Parametric sensitivities,

We illustrate the use of Eq. (4.26) by calculating the maximum achievable temperature in the HDA and vinyl acetate reactors discussed in this book. Both of these are gas-phase, plug-flow systems. 

Starting with the HDA reactor, we find most of the needed information in Chap. 10. The feed temperature is llSOT, the heat of reaction -21,500 Btu/lb mol, and the mole fraction of toluene (limiting component) in the reactor feed is 0.0856. The molar heat capacity of the feed is computed from its composition and standard literature data  

The adiabatic temperature rise for this system is roughly A7 i = 115°F = 64 C. 

The vinyl acetate process is described in Chap. 11. The reactor inlet temperature is 148.5 C and oxygen is the limiting reactant (yA0 = 0.075). The heat of reaction is -42.1 kcal/mol vinyl acetate or -84.2 kcal/mol oxygen. The average heat capacity of the feed is computed from data provided in Chap. 11  

When we consider the main reaction and exclude the side reaction, the estimated maximum temperature is 

The differential equations governing a non-isothermal batch reaction describe the material balance coupled to the heat balance (see also Section 2.4.2)  

By rearranging and expressing the kinetic constant as a function of temperature, one obtains  

Thus the temperature course in a reactor depends on the following terms  

Whereas the cooling capacity depends linearly on temperature, the heat production rate depends exponentially following the Arrhenius law. This may result in extremely high temperature maxima, if the control is not appropriate. Thus, it is important to characterize the effect of temperature on the heat balance. 

This problem was studied by many authors [1-9]. A comprehensive review has been presented by [10] and [11]. 

For given reaction kinetics the thermal behavior of tubular reactors depends on three different parameters  

The characteristic reaction time is calculated with the temperature of the cooling medium T ) respectively the inlet temperature (Tg). In the following discussion the inlet and the cooling temperature are supposed to be identical (Tg = T ). 

To facilitate the discussion on the influence of the above-defined parameters (Equations 5.37-5.39) on the reactor behavior and the parametric sensitivity, Equations 5.34 and 5.36 are given in a dimensionless form. According to the studies of Barkelew [25] the mean residence time is referred to the characteristic reaction time and the temperature is given in the form of a relative temperature difference normalized with the Arrhenius number (Equation 5.41). 

The dimensionless residence time, can be interpreted as a first Damkohler number, defined with the characteristic reaction time calculated with the temperature of the cooling medium. 

With these definitions the conversion and the temperature as function of the dimensionless channel length can be calculated. 

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Whereas there is extensive Hterature on design methods for azeotropic and extractive distillation, much less has been pubUshed on operabiUty and control. It is, however, widely recognized that azeotropic distillation columns are difficult to operate and control because these columns exhibit complex dynamic behavior and parametric sensitivity (2—11). In contrast, extractive distillations do not exhibit such complex behavior and even highly optimized columns are no more difficult to control than ordinary distillation columns producing high purity products (12). 

The variations were mainly due to operating conditions very close to the parametrically sensitive region, i.e., to the incipient temperature runaway. Small errors in the estimation of temperature effects caused runaways and, consequently, large differences. 

Cost Whether measurement or a computation technique is cheaper depends on the situation in question. In small and simple problems, it is usually more profitable to use measurement techniques. In large and complex problems, where parametric/sensitivity study is the objective, computation may be a better alternative. 

A more dramatic comparison of the piston flow and axial dispersion models is shown in Figure 9.12. Input parameters are the same as for Figure 9.11 except that Tin and T aii were increased by 1K. This is another example of parametric sensitivity. Compare Example 9.2. 

Flow reversal performance is controlled weakly by the period r. Flow reversal is an autothermal operation and as such exhibits parametric sensitivity. Greater stability can be ensured by the conventional expedient of providing cooling in the catalyst bed. It can also be done through bypassing part of the reactor effluent gas around the recuperator section. 

The last question pertains to the problem of parametric sensitivity. There is extensive literature dealing with this topic, but it is beyond the scope of this book. 

Parametric sensitivity analysis showed that for nonreactive systems, the adsorption equilibrium assumption can be safely invoked for transient CO adsorption and desorption, and that intrapellet diffusion resistances have a strong influence on the time scale of the transients (they tend to slow down the responses). The latter observation has important implications in the analysis of transient adsorption and desorption over supported catalysts that is, the results of transient chemisorption studies should be viewed with caution, if the effects of intrapellet diffusion resistances are not properly accounted for. 

Parametric Sensitivity in Chemical Systems, Arvind Varma, Massimo Morbidelli and Hua Wu... 

Papayannokos, N. G., C. A. Koufopanos, and A. Karetsou (1993). "Studies on Parametric Sensitivity and Safe Operation Criteria of Batch Processes." Chem. Eng. Technol. 16,318-24. 

Table VIII. ZSM-5 Model - Parametric Sensitivity Effect of Base Octane...

N. G. Karanth and J. E. Bailey, Diffusional influences on the parametric sensitivity of immobilized enzyme catalysts, Biotechnol. Bioeng. 1978, 20, 1817-1831. 

The quality of the available kinetic equation to fit experimental data in a wide range of isothermal and scanning rate conditions (the numerical solution exhibits a very high parametric sensitivity on the values of the activation energies). 

The inquiry showed that the process was operated in the parametric sensitive range. As the batch size was increased to 1100 kg, the maximum temperature during the holding phase increased to 170 °C. Moreover, the thermometer had a range of 200 °C from -30 °C to +170 °C, because the reactor was multi-purpose equipment also equipped with a brine cooling system. Thus, the technical equipment was not adapted to the process conditions. 

A more comprehensive approach consists of studying the variation of the Semenov criterion as a function of the reaction energy. Such an approach is presented in [12], where the reciprocal Semenov criterion is studied as a function of the dimensionless adiabatic temperature rise. This leads to a stability diagram similar to those presented in Figure 5.2 [11, 13]. The lines separating the area of parametric sensitivity, where runaway may occur, from the area of stability is not a sharp border line it depends on the models used by the different authors. For safe behavior, the ratio of cooling rate over heat release rate must be higher than the potential of the reaction, evaluated as the dimensionless adiabatic temperature rise. 

The Villermaux criterion and the Da/Si criterion are dynamic stability criteria, meaning that with a cooling medium temperature above the limit level, 20 resp. 30 °C, the reactor will be operated in the instable region and present the phenomenon of parametric sensitivity. If instead of B12, B is used, both criteria lead to the same result. This should not be surprising since they derive from the same heat balance considerations, that is, the heat release rate of the reaction increases faster with temperature than the heat removal does. 

Runaway behaviour and parametric sensitivity of a batch reactor an experimental study. 

This criterion expresses the fact that the reaction rate must be high enough to avoid the accumulation of reactant, even if the reaction is performed at the temperature of the coolant, and the cooling capacity must be sufficient to control the temperature. If this criterion is fulfilled, the reactor will not be parametric sensitive. The set temperature of the reactor can be chosen using the following criterion ... 

Chemburkar et al. [3]. An extensive discussion of the parametric sensitivity of the CSTR is presented by Varma [4]. In the example in Figure 8.4, A is a working point on the cold branch, B is an instable point, and C is a working point on the hot branch. The consequences of this multiplicity are explained in more detail in Section 8.2.6.1. 

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