Question

Practical steam engines utilize 450ºC steam, which is later exhausted at 270ºC. (a) What is the maximum efficiency that such a heat engine can have? (b) Since 270ºC steam is still quite hot, a second steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its exhaust has a temperature of 150ºC? (c) What is the overall efficiency of the two engines? (d) Show that this is the same efficiency as a single Carnot engine operating between 450ºC and 150ºC. Explicitly show how you follow the steps in the Problem-Solving Strategies for Thermodynamics.

a) 40.0%
b) 33.3%
c) 64.0%
d) none of the above

Answer 1

The maximum efficiency of the first engine is 40%. The maximum efficiency of the second engine is 44.4%. The overall efficiency of the two engines is 17.76%. Option d is the correct answer.

Step-by-step explanation:

(a) Maximum efficiency:

The maximum efficiency of a heat engine can be calculated using the formula:

efficiency = 1 - (Tc/Th)

where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.

For the given steam engine, Th = 450ºC and Tc = 270ºC. Substituting these values into the formula:

efficiency = 1 - (270/450) = 40%

So, the maximum efficiency of the steam engine is 40%.

(b) Efficiency of the second engine:

The exhaust temperature of the first engine is 270ºC, and the exhaust temperature of the second engine is 150ºC.

Using the same formula as above, the efficiency of the second engine can be calculated:

efficiency = 1 - (150/270) = 44.4%

So, the maximum efficiency of the second engine is 44.4%.

(c) Overall efficiency:

To calculate the overall efficiency of the two engines, we need to multiply their efficiencies:

overall efficiency = efficiency of first engine * efficiency of second engine = 40% * 44.4% = 17.76%

So, the overall efficiency of the two engines is 17.76%.

(d) Efficiency of a single Carnot engine:

A Carnot engine is the most efficient heat engine theoretically possible. Its efficiency is given by the formula:

efficiency = 1 - (Tc/Th)

where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.

For a Carnot engine operating between 450ºC and 150ºC:

efficiency = 1 - (150/450) = 66.7%

Comparing this with the overall efficiency of the two engines (17.76%), we can see that they have the same efficiency.