2. Equation of Continuity
If an incompressible liquid is continuously flowing through a pipe or channel,
the quantity of liquid passing per second is the same at all sections. This is
known as the equation of continuity.
The continuity equation is used to relate a fluid’s velocity to a change in the
pipe’s cross-sectional area.
Q= A1V1=A2V2
Where:
Q = the volumetric flow rate
A = the cross sectional area of flow
V = the mean velocity
5. Applications
Equation of Continuity has a vast usage in the field of Hydrodynamics,
Aerodynamics, Electromagnetism, Quantum Mechanics. As it is the fundamental rule
of Bernoulli’s Principle, it is indirectly involved in Aerodynamics principle and
applications.
Apart from this, to check the consistency of Maxwell’s Equation, we also use the
differential form of continuity equation in Electromagnetism.
Also, to check the consistency of Schrodinger Equation, we also use the continuity
equation.
Other applications are General and Special Theory of Relativity, Noether’s Theorem.
7. Bernoulli's Theorem
It states, “For a perfect incompressible liquid, flowing in a continuous
stream, the total energy of a particle remains the same, while the particle
moves from one point to another”. This statement is based on the assumption
that there are no losses due to friction in the pipe.
Z + v2/2g + p/w
10. Energy of Liquid in Motion
Potential Energy
Kinetic Energy
Pressure Energy
11. Potential Energy of a Liquid Particles
It is energy possessed by a liquid particle by virtue of its position. If a liquid
particle is Z meters above the horizontal datum, the potential energy of the
particle will be Z meter-kilogram per kg of the liquid. The potential head of
the liquid, at that point, will be Z meters of the liquid.
P.E= mgh
Or
P.E= WZ in meters
12. Kinetic Energy of a Liquid Particle
It is the energy possessed by a liquid particle by virtue of its motion or
velocity. If a liquid particle is flowing with a mean velocity of “V” meters per
second, then kinetic energy of the particle will by V2/2g meter kg/kg of the
liquid, velocity head of the liquid at that velocity will be V2/2g meter of the
liquid.
K.E= ½ mV2
13. Pressure Energy of a Liquid Particle
It is the energy, possessed by a liquid particle, by virtue of its existing
pressure. If a liquid particle is under a pressure of P KN/m2 (i.e. KPa) then
the pressure energy of the particle will be P/w m-kg/kg of the liquid, where
“w” is the specific weight of liquid.
The total energy of a liquid, in motion, is the sum of its potential energy,
kinetic energy and pressure energy,
E = Z + v2/2g + p/w m of liquid
Total head of a liquid particle:
H = Z + v2/2g + p/w m of liquid
14. Steady Flow
A flow in which the quantity of liquid flowing per second is constant.
15. Velocity & Acceleration in Steady Flow
It should be noted, steady flow does not mean the velocity and accelerations
are constant. Flow in a curved pipe or through a nozzle may be steady, but
the velocity or acceleration is not constant. Also the velocity at any point in
the field may change with time.
a = V ∂V/ ∂s
16. Unsteady Flow
A flow in which the quantity of liquid flowing per second is not constant.
17. Velocity & Acceleration in Unsteady Flow
Unsteady flows can be further divided into periodic flow, nonperiodic flow
and random flow. The graphical representations of these flows are given in
the figure.
a = V ∂V/ ∂s + ∂V/ ∂t
18. Limitation of Flow Net
Can't be applied in the region close to the boundry.
Can't be applied to sharply converging flow.
Can't describe wakes.
19. Assumptions
The fluid is non viscous (i.e., the frictional losses are zero.
The fluid in homogeneous and incompressible.
The flow is continuous, steady and along the streamline.
The velocity of the flow is uniform over the section.
No energy force (except gravity and pressure forces) is involved in the flow.
20. Uses of Flow Net
Determination of quantity of seepage
Determination of hydrostatic pressure.
Determination of seepage pressure.
Determination of exit gradient.
Determination of uplift pressure.
