Ideal nonisothermal reactors

The kinetics of reactions has been studied for different reaction systems in isothermal reactors. The majority of reactions and processes are not isothermal, since the reactions are endothermic or exothermic. Depending on the extent of exothermicity or endothermicity, the thermal effects on conversion, selectivity, or yield are quite pronounced. 

The reactions in liquid phase with low heat capacity can be performed in a reactor operating isothermally. However, reactions with high heat capacity in liquid or gas phase affect significantly the conversion, selectivity, yield, and/or the reactor features, as volume (especially volume). 

The effect of temperature is observed on the reaction rate, because temperature affects the rate constant through the Arrhenius constant. The temperature influences significantly the rate constant due to the exponential term that contains the activation energy and temperature. The temperature effect is lower on the preexponential factor, which takes into account the collision between molecules, but it may become important in catalyzed reactions. 

Thermodynamics show that the equilibrium conversion increases exponentially with increasing temperature in endothermic reactions but decreases in exothermic reactions. 

If the conversion varies with temperature, we have one more unknown variable, i.e., the temperature, and therefore we need to perform an energy balance. 

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Let us consider a special case of adiabatic reactor without dispersion. The reactor can be considered as an ideal nonisothermal reactor. The heat is generated due to chemical reaction, i.e., the process is exothermic. 

Chapter 1 treated single, elementary reactions in ideal reactors. Chapter 2 broadens the kinetics to include multiple and nonelementary reactions. Attention is restricted to batch reactors, but the method for formulating the kinetics of complex reactions will also be used for the flow reactors of Chapters 3 and 4 and for the nonisothermal reactors of Chapter 5. 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

The design formulation of nonisothermal batch reactors consists of + 1 nonlinear first-order differential equations whose initial values are specified. The solutions of these equations provide Z s and 6 as functions of t. The examples below illustrate the design of nonisothermal ideal batch reactors. 

We begin a discussion of scaleup relationships and strategies for tubular reactors. Results are restricted to tubes with a constant cross-sectional area. Chapter 3 discusses only isothermal or adiabatic reactors, but the relationships in Tables 3.1-3.3 include scaleup factors for the nonisothermal reactors that are discussed in Chapter 5. These results assume constant density, but Tables 3.4 and 3.5 give some specialized results for ideal gases when the pressure drop down the tube is significant. 

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... 

We deliberately separate the treatment of characterization of ideal flow (Chapter 13) and of nonideal flow (Chapter 19) from the treatment of reactors involving such flow. This is because (1) the characterization can be applied to situations other than those involving chemical reactors and (2) it is useful to have the characterization complete in the two locations so that it can be drawn on for whatever reactor application ensues in Chapters 14-18 and 20-24. We also incorporate nonisothermal behavior in the discussion of each reactor type as it is introduced, rather than treat this behavior separately for various reactor types. 

In this chapter we are concerned only with the rate equation for the i hemical step (no physical resistances). Also, it will be supposed that /"the temperature is constant, both during the course of the reaction and in all parts of the reactor volume. These ideal conditions are often met in the stirred-tank reactor (see-Se c." l-6). Data are invariably obtained with this objective, because it is extremely hazardous to try to establish a rate equation from nonisothermal data or data obtained in inadequately mixed systems. Under these restrictions the integration and differential methods can be used with Eqs. l-X and (2-5) or, if the density is constant, with Eq. (2-6). Even with these restrictions, evaluating a rate equation from data may be an involved problem. Reactions may be simple or complex, or reversible or irreversible, or the density may change even at constant temperatur (for example, if there is a change in number of moles in a gaseous reaction). These several types of reactions are analyzed in Secs. 2-7 to 2-11 under the categories of simple and complex systems. 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

The most important feature of a CSTR is its mixing characteristics. The idealized model of reactor performance presumes that the reactor contents are perfectly mixed so that the properties of the reacting fluid are uniform throughout. The composition and temperature of the effluent are thus identical with those of the reactor contents. This feature greatly simplifies the analysis of stirred-tank reactors vis-h-vis tubular reactors for both isothermal and nonisothermal... 

In this section, the AR for the CH4 steam reforming and water-gas shift reaction will be investigated. This system involves two independent reactions involving five components. Ordinarily, generation of the AR for this system would involve the construction of a two-dimensional AR. For this example, the interest will also be in understanding the minimum reactor volume achievable. Reactions of all components occur in the gas phase under nonisothermal conditions. The ideal gas equation of state is hence not a suitable one for this system. Instead, the Peng-Robinson equation of state shall be employed for this purpose. 

Tubular reactor The contact time is the same for all molecules or fluid elements along the reactor when the velocity is uniform in the cross section of the tube, satisfying the plug flow. All molecules have the same velocity. Therefore, the concentration is uniform in a cross section of the tube and varies only along the reactor. In the isothermal case, the temperature remains constant in the longitudinal and radial directions. In the nonisothermal case, the temperature varies along the reactor. This reactor will be denominated ideal PFR (plug flow reactor). 

In developing Eqns. (3-26)-(3-28), we assumed that the temperature was constant in any cross section normal to the direction of flow. We did not assume that the temperature was constant in the direction of flow. For a PFR, the reactor is said to be isothermal if the temperature does not vary with position in the direction of flow, e.g., with axial position in a tubular reactor. On the other hand, for nonisothermal operation, the temperature will vary with axial position. Consequently, the rate constant and perhaps other parameters in the rate equation such as an equilibrium constant will also vary with axial position. The design equations for an ideal PFR are valid for both isothermal and nonisothermal operation. 

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