Surface reaction nth-order

For an nth-order surface reaction of species A, the rate and Thiele modulus, respectively are... 

Consider an nth-order surface reaction, represented by A(g) — product(s), occurring in a catalyst particle, with negligible external resistance to mass transfer so that cAs cAg- Then the observed rate of reaction is... 

According to equation 8.5-28, the nth-order surface reaction becomes a reaction for which the observed order is (n + l)/2. Thus, a zero-order surface reaction becomes one of order 1/2, a first-order reaction remains first-order, and second-order becomes order 3/2. This is the result if De is independent of concentration, as would be the case if Knud-sen diffusion predominated. If molecular diffusion predominates, for pure A, DecrcA, and the observed order becomes n/2,with corresponding results for particular orders of surface reaction (e.g., a first-order surface reaction is observed to have order 1/2). 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

The conservation equations for mass and enthalpy for this special situation have already been given with eqs 76 and 62. As there is no diffusional mass transport inside the pellet, the overall catalyst effectiveness factor is identical to the film effectiveness factor i/cxl which is defined as the ratio of the effective reaction rate under surface conditions divided by the intrinsic chemical rate under bulk fluid phase conditions (see eq 61). For an nth order, irreversible reaction we have the following expression ... 

Provided the interphase mass transfer resistance (1 /k() is sufficiently large, the reactant concentration at the external pellet surface will drop almost to zero. Thus, we may neglect the surface concentration cs compared to the bulk concentration q>. With cs — 0 in eq 115, it is obvious that in this case the reaction will effectively follow a first-order rate law. Moreover, it is also clear that the temperature dependence of the effective reaction rate is controlled by the mass transfer coefficient k(. This exhibits basically the same temperature dependence as the bulk diffusivity Dm, since the boundary layer thickness 5 is virtually not affected by temperature (kf = Dm/<5). Thus, we have the rule of thumb that the effective activation energy of an isothermal, simple, nth order, irreversible reaction will be less than 5-lOkJmor1 when the overall reaction rate is controlled by interphase diffusion. 

The concept of effectiveness factor has been developed to calculate the overall reaction rate in terms of the concentration at the external surface C , an nth-order irreversible reaction Vp / (n + 1) kCZ (7-103)... 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

Summary on Surface Kinetics. From this discussion we conclude that it is good enough to use the simplest available correlating rate expression, hence first-order or nth-order kinetics, to represent the surface reaction. 

Suppose that a gas reactant in a flowing fluid reacts on a nonporous catalyst at isothermal conditions. In steady state, the mass transfer rate of the reactant from the bulk of the fluid to the catalytic surface is equal to the reaction rate (nth order) ... 

A heterogeneous reaction A -> 2B with nth order kinetics. /rA = k( A (n > 0) takes place on a catalyst surface. The component A with initial concentration CA0 diffusses through a stagnant film on the catalyst surface at isothermal and isobaric conditions. Assume one-dimensional diffusion, and determine the concentration profile of component A within the film of thickness 8 if the k is constant. 

As mentioned previously, at high temperatures, the denominator of the catalytic rate law approaches 1. Consequently, for the moment, it is reasonable to assume that the surface reaction is of nth order in the gas-phase concentration of A within the pellet. 

In the longer term, the usual types of nth-order kinetic models used for gas phase and solution phase reactions will not suffice to describe the solid or liquid phase reactions at a burning surface. This is because the generally accepted... 

The dimensionless scaling factor in the mass transfer equation for reactant A with diffusion and chemical reaction is written with subscript j for the jth chemical reaction in a multiple reaction sequence. Hence, A corresponds to the Damkohler number for reaction j. The only distinguishing factor between all of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the 7th reaction (i.e., kj) changes from one reaction to another. The characteristic length, the molar density of key-limiting reactant A on the external surface of the catalyst, and the effective diffusion coefficient of reactant A are the same in all the Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the species mass balances. effective is the effective intrapellet diffusion coefficient of species i. If there is only one chemical reaction, or one rate-limiting step in a multiple reaction sequence, that is characterized by nth-order irreversible kinetics, then the rate constant in the numerator of the Damkohler numbers is the same for each A -. Hence, kj is written as k , which signifies that has units of (volume/mole)" /time for... 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

Equation (64) is the fundamental equation for nth order reaction accompanied by diffusion within a spherical catalyst particle, to be solved with boundary conditions Ca = Co for r = R (i.e., at the surface of the pellet) and dCAldx = 0 when r = 0. The general method of solution is given in the Appendix, Section 2, and Thiele s paper (7) may be consulted for further mathematical details. The essential result is that for first order reaction (n = 1) the fraction of surface available works out to be ... 

The intrinsic rate is based on the assumption that the concentration throughout the polymer particle is given by the concentration at the external surface. The effectiveness factor, n, is a measure of the extent to which the concentration is not uniform, and at the surface value, throughout the polymer particle. For simple nth-order reactions (n = 0, 1, 2) analytical relations have been developed in which... 

The first step in heterogenous catalytic processes is the transfer of the reactant from the bulk phase to the external surface of the catalyst pellet. If a nonporous catalyst is used, only external mass and heat transfer can influence the effective rate of transformation. The same situation will occur for very fast reactions, where the reactants are completely consumed at the external catalyst surface. As no internal mass and heat transfer resistances are considered, the overall catalyst effectiveness factor corresponds to the external effectiveness factor, For a simple irreversible reaction of nth order, the following relation results ... 

The analytical integration of the differential equation for diffusion with chemical reaction in a catalyst particle is achievable just for first-order reactions. A generalized modulus has been proposed for extending the use of the // expression in Equations 2.61 and 2.64a to any type of rate expression [17], at least approximately. For irreversible nth order reactions, the generalized modulus for a sphere becomes dependent on the exterior surface concentration ... 

Although the reaction rate is nth order in the absence of internal mass transfer limitations, the order determined from global rate data affected by pore diffusion will be ((n -t l)/2] instead of the true order of the surface reaction, for example, 3/2 instead of 2. The intrinsic reaction order will be observed only when n = 1. 

Aica is a core-to-annulus (interregion) mass transfer coefficient, with all other mass transfer resistances (e.g., from the bulk to the particle surfaces within each region) neglected. The final term in each of these equations accounts for reaction, with. k being the nth-order rate constant. The relevant boundary condition is Cc = Q at z = 0, where Cq is the inlet concentration of the reacting species. For a first-order reaction, it is straightforward to derive an analytical solution. 

In heterogeneous catalysis, nth-order kinetics may be the result of adsorption on a nonideal catalyst surface. In homogeneous systems, nth-order kinetics may represent the overall rate of the underlying elementary reactions, e.g., the classical Rice-Herzfeld mechanism for thermal cracking of hydrocarbonsFor simplicity, n is assumed to be constant for all species. This is not a strong assumption for many petroleum processes. 

A heterogeneous reaction A 2B with nth order kinetics = —kC (n > 0) takes place on a catalyst surface. 

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