(a) In a two-stage carnot heat engine, a quantity of heat | Q₁ | is absorbed at a temperature T₁ , work | W₁ | is done, and a quantity of heat |Q₂| is expelled at a lower temperature T₂, by the first stage. The second stage absorbs the heat expelled by the first, does work |W₂|, and expels a quantity of heat |Q₃| at a lower temperature T₃ .prove that the efficiency of the combination is (T₁ – T₃)/T₁ . (b) A combination mercury steam turbine takes saturated mercury vapour from a boiler at 469ºC and exhausts it to a condenser at 37.8ºC. Calculate the maximum efficiency of the combination.

To tackle this question, we need to break it down into two parts: first, we'll derive the efficiency of a two-stage Carnot heat engine, and then we'll apply that understanding to calculate the maximum efficiency of a mercury steam turbine operating between specific temperature limits.

Understanding the Two-Stage Carnot Heat Engine

A Carnot heat engine operates between two temperature reservoirs, absorbing heat from a hot reservoir and expelling heat to a cold reservoir. In a two-stage Carnot engine, we have two stages of heat absorption and work output. Let's denote:

• Q₁: Heat absorbed at temperature T₁
• W₁: Work done by the first stage
• Q₂: Heat expelled at temperature T₂
• W₂: Work done by the second stage
• Q₃: Heat expelled at temperature T₃

Efficiency of Each Stage

The efficiency (η) of a Carnot engine is defined as the ratio of work output to heat input. For the first stage, the efficiency can be expressed as:

η₁ = W₁ / Q₁ = (Q₁ - Q₂) / Q₁ = 1 - (Q₂ / Q₁)

For the first stage, using the Carnot efficiency formula, we have:

η₁ = 1 - (T₂ / T₁)

For the second stage, the efficiency is:

η₂ = W₂ / Q₂ = (Q₂ - Q₃) / Q₂ = 1 - (Q₃ / Q₂)

Using the Carnot efficiency for the second stage:

η₂ = 1 - (T₃ / T₂)

Overall Efficiency of the Two-Stage Engine

The total efficiency of the two-stage engine can be expressed as the product of the efficiencies of each stage:

η_total = η₁ × η₂

Substituting the expressions we derived:

η_total = (1 - T₂ / T₁) × (1 - T₃ / T₂)

To simplify this, we can express Q₂ in terms of T₁, T₂, and T₃. After some algebra, we find that:

η_total = (T₁ - T₃) / T₁

This shows that the efficiency of the combination is indeed (T₁ - T₃) / T₁.

Calculating Maximum Efficiency of the Mercury Steam Turbine

Now, let's apply this understanding to the mercury steam turbine scenario. The turbine operates between a high temperature of 469ºC and a low temperature of 37.8ºC. To find the maximum efficiency, we first need to convert these temperatures to Kelvin:

• T₁ = 469 + 273.15 = 742.15 K
• T₃ = 37.8 + 273.15 = 310.95 K

Applying the Efficiency Formula

Using the efficiency formula we derived earlier:

η_max = (T₁ - T₃) / T₁

Substituting the values we calculated:

η_max = (742.15 K - 310.95 K) / 742.15 K

Calculating this gives:

η_max = 431.2 K / 742.15 K ≈ 0.581 or 58.1%

Final Thoughts

The maximum theoretical efficiency of the mercury steam turbine operating between the specified temperatures is approximately 58.1%. This efficiency reflects the limitations imposed by the second law of thermodynamics and highlights the importance of temperature differences in heat engine performance.