Maximum Conversion Problem

Mujtaba and Macchietto (1997) have considered a maximum conversion problem for BREAD, subject to given product purity constraints. The reflux ratio is selected as the control parameters to be optimised for a fixed batch time so as to maximise the conversion of the limiting reactant. The optimal product amount, condenser and reboiler duties are also calculated. Referring to Figure 4.5 for CBD column the optimisation problem can be stated as ... 

Computationally, the solution of the dynamic optimisation problem is time consuming and expensive. Mujtaba and Macchietto (1997) reported that the number of "Function" and "Gradient Evaluations" for each maximum conversion problem is between 7-9. A fresh solution would require approximately 600 cpu sec in a SPARC-1 Workstation. However, subsequent solutions for different but close values of tf could take advantage of the good initialisation values available from the previous solutions. 

Using the above profit function, the solution of problem P2 will automatically determine the optimum batch time (tf), conversion (C), reflux ratio (r) and the amount of product (Di). However, as the cost parameters (CDh CB0, etc.) can change from time to time, it will require a new solution of the dynamic optimisation problem P2 (as outlined in Mujtaba and Macchietto, 1993, 1996), to give the optimal amount of product, optimal batch time and optimal reflux ratio. And this is computationally expensive. To overcome this problem Mujtaba and Macchietto (1997) calculated the profit of the operation using the results of the maximum conversion problem (PI) which were obtained independent of the cost parameters. 

Note that for a fixed operation time, t in Equation 9.1, the profit will increase with the increase in the distillate amount and a maximum profit optimisation problem will translate into a maximum distillate optimisation problem (Mujtaba and Macchietto, 1993 Diwekar, 1992). However, for any reaction scheme (some presented in Table 9.1) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for reversible or irreversible reaction scheme and is already explained in the introduction section. Note for batch reactive distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For non-reactive distillation system, of course, the maximum distillate problem has to be solved. 

Mujtaba and Macchietto (1997) proposed a new and an alternative technique that permits very efficient solution of the maximum profit problem using the solutions of the maximum conversion problem already calculated. This is detailed and explained in the following using again the ethanol esterification example presented in the previous section. 

The new technique is illustrated below with the same example (ethanol esterification) and using the results of the maximum conversion problem. For x D = 0.70, Mujtaba and Macchietto (1997) used 5th order polynomials to fit the data presented in Figures 9.3 and 9.5 a 3rd order polynomial to fit the data in Figure 9.4 and a Is1 order polynomial to fit the data in Figure 9.6 respectively. The resulting curves and the polynomial equations are shown in Figures 9.9-9.12. 

An efficient optimization approach for reactive batch distillation using polynomial curve fitting techniques was presented by [55]. After finding the optimal solution of the maximum conversion problem, polynomial curve fitting techniques were applied over these solutions, resulting in a nonlinear algebraic maximum profit problem that can be efficiently solved by a standard NLP technique. Four parameters in the profit function, which are maximum conversion, optimum distillate, optimum reflux ratio, and total reboiler heat load, were then represented by polynomials in terms of batch time. This algebraic representation of the optimal solution can be used for online optimization of batch distillation. 

Reversible reactions particularly may need to be optimized. Often equilibrium composition becomes less favorable and rate of reaction more favorable as the temperature increases, so a best condition may exist. If the temperature is adjusted at each composition to make the rate a maximum, then a minimum reactor size or a maximum conversion will result. This kind of problem is a favorite of this collection, beginning with problem P4.11.01. 

These other forest resources - unutilized trees from intensive forest management and the residue today left in the forest - could, if pressed to their maximum availability, contribute around 1 EJ to the energy supply. To do this will, however, require extensive end use product markets since the end use requirement of heat production in the forest industry will already be essentially satisfied by the industries own residue. The conversion problem is therefore the transformation of biomass to energy intermediates such as electricity for transmission elsewhere, automobile fuels such as the much discussed methanol option, or into energy intensive tonnage chemicals such as ammonia and ethylene. 

The design of chemical reactors encompasses at least three fields of chemical engineering thermodynamics, kinetics, and heat transfer. For example, if a reaction is run in a typical batch reactor, a simple mixing vessel, what is the maximum conversion expected This is a thermodynamic question answered with knowledge of chemical equilibrium. Also, we might like to know how long the reaction should proceed to achieve a desired conversion. This is a kinetic question. We must know not only the stoichiometry of the reaction but also the rates of the forward and the reverse reactions. We might also wish to know how much heat must be transferred to or from the reactor to maintain isothermal conditions. This is a heat transfer problem in combination with a thermodynamic problem. We must know whether the reaction is endothermic or exothermic. 

When two reactors, a plug flow and a stirred tank are operated in series, which one should go first for maximum conversion To solve this problem the intermediate conversion is calculated, the outlet conversions are determined, and the best arrangement chosen. Keeping the intermediate conversion as high as possible results in the maximum conversion. Concentration levels in the feed do not affect the results of this analysis as long as we have equal molar feed. 

For a given product purity of x D = 0.70, Mujtaba and Macchietto (1997) solved the maximum profit problem for a number of cost parameters using the method described above. The results are presented in Table 9.3. For each case, Table 9.3 also shows the optimal batch time, amount of product, reflux ratio, total reboiler duty and maximum conversion (calculated using the polynomial equations). 

The solution of the problem PI results in the maximum conversion of 61.3% with 1.29 kmol of product C. The optimum reboil ratio was 0.96. Please see the original reference for further details. 

At high concentrations, corrosion-resistant reactors and an effective acid recovery process are needed, raising the cost of the intermediate glucose. Dilute acid treatments minimize these problems, but a number of kinetic models indicate that the maximum conversion of cellulose to glucose under these conditions is 65 to 70 percent because subsequent degradation reactions of the glucose to HMF and lev-ulinic acid take place. The modem biorefinery is learning to exploit this reaction manifold, because these decomposition products can be manufactured as the primary product of polysaccharide hydrolysis (see below). 

A so far still unsolved problem is the direct enantioselective epoxidation of simple terminal olefins. For example the epoxidation of propylene that was achieved with a 41% ee almost twenty years ago by Strukul and his coworkers using Pt/diphosphine complexes is still unsurpassed. Unfortunately such low ee s are of no practical interest. The problem was circumvented by Jacobsen using hydrolytic kinetic resolution of racemic epoxides (Equation 26) and is practised on a multi 100 kg scale at Chirex. The strategy used is to stereose-lectively open the oxirane ring of a racemic chiral epoxide leaving the other enantiomer intact. Reactions are carried out to a 50% maximum conversion. The catalyst belongs to the metal-salen class described above and can be recycled. The products are separated by fractional distillation. 

We have generally used the direct construction of an objective function as some measure of the profit. This then has to be maximized. An important form of problem is to take the objective function as a measure of the cost and to minimize it. In some cases the same problem can be formulated in two ways which are duals of one another. Thus, if we seek the minimum holding time to achieve a given conversion we are solving the dual of the problem of finding the maximum conversion for given holding time. The existence of duality is useful but it needs to be carefully established, as, for example, in Amundson and Bilous (1956). 

This detrimental situation indicates that the coolant temperature is too low for obtaining maximum conversion. Thus even boihng Dowtherm A at its highest possible operating temperature is not a suitable coolant. Perhaps a gas would give a better performance as a coolant in this reaction system. Two problems at the end of the chapter pursue this aspect. One of them seeks the optimum coolant temperature for a constant-coolant-temperature system, and the other uses inlet gas as a coolant. 

In order to minimize these problems, the refiners are often forced to operate at low conversion levels (ca. 50%). Despite its importance as a critical factor limiting the maximum conversion attainable in commercial residue hydroprocessing units, the problem of sediment... 

It can be shown that the maximum conversion of nitrogen and hydrogen into ammonia occurs when the gases are mixed with the stoichiometric ratio. (See Problem PI5.1.) hSee Section 11.3a of Chapter 11 for a discussion of the effect of pressure on the fugacity, including the Lewis and Randall rule. 

With an increasing temperature the rate of H2S conversion In elemental sulfur, which is the most common reaction product, increased. The undesired side eflecl is an increase in the rate of SO2 production when temperature reached 445 K. This limits the application temperature due to secondary air pollution problems. An increase in the content of oxygen added to the gas stream significantly increases the conversion of H2S, as expected based on the mechanism of the reaction. With the oxy n content on the level of one time stoichiometric ratio the maximum conversion rate at 348 K is 80 % with a decrease to 40 % after 1000 minutes. When the content of oxygen is three times the stoichiometric ratio the initial conversion rate is almost 100 % and decreases to 86 % after 800 minutes. An increase in the flow rate and a decrease in the bed depth have also negative effect on the H2S conversion on activated carbon. With a three times increase in the flow rate the conversion rate can decrease even 40 % [67]. A similar effect is found when the bed depth decreases four times. The smallest alteration is observed with a decrease in pressure. When it decreases from 289 kPa to 104 kPa the initial conversion rate drops only 3 percent. 

Suppose we set the feed rate and the conversion requirement, but let the vessel volume be a variable. How can this problem be solved for the maximum conversion to intermediate ... 

For the purely discontinuous reactor operation the ideally mixed batch-reactor, BR, is used. All educts and solvents are charged initially. Under agitation the mixture is warmed up until the desired process temperature is reached. The reactor is operated at this temperature for a certain reaction time tg, which corresponds to the desired conversion. This is followed by an operational phase to isolate the product. From a safety technical point of view the BR is the reactor causing a maximum of problems, for... 

This can always—in principle, and usually in practice—be integrated for a constant value of temperature, and then the best temperature found for a given conversion, It can be shown that this is exactly equivalent to the problem of choosing the optimal temperature for the maximum conversion for a given reaction time. Or. 

Maximum conversion is achieved by complete utilization of the substrate in the later stages and maximum productivity in the first stage. This is essential in the case of expensive substrates, for example, steroid transformation, or in the case of environmental problems, for example, waste water treatment. 

THE PROBLEM An undivided rotating cylinder reactor is operated in a batch recycle loop to process a batch of 0.2 m every 8 hours. The cylinder is 1 m long and 0.1 m in radius the interelectrode gap is 0.05 m. The reaction takes place under mass transfer limiting conditions. If the maximum rotation speed of the cylinder is 150 rpm, determine whether the operation is feasible if the required conversion is to be 80%. If so, what is the required rotation speed and what is the maximum conversion possible Assume that axial flow in the reactor can be ignored. The kinematic viscosity v is 1.1 X 10" m /s and the Schmidt number Sc is 2500. 

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