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A heating budget

Heat and matter · level 7, lesson 6 of 6

A question to start

Four people live in a habitat on Mars. How big a heater do they need? The answer may surprise you.

The idea

A heating budget adds up every way heat leaves the habitat, and every source of heat inside. The heater makes up the difference:

\[P_\text{heater} = P_\text{lost} - P_\text{gained inside}.\]

Heat is gained from:

  • people, each giving out about 100 W at rest;
  • every electrical device: lights, computers, pumps, cookers. Almost all the electricity they use ends up as heat in the room.
lost: 2.2 kWgained:peopledevices 1.5 kWheater

Worked examples

1. The heat lost. A habitat is a cylinder 10 m long and 5 m across: its walls and ends have area \(\pi \times 5 \times 10 + 2 \times \pi \times 2.5^2 = 157 + 39 = 196\) m². With the wall of lesson 2, \(U = 0.14\), and 79 K across it: \(P = 0.14 \times 196 \times 79 \approx 2{,}200\) W.

2. Heat gained inside. Four people: 400 W. Suppose lights, computers and life support use 1.5 kW of electricity: 1,500 W of heat. Total 1,900 W.

3. The heater. \(2{,}200 - 1{,}900 = 300\) W. A small heater is enough.

Same habitat, same wall Mars Earth
\(\Delta T\), inside 20 °C 79 K 5 K
Heat lost about 2,200 W about 140 W
Heat gained inside 1,900 W 1,900 W
Needs a 300 W heater about 1,760 W of cooling

On Earth, this well-insulated home would overheat. On Mars, its own electricity almost heats it, and a dust storm that cut the power would also cut the heat.

Try it

  1. Find the heat lost through 300 m² of wall with \(U = 0.1\) and 79 K across it.
  2. Six people and 2 kW of electrical devices are inside. What heat do they give?
  3. What heater does that habitat need?
  4. Why does nearly all the electricity used inside a habitat end up as heat?
Answers
  1. \(0.1 \times 300 \times 79 = 2{,}370\) W.
  2. \(600 + 2000 = 2{,}600\) W.
  3. None: it gains 230 W more than it loses, so it needs a little cooling.
  4. Light is absorbed by walls, motors and computers warm up, and sound fades: the energy ends as heat in the room.

Common mistake

Forgetting the heat from inside. People and machines are heaters too. Leaving them out can size a heater many times too big, and miss that the habitat may need cooling.

Where it leads