The subject of investigation in this case is the transition process which takes
place in a
module - this is a circuit component consisting of two elements: a vessel or another fluid-accumulating device with prominent capacitance
, and an orifice or other device with prominent dissipance
. The transition process is the response
to sudden cutout (at time t=0 ) of input flow (Fig.G-40)
or to sudden opening of the exit in a configuration without any inflow (Fig.G-41). In both cases there is, for t>0, a drained vessel with the liquid discharging through an orifice, such as in the typical experimental arrangement from Fig. G-42 in which it is the liquid level height
- of course porportional to specific energy drop
- studied as a function of time. Fig.G-41 shows that the latter, opened-exit case conditions are farther from the indealised situation described by the present simple model - a better model for these conditions would
 |
Fig.G-42 |
involve
also the inertance. The derivation of the governing equation is in Fig.G-40. Note that it is possible to write it as

Solution by separation of variables, for the initial condition
= 1 (so that the integration constant is -2), leads to

This is presented as a graph in Fig.G-43. To obtain the nondimensional quantity, the relative value of the specific-energy drop across the dissipance element is here related to the value
at the beginning of the discharge process.
It is also possible to solve the transition process by considering the flowrates.
In that case, we start with the same flowrate budget as in Fig.G-40, but then
insert


Note that the definition of the characteristic time in Fig.G-40 is equivalent to

where
is the initial mass flow rate at
= 0. The result may be re-written in nondimensional form as

or, using the derivative with respect to the relative time
as

An important fact is that since the derivative is here constant, output flow decreases linearly with time, as shown in the bottom part of Fig.G-43.
Basically the same approach may be used to solve the more complicated cases with several vessels or several orifices (- restrictors or nozzles, it the investigated process in discharge from a vessel) as shown in Fig.G-44.
The only difference is in the resultant expression for the characteristic time

, which is easily evaluated by using the formulas for parallel or series connection. The same solution is, in fact, also applicable if the vessel is emptied by a pump, Fig.G-45, as long, of course, as the pump behaviour may be described by the model from Fig.G-15.
It is interesting, however, that a different (although not very different) equation results if the order of the devices is changed from CQ to QC, Fig.G-42.
 |
Fig.G-45 |
 |
Fig.G-42 |
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This is page Nr. G12 from textbook
Vaclav TESAR : "BASIC FLUID MECHANICS"
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tesar@fsid.cvut.cz
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