Heat engine

(source: commons.wikimedia.org)

• $W=Q_H-Q_L$ (from the first law of thermodynamics)
• ${\eta }_{\text{th}}=\frac{W}{Q_{\mathrm{H}}}=\frac{Q_{\mathrm{H}}-Q_{\mathrm{L}}}{Q_{\mathrm{H}}}=1-\frac{Q_{\mathrm{L}}}{Q_{\mathrm{H}}}$
• Carnot’s theorem
  • ${\eta }_{\text{max}}=\frac{T_{\mathrm{H}}-T_{\mathrm{L}}}{T_{\mathrm{H}}}=1-\frac{T_{\mathrm{L}}}{T_{\mathrm{H}}}$

COP

• heat pump:
  • $\text{COP}=\frac{Q_{\text{H}}}{W}$
  • ${\text{COP}}_{\text{ideal}}=\frac{T_{\mathrm{H}}}{T_{\mathrm{H}}-T_{\mathrm{L}}}$
• refrigerator/AC:
  • $\text{COP}=\frac{Q_{\text{L}}}{W}$
  • ${\text{COP}}_{\text{ideal}}=\frac{T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}$