Intrinsic reaction rate, equation

We have determined the intrinsic reaction rate equation for the hydrogenation of the 2-butanone into the 2-butanol on RU-AI2O3 in a preliminary kinetic investigation using a perfectly agitated slurry reactor. After some corrections due to a partial diffusional resistance at the liquid-solid interface, this equation takes the form ... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

When the effectiveness factors for both reactions approach unity, the selectivity for two independent simultaneous reactions is the ratio of the two intrinsic reaction-rate constants. However, at low values of both effectiveness factors, the selectivity of a porous catalyst may be greater than or less than that for a plane-catalyst surface. For a porous spherical catalyst at large values of the Thiele modulus s, the effectiveness factor becomes inversely proportional to (j>S9 as indicated by equation 12.3.68. In this situation, equation 12.3.133 becomes... 

The intrinsic kinetics was measured in an isothermal integrated reactor and the reaction rate equations in terms of power function have been established... 

As discussed in Sec. 7, the intrinsic reaction rate and the reaction rate per unit volume of reactor are obtained based on laboratory experiments. The kinetics are incorporated into the corresponding reactor model to estimate the required volume to achieve the desired conversion for the required throughput. The acceptable pressure drop across the reactor often can determine the reactor aspect ratio. The pressure drop may be estimated by using the Ergun equation... 

Replacing the effective reaction rate re in eq 62 with the effectiveness factor, multiplied by the intrinsic reaction rate under bulk fluid phase conditions, and dividing both sides of the resulting equation by 4nR2 yields the relationship... 

The intrinsic reaction rate has been measured in a discontinuous slurry stirred tank reactor and in a continuous microtrickle bed reactor (6 ). This last one is represented in Figure 1 Both methods lead to rather similar expressions for the intrin-sid rate equation r(mol/kg Pd. s)... 

For the particle sizes used in industrial reactors (> 1.5 mm), intraparticle transport of the reactants and ammonia to and from the active inner catalyst surface may be slower than the intrinsic reaction rate and therefore cannot be neglected. The overall reaction can in this way be considerably limited by ammonia diffusion through the pores within the catalysts [211]. The ratio of the actual reaction rate to the intrinsic reaction rate (absence of mass transport restriction) has been termed as pore effectiveness factor E. This is often used as a correction factor for the rate equation constants in the engineering design of ammonia converters. 

The reaction kinetics were studied by many researchers with sometimes different or contradictory results. Reviews are found in [592], [602], [603], [609]. A reason for this is that often diffusional effects were not eliminated, and therefore no real intrinsic reaction rates were obtained [610], [611]. In industrial practice equations are required for dimensioning reaction vessels and catalyst. The mathematical expressions used for this purpose are for the most part not based on theoretical assumptions and research... 

Rostrup-Nielsen found that the intrinsic reaction rate, rj, for methane steam reforming is correlated with the sulphur coverage by equation 6 (2). In the adiabatic prereformer, the sulphur acts as a pore mouth poison and as the reactions are restricted by pore diffusion 2,8), the effective activity of the sulphur poisoned catalyst pellet can be described by an empirical relation, equation 7, between the effective pellet reaction rate, rp, and the average sulphur coverage, 0av (7/... 

The governing mass and heat balance equations were derived in section 5.1.9 which simulate the concentration and temperature gradient between the bulk fluid and the external surface of the catalyst pellet. The effectiveness factors which represent the ratios of the observed actual rates of reactions to the intrinsic reactions rates where there is no mass and heat transfer resistances are computed for different reactions and different components. 

Substrate (and product) profiles are obtained from the numerical resolution of the above differential equations (system of differential equations in the case of product inhibition). The corresponding local effectiveness factors (ratio of effective and intrinsic reaction rates) are then calculated and the global effectiveness factor determined from their profiles, as in the case of simple Michaels-Menten kinetics. Results are represented in three-dimensional plots in Figs. 4.15 to 4.18 respectively. 

And is called kinetic equation or reaction rate law. Here r. is rate of reactions normalized over volume, C.,. is molar concentrations of reac-tants, k. is constant value characterizing the rate reactions at reactants concentration equal to 1, which is called reaction rate constant or intrinsic reaction rate, v.. is stoichiometric coefficient of the component i usually called partial order of reaction. Sum of one reaction partial order determines order of the reaction overall or order of its rate law. Elementary reactions (acts) dominate, which are subject to the rate law of zero, first and second order. For instance, for an elementary direct reaction... 

In the absence of substrate transport limitations, the substrate concentration is Independent of radial position in the polymer. The rate measured under these circumstances would be an intrinsic reaction rate. The substrate material balance is greatly simplified because Equation 8 no longer needs to be solved. A specific example of this can be found in the literature (16). 

If the intrinsic reaction rate is fast compared to the internal and/or external mass transfer processes, the reactant concentration within the porous catalyst and on its outer surface is smaller compared to the bulk concentration, whereas the concentration of the intermediate will be higher. Consequently, the consecutive reaction is promoted and the yield diminishes. The degree of yield losses depends on the ratio between transfer time and the intrinsic rate of the consecutive reaction, which is characterized by the corresponding Thiele moduli and Damkohler numbers referred to the consecutive reaction. For irreversible first-order reactions, the equations are as follows ... 

A further increase of the intrinsic reaction rate at constant volumetric mass transfer coefficient (/rg, ) results in Hatta numbers greater than 3 Ha > 3). The reaction rate can be considered as very fast compared to the mass transfer rate. As a consequence, the reactants do not reach the bulk phase (Cg jj 0) the reaction takes place only in the boundary layer (Figure 2.12c). Under these conditions, the reaction rate increases proportionally with the specific interfacial area between the phases ( ), the square root of the reaction rate constant, and the catalyst concentration as indicated in Equation 2.93. 

The kinetics of steam reforming has been widely studied, and there are a number of rate equations available in the literature (6).To design a steam reformer, equations expressing the intrinsic reaction rates in the following form are required ... 

In this sequence, steps 3-5 are the chemical rate processes laboratory analysis of these steps in the absence of physical effects yields the intrinsic reaction rate. Steps 1 and 7 are external physical rate processes separated from and in series with the chemical rate processes, while steps 2 and 6 are internal physical rate processes occurring simultaneously with chemical rate processes. The external and internal physical transport effects existing in a particular system are superimposed on the intrinsic reaction rate to obtain the global reaction rate, which is used in the macroscopic mass and energy transport equations required for reactor design. 

This Hougen-Watson type equation considers the inhibiting influence of H2O on the reaction rate, whereby the intrinsic reaction rate constant k, h2,hw and the coefficient Khw are given by... 

There is a second important reason for introducing the concept of an effectiveness factor. In the ordinary course of events, concentrations within a catalytic reactor packed with catalyst particles will vary both axially in the direction of flow as well as radially within the catalyst pellets. The model mass balance for such a system would consequently lead to a partial differential equation (PDE). By using an effectiveness factor, we reduce the PDE to an equivalent set of two ODEs — one the pellet mass balance in the radial direction and the other the reactor mass balance in the direction of flow. The reaction rate, which previously varied in two directions rj r, z), is now a function of the axial distance only We replace r (z, r) by E r, i(z), where is the so-called intrinsic reaction rate measiued experimentally on a fine powder and excludes diffusional effects. The latter are lumped into the effectiveness factor that now acts as a fractional efficiency on the intrinsic rate r, . This product of Er fz) is used in the reactor mass balance. 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Before deriving the rate equations, we first need to think about the dimensions of the rates. As heterogeneous catalysis involves reactants and products in the three-dimensional space of gases or liquids, but with intermediates on a two-dimensional surface we cannot simply use concentrations as in the case of uncatalyzed reactions. Our choice throughout this book will be to express the macroscopic rate of a catalytic reaction in moles per unit of time. In addition, we will use the microscopic concept of turnover frequency, defined as the number of molecules converted per active site and per unit of time. The macroscopic rate can be seen as a characteristic activity per weight or per volume unit of catalyst in all its complexity with regard to shape, composition, etc., whereas the turnover frequency is a measure of the intrinsic activity of a catalytic site. 

For situations where the reaction is very slow relative to diffusion, the effectiveness factor for the poisoned catalyst will be unity, and the apparent activation energy of the reaction will be the true activation energy for the intrinsic chemical reaction. As the temperature increases, however, the reaction rate increases much faster than the diffusion rate and one may enter a regime where hT( 1 — a) is larger than 2, so the apparent activation energy will drop to that given by equation 12.3.85 (approximately half the value for the intrinsic reaction). As the temperature increases further, the Thiele modulus [hT( 1 — a)] continues to increase with a concomitant decrease in the effectiveness with which the catalyst surface area is used and in the depth to which the reactants are capable of... 

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