Question

During January, at a location in Alaska winds at -30°C can be observed, However, several meters below ground the temperature remains at 15°C. An inventor claims to have devised a power cycle working between these temperatures having a thermal efficiency of 10%. Investigate this claim.

Part A
What is the maximum thermal efficiency for these condition?
r/max =
the tolerance is +/-2%

Answer 1

The maximum theoretical efficiency of a heat engine operating between -30°C and 15°C is approximately 15.6%. The inventor's claim of a 10% thermal efficiency is plausible as it falls below the theoretical maximum, considering the +/-2% tolerance.

Step-by-step explanation:

The maximum thermal efficiency of a heat engine can be determined using the Carnot efficiency, which is defined as 1 - (Tc/Th), where Tc is the cold reservoir temperature and Th is the hot reservoir temperature, both in Kelvin. To convert the Alaska temperatures from degrees Celsius to Kelvin, add 273.15 to each temperature. So, -30°C becomes 243.15 K and 15°C becomes 288.15 K. Therefore, the maximum theoretical efficiency (ηmax) is 1 - (243.15/288.15) ≈ 0.156, or 15.6%. Given the +/-2% tolerance, the efficiency claim of 10% by the inventor falls well below the theoretical limit and could be considered possible.

The maximum thermal efficiency for a power cycle operating between temperatures of -30°C and 15°C in Alaska can be calculated using the Carnot efficiency formula, which is given by:

Emax = 1 - Tcold/T

where Tcold is the temperature of the cold reservoir (-30°C) and T is the temperature of the hot reservoir (15°C).

Substituting the values into the formula, we get:

Emax = 1 - (-30+273)/(15+273) = 1 - 243/288 = 0.1556

Therefore, the maximum thermal efficiency for these conditions is 15.56%.