Question

A turbofan engine operating on an aircraft flying at \(200 \mathrm{m} / \mathrm{s}\) at an altitude where the air is at \(50 \mathrm{kPa}\) and \(-20^{\circ} \mathrm{C}\) is to produce \(50,000 \mathrm{N}\) of thrust. The inlet diameter of this engine is \(2.5 \mathrm{m} ;\) the compressor pressure ratio is \(12 ;\) and the mass flow rate ratio is \(8 .\) Determine the air temperature at the fan outlet needed to produce this thrust. Assume ideal operation for all components and constant specific heats at room temperature.

Short answer

Given the parameters of an ideal turbofan engine, the required air temperature at the fan outlet to produce a thrust of 50,000 N is approximately 21.103 K.

Step-by-step solution

Find mass flow rate of primary and secondary flow

We are given the mass flow rate ratio, which is defined as the ratio of mass flow rate in the primary flow to the mass flow rate in the secondary (bypass) flow. If we let \(m_1\) represent the primary flow mass flow rate and \(m_2\) represent the secondary flow mass flow rate, then the mass flow rate ratio is given by: $$ m_2 = 8m_1 $$ Moreover, we know that the total mass flow rate through the inlet can be obtained by multiplying the inlet area with the freestream flow velocity and the freestream air density. Now, let's calculate the inlet area and find the total mass flow rate.

Calculate the inlet area and total mass flow rate

The inlet area (A) can be calculated by using the formula for the area of a circle: $$ A = \pi\left(\frac{D}{2}\right)^2 $$ Where \(D = 2.5\mathrm{m}\) is the diameter of the inlet. Substituting the values, we get: $$ A = \pi\left(\frac{2.5}{2}\right)^2 \approx 4.910\,\mathrm{m^2} $$ Next, we need to find the air density at an altitude with a pressure of \(50\,\mathrm{kPa}\) and a temperature of \(-20^{\circ}\mathrm{C}\). Since the specific heats are assumed to be constant, we can use the ideal gas law: $$ \rho = \frac{p}{RT} $$ Where \(\rho\) is the air density, \(p = 50\,\mathrm{kPa} = 50,000\,\mathrm{Pa}\) is the pressure, \(R = 287\,\mathrm{J/(kg\cdot K)}\) is the specific gas constant for air, and \(T = -20^{\circ}\mathrm{C} + 273.15 = 253.15\,\mathrm{K}\) is the absolute temperature. Plug in the values and get the air density: $$ \rho = \frac{50,000}{287 \cdot 253.15} \approx 0.684\,\mathrm{kg/m^3} $$ We can now calculate the total mass flow rate (\(m_{total}\)) using the formula: $$ m_{total} = \rho VA $$ Where V is the freestream flow velocity (200 m/s) and A is the inlet area. Substitute the values: $$ m_{total} = 0.684 \cdot 200 \cdot 4.910 \approx 672.482\,\mathrm{kg/s} $$

Compute the primary and secondary mass flow rates

From the mass flow rate ratio equation previously determined, we can solve for the primary and secondary mass flow rates. The sum of the primary and secondary mass flow rates should equal the total mass flow rate: $$ m_1 + m_2 = m_{total} $$ Substitute \(m_2 = 8m_1\): $$ m_1 + 8m_1 = 672.482 $$ Solve for \(m_1\): $$ m_1 = \frac{672.482}{9} \approx 74.720\,\mathrm{kg/s} $$ Now we can compute the secondary mass flow rate: $$ m_2 = 8m_1 = 8 \times 74.720 \approx 597.762\,\mathrm{kg/s} $$

Utilize the momentum equation to find the exhaust velocities

The momentum equation for a turbofan engine can be written as: $$ F = m_1(v_{e1} - V) + m_2(v_{e2} - V) $$ Where \(F = 50,000\,\mathrm{N}\) is the thrust, \(v_{e1}\) is the exhaust velocity of primary flow, \(v_{e2}\) is the exhaust velocity of secondary flow, and \(V = 200\,\mathrm{m/s}\) is the freestream flow velocity. Additionally, the energy equation for the turbofan engine can be written as: $$ Q_{in} = \frac{m_1}{2}(v_{e1}^2 - V^2) + \frac{m_2}{2}(v_{e2}^2 - V^2) $$ Where \(Q_{in}\) is the heat input.

Calculate the exhaust velocities

Since we know the pressure ratio of the compressor, we can determine the temperature ratio: $$ \frac{T_{e1}}{T_1}=\left(\frac{p_{e1}}{p_1}\right)^{(\gamma-1)/\gamma} $$ Where \(\gamma=1.4\) is the specific heat ratio, \(T_{e1}\) is the exhaust temperature of primary flow, \(T_1 = 253.15\,\mathrm{K}\) is the given inlet temperature, and \(p_{e1}/p_1 = 12\). Calculate the temperature ratio: $$ \frac{T_{e1}}{253.15} = 12^{\frac{1.4-1}{1.4}} \implies T_{e1} \approx 719.604\,\mathrm{K} $$ Now, using the ideal gas law, we can determine the exhaust velocity of the primary flow: $$ v_{e1} = \sqrt{\frac{2\gamma RT_{e1}}{\gamma-1}} \approx 728.08\,\mathrm{m/s} $$ To find the exhaust velocity of the secondary flow, we can use the momentum equation: $$ v_{e2} = \frac{F - m_1(v_{e1} - V)}{m_2}+V \approx 284.052\,\mathrm{m/s} $$

Calculate the air temperature at the fan outlet

Finally, we can use the energy equation to find the air temperature at the fan outlet. Solving the energy equation for the heat input: $$ Q_{in} = \frac{m_1}{2}(v_{e1}^2 - V^2) + \frac{m_2}{2}(v_{e2}^2 - V^2) \approx 4.785 \times 10^7\,\mathrm{W} $$ Now, using the energy equation for the secondary flow, we can write: $$ Q_{in} = \frac{m_2}{2}(v_{e2}^2 - V^2) + m_2C_p(T_{e2} - T_1) $$ Where \(T_{e2}\) is the air temperature at the fan outlet and \(C_p = 1005\,\mathrm{J/(kg\cdot K)}\) is the specific heat at constant pressure. We can now solve for the fan outlet temperature: $$ T_{e2} = T_1 + \frac{1}{m_2C_p}\left[Q_{in} - \frac{m_2}{2}(v_{e2}^2 - V^2)\right] \approx 21.103\,\mathrm{K} $$ Thus, the air temperature at the fan outlet needed to produce a thrust of \(50,000\,\mathrm{N}\) is approximately \(21.103\,\mathrm{K}\).

Key concepts

Mass Flow Rate Ratio

Understanding the mass flow rate ratio in a turbofan engine is essential in determining individual component performance and overall engine thrust. The mass flow rate ratio is the proportion of air mass flowing through the bypass duct (secondary flow) to the air mass flowing through the core engine (primary flow). It is a critical factor in a turbofan engine because it impacts the engine's thrust, efficiency, and noise.

For example, in the given problem, the mass flow rate ratio is 8, indicating the secondary flow is eight times the mass rate of the primary flow. This concept is fundamental when calculating mass flow rates for each engine section. The equation provided in the solution \(m_2 = 8m_1\) helps to calculate the primary and secondary mass flows given the engine's total mass flow rate, which is derived from the inlet air density and velocity. A high bypass ratio, where the mass flow rate ratio is large, is often used for commercial aircraft engines to improve fuel efficiency and reduce noise.

Inlet Area Calculation

The inlet of a turbofan engine is where atmospheric air enters the engine, and its area plays a significant role in determining how much air the engine can ingest. Calculating the correct inlet area is fundamental for assessing the engine's mass flow rate. The area can be calculated using basic geometry, typically the area of a circle, as illustrated in the step-by-step solution \(A = \pi\left(\frac{D}{2}\right)^2\), where \(D\) is the diameter of the inlet.

For an engine with a given diameter, like the \(2.5\mathrm{m}\) in the exercise, using the area formula allows us to calculate the precise inlet area which is essential for further calculations such as the determination of the total mass flow rate through the engine. The area must be accurately calculated to avoid discrepancies in later calculations, which base on the inlet area to find other important parameters like total mass flow rate and subsequently the desired thrust.

Ideal Gas Law Application

The ideal gas law is a key equation in thermodynamics, helping to relate the pressure, volume, temperature, and quantity of a gas. In the context of a turbofan engine, we apply this basic equation to determine air density at different conditions which influence engine performance. The equation \(\rho = \frac{p}{RT}\) demonstrates how the air density \(\rho\) is inversely proportional to the temperature and directly proportional to the pressure. Here, \(R\) is the specific gas constant for air.

Using the ideal gas law as per the step-by-step solution, given the atmospheric conditions at the flight altitude, we can determine the air density which feeds into calculating the mass flow rate, a critical step in engine analysis. It is essential to consider real gas effects at different temperatures and pressures but for many practical cases, like the given exercise assuming constant specific heats at room temperature, the ideal gas law provides sufficient accuracy. Proper application of this law is important for accurate engine inlet and exit calculations.