# Adaptive Thermal Comfort Arctangent: Thermal Joy, Indoors and Out

According to research done by ASHRAE-55 2010 optimal indoor temperature depends on outdoor temperature. However they defined the minimum outdoor temperature to be 10C and maximum 33.5C.  The relationship between outdoor temperature and indoor temperature is not linear, as can be seen in this graph, where at 10C outdoor, 90% are satisfied with 18.2C indoors, and at 32C outdoors, 90% are satisfied with 30C indoors.

Optimal indoor outdoor temperature chart, for 90 and 80% satisfaction.

What I would like to do is figure out a formula for use in a dynamic thermostat to extend this downwards for the frigid temperatures of Canada, where it gets down to -50C or less in some places.  Though even in Toronto it’s been -30C, and in Owen Sound -34C.

Making a dynamic thermostat with such a formula may both save a lot of money, and increase thermal alliesthesia (temperature related joy),  while being  easier for people to both enter and exit the home, since there will be smaller temperature differences.

Ideally it would an s-shaped function, that would have indoor limits at 10C and 32C, passing through 24C as neutral point where body doesn’t need to do any work to heat up or cool down.

Here is an example arctangent that covers the world’s temperatures, from -80C in Antarctica, to +80C which it might reach some places due to climate change, recently reached 73C in part of Iran.

You can see the formula I use atan((x-24)/16)*8+24 = indoor-temp it is the same for all of the graphs.  the main difference is the range of temperatures.

I’ll quickly brake down the parts of it for you:

atan (arctangent) gives the s-shape I was talking about.  the x is where the outdoor temperature is plugged in, the result being the indoor temperature.   The -24 shifts the center of the s-shape to be at outdoor 24C,   the +24 shifts the center of the indoor temperature to be at 24C.  dividing by 16 stretches the S horizontally, and multiplying by 8 stretches the S vertically.

Here is a more local one for extreme temperatures encountered in Owen Sound, Canada

Here is one for average temperatures in hottest month in Owen Sound

This kind of shows, that for average temperatures in Owen Sound, most of the time the thermal mass of the building itself would probably suffice, no air conditioning is required.

here is for average coldest month in Owen Sound:

Here the temperature hovers just under 16C, which is around the lowest which is allowed for sedentary office environments at least in the UK from-source http://www.hse.gov.uk/temperature/faq.htm . 13C being the lowest for work places involving physical labour.

One reason to keep it chronically low is that it helps grow brown-fat which burns white-fat to keep the body warm.  Temperatures at or above 24C have been shown to reduce brown-fat, while temperatures 19C and below have been shown to increase it from-source http://www.nih.gov/researchmatters/july2014/07282014fat.htm

If you decide to try this at home, you’ll need a properly functioning metabolism and thus thyroid so remember to meet or exceed your iodine requirements every day, from-source https://ods.od.nih.gov/factsheets/Iodine-HealthProfessional/ .  the minimum for an adult old can be achieved by either with a teaspoon of dulse or 3 sheets of nori seaweed per day, but since it may be cooler than usual may wish to consume more than that though no more than 7 times that which is upper limit.

Another reason of course is that it would save on the heating bill,  though may increase the food bill a little, as more energy would be needed to keep bodies warm.

In terms of putting this system into practice,  with a regular thermostat there are options based on the complexity of your thermostat.

1. if you just have a one-setting heater, then set it based on the lowest average temperature that month.
2. If you also have a one setting air-conditioner, then set it based on the highest average temperature that is likely for that month.
3. If you can set different day time and night time temperatures, then set night-time by average low, and daytime by average high, with twilight at average temperature.

Here is an example table for Owen Sound.  We’ll be moving in September, so I’ll start with that.  Average low in Owen Sound for September is 11.4C, which we can plug into the formula:

(atan((11.4 – 24) / 16) * 8) + 24 = approx. 18.663421

so thermostat heater for night can be set to 18.6C or 19C depending on it’s resolution.

Times of day based on local apparent time:
night 20:00 (8pm) til 04:00 (4am),
dawn twilight 04:00 til 08:00,
day 08:00 till 16:00 (4pm),
dusk twilight 16:00 till 20:00 (8pm).

Owen Sound summer meridian (when sun is highest point in sky) is 13:29 day light savings time from-source http://www.timeanddate.com/astronomy/canada/owen-sound , so would have to add hour and 29 minutes to all of those times if thermostat in daylight savings time, or just 29 minutes if it is in standard time.

Adaptive Thermal Comfort temperatures for Owen Sound
Thermostat setting for night for twilight for day
August 19.9C 21.9C 24.0C
September 18.6C 20.2C 22.1C
October 17.2C 18.1C 19.3C
November 16.2C 16.8C 17.5C
December 15.5C 15.9C 16.3C
January 15.0C 15.4C 15.9C
February 15.0C 15.5C 15.9C
March 15.4C 16.0C 16.7C
April 16.3C 17.2C 18.3C
May 17.3C 18.7C 20.5C
June 18.7C 20.5C 22.9C
July 20.0C 22.0C 24.4C

We’ll be trying this out,  I’ll either make new posts or comment on this one to talk about it’s progress.