Reactor time constant

Luyben et al. Q996) explored this question in detail by developing a rigorous simulation of such a process. Their results demonstrate that the proposed control structure does provide effective control for processes with fast reactor dynamics. The time constant of the separation section is about 30 minutes. The reactor time constant was reduced to 3 minutes, and control was still good. 

Reactor time constant Reactor time constant is the residence time defined by... 

Consider a simple dynamic system, the reactor/ column plant described in Table G.l, and assume that the column dynamics are fast compared to the reactor dynamics. Table G.3 indicates that the holdups in these two units are Hr = 2,400 lb-moles and Hr + 20 Hs + Hj) = 930 lb-moles. Because each unit has the same flow rate F, the mean residence times for the two units are in the ratio of 2,400/930, or approximately 2.5. The effect of chemical reaction normally is to make the reactor time constant somewhat smaller than its mean residence time (see Eq. 4-89) however, the portion of column holdup located directly in the recycle loop, that is, the reflux drum plus the stripping stages, is only about one-half the total column holdup. Thus, the actual ratio of the basic time constants for the two units is... 

Note that the plant time constant without recycle (D = 0) reduces to the reactor time constant... 

Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flowthrough a packed bed reactor. The time constants for different phenomena are ... 

In the SFM the reactor is divided into three zones two feed zones fj and (2 and the bulk b (Figure 8.1). The feed zones exchange mass with each other and with the bulk as depicted with the flow rates mi 2, i,3 and 2,3 respectively, according to the time constants characteristic for micromixing and mesomix-ing. As imperfect mixing leads to gradients of the concentrations in the reactor, different supersaturation levels in different compartments govern the precipitation rates, especially the rapid nucleation process. 

This scale-up criterion is based on achieving a constant pumping rate per unit volume with scale-up and therefore leads to similar macromixing on different scales, as the circulation time in the reactor remains constant. 

Peaking and Non-isothermal Polymerizations. Biesenberger a (3) have studied the theory of "thermal ignition" applied to chain addition polymerization and worked out computational and experimental cases for batch styrene polymerization with various catalysts. They define thermal ignition as the condition where the reaction temperature increases rapidly with time and the rate of increase in temperature also increases with time (concave upward curve). Their theory, computations, and experiments were for well stirred batch reactors with constant heat transfer coefficients. Their work is of interest for understanding the boundaries of stability for abnormal situations like catalyst mischarge or control malfunctions. In practice, however, the criterion for stability in low conversion... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

The terms space time and space velocity are antiques of petroleum refining, but have some utility in this example. The space time is defined as F/2, , which is what t would be if the fluid remained at its inlet density. The space time in a tubular reactor with constant cross section is [L/m, ]. The space velocity is the inverse of the space time. The mean residence time, F, is VpjiQp) where p is the average density and pQ is a constant (because the mass flow is constant) that can be evaluated at any point in the reactor. The mean residence time ranges from the space time to two-thirds the space time in a gas-phase tubular reactor when the gas obeys the ideal gas law. 

A simpler method arbitrarily picks values for oq and reacts this material in a batch reactor at constant V and T. When the reaction is complete, P is calculated from the molar density of the equilibrium mixture. As an example, set = 22.2 (P=l atm) and react to completion. The long-time results from integrating the constant-volume batch equations are a = 5.53, 5 = c= 16.63, = 38.79mol/m, and y =0.143. The pressure at equili-... 

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

Mass transfer in a gas-liquid or a liquid-liquid reactor is mainly determined by the size of the fluid particles and the interfacial area. The diffusivity in gas phase is high, and usually no concentration gradients are observed in a bubble, whereas large concentration gradients are observed in drops. An internal circulation enhances the mass transfer in a drop, but it is still the molecular diffusion in the drop that limits the mass transfer. An estimation, from the time constant, of the time it wiU take to empty a 5-mm drop is given by Td = d /4D = (10 ) /4 x 10 = 6000s. The diffusion timescale varies with the square of the diameter of the drop, so... 

The dynamic error existing between and Cr depends on the relative magnitudes of the respective time constants. For the reactor, assuming a first-order, constant volume reaction... 

At steady state, the temperature and concentration in the reactor are constant with respect to time and... 

There are circumstances when a complex process may involve two competing (i.e., opposing) dynamic effects that have different time constants. One example is the increase in inlet temperature to a tubular catalytic reactor with exothermic kinetics. The initial effect is that the exit temperature will momentarily decrease as increased conversion near the entrance region depletes reactants at the distal, exit end. Given time, however, higher reaction rates lead to a higher exit temperature. 

It is important to understand that the time constant xp of a process, say, a stirred tank is not the same as the space time x. Review this point with the stirred-tank heater example in Chapter 2. Further, derive the time constant of a continuous flow stirred-tank reactor (CSTR) with a first-order chemical reaction... 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

Within the accuracy of the experimental data the galvanostat-ic transient response of AV is identical to the transient rate response Ari andAr2, i.e. t = x where x is the relaxation time constant for the two rates (17. This s shown in figure 5 for two different reactors under similar operating conditions and also in figure 6 where the transient and the steady state Ar values from four reactors are plotted vs. the cell overvoltage AV. In view of the fact that r. is proportional to the surface area Q it follows from figure 6 tftat for constant gas phase composition... 

It is assumed that all the tank-type reactors, covered in this and the immediately following sections, are at all times perfectly mixed, such that concentration and temperature conditions are uniform throughout the tanks contents. Figure 3.8 shows a batch reactor with a cooling jacket. Since there are no flows into the reactor or from the reactor, the total material balance tells us that the total mass, within the reactor, remains constant. 

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