Diffusion effects first order reactions

The importance of diffusion in a tubular reactor is determined by a dimensionless parameter, 3>Ai/R2 = aL/(uR2), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SCff/- 2 is small, then the effects of diffusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin1 studied the effects of diffusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

Catalyst deactivation in large-pore slab catalysts, where intrapaiticle convection, diffusion and first order reaction are the competing processes, is analyzed by uniform and shell-progressive models. Analytical solutions arc provid as well as plots of effectiveness factors as a function of model parameters as a basis for steady-state reactor design. 

The catalyst surface area per unit volume, g(Ss), can be made to vary with pellet coordinate by choosing an appropriate impregnation method. Hence, this function represents not only the level of dispersion but also activity distribution. A partially impregnated (or equivalently hollow) pellet is a typical example of a pellet with a certain activity distribution. The motivation for making such a pellet becomes obvious if it is recognized that the reactant concentration becomes almost zero at some point in the pellet when the reaction is diffusion-limited. The fraction of the volume of the pellet for which the concentration is zero is not utilized at all. If this fraction is made hollow or inert, then the observed rate on a per pellet basis should be the same as the fully impregnated pellet. Let us examine this further. Suppose that a pellet is hollow or partially impregnated for a distance Li from the center. Consider a diffusion-limited, first-order reaction. The internal effectiveness factor for this hollow pellet is ... 

Equations of (membrane type) unidirectional diffusion reactions in the steady state with one Michaelis type enzyme. Fundamental cases no regulatory effects firsts order reaction v k S k = V/(Km- -s) assumption S[Pg.434]

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The Merrill and Hamrin criterion was derived for a first-order reaction. It should apply reasonably well to other simple reactions, but reactions exist that are quite sensitive to diffusion. Examples include the decomposition of free-radical initiators where a few initial events can cause a large number of propagation reactions, and coupling or cross-linking reactions where a few events can have a large effect on product properties. 

Comparing this equation with Equation (8.34) shows that 3At/Y is the flat-plate counterpart of aAII - We thus seek a value for t/T below which diffusion has a negligible effect on the yield of a first-order reaction. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Effectiveness factor plot for spherical catalyst particles based on effective diffusivities (first-order reaction). 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

An exothermic first-order reaction A—h B is conducted in an FBCR, operating adiabatically and isobarically. The bed has a radius of 1.25 m and is 4 m long. The feed contains pure A at a concentration of 2.0 mol m-3, and flowing at q = 39.3 m3 s 1. The reaction may be diffusion limited assume that the relationship between r) and is 7] = (tanh The diffusivity is proportional to Tia, and Le for the particles is 0.50 mm. Determine the fractional conversion of A and the temperature at the bed outlet. How would your answer change, if (a) diffusion limitations were ignored, and (b) a constant effectiveness factor, based on inlet conditions, was assumed. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

A reactant in liquid will be converted to a product by an irreversible first-order reaction using spherical catalyst particles that are 0.4cm in diameter. The first-order reaction rate constant and the effective diffusion coefficient of the reactant in catalyst particles are 0.001 s and 1.2 X 10 ( ii s , respectively. The liquid film mass transfer resistance of the particles can be neglected. 

The overall reaction rate may be affected by diffusion if the mass transfer from the bulk fluid to the coal surface and/or the pore diffusion steps are relatively slow. I have made tentative estimates of both these diffusion effects. The following equation can be written (2) for the rate of oxygen uptake of a porous solid such as coal, assuming the oxygen to be consumed in an irreversible first-order reaction. 

The absorption of ozone from the gas occurred simultaneously with the reaction of the PAH inside the oil droplets. In order to prove that the mass transfer rates of ozone were not limiting in this case, the mass transfer gas/water was optimized and the influence of the mass transfer water/oil was studied by ozonating various oil/water-emulsions with defined oil droplet size distributions. No influence of the mean droplet diameter (1.2 15 pm) on the reaction rate of PAH was observed, consequently the chemical reaction was not controlled by mass transfer at the water/oil interface or diffusion inside the oil droplets. Therefore, a microkinetic description was possible by a first order reaction with regard to the PAH concentration (Kornmuller et al., 1997 a). The effects of pH variation and addition of scavengers indicated a selective direct reaction mechanism of PAH inside the oil droplets... 

It should be stressed that the reversible chemical reactions give us better chance to observe many-particle effects since there is no need here to monitor vanishing particle concentrations over many orders of magnitude. Indeed, the fluctuation-controlled law of the approach to the reaction equilibrium similar to (2.1.61) was observed recently experimentally [85] for the pseudo-first-order reaction A + B AB of laser-excited ROH dye molecules which dissociate in the excited state to create a geminate proton-excited anion pair. The solvated proton is attracted to the anion and recombines with it reversibly. After several dissociation-association cycles it finally diffuses to long distances and further recombination becomes unobservable. 

The pure compound rate constants were measured with 20-28 mesh catalyst particles and reflect intrinsic rates (—i.e., rates free from diffusion effects). Estimated pore diffusion thresholds are shown for 1/8-inch and 1/16-inch catalyst sizes. These curves show the approximate reaction rate constants above which pore diffusion effects may be observed for these two catalyst sizes. These thresholds were calculated using pore diffusion theory for first-order reactions (18). Effective diffusivities were estimated using the Wilke-Chang correlation (19) and applying a tortuosity of 4.0. The pure compound data were obtained by G. E. Langlois and co-workers in our laboratories. Product yields and suggested reaction mechanisms for hydrocracking many of these compounds have been published elsewhere (20-25). 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first order, whereas a second-order reaction has an apparent order of 3/2 when strongly influenced by diffusional effects. Because k and n are modified in the diffusion controlled region then, if the rate of the overall process is estimated by multiplying the chemical reaction rate by the effectiveness factor (as in equation 3.8), it is imperative to know the true rate of chemical reaction uninfluenced by diffusion effects. 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

Newson (1975) was among the first to develop a pore plugging model of demetallation to predict catalyst life. By using the pore structure model of Wheeler (1951), the pellet was assumed to have N pores of identical length but with a specified distribution of pore radii. Metal deposition was assumed to be a first-order reaction over an outer fraction of the pore length and to have a uniform thickness. This model showed that the broadness of the size distribution had little effect on the catalyst life for the same average radii, but that increasing the radii from 45 to 65 A more than doubled the catalyst life. The restricted form of the diffusivity (see Section IV,B,5) was not employed in this model. 

The model formulated by Ahn and Smith (1984) considered partial surface poisoning for HDS and pore mouth plugging for HDM reactions. The conservation equations with first-order reactions for metal-bearing and sulfur-bearing species were based on spherical pellet geometry rather than on single pores. Hence, a restricted effective diffusivity was employed... 

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed as a first-order reaction. 

After the absence of film diffusion effects has been verified and if the reaction order n is known, the expression for the rate equation r = r) kcat[E][S]buik/KM (first-order reaction assumed) can be inserted into the definition for 7j and the unknown rate constant k can be eliminated (Weisz, 1954) [Eq. (5.66)]. 

We have used CO oxidation on Pt to illustrate the evolution of models applied to interpret critical effects in catalytic oxidation reactions. All the above models use concepts concerning the complex detailed mechanism. But, as has been shown previously, critical. effects in oxidation reactions were studied as early as the 1930s. For their interpretation primary attention is paid to the interaction of kinetic dependences with the heat-and-mass transfer law [146], It is likely that in these cases there is still more variety in dynamic behaviour than when we deal with purely kinetic factors. A theory for the non-isothermal continuous stirred tank reactor for first-order reactions was suggested in refs. 152-155. The dynamics of CO oxidation in non-isothermal, in particular adiabatic, reactors has been studied [77-80, 155]. A sufficiently complex dynamic behaviour is also observed in isothermal reactors for CO oxidation by taking into account the diffusion both in pores [71, 147-149] and on the surfaces of catalyst [201, 202]. The simplest model accounting for the combination of kinetic and transport processes is an isothermal continuously stirred tank reactor (CSTR). It was Matsuura and Kato [157] who first showed that if the kinetic curve has a maximum peak (this curve is also obtained for CO oxidation [158]), then the isothermal CSTR can have several steady states (see also ref. 203). Recently several authors [3, 76, 118, 156, 159, 160] have applied CSTR models corresponding to the detailed mechanism of catalytic reactions. 

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

It is convenient to express the effect of diffusion by means of a diffusion factor rj, which is defined as the ratio of the observed reaction rate to the reaction rate when no diffusion effects are present. The diffusion factor r/ is a function of a parameter which contains all the independent variables of the system. For example, for a first-order reaction... 

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