Question

Two air streams of mass flow rates \( \dot{m_1} \) and \( \dot{m_2} \) enter a mixing chamber and exit after perfect mixing. The corresponding temperatures of the inlet streams are \( T_1 \) and \( T_2 \), respectively. Heat loss rate from the mixing chamber to the surrounding is \( \dot{Q} \). Assume that the process is steady, specific heat capacity is constant, and air behaves as an ideal gas. Identify the correct expression for the final exit temperature \( T_3 \) after mixing. The mass specific heat capacities of the gas at constant volume and constant pressure are \( c_v \) and \( c_p \), respectively. Neglect the bulk kinetic and potential energies of the streams.

• A. \( T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} - \frac{\dot{Q}}{c_v(\dot{m_1} + \dot{m_2})} \)
• B. \( T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} + \frac{\dot{Q}}{c_p(\dot{m_1} + \dot{m_2})} \)
• C. \( T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} - \frac{\dot{Q}}{c_p(\dot{m_1} + \dot{m_2})} \)
• D. \( T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} + \frac{\dot{Q}}{c_v(\dot{m_1} + \dot{m_2})} \)

Hint:

In mixing processes, use the first law of thermodynamics, expressing enthalpy in terms of temperature, to derive the final temperature after mixing.

Answer 1

The Correct Option is C

Step 1: Use the first law of thermodynamics.
For the steady-state process, the energy balance equation can be written as: \[ \dot{m_1} h_1 + \dot{m_2} h_2 - \dot{Q} = (\dot{m_1} + \dot{m_2}) h_3 \] where \( h_1 \), \( h_2 \), and \( h_3 \) are the specific enthalpies of the air streams at inlet and exit, respectively.

Step 2: Express specific enthalpy in terms of temperature.
For an ideal gas, specific enthalpy is related to temperature by: \[ h = c_p T \] Thus, the energy balance equation becomes: \[ \dot{m_1} c_p T_1 + \dot{m_2} c_p T_2 - \dot{Q} = (\dot{m_1} + \dot{m_2}) c_p T_3 \]

Step 3: Solve for the final temperature \( T_3 \).
Rearranging the equation to solve for \( T_3 \), we get: \[ T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} - \frac{\dot{Q}}{c_p (\dot{m_1} + \dot{m_2})} \]

Step 4: Conclusion.
Thus, the correct expression for the final temperature \( T_3 \) after mixing is option (C).

Final Answer: (C) \( T_3 = \frac{\dot{m_1} T_1 + \dot{m_2} T_2}{\dot{m_1} + \dot{m_2}} - \frac{\dot{Q}}{c_p(\dot{m_1} + \dot{m_2})} \)