VIPSolutionsGive step-by-step solution with explanation and final answer: Refrigeration & Air Conditioning in Mechanical Engineering - Numerical Practice An air conditioner operates between temperatures of 20\deg C and 40\deg C. Calculate the theoretical COP of the system.
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Give step-by-step solution with explanation and final answer: Refrigeration & Air Conditioning in Mechanical Engineering - Numerical Practice An air conditioner operates between temperatures of 20\deg C and 40\deg C. Calculate the theoretical COP of the system.
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Asked by: Rahul
Created at: 2026-01-20 03:48:48
Asked by: Rahul
Created at: 2026-01-20 03:48:48
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# Step-by-Step Solution: Theoretical COP of Air Conditioner
## 1. Understanding the Problem
- **Air conditioner**: Works as a heat pump, transferring heat from a cold space (inside) to a hot space (outside).
- **Given temperatures**:
- Cold reservoir (\( T_L \)): \( 20^\circ C \)
- Hot reservoir (\( T_H \)): \( 40^\circ C \)
- **To find**: Theoretical Coefficient of Performance (COP) for a refrigerator/air conditioner.
## 2. Converting Temperatures to Kelvin
The Carnot COP formula uses **Kelvin**. Convert Celsius to Kelvin:
\[
T(K) = T(^\circ C) + 273.15
\]
- \( T_L = 20^\circ C + 273.15 = 293.15\,K \)
- \( T_H = 40^\circ C + 273.15 = 313.15\,K \)
## 3. COP Formula for a Refrigerator (Carnot Cycle)
\[
\text{COP}_{\text{ref}} = \frac{T_L}{T_H - T_L}
\]
## 4. Substitute the Values
\[
\text{COP}_{\text{ref}} = \frac{293.15}{313.15 - 293.15}
\]
\[
\text{COP}_{\text{ref}} = \frac{293.15}{20}
\]
## 5. Calculate the Value
\[
\text{COP}_{\text{ref}} = 14.6575
\]
## 6. Final Answer
**Theoretical COP of the air conditioning system is:**
\[
\boxed{14.66}
\]
---
**Summary Table:**
| Parameter | Value |
|-----------|------------------|
| \(T_L\) | 293.15 K |
| \(T_H\) | 313.15 K |
| COP | 14.66 |
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