Problem Statement
A gas with a velocity of 350 m/s is flowing through a horizontal pipe at a section where pressure is 8 N/cm² (absolute) and temperature is 30°C. The pipe changes in diameter and at this section the pressure is 12 N/cm² (absolute). Find the velocity of the gas at this section if the flow of the gas is adiabatic. Take R = 287 J/kg K and k = 1.4.
Given Data & Constants
- Gas constant, \(R = 287 \, \text{J/kg K}\)
- Adiabatic index, \(k = 1.4\)
- Section 1:
- Velocity, \(V_1 = 350 \, \text{m/s}\)
- Absolute Pressure, \(P_1 = 8 \, \text{N/cm}^2 = 80,000 \, \text{N/m}^2\)
- Temperature, \(T_1 = 30^\circ\text{C} = 303.15 \, \text{K}\)
- Section 2:
- Absolute Pressure, \(P_2 = 12 \, \text{N/cm}^2 = 120,000 \, \text{N/m}^2\)
Solution
1. Calculate Temperature at Section 2 (\(T_2\))
For a reversible adiabatic (isentropic) process, the relationship between pressure and temperature is:
$$ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} $$
$$ T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{1.4-1}{1.4}} = 303.15 \times \left(\frac{120000}{80000}\right)^{0.2857} $$
$$ T_2 = 303.15 \times (1.5)^{0.2857} \approx 303.15 \times 1.1255 \approx 341.2 \, \text{K} $$
2. Apply the Steady Flow Energy Equation
For an adiabatic flow in a horizontal pipe, the energy equation relates the change in enthalpy to the change in kinetic energy.
$$ h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2} $$
$$ \text{Since } h = C_p T \text{ and } C_p = \frac{k R}{k-1}: $$
$$ \frac{k R T_1}{k-1} + \frac{V_1^2}{2} = \frac{k R T_2}{k-1} + \frac{V_2^2}{2} $$
$$ \frac{V_2^2}{2} = \frac{k R}{k-1}(T_1 - T_2) + \frac{V_1^2}{2} $$
3. Solve for the Velocity at Section 2 (\(V_2\))
$$ C_p = \frac{1.4 \times 287}{1.4 - 1} = 1004.5 \, \text{J/kg K} $$
$$ \frac{V_2^2}{2} = 1004.5 \times (303.15 - 341.2) + \frac{350^2}{2} $$
$$ \frac{V_2^2}{2} = 1004.5 \times (-38.05) + 61250 $$
$$ \frac{V_2^2}{2} = -38221 + 61250 = 23029 $$
$$ V_2^2 = 46058 \implies V_2 \approx 214.6 \, \text{m/s} $$
Final Result:
The velocity of the gas at the second section is approximately 214.6 m/s.
