# Fog, concrete, plus no heating for two days

We were away this weekend – in sunny Whakatane. That was a smart move weather-wise. We arrived back home about 2.30pm yesterday afternoon, just as the fog was lifting from Cambridge. An hour later, it was settling down again for the night. A glance at the MetService website shows that, while Whakatane and Rotorua basked in 12 degree maximum temperatures (and somehow Tauranga hit 15 C), Hamilton made it to 7 Celsius.

A consequence of this was that our house was COLD when we came back to it.

Now, those of you who have seen our house will know that it's built to catch the winter sun (pity there wasn't any yesterday) via lots of glass and soak up the warmth with lots of concrete. And it works really well on cold winter days when the sun's out. But on cold winter days when the sun is not out (i.e. Waikato fog) a fundamental design flaw is evident. With no heating on for two days with morning temperatures well below zero, the house got cold. About 7 degrees inside. That was no surprise. The first thing we did was to turn the heat pumps on. An hour later it was still cold. Three hours later it was cold. About six hours later we had the temperature up to 15 degrees. The poor heat pump was blatting out hot air as powerfully as it could, but the house was just not getting hotter quickly at all.

The reason is that concrete again. Just as its designed to soak up the heat from the sun and re-radiate it slowly, keeping some warmth in the evening, so it takes a long time to warm up. Here's a quick calculation.

We have internal concrete block walls about 20 cm thick. Let's suppose they have cooled to 7 degrees. How long does it take (with warm air on both surfaces) to get that concrete back up to temperature?  It's a question of diffusion of heat. The heat gets to the centre of the concrete through conduction. This process can be described by the diffusion equation. See here for the lovely maths. The key parameter is something called the thermal diffusivity – often called 'D' – it depends on the thermal conductivity, density and heat capacity of the concrete.  For concrete the thermal diffusivity is about 5 x 10-7 m2 s-1.   In a time t, heat will diffuse a distance of approximately the square-root of D t.  In six hours (about 20 000 seconds), this comes to about 0.1 metres – that is 10 cm. After this time we can consider the middle of that concrete block as having warmed up. Approximately speaking, the concrete wall will then no longer be soaking up the heat from the air, and instead, the air can stay warm.

I have to say that this was about our experience last night. By bedtime, the house was tolerably warm again.

Incidentally, I have also done a quick estimate of the number of kWh of electricity we used last night. Ouch.