Page 38 - Mechanical Engineers Reference Book

Mechanics of fluids 1/27
integration. This may be approximated to by dividing the cross For a rectangular notch of width B:
section into a series of concentric annuli of equal thickness,
measuring the velocity at the middle of each annulus, mul- V = 2.953CdBH'.' (1.64.)
tiplying by the corresponding annulus area and adding to give The empirical Francis formula may be applied to sharp-
the total flow rate. edged weirs and rectangular notches:
Current meters, torpedo-shaped devices with a propeller at
the rear, may be inserted into pipes. The number of rotations V = 2.953Cd(B - 0.2H) el5 - id)'I (1.65)
of the propeller are counted electrically. This number together
with coefficients peculiar to the propeller are used in empirical
equations to determine the velocity. These meters are more or neglecting the approach velocity:
often u:;ed in open channels (see Section 1.5.4.2). V = 2.953C,j(B - 0.2H)H" (1.66)
Velorneters, vaned anemometers and hot wire anemometers
are not usually used to measure the velocities of in- For a venturi flume of throat width B, or a broad-crested weir
compressible fluids in pipes, and will be discussed in Section of width B. the same equation applies:
1.5.8. V = 1.705CdB el5 (1.67)
Since the value of e depends on the approach velocity v,
which in turn depends on the flow rate V, equations (1.65)
1.5.4.2 Open-channel flow and (1.67) are usually solved by an iterative method in which
(a) Velocity meters In channels of regular or irregular cross the first estimation of the approach velocity v is zero. Success-
section the flow may be measured using the velocity meters ive values of v are found from the upstream flow cross-
described in Section 1.5.4.1(d) (current meters are often used sectional area and the preceding value of $'. the resulting
in rivers or large channels). For this method the cross section value of e is then used in equation (1.67) for p. This is
is divided into relatively small regular areas, over which the repeated until there is little change in the required values. The
velocity is assumed to be constant. The velocity meter is then discharge coefficient Cd in each of the flow equations (1.62) to
placed at the centre of each small area, and from the velocity (1.67) has a value of about 0.62.
and area the flow rate may be calculated. Adding together the As before, it is much more accurate to calibrate the device.
flow rates for all the small areas gives the flow rate for the For convenience, the calibration curves often plot the flow
channel. rate against the upstream depth.
It should be noted that in open channels the velocity varies
with depth as well as with distance from the channel walls. (c) Floats In large rivers, where it is incoilvenient to install
Selection of the shape and location of the small areas need to flumes or weirs, or to use velocity meters, floats may be used.
take this into account. The timing of the passage of the floats over a measured
distance will give an indication of the velocity. From the
(b) Notches, flumes and weirs As in pipe flow, flow rates in velocity, and as accurate a value of cross-sectional area as
channels may be related to changes in head produced by possible, the flow can be estimated.
obstructions to the flow. These obstructions may be in the
form of notches, flumes or weirs and change in head observed (d) Chemical dilution In large, fast-flowing rivers chemical
as a ch:mge in depth of fluid. Notches may be rectangular, dilution may be the only acceptable method of flow measure-
V-shaped, trapezoidal or semi-circular. Weirs may be sharp- ment. The water is chemically analysed just upstream of the
edged or broad-crested. Flumes are similar to venturis, with a injection point and the natural concentration C1 of the se-
controlled decrease in width to a throat followed by a gradual lected chemical in the water established. The concentration of
increase to full channel wdith. They are often known as the chemical injected is C, and the injection rate is R,. Analysis
venturi flumes. For most of these devices there is a simplified of the water again at some distance downstream of the
relationship between the flow rate T/ and the upstream specific injection point determines the new concentration C, of the
energy e: chemical. The flow rate V along the river may be estimated
from
V=Ke" (1.60)
(: :
I/=&----- ::) (1.68)
where K is a coefficient which may be constant for a particular
type of device (and for a specific device). The index n is
approxiimately 1.5 for rectangular notches, weirs and flumes: 1.5.5 Open-channel flow
and 2.5 for V-notches. The specific energy e is the sum of the
depth and the velocity head: An open channel in this context is one containing a liquid with
a free surface, even though the channel (or other duct) may or
may not be closed. A pipe which is not flowing full is treated as
(1.61) an open channel.
In many applications, particularly at the exit of large tanks 1.5.5.1 Normal flow
or reservoirs, the upstream (or approach) velocity may be Normal flow is steady flow at constant depth along the
negligiblle and e becomes equal to either the depth D or the channel. It is not often found in practice, but is widely used in
head above the base of the notch or weir H. the design of channel invert (cross section) proportions.
For a V-notch of included angle 28:
= 2.36Cd(tan 0)H' (1.62) (a) Flow velocity The average velocity, v. of flow in a
channel may be found by using a modified form of the D'Arcy
For a 90" notch: head loss equation for pipes, known as the Chezy equation:
$' = 2.36C,H2' (1.63) v = C(mi)'.' (1.69)