Show how Bernoulli’s energy equation for a fluid can be derived by
considering how the work done by the fluid moving from point 1 to
point 2 is equal to the energy gained or lost by the fluid. You can
assume the potential energy PE=mgh and the kinetic energy
KE=0.5mv
2
, where the terms have their usual meanings.
To derive Bernoulli's energy equation for a fluid, we need to consider the work done by the fluid moving from point 1 to point 2, and equate it to the energy gained or lost by the fluid.
First, let's consider the work done by the fluid. Work is defined as the force applied over a distance:
Work = Force x Distance
In the case of a fluid, the force can be represented as the pressure multiplied by the area:
Force = Pressure x Area
The distance in this case is the displacement of the fluid from point 1 to point 2.
Now let's look at the energy gained or lost by the fluid. The fluid can have two forms of energy: potential energy (PE) and kinetic energy (KE).
The potential energy of the fluid at a certain height h is given by:
PE = mgh
where m is the mass of the fluid and g is the acceleration due to gravity.
The kinetic energy of the fluid with a velocity v is given by:
KE = 0.5mv^2
Next, we need to consider the conservation of energy principle, which states that the total energy of a system remains constant if there are no external forces acting on it.
At point 1, the fluid has potential energy (PE1) and kinetic energy (KE1), and at point 2, it has potential energy (PE2) and kinetic energy (KE2).
Applying the conservation of energy principle, we have:
PE1 + KE1 + Work = PE2 + KE2
Substituting the expressions for potential energy and kinetic energy, we get:
mgh1 + 0.5mv1^2 + Work = mgh2 + 0.5mv2^2
Now, let's substitute the expression for work in terms of pressure and area:
Pressure1 x Area1 x Distance + 0.5mv1^2 = Pressure2 x Area2 x Distance + 0.5mv2^2
Dividing both sides of the equation by the mass, we get:
gh1 + 0.5v1^2 + (Pressure1 / ρ) = gh2 + 0.5v2^2 + (Pressure2 / ρ)
Here, ρ represents the density of the fluid.
Finally, rearranging the terms and grouping them, we arrive at Bernoulli's energy equation for a fluid:
P + 0.5ρv^2 + ρgh = constant
where P represents the pressure, ρ represents the density, v represents the velocity, and h represents the height.