Question

Water of density \(998.2 \mathrm{~kg} / \mathrm{m}^{3}\) is moving at negligible speed under a pressure of 101.3 kPa but is then accelerated to a 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. (Vapor pressure is the pressure of the vapor resulting from evaporation of a liquid above a sample of the liquid in a closed container.).

Short answer

Answer: The water will begin to experience cavitation at a flow speed of approximately 0.444 m/s.

Step-by-step solution

Write down Bernoulli's Equation

First, write down Bernoulli's equation for the initial and final points of the water flow: \(P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2\) Since the water initially has negligible speed, we can assume \(v_1 = 0\). Also, we can assume no difference in height and cancel out the height terms: \(h_1 = h_2\). So, the equation simplifies to: \(P_1 = P_2 + \frac{1}{2} \rho v_2^2\)

Write the known values

Now, write the known values: \(P_1 = 101.3\) \(\rho = 998.2 \mathrm{~kg} / \mathrm{m}^{3}\) \(P_2 = 2.3388\) Plug these values into the simplified Bernoulli's equation: \(101.3 = 2.3388 + \frac{1}{2}(998.2)v_2^2\)

Solve for \(v_2\)

Solve the equation for \(v_2\): \((101.3 - 2.3388) = \frac{1}{2}(998.2)v_2^2\) \(v_2^2 = \frac{2(101.3 - 2.3388)}{998.2}\) \(v_2^2 = 0.197\) \(v_2 = \sqrt{0.197}= 0.444 \mathrm{~m/s}\) The water will begin to boil (experience cavitation) when the flow speed reaches approximately \(0.444 \mathrm{~m/s}\).

Key concepts

Bernoulli's Equation

Understanding Bernoulli's equation is essential in tackling problems related to fluid dynamics. It describes the conservation of energy in a flowing fluid. According to the equation, for an incompressible, frictionless fluid, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

The simplified form of Bernoulli's equation can often be used when certain conditions apply, such as negligible changes in height or speed. In the given problem, we focus on the pressure and kinetic energy terms because the height change is considered nil, and the initial velocity is very small. Focusing on the essential terms simplifies our calculations and helps us directly relate changes in fluid pressure to changes in velocity.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. It is a specific kind of pressure that results from the evaporation of a liquid. An important feature of vapor pressure is its dependency on temperature; as the temperature increases, so does the vapor pressure.

When the vapor pressure reaches the surrounding pressure of a liquid, boiling occurs. In the case of a propeller accelerating water, the reduced pressure can cause the water to reach its boiling point prematurely, a phenomenon known as cavitation. Understanding this concept is pivotal as it links temperature, phase changes, and the dynamic behavior of fluids.

Fluid Acceleration

Fluid acceleration refers to the change in velocity of a fluid as it flows through different conditions. In the context of our problem, water experiences acceleration when it moves from a region of higher pressure to a region of lower pressure, caused by the propeller blades. When water is accelerated, the fluid speed increases, and according to Bernoulli's equation, the pressure within the fluid must decrease.

Such changes in pressure and speed are not only significant in calculating when cavitation occurs but are also crucial in engineering applications like designing pumps, propellers, and predicting the behavior of fluids in motion. The challenge is to balance the acceleration to prevent reaching the boiling point, causing cavitation and potential damage to machinery.

Boiling Point

The boiling point of a substance is the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure. For water, this commonly occurs at 100 degrees Celsius at sea level atmospheric pressure. However, the boiling point can change when the surrounding pressure changes. In fluid dynamics, particularly in the phenomenon of cavitation, a rapid decrease in pressure due to fluid acceleration can cause the boiling point to drop significantly.

Hence, when a liquid such as water is subjected to rapid decreases in pressure, like from the action of propeller blades, it can reach its boiling point at relatively low temperatures; boiling then occurs not from heating, but from a pressure drop. Recognizing this relationship allows for a deeper understanding of the conditions under which cavitation can happen, and how it limits the performance of hydraulic equipment.