Problem — Minimum temperature of a power plant

Problem

A power plant reaches an efficiency of $40\%$ with a cold fluid at $T_f = 320$ K. What minimum temperature must $T_c$ reach, for an ideal Carnot cycle?

Solution

Setup. Treating the plant as an ideal Carnot engine, the efficiency constrains the two temperatures:

$\eta = 1 - \frac{T_f}{T_c}$

Isolate $T_c$. Rearranging:

$\frac{T_f}{T_c} = 1 - \eta \quad\Rightarrow\quad T_c = \frac{T_f}{1 - \eta}$

Numerical substitution. With $\eta = 0{,}40$ and $T_f = 320$ K:

$T_c = \frac{320}{1 - 0{,}40} = \frac{320}{0{,}60} \approx 533 \ \text{K}$

$\ev{T_c \approx 533 \ \text{K} \ (\approx 260\,^\circ\text{C})}$

This is the minimum temperature of the hot reservoir: since Carnot is the ideal limit, a real plant with the same $\eta = 40\%$ would have to operate at an even higher $T_c$ to compensate for irreversibilities.

Collegamenti

Argomenti: Heat engines Concetti: Carnot cycle · Efficiency Competenze: Symbolic setup Oggetti: Heat engine