CSTR heat balance

In particular cases, considering extremely exothermic or endothermic processes, a global CSTR heat balance may be employed to determine a uniform operating temperature and the necessary heating/cooling capacity. 

FIGURE 5.6 Heat balance in a CSTR (a) heat generated by reaction (b) heat removed by flow and transfer to the environment (c) superposition of generation and removal curves. The intersection points are steady states, (d) Superposition of alternative heat removal curves that give only one steady state. 

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. 

There is one significant difference between batch and continuous-flow stirred tanks. The heat balance for a CSTR depends on the inlet temperature, and Tin can be adjusted to achieve a desired steady state. As discussed in Section 5.3.1, this can eliminate scaleup problems. 

In cases where the isothermal CSTR is cooled by the jacket, the heat balance comprises three terms ... 

Figu re 8.2 Heat balance of a cooled CSTR, with the cooling term (straight line) and the S-shaped heat release rate curve. The working point is at the intercept. 

The heat balance can be written either globally over the whole reactor volume or locally for a differential element dV. The global heat balance is similar to the CSTR ... 

They treated the system much like a CSTR, with the balance for the gas-phase concentration substituted by the coverage equation for the catalyst. Ray and Hastings then applied the analytical treatment that they had developed for the CSTR in this same publication. Stability analysis revealed that the critical Lewis numbers for oscillations were in a range that did not allow for oscillations on normal nonporous catalytic surfaces. However, as Jensen and Ray 243) showed, a certain model for catalytic surfaces, the fuzzy wire model, with the assumption of a very rough surface with protrusions is able to produce Lewis numbers in the proper range for the occurrence of oscillations. This model, however, included both mass and heat balances as well as coverage equations, thus combining the two classes of reactor-reaction models discussed above. 

To obtain a plot of heat generated, G(T), as a function of temperature, we must solve for X as a function of T using the CSTR mole balance, the rate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes... 

The evolution of the local temperature in a reacting system is governed by the heat balance equation. This may become quite complex in unstirred systems, especially if convective heat transfer processes develop as a consequence of local heating. For the simple CSTR described earlier we can proceed with an ordinary differential equation of the form... 

The reactor models considering complete mixing may be subdivided into batch and continuous types. In the continuous stirred tank reactor (CSTR) models, an entering fluid is assumed to be instantaneously mixed with the existing contents of the reactor so that it loses its identity. This type of reactor operates at uniform concentration and temperature levels. For this reason the species mass balances and the temperature equation may be written for the entire reactor volume, not only over a differential volume element. Under steady-state conditions, the species mass and heat balances reduce to algebraic equations. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

They do require knowledge of Qin, Gout, and V, and determination of these quantities is not immediate when the density of the reacting mixture varies with composition. The inlet temperature and the inlet and outlet pressures in a CSTR are usually determined or controlled independently of the extent of reaction. The outlet temperature can be set arbitrarily in small laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors, for example, if the heat of reaction is small or if the contents are boiling. Chapter 5 discusses the case where the outlet temperature is found from a heat balance. The design equations do not depend on Pi and Tjn, and we assume Gin is known. This chapter considers the case where both Pout and Tout are known. The outlet density Pout will vary with composition and is determined from an equation of state. The volumetric flow rate at the outlet. Gout, is found from a steady-state material balance ... 

Piston flow reactors lack any internal mechanisms for memory. There is no axial dispersion of heat or mass. What has happened previously has no effect on what is happening now. Given a set of inlet conditions (flin, 7i , Text), only one output (flout, 7 out)is possible. A PFR cannot exhibit steady-state multiplicity unless there is some form of external feedback. External recycle of mass or heat can provide this feedback and may destabilize the system. Figure 14.7 shows an example of external feedback of heat that can lead to the same multiple steady states possible with a CSTR. Another example is when the vessel walls or packing has significant thermal capacity. In such cases, a second heat balance must be added to supplement Equation 14.16. See Section 10.6 for a comparable result. 

A dynamic model of a CSTR can be derived based on unsteady mass and energy balance (see Chapters 4 and 8). The model contains a few nonlinear differential equations, being amenable to analytic or numerical investigation. When n -order reaction is considered, the mass and heat balance equations can be written in the following dimensionless form ... 

Figure 5.2 Steady-state heat balance for a CSTR.

Figure 5.3 Heat balance for a CSTR that may exhibit a limit cycle.

The following deduction of the heat balance assumes that the characteristic of a constant reactor volume is valid for the steady state as well as for the dynamic operating conditions. Such a boundary condition remains valid in practice also if the reactor is filled with an amount of either pure solvent or the product mixture of a foregoing batch, equal to the reactor volume under stationary operating conditions before the start-up of the CSTR is initialized. 

The CSTR is characterized by a continuous input and output mass flow. Consequently, the heat balance has at least to account for the convective heat transport as well as for the heat transfer through the reactor wall. 

This temperature To is called reference temperature of the CSTR. Its introduction serves to simplify the final heat balance equation as well as the discussion of different operating conditions. To illustrate the latter statement the reactor behaviour for a certain process performed in a reactor with well-known properties will always be identical if initial and reference temperature as well as mean residence time are kept constant, independent of the combination of feed and coolant temperature which results in this reference temperature. The terms for convective heat transport and for the heat exchange of the reactor wall can now be combined into one heat removal term using this reference temperature. 

With its use the unsteady heat balance of the cooled CSTR using dimensionless numbers is obtained ... 

In analogy to the CSTR, the insertion of both heat flux terms into the general heat balance results in the mixing temperature ... 

The easiest way to obtain such plots is to use the so-called coupling equation and the stationary mass balance of the CSTR given in Equ. (4-29). The coupling equation is derived by first transforming the imsteady heat balance of the cooled CSTR into the stationary balance ... 

The necessary procedure can be derived from the classical stability theory for oscillating systems. The unsteady mass and heat balances for the cooled CSTR may be expressed in a very general form... 

The heat balance at the steady state of the CSTR is expressed as ... 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

Figure 5.1 Heat balance multiplicity during exothermic reaction in a CSTR (from [6]).

An equation relating temperature T and conversion Xa is required to design the non-isothermal reactors. This relationship between temperature T and conversion is obtained by setting up a heat balance equation around the reactor (Section 3.1.5.3). In certain cases, reactor temperature T is deliberately varied with conversion by regulating the heat supply to the reactor or heat removal from the reactor. One such case is the non-isothermal reactor in which a reversible exothermic reaction is carried out. In the case of a reversible exothermic reaction, there is an optimum temperature T for every value of conversion x at which the rate is maximum. A specified conversion Xaj will be achieved in a CSTR or a PFR with the smallest volume or in a batch reactor in the shortest reaction time if the temperature in the reaction vessel is maintained at the optimum level. This optimal temperature policy in which temperature is varied as a function of conversion x,i is known as the optimal progression of temperature presented in the following section. 

The general steady-state heat balance of a CSTR with an exothermic reaction reads ... 

The instability follows from eqs. (8.1) and (8.2). A small variation in the conditions, e.g., the feed concentration may cause the temperature to rise slightly, so that the rate constant goes up also. This results in a higher reaction rate, causing the temperature to rise more, and so forth. However, the temperature rise will be check by the decrease of the reactant concentration. This is demonstrated by the the non-steady state heat balance for a first order reaction in a CSTR ... 

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