Falling Off the Steady State

We now consider what can happen to a CSTR operating at an upper steady state when an upset occurs in either the ambient temperature, the entering femperature, the flow rate, reactor temperature, or some other variable. To illustrate, let s reconsider the production of propylene glycol in a CSTR. 

In Example 9-4 we saw how a 500-gal CSTR used for the production of propylene glycol approached steady-state. For the flow rates and conditions (e.g., Tq = 75°F, = 60° ), the steady-state temperature was 138°Fand the corresponding conversion was 75.5%. Determine the steady-state temperature and conversion that would result if the entering temperature were to drop from 75°F to 70°F, assuming that all other conditions remain the same. First, sketch the steady state conversions calculated from the mole and energy balances as a function of temperature before and after the drop in entering temperature occurred. Next, plot the conversion, concentration of A, and the temperature in the reactor as a function of time after the entering temperature drops from 75°F to 70°F. 

The steady-state conversions can be calculated from the mole balance. 

We see that for Fq =70°F the reactor has dropped below the extinetion temperature and can no longer operate at the upper steady state. In Problem P9-16, we will see it is not always necessary for the temperature to drop below the extinetion temperature in order to fall to the lower steady state. The equations deseribing the dynamic drop from the upper steady state to the lower steady state are identieal to those given in Example 9-4 only the initial conditions and entering temperature are different. Consequently, the same POLYMATH and MATLAB programs ean be used with these modifications. (See CD-ROM) 

Initial conditions are taken from the final steady-state values given in Example 9-4. 

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Example CD9-1 Startup of a CSTR Example CD9-2 Falling Off The Steady State Example CD9-3 Proportional Integral I) Control Living Example Problems... 

Falling Off the Steady State 623 Nonisothermal Multiple Reactions 625 Unsteady Operation of Plug-Flow Reactors... 

Example CD 1.3-1 Use of the ARSST Example CD 13-2 Startup of a CSTR Example CD13-3 Falling Off the Steady State Example CDl.3-4 Proportional-Integral (PI) Control... 

The observation of negative apparent activation energy can most simply be interpreted in terms of the competition between the adsorption and desorption of methylacetylene on the surface. This qualitative explanation is illustrated in Figure 3, where the steady-state production of trimethylbenzene is compared with the TPD trace of methylacetylene. The fall off in steady state cyclotrimerization rate matches the tail of the desorption spectrum and illustrates the role of reactant desorption at higher temperatiu-es controlling the availability of alkyne monomers and thus the overall cyclotrimerization rate in this temperatime/pressure regime. 

The attrition rate, i.e., the rate of generation of fines, 0-d microns, at the submerged jets in a fluidized bed, tends to fall off asymptotically with time to a steady-state rate as shown in Fig. 9. Initially the attrition rate is high due to the wearing off of angular comers. Typically, it takes long time, hours to days, for the particles to reach steady-state (equilibrium) where the particles tend to be more rounded. For most catalytic fluidized bed processes, the bed operates at equilibrium. That means the most significant part of the attrition rate curve is the steady-state rate. 

Figures 57 and 58 shows the estimation results for the intervals of the unmeasured states Cti and Z. Notice how the interval bounds estimated by the interval observer envelop correctly these unmeasured states. For all the other unmeasured states, notice that although the interval observer design did not allow us to tune the convergence rate, the interval observer showed excellent robustness and stability properties and provided satisfactory estimation results in the event of highly corrupted measurements and operational failures. Notice in particular, the robustness of the interval observer around day 25 when the inlet concentrations drastically increased and when a major disturbance occurred at day 31, due to an operational failure, resulting in a rapid fall of both, the dilution rate (which actually fell to zero) and the substrate concentration readings. Off-line readings of Cti and Z (not used in the state estimation calculations) were also added to validate the proposed interval observer design (see Figures 57 and 58). It should be noticed that the compromise between the convergence rate and robustness was not fully achieved until the estimation error dynamics reached the steady state.

Whenever multiple steady states in a reactor are possible, we must be very concerned that we are operating on the desired steady-state branch. This requires a proper startup procedure to attain the desired steady state and suitable operation limits to make sure that we never exhibit a sufficiently large transient to cause the system to fall off the desired conversion branch. We will consider transients in the CSTR in the next section. 

In practice, great care must be taken that the reaction is followed for at least 10 half-lives of the exponential phase to ensure that the steady state rate is reached. During that time, the substrate should not be so depleted that the rate falls off as a consequence or there is onset of product inhibition. Even so, it may still be difficult to distinguish between the kinetics of mechanism A and variants of mechanism B if the free enzyme and the two forms of enzyme inhibitor complex are in not in steady state equilibrium.28... 

If the rate of evaporative loss per unit gross area of surface— negative rainfall—is v (cm./day) and the effective total cross-sectional area of water in continuous soil capillaries is s, the mean linear rate of upflow in the soil water is v/s. If diffusion coefficient of the solute is D (sq. cm./sec.), it can be shown that the concentration at the steady state falls off with distance x (cm.) from the surface according to... 

We will then discuss reactor start-up, falling off the upper-steady state, the control of chemical reactors, and multiple reactions with heat effects. 

The fall-off of the unimolecular rate constant as a function of pressure arises because of the way in which the competition between reaction and deactivation depends on pressure. The rate of reaction is equal to /t2[A ], and so its pressure dependence follows the pressure dependence of [A ], given by Eq. (8). As the pressure is reduced, the importance of k2 in the denominator relative to A i[M] increases, and so the steady-state concentration [A ] is reduced. A useful way of looking at this is that [A ] is depleted by reaction, and is lower than its Boltzmann value, A i[A]/fc i. This reactive depletion is most important at low pressures, where the collisional activation is too slow to replenish it. 

The behavior of HO2 and OH in the upper troposphere is dominated by CO chemistry. (Because of its 1 to 3 month lifetime, CO is more or less uniformly mixed up to the tropopause. Above the tropopause, CO falls off with increasing altitude. Because of the much slower vertical transport rate in the stratosphere, the rate of the CO-OH reaction competes with the rate of vertical mixing.) Tropospheric CO oxidation proceeds according to reactions 5.24 and 5.25, coupled to reactions 5.1 to 5.3. (Note that 4.36 and 5.25 are the same reaction.) From Section 5.2 we can obtain an expression for the HO2/OH ratio in the upper troposphere. Based on the steady-state relation for HO2, we obtain... 

The two isomers n-CsHy and FC3H7 are separated by a barrier of 37kcal/mol (measured with respect to n-CsH ) and they can easily interconvert at sufficiently high temperatures. Although in reality both radicals dissociate to propene + H and ethylene+ CH3 (see Fig. 8), we will ignore these channels here and focus exclusively on the isomerization part. The steady-state analysis with CARRA yields one apparent pressure-dependent rate constant, since the rate constant for the reverse reaction is determined by the equilibrium constant. The predictions with both versions (MSC and ME) for T = 1200 K and various pressures are shown in Fig. 9. The results are very similar and show the expected fall-off behavior. The MSC treatment—despite its simplicity—captures the pressure dependence well. 

PRS R13.2 Example CD 13-5. Load the Living Example Problem for Falling Off the Upper Steady State. Try varying the entering temperature, Tq, between 80 and 68°F and plot the steady-state conversion as a function of Tj), Vary the coolant rate between 10,000 and 4(H) mol/h. Plot conversion and reactor temperature as a function of coolant rate. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

This procedure is usually not viable for processes with low air turn-over rate and/or poor filtration. One key feature is that the humidifier "off" condition should approach a reasonable steady-state character. However, it may be unnecessary to achieve steady-state if first it is clearly shown that dust levels fall well below the PEL. 

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