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Tests on a pump under standard atmospheric conditions show that when water at \(20^{\circ} \mathrm{C}\) is pumped at \(60 \mathrm{~L} / \mathrm{s}\) and the head added by the pump is \(40 \mathrm{~m},\) cavitation occurs when the pressure head plus velocity head on the suction side of the pump is \(3.9 \mathrm{~m}\). (a) Determine the required net positive suction head and the cavitation number of the pump. (b) If this same pump is operated on a mountain under the same flow rate and added head condition but the temperature of the water is \(5^{\circ} \mathrm{C}\) and the atmospheric pressure is \(90 \mathrm{kPa}\), by how much must the elevation of the pump above the sump reservoir be reduced compared with the test condition? Assume that the friction loss in the suction pipe remains approximately the same and that the sump reservoir is open to the atmosphere in both cases.

Step by step solution

01

Calculate the Vapor Pressure Head for water at 20°C

Lookup the vapor pressure of water at 20°C, which is approximately 2.34 kPa, and use the equation for converting pressure to pressure head: \[ h_v = \frac{P_v}{\rho g} = \frac{2.34 \times 10^3}{1000 \times 9.81} \approx 0.238 \text{ m} \]

02

Calculate the Required Net Positive Suction Head (NPSH)

Using the formula for NPSH: \[ NPSH = H_s + H_v \] where \(H_s = 3.9 \text{ m}\) (given) and \(H_v = 0.238 \text{ m}\) from Step 1. Thus, \[ NPSH = 3.9 + 0.238 = 4.138 \text{ m} \]

03

Calculate the Cavitation Number (σ)

The cavitation number is given by the formula: \[ \sigma = \frac{P_a - P_v}{\rho g H} \] where \(P_a = 101.3 \text{ kPa}\) (standard atmospheric pressure), \(P_v = 2.34 \text{ kPa}\) (vapor pressure at 20°C), \(H = 40 \text{ m}\). Substitute these values to find \(\sigma\): \[ \sigma = \frac{(101.3 - 2.34) \times 10^3}{1000 \times 9.81 \times 40} \approx 0.253 \]

04

Calculate the Vapor Pressure Head for water at 5°C

Lookup the vapor pressure of water at 5°C, which is approximately 0.87 kPa, and convert to pressure head: \[ h_v' = \frac{0.87 \times 10^3}{1000 \times 9.81} \approx 0.089 \text{ m} \]

05

Determine New NPSH Required on the Mountain

Process the new conditions on the mountain: \[ h_s' = NPSH - h_v' = 4.138 - 0.089 = 4.049 \text{ m} \]

06

Calculate Required Atmospheric Head on the Mountain

Since the atmospheric pressure is 90 kPa on the mountain, determine the equivalent head: \[ P_a' = 90 \text{ kPa}, \quad h_a' = \frac{90 \times 10^3}{1000 \times 9.81} \approx 9.17 \text{ m} \]

07

Calculate Required Height Reduction

Use the NPSH equation to find the reduction in elevation: \[ h_s' = h_a' - h_f - h_v', \quad \text{where } h_f \text{ is the friction loss head (assumed constant)}. \] If \(h_s' < NPSH\), find \(\Delta h = NPSH - h_s' \approx 0.14 \text{ m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cavitation

Cavitation occurs when the pressure in a liquid falls below its vapor pressure, causing the liquid to form vapor bubbles. These bubbles can collapse violently, leading to damage in pumps and other hydraulic equipment.
Cavitation is undesirable as it reduces pump performance and efficiency. It can cause vibrations and noise, leading to mechanical damage. To predict and prevent cavitation, it’s essential to calculate the Net Positive Suction Head (NPSH). When liquid is drawn into the pump, maintaining the appropriate pressure limits bubble formation inside the pump.
To avoid cavitation, ensure the pump's suction head is correctly maintained above the vapor pressure of the liquid. This prevents the formation of vapor bubbles and potential equipment damage.

Net Positive Suction Head (NPSH)

Net Positive Suction Head (NPSH) is a critical concept in ensuring a pump functions without cavitation. It represents the absolute pressure above the vapor pressure at the pump's suction. There are two main types:

• NPSH Required (NPSHr): This is the minimum pressure required at the pump's suction to avoid cavitation. It's provided by the manufacturer.
• NPSH Available (NPSHa): This is the actual pressure available at the pump's suction. NPSHa should exceed NPSHr for proper operation.

The NPSH is calculated using the equation: \( NPSH = H_s + H_v \), where \( H_s \) is the pressure head at the pump's suction and \( H_v \) is the vapor pressure head.
Ensuring NPSHa is greater than NPSHr is crucial for preventing cavitation. Improving NPSH can be achieved by lowering the pump’s elevation above the water source or increasing the atmospheric pressure.

Pump Performance

Pump performance under various conditions is determined by factors including flow rate, head, efficiency, and cavitation. A pump's ability to move fluid effectively is described by its performance curves.
Factors affecting pump performance include the pump design, liquid properties, and operating conditions such as temperature and pressure variations. The head added by the pump and the flow rate are key indicators of performance. Pumps are tested under controlled conditions to measure efficiency and detect potential cavitation issues.
Operating conditions, such as changes in altitude or temperature, affect a pump's NPSH and cavitation risk. Adjustments may be necessary when the pump is used in different environments to maintain optimal performance.

Vapor Pressure

Vapor pressure is a defining property of liquids, representing the pressure at which a liquid begins to vaporize at a given temperature. Each liquid has a unique vapor pressure curve, critical in predicting boiling points in various conditions.
Understanding vapor pressure is essential in fluid mechanics to avoid cavitation. As temperature rises, vapor pressure increases, potentially leading to cavitation if the pump suction head doesn't compensate for this change.
In practical terms, it's crucial to monitor the relationship between operational temperature and vapor pressure. This ensures that pumps operate within safe pressure margins, avoiding the transition to vapor phase inside the pump, which can damage the system and decrease efficiency.