning from the inside-out for series deactivation. 

Finally, consider side-by-side deactivation. Whatever the concentration of reactants and products may be, the rate at which the poison from the feed reacts with the surface determines where it deposits. For a small poison rate constant the poison penetrates the pellet uniformly and deactivates all elements of the catalyst surface in the same way. For a large rate constant poisoning occurs at the pellet exterior, as soon as the poison reaches the surface. 

The above discussion shows that the progress of deactivation may occur in different ways depending on the type of decay reaction occurring and on the value of the pore diffusion factor. For parallel and series poisoning, the Thiele modulus for the main reaction is the pertinent pore diffusion parameter. For side-by-side reactions, the Thiele modulus for the deactivation is the prime parameter. 

Nonisothermal effects within pellets may also cause variations in deactivation with location, especially when deactivation is caused by surface modifications due to high temperatures. 

Additional Factors Influencing Decay. Numerous other factors may influence the observed change in activity of catalyst. These include pore mouth blocking by deposited solid, equilibrium, or reversible poisoning where some activity always remains, and the action of regeneration (this often leaves catalyst with an active exterior but inactive core). 

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The second term in brackets in equation 36 is the separative work produced per unit time, called the separative capacity of the cascade. It is a function only of the rates and concentrations of the separation task being performed, and its value can be calculated quite easily from a value balance about the cascade. The separative capacity, sometimes called the separative power, is a defined mathematical quantity. Its usefulness arises from the fact that it is directly proportional to the total flow in the cascade and, therefore, directly proportional to the amount of equipment required for the cascade, the power requirement of the cascade, and the cost of the cascade. The separative capacity can be calculated using either molar flows and mol fractions or mass flows and weight fractions. The common unit for measuring separative work is the separative work unit (SWU) which is obtained when the flows are measured in kilograms of uranium and the concentrations in weight fractions. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

In order to evaluate the catalytic characteristics of colloidal platinum, a comparison of the efficiency of Pt nanoparticles in the quasi-homogeneous reaction shown in Equation 3.7, with that of supported colloids of the same charge and of a conventional heterogeneous platinum catalyst was performed. The quasi-homogeneous colloidal system surpassed the conventional catalyst in turnover frequency by a factor of 3 [157], Enantioselectivity of the reaction (Equation 3.7) in the presence of polyvinyl-pyrrolidone as stabilizer has been studied by Bradley et al. [158,159], who observed that the presence of HC1 in as-prepared cinchona alkaloids modified Pt sols had a marked effect on the rate and reproducibility [158], Removal of HC1 by dialysis improved the performance of the catalysts in both rate and reproducibility. These purified colloidal catalysts can serve as reliable... 

The experimental study of solid catalyzed gaseous reactions can be performed in batch, continuous flow stirred tank, or tubular flow reactors. This involves a stirred tank reactor with a recycle system flowing through a catalyzed bed (Figure 5-31). For integral analysis, a rate equation is selected for testing and the batch reactor performance equation is integrated. An example is the rate on a catalyst mass basis in Equation 5-322. 

First, the role of rubber modification in high rate impact is to suppress spallation by inducing the material to yield in the presence of dynamic tensile stresses arising from impact. Second, this yield-spall transition occurs at different strain rates for different rubber contents and may be predictable using quasistatic, low temperature tests of this type. These tests can also provide information concerning the basic nature of the yield process in these materials through the activation parameters which are obtained. Third, the Bauwens-Crowet equation seems to be a good model for the rate and temperature sensitive behavior of the American Cyanamid materials and is therefore a likely candidate for a yield criterion to use in the analytical code work on these materials which we hope to perform as a continuation of this work. 

Using the experimentally determined values of the rate constants, WGS calculated to be 25.4 by equation 11. The actual value of the equilibrium constant at 637 K is 17.6, and this agreement is acceptable. The second check was performed by predicting both the rate and the kinetic rate expression for WGS and comparing these predictions to reported values. 

The rate and mechanism of this hydrolysis was performed by Naiditch and Yost (13). These authors found that hydrolysis is catalyzed by acid as well as water, but the effect of water is much less than that of acid. The rate equation can be expressed... 

As a first step, based on laboratory-scale data at different times, a rate equation can be developed and then the batch reactor performance equation... 

The multi-physics of SOFCs are governed by the mass, momentum, and energy conservation equations, and the chemistry and electrochemistry. The governing equations of SOFCs are tightly coupled and changes to one aspect of the fuel cell can drastically affect another. For example, the rate and composition of the fuel flow in the anode will affect the temperature distributions in the cell, which can induce stresses due to mismatches between the coeSicients of thermal expansion of the various layers in the SOFC. The fuel flow wiU also affect the overall performance of the fuel cell based on the distribution of species in the anode and the electrochemical reactions. 

Table 4.10.3 gives the conversion in a batch reactor, a PFR, and a CSTR for different values of 8v for the example of Do = 1. The data indicate that, in contrast to a batch reactor, Xa decreases for a reaction with increasing volume both in a CSTR and in a PFR, which is in general true for a reaction order >0 [see Levenspiel (1996, 1999)]. For a reaction with decreasing volume rate, this is reversed. In both flow reactors (PFR, CSTR), the residence time changes compared to a constant volume reaction, while in a batch reactor the reaction time does not. Thus for reactions with changing volume, the batch and the plug flow performance equations are different. 

The spatial discretization is performed on the basis finite-volume approach. It means, the mass and energy equations are solved within control volumes, and the momentum equations are solved over flow paths — or junctions — connecting the centres of control volumes. The solution variables are the pressure, vapour temperature, liquid temperature and mass quality within a control volume, as well as the mass flow rate at a junction. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

The clinical performance of a hemodialy2er is usually described in terms of clearance, a term having its roots in renal physiology, which is defined as the rate of solute removal divided by the inlet flow concentration as shown in equation 7, where Cl is clearance in ml,/min and all other terms are as defined previously except that, in deference to convention, flow rates are now expressed in minutes rather than seconds and feed side (/) is now synonymous with blood flow on the luminal side. 

This equation has two unknowns Xq and Xe), and an empirical relation between them is needed. Many have been tried, and one of the best is to assume that the excess of To over Te expressed as a ratio to Tp (zero for a perfectly stirred chamber) is a constant A [ (Tg — Tg)/ Tp]. Although A should vaiy with burner type, the effects of firing rate and percent excess air are small. In the absence of performance data on the land of furnace under study, assume A = 300/Tp, °R or 170/Tp, K. The left side of Eq. (5-178) then becomes D 1 —Xc + A), and with coefficients of Xc and Xc collected, the equation becomes... 

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