A steam engine is being planned that is to use steam at 400 ˚F and whose efficiency is to be 20 percent. Find the maximum temperature at which the spent steam can emerge.
How to solve
To find the maximum temperature at which the spent steam can emerge, we need to use the concept of Carnot efficiency and the given information.
The Carnot efficiency (η) of a heat engine is given by the formula:
η = 1 - (Tc/Th)
Where:
η is the efficiency of the engine
Tc is the absolute temperature of the cold reservoir (in Kelvin)
Th is the absolute temperature of the hot reservoir (in Kelvin)
In this case, we know that the efficiency of the engine is 20%, which can be written as 0.20. So, we can rewrite the equation as:
0.20 = 1 - (Tc/Th)
We are also given that the steam temperature is 400 ˚F. To convert this temperature to Kelvin, we use the formula:
T(K) = (T(°F) + 459.67) × 5/9
So, T(400 ˚F) = (400 + 459.67) × 5/9 = 477.59 K (approximately)
Let's substitute the known values into the Carnot efficiency equation and solve for the cold temperature (Tc):
0.20 = 1 - (Tc/477.59)
Rearranging the equation, we get:
Tc/477.59 = 1 - 0.20
Tc/477.59 = 0.80
Tc = 0.80 * 477.59
Tc ≈ 382.07 K
Therefore, the maximum temperature at which the spent steam can emerge is approximately 382.07 Kelvin.