Optimal Single-Stage Operation

From equation 21.5-4, the amount of catalyst is a minimum, Wmin, if (-rA) is the maximum rate at /A. For an exothermic, reversible reaction, this means operating 

1 Multistage Operation with Znterstage Heat Transfer 

Equation 21.5-6 is in a form to determine the axial temperature profile through the catalyst bed. To determine W firm equation 21.54, however, we transform 21.5-6 to relate /A and T, corresponding to equation (D) in Example 15-7 for adiabatic operation 

On integration, with /A = 0 at Tg, and the coefficient of dT constant, this becomes 

Integration of equation 21.5-8 from the inlet to the outlet of the ith stage of a multistage arrangement, again with the coefficient of dT constant, results in 

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Optimal for single-hatch operation. For the sake of simplicity suppose that (1) the performance of an equipment unit is the fraction of the feed material converted to the material that is suitable for the next stage (e.g. the yield of the desired intermediate or final product), and (2) that the objective function is the amount of suitable material produced per unit time. Let us consider the situation shown in Fig. 7.4-5. On completion of cleaning at time ta processing of a batch begins. This processing is characterized by the performance curve/(r), e.g. the yield or conversion versus time relationship. The objective function F is defined as ... 

Because most single-stage continuous fer-mentors are used to produce biomass, they are usually operated to optimize biomass productivity. The unit volume biomass productivity of such a reactor is defined as DX. This unit volume productivity can be expressed as... 

The WGS reactor is positioned directly downstream of the reformer and, due to its exothermic nature, benefits from lower operating temperatures. The optimal temperature of operation is a balance between the thermodynamics of the reaction and the activity of the catalyst for a single-stage WGS reactor it is usually in the range of 200-280 °C. A well-designed WGS reactor should be able to reduce the CO content from a reformer down from around 10% to less than 5000 ppm. 

The selectivity of a catalyst is typically optimized towards a reaction type, but some operations required a high level of removal for more than one contaminant. In fact, the treatment of a VGO, for instance, involves the removal of metals, S and N. Depending on the quality of the feed and on the specifications of the desired product, the hydrotreatment may require more than one catalyst. The catalyst can be stacked in a single reactor or disposed in sequential stages, when more than one reactor is available. Stacked-bed reactors with more than one catalyst type are a common practice in HDT. 

In a 1991 study by van Reis et al. (5), a filtration operation as applied to harvest of animal cells was optimized by the use of dimensional analysis. The fluid dynamic variables used in the scale-up work were the length of the fibers (L, per stage), the fiber diameter (D), the number of fibers per cartridge (k), the density of the culture (p), and the viscosity of the culture (p). From these variables, scale-up parameters such as wall shear rate (y ) and its effect on flux (L/m /h) were derived. Based on these calculations, an optimum wall shear rate for membrane utilization, operating time, and flux was found. However, because there is no single mathematical expression relating all of these parameters simultaneously, the optimal solution required additional experimental research. 

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