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 C_d 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 H^3/2 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. C_d=0.61.
[1.24m]

6.3
Show that the rate of flow across a triangular notch is given by Q=C_dKH^5/2 cumecs, where C_d 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 C_d 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=1.77BH^3/2 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 C_d=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]