Slope condition

The above inequality, called a slope condition, is the requirement for thermal insensitivity, expressed here for first order reactions. This form was derived by Perlmutter in (1972.) In most cases it is adequate to define the condition for a stable reactor, but not always. The area of sensitive domain was defined by Van Heerden (1953.)... 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

The concentration is continuous at the reactor exit for all values of D and this forces the zero-slope condition of Equation (9.17). The zero-slope condition may also seem counterintuitive, but recall that CSTRs behave in the same way. The reaction stops so the concentration stops changing. 

Constant conduction heat pipes, 13 227 Constant failure rate, 13 167 Constant-field scaling, of FETs, 22 253, 254 Constant-modulus alloys, 17 101 Constant of proportionality, 14 237 Constant pressure heat capacity, 24 656 Constant rate drying, 9 103-105 Constant rate period, 9 97 23 66-67 Constant retard ratio (CRR) mode, 24 103 Constant slope condition, 24 136-137 Constant stress test, 13 472 19 583 Constant-voltage scaling, of FETs, 22 253 Constant volume heat capacity, 24 656 Constant volume sampling system (CVS), 10 33... 

Assuming identical detection coefficients for the two species, the radioactivity ratio obviously reduces to 1. This condition, known as secular equilibrium, is illustrated in figure 11.8C for ty2, = °° and ti/2 2 = 0.8 hr. Secular equilibrium can be conceived of as a limiting case of transient equihbrium with the angular coefficient of decay curves progressively approaching the zero slope condition attained in figure 11.8C. 

Such was the state of the art when Amundson and Bilous s paper was published in the first volume of the newly founded A.I.CH.E. Journal (Bilous and Amundson, 1955). This for the first time treated the reactor as a dynamical system and, using Lyapounov s method of linearization, gave a pair of algebraic conditions for local stability. One of these corresponded to the slope condition of previous analyses, and there was a brief flurry of attempts to invest the other with a similarly physical explanation. For the global picture they introduced the phase plane (another feature of the theory of dynamical systems) and, with consummate skill, Bilous conjured the now classic figures from a Reeves electronic analogue computer. Even in this early paper, they had touched upon the consecutive reaction scheme A - B - C and had shown that up to five steady states might be expected under some conditions. 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

When two steady states exist, the low-conversion one is unstable. This can be demonstrated showing that a slope condition is not fulfilled ... 

Figure 4.9 shows the results of a dynamic simulation we performed featuring the open-loop behavior of a backmixed reactor that satisfies the slope condition for steady-state stability but has dynamically unstable roots. Table 4.1 contains the reactor parameters and operating conditions used in the model, as listed by Vleeschhouwer et al. (1992). 

Equation (5.11) represents a straight line in the diagram of fractional temperature rise versus reactor feed temperature. We show three such lines in Fig. 5.21. All lines intersect the temperature rise curve at least once (at a low temperature not shown in Fig. 5.21). It therefore appears that the reactor FEHE can have one, two, or three steady-state solutions for this particular set of reaction kinetics. Furthermore, the intermediate steady state, in the case of three solutions, is open-loop unstable due to the slope condition discussed in Chap. 4. This was verified by Douglas et al. (1962) in a control study of a reactor heat exchange system. 

Exercise 7.5.2. Show that for a single endothermic reaction the steady state is always stable. Show also that in the case of an adiabatic reactor the slope condition LM > JV is sufficient as well as necessary. 

We have already considered the stability of the adiabatic stirred tank (Sec. 7.5) and have observed that since M = 1, the slope condition L> N suffices to ensure stability. Now recalling the definitions of L and N this is... 

Ti = Ti-i, but the temperature analog of Equation 8.32 provides a smoother estimate of the zero-slope condition. 

Equation 8.58 reduces to Twaii = Tixt if both ho and /Cwaii are large and reduces to a smooth, zero-slope condition when either/ o or/Cwaii are zero (compare Equation 8.32). 

The zero-slope condition may seem counterintuitive. CSTRs behave in this way, but PFRs do not. The reasonableness of the assumption can be verified by a limiting process on a system with an open outlet as discussed in Example 9.2. The Danckwerts boundary conditions are further explored in Example 9.2, which treats open systems. The end result is that the boundary conditions are somewhat unimportant in the sense that closed and open systems behave identically as reactors. 

The boundary conditions are provided by the mass-transfer and heat-transfer rates at the pellet exterior surface, and the zero slope conditions at the pellet center... 

Given diffusion in the tube, the inlet boundary condition is no longer just the feed condition, c(0) = c/, that we used in the PFR. To derive the boundary condition at the inlet, we write a material balance over a small region containing the entry point, and consider diffusion and convection terms. The diffusion term introduces a second-order derivative in Equation 8.27, so we now require two boundary conditions. We specify a zero slope condition at the tube exit. 

Figure 8.18 Stable and unstable steady states the slope condition...

In Fig. 13.23 the lower steady state is unstable and has unusual behaviour larger reactor gives lower conversion. The instability can be proved based on steady state considerations only, showing that the analogue of CSTR s slope condition is not fulfilled. Note that the low-conversion state is closed-loop unstable. Moreover, it is independent on the dynamic separation model. Because this instability cannot be removed by control, operation is possible when the following requirements, necessary but not sufficient, are met ... 

On a mesoscale (100 m) level, no significant accumulation of radiocesium in relation to different slope conditions was found. Small scale (m) hollows, however, contained a higher amount of Chernobyl-derived Cs than. small scale collines, hut there was no. statistically significant differences found for radiocesium from global fallout. It is concluded that the inhomogenous distribution of radiocesium is caused by a run-off phenomenon during the deposition of Chernobyl-derived Cs in May 1986 rather than by sediment redistribution in the last 35 years. The consequences of these results for soil-to-plant radiocesium transfer models based on soil/plant radiocesium concentrations or transfer factors are discussed. 

The latter two conditions indicate that reactant concentration within the catalyst vanishes at the critical spatial coordinate when 0 < criticai < H and it does so with a zero slope. Conditions 2a and 3 are reasonable because reactant A will not diffuse further into the catalyst, to smaller values of r), if it exhibits zero flux at ]criticai. When / critical < 0, couditiou 2b must be employed, which is consistent with the well-known symmetry condition at the center of the catalyst for kinetic rate laws where lEl constant. Zeroth-order reactions are unique because they require one to implement a method of turning ofF the rate of reaction when no reactants are present. Obviously, a zeroth-order rate law always produces the same rate of reaction because reactant molar densities do not appear explicitly in the chemical reaction term. Hence, the mass balance for homogeneous onedimensional diffusion and zeroth-order chemical reaction is solved only over the following range of the independent variable criticai < < 1. when Jiciiacai is... 

Figure 14.8. Summary of granite rock slope conditions and possible remedial measures, housing scheme, Kuala Lumpur, Malaysia.

A code-based system for the evaluation and classification of slope condition and instability... 

For gas-solid systems with ai,/r.3 >> 1, the first two conditions in [11] imply the slope condition (6, , 9 ). Condition [7] is stronger than the slope condition if... 

This is another form of the slope condition. Since this is a necessary condition, it is not sufficient to guarantee stability. In fact, Lee and cowprkers (1972) found that a change in the ratio of ht/fi that appears in the dimensionless version of Eq. 8.42 may destabilize a steady state even when Eq. 8.43 is satisfied. However, they found that instabilities exist only for unrealistically low Values of Le/i3. 

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