The nth-Order Reaction

Relatively few kinetic schemes admit analytical solutions. The present section is concerned with the special cases that do and also with some cases where preliminary analytical work will ease the subsequent numerical studies. We begin with the nth-order reaction. 

This reaction can be elementary if n = 1,2. More generally, it is complex. Noninteger values for n are often found when fitting rate data empirically, sometimes for sound kinetic reasons, as will be seen in Section 2.5.3. For an isothermal, constant-volume 

The first-order reaction is a special case mathematically. For n= 1, the solution has the exponential form of Equation 1.24  

If the reaction order does not change, reactions with n I will go to completion in the finite time tmax- This is sometimes observed. Solid rocket propellants or firecracker fuses can burn at an essentially constant rate (a zero-order reaction) until all reactants are consumed. These are typically multiphase reactions limited by heat transfer and are discussed in Chapter 11. For single-phase systems, a reaction with n 1 can be expected to slow and become first or second order in the limit of low concentration. 

For n 1, the reaction rate of Equation 2.14 should be supplemented by the condition that 

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Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

Comparison of performance of single CSTR and plug flow reactor for the nth-order reactions... 

When the rate expression for a homogeneous chemical reaction is written in the form of Eq. 4, the dimensions of the rate constant k for the nth-order reaction are... 

For the nth order reaction with respect to one reactant, the general solution for rate expression is ... 

This is due to the fact that under isothermal conditions, the nth-order reaction presents its maximum heat release rate at the beginning of the exposure to initial temperature, whereas the autocatalytic reaction presents no heat release rate at this time. Thus, temperature increase is delayed and only detected later after an induction period, as the reaction rate becomes sufficiently fast. Hence acceleration, due to both product concentration and temperature increase, becomes very sharp. 

Figure 5. Comparison of performance of single mixed and plug flow reactors for the nth-order reactions. (Reprinted with permission from Ref 13, Fig. 6.1, 1972, John Wiley and Sons.)...

Effect of Intraparticle Diffusion on Observed Order and Activation Energy Taking the nth-order reaction case in the limit of intraparticle diffusion control, i.e., large Thiele modulus, the effectiveness factor is... 

The form of the function Fy = FAC) depends on the reaction kinetics moreover, FA0) = 0. The reaction most frequently found in scientific literature is the nth-order reaction, for which [18, 103, 270]... 

For example, the nth-order reaction-rate expression r = c , is convex ifn > 1 and concave ifn < 1. Note that first-order rate functions are both convex and concave,... 

Tavera EM. Analytical expression for the non-isothermal effectiveness factor The nth-order reaction in a slab geometry. Chemical Engineering Science 2005 60 907-916. 

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