Problem — Polar tent

Problem

A polar team in a tent keeps the interior at $15 \ ^{\circ}\text{C}$ with the outside at $-40 \ ^{\circ}\text{C}$. The tent has a total surface area $A \approx 12 \ \text{m}^2$ and an equivalent wall of $0{,}05 \ \text{m}$ of insulating material with $k \approx 0{,}04 \ \text{W/(m·K)}$. Estimate the power needed to keep the interior warm and compare it with a camping stove (about $2 \ \text{kW}$).

Diagram of the tent: heat $\dot Q$ flows from the warm interior towards the cold exterior through the insulating wall.

Solution

In steady state, the heat flow through an insulating wall is given by the conduction law $\dot Q = \dfrac{k A \, \Delta T}{d}$, where $\Delta T$ is the temperature difference between inside and outside. With $\Delta T = 15 - (-40) = 55 \ \text{K}$ we get $\dot Q = \frac{k A \, \Delta T}{d} = \frac{0{,}04 \cdot 12 \cdot 55}{0{,}05} = 0{,}04 \cdot 12 \cdot 1100 = 528 \ \text{W}$ A $2 \ \text{kW}$ stove is therefore more than sufficient: it supplies about four times the power lost and also compensates for air exchange and additional losses. $\ev{\dot Q \approx 530 \ \text{W}}$

Collegamenti

Argomenti: Termologia Concetti: Trasmissione del calore Competenze: Stime di Fermi