The subject of investigation in this case is the transition process which takes place in a module - this is a circuit component consiting of two elements: a pipe sufficiently long to possess non-negligible inertance , and a restrictor element - an orifice, nozzle, valve, or other device with prominent dissipance (the schematic
Fig.G-52
representation in Fig.G-52 shows a nozzle, from the exit of which fluid leaves to atmosphere). The transition process is the response to sudden connection to a source of constant specific energy (= capable to perform constant specific work ).
Derivation of the governing equation starts from the fact that the specific work provided by the source is partly dissipated in the restrictor device and the rest part is used to overcome the inertial effects. The energetic drop across the restrictor (nozzle) is equal, as it does in a steady state, to . Of course, this value gradually increases with increasing flow rate . The inertial effects require energetic drop .
As a result, the equation is = +
The mass flow rate is assumed to increase from the zero initial flow = 0 to the asymptotic value, theoretically attained at infinite time, = . Note that at the final, asymptotic state is steady state, where the second term is zero, so that . Dividing the above equation by this expression leads to
1 = +
.. where there is = / and use is made of the definition of characteristic time in
Fig.G-53
Fig. G-52. Using the definition of the relative time then leads to the equation of the transition process in dimensionless co-ordinates:

This equation is solved by separation of variables. The result
= tgh
is shown plotted in Fig.G-53 in the two diagrams, the top one presenting the dependence of relative energy and the bottom one dependence of relative mass flow rate on the relative time .


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This is page Nr. G13 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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