Question

A perfect gas flows through a frictionless constant-area duct with heat addition. The inlet conditions are as follows: pressure 100 kPa, density 1 kg/m\(^3\), and velocity 100 m/s. At a particular downstream location, the gas velocity is 200 m/s. The static pressure at the downstream location is __________ kPa (answer in integer).

Hint:

For a frictionless, constant-area duct with heat addition, use the continuity equation to relate the densities and velocities at the inlet and downstream locations. Then, apply the ideal gas law to find the static pressure.

Answer 1

Assumptions
Perfect gas behavior
Frictionless flow (\( \tau_w = 0 \))
Constant-area duct (\( A_1 = A_2 \))
Steady flow

Step 1: Apply Continuity Equation
For a constant-area duct:
\[ \rho_1 V_1 = \rho_2 V_2 \]
\[ 1 \times 100 = \rho_2 \times 200 \]
\[ \rho_2 = \frac{100}{200} = 0.5 \, \text{kg/m}^3 \]
Step 2: Apply Momentum Equation
The momentum equation for frictionless flow:
\[ p_1 + \rho_1 V_1^2 = p_2 + \rho_2 V_2^2 \]
Substitute known values:
\[ 100 \times 10^3 + 1 \times (100)^2 = p_2 + 0.5 \times (200)^2 \]
\[ 100,000 + 10,000 = p_2 + 20,000 \]
\[ 110,000 = p_2 + 20,000 \]
\[ p_2 = 110,000 - 20,000 = 90,000 \, \text{Pa} = 90 \, \text{kPa} \]
Final Answer
The static pressure at the downstream location is \( \boxed{90} \, \text{kPa} \).