Adiabatic reactor trajectories

These data are plotted in Figure 5-1 1. Thus we have a very serious problem if this reaction is reversible because the adiabatic reactor trajectory intersects the equilibrium curve at a low conversion. For these kinetics, equilibrium limits the process to a very low conversion at high temperatures. 

In Figure 5-1 3 are plotted the possible trajectories for the exothermic reaction. These are the limiting cases of the trajectory in a wall-cooled reactor, and any waU-cooled reactor win have a trajectory between these two straight lines. The trajectory cannot go above the equihbrium curve X,(T). For an adiabatic reactor the curve stops there, and for finite UA, the curve finishes at Xe at Tq. 

The reactor configurations and possible trajectories for three adiabatic reactors with interstage cooling are shown in Figure 5-20. 

Fig. 3.8. Reactor trajectories in adiabatic and cooled catalyst beds...

If the reaction is split into many small pieces, with cooling between each piece, then each reactor would traverse along a small segment on the boundary ABCDE in Eigure 7.26. As these sections are made smaller, we approach the structure of a differentially cooled reactor (DCR), and thus the candidate region boundary associated with the DCR is therefore given by an envelope of adiabatic PFR trajectories with different feed temperatures. 

Assume that two adiabatic reactors are available with the same kinetics given in Section 7.3.3.1. Using an inlet temperature of 300 K, plot the PFR trajectory for the first reactor as well as the possible PFR trajectories for the second reactor if cold-shot cooling is employed. Use mixing fractions of A = 0.25, 0.5, and 0.75. Assume that the feed vector is given by Cj = [c j, t°] = [1,0], and that the exit concentration from the first reactor is Ca = 0.2 mol/L. 

It will be recalled that we already defined J earlier in Chapter 1. We note from Equation 8.17 that a trajectory of slope Cp /-AH on an plot, as shown by line A in Figure 8.6a, represents the path of an adiabatic reaction. The and curves along with the adiabatic line A and the rate contours shown in the X -T plane are central to the design of an adiabatic reactor. Note that this relationship between conversion and temperature is unique to adiabatic reactors and cannot be used for any nonadiabatic situation (i.e., where heat is supplied or ranoved). 

To demonstrate the main features of the flow in horizontal CVD reactors, the deposition of silicon from silane is used as an example (87). The conditions are as follows an 8-cm-wide reactor with either adiabatic side walls or side walls cooled to the top wall temperature of 300 K, a 1323 K hot susceptor (bottom wall), a total pressure of 101 kPa, and an initial partial pressure of silane in H2 of 101 Pa. The growth rate of silicon is strongly influenced by mass transfer under these conditions. Figure 8 shows fluid-particle trajectories and spatially varied growth rates for three characteristic cases. 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

The dynamic stability of the reactor can be studied by using the temperature-conversion trajectory, as represented in Figure 6.9. During the adiabatic period, the trajectory is linear with a slope equal to the adiabatic temperature rise. If no cooling is applied, the maximum temperature Tlnax would be reached for a conversion %A,max ... 

Since this initial work, analysis of these batch systems has been further expanded to include reactant consumption, beginning with the work of Rice, Allen, and Campbell. Furthermore, an excellent study of stability with a generalized nth order reaction rate and the effect of the heat capacity of the reactor walls (when Le 1) was presented by Balakotaiah, Kodra, and Nguyen. They verified previous work which showed the boundary to runaway behavior occurs when two inflection points appear in the reaction trajectory between the initial and final states. In the limit of 7 oo and 6c = 0, the safe criteria under adiabatic conditions (a = 0) is given as B < Le + /n) and for highly exothermic reactions (B 1) with large cooling (a > 1) the safe criteria approaches Semenov s classical result x/B > e. 

Batch reactors operated adiabatically are often used to determine the reaction orders, activation energies, and specific reaction rates of exothermic reactions by monitoring the temperature-time trajectories for different initial conditions. In the steps that follow, we will derive the temperature-cons crsion relationship for adiabatic operation. 

Trajectories of adiabatic and wall-cooled reactors Consider the reaction... 

If a constant wall temperature is maintained by cooling, then the reactor is no longer adiabatic and the trajectories will lie between the limits shown in Figure 12.4b where Ty, represents the wall temperature. The behavior of an endothermic reaction (where heat has to be supplied) is sketched in Figure 12.4c. 

AR Construction Since the reactor temperature has been expressed explicitly in terms of components x and y, construction of the AR for the adiabatic system from this point onward follows the same approach as that for an isothermal system. The system is two-dimensional as there are two independent reactions participating in the system, and therefore we need only consider combinations of PFRs and CSTRs in the construction of the candidate region. It is customary to begin by generating the PER trajectory and CSTR locus from the feed. Due to the nonlinear nature of the kinetics introduced by the adiabatic constraint, the system exhibits multiple CSTR solutions from the feed point. We... 

Point X is also an intersection point with the PFR profile with an inlet temperature of Tjjj = 320K. Since the shape of the adiabatic profiles is fixed for an inlet temperature, we may traverse along the second PFR trajectory XCY if the inlet temperature to the second reactor is equal to T2. In this way, we transition from the adiabatic curve corresponding to 350 K to the curve corresponding to 320 K and achieve a slightly higher conversion in the process. Since the exit temperature from the first reactor is Tj = 499.84 K, one must cool this stream down to a temperature that corresponds to the temperature at X for the second reactor with an inlet temperature of T2 = 469.84 K. 

Very frequently non-optimal setpoint trajectories are used for controlling reactor temperatures in batch reactors [25,39,179,180]. Reactor temperatures maybe allowed to increase from ambient temperatures up to a maximum temperature value, in order to use the heat released by reaction to heat the reaction medium and save energy (reduce energy costs). The temperature increase is almost always performed linearly, because of hardware limitations and simplicity of controller programming. After reaching the maximum allowed temperature value, reactor temperature is kept constant for a certain time interval, for production of polymer material at isothermal conditions. At the end of the batch, the reaction temperature is increased in order to reduce the residual monomer content of the final resin, usually with the help of a second catalyst. Heuristic optimum temperature trajectories were also formulated for batch polymerizations of acrylamide and quaternary ammonium cationic monomers, in order to use the available heat of reaction [181]. The batch time was split into two batch periods an isothermal reaction period and an adiabatic reaction period. 

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