Question

Devise a refrigeration cycle that works on the reversed Stirling cycle. Also, determine the COP for this cycle.

Short answer

Answer: The formula for the COP is given by: \(COP = \frac{T_3 - T_2}{(T_2 - T_1) + (T_4 - T_3)}\)

Step-by-step solution

Understand the Stirling Cycle

The Stirling cycle consists of the following processes: 1. Isothermal compression 2. Isochoric heat removal 3. Isothermal expansion 4. Isochoric heat addition

Devise the Reversed Stirling Cycle

In a reversed Stirling cycle, the processes will be reversed. Thus the processes are: 1. Isothermal expansion 2. Isochoric heat addition 3. Isothermal compression 4. Isochoric heat removal The reversed Stirling cycle can be used to model a refrigeration cycle since it involves heat absorption (Isochoric heat addition) and heat rejection (Isochoric heat removal), along with thermodynamic work input (Isothermal compression).

Apply the First Law of Thermodynamics for Each Process

We will apply the first law of thermodynamics for each process to calculate the heat transfers and work inputs or outputs: 1. Isothermal expansion: \(\Delta Q_1 = W_1 = mC_v(T_2 - T_1)\) 2. Isochoric heat addition: \(\Delta Q_2 = mC_v(T_3 - T_2) = W_2 = 0\) 3. Isothermal compression: \(\Delta Q_3 = W_3 = mC_v(T_4 - T_3)\) 4. Isochoric heat removal: \(\Delta Q_4 = mC_v(T_4 - T_1) = W_4 = 0\)

Calculate the COP

The COP for the refrigeration cycle is given by: $$ COP = \frac{\text{Desired Output (Heat Absorption)}}{\text{Required Input (Work)}} = \frac{Q_2}{W_1 + W_3} $$ Substitute the heat transfers and work values from Step 3 into the equation: $$ COP = \frac{mC_v(T_3 - T_2)}{mC_v(T_2 - T_1) + mC_v(T_4 - T_3)} $$ To simplify the equation, observe that \(mC_v\) appears in all terms, and hence can be canceled out: $$ COP = \frac{T_3 - T_2}{(T_2 - T_1) + (T_4 - T_3)} $$ Now, we have derived the COP for the refrigeration cycle based on the reversed Stirling cycle.