ENERGY EQUATION
The Bernoulli's equation is composed of Kinetic Energy, Potential Energy, and Internal Energy (which includes thermal, nuclear, chemical and electrostatic which we will not be discussing)
Bernoulli's Theorem (constant along a
streamline):
where p/g, is the pressure head, which
is energy per unit weight; z is potential head, which distance from an assigned
datum v2/2g is velocity head, which is kinetic energy per unit
weight.
When using the Bernoulli's equations it is important to keep these guidelines in mind:
1.
It assumes viscous (friction) effects are negligible;
2.
It assumes the flow is steady;
3.
The equation applies along a streamline;
4.
It assumes the fluid to be incompressible; and
5.
It assumes no energy is added to or removed from the fluid along the
streamline.
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Example 1:
Water
flows through a water treatment plant. Find
the pressure at the second point using the given information.
Neglect head loss.
Set your datum at (1)
P2=263.9kPa
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FRICTION
and Bernoulli's Equation:
Bernoulli's equation take into account friction losses within the system or friction head loss (energy loss due to friction such as water rubbing against the pipe). It also takes into account entry losses (due to suction) or loss due to the diameter change within the pipe system.
Wall
friction Head loss: l,
shear force against wall, boundary or surface of the element of area
of PL, A is the cross sectional area of pipe.
With
friction taken into account the Bernoulli's equation is as shown:
Fluid friction loss from any such cause, including wall or pipe friction, we commonly refer to as head loss (hf )
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Water flows through a water treatment plant. Find the head loss due to friction using the given information.
Set your datum at (1)
hf= 2.42m
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The
Bernoulli's equation can also incorporate energy losses or gains from other
sources such as machines, ( hm ). If the machine is a
pump, which adds energy to the fluid, hm is positive; if the
machine is a turbine, which removes energy from the fluid, hm
is negative.
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Example
3
Water is pumped through a water treatment plant at a rate of 2.5n*m/m. Find the pressure at the first point using the given information. Head loss due to friction is –0.00052m.
Set your datum at (1)
P1= 299.2kPa
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Note: The follow are more types of head losses, which will not be explained in too much detail.
Other Head Losses
A more technical type of head loss
due to friction is in a circular pipe, flowing full (laminar or turbulent flow)
for equation (1): f=friction factor; L=length of pipe; D=diameter of pipe.
For equation (2): R=Reynolds's number; D=diameter of pipe; V=velocity of fluid
through pipe: u=kinematic
viscosity.
(1)
(2)
Equations (1) and (2) are commonly used simultaneously with the Moody Diagram (shown below).
More Head Losses
k = loss coefficient which is unit less and is used in the following cases:
Head loss
due to entrance:
Head loss due to :
submerged
discharge into still water: 
submerged
discharge into moving water:
sudden
contraction:
expansion:
References:
Fluid Mechanics with Engineering Applications, E. John Finnemore and Joseph B. Franzini
Essentials of Engineering Fluid Mechanics, Reuben M. Olson and Steven J. Wright
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