Question

Nitrogen gas at \(60 \mathrm{kPa}\) and \(7^{\circ} \mathrm{C}\) enters an adiabatic diffuser steadily with a velocity of \(275 \mathrm{m} / \mathrm{s}\) and leaves at 85 \(\mathrm{kPa}\) and \(27^{\circ} \mathrm{C}\). Determine \((a)\) the exit velocity of the nitrogen and \((b)\) the ratio of the inlet to exit area \(A_{1} / A_{2}\).

Short answer

In summary, for an adiabatic diffuser with the given initial and final pressure and temperature conditions, the exit velocity of the nitrogen gas is 209.4 m/s, and the ratio of the inlet to exit area is 0.758.

Step-by-step solution

Calculate specific enthalpy at inlet and outlet points#

First, we must find the specific enthalpy at both the inlet (point 1) and outlet (point 2). We can use the ideal gas equation to calculate the specific volume and the specific heat capacity at constant pressure (\(C_p\)) for nitrogen gas to calculate the enthalpy change: \(H_1 =C_{p1}T_1\) \(H_2 =C_{p2}T_2\) For nitrogen, \(C_p = 1.04 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}}\). Now, calculate the specific enthalpy at point 1 and 2: \(H_1 = (1.04 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}})(7 + 273.15) = 290.6 \mathrm{kJ/kg}\) \(H_2 = (1.04 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}})(27 + 273.15) = 312.6 \mathrm{kJ/kg}\)

Apply the steady-flow energy equation to calculate exit velocity#

Now, apply the steady-flow energy equation: \(H_1 + \frac{v_1^2}{2} = H_2 + \frac{v_2^2}{2}\) We're given the inlet velocity \(v_1 = 275 \mathrm{m}/\mathrm{s}\), and the specific enthalpy values \(H_1\) and \(H_2\) from Step 1. Plug in the values to find the exit velocity \(v_2\): \(290.6 \mathrm{kJ/kg} + \frac{(275 \mathrm{m}/\mathrm{s})^2}{2} = 312.6 \mathrm{kJ/kg} + \frac{v_2^2}{2}\) Solving for \(v_2\): \(v_2 = 209.4 \mathrm{m}/\mathrm{s}\) This is the answer for part (a): The exit velocity of nitrogen gas is \(209.4 \mathrm{m}/\mathrm{s}\).

Apply the continuity equation to find the ratio of the inlet to exit area#

The continuity equation is: \(\rho_1 v_1 A_1 = \rho_2 v_2 A_2\) Since it's an adiabatic process, we can assume the mass flow rate is constant, so we can rearrange the equation to solve for the ratio of inlet to exit area \(A_1/A_2\): \(\frac{A_1}{A_2} = \frac{\rho_2 v_2}{\rho_1 v_1}\) We can calculate the specific volumes at point 1 and 2 using the ideal gas equation: \(v_1 = \frac{R T_1}{P_1}\) \(v_2 = \frac{R T_2}{P_2}\) For nitrogen, \(R=0.2968 \frac{\mathrm{kJ}}{\mathrm{kg}\cdot\mathrm{K}}\). Calculate the specific volumes at point 1 and 2: \(v_1 = \frac{(0.2968 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}})(7^\circ \mathrm{C} + 273.15)}{60 \mathrm{kPa}} = 0.1791 \mathrm{m}^3/\mathrm{kg}\) \(v_2 = \frac{(0.2968 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}})(27^\circ \mathrm{C} + 273.15)}{85 \mathrm{kPa}} = 0.1807 \mathrm{m}^3/\mathrm{kg}\) We can now find the densities at point 1 and 2: \(\rho_1 = \frac{1}{v_1} = 5.586 \mathrm{kg/m^3}\) \(\rho_2 = \frac{1}{v_2} = 5.537 \mathrm{kg/m^3}\) Now, we can plug the densities and velocities into the equation to find the ratio of the inlet to the exit area: \(\frac{A_1}{A_2} = \frac{5.537 \mathrm{kg/m^3} \cdot 209.4 \mathrm{m}/\mathrm{s}}{5.586 \mathrm{kg/m^3} \cdot 275 \mathrm{m}/\mathrm{s}} = 0.758\) This is the answer for part (b): The ratio of the inlet to exit area (\(A_1 / A_2\)) is \(0.758\).

Key concepts

Adiabatic Process

An adiabatic process is a term used in thermodynamics to describe a process where no heat is transferred into or out of the system. When dealing with gasses, such as nitrogen in the exercise, it implies that all the work done on or by the gas results in a change in internal energy, typically observed as changes in temperature and pressure. Unlike isothermal processes where temperature remains constant, in an adiabatic process, compression results in heating and expansion in cooling.
In the exercise example, nitrogen gas entering and leaving the adiabatic diffuser undergoes changes in pressure and temperature. The conservation of energy within this adiabatic system allows us to relate the change in kinetic energy, represented by the change in velocity of the gas, with the change in enthalpy, without worrying about heat exchange with the surroundings.

Steady-Flow Energy Equation

The steady-flow energy equation is a representation of the First Law of Thermodynamics tailored for a control volume where fluid flows in and out steadily. This implies that the mass flow rate and energy content of the fluid remain constant over time, which is essential when solving problems related to turbines, compressors, diffusers, and nozzles.
In our nitrogen gas example, we applied this principle to equate the sum of the enthalpy and kinetic energy at the entrance and exit of the adiabatic diffuser. By doing so, we found that the kinetic energy of the gas (in the form of velocity) decreases as its enthalpy (and therefore temperature) increases from the inlet to the outlet. This is a classic application of the steady-flow energy equation, showing how energy is conserved as it transforms from one form to another inside a flowing system.

Continuity Equation

The continuity equation, a fundamental principle in fluid mechanics and thermodynamics, expresses the conservation of mass in a flow system. For steady, incompressible flow, it can be boiled down to the simple statement that what flows into a system must flow out, assuming there's no accumulation.
Our exercise applies the continuity equation to a compressible gas (nitrogen), undergoing changes in velocity and cross-sectional area in the diffuser. By assuming a constant mass flow rate, despite changes in pressure and temperature, we can equate the product of density, velocity, and area at both the inlet and outlet. We can easily handle the variations in density by referring back to the ideal gas law, essentially connecting the dots between density, temperature, and pressure. After determining these variables at both ends of the process, we solved for the ratio of the inlet to the exit area. By doing so, we illustrated how different sections of a system must be designed to accommodate variations in flow properties to maintain a steady flow of mass through the system.