During a laboratory test on a water pump appreciable cavitation began when the pressure plus velocity head at inlet was reduced to \(3.26 \mathrm{~m}\) while the total head change across the pump was \(36.5 \mathrm{~m}\) and the discharge was \(48 \mathrm{~L} \cdot \mathrm{s}^{-1} .\) Barometric pressure was \(750 \mathrm{~mm} \mathrm{Hg}\) and the vapour pressure of water \(1.8 \mathrm{kPa}\). What is the value of \(\sigma_{\mathrm{c}}\) ? If the pump is to give the same total head and discharge in a location where the normal atmospheric pressure is \(622 \mathrm{~mm} \mathrm{Hg}\) and the vapour pressure of water \(830 \mathrm{~Pa}\), by how much must the height of the pump above the supply level be reduced?

Net Positive Suction Head (NPSH) is a critical concept when considering pump operation, particularly to avoid cavitation. NPSH is a measure of the absolute pressure at the suction port of the pump and is given by the equation: \( \text{NPSH} = h_{atm} + h_{s} - h_{v} - h_{f} \). In this formula: - \( h_{atm} \): Ambient pressure head (converted from barometric pressure).
- \( h_{s} \): Suction head.
- \( h_{v} \): Vapor pressure head of the liquid being pumped.
- \( h_{f} \): Friction head loss, often considered negligible in many cases.
A higher NPSH ensures that the pump can avoid cavitation, which can cause serious damage. To calculate NPSH, each pressure must often be converted to consistent units such as meters of liquid, which can involve converting atmospheric pressures from units like mm Hg and vapor pressures from kPa or Pa.

The cavitation number, represented by \( \sigma \), is another key concept in understanding pump performance. It is a dimensionless number that describes the likelihood of cavitation occurring within the pump. Mathematically, it can be expressed as: \( \sigma = \frac{NPSH}{\text{Total head}} \). Here, NPSH is the Net Positive Suction Head discussed earlier, and the total head is the height increase provided by the pump. Cavitation number helps engineers assess whether a pump can operate safely without cavitation. Lower values of \( \sigma \) indicate a higher risk of cavitation, whereas higher values suggest safer operating conditions. Proper calculation and consideration of this number ensure the longevity and reliability of the pump system.

A vital step in many pump and fluid dynamics calculations is converting barometric pressure into a compatible form, such as meters of water head. This conversion is essential when calculating NPSH. The barometric pressure is often provided in units like mm Hg (millimeters of mercury), which need to be converted using the relationship: \( 1 \text{ mm Hg} = 0.0136 \text{ m of water head} \). For example, a barometric pressure of 750 mm Hg can be converted as follows: \( 750 \times 0.0136 = 10.2 \text{ m} \). This value can then be incorporated into the NPSH formula. Appropriate conversion ensures accurate and meaningful results in your calculations, aiding in the proper design and operation of pump systems.

Vapor pressure is the pressure at which a liquid begins to vaporize and is critical in avoidance of cavitation in pumps. When vapor pressure is reached, the liquid can form vapor bubbles, leading to cavitation. To include vapor pressure in NPSH calculations, it must often be converted to the same units as the other pressures, typically meters of liquid. This can be done using the conversion: \( \text{head} = \frac{ \text{Vapor Pressure (Pa)} }{ 9810 } \).
For instance, a water vapor pressure of 1.8 kPa (or 1800 Pa) can be converted as: \( \frac{1800}{9810} = 0.18 \text{ m} \). Such conversions ensure vapor pressure is accurately accounted for in NPSH calculations, helping to prevent cavitation by ensuring the liquid pressure does not drop below this point.

Adjusting the height of a pump relative to the water supply level can directly influence cavitation occurrence. Proper height adjustment ensures that the NPSH is sufficient under varying operational conditions. For instance, in the example problem, when the atmospheric pressure and vapor pressure conditions at a new location are different, the required suction head (\( h_{s} \)) changes. Calculating the new NPSH: \( \text{NPSH}_{new} = h_{atm} + h_{s} - h_{v} \), and solving for the correct \( h_{s} \), helps determine by how much the pump’s height above the supply level must be adjusted. Proper calculations ensure operational efficiency and extend the pump’s lifespan by preventing cavitation under altered environmental conditions.