Page 42 - Mechanical Engineers Reference Book

Mechanics of fluids 1/31
At t = 3L/c, the rarefaction reaches the reservoir and the
pressure instantaneously rises to reservoir pressure po. The
resulting pressure wave travels towards the valve and fluid
R
flows away from the reservoir at velocity vo.
At t = 4L/c, the situation is the same as when the valve first
L
(a)O<t< - closed at t = 0, and the cycle is repeated.
C In practice, friction quickly dampens out the pressure waves
and cavitation reduces the pressure decrease during the rare-
factions.
A typical plot of pressure against time at a valve following
Po - rapid valve closure is superimposed on the theoretical plot in
R vo Figure 1.46.
1.5.8 Gas flow
L 2L
(b) - <t<-
C C 1.5.8.1 General relationships
The behaviour of gases during processes involving thermal
energy interactions and exchanges fits more properly into a
study of thermodynamics. However, if only the flow mecha-
nics are considered, the thermal and temperature effects may
R
be restricted to those mainly relating to pressure and density.
The most straightforward approach is to consider zero
2L 3L thermal energy transfer (heat transfer) to or from the fluid, or
(6) - <t< -
C C adiabatic flow. If, in addition, the changes in the fluid's
properties are assumed to be reversible, then the flow be-
r- comes isentropic and the relationship between pressure and
density the simple and well-known one for an idea! gas:
_- (1.96)
- constant
PY
3L
4L
(d) -<I'< - Also, the ideal gas law:
C C
(1.97)
Figure 1.45 Progress of a pressure wave P
applies. Other useful relationships are:
Equation (1.94a) is often written as c = [C,/p]o.5, where G,
is the effective bulk nodulus of the fluid and pipe combina- 1. The ratio of specific heats y = (1.98)
tions: CV
1
1
__ - __ D (1.95) 2. The gas constant R = cp - (1.99)
- + ~((1.25 - U)
6, G Ex 3. The universal gas constant R, = R X molecular mass
If the valve is at the entrance to the pipe, then rapid valve = 8.3143 kJ kg,& K-l
closure results in a rarefaction (pressure drop) at the valve. In
other words, the pressure change is "t
ap = -pcvo (1.93')
1.5.7.4 The progress of a pressure wave
Assumiing no friction and no cavitation in the fluid, the
progress of a pressure wave aiong a pipe between a valve and a
reservoir following valve closure is as shown in Figure 1.45.
The fluid in the pipe is successively brought to rest by the
passage of the pressure wave.
At a time t = L/c after valve closure, the pressure wave
reaches the reservoir. The whole olf the fluid in the pipe is at
rest at a pressure p = po + Ap, vvhich at the reservoir end
instantaneously drops to reservoir pressure PO. The resulting
pressure wave travels along the pipe towards the valve and the
fluid at the higher pressure in the pipe flows towards the
reservoir at its initial velocity v".
At t == 2L/c, the situation is the same as for a rapid closure
of a valve downstream of the flow, producing an instantaneous
pressure drop to po - Ap and a rarefaction which travels
towards the reservoir. The passage of the rarefaction success-
ively brings the fluid to rest along the pipe. Figure 1.46 Pressure versus time at a valve