Question

An air-conditioner with refrigerant-134a as the working fluid is used to keep a room at \(23^{\circ} \mathrm{C}\) by rejecting the waste heat to the outdoor air at \(34^{\circ} \mathrm{C}\). The room gains heat through the walls and the windows at a rate of \(250 \mathrm{kJ} / \mathrm{min}\) while the heat generated by the computer, \(\mathrm{TV}\) and lights amounts to \(900 \mathrm{W}\). The refrigerant enters the compressor at \(400 \mathrm{kPa}\) as a saturated vapor at a rate of \(80 \mathrm{L} / \mathrm{min}\) and leaves at \(1200 \mathrm{kPa}\) and \(70^{\circ} \mathrm{C}\). Determine \((a)\) the actual \(\mathrm{COP},(b)\) the maximum \(\mathrm{COP}\), and \((c)\) the minimum volume flow rate of the refrigerant at the compressor inlet for the same compressor inlet and exit conditions.

Short answer

In summary: (a) The actual Coefficient of Performance (COP) of the air-conditioner is 6.23. (b) The maximum COP (Carnot COP) is 27.43. (c) The minimum volume flow rate of the refrigerant at the compressor inlet under the same compressor inlet and exit conditions is 18.1 L/min.

Step-by-step solution

Calculate Total Heat Transfer Rate

To calculate the total heat transfer rate (Q) to be removed from the room, we need to consider both the heat transfer through the walls and windows, and the heat generated by the computer, TV and lights. The heat transfer through the walls and windows is given, and we can easily convert the heat generation rate from watts to kJ/min using the relation 1 W = 60 kJ/min. Heat transfer through walls and windows: 250 kJ/min Heat generated by computer, TV and lights: 900 W * 60 kJ/min/W = 54 kJ/min Total heat transfer rate (Q) = 250 kJ/min + 54 kJ/min = 304 kJ/min

Calculate Work Input to the Compressor

Using the given information about the refrigerant conditions at the compressor inlet and exit, we can find the difference in enthalpy (Delta h) between these two points. From the refrigerant-134a tables or software, we can find the values of enthalpy at the given temperature and pressure conditions. At the compressor inlet (Point 1): - Pressure, P1 = 400 kPa - Temperature, T1 = 23 °C - Enthalpy, h1 = 412.92 kJ/kg (saturated vapor) At the compressor exit (Point 2): - Pressure, P2 = 1200 kPa - Temperature, T2 = 70 °C - Enthalpy, h2 = 463.83 kJ/kg Delta h = h2 - h1 = 463.83 kJ/kg - 412.92 kJ/kg = 50.91 kJ/kg Now, we can calculate the total mass flow rate (m_dot) of the refrigerant using the given volume flow rate (80 L/min) and the specific volume (v1) of the refrigerant at the compressor inlet. Specific volume at point 1, v1 = 0.08348 m^3/kg (refrigerant-134a tables) Volume flow rate = 80 L/min = 0.08 m^3/min m_dot = (volume flow rate) / v1 = 0.08 m^3/min / 0.08348 m^3/kg = 0.958 kg/min The work input to the compressor (W_dot) can be calculated by multiplying the mass flow rate (m_dot) and the difference in enthalpy (Delta h). W_dot = m_dot * Delta h = 0.958 kg/min * 50.91 kJ/kg = 48.76 kJ/min

Calculate the Actual COP

The Coefficient of Performance (COP) of the air-conditioner is defined as the heat transfer rate (Q) divided by the work input to the compressor (W_dot). Using the values calculated in steps 1 and 2: Actual COP = Q / W_dot = 304 kJ/min / 48.76 kJ/min = 6.23

Calculate the Maximum COP

The maximum COP of the air-conditioner occurs under the Carnot cycle conditions. The Carnot COP can be calculated using the given room and outdoor temperatures. Room temperature, T_room = 23 °C = 296.15 K Outdoor temperature, T_outdoor = 34 °C = 307.15 K Maximum COP (Carnot COP) = T_room / (T_outdoor - T_room) = 296.15 K / (307.15 K - 296.15 K) = 27.43

Calculate the Minimum Volume Flow Rate

To calculate the minimum volume flow rate of the refrigerant at the compressor inlet, we need to make use of the maximum COP (Carnot COP) calculated in step 4. By equating the actual COP to the maximum COP, we can find the minimum work input to the compressor (W_dot_min). Carnot COP = Q / W_dot_min W_dot_min = Q / (Carnot COP) = 304 kJ/min / 27.43 = 11.08 kJ/min Using the minimum work input to the compressor (W_dot_min) and the difference in enthalpy (Delta h), we can calculate the corresponding minimum mass flow rate (m_dot_min). m_dot_min = W_dot_min / Delta h = 11.08 kJ/min / 50.91 kJ/kg = 0.217 kg/min Finally, we can calculate the minimum volume flow rate at the compressor inlet using the specific volume (v1) of the refrigerant at the compressor inlet. Minimum volume flow rate = m_dot_min * v1 = 0.217 kg/min * 0.08348 m^3/kg = 0.0181 m^3/min = 18.1 L/min The minimum volume flow rate of the refrigerant at the compressor inlet is 18.1 L/min under the same compressor inlet and exit conditions.