Question

Refrigerant-134a enters a 28-cm-diameter pipe steadily at \(200 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) with a velocity of \(5 \mathrm{m} / \mathrm{s}\). The refrigerant gains heat as it flows and leaves the pipe at \(180 \mathrm{kPa}\) and \(40^{\circ} \mathrm{C}\). Determine (a) the volume flow rate of the refrigerant at the inlet, \((b)\) the mass flow rate of the refrigerant, and \((c)\) the velocity and volume flow rate at the exit.

Short answer

Question: Determine the volume flow rate at the inlet, the mass flow rate, and the velocity and volume flow rate at the exit for Refrigerant-134a in a given pipe. Answer: The volume flow rate at the inlet is 0.3079 m³/s, the mass flow rate is 1.673 kg/s, the exit velocity is 5.636 m/s, and the volume flow rate at the exit is 0.347 m³/s.

Step-by-step solution

Calculate the volume flow rate at the inlet

To calculate the volume flow rate at the inlet, we can use the equation: \(Q_{in}=A_{in} \times v_{in}\). Given the diameter, we can calculate the area with the following equation: \(A_{in}=\pi \frac{D^2}{4}\). Using the information from the exercise, the area is: \(A_{in}=\pi \frac{(0.28\,\text{m})^2}{4}=0.06157\,\text{m}^2\) Now, we can calculate the volume flow rate by multiplying the area with the velocity. \(Q_{in}=0.06157\,\text{m}^2 \times 5\,\text{m/s} = 0.3079\,\text{m}^3/\text{s}\)

Calculate the mass flow rate

To calculate the mass flow rate, we can use the equation: \(\dot{m} = \rho_{in} \times Q_{in}\). First, we need to find the density of the refrigerant at the entrance conditions, \(200\, \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\). Using a refrigerant-134a table or an online calculator, the density is found to be \(\rho_{in}= 5.436\,\text{kg/m}^3\). Now, we can find the mass flow rate: \(\dot{m}= 5.436\,\text{kg/m}^3 \times 0.3079\,\text{m}^3/\text{s}= 1.673\,\text{kg/s}\)

Calculate the exit velocity

The conservation of mass principle states that the mass flow rate at the entrance is equal to the mass flow rate at the exit. Thus, we can write the equation: \(\rho_{in} A_{in} v_{in}=\rho_{out} A_{out} v_{out}\) We are given the properties at the exit as \(180 \mathrm{kPa}\) and \(40^{\circ} \mathrm{C}\). Using a refrigerant-134a table or an online calculator, the density at the exit is found to be \(\rho_{out}= 4.826\,\text{kg/m}^3\). The area of the pipe remains unchanged. We can now solve for the exit velocity: \(v_{out}=\frac{\rho_{in} A_{in} v_{in}}{\rho_{out} A_{out}}=\frac{5.436\,\text{kg/m}^3 \times 0.06157\,\text{m}^2 \times 5\,\text{m/s}}{4.826\,\text{kg/m}^3 \times 0.06157\,\text{m}^2} = 5.636\,\text{m/s}\)

Calculate the volume flow rate at the exit

To calculate the volume flow rate at the exit, we can use the equation: \(Q_{out}=A_{out} \times v_{out}\). \(Q_{out}=0.06157\,\text{m}^2 \times 5.636\,\text{m/s} = 0.347\,\text{m}^3/\text{s}\) In conclusion, the volume flow rate at the inlet is \(0.3079\,\text{m}^3/\text{s}\), the mass flow rate is \(1.673\,\text{kg/s}\), the exit velocity is \(5.636\,\text{m/s}\) and the volume flow rate at the exit is \(0.347\,\text{m}^3/\text{s}\).

Key concepts

Volume Flow Rate

Volume flow rate is a crucial concept in thermodynamics and fluid mechanics, representing the quantity of fluid that passes a given point within a pipe or a channel per unit time. It's denoted by the symbol \(Q\) and commonly measured in cubic meters per second (\(m^3/s\)). To calculate the volume flow rate, one needs to know the cross-sectional area of the pipe (\(A\)) and the velocity of the fluid (\(v\)) at that point, with the relationship defined as \(Q = A \times v\).

In our exercise involving Refrigerant-134a, we utilized this principle to find the flow rate at the inlet by measuring the pipe's diameter to determine its cross-sectional area and multiplying it by the fluid's velocity. Ensuring accurate measurements and understanding the relationship between these variables is essential for engineers and students alike in the field.

Mass Flow Rate

Mass flow rate is a measure that provides insight into the amount of mass moving through a cross-section of a pipe or channel per unit time. Represented by \(\dot{m}\), it is calculated by multiplying the density of the fluid (\(\rho\)) by the volume flow rate (\(Q\)), given by the equation \(\dot{m} = \rho \times Q\).

In practical scenarios like our textbook example, where the mass flow rate of Refrigerant-134a was needed, you must first determine the fluid's density at the given pressure and temperature. With this information, it's possible to calculate how much mass of refrigerant flows per second, which is fundamental for applications in heating, cooling, and refrigeration systems where precise control of mass is required for system efficiency and effectiveness.

Conservation of Mass

The conservation of mass, also known as the principle of mass conservation, asserts that mass cannot be created or destroyed within a closed system; it's conserved over time. This fundamental concept is leveraged in fluid dynamics to equate the mass entering a system to the mass exiting it, when there are no leaks or accumulation, described mathematically as \(\rho_{in} A_{in} v_{in} = \rho_{out} A_{out} v_{out}\).

In the context of our exercise, we applied the principle to determine the velocity and volume flow rate of the refrigerant at the exit of the pipe. By using the densities and area of the pipe, we ensured the mass remained constant from the inlet to the outlet. Understanding how this principle operates is invaluable for engineers in designing systems where changes in fluid properties and flow characteristics are expected.

Refrigerant Properties

Refrigerants are substances used in heat pumps and refrigeration cycles, with properties that can change significantly under varying temperatures and pressures. The thermodynamic properties of refrigerants, including density, specific heat, and boiling point, are essential for calculating the flow rates and for the design process of refrigeration systems.

For our exercise solution, we referenced refrigerant-134a property tables to find the exact density at specified conditions. It's critical to have accurate data about refrigerant properties, as these directly influence calculations related to mass and volume flow rates, and consequently impact the performance of refrigeration and HVAC systems. Professionals in the field often consult standardized tables or use software tools to obtain such detailed information about refrigerant properties.