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The results in the first columns of each a390 case are the optimum designs with the assumption that feed locations NFi and are at the bottom and top of the reactive zone, respectively. The results in the second column of each (X390 case are the optimum designs where feed locations are also taken as design optimization variables while other design variables are kept at their optimum values.

The results in the first columns of each a390 case are the optimum designs with the assumption that feed locations NFi and are at the bottom and top of the reactive zone, respectively. The results in the second column of each (X390 case are the optimum designs where feed locations are also taken as design optimization variables while other design variables are kept at their optimum values.

16.2.5 Reactive Column with Optimum Feed Tray Locations

The results shown in Figure 16.7 for the reactive distillation column are based on designs in which the two fresh feedstreams are introduced at the two ends of the reactive zone. The lighter reactant is fed at the bottom reactive tray, and the heavier reactant is fed at the top reactive tray.

Figure 16.8 and Table 16.2 give results for when this assumption is relaxed. The optimum locations of the two fresh feeds that minimize TAC are found to be inside the reactive zone. For example, as shown in Table 16.2 for the base case a390 = 2, the lower feedstream is moved from tray 6 up to tray 7 and the upper feedstream is moved from

tray 12 down to tray 11. There is a reduction in the vapor boilup from 28.78 to 26.65 mol/s, which leads to a significant reduction in TAC from 240.1 to 227.0 103 $/year.

For the case with a390 = 0.95, the lower feedstream is moved from tray 15 up to tray 29 and the upper feedstream is moved from tray 82 down to tray 68. There is a reduction in the vapor boilup from 99.15 to 94.76 mol/s, which leads to a significant reduction in TAC from 852.2 to 823.5 103 $/year.

Comparing the lower graphs in Figures 16.7 and 16.8 shows that the break-even point between the reactive distillation and column/side reactor process moves from an a390 of about 1.5 down to about 1.4.

16.3 CONTROL OF QUATERNARY IDEAL SYSTEM 16.3.1 Dynamic Tubular Reactor Model

In the steady-state design discussed earlier, the tubular reactors are modeled using ordinary differential equations that describe the variation of the temperature and compositions with axial length. Because we are interested in the dynamic behavior of the system for this control study, the tubular reactor model that is used must describe the changes in variables in both time and axial position. A rigorous approach is to use partial differential equations. A more simple method is to use a "lumped" model in which the spatial variations in temperature and composition variables are approximated with a series of CSTRs. With a reasonable number of lumps, this simple model gives a steady-state profile similar to the real steady-state profile obtained when using the rigorous steady-state equations. Each lump has its dynamic component and energy balances with variables needed to be numerically integrated in time: dT/dt, dx/dt, and so forth.

In these reactors, both the process liquid and the catalyst have thermal capacitance, so the temperatures of these two phases can be dynamically different in each lump with heat transfer between the solid and liquid phases. Of course, the liquid compositions change dynamically from lump to lump because of the convective flows in and out and because of the reaction. The kinetics are based on the volume of the liquid. This brings up the issue of specifying the amount of liquid and the amount of solid catalyst in a given total vessel volume. It is assumed that the void volume of the catalyst is 0.5, so the vessel is half filled with liquid and half filled with liquid.

The reaction occurs at the catalyst site, so that is where the heat of reaction affects the catalyst temperature. There is also heat transfer from the catalyst to the liquid. This means that there will be a temperature difference between the liquid and the solid even under steady-state conditions. The energy equation for the solid catalyst in the nth lump is dTcat,n _ — 1-PliqVliq/Mliq A , \ UA ^ n,

-~t— — -7}—-\kFxK,nxb,n — kBxc,nxd,n)--7}—-i/cat,n — T liq,nj (16- 7)

dt Pcat VcatCPcat Pcat VcatCPcat

The energy equation for the liquid in each lump includes the convective terms associated with the flow of liquid in and out of the lump at their respective temperatures.

hp — pliq V 'iq,n (Tliq,n—1 — Tliq,n) + p V C (Tcat,n — Tliq,n) (16-8)

dt Pliq Vliq Pliq Vliq CPliq

The last term in both of these energy balances comes from the heat transfer between the catalyst and the liquid.

Two assumptions have been made to calculate the values of the heat transfer coefficient U and area A needed for these equations: the temperature difference between the liquid and solid is ^2 K at steady state, and the catalyst particles are spherical with a 0.002-m diameter.

The reaction directly affects the compositions in the liquid phase. The forward and reverse specific reaction rates follow the Arrhenius law:

The overall reaction rate is based on the concentrations in mole fractions and liquid holdups in moles. To avoid confusion, the specific reaction rates used in this control section (with the reactor half full of catalyst) are twice those used in the earlier design section in which catalyst was not considered. Thus, the total reactor volumes are the same as those used in the steady-state design.

Because the reaction is exothermic, some vapor is produced as the liquid from the high pressure reactor flashes into the low pressure column. This results in an increase of the vapor flowrate at each external reactor location and a corresponding decrease of the liquid rate below the external reactor location. The quantity of this vapor is calculated from the heat generated by the reaction occurring in the external reactor, the flowrate of material into the reactor, and the thermal properties. The steady-state vapor and liquid rates are constant through the stripping and rectifying sections because equimolal overflow is assumed. However, there are different liquid and vapor rates in the middle sections of the column.

Figure 16.9 shows the system that was studied, and Figure 16.10 gives the composition and temperature profiles for the column. The left graphs are steady-state results using rigorous ordinary differential equation models for the three reactors. The right graphs are steady-state results using the lumped reactor model. The column profiles are essentially identical. Figure 16.11 gives composition and temperature profiles for each of the three reactors using either the rigorous reactor model (solid lines) or the lumped reactor model (dashed lines). These profiles are quite similar.

16.3.2 Control Structures

In practical applications it is desirable to use inferential temperature measurements whenever possible instead of direct composition measurements. Composition analyzers have higher cost, require more maintenance, and can introduce dead time into the control loop. Therefore, we first explore a control structure that does not have any composition analyzer. The performance of this two-temperature control structure will then be compared with a structure that uses a composition analyzer.

The control structures are single-input, single-output structures with PI controllers for temperature and composition and proportional-only controllers for levels with gains of 2. The Tyreus-Luyben tuning method is used to tune the temperature and composition kF = aPe~Ep/RT«

kB = aBe~Eii/RTcat

-T, — (xi,n-1 ~ xi,n) + (kFxa,nxb,n _ kRxc,nxd,n)

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