How to predict the temperature of a car cabin parking in the very cold weather?

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The result of previous tests

Result 1

Temperature of inside of the car will be higher than outside by 3 degrees Celsius until midnight with following items.

  1. Electric blankets or 1 person ( 120W heat generator)
  2. Fiberglass insulator (Installed inside of inner linings.)

Result 2

If you add the items below, temperature of inside of the car will be higher than outside by 4 degrees Celsius until midnight.

  1. Styrofoam window panels
  2. Curtains between the passenger cabin and cargo room

Hereinafter, I shall omit “Celsius”.

Heat insulating items I brought to Hokkaido

Taking them into consideration, I went to Hokkaido with following items.

  1. Electric blankets
  2. Fiberglass insulator (Installed inside of inner linings.)
  3. Styrofoam window panels
  4. Curtains between the passenger cabin and cargo room
  5. Heat insulation mats

Temperature graphs in Hokkaido

Typical 3 types of difference variations are found.

Feb/27 Very warm => Very cold

I arrived at Tomakomai, Hokkaido by ferry at 21:00 on Feb/27/2018. Then, drove to Shimukappu village, and reached there around 23:00. At that time, outside temperature was -10 degrees, and inside temperature was more than 15 degrees. So the difference was almost 30 degrees at first.

I stopped the engine, and started again around 5:10 next morning. While the car was parking, outside temperature fell rapidly and was below -20 degrees, and difference also fell rapidly as well, but still it was about 8 degrees 6 hours after engine stopped.

At 5:00 next morning, inside of the car temperature was -15 degrees, and it was very cold, but still difference was relatively high compared with previous tests.

Predict the temperature of car cabin

Feb/28 Very warm => Cold

This graph is the next day’s temperature graph. After I left Shimukappu, drove around Furano city and Biei town. Then, arrived at Minami Frano town Roadside Station. (Roadside Station is public rest area along main roads.)

It was still 16:00 and outside was a bit warm, and inside was very warm. Then, engine stopped, and inside temperature fell, but outside temperature was not as cold as last night at Shimukappu.

Predict the temperature of car cabin

Mar/1 A bit warm => A bit cold

Again, this is the next day of above graph. Heavy snow was falling all day, and the roads to other cities were all closed. So I must stay in Frano town Roadiside Station all day. I drove the car several times to go for shopping or make space for snow removing machine. Difference never exceeded 10 degrees all day, and it reached almost zero at night.

Predict the temperature of car cabin

What is common among these graphs?

Since I have made these graphs, have been thinking there must be something common, and they must be part of a certain curve.

And a few days ago, I found what I was looking for.

Exponential decay 

The below graph is reproduced from Wikipedia. This is exponential decay graph.

This graph is described by below equation.

dN/dt=-λN

This means the reduction rate of N is in direct proportion to its value. So reduction rate becomes smaller when N decreases.

If you put a car in the cold weather, heat of the car is discharged by heat conduction, and the amount of heat to be discharged is proportional to the difference between inside and outside temperature. So I guess this equation can explain the variation of difference.

I drew a graph of exponential decay  with N=-90, λ=8.

Predict the temperature of car cabin

N and λ are adjusted to fit for above temperature graphs.

Is the assumption correct?

Are these temperature graphs are part of exponential decay graph?

I overlaid temperature graphs and exponential decay graph as below.

Feb/27 Very warm => Very cold

I overlaid the graphs by Photoshop. I made exponential decay graph transparent and put on the temperature graph. You can see that the temperature graph is left part of exponential decay graph.

Predict the temperature of car cabin

Feb/28 Very warm => Cold

The temperature graph is central part of exponential decay graph.

Predict the temperature of car cabin

Mar/1 A bit warm => A bit cold

The temperature graph is right part of exponential decay graph.

If insulation power changes, λ changes

As shown in above, if conditions such as heat generation inside of the car and heat insulation are same, difference graph follows the same curve. It is not affected by outside temperature, only by value of difference.

If you add or remove insulation, λ changes.

As shown below, if λ increases, temperature falls rapidly, and it represents removing insulation.

Graph 1 N=90, λ=8

Predict the temperature of car cabin

Graph 2 N=90, λ=14

Predict the temperature of car cabin

Conclusion

If you find N and λ of your car, you can predict value of difference (= inside temperature – outside temperature) after given time.

Example

If you heat inside of the car up to 40 degrees warmer than outside, it will take

(0.33-0.12)= 0.21 day = about 5 hours until difference goes down to 10 degrees.

On the other hand, if you heat up to 20 degrees warmer, then it will take

(0.33-0.22)= 0.11 day= about 2.6 hours until difference goes down to 10 degrees.

Predict the temperature of car cabin

Provided that radiant heat like sunlight does not come in.

Thank you for reading my article.

Note

The original article was written and published on Mar/18/2018.

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