Question

Show that \(\mathrm{COP}_{\mathrm{HP}}=\mathrm{COP}_{\mathrm{R}}+1\) when both the heat pump and the refrigerator have the same \(Q_{L}\) and \(Q_{H}\) values.

Short answer

Question: Show that the Coefficient of Performance (COP) for a heat pump is equal to the COP for a refrigerator plus one when both the heat pump and the refrigerator have the same heat transfer values at low temperatures (\(Q_L\)) and high temperatures (\(Q_H\)). Answer: The relationship between the COP for a heat pump (COP_HP) and the COP for a refrigerator (COP_R), when they have the same heat transfer values at low and high temperatures, can be shown as \(COP_{HP} = COP_{R} + 1\).

Step-by-step solution

Equation for COP_HP

\(COP_{HP} = \frac{Q_H}{W}\)

Equation for COP_R

\(COP_{R} = \frac{Q_L}{W}\) Now, since both the heat pump and refrigerator have the same heat transfer values, we can use energy conservation to relate \(Q_L\), \(Q_H\), and \(W\):

Energy Conservation Equation

\(Q_H = Q_L + W\) We must eliminate W from the COP equations and show that \(COP_{HP} = COP_{R} + 1\). Rearrange the energy conservation equation to express \(W\) in terms of \(Q_H\) and \(Q_L\):

Rewrite the Energy Conservation Equation to Isolate W

\(W = Q_H - Q_L\) Now, substitute this expression for \(W\) into the equations for COP_HP and COP_R:

Substitute W in the COP_HP Equation

\(COP_{HP} = \frac{Q_H}{Q_H-Q_L}\)

Substitute W in the COP_R Equation

\(COP_{R} = \frac{Q_L}{Q_H-Q_L}\) Now we need to show that \(COP_{HP} = COP_{R} + 1\). To do this, let's manipulate the COP_R equation to isolate \(COP_R + 1\):

Manipulate COP_R Equation

\((COP_{R} + 1) = \frac{Q_L}{Q_H-Q_L} + 1 = \frac{Q_L + (Q_H-Q_L)}{Q_H-Q_L} = \frac{Q_H}{Q_H-Q_L}\) As we can see, \((COP_{R} + 1)\) ends up equaling the same expression as \(COP_{HP}\). Therefore, we have proven that:

Show the Relationship

\(COP_{HP} = COP_{R} + 1\) And since we are given in the problem that both the heat pump and the refrigerator have the same \(Q_L\) and \(Q_H\) values, the relationship holds true.