Quiescence interval

In this section, we are concerned with simulation algorithms based on the approach of a fixed discrete time step for the quiescence interval. The concentrations in each drop can be updated using their initial values and reaction rates without interruption by drop entry, exit, or coalescence-redispersion. At the end of the interval, however, by generating a suitably calculated random number (to be presented subsequently), the process which disturbed the quiescence may be identified. The state of the population is now readily updated for further continuation of the simulation. 

The cumulative distribution function for the quiescence interval is thus known from knowledge of the state of the population at time t. The random number for the interval of quiescence must be generated so that the distribution function (4.6.6) is satisfied. If the particles in the population did not grow so that the states identified at time t remained the same with the passage of time, expression (4.6.6) becomes... 

The probabilities represented above are conditional not only on the state Aj, but also on the quiescence interval t. Since the four discrete events that can disturb quiescence have probabilities as listed in (4.6.9), a random number can be generated to identify the event that occurs at this stage in the sample path. ... 

In either case, a sample path of the simulation produces a sequence of quiescence intervals t, T2 initial time t = 0 and ending the... 

It is convenient to use the Monte Carlo technique based on the quiescence interval technique discussed in this section on steady State continuous systems. For example, consider the continuous extraction process in Section 3.2.4 retaining all the assumptions therein. 

Although the single particle process has been identified, we are faced with the inconvenient circumstance of having to increment the particle size by an amount whose distribution is itself the quantity to be calculated. Thus, the situation here is not as desirable as that in the breakage process considered earlier. However, let us proceed to consider the simulation of this process using the technique of Shah et al (1977). The cumulative distribution function for the quiescence interval T during which time no increment occurs in the particle is given by... 

Show that the single particle simulation of a breakage process can be extended to the case where particle growth occurs in accord with the function X(x). Elucidate the simulation strategy by calculating the quiescence interval distribution. (See Ramkrishna et al 1995 for application to a mass transfer problem in a stirred liquid-liquid contactor). 

Shah, B.H., Borwanker, J.D. and Ramkrishna, D., 1977. Simulation of particle systems using the concept of the interval of quiescence. American Institute of Chemical Engineers Journal, 23, 897-904. 

Equation (7.135) allows us to derive the following expression for the cumulative distribution for the probability of the interval of quiescence ... 

The corresponding expected interval of quiescence conditioned on the initial states of the particles can readily be found by calculating first the probability distribution /t = dFx/dr and then its first moment with respect to t, resulting in... 

Shah, B. H., Ramkrishna, D. Borwanker, J. D. 1977 Simulation of particulate systems using concept of interval of quiescence. AIChE Journal 23, 897-904. 

Song, M. Qiu, X.-J. 1999 An alternative to the concept of the interval of quiescence (IQ) in the Monte Carlo simulation of population balances. Chemical Engineering Science 54, 5711-5715. 

If not is the total probability for reaction, is the probability that a reaction has not occurred during time interval r, which leads directly to Equation 4.94 for choosing the time of the next reaction. We wlU derive this fact in Chapter 8 when we develop the residence-time distribution for a CSTR. Shah, Ramkrishna and Borwanker call this time the interval of quiescence, and us.e it to develop a stochastic. simulation algorithm for particulate. system.dyii.amic.s rather than. c.bem.ical. kinetics.114]. 

Simple periodic behaviour is far from being the only mode of oscillation observed in chemical and, even more, biological systems. For many nerve cells, indeed, particularly in molluscs, oscillations take the form of bursts of action potentials, recurring at regular intervals representing a phase of quiescence. The best-characterized example of this mode of oscillatory behaviour known as bursting is provided by the R15 neuron of Aplysia (Alving, 1968 Adams Benson, 1985). Neurons of the central nervous system of mammals (Johnston Brown, 1984) also present this type of oscillations. In addition, complex oscillations have been observed and modelled in chemical systems (see, for example, Janz, Vanacek Field, 1980 Rinzel Troy, 1982, 1983 Petrov, Scott Sho waiter, 1992). 

Geomagnetic snbstorm Interval of approximately 1 to 3 hr of auroral, geomagnetic, and magnetospheric activity, usually followed by a several-hour interval of relative quiescence. 

In the foregoing example, the deterministic event is the chemical reaction in the drops while the random events are those of drop entry into and exit from the reactor, and coalescence-redispersion within the reactor. The interval of quiescence, therefore, represents the period in which none of the following processes occur (i) addition of drops with the feed, (ii) loss of... 

In expounding the technique of simulation, our strategy will be first to stipulate the state of the system (i.e., in the domain of volume Vj.) at any instant t and then to show how to simulate the change at a later instant (an interval of quiescence away). Clearly, once such a strategy is available, it is only necessary to show how the initial state of the system is to be specified. If the initial state is known exactly, the issue is immediately resolved. If it is known probabilistically, as by specification of its distribution function, then a particular realization of the initial state can be created by the technique of random number generation that is indicated in footnote 33. 

Since particle entry, exit, breakup, or aggregation disturbs quiescence, we may write for the probability that no quiescence disturbing events occur during the interval... 

Equation (4.6.5) reflects the fact that the interval of quiescence time is strictly greater than 0. Equation (4.6.4) can be readily solved subject to the initial condition (4.6.5) to yield the cumulative distribution function for the interval of quiescence as... 

We now briefly pause to examine the simulation strategies in Section 4.6.1 which use a fixed discretization interval for the quiescence period in the light of the statistically exact technique presented in Section 4.6.2. The discussion is considerably simpler for the case in which no particle growth occurs. 

If the discretization interval, say h, is considerably smaller than the average quiescence time as given by (4.6.8), i.e., 1, then... 

The foregoing condition may be used to determine t) in (7.4.5). Equations (7.4.5) and (7.4.6) then consist of the population balance model for aggregating systems with a local volume of mixing as defined in this section. It is our objective next to compare the prediction of this population balance model with Monte Carlo simulations using the interval of quiescence in Section 4.6 as well as with predictions made from the product density equations using various closure approximations. The Monte Carlo approach provides one... 

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