Energy balance expressions CSTRs

Eq.(l) and (9) can be utilized for modeling a CSTR considering the material and energy balances as well as the expression for the rate flow of heat removed, Q. This heat rate is obtained from the overall heat transfer coefficient U and the transmission area A by the equation Q = UA(T — Tj) [1], [9], [13], [14], [18], [22],... 

For the corresponding energy balance in a CSTR we write an analogous expression [accumulation of heat] = [heat flow in] — [heat flow out]... 

Now we substitute this expression for X(T) into the energy-balance equation to yield a combined energy- and mass-balance equation for a first-order irreversible reaction in a CSTR,... 

The energy balance on the CSTR can be written as [from Equation (9.5.1) with a first-order reaction rate expression] ... 

In the remainder of the chapter, we discuss how to apply the design equations and the energy balance equations to determine various quantities related to the operations of CSTRs. In Section 8.2 we examine isothermal operations with single reactions to illustrate how the rate expressions are incorporated into the design equation and how rate expressions are determined. In Section 8.3, we expand the analysis to isothermal operations with multiple reactions. In Section... 

When more than one chemical reaction takes place in the reactor, we have to determine how many independent reactions there are (and how many design equations are needed) and select a set of independent reactions. Next, we have to identify all the reactions that actually take place (including dependent reactions) and express their rates. We write Eq. 8.1.1 for each independent chemical reaction. To solve the design equations (obtain relationships between Z s and t), we express the rates of the individual chemical reactions in terms of the Zm Js and t. Since the temperature is constant, the energy balance equation is used to determine the heating load. The procedure for designing isothermal CSTRs with multiple reactions goes as follows ... 

Multiple Steady States and Local Stability in CSTR.—In the two decades since the seminal work of van Heerden and Amimdson, there has been vast output of papers conoemed with the dynamic behaviour of stirred-tank reactors. Bilous and Amundson put the van He den analysis of local stability of the equilibrium state on a rigorous basis by use of linear stability theory. Their method is similar to the phase-plane treatments of thermokinetic ignitions and oscillations discussed here in Sections 4 and 3 (and preceded them dironologically). The mass and energy balance for the CSTR having a single reactant as feedstock may be expressed as ... 

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. 

A first principle mathematical description of a CSTR is based on balance equations expressing the general laws of conservation of mass and energy. Assuming that n components are mixed, the material balance of the i-component, taking into account all forms of supply and discharge in the volume V of the... 

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