Adiabatic energy balance

In the previous section, we derived Equation (11-28), which relates conversion to temperature and the heat added to the reactor, Q. Let s stop a minute and consider a system with the special set of conditions of no work, IPj = 0, adiabatic ( eration = 0, let 7 = Tq and then rearrange (11-28) into the form 

Equation (11-29) applies U a CSTR, PFR, or PBR, and also lo a batch (as will be shown in Chapter 13). For 0 = 0 and IP, = 0 Equation (11-29) gives us the explicit relationship between X and T needed to be used in conjunction with (he mole balance to solve reaction engineering problems as discussed in Section 11.1. 

The final mixture temperature T may be expressed as a function of the properties of the individual beakers by performing an adiabatic energy balance for the process. For simplicity, assume that the specific heat capacity (Cp) of the components in beakers 1 and 2 and the mixture may be described by a single bulk Cp term. That is, the Cp of the bulk mixture for beakers 1 and 2 and the total are given by Cpi, Cp2, and Cp, respectively. Assume also that the Cps are not functions of temperature. A simphfied adiabatic energy balance may be written over the system, giving the following result  

The term AH, ij represents the enthalpy of mixing for the mixture at the mixture temperature. Tq is a basis temperature, from which the enthalpy of each beaker is measured (typically 25 °C). A similar substitution may be performed, as described in Chapter 2, to express the mixing process in terms of the ratio of masses in beakers 1,2, and 3. If the substitution A = mj/m is made, then the resulting expression is obtained  

The goal here is to express T as a linear combination of Ti and T2 in the following form  

That the bulk heat capacities of the individual beakers are identical to the mixture, that is, Cpi = Cp2 = Cp  

From these assumptions, it is evident that expressing T as a linear combination of Tj and T2 only occurs under extraordinary constraints. It is therefore unlikely that temperature may be mixed in the same manner as concentration or residence time for the purpose of AR constructions. The construction of candidate regions using temperature are generally not considered with temperature on a dedicated axis, such as with concentration and residence time. The role 

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Finally, if from an adiabatic energy balance we assume that the power goes into heat generation, we can estimate the temperature rise within the material to be... 

We now consider an adiabatic reactor of fixed sire or catalyst weight and investigate what happens as the feed temperature is varied. ITie reaction is reversible and exothermic. At one temperature extreme, using a very high feed temperature, the specific reaction rate will be large and the reaction will proceed rapidly, but the equilibrium conversion will be close to zero. Consequently, very little product will be formed. A plot of the equilibrium conversion and the conversion calculated from the adiabatic energy balance,... 

The adiabatic energy balance (12.4.13) can be expressed in the form of a lever rule in enthalpies. 

An adiabatic constraint An AR will be constructed when an adiabatic energy balance is introduced. The implications of how this constraint impacts AR construction will provide for an interesting discussion. Temperature will be an important consideration in this instance, and hence it is important to understand how temperature may be accommodated in AR constructions. 

We will assume that a feed containing pure A is available, so that Cf = [x, y]T = [1,0]. Isothermal operation is no longer assumed. Rather, we will use an expression to describe an adiabatic energy balance around the reactor network. 

The Energy Balance Since the system is assumed to operate adiabatically, an adiabatic energy balance may be written over the reactor system, and an expression for temperature in terms of species concentrations x and y may then be obtained. This expression may be substituted into the temperature dependent rate constants, found in Equation 7.12. Due to the nonlinear nature of both the kinetics and energy balance terms, the final kinetic expression is typically a nonlinear function. For the purposes of demonstration, we shall assume that the resulting energy balance expression, after... 

Since the system is now isothermal, the effect of the adiabatic energy balance developed previously shall be... 

Substituting the above equation into the adiabatic energy balance gives us... 

The operating point of an adiabatic CSTR at steady state must lie somewhere on the line that represents the adiabatic energy balance. For an adiabatic PFR at steady state, or for an adiabatic batch reactor, the energy balance line describes the path of the reaction, including the exit condition for a PFR and the final condition for a batch reactor. For any type of adiabatic reactor, if a given point (x, T) does not lie on the line, the energy balance is not satisfied. 

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