Question

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:

• A. $\dfrac{7}{3}$
• B. $\dfrac{3}{7}$
• C. $\dfrac{2}{7}$
• D. $\dfrac{4}{7}$

Hint:

In adiabatic mixing chambers, energy balance reduces to a simple enthalpy balance. Always set the weighted-average temperature equal to the exit temperature.

Answer 1

The Correct Option is B

Since the mixing chamber is insulated and rigid, the steady-flow energy equation reduces to enthalpy balance:
\[ \dot{m}_c c_p T_c + \dot{m}_a c_p T_a = (\dot{m}_c + \dot{m}_a)c_p T_{\text{mix}} \] Cancel $c_p$ throughout:
\[ \dot{m}_c T_c + \dot{m}_a T_a = (\dot{m}_c + \dot{m}_a)T_{\text{mix}} \] Substitute the given temperatures: \[ T_c = 10^\circ\text{C}, \quad T_a = 30^\circ\text{C}, \quad T_{\text{mix}} = 24^\circ\text{C} \] Insert values:
\[ 10\dot{m}_c + 30\dot{m}_a = 24(\dot{m}_c + \dot{m}_a) \] Expand the right-hand side:
\[ 10\dot{m}_c + 30\dot{m}_a = 24\dot{m}_c + 24\dot{m}_a \] Rearrange:
\[ 30\dot{m}_a - 24\dot{m}_a = 24\dot{m}_c - 10\dot{m}_c \] \[ 6\dot{m}_a = 14\dot{m}_c \] Thus, the required ratio is:
\[ \frac{\dot{m}_c}{\dot{m}_a} = \frac{6}{14} = \frac{3}{7} \]