Rate law nth-order

Figure 3.4 Linear integrated form of nth-order rate law ( rA) = k c for constant-volume BR (n 1)...

For a given duty the ratio of sizes of mixed and plug flow reactors will depend on the extent of reaction, the stoichiometry, and the form of the rate equation. For the general case, a comparison of Eqs. 5.11 and 5.17 will give this size ratio. Let us make this comparison for the large class of reactions approximated by the simple nth-order rate law... 

For reactions that obey a simple nth-order rate law and for which volumetric expansion effects may be significant, equation (8.1.8) becomes... 

If the reaction follows a simple nth order rate law and the temperature dependence of the rate constant is described by Arrhenius equation the thermogravimetric loss rate is given by the slope equation ... 

A This rate law does not have the general form of equation (20.6), or the usual nth order rate law seen in Table 20.5. 

This assumes that the concentration at any value of x is not a function of radius. Ca is the concentration of reacting species A, u the mean convective velocity, which is assumed to be neither a function of axial or radial position, and Ta is the reaction rate of A based on unit volume. If nth-order power law kinetics pertain, i.e. 

For a single reaction A — P following an nth-order kinetic law, the reaction rate is given by... 

Reactions often follow nth-order kinetic law. Under isothermal conditions, for example, under conditions where the sample temperature remains constant, the heat release rate decreases uniformly with time. In the case of autocatalytic decomposition, the behavior is different an acceleration of the reaction with time is observed. The corresponding heat release rate passes through a maximum and then decreases again (Figure 12.1), giving a bell-shaped heat release rate curve or... 

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. 

A chemical reaction with stoichiometry A — products is said to follow an n -order rate law if A is consumed at a rale proportional to the nth power of its concentration in the reaction mixture. If is the rale of consumption of A per unit reactor volume, then... 

However, when the view is restricted to simple, irreversible reactions obeying an nth order power rate law and, if additionally, isothermal conditions arc supposed, then—together with the results of Section 6.2.3—it can be easily understood how the effective activation energy and the effective reaction order will change during the transition from the kinetic regime to the diffusion controlled regime of the reaction. 

Table 2 lists most of the available experimental criteria for intraparticle heat and mass transfer. These criteria apply to single reactions only, where it is additionally supposed that the kinetics may be described by a simple nth order power rate law. The most general of the criteria, 5 and 8 in Table 2, ensure the absence of any net effects (combined) of intraparticle temperature and concentration gradients on the observable reaction rate. However, these criteria do not guarantee that this may not be due to a compensation of heat and mass transfer effects (this point has been discussed in the previous section). In fact, this happens when y/J n [12]. 

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. 

Upon looking back at these sections on descriptive kinetics, one may be struck by the fact that all rate laws describe graphical patterns which, in fact, have been employed in the analysis of Section 1.9.2. There is one further thing we can do here to exploit this in interpretation of kinetics. Suppose we have a normal nth-order irreversible... 

The reversible kinetic rate law for nth-order chemical reaction is... 

Describe the hnear least-squares analysis (LLSA) procedure that allows one to calculate the reaction order n from a set of discrete data points for reaction half-time fi/2 vs. the initial concentration of reactant A, Cao- The kinetics are irreversible and nth-order, and the rate law is only a function of the molar density of reactant A. Answer this qnestion by providing the following information ... 

The Hougen-Watson rate law lEinw, with units of moles per area per time, is written on a pseudo-volumetric basis using the internal surface area per mass of catalyst S , and the apparent mass density of the pellet Papp. k is the nth-order kinetic rate constant with units of (volume/mole)" per time when the rate law is expressed on a volumetric basis using molar densities. 

For reversible chemical reactions in which 100% conversion of reactants to products cannot be achieved, the upper integration limit is XequiBbrium and the factor of 3 in (15-19) must be replaced by 3/[l — (1 — Xequmbnum) ]- Equation (15-19) is evaluated for irreversible nth-order chemical kinetics when the rate law is only a function of the molar density of the key-limiting reactant. Under these conditions. 

Step 2. Use the general expression in step 1 to determine the best value of in terms of k for irreversible nth-order chemical kinetics where the rate law is expressed in terms of the molar density of only one key-limiting reactant. 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

The mass balance with homogeneous one-dimensional diffusion and irreversible nth-order chemical reaction provides basic information for the spatial dependence of reactant molar density within a catalytic pellet. Since this problem is based on one isolated pellet, the molar density profile can be obtained for any type of chemical kinetics. Of course, analytical solutions are available only when the rate law conforms to simple zeroth- or first-order kinetics. Numerical techniques are required to solve the mass balance when the kinetics are more complex. The rationale for developing a correlation between the effectiveness factor and intrapellet Damkohler number is based on the fact that the reactor design engineer does not want to consider details of the interplay between diffusion and chemical reaction in each catalytic pellet when these pellets are packed in a large-scale reactor. The strategy is formulated as follows ... 

Since is only a function of spatial coordinate r), the partial derivative in equation (20-7) is replaced by a total derivative, and the dimensionless concentration gradient evaluated at the external surface (i.e = 1) is a constant that can be removed from the surface integral in the numerator of the effectiveness factor (see equation 20-6). For simple nth-order irreversible chemical kinetics in catalytic pellets, where the rate law is a function of the molar density of only one reactant. 

If the kinetics are not zeroth-order, then these integral expressions are more tedions to nse than the ones developed earlier in this chapter based on mass transfer across the external snrface of the catalyst. The preferred expressions for the effectiveness factor are summarized below for nth-order irreversible chemical kinetics when the rate law is only a function of the molar density of one reactant ... 

The following expressions for the effectiveness factor E have been derived for nth-order irreversible chemical kinetics (n = j0) based on a volumetric average of the rate law and diffusion across the external surface of the catalyst ... 

In Chapter 10, the dimensionless scaling factor in the mass transfer equation with diffusion and chemical reaction was written with subscript j for the jth chemical reaction in a multiple-reaction sequence (see equation 10-10). In the absence of convective mass transfer, the number of dimensionless scaling factors in the mass transfer equation for component i is equal to the number of chemical reactions. Hence, corresponds to the Damkohler number for reaction j. The only distinguishing factor between aU of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the jth reaction (i.e., kj), for a volumetric rate law based on molar densities, changes from one reaction to another. The characteristic length L, the molar density of key-limiting reactant A on the external surface of the catalyst CA.sur ce, and the effective diffusion coefficient of reactant A, a. effective, are the same in aU 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 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... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Consider one-dimensional (i.e., radial) diffusion and multiple chemical reactions in a porous catalytic pellet with spherical symmetry. For each chemical reaction, the kinetic rate law is given by a simple nth-order expression that depends only on the molar density of reactant A. Furthermore, the thermal energy generation parameter for each chemical reaction, Pj = 0. 

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