Solving the Heatwave Puzzle: How Effective Are Double Layers of Curtains and Blinds?

The Puzzle Setup

Earlier today, Alex Bellos presented a challenge involving keeping homes cool during a heatwave by understanding how double-layered window coverings—specifically roller blinds and curtains—work to prevent solar radiation from entering the room. The scenario considered both layers of material to be made of homogeneous substances, allowing for an analysis based on basic principles of thermal radiation.

Imagine a room with a set of curtains. When the Sun is shining, its energy comes through the windows where it is absorbed by the curtains. As the temperature rises, the curtains continue to absorb heat until they reach a stable state where they are radiating as much heat as they are absorbing. The system reaches an equilibrium where half of the heat radiated by the curtain returns towards the window and the other half is directed into the room. Therefore, 50% of the Sun's energy does not pass through the curtains.

Understanding the Solution

The key insight in this puzzle is that each layer of material (blinds and curtains) reaches a stable temperature where it radiates heat in equal amounts outwards. If we denote the total energy absorbed by the curtain as 2h, then it must have received this from the blind, meaning the blind is emitting 4h of energy.

Breaking down the thermal radiation further: The blinds are also a two-layer system with each layer radiating an equal amount inwards and outwards. Since the blinds receive h from the curtain and must emit 4h to reach equilibrium, only 3h comes directly from solar radiation. Thus, of the total energy received by the blinds (4h), 3h is actually from the Sun.

Therefore, the percentage of sunlight that enters the room after passing through both layers can be calculated as follows:

• Curtains radiate half their absorbed energy back towards the window and the other half into the room. This means 50% of the solar energy is stopped by the curtains.
• The blinds, which also receive h from the curtain, emit 2h forwards and 2h backwards to reach equilibrium.

Given this, only 3h out of the 4h emitted by the blinds comes directly from solar radiation. Since one-third (3h) enters the room, the percentage of sunlight that actually penetrates both layers is calculated as 33%.

The Mathematical Breakdown

Using a diagram to visualize the flow of heat, we can see how each layer interacts with the incoming solar radiation. The curtain radiates half its received energy back towards the blind and the other half into the room. Similarly, the blind emits equal amounts both ways.

The calculations go as follows:

• Curtains absorb 2h of heat from the Sun.
• This absorbed heat must be radiated outwards, so they emit h towards the window and h into the room. Therefore, curtains stop 50% of the solar energy.
• Blinds receive h from the curtain and must emit 4h to reach equilibrium.
• Out of this 4h emitted by the blinds, only 3h comes directly from solar radiation because a portion is also emitted towards the room.

Thus, if 3h enters the room out of the total 4h emitted by the blinds, the percentage of sunlight that actually passes through both layers and reaches the room is calculated as 33%.

Conclusion

This puzzle not only demonstrates practical applications of physics but also offers a deeper understanding of thermal dynamics within enclosed spaces. Such puzzles are regularly featured by Bellos in his column, serving as engaging ways to explore scientific concepts through everyday scenarios. Future challenges will continue to challenge readers' intellects, ensuring an ongoing exploration of the fascinating world of science.

Alex Bellos has been setting a puzzle here on alternate Mondays since 2015 and is always on the lookout for great puzzles. If you would like to suggest one, feel free to email him. The next puzzle will be published in two weeks.

Source: The Guardian