Steady-state nonisothermal energy balance

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. 

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 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. 

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... 

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