Fall transient, acceleration

Chapter VI illustrates what a mesoscopic level of description is meant to contain. It involves the variables of interest (usually assumed to be slow) plus a suitable set of auxiliary variables, whose role is to mimic the influence of the thermal bath on the variables of interest themselves. This level of description (reduced model theory) is less detailed than the truly microscopic one, because an overwhelming number of microscopic degrees of freedom are simulated with fluctuation-dissipation processes of standard type. The mesoscopic level, however, is still detailed enough to preserve the essential information without which the theoretical investigation becomes difficult and obscure. A new class of non-Gaussian equilibrium properties is proven to be responsible for the acceleration of the fall transient described in Chapter V. To obtain these results, use is made both of the theoretical tools already mentioned and of computer simulation (one-dimensional for translation and two-dimensional for rotation). 

The hulk liquid flow considered so far has heen at steady state. The liquid was flowing either up or down the device at a constant rate. Further, the particles or drops were assumed to he falling with a steady terminal velocity the initial transient acceleration period was not considered relevant. Correspondingly, the d z/df term in equation (3.1.61) was neglected. In the industrial operation of a hydraulic jig, the initial transient is quite important. The principle of operation of such a device (shown in Figure 6.3.3) is as follows. 

The introduction of the valve into the pipe described in Section 6.7 has decreased the flow rate, even at fully open, from the value 2.626 kg/s calculated in Section 6.7 to 2.28 kg/s now. However, the flow at the outlet from the pipe is still sonic, and hence the pressure just inside the pipe outlet, p3, is greater than the atmospheric pressure that exists just outside the pipe, p4. As the valve is closed, however, the pressure drop across it increases, until, 17 seconds into the transient, the throat to inlet pressure ratio falls to the critical value needed for sonic flow in the valve. At this point we have the interesting phenomenon that the flow is sonic in the valve throat, then reduces to subsonic at the valve outlet, only to accelerate to sonic velocity at the pipe outlet. This is shown most clearly in Figure 10.3, which plots the Mach numbers at various points in the pipe. For about 3 seconds in the middle of the transient, the Mach number at the throat of the valve is equal to unity, as is the Mach number at the pipe outlet ... 

Figures 5 and 6 show the responses of shaft speed and Brayton loop rrrass flow rates during the over speed transient. The shaft speed in the normally operating loop is held corrstartt. This is consistent with the shaft speed control that woitld be employed. The shaft speed in the loop losing its electrical load accelerates from 45,000 rpm to about 59,000 rpm (Figitre 5). Mass flow rate in that loop increases slightly at the start of the transient then falls shghtly below the normally operating loop mass flow rate (Figure 6). Although the shaft speed has increased by 30 percent, mass flow rate ends at about the same value as at the start. This is because the density at the compressor has decreased by 20 percent because pressure downstream of the turbine is lower.

The overall picture (following the initial period of particle acceleration, which usually amounts to a very small fraction of the total transient response time) is thus of two zones, both in fluid particle equilibrium an upper zone at void fraction ei, in which the particles are all falling at a constant velocity, and a lower zone of stationary particles at void fraction 2- There are thus two travelling interfaces the falling surface of the bed, and the rising discontinuity, or shock wave, that separates the two zones (Figure 5.1). When these meet the whole bed will have attained the new equilibrium condition U2, 2-... 

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