Question

Air enters a \(28-\mathrm{cm}\) diameter pipe steadily at \(200 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) with a velocity of \(5 \mathrm{m} / \mathrm{s}\). Air is heated as it flows, and leaves the pipe at \(180 \mathrm{kPa}\) and \(40^{\circ} \mathrm{C}\). Determine (a) the volume flow rate of air at the inlet, \((b)\) the mass flow rate of air, and \((c)\) the velocity and volume flow rate at the exit.

Short answer

Question: Calculate (a) the volume flow rate at the inlet, (b) the mass flow rate of the air, and (c) the velocity and volume flow rate at the exit of a pipe with a diameter of 28 cm and the air entering with a velocity of 5 m/s. The temperature and pressure at the inlet are 20°C and 200 kPa, while at the exit, they are 40°C and 180 kPa. Use the Ideal Gas Law and the relationship between mass flow rate, volume flow rate, and velocity. Answer: (a) The volume flow rate at the inlet is \(Q = A \times v\), where \(A\) is the cross-sectional area of the pipe and \(v\) is the velocity of the fluid. (b) The mass flow rate of air can be calculated using the Ideal Gas Law (\(\rho = \frac{P}{RT}\)) and the volume flow rate to find the density (\(\rho\)). (c) To calculate the velocity and volume flow rate at the exit, use the Ideal Gas Law and the definition of density to determine the density of air at the exit and then solve for velocity and volume flow rate using the mass flow rate.

Step-by-step solution

(a) Calculate the volume flow rate at the inlet

To calculate the volume flow rate at the inlet, we can use the following formula: \(Q = A \times v\) where \(Q\) is the volume flow rate, \(A\) is the cross-sectional area of the tube, and \(v\) is the velocity of the fluid. First, let's calculate the cross-sectional area of the pipe, given its diameter (\(D = 28 \mathrm{cm}\)): \(A = \pi \left(\frac{D}{2}\right)^2\) Now, we can plug in the given velocity (\(v=5 \mathrm{m/s}\)) to calculate the volume flow rate at the inlet: \(Q = A \times v\)

(b) Calculate the mass flow rate of air

To calculate the mass flow rate of air, we will use the Ideal Gas Law and the definition of density: \(\rho = \frac{m}{V} = \frac{P}{RT}\) where \(\rho\) is the density of the air, \(m\) is the mass, \(V\) is the volume, \(P\) is the pressure, \(R\) is the gas constant for air, and \(T\) is the temperature in Kelvin. First, we need to convert the given temperature and pressure at the inlet to Kelvin and Pascal, respectively: \(T = 20^{\circ} \mathrm{C} + 273.15 = 293.15 \mathrm{K}\) \(P = 200 \mathrm{kPa} \times 1000 = 200000 \mathrm{Pa}\) The gas constant for air is \(R = 287.058 \mathrm{J/(kg \cdot K)}\), so we can now calculate the density of air at the inlet: \(\rho = \frac{P}{RT}\) Once we have the density of air at the inlet, we can calculate the mass flow rate using the definition of volume flow rate and the velocity: \(\dot{m} = \rho \times Q\)

(c) Calculate the velocity and volume flow rate at the exit

To calculate the velocity and volume flow rate at the exit, we will once again use the Ideal Gas Law and the definition of density: First, convert the given temperature and pressure at the exit to Kelvin and Pascal, respectively: \(T = 40^{\circ} \mathrm{C} + 273.15 = 313.15 \mathrm{K}\) \(P = 180 \mathrm{kPa} \times 1000 = 180000 \mathrm{Pa}\) Now, we can calculate the density of air at the exit the same way we did for the inlet: \(\rho_{exit} = \frac{P}{RT}\) Since the mass flow rate of air is constant throughout the pipe, we can use the following relationship: \(\dot{m} = \rho_{exit} \times Q_{exit}\) We can rearrange this equation to solve for \(Q_{exit}\) (the volume flow rate at the exit): \(Q_{exit} = \frac{\dot{m}}{\rho_{exit}}\) Now, to find the velocity at the exit, we can use the relationship between volume flow rate and velocity: \(v_{exit} = \frac{Q_{exit}}{A}\)

Key concepts

Volume Flow Rate

Volume flow rate, often represented by the symbol \(Q\), is a crucial concept in fluid dynamics and refers to the volume of fluid that passes through a given surface per unit time. It's generally measured in cubic meters per second \(\text{m}^3/s\) or liters per second \(L/s\).

To compute the volume flow rate, we use the formula \(Q = A \times v\), where \(A\) is the cross-sectional area of the pipe and \(v\) is the fluid velocity. If we're dealing with a circular pipe, which is the case in the problem provided, the area can be calculated using the formula \(A = \pi \left(\frac{D}{2}\right)^2\), with \(D\) being the diameter of the pipe. Subsequently, by multiplying the area by the velocity of the fluid, we find the volume flow rate.

Mass Flow Rate

The mass flow rate measures the mass of a substance that passes through a given surface per unit time. Its units are typically kilograms per second \(kg/s\). Understanding the mass flow rate is essential for many engineering applications, such as calculating the fuel consumption in engines or the mass of air flowing through the HVAC systems.

For gases, the mass flow rate can be found by utilizing the ideal gas law in combination with the volume flow rate. The ideal gas law relates the pressure \(P\), volume \(V\), and temperature \(T\) of the gas with its amount in moles \(n\) as expressed in the equation \(PV=nRT\), where \(R\) is the specific gas constant. Using the idea that \(\rho = m/V\) and rearranging the ideal gas law gives us the density, \(\rho = \frac{P}{RT}\). We then multiply the density by the volume flow rate, \(\dot{m} = \rho \times Q\), to determine the mass flow rate.

Ideal Gas Law

The ideal gas law is a pivotal equation in thermodynamics that connects the pressure, volume, and temperature of an ideal gas with its amount in moles. The equation is written as \(PV = nRT\), with \(P\) representing pressure, \(V\) volume, \(n\) the number of moles, \(R\) the ideal gas constant, and \(T\) temperature in Kelvin.

For practical applications, the ideal gas law is often reformed in terms of the density \(\rho\), allowing for calculations regarding the physical properties of gases under changing conditions. By setting the equation in terms of mass and volume, you get \(PV = mRT\), from which density can be expressed as \(\rho = \frac{P}{RT}\). This form is especially useful when you are dealing with problems in fluid flow, allowing you to compute changes in density due to variations in temperature and pressure, as seen in the problem where air expands and heats up as it flows through the pipe.

Fluid Velocity

Fluid velocity is the speed at which a fluid particle moves in a given direction and is typically expressed in meters per second \(m/s\). It's a vector quantity, meaning that it has both magnitude and direction. In many cases of fluid flow in pipes, we're concerned with the average velocity, where the flow is assumed to be steady and uniform across any cross-section.

The relationship between fluid velocity, volume flow rate, and pipe cross-sectional area is key to many calculations involving fluid dynamics. If you have the volume flow rate \(Q\) and the cross-sectional area \(A\) (as in a pipe), you can find the average velocity using the formula \(v = \frac{Q}{A}\). It's important to note that changes in cross-sectional area along a pipe, without any change in the volume flow rate, will result in changes in velocity. In our problem scenario, the mass flow rate stays constant, but when the gas expands due to heating, this affects the volume flow rate and consequently the velocity.