Question

Suppose you want to design a steam engine that has heat transfer to the environment at 270ºC and has a Carnot efficiency of 0.800. What temperature of hot steam must you use?

Answer 1

To determine the temperature of the hot steam that must be used in order to achieve a Carnot efficiency of 0.800 and have heat transfer to the environment at 270ºC, we can use the Carnot efficiency formula.

The Carnot efficiency (η) is given by the formula:

η = 1 - (Tc/Th)

Where:

- η is the Carnot efficiency,

- Tc is the temperature of the cold reservoir (environment) in Kelvin,

- Th is the temperature of the hot reservoir (hot steam) in Kelvin.

We are given that the Carnot efficiency (η) is 0.800, and the temperature of the cold reservoir (Tc) is 270ºC. To convert this temperature to Kelvin, we need to add 273.15 to it, since Kelvin is an absolute temperature scale.

Using the Carnot efficiency formula, we can rearrange it to solve for Th:

Th = Tc / (1 - η)

Now, let's plug in the values:

Tc = 270ºC + 273.15 = 543.15 K (converted to Kelvin)

η = 0.800

Th = 543.15 K / (1 - 0.800) = 543.15 K / 0.200 = 2715.75 K

Therefore, to achieve a Carnot efficiency of 0.800 and have heat transfer to the environment at 270ºC, the temperature of the hot steam should be approximately 2715.75 Kelvin.