An isentropic air-turbine is used to supply 0.1 kg/s of air at 0.1 MN/m² and 285 K to a cabin. The pressure at the inlet to the turbine is 0.4 MN/m². What will be the power developed by the turbine? [Consider \(C_p = 1 \, \text{kJ/kg-K}, \gamma = 1.41, 4^{\frac{(0.41)}{(1.41 ) }} = 1.5\) ]

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SJVN ET Mechanical 2019 Official Paper

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Answer 1

Concept:

We use isentropic expansion relations in a turbine to determine the power developed when supplying air to a cabin.

Given:

Mass flow rate, \( \dot{m} = 0.1 \, \text{kg/s} \)
Inlet pressure, \( P_1 = 0.4 \, \text{MN/m²} = 400 \, \text{kPa} \)
Exit pressure, \( P_2 = 0.1 \, \text{MN/m²} = 100 \, \text{kPa} \)
Exit temperature, \( T_2 = 285 \, \text{K} \)
Specific heat at constant pressure, \( C_p = 1 \, \text{kJ/kg·K} \)
Ratio of specific heats, \( \gamma = 1.41 \)
Given relation: \( 4^{\frac{(\gamma - 1)}{\gamma}} = 1.5 \)

Step 1: Determine the inlet temperature (T₁)

For an isentropic process:

\( \frac{T_1}{T_2} = \left(\frac{P_1}{P_2}\right)^{\frac{\gamma - 1}{\gamma}} = 1.5 \)

\( T_1 = 285 \times 1.5 = 427.5 \, \text{K} \)

Step 2: Calculate the work output (power)

The power developed by the turbine is given by:

\( W = \dot{m} \times C_p \times (T_1 - T_2) \)

\( W = 0.1 \times 1 \times (427.5 - 285) = 14.25 \, \text{kW} \)