---
author: Akbar Rahman
date: \today
title: MMME2047 // Turbomachinery
tags: [ turbomachinery ]
uuid: 11f0f745-2364-4594-8e47-127a4af39417
lecture_slides: [ ./lecture_slides/T5 - Turbomachinery - with solutions.pdf ]
lecture_notes: [ ./lecture_notes/turbomachinery lecture notes(H Power).pdf ]
exercise_sheets: [ ./exercise_sheets/Turbomachinery-problems.pdf]
---

Turbomachinery are rotating devices that add (pump for liquids; fan, blower, or compressor for gases at <0.02, <1 bar, and > 1 bar respectively) or extract (turbine) energy from a fluid.

# Positive Displacement (PD) Pumps

- PD pumps force fluid along using volume changes (e.g. bike pumps, the heart)
- All PD pumps deliver a periodic flow
- They deliver any fluid regardless of viscosity (dynamic pumps struggle with viscous fluids)
- They are self priming (will be filled automatically)
- They can operate under high pressures (300 atm) but low flow rates (25 m$^3$h$^{-1}$)
- flow rate can be only be changed by vary speed or displacement

# Dynamics Pumps

- add momentum to fluid by fast moving blades or vanes
- classified based on direction of flow at exit:

    - centrifugal
    - axial
    - mixed flow

- fluid increases momentum while moving through open passages and extra velocity is converted to
  pressure through exiting it into a diffuser section
- provide high flow rates (up to 70000 m$^3$h$^{-1}$) but usually at moderate pressure rises (a few atm)
- require priming

# Centrifugal Pumps

- fluid enters through eye of casing and gets caught in impeller blades
- fluid is whirled outwards until it leaves via the expanding area section, known as the diffuser or volute

## Blades

- backward inclined blades - most common and efficient, intermediate pressure rise, less robust
- straight blades - simplest geometry, high pressure rise, less robust
- forward inclined blades - more blades but smaller, lowest pressure rise, lowest efficiency, more robust


## Integral Analysis of Centrifugal Pumps

Derivation in slides (p. 23-25).

\begin{align}
\frac{w_s}{g} - \left(\frac{u_2-_1-q}{g}\right) = \left(\frac{p_2}{\rho g} + z_2 + \frac{v_2^2}{2g}\right) -\left(\frac{p_1}{\rho g} + z_1 + \frac{v_1^2}{2g}\right) \\
H_s - H_f &= H = H_{T,2} - H_{T,1} \nonumber
\end{align}

where $H_s$ is supplied head to pump, $H_f$ friction loss head, $H$ is head supplied to fluid, $H_{T,1}$ is total head at inlet, and $H_{T,2}$ is total head at outlet.

Assuming that $z_1 \approx z_2$, $v_1 \approx v_2$ (from inlet and outlet diameters are equal) then:

\begin{equation}
H \approx \frac{p_2-p_1}{\rho g}
\end{equation}

and the power to the fluid (water horsepower) is:

\begin{equation}
P_w = \rho QgH
\end{equation}

where $Q$ is volumetric flow rate.

Power supplied to the pump (brake horsepower), $P = \omega T$, lets us find the overall pump efficiency:

\begin{equation}
\eta = \frac{P_w}{P} = \frac{\rho QgH}{\omega T} = \eta_h \eta_m \eta_v
\end{equation}

where:

- $\eta_h = 1 - \frac{H_f}{H_s}$ is hydraulic efficiency
- $\eta_m = 1- \frac{P_f}{P}$ is mechanical efficiency
- $\eta_v = \frac{Q}{Q+Q_L}$ (where $Q_L$ is loss due to leakage flow) is the volumetric efficiency

## Performance

![](./images/vimscrot-2023-03-13T10:07:58,750255090+00:00.png)

# Cavitation

Cavitation is when bubbles form in liquid by sudden pressure drop, followed by their implosion when
original pressure is restored.
The implosion generates a high pressure wave that can damage nearby solid surfaces.

In a centrifugal pump, the fluid pressure drops at the impeller's eye, where it has the minimum
value.
If pressure falls below saturation pressure, bubbles appear.
Pressure grows as the fluid flows between the blades as the ducts are diverging.
Pressure is maximum at the trailing edge of the blades, on their front side.
This is where cavitation occurs and causes wear on the blade.

<https://www.michael-smith-engineers.co.uk/resources/useful-info/pump-cavitation>  
<https://www.youtube.com/watch?v=g1o5Z9o7b0>  
<https://www.youtube.com/watch?v=eMDAw0TXvUo>  
<https://www.youtube.com/watch?v=1Lbxtjfdat4>

![](./images/cavitation.png)

# Net Positive Suction Head (NPSH)

The following conditions must be satisfied to prevent cavitation:

\begin{equation}
H_i - \frac{p_v}{\rho g} > \text{NSPSH}
\end{equation}

where $H_i = \frac{p_i}{\rho g} + \frac{v_i^2}{2g}$ is total head at inlet, $p_v$ is saturation
pressure at $T_i$.

It is important that the inlet pressure is as high as possible.
To do this, one can reduce frictional losses (e.g. shorter smoother pipes) or install the pump lower down
(even below the reservoir) (slides p. 36).