Total energy of a liquid
A liquid in motion possesses pressure energy, kinetic energy and potential energy.
(i) Pressure energy
It is the energy possessed by a liquid by virtue of its pressure.
Consider a liquid of density ρ contained in a wide tank T having a side tube near the bottom of the tank as shown in Fig. . A frictionless piston of cross sectional area 'a' is fitted to the side tube. Pressure exerted by the liquid on the piston is P = h ρ g where h is the height of liquid column above the axis of the side tube.
If x is the distance through which the piston is pushed inwards, then
Volume of liquid pushed into the tank = ax
∴ Mass of the liquid pushed into the tank = ax ρ
As the tank is wide enough and a very small amount of liquid is pushed inside the tank, the height h and hence the pressure P may be considered as constant.
Work done in pushing the piston through the distance x = Force on the piston × distance moved
(i.e) W = Pax
This work done is the pressure energy of the liquid of mass axρ.
∴ Pressure energy per unit mass of the liquid = Pax /axp = P/p
(ii) Kinetic energy
It is the energy possessed by a liquid by virtue of its motion.
If m is the mass of the liquid moving with a velocity v, the kinetic energy of the liquid = ½ mv2
Kinetic energy per unit mass = 1/2mv2 / m = v2 / 2
(iii) Potential energy
It is the energy possessed by a liquid by virtue of its height above the ground level.
If m is the mass of the liquid at a height h from the ground level, the potential energy of the liquid = mgh
Potential energy per unit mass = mgh/ m = gh
Total energy of the liquid in motion = pressure energy + kinetic energy + potential energy.
Total energy per unit mass of the flowing liquid = P/p + v2/2 + gh
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