Optimal Temperatures for Isothermal Reactors

Reaction rates usually increase with temperature. Thus the best temperature for a single, irreversible reaction, whether elementary or complex, is the highest possible temperature. Practical reactor designs must consider materials of construction 

Multiple reactions, and reversible reactions since these are a special form of a multiple reaction, usually exhibit an optimal temperature with respect to the yield of a desired product. The reaction energetics are not trivial even if the reactor is approximately isothermal. One must specify the isotherm at which to operate. Consider the elementary, reversible reaction 

The outlet concentration from the stirred tank, assuming constant physical properties and bin = 0, is given by 

We assume the forward and reverse reactions have Arrhenins temperature dependences with AEf AEr. Setting dboui/cU = 0 gives 

The reader who duplicates the algebra needed for this analytical solution will soon appreciate that a CSTR is the most complicated reactor and Equation 5.3 is the most complicated reaction for which an analytical solution for Toptimai is likely. The same reaction occurring in a PER with bin = 0 leads to 

Reaction rates almost always increase with temperature. Thus, the best temperature for a single, irreversible reaction, whether elementary or complex, is the highest possible temperature. Practical reactor designs must consider limitations of materials of construction and economic tradeoffs between heating costs and yield, but there is no optimal temperature from a strictly kinetic viewpoint. Of course, at sufficiently high temperatures, a competitive reaction or reversibility will emerge. 

Suppose this reaction is occurring in a CSTR of fixed volume and throughput. It is desired to find the reaction temperature that maximizes the yield of product B. Suppose Ef Ef, as is normally the case when the forward reaction is endothermic. Then the forward reaction is favored by increasing temperature. The equilibrium shifts in the desirable direction, and the reaction rate increases. The best temperature is the highest possible temperature and there is no interior optimum. 

For Ef Ef, increasing the temperature shifts the equilibrium in the wrong direction, but the forward reaction rate still increases with increasing temperature. There is an optimum temperature for this case. A very low reaction temperature gives a low yield of B because the forward rate is low. A very high reaction temperature also gives a low yield of B because the equilibrium is shifted toward the left. 

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If the integral in Eq. (9.6.2) can be worked out explicitly, then it may be possible to obtain an equation for the optimal temperature. For example, consider the first order reversible reaction A B, with rate law r = ka — taking place in an isothermal reactor whose feed is pure A. If the required fractional conversion is Y, then the feed concentration can be written the current concentrations are a = Gq — and 6 = f, and... 

Example 6.7 Determine optimal reactor volumes and operating temperatures for the three ideal reactors a single CSTR, an isothermal PER, and an adiabatic PER. 

In this paper we formulated and solved the time optimal problem for a batch reactor in its final stage for isothermal and nonisothermal policies. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and optimum time was studied. It was shown that the optimum isothermal policy was influenced by two factors the equilibrium monomer concentration, and the dead end polymerization caused by the depletion of the initiator. When values determine optimum temperature, a faster initiator or higher initiator concentration should be used to reduce reaction time. 

What are the optimal isothermal temperature for operation and the corresponding reactor volume for a final fractional conversion Zf of 0.68. Is this the best way of operating the reactor ... 

Simulation results with different time interval (P) are reported in Table 4. Optimal control policy in reactor temperature for each case is shown in Fig. 4. As shown in Table 4, when one time interval (P = 1) is used, the amount of product C obtained at the final time (tf = 200 min) is 7.0171 kmol and the optimal temperature (isothermal operation) setpoint is 88.01 °C whereas usingP = 20, the amount of product C achieved is 7.0379 kmol. It was found from... 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

We shall recapitulate the governing equations in the next section and discuss the economic operation in the one following. The results on optimal control are essentially a reinterpretation of the optimal design for the tubular reactor. We shall not attempt a full derivation but hope that the qualitative description will be sufficiently convincing. The isothermal operation of a batch reactor is completely covered by the discussion in Chap. 5 of the integration of the rate equations at constant temperature. The simplest form of nonisothermal operation occurs when the reactor is insulated and the reaction follows an adiabatic path the behavior of the reactor is then entirely similar to that discussed in Chap. 8. 

The optimal temperature policy in a batch reactor, for a first order irreversible reaction was formulated by Szepe and Levenspiel (1968). The optimal situation was found to be either operating at the maximum allowable temperature, or with a rising temperature policy, Chou el al. (1967) have discussed the problem of simple optimal control policies of isothermal tubular reactors with catalyst decay. They found that the optimal policy is to maintain a constant conversion assuming that the decay is dependent on temperature. Ogunye and Ray (1968) found that, for both reversible and irreversible reactions, the simple optimal policies for the maximization of a total yield of a reactor over a period of catalyst decay were not always optimal. The optimal policy can be mixed containing both constrained and unconstrained parts as well as being purely constrained. 

Other recent work in the field of optimization of catalytic reactors experiencing catalyst decay includes the work of Romero e/ n/. (1981 a) who carried out an analysis of the temperature-time sequence for deactivating isothermal catalyst bed. Sandana (1982) investigated the optimum temperature policy for a deactivating catalytic packed bed reactor which is operated isothermally. Promanik and Kunzru (1984) obtained the optimal policy for a consecutive reaction in a CSTR with concentration dependent catalyst deactivation. Ferraris ei al. (1984) suggested an approximate method to obtain the optimal control policy for tubular catalytic reactors with catalyst decay. 

The problem is best solved in the time domain, t = z/u, since the results will then be independent of tube diameter and flow rate. Divide the reactor into Azones equal length zones each with residence time f/Azones- Treat each zone as an isothermal reactor operating at temperature n = 1, 2,. .. Azones- The problem in functional optimization has been converted to a problem in parameter optimization, the parameters being the various T . The profile for 10-zone optimization is shown in Figure 6.2a. These results were generated by the Code for Example 6.5 in less than a minute. 

Very frequently non-optimal setpoint trajectories are used for controlling reactor temperatures in batch reactors [25,39,179,180]. Reactor temperatures maybe allowed to increase from ambient temperatures up to a maximum temperature value, in order to use the heat released by reaction to heat the reaction medium and save energy (reduce energy costs). The temperature increase is almost always performed linearly, because of hardware limitations and simplicity of controller programming. After reaching the maximum allowed temperature value, reactor temperature is kept constant for a certain time interval, for production of polymer material at isothermal conditions. At the end of the batch, the reaction temperature is increased in order to reduce the residual monomer content of the final resin, usually with the help of a second catalyst. Heuristic optimum temperature trajectories were also formulated for batch polymerizations of acrylamide and quaternary ammonium cationic monomers, in order to use the available heat of reaction [181]. The batch time was split into two batch periods an isothermal reaction period and an adiabatic reaction period. 

Various parameters must be considered when selecting a reactor for multiphase reactions, such as the number of phases involved, the differences in the physical properties of the participating phases, the post-reaction separation, the inherent reaction nature (stoichiometry of reactants, intrinsic reaction rate, isothermal/ adiabatic conditions, etc.), the residence time required and the mass and heat transfer characteristics of the reactor For a given reaction system, the first four aspects are usually controlled to only a limited extent, if at aH, while the remainder serve as design variables to optimize reactor performance. High rates of heat and mass transfer improve effective rates and selectivities and the elimination of transport resistances, in particular for the rapid catalytic reactions, enables the reaction to achieve its chemical potential in the optimal temperature and concentration window. Transport processes can be ameliorated by greater heat exchange or interfadal surface areas and short diffusion paths. These are easily attained in microstructured reactors. 

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. 

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