Question

An engine pump is used to pump a liquid of density \(\rho\) continuously through a pipe of cross-sectional area \(\mathrm{A}\). If the speed of flow of the liquid in the pipe is \(\mathrm{v}\), then the rate at which kinetic energy is being imparted to the liquid is (A) \((1 / 2) \mathrm{A} \rho \mathrm{V}^{3}\) (B) \((1 / 2) \mathrm{A} \rho \mathrm{V}^{2}\) (C) \((1 / 2) \mathrm{A} \rho \mathrm{V}\) (B) \(\mathrm{ApV}\)

Short answer

The correct answer is (A) \(\frac{1}{2} \rho A v^3\).

Step-by-step solution

Determining the mass flow rate

In this case, the mass flow rate can be found by multiplying the density (ρ) of the liquid by the cross-sectional area (A) of the pipe and the flow velocity (v). The formula for mass flow rate is given by: Mass flow rate = ρAv

Applying the kinetic energy formula

Now, let's substitute the mass flow rate into the kinetic energy formula to determine the rate of imparted kinetic energy. Rate of imparted kinetic energy = (1/2)(Mass flow rate)v^2 Substitute the mass flow rate (ρAv) into the formula: Rate of imparted kinetic energy = (1/2)(ρAv)v^2

Simplifying the expression

Now, we'll simplify the expression: Rate of imparted kinetic energy = (1/2)ρAv^3

Comparing the expression with given options

After simplifying the expression, we have found that the rate of imparted kinetic energy to the liquid is (1/2)ρAv^3. Comparing this expression with the given options, we can see that the correct answer is: (A) (1 / 2) ρAV^3

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In fluid mechanics, especially when dealing with liquid flow through pipes, understanding the kinetic energy imparted to the fluid is crucial. This is because it is the amount of energy used to move the liquid at a certain velocity within a system, like in a pump or a pipeline.
The kinetic energy (KE) of a moving object can be determined using the formula:

• KE = \( \frac{1}{2} mv^2 \)

where:

• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

In scenarios involving flowing liquids, we often deal with the kinetic energy rate, meaning how much energy per second is imparted to the liquid by the engine or pump. To find this, we multiply the mass flow rate (amount of mass flowing per second) by the square of the velocity and adjust by the factor of \( \frac{1}{2} \). This gives us the rate at which kinetic energy is imparted to the fluid, helping engineers design systems capable of meeting specific energy requirements.

Mass Flow Rate

Mass flow rate is a measure of the amount of mass passing through a cross-sectional area of a pipe per unit time. It is a central concept in fluid mechanics and is essential in determining how much liquid a system can transport over a period. The formula for calculating mass flow rate (\dot{m}) is given by:

• \(\dot{m} = \rho A v\)

where:

• \( \rho \) is the fluid density, measured in mass per unit volume (e.g., kg/m³),
• \( A \) is the cross-sectional area of the pipe (e.g., m²),
• \( v \) is the velocity of the fluid (e.g., m/s).

This formula reflects how density, cross-sectional area, and velocity combine to determine the rate at which mass flows through a system. Understanding and calculating mass flow rate is vital for ensuring that systems like pumps operate efficiently and deliver the correct amount of fluid without overloading or causing disruption.

Continuity Equation

The Continuity Equation is a fundamental principle in fluid mechanics, expressing the conservation of mass in a fluid flow system. It ensures that the amount of mass entering a system is equal to the amount leaving, provided the system is at a steady state. The equation is particularly important when analyzing fluid flow through pipes of varying diameters, because it helps maintain the consistency of mass flow throughout the system.
This principle can be mathematically expressed as:

• \(\rho_1 A_1 v_1 = \rho_2 A_2 v_2\)

In cases where the density of the fluid remains constant, such as with incompressible liquids like water, the equation simplifies to:

• \(A_1 v_1 = A_2 v_2\)

This simplified form highlights how changes in the cross-sectional area of the pipe result in changes in velocity to maintain the same mass flow rate. If the area decreases, velocity increases and vice versa, but the mass flow rate stays constant. The Continuity Equation is crucial for designing effective piping systems that can handle varying conditions and flow rates without failing or reducing efficiency.