Question

Consider an isentropic flow of air (ratio of specific heats = 1.4) through a duct as shown in the figure. The variations in the flow across the cross-section are negligible. The flow conditions at Location 1 are given as follows:
\(P_1 = 100\) kPa, \(\rho_1 = 1.2\) kg/m³, \(u_1 = 400\) m/s
The duct cross-sectional area at Location 2 is given by \(A_2 = 2A_1\), where \(A_1\) denotes the duct cross-sectional area at Location 1. Which one of the given statements about the velocity \(u_2\) and pressure \(P_2\) at Location 2 is TRUE?

• A. \(u_2<u_1 , P_2<P_1\)
• B. \(u_2<u_1 , P_2>P_1\)
• C. \(u_2>u_1 , P_2<P_1\)
• D. \(u_2>u_1 , P_2>P_1\)

Hint:

Remember the counter-intuitive behavior of supersonic flow:
- {Diverging duct (Diffuser for subsonic, Nozzle for supersonic):} Subsonic flow decelerates, Supersonic flow accelerates.
- {Converging duct (Nozzle for subsonic, Diffuser for supersonic):} Subsonic flow accelerates, Supersonic flow decelerates.
In both regimes, pressure change is always opposite to velocity change for isentropic flow.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem deals with one-dimensional isentropic compressible flow. The key is to first determine if the flow is subsonic or supersonic at the inlet (Location 1). The behavior of the flow properties (velocity, pressure) in a variable-area duct depends critically on the Mach number.
Step 2: Key Formula or Approach:
1. Calculate the speed of sound \(c_1\) at Location 1 using the formula \(c = \sqrt{\gamma P / \rho}\).
2. Calculate the Mach number \(M_1 = u_1 / c_1\).
3. Use the differential area-velocity relation for isentropic flow: \[ \frac{dA}{A} = (M^2 - 1) \frac{du}{u} \] This equation tells us how velocity changes with area for different Mach regimes.
4. From the change in velocity, determine the change in pressure using the energy equation (or isentropic relations). For an isentropic process, an increase in velocity corresponds to a decrease in pressure, and vice versa.
Step 3: Detailed Calculation:
1. Calculate the speed of sound at Location 1: \[ c_1 = \sqrt{\frac{\gamma P_1}{\rho_1}} = \sqrt{\frac{1.4 \times (100 \times 10^3 \text{ Pa})}{1.2 \text{ kg/m}^3}} = \sqrt{\frac{140000}{1.2}} \approx 341.6 \text{ m/s} \] 2. Calculate the Mach number at Location 1: \[ M_1 = \frac{u_1}{c_1} = \frac{400 \text{ m/s}}{341.6 \text{ m/s}} \approx 1.17 \] Since \(M_1>1\), the flow at the inlet is supersonic. 3. Analyze the flow through the diverging duct: The duct is diverging, which means the area is increasing (\(dA>0\)). The flow is supersonic (\(M>1\)), which means the term \((M^2 - 1)\) is positive. Using the area-velocity relation: \[ \frac{dA}{A} = (M^2 - 1) \frac{du}{u} \] Since \(\frac{dA}{A}\) is positive and \((M^2 - 1)\) is positive, the term \(\frac{du}{u}\) must also be positive. A positive \(\frac{du}{u}\) implies that the velocity increases. Therefore, \(\mathbf{u_2>u_1}\). 4. Determine the change in pressure:
For any isentropic flow (subsonic or supersonic), an increase in velocity is always accompanied by a decrease in static pressure and temperature. This can be seen from the differential form of the energy equation or Euler's equation (\(dP = -\rho u du\)). Since \(du\) is positive, \(dP\) must be negative.
Therefore, the pressure decreases: \(\mathbf{P_2<P_1}\).
Step 4: Final Answer:
The velocity at Location 2 is greater than at Location 1 (\(u_2>u_1\)), and the pressure at Location 2 is less than at Location 1 (\(P_2<P_1\)). This corresponds to option (C).
Step 5: Why This is Correct:
The calculations correctly identify the inlet flow as supersonic. For supersonic isentropic flow, a diverging duct acts as a nozzle, accelerating the flow and decreasing its pressure. This is the principle behind the diverging section of a de Laval nozzle used to produce supersonic jets.