Unsteady steady periodic

An inherent property of the LES approach is that the simulated flow field is no longer steady, but exhibits a transient character due to the presence and motion of large-scale eddies. The LES methodology has proven to be a powerful tool for studying and visualizing stirred tank flows (Eggels, 1996 Derksen et al. 1999 Bakker et al., 2000 Derksen, 2001 Bakker and Oshinowo, 2004), as it inherently takes the unsteady and periodic behavior of the flow (around impeller and baffles) into account. 

Information about transport diffusion in catalyst particles can also be deduced during the initial, unsteady state period of a permeation experiment. In this stage, the number of molecules passing the plug of catalyst per unit time will increase from zero until the rate of permeation characterizing the steady state behavior is attained. In the limit t - oo, the total amount of molecules which have permeated in the time interval 0... t is given by the relation [1,2, 12]... 

The curved portion, AB, is representative of the unsteady state period during which the solid temperature reaches its steady state value, ts. AB may occur at decreasing rate as well as at the increasing rate shown. 

In Section 3.2 we focused on the unsteady solution and its steady part for periodic lumped problems. We learned then the practical importance of steady periodic solutions and, in terms of the method of complex temperature, an easy way of obtaining only the steady part of periodic solutions. In this section we apply the method of complex... 

When the toner initiates the detonation, in its first stage, detonation rate varies and increases gradually. This stage is unsteady detonation. Detonation experiences unsteady acceleration period and reaches the maximum steady velocity/rate/speed, and this maximum velocity is maintained until all explosives react completely. 

Typical simulation curves for steady-state and unsteady-state (periodical) operation conditions under the same time-average conditions can be seen. 

Unsteady-State Direct Oxidation Process. Periodic iatermption of the feeds can be used to reduce the sharp temperature gradients associated with the conventional oxidation of ethylene over a silver catalyst (209). Steady and periodic operation of a packed-bed reactor has been iavestigated for the production of ethylene oxide (210). By periodically varyiag the inlet feed concentration of ethylene or oxygen, or both, considerable improvements ia the selectivity to ethylene oxide were claimed. 

Fig. 17. Comparison of the variation of the time-average S02 conversion and the maximum bed temperature predicted for stationary cycling condition by an unsteady-state and a steady-state kinetic model for a packed-bed S02 converter operating with periodic flow reversal...

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

A record of the axial velocity component vx for steady turbulent flow in a pipe would look like the trace shown in Figure 1.22. The trace exhibits rapid fluctuations about the mean value, which is determined by averaging the instantaneous velocity over a sufficiently long period of time. Figure 1.22 shows the case in which the mean velocity remains constant this is therefore known as steady turbulent flow. In unsteady turbulent flow, the mean value changes with time but it is still possible to define a mean value because, in practice, the mean will drift slowly compared with the frequency of the fluctuations. 

The extruder would operate for several hours to days at steady state, and then for no apparent reason It would flow surge for several hours. After a period of time, the extruder would return to a steady-state operation and would remain there until the cycle repeated. Problem diagnosis was impossible without transient process data. Moreover, molten resin would frequently flow out the vent, especially during times of unsteady-state operation. 

The duration of the unsteady period, denoted by C, the time required for Sh to come within 100x% of the steady value, is different for rigid and fluid spheres. For a rigid sphere at high Pe, T ocPe "/". From Stokes s law, Eq. (3-18), Uj oc a" hence C is independent of particle size for a given fluid. However, for a fluid sphere, oc (1 + K )/Pe thus Ujt /a is a constant, and a given fractional approach to steady state is achieved when the particle has moved a fixed number of radii, e.g.. 

Approximate values of Sh for intermediate Pe may be obtained by using Eq. (3-61) until Sh equals the steady-state value of Fig. 3.10. For larger t this steady-state value is used. Although this approximation underestimates the duration of the unsteady period, the error in Sh is not large. In terms of the time-averaged Sh or the total mass transferred, the error is less than 15% for all times. 

In Figs. 5.4.1a and 5.4.2a, we present the steady VC curves, corresponding to Vcr — oo, computed, respectively, for e = 10-4, 10-6. For Vct finite the above procedure is carried out until V r is reached. At this point the modulation is switched on and the unsteady computation is performed for a few tens of periods T until the transients die out. The computed current density... 

Chrono amperom etry, Chronopotentiometry, Chronopolarography and Oscillographic Pol orography. A "steady state is said to be reached when the current-potential curve of a soln examined by the usual electrochemical methods (such as potentiometric titration, voltammetry, etc) becomes independent of time (at least for the period of time needed to carry out the measurements). However there are electrolitic phenomena which change with time,resulting in the existence of an unsteady... 

Steady-state periodic heating and unsteady-state methods can be applied to measure the thermal conductivity and diffusivity of coal. Methods such as the compound bar method and calorimetry have been replaced by transient hot-wire/line heat source, and transient hot plate methods that allow very rapid and independent measurements of a and X. In fact, such methods offer the additional advantage of measuring these properties not only for monolithic samples but also for coal aggregates and powders under conditions similar to those encountered in coal utilization systems. 

A set of the possible unsteady or cyclic states of the dynamic system comprises a narrower set of steady states. Therefore, the maximum value of an object or performance function /max obtained under optimal unsteady-state conditions may not be lower than that for optimal steady-state JJ ", i.e. 7max > "ax. However, efficient or proper periodic operation includes only the regimes which provide higher performance in comparison with the optimal steady-state, Jmax >... 

Sufficient conditions for optimality of forced unsteady-state operation which provides J > Js, can be determined on the basis of analysis of two limiting types of periodic control [10]. The first limiting type is a so-called quasisteady operation which corresponds to a very long cycle duration compared to the process response time t. In this case the description of the process dynamics is reduced to the equations x(t) = /t(u(t)), where h is defined as a solution of the equation describing a steady-state system 0 = f(/t(u(t),u(t))). The second limiting type of operation, the so-called relaxed operation, corresponds to a very small cycle time compared to the process response time (tc t). The description of the system is changed to ... 

Can be operated at a steady state (if the catalyst does not decay rapidly) or as a pulsed (or periodic-flow) reactor which is operated under unsteady-state conditions. 

Figures 3 through 6 show conversion-time data for a number of vinyl acetate runs. The start-up procedure for these experiments consisted of filling the reaction vessel with degassed water prior to introducing any feed streams. Periodic samples were taken and the monomer conversion measured gravimetrically. As can be seen, some of the conversion transients did not reach a steady state. Tendency toward unsteady behavior and the magnitude of the oscillations seemed to increase with increasing initiator concentration and mean residence time. The influence of changing the emulsifier concentration is not clear.

---