Discussion
An air-conditioning system consists of an insulated rigid mixing chamber designed to supply air at 24 °C to a building. The mixing chamber mixes two air streams: (i) a cold air stream at 10 °C and mass flow rate \(\dot{m}_{c}\) (kg/s), and (ii) a stream of fresh ambient air at 30 °C and mass flow rate \(\dot{m}_{a}\) (kg/s). Assume air to be an ideal gas with constant specific heat (\(c_{p} = 1.005\) kJ/(kg K), \(\gamma = c_{p}/c_{v} = 1.4\)). Neglect change in kinetic and potential energies as compared to change in enthalpy. Under the steady state condition, the ratio of the mass flow rates of the two streams (\(\dot{m}_{c}/\dot{m}_{a}\)) is
Answer: B
The ratio is \(\frac{3}{7}\).
For an insulated steady mixing chamber with constant \(c_{p}\), the enthalpy given up by the warm stream equals that gained by the cold stream:
\[ \begin{aligned} \dot{m}_{c}c_{p}(24 - 10) &= \dot{m}_{a}c_{p}(30 - 24) \\ \frac{\dot{m}_{c}}{\dot{m}_{a}} &= \frac{6}{14} = \frac{3}{7} \end{aligned} \]
\(7/3\) is the inverse ratio, \(\dot{m}_{a}/\dot{m}_{c}\). \(2/7\) and \(4/7\) do not satisfy the energy balance. The values of \(c_{p}\) and \(\gamma\) cancel and are not needed.
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