Steady criterion

Naman s spectral steadiness criterion was used for analysis of the accuracy and steadiness of the finite difference scheme (2.44) and for estimation of the approximation error. According to this criterion, the scheme must be steady for any elementary conditions, including for the first harmonic Yq = where p is the... 

For these cases, the conservation statement is made around the outside of the catalyst. In steady-state, everything that is consumed or produced inside the catalyst must go through the outside boundary layer of the fluid surrounding the catalyst. In case of serious selectivity problems with a desired and reactive intermediate, the criterion should be calculated for that component. 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

The final estimation of the value of ay may appear tedious and several assumptions are made in its derivation, but experimental evidence suggests that it may be used with reasonable accuracy to assess the levels of potentially damaging cavitation erosion. In small valves with nominal bores up to 65 mm cavitation inception occurs in intermittent bursts when the value oy is approximately unity. The cavitation becomes continuous and audible as Oy is reduced to about 0.6, but the risk of damage does not become significant until the value falls below 0.4. As a design criterion the condition of light, steady noise has been described by Tullis as the critical level and is sug-... 

Step 2. The qualitative value of the desired change is propagated through the steady-state model equations of the plant equipment, following the constraint propagation procedure of Steele (1980). Manipulations that cause the desired change and that are feasible are identified as White Knights and are constrained to lie before the situation of interest s, in accordance with the truth criterion. 

Usually, one has obtained an estimate for the elimination constant and the distribution volume Vp from a single intravenous injection. These pharmacokinetic parameters, together with the interval between administrations 0 and the single-dose D, then allow us to compute the steady-state peak and trough values. The criterion for an optimal dose regimen depends on the minimum therapeutic concentration (which must be exceeded by and on the maximum safe... 

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

The program REFRIG 1 calculates the steady-state heat generation and heat loss quantities, HG and HL, as functions of the reactor temperature, TR, over the range 320 to 410 K. It is shown that, according to the van Heerden steady-state stability criterion that the simple loop control response, KP=0 is unstable. 

The classification of methods for studying electrode kinetics is based on the criterion of whether the electrical potential or the current density is controlled. The other variable, which is then a function of time, is determined by the electrode process. Obviously, for a steady-state process, these two quantities are interdependent and further classification is unnecessary. Techniques employing a small periodic perturbation of the system by current or potential oscillations with a small amplitude will be classified separately. 

Since the branching parameter a is greater than unity (usually it is 2), it is conceivable that under certain circumstances the denominator of the overall rate expression could become zero. In principle this would lead to an infinite reaction rate (i.e., an explosion). In reality it becomes very large rather than infinite, since the steady-state approximation will break down when the radical concentration becomes quite large. Nonetheless, we will consider the condition that Mol - 1) is equal to (fst T fgt) to be a valid criterion for an explosion limit. 

Replacing the nitrile group by a benzothiazole produces an important subclass of fluorescent compounds represented by thioflavin T (25, Fig. 10). It is not clear if this compound undergoes deactivation via intramolecular rotation that would meet the criterion for a molecular rotor. The steady-state absorption and emission properties of thioflavin T has been attributed to micelle formation [53, 54], dimer and excimer formation [55, 56], and deactivation through intramolecular rotation [57]. 

For a step change, a material-balance criterion, analogous to equation 19.3-2 for a pulse input, is that the steady-state inlet and outlet tracer concentrations must be equal, both before and after the step change. Then, it may be concluded that the response of the system is linear with respect to the tracer, and that there is no loss of tracer because of reaction or adsorption. 

In order to apply this extinction criterion, it will further be assumed that the steady state form of Equation (9.112) applies with no char, no water and no backface loss. As a consequence the burning rate can be described as... 

This expression could be regarded as a general lability criterion for the steady-state supply of M in spherical geometry. For the particular case that eKc 3> 1, one can recover the condition [57] ... 

These multiple branches are not equivalent. From the contour lines in Figure 7.8, we can deduce which branch is stable, and which branch is not. The middle branch (C-C1) lies in a range where, for a given value of p, F increases with u. The derivative of F with respect to u is therefore positive which is just the criterion we found for an unstable equilibrium. Any fluctuation of u at constant p will drive the system away from the branch C-C. The opposite holds for the upper and lower branches A-C and A -C that lie in a range where F decreases when u increases. The derivative of F with respect to u is therefore negative and any concentration fluctuation around an equilibrium state along these branches dies out rapidly. The branches A-C and A -C are stable steady-states. 

Solving for yields an estimated steady-state temperature of 249°C (480°E). Because this temperature exceeds the first-order failure criterion for fiberglass of 150°C and the fire burns for 10 minutes, which is longer than the reported 2-6 minutes failure time for empty fiberglass pipes, the process line will be compromised. 

The criterion for a steady state is that the rate of reduction of the cathodic current density i m, is equal to the rate of oxidation of the reducing agent Red, the anodic current density... 

The short-time criterion is the more stringent except when H 1 and p/ 1. External resistance controls and the concentration within the particle is uniform when the inequalities in Eqs. (3-74) and (3-75) are reversed. Even if the external resistance is not negligible relative to the internal resistance, it may be possible to assume constant external resistance, i.e., quasi-steady behavior. Comparison... 

S.S. Novikov Yu.S. Ryazantsev, Zh-Prikl i TekhnFiz (1965) No 1, 56-61 (On the theory of combustion stability of propellants) 9) B.V. Novozhilov, Ibid, (1965),No 4, 157-60(1965) (Stability criterion for steady-state propellant combustion) 10) I.G. Cameron H.H.M. Pike, "The Instability of an Interface Between Two Fluids under Variable Normal Acceleration , 4thONRSympDeton (1965), PP 305-15 10a) L.A. Elliot, "Calculation of the Growth... 

This ratio provides the criterion for the applicability of the quasi-steady-state approximation for the concentration as shown in Table V. 

This is the so-called evolution criterion. The equality sign applies only to steady states. Two cases may then arise ... 

A stability criterion for nonequilibrium steady states can readily be deduced from (9). Suppose that for all small departures from the steady state considered we have... 

Since the velocity profile approaches the Hagen-Poiseuille profile asymptotically, the factor 0.05 depends on a criterion for deciding how close the profile needs to be to the Hagen-Poiseuille profile (e.g., the maximum velocity is within 1% of the steady state maximum velocity). In any case, the entry length scales linearly with the Reynolds number. 

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