# Fundamentals of Turbomachinery

Turbomachinery devices inject life into fluids. Principles of turbomachinery form the preliminary design tools in design of them.

## Turbomachinery principles

Consider following turbo machines. An axial turbine, a centrifugal machine or a pelton wheel, you can predict performance of all of these from same turbo machinery fundamentals.

## Euler Turbomachinery equation

To develop turbo machinery fundamentals consider fluid flow through channel shown below. The inlet velocity, V1 gets changed to outlet velocity V2. Velocity of fluid can be split into tangential and radial components. This is shown in following figure.

To make the fluid flow there should be an external torque acting on it. This torque can be derived from Newton’s 2nd law of motion, which acts as fundamental equation of turbo machinery. The torque is given by following equation, which is also called as Euler turbomachine equation.

Torque = m(r2Vθ2 − r1 Vθ1)

This is the most important equation in turbo machinery.

## Pump or Turbine?

If the channel is rotating at an angular velocity omega, power required to maintain the fluid flow will be torque multiplied by angular velocity.

Torque × ω = m(r2Vθ2 − r1Vθ1) × ω

ω times radius becomes channel velocity or blade velocity. So power required for this fluid motion can be taken as difference in product of blade velocity times tangential fluid velocity.

Powerreq = m(U2Vθ2 − U1Vθ1)

Vθ is positive, if it is in same direction of blade velocity. Otherwise it is negative.

If we divide power by weight of fluid flowing, we will get what’s the energy head required to maintain this flow.

If power required by the fluid is positive, that means fluid is absorbing energy. Or the device is acting as a compressor. Otherwise fluid is losing energy, so the device acts as a turbine.
Vθ more precisely means, component of fluid velocity which is parallel to blade velocity.

But determination of fluid velocity is a tricky affair in turbo machinery, since we are dealing with rotating components. For this purpose we should understand concept of velocity triangles.

## Concept of relative velocity and velocity triangle

The key idea in turbo machinery is concept of relative velocity. Suppose you are standing on this rotating turbo machine. The velocity of fluid you experience while moving with it is called as relative velocity.
If fluid is having an absolute velocity V, and the blade is moving with a velocity U, then relative velocity experienced by you will be as follows.

For a stationary device in order to have smooth operation, flow should be tangential to the blade. Similarly in a moving device relative velocity should be tangential to blade profile. With knowledge of direction of relative velocity and the vectorial representation of relative velocity, these 3 velocities could be drawn as shown below. This is known as a velocity triangle.

Similar velocity triangle can be made on inlet of turbo machine. The beauty of turbo machinery is that using relatively simple analysis of inlet and outlet velocities you can predict performance of any turbo machine.

## Performance of Centrifugal pump

We will see, how to predict performance of a centrifugal pump using the concepts we developed. Here we have shown impeller of a centrifugal pump. If you know the blade geometry, you can find out blade angle at inlet and outlet. Blade angle is defined as angle opposite of blade velocity. So we can easily fix direction of relative velocity. This is shown in Fig. 6(b).Radial component of flow velocity determines how much the volume flow rate is leaving the impeller. So you can determine Vr at outlet from this equation. Here b2 means width of the impeller.

Now construction of velocity triangle is easy. You can draw lines parallel to these velocity components. These lines are drawn in grey color in Fig. 6(b). From parallelogram constructed in Fig. 6(b)absolute velocity of flow can be easily drawn. This is shown in Fig. 6(c). From this we can find out tangential component of flow velocity, this is marked as Vθ2 in Fig. 6(c).

At inlet of centrifugal pump flow velocity will be radial. So tangential component of velocity is zero.

#### V01=0

So energy head developed by the pump simplifies like this.

#### h = 1 ∕ g U2V02

From Fig.6(c) outlet blade angle can be easily represented as follows.

You can substitute values of Vθ2 from this equation, into head equation. After substituting value of Vr2 also in that, we get most important performance equation of centrifugal pump. How energy head is varied with flow rate.