Question

Nitrogen enters a steady-flow heat exchanger at \(150 \mathrm{kPa}, 10^{\circ} \mathrm{C},\) and \(100 \mathrm{m} / \mathrm{s},\) and it receives heat in the amount of \(150 \mathrm{kJ} / \mathrm{kg}\) as it flows through it. The nitrogen leaves the heat exchanger at \(100 \mathrm{kPa}\) with a velocity of \(200 \mathrm{m} / \mathrm{s}\) Determine the stagnation pressure and temperature of the nitrogen at the inlet and exit states.

Short answer

Question: Calculate the stagnation pressure and temperature of nitrogen at the inlet and exit states of a steady-flow heat exchanger, given the inlet pressure \(P_1 = 150\,\text{kPa}\), inlet temperature \(T_1 = 10^\circ\text{C}\), inlet velocity \(V_1 = 100\,\text{m/s}\), exit pressure \(P_2 = 100\,\text{kPa}\), exit velocity \(V_2 = 200\,\text{m/s}\), and the amount of heat received per mass unit \(Q = 150\,\text{kJ/kg}\). Assume nitrogen behaves as an ideal gas and neglect the potential energy. Answer: To calculate the stagnation pressure and temperature at the inlet and exit states of the heat exchanger, follow these steps: 1. Find the properties of nitrogen at the inlet and exit states using the ideal gas equation and property tables or software. 2. Apply the conservation of mechanical energy principle to find the temperature of nitrogen at the exit state (\(T_2\)). 3. Use the relationships for stagnation pressure and temperature to calculate \(T_{01}\), \(P_{01}\), \(T_{02}\), and \(P_{02}\). After following these steps and plugging in the given values, you can calculate the stagnation pressure and temperature of nitrogen at the inlet and exit states.

Step-by-step solution

Determine the properties of nitrogen at the inlet and exit states

Using the given pressures and temperatures, and assuming that nitrogen behaves as an ideal gas, you can find the specific volumes, internal energies, and enthalpies of nitrogen at the inlet and exit states using the ideal gas equation of state and the property tables or software like REFPROP. At the inlet state: - Pressure: \(P_1 = 150\,\text{kPa}\) - Temperature: \(T_1 = 10^\circ\text{C} = 283.15\,\text{K}\) At the exit state: - Pressure: \(P_2 = 100\,\text{kPa}\) - Temperature: Determine \(T_2\) in Step 3.

Conservation of mechanical energy principle

The conservation of mechanical energy principle states that the sum of kinetic energies, potential energies, and internal energies remains constant in a process. However, in this case, potential energy can be neglected, as there is no significant change in height. Therefore, use the conservation of mechanical energy equation to find the temperature of the nitrogen at the exit state: \(\frac{V_1^2}{2} + C_v T_1 + Q = \frac{V_2^2}{2} + C_v T_2\) Plug in the given values and the specific heat at constant volume, \(C_v\), for nitrogen: \(\frac{(100\,\text{m/s})^2}{2} + C_v (283.15\,\text{K}) + 150\,\text{kJ/kg} = \frac{(200\,\text{m/s})^2}{2} + C_v T_2\) Solve for \(T_2\), and you'll find the temperature of nitrogen at the exit state.

Find the stagnation pressure and temperature at the inlet and exit states

Now, you can find the stagnation pressure and temperature of nitrogen at the inlet and exit states using the following relationships: \(T_{01} = T_1 + \frac{V_1^2}{2C_p} = 283.15\,\text{K} + \frac{(100\,\text{m/s})^2}{2\cdot C_p}\) \(P_{01} = P_1 \cdot \left(\frac{T_{01}}{T_1}\right)^{\frac{\gamma}{\gamma - 1}}\) \(T_{02} = T_2 + \frac{V_2^2}{2C_p}\) \(P_{02} = P_2 \cdot \left(\frac{T_{02}}{T_2}\right)^{\frac{\gamma}{\gamma - 1}}\) Where \(C_p\) represents the specific heat at constant pressure, and \(\gamma\) represents the ratio of specific heats for nitrogen. Plug in the values and calculate \(T_{01}\), \(P_{01}\), \(T_{02}\), and \(P_{02}\) to find the stagnation pressure and temperature of nitrogen at the inlet and exit states.

Key concepts

Steady-Flow Heat Exchanger

A steady-flow heat exchanger is a device that transfers heat from one fluid to another without any accumulation of heat within the exchanger itself. In such systems, properties like mass flow rate, energy, and momentum remain constant over time.

In our exercise, nitrogen gas enters a heat exchanger and gains energy in the form of heat, which increases its internal energy and possibly its temperature and velocity. This process is guided by the first law of thermodynamics, particularly for open systems, where energy is exchanged but mass remains constant. It's crucial to consider that heat exchangers do not generate energy; they merely transfer it, altering the state properties of the working fluid, such as temperature, pressure, and velocity.

Ideal Gas Equation of State

The ideal gas equation of state is a useful approximation for the behavior of many gases under a variety of conditions, defined by the formula:
\( PV = nRT \)
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of substance (in moles), \(R\) is the ideal gas constant, and \(T\) is the absolute temperature.

In our exercise, we presume nitrogen acts as an ideal gas, allowing us to relate its pressure and temperature to determine other properties like specific volumes and internal energies. The ideal gas approximation greatly simplifies calculations in thermodynamics by treating complex interactions within a gas as negligible, thus making it feasible for students to understand and solve related problems effectively.

Conservation of Mechanical Energy

The conservation of mechanical energy principle indicates that in the absence of non-conservative forces (like friction or resistances), the total mechanical energy of a system remains constant. This is a specific case of the conservation of energy principle, focusing on kinetic and potential energies.

In the context of the heat exchanger exercise, we consider nitrogen's kinetic energy along with its internal energy. Since potential energy change is dismissed due to negligible height difference, we apply the mechanical energy conservation to compute the exit temperature of nitrogen by setting the sum of kinetic and internal energy at the entry equal to that at the exit, adjusted for the heat input. This gives students a concrete method to determine changes in energetic properties as fluids move through systems.

Specific Heat Ratio

The specific heat ratio, denoted as \(\gamma\), is a dimensionless number representing the ratio of specific heat at constant pressure \(C_p\) to specific heat at constant volume \(C_v\). For an ideal gas, this ratio is a measure of the gas's ability to convert heat energy into work and vice versa, during isentropic processes.
\( \gamma = \frac{C_p}{C_v} \)

In our exercise, knowing \(\textstyle \gamma\) for nitrogen helps us determine the stagnation pressure and temperature, which are conditions of a fluid where its kinetic energy has been 'converted' into pressure and temperature. It's noteworthy for students to recognize that \(\gamma\) is crucial in compressible flow calculations, such as those involving high-speed flows where changes in density and temperature are significant.