Steady-state nonisothermal balances

Consider the nonisothermal porous spherical catalyst particle of radius R in which a single, irreversible, first-order reaction takes place at steady state (Figure 2.11). Taking the same spherical shell of thickness Ar at a radius r from the center, the steady-state energy balance over a differential shell of volume 4nt Ar includes conduction into and out of the control volume in the radial direction as well as heat release by reaction within the control volume ... 

Nonisothermal Reactor Design-The Steady State Energy Balance and Adiabatic PER Applications... 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

Assuming a steady state, for first-order reaction-diffusion system A -> B under nonisothermal catalyst pellet conditions, the mass and energy balances are... 

Up to now we have focused on the steady-state operation of nonisothermal reactors. In this section the unsteady-state energy balance wtU be developed and then applied to CSTRs, plug-flow reactors, and well-mixed batch and semibateh reactors. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation 5.27. The same basic procedure can be used for Equation 5.24. The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from //out after each time step. The examples that follow assume constant physical properties and use Equation 5.27. Their purpose is to explore nonisothermal reaction phenomena rather than to present detailed design calculations. 

Nonisothermal stirred tanks are governed by an enthalpy balance that contains the heat of reaction as a significant term. If the heat of reaction is unimportant so that a desired Tout can be imposed on the system regardless of the extent of reaction, then the reactor dynamics can be analyzed by the methods of the previous section. This section focuses on situations where Equation 14.3 must be considered as part of the design. Even for these situations, it is usually possible to control a steady-state CSTR at a desired temperature. If temperature control can be achieved rapidly, then isothermal design techniques again become applicable. Rapid means on a time scale that is fast compared to reaction times and composition changes. 

Nonisothermal reactor design requires the simultaneous solution of the appropriate energy balance and the species material balances. For the batch, semi-batch, and steady-state plug-flow reactors, these balances are sets of initial-value ODEs that must be solved numerically, in very limited situations (constant thermodynamic properties, single... 

This example demonstrates how multiple CSTR steady states may arise in nonisothermal systems, even when the associated kinetics are simple, and temperature is assumed to be linear in terms of concentration. The inherent nonlinear nature of rate expressions in general thus often leads to complex behavior even when the energy balance is of a simple form. Multiple steady states must be included in the AR in order to understand the true bounds of achievability. Omission of these states may have important implications on subsequent optimizations, such as if we wish to maximize the concentration of component B. 

It is impossible to create an isothermal process in plug flow reactors as it requires the variation of thermal transfei along the reactor length, according to the kinetics of heat emission. Therefore, plug flow reactors run under adiabatic conditions or at least imder nonisothermal mode conditions with external heat removal. The heat balance equation for steady state conditions for the micro volume of a reactor can be written in the form [4] ... 

This may be seen better by referring to Figure 17.4, where monomer conversion has been plotted versus reactor residence time. (A similar plot will result from the heat balance multiplicity in a nonisothermal CSTR.) It may be seen that over a range of residence times, three values of monomer conversion are possible. As before, the upper and lower steady states are usually stable, while the middle steady state is not. 

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