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
.
Going to another page:
click
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
WWW server administrators: Jiri Kvarda, Zdenek Maruna
...... Contact: webmaster@vc.cvut.cz
Last change : 13.02.1998