Thermal Conductivity

This article has not been revised for the current version (***U50-587362***). It was last updated for *U47-562984*. It may contain inaccuracies.
This article has not been revised for the current version (***U50-587362***). It was last updated for *U47-562984*. It may contain inaccuracies.
**Thermal Conductivity** is the property of a material that determines how quickly it heats or cools as it comes into contact with objects of different temperatures. Although the game states that between two objects, the lowest thermal conductivity is used, this is not true for all cases.

## Equations

The calculation of Heat Transfer

q

{\displaystyle q}

 in

D
T
U

{\displaystyle DTU}

 is mainly a product of:

* Δ
T

{\displaystyle \Delta T}

 the Temperature difference in °C
* Δ
t

{\displaystyle \Delta t}

 the passing time, which is always one tick,

0.2
s

{\displaystyle 0.2s}
* k

{\displaystyle k}

 the applicable thermal conductivity in

D
T
U

/

(
m
⋅
s
⋅
°
C
)

{\displaystyle DTU/(m \cdot s \cdot \text{°}C)}

	+ k

	l
	o
	w
	e
	s
	t

	{\displaystyle k\_{lowest}}

	 is the lower of the two
	+ k

	a
	v
	e
	r
	a
	g
	e

	{\displaystyle k\_{average} }

	 the average of the two:

	0.5
	(

	k
	1

	k
	2

	)

	{\displaystyle 0.5 (k\_1 + k\_2)}
	+ k

	g
	e
	o
	m

	{\displaystyle k\_{geom}}

	 is the geometric mean of the two:

	k
	1

	⋅

	k
	2

	{\displaystyle \sqrt{k\_1 \cdot k\_2}}

For heat transfer with buildings, additional factors include:

* C

h
o
t

{\displaystyle C\_{hot} }

 the thermal mass of the hotter object:

m
⋅
c
⋅

[

1
5

]

if building

{\displaystyle m \cdot c \cdot \left[ \frac{1}{5} \right]\_\text{if building}}

 where

m

{\displaystyle m}

 is mass,

c

{\displaystyle \,c}

 is specific heat capacity (SHC), and divide by 5 if the hotter object is a building.
* A

{\displaystyle A}

 the building's area

Equations

# q
Δ
T
⋅
Δ
t
⋅

k

g
e
o
m

⋅
1000

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{geom} \cdot 1000}
 Solid ↔ Gas

# q
Δ
T
⋅
Δ
t
⋅

k

g
e
o
m

⋅
1000
⋅
25

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{geom} \cdot 1000 \cdot 25}
 Liquid ↔ Liquid

# q
Δ
T
⋅
Δ
t
⋅

k

g
e
o
m

⋅
1000
⋅
625

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{geom} \cdot 1000 \cdot 625}
 Entity inside of a cell

# q
Δ
T
⋅
Δ
t
⋅

k

l
o
w
e
s
t

⋅
1000

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{lowest} \cdot 1000}
 Entity on top of a solid cell

# q
Δ
T
⋅
Δ
t
⋅

k

l
o
w
e
s
t

⋅
62.5

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{lowest} \cdot 62.5 }
 Insulated pipe and its contents

# q
Δ
T
⋅
Δ
t
⋅

k

l
o
w
e
s
t

⋅
50

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{lowest} \cdot 50}
 Other pipes and their contents

# q
Δ
T
⋅
Δ
t
⋅

k

a
v
e
r
a
g
e

⋅
50

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{average} \cdot 50}
 Building and the cells it occupies(only divide by A if the building is hotter)

# q
Δ
T
⋅
Δ
t
⋅

k

b
u
i
l
d
i
n
g

⋅

k

c
e
l
l

⋅

1
2

⋅

C

h
o
t

A

{\displaystyle q = \Delta T \cdot \Delta t \cdot k\_{building} \cdot k\_{cell} \cdot \frac{1}{2} \cdot \frac{C\_{hot}}{A} }

Any Tiles, be they normal, insulated, metal, plastic, bunker, glass, or carpeted tiles, as well as doors, Farm Tiles, Heavi-Watt (Conductive) Joint Plates, and Tube Crossings count as Solid Cells, not as buildings.

Insulated Tiles reduce the thermal conductivity of their building material by (2/255)² (or 16 256) instead of 100 as stated in the game. It also uses

k

l
o
w
e
s
t

{\displaystyle k\_{lowest}}

instead of

k

g
e
o
m

{\displaystyle k\_{geom}}

for the purpose of cell to cell conductivity, which is mostly going to be the insulated tile conductivity. Solid to gas multiplier still applies.

**Note:** in real life, the units would cancel out, but surface area plays no role in ONI's calculation. Therefore, an additional

[
m
]

{\displaystyle [m]}

 meter needs to be multiplied to the product to have the equation result in DTU. (Not just meter, but a large string of units when calculating buildings.)

Another way to interpret this is that the units of Thermal Conductivity are given in-game as

D
T
U

/

(
m
⋅
s
⋅
°
C
)

{\displaystyle DTU/(m \cdot s \cdot \text{°}C)}

 while the unit is actually interpreted as

D
T
U

/

(
s
⋅
°
C
)

{\displaystyle DTU/(s \cdot \text{°}C)}

.

Recap of Cell to Cell multipliers
 Liquid
 Solid

Because of the Gas to Solid x25 multiplier, it's recommended to add a double tiles layer or a thin liquid layer when trying to insulate two rooms, to instead get a x1 multiplier.

## Limits of Heat Transfer

### Lower Limits

Heat Transfer will not occur if:

* the temperature difference is less than 1 °C
* the calculated thermal flow is less than 0.1 DTU
* either of the masses is less than 1g

### Upper limits

Heat Transfer between cells has the following cap:

* If the calculated heat transfer would result in a temperature jump of more than a fourth of their temperature difference

(

T
1

−

T
2

)

/

4

{\displaystyle (T\_1-T\_2)/4}

 in either material.

q

max
1
a

=

T
1

−

T
2

4

⋅

m
1

⋅

c
1

or

q

max
1
b

=

T
1

−

T
2

4

⋅

m
2

⋅

c
2

{\displaystyle q\_{\text{max }1a}=\frac{T\_1-T\_2}{4}\cdot m\_1 \cdot c\_1 \quad \text{or}\quad q\_{\text{max }1b}=\frac{T\_1-T\_2}{4}\cdot m\_2 \cdot c\_2}

Simply said: if the temperature difference is 40 °C, a materials temperature can change at most 10 °C per tick

### Building Limits

Heat transfer between a building and a cell has different limits, the lower limits which are applied to cells do not apply to buildings, but the upper limit is conceptually similar.

A building exchanges heat with all cells it covers simultaneously. In order to ensure that thermodynamics will not be violated the game limits heat transfer per cell such that at most the final temperature of the building would be the equilibrium temperature, assuming that the building completely covers such cells:

T

e
q

=

T

b
u
i
l
d
i
n
g

⋅

C

b
u
i
l
d
i
n
g

T

c
e
l
l

⋅

C

c
e
l
l

⋅
A

C

b
u
i
l
d
i
n
g

C

c
e
l
l

⋅
A

{\displaystyle T\_{eq} = \frac{T\_{building} \cdot C\_{building} + T\_{cell} \cdot C\_{cell} \cdot A}{C\_{building} + C\_{cell} \cdot A}}

The maximum permitted heat transfer per cell is the difference between the building's temperature and the equilibrium temperature divided by the Area of the building.

q

m
a
x

=

C

b
u
i
l
d
i
n
g

⋅

T

b
u
i
l
d
i
n
g

−

T

e
q

A

{\displaystyle q\_{max} = C\_{building} \cdot \frac{T\_{building} - T\_{eq}}{A}}

If the thermal mass of the cell is very large relative to the building, then the maximum temperature change can be approximated as simply

Δ
T

A

{\displaystyle \frac{\Delta T}{A}}

### Floating Point Calculation Limits

While the above limits are deliberately implemented, it is also possible for heat exchange to fail to happen due to limitations of the floating point calculations used to calculate temperature changes.

Internally ONI uses 32 bit floating point numbers to represent temperatures, and due to the limited precision of floating point numbers it is possible for the result of a floating point calculation to be the same number.
For example with 32 bit floats, the calculation: 300.0 + 0.00001 = 300.0

The game has a rule that if either tile fails to change temperature, then no heat exchange is allowed to take place, this prevents an exploit where a large tile, especially an unnaturally large tile, infinitely dumps heat/cooling into a smaller tile without itself changing temperature.

In real games this limit comes up all the time when the temperature difference between an Insulated Tile and a solid or liquid tile is relatively small. For example an Igneous Rock Insulated Tile which is itself at 20 C, will not exchange heat with a solid or liquid tile unless the temperature delta is at least 248.05 C, and won't exchange heat with a gas tile unless the temperature delta is at least 9.92 C. This makes it quite easy to achieve actually zero heat transfer without resorting to the Insulation material or vacuum. The exact formulas governing this are:

Δ

T

i
g
n
o
r
a
b
l
e

=

2

⌊

log
2

⁡
(
T
)
−
24
⌋

{\displaystyle \Delta T\_{ignorable}=2^{\lfloor \log\_2(T)-24 \rfloor}}

 and

q

m
a
x

=
Δ

T

i
g
n
o
r
a
b
l
e

⋅
m
⋅
S
H
C

{\displaystyle q\_{max}=\Delta T\_{ignorable} \cdot m \cdot SHC}

, where

T

{\displaystyle T}

,

m

{\displaystyle m}

, and

S
H
C

{\displaystyle SHC}

 are for the cell holding everything constant, and

q

m
a
x

{\displaystyle q\_{max}}

 is the heat-exchange formula relevant between the two cells, which can be reversed to find

Δ

T

m
a
x

{\displaystyle \Delta T\_{max}}

.

It is also readily observed with liquid tiles, Magma and Water can have immense thermal mass which means relatively large DTU inputs are required to cause a temperature change. This results in the paradoxical outcome where full magma tiles don't exchange heat with insulated tiles, but partial magma tiles can exchange heat if they are sufficiently low mass. Using the above formula but applied to the Magma tile instead of the insulated tile, we can see that a cell with 715.6 kg or more magma will be unable to exchange temperature with an Igneous Rock Insulated Tile at 0C or higher, regardless of the magma temperature. For Mafic Rock, which has half the conductivity, only 357.8 kg of magma are needed.

Suffice to say that while floating point imprecision sometimes causes heat exchange to not happen at all, when temperature changes are small it also causes the actual temperature change to deviate quite significantly from what higher precision calculations would suggest.

## Thermal descriptors

There are 4 thermal descriptors in the game, and they get attached to elements when their thermal characteristic reach a certain threshold. These descriptor does not affect the element any further.

* **Thermally Reactive**: Elements have a specific Heat Capacity of less than or equal to 0.2
* **Slow heating**: Elements have a specific Heat Capacity of greater than or equal to 1.0
* **Insulator**: Elements have a thermal conductivity of less than or equal to 1.0
* **High Thermal Conductivity**: Elements have a thermal conductivity of greater than or equal to 10.0

## Solid Tiles list

**Important**: For Insulated Tiles, these numbers will not match what is seen in-game. This is because the value displayed in-game is

b
a
s
e

100

{\displaystyle \frac{base}{100}}

, but the actual value used by calculations (and shown here) is

b
a
s
e
⋅
(

2
255

)
2

{\displaystyle base \cdot (\frac{2}{255})^2}

.

## Tips

* When cooling or heating an area it's better to run pipes through tiles than through atmosphere. In both cases the equation for "Building and the cells it occupies" is used which multiplies both Thermal conductivities, and in general, gases have a much lower thermal conductivity than liquids, which have lower conductivity than solids.
	+ However, if drastic cooling is desired, then Steam Turbines and Aquatuners will have to be involved, which means a cavity filled with Steam will have to be used.
* Since Insulated Tiles have a factor of 1/16256, and pipes a factor of 1/32, much less heat is transferred if a regular pipe goes through an insulated tile than when an insulated pipe goes through a regular Tile. Though, of course, insulating both has an even better insulating effect.
* Even though Insulation has a lower thermal conductivity than any Insulated Tile, the change in formula from

k

g
e
o
m

{\displaystyle k\_{geom}}

 to

k

l
o
w
e
s
t

{\displaystyle k\_{lowest}}

 makes insulated tiles much more practical insulators than a regular Tile made from insulation. Indeed, they are so good that even using regular rock is often sufficient to shut down heat transfer completely, or to practically unnoticeable levels.

## References

https://forums.kleientertainment.com/forums/topic/84275-decrypting-heat-transfer/

1. ↑ forum post on empirically derived heat transfer equations