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Background

I am designing a simulator for building designs, and need to track the interior temperature of the constructed building. For simplicity, this building will be one room; meaning the only walls are the 4 exterior facing walls.

Variables

I am starting with the outdoor temperature, the area of the wall that is being hit by sunlight (ray casting), and the material the wall is made out of. I can also get time since simulation started for delta time, or calculate the "in world" 24 day equivalent of time passed.

Attempts

I know that using the material I can get the thermal conductivity, and I believe I can calculate resistance from wall thickness / (conductivity * area).

My issue is that all formulas I can find use the unknown temperature to solve for the heat transfer Q.

My experimentation with providing a random default interior temperature results in the interior immediately jumping to whatever the exterior is with no gradual change (and no interior heat creation Qi). When I add something to mock an internal heater the internal temperature rises perpetually with seemingly no rhyme or reason.

The formula I have been mainly using is

// T[in](Time) = T[in](initial) * e ^((-k/c) * t) + [T[out] + Q[i]/K] [1-e ^((-k/c) * t)]

$$T_{in}(t) = T_{in}(t=0) * e ^{(-k/c) * t} + \left(T_{out} + \frac{Q(i)}{K}\right)\cdot \left(1-e ^{(-k/c) * t}\right)$$

I have also been trying to make use of the following info:

$$Q_x = -\frac{K}{L}\cdot A \cdot dT$$

where:

  • K = thermal conductivity of wall [W/M c or W/M k]
  • A = area of wall [m^2]
  • dT = temp difference [C or K]
  • L = wall thickness in m
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    $\begingroup$ What about the roof? $\endgroup$
    – Solar Mike
    Commented Feb 10, 2022 at 4:12
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    $\begingroup$ @TempQuestion I tried to rewrite the equation to a form that IMHO is more easy to read. However I had problems with the main formula you are using. Could you explain where you obtained that formula and what is the physical reasoning behind the decaying exponential? $\endgroup$
    – NMech
    Commented Feb 10, 2022 at 9:50
  • $\begingroup$ What you are trying to develop is a FEA thermal model. I suggest you do some research. it isn't hard to make your own but you've made some mistakes as others have pointed out. Maybe start by modeling a bucket of water, and you will get ideas on what you missed. $\endgroup$
    – Abel
    Commented Feb 10, 2022 at 12:10
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    $\begingroup$ Building heat loads are usually solved emperically using HVAC methods. The ASHRAE Fundamentals book is the gold standard here. Those calcs are always used to calculate the maximum load. There aren't many buildings IRL that are sealsed with nothing in them where you'd want to know the temp inside. There's just so much missing here, so many more sources of heat that are left out. $\endgroup$
    – Tiger Guy
    Commented Feb 10, 2022 at 13:09

1 Answer 1

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There are a few things that IMHO are issues in this problem:

  1. I don't see any physical reasoning behind the main equation you are using.
  2. Since this is about a transient problem (i.e. how does the temperature change with respect to time), you should also considered the the heat capacity $c_p$ of the contents of the room (e.g. air, furniture).
  3. There is no indication of thermal equilibrium being considered. E.g. you mention sunlight heat a wall but there are is radiation influx, or losses to the environment. I.e. one wall will be warmer but the remaining walls will be losing heat.

Even if the above are met, then the distribution on the temperature in the room will differ from point to point. The bare minimum would be a inite volume scheme to estimate the temperature at a specific point in the room.

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