Problem 49 - Aircraft Propulsion

Question

Consider the flow of perfect gas in a duct with \(T_{2} / T_{1}=1.01, p_{2} / p_{1}=1.038\) and \(M_{2} / M_{1}=0.88\). Assuming gas properties are \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) the density ratio, \(\rho_{2} / \rho_{1}\) (b) the velocity ratio, \(V_{2} / V_{1}\) (c) the duct area ratio, \(A_{2} / A_{1}\)

Step-by-Step Solution

Answer

The required ratios are: (a) Density Ratio, \(\rho_{2} / \rho_{1}= 1.02772\), (b) Velocity Ratio, \(V_{2} / V_{1}=0.884705\), and (c) Duct Area Ratio, \(A_{2} / A_{1}=1.086812\)

Step 1: Density Ratio Calculation

The density ratio can be calculated using the ideal gas law. From the definition of the ideal gas law, \(p = \rho R T\), we have \(\rho_{2} / \rho_{1} = p_{2} / p_{1} * T_{1} / T_{2}\). Now, substitute the values \(p_{2} / p_{1}=1.038\) and \(T_{2} / T_{1}=1.01\), we get \(\rho_{2} / \rho_{1}=1.038 / 1.01 = 1.02772\)

Step 2: Velocity Ratio Calculation

To calculate the velocity ratio, we can use the relation \(M = V / (\gamma R T)^{1/2}\), where M is the Mach number, \(\gamma\) is the ratio of specific heats, R is the gas constant, and T is the temperature. Thus, we have \(V_{2} / V_{1} = M_{2} / M_{1} * (T_{2} / T_{1})^{1/2}\). Substituting the given values \(M_{2} / M_{1}=0.88\) and \(T_{2} / T_{1}=1.01\), we find \(V_{2} / V_{1}=0.88 * (1.01)^{1/2} = 0.884705\)

Step 3: Duct Area Ratio Calculation

By definition, the rate of mass flow is constant for duct flow, that is, \(\rho_{1}V_{1}A_{1} = \rho_{2}V_{2}A_{2}\). Re-arranging gives us \(A_{2} / A_{1} = \rho_{1} / \rho_{2} * V_{1} / V_{2}\). Plugging in the results from Step 1 and Step 2 where we found \(\rho_{2} / \rho_{1} = 1.02772\) and \(V_{2} / V_{1}= 0.884705\), we find \(A_{2} / A_{1} = 1/1.02772 * 1 / 0.884705 = 1.086812\)

Key Concepts

Density Ratio Calculations

When dealing with gas flow in ducts, one important aspect to consider is the density ratio. Understanding how to calculate this ratio is key to analyzing the behavior of gas in the duct.

The ideal gas law, which states that pressure is the product of density, gas constant, and temperature (\( p = \rho R T \)), is crucial for finding the density ratio.

Given that the ideal gas law relates density, pressure, and temperature, we can derive the density ratio (\( \rho_2 / \rho_1 \)) using the relation:

• \( \rho_{2} / \rho_{1} = (p_2 / p_1) \times (T_1 / T_2) \)

This formula comes from the rearrangement of the ideal gas law, ensuring that both sides of the equation are dimensionally consistent.

From the given values, substituting \( p_{2} / p_{1} = 1.038 \) and \( T_{2} / T_{1} = 1.01 \), you calculate a density ratio of \( \rho_{2} / \rho_{1} = 1.038 / 1.01 \approx 1.02772 \). This means the density of the gas at the second point is slightly higher than at the first point.

Velocity Ratio Calculations

Velocity ratio is another important concept in analyzing gas flow in ducts. It's directly related to the Mach number, which is the ratio of the velocity of the gas to the speed of sound in its medium.

To find the velocity ratio, one must understand the relationship expressed in the formula:

• \( V_{2} / V_{1} = (M_{2} / M_{1}) \times (T_{2} / T_{1})^{1/2} \)

This formula links the Mach number at two different points, along with the temperatures at those points, to find how the velocity changes along the duct. \( \gamma \) is the ratio of specific heats, and \( R \) is the specific gas constant.

Plugging in the given values \( M_{2} / M_{1} = 0.88 \) and \( T_{2} / T_{1} = 1.01 \), you find \( V_{2} / V_{1} = 0.88 \times (1.01)^{1/2} \approx 0.884705 \). This indicates that the velocity of the gas decreases as it moves through the duct.

Duct Area Ratio Calculations

In the analysis of duct flows, understanding how the cross-sectional area of the duct changes with the gas conditions is crucial. The duct area ratio, \( A_{2} / A_{1} \), can be calculated from the mass flow rate equation.

Since the mass flow rate is conserved in steady flow, we have the relationship:

• \( \rho_{1} V_{1} A_{1} = \rho_{2} V_{2} A_{2} \)

This equation can be rearranged to solve for the duct area ratio:

• \( A_{2} / A_{1} = (\rho_{1} / \rho_{2}) \times (V_{1} / V_{2}) \)

By using the previously calculated values \( \rho_{2} / \rho_{1} = 1.02772 \) and \( V_{2} / V_{1} = 0.884705 \), we find \( A_{2} / A_{1} = (1/1.02772) \times (1/0.884705) \approx 1.086812 \).

This suggests that, assuming a constant mass flow rate, the duct must widen, or the area must increase, as the gas flows through it.

Ideal Gas Law

The ideal gas law is a fundamental principle used extensively in the study of gas flow in ducts. It relates the pressure, volume, and temperature of a gas with its density and is given by the equation: \( p = \rho R T \).

In this equation:

• \( p \) is the pressure,
• \( \rho \) is the density,
• \( R \) is the specific gas constant, and
• \( T \) is the temperature.

This law assumes an ideal gas, one that perfectly obeys these rules without interactions between molecules or volume occupied by the molecules themselves.

In calculations, the ideal gas law helps find out how changes in temperature and pressure affect other properties like density or volume, especially across different sections of duct flow.

Mach Number

The Mach number is a dimensionless unit that represents the ratio of the speed of a gas flow to the speed of sound in the same medium. It is calculated as:

• \( M = V / c \)

where \( V \) is the velocity of the gas and \( c \) is the speed of sound. The speed of sound depends on the temperature and the gas properties, calculated using:

• \( c = (\gamma R T)^{1/2} \)

The Mach number helps identify the flow regime of the gas; whether it’s subsonic (\( M < 1 \)), sonic (\( M = 1 \)), or supersonic (\( M > 1 \)). This identifier is essential in determining how flow might be affected by pressure changes and duct geometry.

When used in conjunction with other ratios, the Mach number provides valuable insights into how the velocity of gas changes in relation to varying duct conditions.