**In this article:**

- we will discuss about how to derive the Bernoulli’s Equation from the energy equations.
- we will also look at how the Bernoulli’s Equation can be used to compute for the Total Dynamic Head,
- we will use the TDH to get the required pump brake power at a given pumping installation.

**Energy Equations:**

In physics, we learned about potential energy and kinetic energy.

PE = m x g x h

*where*: PE: potential energy,

m: Â mass,

g: acceleration due to gravity, and

h: height

KE = 1/2 m(v^{2})

*where*: KE: kinetic energy

m: mass, and

v: velocity.

**Work**

Also, we learned in physics that work done is equal to the force multiplied by its displacement.

W = Fx

*where*: W: work

F: force, and

x: displacement.

Since pressure equals force per unit area, then F = PA. Substituting this in the work formula, we come up with W = PAx. We can say that area multiplied by displacement is the volume, therefore W = PV.

**Work-Energy Theorem**

The law of conservation of energy states that the total energy remains constant in an isolated system. Energy is neither created nor destroyed, but can be transformed from one form to another. Thus, we can apply the work-energy theorem wherein the change in work is equal to the change in energy.

W_{1} – W_{2} = (PE_{2} – PE_{1}) + (KE_{2} – KE_{1})

P_{1}V_{1} – P_{2}V_{2} = mgh_{2} – mgh_{1} + 1/2 mv_{2}^{2} â€“ 1/2 mv_{1}^{2}

Dividing by mass and acceleration due to gravity, we can simplify the equation.

Since density is mass per unit volume, and specific weight is density multiplied by gravity, we come up with the Bernoulli’s Equation.

When using Bernoulli’s Equation, we should consider the following assumptions:

- The fluid is incompressible, thus the density is constant (e.g. water)
- The fluid friction is negligible

**Total Dynamic Head (TDH)**

Total dynamic head is the total amount of head that a fluid needs to be pumped in a given installation, taking into account the friction losses.

This is the equation for total dynamic head which is derived from the Bernoulli’s Equation, where H is the total dynamic head and hf are head losses due to friction. The subscripts s & d stand for suction and discharge.

The head losses due to friction are additive for both suction and discharge. Note that the elevation head is also called static head. When the suction tank is below the elevation of the pump, then the elevation head is called static lift and has a negative value.

**Darcy-Weisbach Equation**

The friction head is computed using the Darcy-Weisbach Equation.

h_{f} = fLv^{2 }/ 2gD

*where*: f = coefficient of friction from the Moody Diagram

L = total length of pipe including equivalent length of fittings

D = inside diameter of pipe

(Note that the Moody Diagram and equivalent length of fittings are not discussed here.)

**Hydraulic Power (or Water Power)**

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The hydraulic power can then be computed using the flow rate, specific weight, and total dynamic head.

Pw = QgH

*where*: Pw = hydraulic power

Q = flow rate

**Brake Power**

**Â **We can now determine the pump brake power. Brake power is the input power of the pump which is computed using pump efficiency.

*Photo credit to SupakitmodÂ of freedigitalphotos.net*

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