Page 101 - Mechanical Engineers' Handbook (Volume 4)

90 Fluid Mechanics

p
2 (k 1) / k 1
0
M
k 1 p
For supersonic flow the stagnation pressure p 0y is downstream of a shock, which is
detached and ahead of the open stagnation tube, and the static pressure p is upstream of
x
the shock. In a wind tunnel the static pressure could be measured with a pressure tap in the
tunnel wall. The Mach number M of the flow is
k /(k 1)
p 0y k 1 M 2k M 1 / (1 k)
k 1
2
2
p 2 k 1 k 1
which is tabulated in gas tables.
In a mixture of gas bubbles and a liquid for gas concentrations C no more than 0.6 by
volume, the velocity of the mixture with the pitot tube and manometer free of bubbles is
2(p p ) 2gh m m 1
1
0
V mixture (1 C) (1 C)
liquid liquid
where h is the manometer deflection in meters for a manometer liquid of specific weight
m
. The error in this equation from neglecting compressible effects for the gas bubbles is
m
shown in Fig. 39. A more correct equation based on the gas–liquid mixture reaching a
stagnation pressure isentropically is
V 2 1 p p 1 C (k 1) / k 1
p
p
p
k
0
0
1
0

2 (1 C) 1 C k 1 p k 1 p
u u 1 1
but is cumbersome to use. As indicated in Fig. 39 the error in using the first equation is
very small for high concentrations of gas bubbles at low speeds and for low concentrations
at high speeds.
If n velocity readings are taken at the centroid of n subareas in a duct, the average
velocity V from the point velocity readings u is
i
V u
n
1
n i 1 i
In a circular duct, readings should be taken at (r/R) 0.055, 0.15, 0.25,..., 0.95.
2
2
Velocities measured at other radial positions may be plotted versus (r/R) , and the area under
the curve may be integrated numerically to obtain the average velocity.

Figure 39 Error in neglecting compressibility of air in measuring velocity of air–water mixture with
a combined pitot tube.