Let's solve the heat pump problem **step by step**. --- ## **Given Data** - **Working fluid:** R-134a - **Low pressure, \( P_L \):** 140 kPa - **High pressure, \( P_H \):** 900 kPa - **Mass flow rate, \( \dot{m} \):** .06 kg/s - **Cycle:** Ideal vapor-compression --- ## **Step 1: Draw a Simple Schematic** 1. **Compressor:** (1 → 2: Compression, isentropic) 2. **Condenser:** (2 → 3: Heat rejection at constant \( P_H \)) 3. **Expansion Valve:** (3 → 4: Expansion at constant enthalpy) 4. **Evaporator:** (4 → 1: Heat absorption at constant \( P_L \)) --- ## **Step 2: Find State Points (using R-134a tables)** We'll need **enthalpies** at each state: ### **State 1: Inlet to Compressor (Evaporator exit)** - \( P_1 = 140\, \text{kPa} \) - Saturated vapor: \( x_1 = 1 \) **From R-134a tables:** - At 140 kPa, \( h_1 \approx 239.15\, \text{kJ/kg} \) - \( s_1 \approx .9172\, \text{kJ/kg}\cdot K \) ### **State 2: Exit Compressor (Condenser inlet)** - \( P_2 = 900\, \text{kPa} \) - Find \( h_2 \): Isentropic compression (\( s_2 = s_1 \)) From R-134a superheated tables at \( P_2 = 900\, \text{kPa} \): Look up \( s_1 = .9172 \) at 900 kPa: - At 900 kPa: - \( T = 40^\circ C \): \( s = .8997, h = 263.86 \) - \( T = 50^\circ C \): \( s = .9305, h = 271.11 \) Interpolate for \( s_2 = s_1 = .9172 \): \[ h_2 = h_{40^\circ C} + \frac{(s_1 - s_{40^\circ C})}{(s_{50^\circ C} - s_{40^\circ C})} \times (h_{50^\circ C} - h_{40^\circ C}) \] \[ h_2 = 263.86 + \frac{(.9172 - .8997)}{(.9305 - .8997)} \times (271.11 - 263.86) \] \[ = 263.86 + \frac{.0175}{.0308} \times 7.25 \] \[ = 263.86 + .568 \times 7.25 = 263.86 + 4.12 = 267.98 \, \text{kJ/kg} \] ### **State 3: Exit Condenser (Expansion valve inlet)** - \( P_3 = 900\, \text{kPa} \) - Saturated liquid: \( x_3 = \) - \( h_3 \) at 900 kPa, sat liq: \( h_3 \approx 95.50\, \text{kJ/kg} \) ### **State 4: Exit Expansion Valve (Evaporator inlet)** - \( P_4 = 140\, \text{kPa} \) - \( h_4 = h_3 \) (isenthalpic expansion) - \( h_4 = 95.50\, \text{kJ/kg} \) --- ## **Step 3: Calculate Required Quantities** ### **(a) Heating load (condenser heat output):** \[ \dot{Q}_{out} = \dot{m} (h_2 - h_3) \] \[ \dot{Q}_{out} = .06 \times (267.98 - 95.50) = .06 \times 172.48 = \boxed{10.35\, \text{kW}} \] ### **(b) Power input (compressor work):** \[ \dot{W}_{in} = \dot{m} (h_2 - h_1) \] \[ \dot{W}_{in} = .06 \times (267.98 - 239.15) = .06 \times 28.83 = \boxed{1.73\, \text{kW}} \] ### **(c) Coefficient of performance (COP):** \[ COP_{HP} = \frac{\dot{Q}_{out}}{\dot{W}_{in}} = \frac{10.35}{1.73} = \boxed{5.98} \] ### **(d) Warmest outside temperature at which cycle is unable to operate** This is when the evaporator is at the **saturated temperature** corresponding to \( P_{low} = 140\, \text{kPa} \): - From R-134a tables, at 140 kPa: - \( T_{sat} \approx -18.8^\circ C \) So, **outside temperature cannot be higher than \( -18.8^\circ C \)** for heat absorption. \[ \boxed{-18.8^\circ C} \] --- ## **Summary Table** | Part | Answer | Units | |------|---------------|----------| | (a) | 10.35 | kW | | (b) | 1.73 | kW | | (c) | 5.98 | | | (d) | -18.8 | °C | --- ### **Let me know if you want details for each calculation or help with R-134a tables!**