Jump equations

For motion by discrete jumps between planes a0 apart, the velocity is the product of the number of jumps per second, T, and the distance aQ of each jump. Equation 13 may now be written as... 

Since the Hugoniot represents the locus of all possible states behind the shock front, then a line joining the initial and final states on the P-v Hugoniot represents the jump condition. This line is called the Raleigh line and is shown in Figure 17.3. If we eliminate the particle velocity term u by manipulating the mass- and momentum-jump equations, and let mq = 0, we get... 

Having previously developed the three jump equations, we now have added a fourth relationship, the Hugoniot. This leaves only one variable out of the original five U, u, p, P, and e) to be specified by a boundary condition. 

We now will treat the rarefaction as if it were a shock that is, we will apply the jump equations such that we will let the high-pressure material Jump down to a lower pressure state. We also are going to allow this to happen in two steps. The first step, or rarefaction wavelet, relieves the material from state P, v (the shock pressure) to 2, 2 (half way down to ambient). 

Although the effect is small, raising the initial temperature of an explosive decreases its detonation velocity and vice versa. Data for some explosives have been measured and are presented in Table 21.8. There are sufficient data on such properties as Cp, a (linear temperature expansion coefficient), A//d (heat of detonation), po, P, and D to calculate this effect using the jump equations and simple thermophysics (Ref. 5 lists such data) for many explosives. Typically, AD/AT), the change in detonation velocity per unit change in temperature, will be found to be in the range of from -0.4 X 10 to -4 x 10 (mm/p,s)/(°C). 

The Prandtl number is important, sinee it direetly influences the magnitude of the temperature jump. Looking at the temperature jump equation, as Pr increases, the difference between wall and fluid temperature at the wall deereases. Therefore, greater Nu values for large Pr are observed. 

This is messy problem analytically. It is fairly easy if you take the viewpoint of someone riding on the jump (the lagrangian viewpoint) and solve by trial and error for the jump velocity that satisfies the hydraulic jump equation in the moving frame of reference. 

The downstream fluid is considered to be an equilibrium state, which could be either liquid, a liquid-vapor mixture, or pure vapor depending on the solution branch and fluid type. For single-phase downstream states, the jump equations (1-3) must be solved numerically by an iterative procedure. This is most conveniently performed by combining equations 1-3 to obtain the Rankine-Hugoniot equation. 

An exact calculation by Akcasu [14] of the reactor response to the sinusoidal reactivity input to a critical reactor shows that the amplitude of the SN/No vibration increases steadily with time instead of remaining constant as stated by the linearized result of Equation (120). A related result has been obtained by Potter [15] by considering departures from stable operation of the reactor at positive reactivities. He showed that the reactor is less stable for such departures than for departures from critical operation. These results are consequences of the fact that the reactor is more responsive with increasing reactivity, which may be seen to be the case from the prompt jump Equation (103). 

The solution is sketched as follows. Using the Pop plot and the shock jump equations, solve for P, V, I and then using the flow equations behind the shock, solve for TV, W, and Ir- Then solve for W and WV from... 

Salis and Kaznessis separated the system into slow and fast reactions and managed to overcome the inadequacies and achieve a substantial speed up compared to the SSA while retaining accuracy. Fast reactions are approximated as a continuous Markov process, through Chemical Langevin Equations (CLE), discussed in Chapter 13, and the slow subset is approximated through jump equations derived by extending the Next Reaction variant approach. 

On the other hand, the time evolution of the subset of slow reactions is propagated in time using a slight modification of the Next Reaction variant of SS A, developed by Gibson and Bruck. A system of differential jump equations is used to calculate the next jump of any slow reaction. The jump equations are defined as follows. 

Equations 18.7 are also ltd differential equations even though they do not contain any Wiener process, because the propensities of the slow reactions depend on the state of the system, which in turn depends on the system of CLEs. Due to the coupling between the system of CLEs and the differential jump equations, a simultaneous numerical integration is necessary. If there is no coupling between fast and slow subsets or there are only slow reactions the system of differential jump equations simplifies to the Next Reaction variant. 

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