Question

Water of density \(998.2 \mathrm{~kg} / \mathrm{m}^{3}\) is moving at negligible speed under a pressure of \(101.3 \mathrm{kPa}\) but is then accelerated to high speed by the blades of a spinning propeller. The vapor pressure of the water at the initial temperature of \(20.0^{\circ} \mathrm{C}\) is \(2.3388 \mathrm{kPa}\). At what flow speed will the water begin to boil? This effect, known as cavitation, limits the performance of propellers in water

Short answer

Answer: The water will begin to boil when its flow speed reaches approximately 3.05 m/s due to cavitation.

Step-by-step solution

Recall Bernoulli's equation

The Bernoulli's equation for an incompressible fluid can be written as: $$ P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 $$ where \(P_1\) and \(P_2\) are pressures of fluid at points 1 and 2, \(\rho\) is the fluid density, \(v_1\) and \(v_2\) are the fluid speeds at points 1 and 2, \(h_1\) and \(h_2\) are the height of the fluid at points 1 and 2, respectively. Since the problem states that the initial flow speed is negligible, we can assume \(v_1 = 0\). And since its the same fluid and the heights are not given, the term \(\rho g h\) can be dropped (as it would be equal for both the points). The equation will then reduce to: $$ P_1 = P_2 + \frac{1}{2} \rho v_2^2 $$

Calculate the boiling pressure

To find the flow speed at which water begins to boil, we will first calculate the boiling pressure by adding the vapor pressure to the initial pressure (remember that the boiling pressure is the pressure at which the liquid transitions into a vapor). The vapor pressure is given to be \(2.3388 \mathrm{kPa}\), and the initial pressure is \(101.3 \mathrm{kPa}\). Therefore, $$ P_{boil} = P_1 - P_{vapor} = 101.3 - 2.3388 = 98.9612\, \mathrm{kPa} $$

Use Bernoulli's equation to find the flow speed

Now that we have the boiling pressure, we can use Bernoulli's equation to calculate the flow speed at the point where cavitation begins. Substitute the boiling pressure for \(P_2\) in the equation from step 1: $$ P_1 = P_{boil} + \frac{1}{2} \rho v_2^2 $$ We are given the initial pressure \(P_1 = 101.3 \mathrm{kPa}\), the density of water \(\rho = 998.2 \mathrm{~kg} / \mathrm{m}^{3}\), so we can solve for the flow speed \(v_2\). Rearrange the equation to solve for \(v_2\): $$ v_2 = \sqrt{\frac{2(P_1 - P_{boil})}{\rho}} $$ Plug in the known values: $$ v_2 = \sqrt{\frac{2(101.3 - 98.9612) \times 10^3}{998.2}} = \sqrt{\frac{4676}{998.2}} $$ Calculate the flow speed: $$ v_2 \approx 3.05 \mathrm{m/s} $$ The water will begin to boil when its flow speed reaches \(3.05 \mathrm{m/s}\) due to cavitation.

Key concepts

Fluid Dynamics

Fluid dynamics is the branch of physics concerned with the study of fluids (liquids and gases) in motion. It involves various principles and equations that help describe and predict the behavior of fluid flow in different scenarios. Bernoulli's Principle is a key concept in fluid dynamics, particularly when dealing with incompressible fluid flow, such as water. It relates the pressure, velocity, and height at different points along a streamline.
Some important concepts in fluid dynamics include:

• Continuity equation: For incompressible flow, the mass flow rate must remain constant from one cross-section of a pipe to another. This is expressed as \(A_1v_1 = A_2v_2\), where \(A\) is the cross-sectional area and \(v\) is the velocity of the fluid.
• Bernoulli’s equation: This derives from the conservation of energy and is expressed as \( P + \frac{1}{2} \rho v^2 + \rho g h = \ ext{constant}\). It demonstrates how, within a streamline, an increased speed results in decreased pressure or potential energy.

Understanding fluid dynamics is crucial in various fields including engineering, meteorology, and oceanography.

Cavitation

Cavitation is a phenomenon in fluid dynamics where vapor bubbles form in a liquid due to rapid changes in pressure. It typically occurs when the pressure in a liquid falls below its vapor pressure, leading to the formation of vapor cavities or bubbles. These bubbles can collapse violently when they move to areas of higher pressure, causing shock waves that may damage surfaces like propeller blades.
Cavitation is significant because:

• It affects the performance and efficiency of fluid machinery such as pumps and propellers.
• It can lead to erosion and damage if not properly managed.
• Understanding cavitation can help design better equipment to withstand its effects.

To minimize cavitation, engineers often design equipment to ensure that pressure does not drop significantly, using streamlined designs and appropriate materials.

Boiling Pressure

Boiling pressure is the specific pressure at which a liquid turns to vapor at a given temperature. For water, this means the vapor pressure reaches the surrounding pressure, allowing bubbles to form within the liquid. In a flowing fluid, like in the exercise, this occurs when the local pressure drops due to increased fluid velocity (as explained by Bernoulli's Principle).
Key points about boiling pressure include:

• Vapor pressure: At a given temperature, it is the pressure exerted by the vapor in equilibrium with its liquid phase. As temperature increases, vapor pressure increases, leading to boiling.
• Impact of cavitation: If a liquid's pressure falls below its vapor pressure in any part of a flowing system, cavitation can occur, leading to potential operational problems.

Recognizing the conditions that lead to boiling pressure helps in designing systems to avoid cavitation, by maintaining pressures above the vapor pressure point in all parts of a system.