Question

Show that the coefficient of performance for a reversible heat pump is $1 /\left(1-T_{\mathrm{C}} / T_{\mathrm{H}}\right)$.

Short answer

Question: Show that the coefficient of performance (COP) for a reversible heat pump is given by the formula: \[COP = \frac{1}{1-\frac{T_C}{T_H}}\]

Step-by-step solution

Define the coefficient of performance for a heat pump

The coefficient of performance (COP) for a heat pump is the ratio of useful heat energy transferred to the cold reservoir (Q_C) to the work input (W) required to run the heat pump. Mathematically, it is given by: \[COP = \frac{Q_C}{W}\]

Define the efficiency of a reversible heat engine

The efficiency of a heat engine is the ratio of the work output (W) to the heat energy input (Q_H). A reversible heat engine has the maximum efficiency possible and the efficiency is given by: \[\eta_{max} = 1 - \frac{T_C}{T_H}\]

Express work (W) in terms of heat energies (Q_H and Q_C)

For a reversible heat engine, we have the principle of energy conservation, meaning that the total energy balance is given by: \[Q_H = Q_C + W\] Now, we can solve for W: \[W = Q_H - Q_C\]

Substitute the relation of W in COP equation

Replace W in the COP equation with the expression we found in Step 3: \[COP = \frac{Q_C}{Q_H - Q_C}\]

Express Q_C in terms of Q_H and T_C and T_H

From the efficiency equation, we can get the relationship between heat energies and temperatures as follows: \[\eta_{max} = 1 - \frac{T_C}{T_H} = 1 - \frac{Q_C}{Q_H}\] Now, we can solve for Q_C: \[Q_C = Q_H \frac{T_C}{T_H}\]

Substitute the relation of Q_C in the COP equation

Replace Q_C in the COP equation with the expression we found in Step 5: \[COP = \frac{Q_H \frac{T_C}{T_H}}{Q_H - Q_H \frac{T_C}{T_H}}\]

Simplify the COP equation

Now, we can simplify the equation to get the desired COP formula: \[\begin{aligned} COP &= \frac{Q_H \frac{T_C}{T_H}}{Q_H \left(1 - \frac{T_C}{T_H}\right)} \\ &= \frac{\frac{T_C}{T_H}}{1 - \frac{T_C}{T_H}} \\ &= \frac{1}{\frac{1}{(\frac{T_C}{T_H})} - 1} \\ &= \frac{1}{1-\frac{T_C}{T_H}} \end{aligned}\] So, the coefficient of performance for a reversible heat pump is: \[COP = \frac{1}{1-\frac{T_C}{T_H}}\]