Energy Balance in a CSTR

We first derive the energy balance in a CSTR. For the mass balance in a constant-density reactor we wrote an integral balance on the rate of change of the number of moles Nj of species j in the reactor to obtain... 

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

The steady-state mass and energy balances in a CSTR are given by Equations (2.20) and (2.21), respectively, which are reproduced here for ready reference ... 

For the first-order reaction, the steady-state equations for mass and energy balance in a CSTR can be combined into a single equation represented as... 

The mathematical problem we encounter here is very similar to the temperature runaway problem we have seen when solving the energy balance across a CSTR in Chapter 1. 

Example (h) In terms of fractional conversion,/ = 1 — C/Cj, the material and energy balances for a first-order CSTR are ... 

Figure 14.8 Illustration of solution of material and energy balances for an endothermic reaction in a CSTR (no multiple stationary-states possible)...

The energy balance for a PFR, as an enthalpy balance, may be developed in a manner similar to that for a CSTR in Section 14.3.1.2, except that the control volume is a differential volume. This is illustrated in Figure 15.3, together with the symbols used. 

Figure 5-1 Energy balance in chemical reactors. The volume shown could be the total reactor volume in a CSTR or a differential volume in a PFTR.

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,... 

Recall from Chapter 6 that the steady-state energy-balance equations in a CSTR can be reduced to a single equation, which we wrote by considering the rates of heat generation and removal. We wrote these as... 

The term Wj, often referred to as the sha/t work, could be produced from such things as a stirrer in a CSTR or a turbine in a PFR. In most instances, the flow work term is combined with those terms in the energy balance that represent the energy exchange by mass flow across the system boundaries. Substituting Equation (8-4) into (8-3) and grouping terms, we have... 

In Example 9-4 we saw how a 500-gal CSTR used for the production of propylene glycol approached steady-state. For the flow rates and conditions (e.g., Tq = 75°F, = 60° ), the steady-state temperature was 138°Fand the corresponding conversion was 75.5%. Determine the steady-state temperature and conversion that would result if the entering temperature were to drop from 75°F to 70°F, assuming that all other conditions remain the same. First, sketch the steady state conversions calculated from the mole and energy balances as a function of temperature before and after the drop in entering temperature occurred. Next, plot the "conversion,"concentration of A, and the temperature in the reactor as a function of time after the entering temperature drops from 75°F to 70°F. 

Equation (8.15) gave the mass balance for a CSTR. A similar equation can be written as an energy balance. This example considers a CSTR in which a first-order reaction occurs. 

Consideration of the coupled mass and energy balances for the CSTR have led to possible behaviors that may seem surprisingly complex for even the simplest kinetic mechanism, an irreversible first-order reaction. Just because these behaviors are possible does not mean that they are normally observed in reactor operation for something as simple as A goes to B. 

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 ... 

The dynamics of temperature and conversion within a cooled continuous-flow stirred tank reactor (CSTR) can be obtained from the material and energy balances. For a simple first order chemical reaction they are in a dimensionless form... 

If a single reaction takes place in a CSTR, the energy balance on the whole reactor is given by... 

Material and energy balances of common types of reactors are summarized in several tables of Sec. 7. For review purposes some material balances are restated here. For the /ith stage of a CSTR batteiy,... 

Use Scalable Heat Transfer. The feed flow rate scales as S and a cold feed stream removes heat from the reaction in direct proportion to the flow rate. If the energy needed to heat the feed from to Tout can absorb the reaction exotherm, the heat balance for the reactor can be scaled indefinitely. Cooling costs may be an issue, but there are large-volume industrial processes that have Tin —40°C and Tout 200°C. Obviously, cold feed to a PFR will not work since the reaction will not start at low temperatures. Injection of cold reactants at intermediate points along the reactor is a possibility. In the limiting case of many injections, this will degrade reactor performance toward that of a CSTR. See Section 3.3 on transpired-wall reactors. 

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