Consider two Carnot engines of efficiencies ₁ and ₂ . The first engine absorbs heat Q₁ from a heat reservoir A and releases heat Q₂ to a heat reservoir B . The second engine takes heat Q₂ from B and releases heat Q₃ to a heat reservoir C . If Q₁ > Q₂ > Q₃ , what is the net efficiency of this combination of the two Carnot engines?

Step-by-step solution

The efficiency of the first Carnot engine is given by ₁ = 1 - Q₂ Q₁ . This can be rewritten as Q₂ Q₁ = 1 - ₁ . The efficiency of the second Carnot engine is given by ₂ = 1 - Q₃ Q₂ . This can be rewritten as Q₃ Q₂ = 1 - ₂ . The net efficiency of the combination of the two engines is the total work done divided by the total heat absorbed from the source. Total work done is W = W₁ + W₂ = (Q₁ - Q₂) + (Q₂ - Q₃) = Q₁ - Q₃ . The net efficiency is = W Q₁ = Q₁ - Q₃ Q₁ = 1 - Q₃ Q₁ . Multiplying the ratios of heat gives Q₃ Q₁