Question

A pump increases the water pressure from 100 kPa at the inlet to \(900 \mathrm{kPa}\) at the outlet. Water enters this pump at \(15^{\circ} \mathrm{C}\) through a 1 -cm-diameter opening and exits through a 1.5-cm-diameter opening. Determine the velocity of the water at the inlet and outlet when the mass flow rate through the pump is \(0.5 \mathrm{kg} / \mathrm{s}\). Will these velocities change significantly if the inlet temperature is raised to \(40^{\circ} \mathrm{C} ?\)

Short answer

Answer: No, raising the inlet temperature to 40°C will not significantly change the water velocities at the inlet and outlet of the pump. The calculated velocities at 15°C are 8.02 m/s and 2.84 m/s, and at 40°C, the velocities are 8.10 m/s and 2.87 m/s, which are not significantly different.

Step-by-step solution

Calculate Area at Inlet and Outlet

To determine the water velocity, you need to know the area at the pump's inlet and outlet. To find the area, use the formula for the area of a circle: \(area = \pi (diameter/2)^2\). Inlet diameter (di) = 1 cm = 0.01 m Outlet diameter (do) = 1.5 cm = 0.015 m Inlet area (Ai) = \(\pi(0.01/2)^2 = 7.854 \times 10^{-5} m^2\) Outlet area (Ao) = \(\pi(0.015/2)^2 = 1.7668 \times 10^{-4} m^2\)

Calculate the density of water

The water density varies with temperature, but we will assume it to be constant for this problem. We will use the density value at \(15^{\circ} \mathrm{C}\): \(\rho = 999 \mathrm{kg} / m^3\).

Find the water velocities at Inlet and Outlet

Now use the formula to find the water velocities at the inlet and outlet: \(mass\ flow\ rate = \rho \times velocity \times area\). Mass flow rate (\(\dot{m}\)): 0.5 \(\mathrm{kg}/\mathrm{s}\) Velocity at inlet \(v_i = \frac{mass\ flow\ rate}{\rho \times A_i}\) \(v_i = \frac{0.5}{999 \times 7.854 \times 10^{-5}} = 8.02\ \mathrm{m/s}\) Velocity at outlet \(v_o = \frac{mass\ flow\ rate}{\rho \times A_o}\) \(v_o = \frac{0.5}{999 \times 1.7668 \times 10^{-4}} = 2.84\ \mathrm{m/s}\)

Consider temperature change and re-calculate velocity

Now let's consider the temperature change and see if it affects the velocities significantly. At \(40^{\circ} \mathrm{C}\), the density of water is \(\rho = 992 \mathrm{kg}/\mathrm{m}^3\). Re-calculate the inlet and outlet velocities with the new water density value: \(v'_i = \frac{0.5}{992 \times 7.854 \times 10^{-5}} = 8.10\ \mathrm{m/s}\) \(v'_o = \frac{0.5}{992 \times 1.7668 \times 10^{-4}} = 2.87\ \mathrm{m/s}\) The changes in velocity are minimal, and these values are not significantly different from the initial velocities calculated (8.02 m/s and 2.84 m/s). Thus, raising the inlet temperature to \(40^{\circ} \mathrm{C}\) will not significantly change the water velocities at the inlet and outlet.

Key concepts

Pressure Increase in Pumps

Understanding how pumps increase pressure is essential in fluid mechanics and applications involving fluid transport systems. A pump's primary function is to add energy to a fluid, which typically results in an increase in fluid pressure. This is achieved by converting mechanical energy from a motor or another external source into hydraulic energy.

The pressure at the pump inlet is often at a lower level; through the operation of the pump, energy is imparted onto the water, causing an increase in pressure. This phenomenon is described by Bernoulli's principle, which states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Pumps essentially work to overcome this principle by adding energy to the system, thus increasing pressure to move the fluid against gravity, friction, or other resistances.

For example, in the given exercise, a pump increases the water pressure from 100 kPa to 900 kPa. This pressure increase enables the water to travel from the pump's inlet to the outlet, overcoming any resistance within the pipe or system it is a part of.

Mass Flow Rate Calculation

Mass flow rate is a critical parameter in thermodynamics and fluid mechanics that quantifies how much mass of fluid passes through a given surface per unit time. This is usually measured in kilograms per second (\text{kg/s}) and can be calculated using the formula:
\[ mass\text{ flow rate } = \rho \times velocity \times area \]
where \(\rho\) is the fluid density, velocity is the speed at which the fluid is moving, and area is the cross-sectional area through which the fluid is passing.

In the solution to the provided exercise, with a known mass flow rate of 0.5 \text{kg/s}, the velocities at the inlet and outlet were found by rearranging the formula to solve for velocity. This is a useful calculation that can help engineers and scientists design systems to handle the required flow and ensure the structural integrity of the pipes and conduits in the system.

Water Velocity in Pump

The velocity of water or any fluid within a pump is a fundamental characteristic that influences the overall performance of the pumping system. Water velocity is indicative of how quickly the fluid is being moved from one point to another within the system. It is influenced by several factors, including pump design, fluid properties, and operational conditions.

The water velocity in a pump is inversely proportional to the cross-sectional area of the pipe through which the fluid flows. As shown in the solution steps, where the cross-sectional area of the outlet is larger than that of the inlet, the velocity at the outlet will be lower than that at the inlet. This is consistent with the continuity equation—a principle in fluid dynamics stating that the mass flow rate must be constant in a pipe, leading to an inverse relationship between velocity and cross-sectional area when there are no leaks or additions to the flow.

Effect of Temperature on Water Density

Temperature plays a prominent role in determining the density of water, which is the mass per unit volume of the water. As a substance's temperature increases, the molecules within begin to move more vigorously, causing them to occupy a larger volume and hence decrease in density. This relationship is vital in various fields, including meteorology, oceanography, and chemical engineering.

In the context of pump operation, as the exercise illustrates, a change in input water temperature from \(15^{\text{\(\backslash\backslash\)circ}} \text{C}\) to \(40^{\text{\(\backslash\backslash\)circ}} \text{C}\) results in a decrease in water density from 999 \text{kg/m}^\(3\) to 992 \text{kg/m}^\(3\). However, the exercise concludes that the change in temperature, and consequently in density, does not significantly affect the mass flow rate or the velocity in the pump under these conditions, which is an important consideration for systems that operate over a range of temperatures.