Fermi estimate: air in a room

Problem

Estimate the heat needed to warm the air in a $50\ \text{m}^3$ room from $15\ ^\circ\text{C}$ to $22\ ^\circ\text{C}$, assuming constant atmospheric pressure. Compare with the power of a 1 kW radiator.

Solution

Mass of the air. With density $\rho\approx 1{,}2\ \text{kg/m}^3$:

$m=\rho V=1{,}2\cdot 50=60\ \text{kg}$

Heat at constant pressure. With $c_p\approx 1005\ \text{J/(kg·K)}$ and $\Delta T = 7\ \text{K}$:

$Q=m\,c_p\,\Delta T=60\cdot 1005\cdot 7\approx 4{,}2\cdot 10^5\ \text{J}$

$\ev{Q\approx 4{,}3\cdot 10^5\ \text{J}}$

Comparison with the radiator. A radiator of $P = 1\ \text{kW}$ would supply this energy in:

$t=\frac{Q}{P}\approx\frac{4{,}2\cdot 10^5}{1000}\approx 420\ \text{s}\approx 7\ \text{min}$

This estimate covers the air alone: realistically about 10 times more is needed to warm the walls and furniture and to make up for losses to the outside.

Links

Topics: Thermodynamics Concepts: Specific heat and heat capacity Skills: Fermi estimates