Analytical solutions, adiabatic reactors

The next example treats isothermal and adiabatic PFRs. Newton s method is used to determine the throughput, and Runge-Kutta integration is used in the Reactor subroutine. (The analytical solution could have been used for the isothermal case as it was for the CSTR.) The optimization technique remains the random one. 

Although semi-analytical solutions are available in some cases [5], these are cumbersome and it is more usual to employ a numerical method. A simple example is presented below which illustrates the solution of the design equation for a batch reactor operated isothermally the adiabatic operation of the same system is then examined. 

The resulting equations are linear in the unknown variables, so an analytical solution is easily obtained. For example, for the case with three adiabatic beds (NR = 3) in the reactor vessel, there are six equations and six unknowns (T ci r(2, 7 c3> Fj, Fos,2 and Fes,3) ... 

J. M. Douglas and L. C. Eagleton, "Analytical Solutions for some Adiabatic Reactor Problems," Ind. Eng. Chem. Fundamentals, 1, 116 (1962). 

Transformation of step and impulse response of tracer concentrations are used to compare mixing models 29). Other analytical procedures can be cited. Douglas and Eagleton 30) have given analytical solutions for the dynamics of adiabatic unpacked reactors. Douglas 31) has developed in detail an analytical procedure for determining the frequency response of a simple nonlinear reactor. 

Table 4.4 Example Analytical Solutions for Adiabatic Plug-Flow Reactors at Constant Pressure ...

Some analytical solutions are even possible for simple-order rate forms—they are given for the analogous situation for plug flow reactors in Chapter 9. Finally, the maximum adiabatic temperature change is found for x = 1.0, and then (for Xao = 0) ... 

In catalytic channels, the flat plate surface temperature in Eq. (3.32) is attained at the channel entry (x O). As the catalytic channel is not amenable to analytical solutions, simulations are provided next for the channel geometry shown in Fig. 3.3. A planar channel is considered in Fig. 3.3, with a length L = 75 mm, height 21) = 1.2 mm, and a wall thickness 5s = 50 pm. A 2D steady model for the gas and solid (described in Section 3.3) is used. The sohd thermal conductivity is k = 6W/m/K referring to FeCr alloy, a common material for catalytic honeycomb reactors in power generation (Carroni et al., 2003). Surface radiation heat transfer was accounted for, with an emissivity = 0.6 for each discretized catalytic surface element, while the inlet and outlet sections were treated as black bodies ( = 1.0). To illustrate differences between the surface temperatures of fuel-lean and fuel-rich hydrogen/air catalytic combustion, computed axial temperature profiles at the gas—wall interface y=h in Fig. 3.3) are shown in Fig. 3.4 for a lean (cp = 0.3) and a rich cp = 6.9) equivalence ratio, p = 1 bar, inlet temperature, and velocity Tj = 300 K and Uin = 10 m/s, respectively. The two selected equivalence ratios have the same adiabatic equilibrium temperature, T d=1189 K. 

Crider, J.E. and Foss, A.S. (1968) An analytic solution for the dynamics of apacked adiabatic chemical reactor. American Institute of Chemical Engineers Journal, 14 (1), 77-84. 

Effective axial transport properties can be determined using an adiabatic reactor. Steady state mass and heat balances result in second-order ordinary differential equations when the axial dispersion is taken into consideration, solutions of which can readily be obtained. Based on these solutions and temperature or concentration measurements, the effective transport properties can be calculated in a manner similar to the procedures used for the radial transport properties. As indicated earlier, a transient experiment can also be used for the determination. Here, experimental and analytical procedures are illustrated for the determination of the effective axial transport property for mass. An unsteady state mass balance for an adiabatic reactor can be written as ... 

Nonanalytical Methods and Approximations for Adiabatic Reactions. A wide variety of techniques and associated numerical methods exist for the solution of non-isothermal reactions and reactor problems which are not tractable to analytical... 

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