ANALYTICAL SOLUTIONS FOR MASS AND ENERGY BALANCES

In some simple cases of reaction kinetics, it is possible to solve the balance equations of the ideal, homogeneous reactors analytically. There is, however, a precondition isother-micity if nonisothermal conditions prevail, analytical solutions become impossible or, at least, extremely difficult to handle, since the energy and molar balances are interconnected through the exponential temperature dependencies of the rate and equilibrium constants (Sections 2.2 and 2.3). Analytical solutions are introduced in-depth in the literature dealing 

TABLE 3.2 Analytical Solution for a First-Order Reaction in Various Ideal Reactors 

In Table 3.3, the concentration expressions of the key components are given for some common types of reaction kinetics. The concentration expressions in Table 3.3 are valid for isothermal conditions, for systems in which the volumetric flow rate and the reaction volume are kept constant. In practice, this implies isothermal liquid-phase reactions in a BR, PFR, and CSTR, isothermal gas-phase reactions in a BR with a constant volume, and isothermal gas-phase reactions with a constant molar amount (J2 ij = O) in a PFR or CSTR. The equations in Table 3.3 were developed from balance equations introduced in Sections 3.2 through 3.4 as well as from the stoichiometric relationships that were presented in Sections 3.5 and 3.6. The equations are written in such a way that the concentrations of key components are expressed explicitly for simple reactions, the corresponding design expressions are obtained by solving the space time (t) or the reaction time (t) in the equations. 

For isothermal gas-phase reactions, in CSTRs and PFRs, analytical design equations can sometimes be derived. In this case, we should begin with expressions such as Equations 3.136 and 3.138 or Equations 3.185 and 3.148. Further, we should utilize a correction term for the volumetric flow rates, such as Equation 3.97. Usually, these kinds of expressions become so complicated that a numerical solution of the balance equations is preferable. In Section 3.8, the analytical solutions of some simple systems are considered in detail. 

Note X = space time in a PFR and CSTR and x = t = reaction time in a BR. 

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