Dynamics
Application of the Bernoulli Equation
4.1
In a vertical pipe carrying water, pressure gauges are inserted at points
A and B where the pipe diameters are 0.15m and 0.075m respectively. The
point B is 2.5m below A and when the flow rate down the pipe is 0.02
cumecs, the pressure at B is 14715 N/m2 greater than that at A.
Assuming the losses in the pipe between A and B can be expressed as 
where v is the velocity at A, find the value of k.
If the gauges at A and B are replaced by tubes filled with water and
connected to a U-tube containing mercury of relative density 13.6, give a
sketch showing how the levels in the two limbs of the U-tube differ and
calculate the value of this difference in metres.
[k = 0.319, 0.0794m]
4.2
A Venturimeter with an entrance diameter of
0.3m and a throat diameter of 0.2m is used to measure the volume of gas
flowing through a pipe. The discharge coefficient of the meter is 0.96.
Assuming the specific weight of the gas to be constant at 19.62 N/m3,
calculate the volume flowing when the pressure difference between the
entrance and the throat is measured as 0.06m on a water U-tube manometer.
[0.816 m3/s]
4.3
A Venturimeter is used for measuring flow of water along a pipe. The
diameter of the Venturi throat is two fifths the diameter of the pipe. The
inlet and throat are connected by water filled tubes to a mercury U-tube
manometer. The velocity of flow along the pipe is found to be 
m/s, where H is the manometer reading in metres of mercury.
Determine the loss of head between inlet and throat of the Venturi when H
is 0.49m. (Relative density of mercury is 13.6).
[0.23m of water]
4.4
Water is discharging from a tank through a convergent-divergent
mouthpiece. The exit from the tank is rounded so that losses there may be
neglected and the minimum diameter is 0.05m.
If the head in the tank above the centre-line of the mouthpiece is 1.83m.
a) What is the discharge?
b) What must be the diameter at the exit if the absolute pressure at the
minimum area is to be 2.44m of water? c) What would the discharge be if
the divergent part of the mouth piece were removed. (Assume atmospheric
pressure is 10m of water).
[0.0752m, 0.0266m3/s, 0.0118m3/s]
4.5
A closed tank has an orifice 0.025m diameter in one of its vertical sides.
The tank contains oil to a depth of 0.61m above the centre of the orifice
and the pressure in the air space above the oil is maintained at 13780 N/m2
above atmospheric. Determine the discharge from the orifice.
(Coefficient of discharge of the orifice is 0.61, relative density of oil
is 0.9).
[0.00195 m3/s]
4.6
The discharge of a Venturimeter was found to be constant for rates of flow
exceeding a certain value. Show that for this condition the loss of head
due to friction in the convergent parts of the meter can be expressed as KQ2
m where K is a constant and Q is the rate of flow in
cumecs.
Obtain the value of K if the inlet and throat diameter of the
Venturimeter are 0.102m and 0.05m respectively and the discharge
coefficient is 0.96.
[K=1060]
4.7
A Venturimeter is to fitted in a horizontal pipe of 0.15m diameter to
measure a flow of water which may be anything up to 240m3/hour.
The pressure head at the inlet for this flow is 18m above atmospheric and
the pressure head at the throat must not be lower than 7m below
atmospheric. Between the inlet and the throat there is an estimated
frictional loss of 10% of the difference in pressure head between these
points. Calculate the minimum allowable diameter for the throat.
[0.063m]
4.8
A Venturimeter of throat diameter 0.076m is fitted in a 0.152m diameter
vertical pipe in which liquid of relative density 0.8 flows downwards.
Pressure gauges are fitted to the inlet and to the throat sections. The
throat being 0.914m below the inlet. Taking the coefficient of the meter
as 0.97 find the discharge
a) when the pressure gauges read the same b)when the inlet gauge reads
15170 N/m2 higher than the throat gauge.
[0.0192m3/s, 0.034m3/s]
Tank emptying
5.1
A reservoir is circular in plan and the sides slope at an angle of tan-1(1/5)
to the horizontal. When the reservoir is full the diameter of the water
surface is 50m. Discharge from the reservoir takes place through a pipe of
diameter 0.65m, the outlet being 4m below top water level. Determine the
time for the water level to fall 2m assuming the discharge to be 
cumecs where a is the cross sectional area of the pipe in m2
and H is the head of water above the outlet in m.
[1325 seconds]
5.2
A rectangular swimming pool is 1m deep at one end and increases uniformly
in depth to 2.6m at the other end. The pool is 8m wide and 32m long and is
emptied through an orifice of area 0.224m2, at the lowest point
in the side of the deep end. Taking Cd for the orifice
as 0.6, find, from first principles,
a) the time for the depth to fall by 1m b) the time to empty the pool
completely.
[299 second, 662 seconds]
5.3
A vertical cylindrical tank 2m diameter has, at the bottom, a 0.05m
diameter sharp edged orifice for which the discharge coefficient is 0.6.
a) If water enters the tank at a constant rate of 0.0095 cumecs find the
depth of water above the orifice when the level in the tank becomes
stable.
b) Find the time for the level to fall from 3m to 1m above the orifice
when the inflow is turned off.
c) If water now runs into the tank at 0.02 cumecs, the orifice remaining
open, find the rate of rise in water level when the level has reached a
depth of 1.7m above the orifice.
[a) 3.314m, b) 881 seconds, c) 0.252m/min]
5.4
A horizontal boiler shell (i.e. a horizontal cylinder) 2m diameter and 10m
long is half full of water. Find the time of emptying the shell through a
short vertical pipe, diameter 0.08m, attached to the bottom of the shell.
Take the coefficient of discharge to be 0.8.
[1370 seconds]
5.5
Two cylinders standing upright contain liquid and are connected by a
submerged orifice. The diameters of the cylinders are 1.75m and 1.0m and
of the orifice, 0.08m. The difference in levels of the liquid is initially
1.35m. Find how long it will take for this difference to be reduced to
0.66m if the coefficient of discharge for the orifice is 0.605. (Work from
first principles.)
[30.7 seconds]
5.6
A rectangular reservoir with vertical walls has a plan area of 60000m3.
Discharge from the reservoir take place over a rectangular weir. The flow
characteristics of the weir is Q = 0.678 H3/2 cumecs
where H is the depth of water above the weir crest. The sill of the
weir is 3.4m above the bottom of the reservoir. Starting with a depth of
water of 4m in the reservoir and no inflow, what will be the depth of
water after one hour?
[3.98m]
Notches and weirs
6.1
Deduce an expression for the discharge of water over a right-angled sharp
edged V-notch, given that the coefficient of discharge is 0.61.
A rectangular tank 16m by 6m has the same notch in one of its short
vertical sides. Determine the time taken for the head, measured from the
bottom of the notch, to fall from 15cm to 7.5cm.
[1399 seconds]
6.2
Derive an expression for the discharge over a sharp crested rectangular
weir. A sharp edged weir is to be constructed across a stream in which the
normal flow is 200 litres/sec. If the maximum flow likely to occur in the
stream is 5 times the normal flow then determine the length of weir
necessary to limit the rise in water level to 38.4cm above that for normal
flow. Cd=0.61.
[1.24m]
6.3
Show that the rate of flow across a triangular notch is given by Q=CdKH5/2
cumecs, where Cd is an experimental coefficient, K
depends on the angle of the notch, and H is the height of the
undisturbed water level above the bottom of the notch in metres. State the
reasons for the introduction of the coefficient.
Water from a tank having a surface area of 10m2 flows over a 90
notch. It is found that the time taken to lower the level from 8cm to 7cm
above the bottom of the notch is 43.5seconds. Determine the coefficient Cd
assuming that it remains constant during his period.
[0.635]
6.4
A reservoir with vertical sides has a plan area of 56000m2.
Discharge from the reservoir takes place over a rectangular weir, the flow
characteristic of which is Q=1.77BH3/2 m3/s.
At times of maximum rainfall, water flows into the reservoir at the rate
of 9m3/s. Find a) the length of weir required to discharge this
quantity if head must not exceed 0.6m; b) the time necessary for the head
to drop from 60cm to 30cm if the inflow suddenly stops.
[10.94m, 3093seconds]
6.5
Develop a formula for the discharge over a 90 V-notch weir in terms of
head above the bottom of the V.
A channel conveys 300 litres/sec of water. At the outlet end there is a 90
V-notch weir for which the coefficient of discharge is 0.58. At what
distance above the bottom of the channel should the weir be placed in
order to make the depth in the channel 1.30m? With the weir in this
position what is the depth of water in the channel when the flow is 200
litres/sec?
[0.755m, 1.218m]
6.6
Show that the quantity of water flowing across a triangular V-notch of
angle 2q is 
. Find the
flow if the measured head above the bottom of the V is 38cm, when q=45 and
Cd=0.6. If the flow is wanted within an accuracy of 2%,
what are the limiting values of the head.
[0.126m3/s, 0.377m, 0.383m]
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