Radiative Cooling Calculator

Cooling of an infinite slab to deep space (0 K) by radiation
Assuming no conduction, no convection, and no radiation from other sources
Initial Temperature K   C   F
Final Temperature K   C   F
Compute this when values change  
 
Elapsed time sec days hr min sec
Specific Heat J/gm K  
Density gm/cc
Emissivity
Thickness mm    


Overview

Using just radiation - How fast do objects cool?

This calculator simulates cooling to space (at 0K) by radiation alone - it ignores conduction and convection, assumes no internal heat source, and assumes that no heat is added from any outside source. Using a simple formula (which has a slightly confusing derivation), it is possible to compute either

The calculator on this page is similar to that provided by Hyperphysics except that I have


Water

Standing water cools much slower than a solid non-metallic surface. This is because the effective thermal conductivity is significantly different.

With low thermal conductivity, there is a large internal thermal gradient because the surface cools much faster than the bulk of the material. To simulate this for materials like sand or concrete, set the thickness to 1mm or less. Smaller values simulate better insulators.

The thermal conductivity of water is not a lot greater than that of non-metallic solids. However, above about 4C, as water cools, its density increases and the cooler surface layer will sink. As a result, the surface layer is mixed to some depth as it cools. This means that a larger volume of material must cool during the same amount of time that a very small volume of a solid material. This is what I mean by effective thermal conductivity.

For calm water in a lake or ocean, I suggest setting the thickness to a value between 1 and 10 cm (10 and 100 mm). For streams, set it to the depth of the stream.

When adding heat to water via IR radiation from above, the opposite is true - Only the top millimeter or so warms and a significant amount of the absorbed heat produces evaporation and, therefore, does not actually increase the temperature.


Future Improvements


Validation against an existing model

When I create a new model, I like to validate the results against existing models. For this calculator, I tried to validate it against the one provided by hyperphysics which uses a sphere. Initially, I got radically different results - but I eventually found out why. While their method provides good values for monatomic gases, it produces the wrong results for other substances.(ref)

As a result, I decided to check the results with argon - 40 gm/mol.

For an object with no internal thermal gradients, only the surface area and mass (volume) affect the rate of cooling via IR radiation. As a result, it is possible to emulate a sphere with a surface area of one square meter by simply evaluating a slab of the appropriate thickness.

For a Sphere

Remember, this value assumes that the temperature of the sphere is uniform (no thermal gradients). Otherwise, the model simply gives a lower (or upper) bound (depending on what is being calculated).

In the hyperphysics calculator

In the calculator above Set Density and Emissivity to the same values in both calculators. (The exact values don't matter as long as they are the same.)

The Final temperature is

Close enough to validate the model.

However, and this is important, the specific heat of water is 4.1813 J/gm K, but the hyperphysics calculator uses

Not even close!!!


References


Author: Robert Clemenzi
URL: http:// mc-computing.com / Science_Facts / RadiationBalance / CoolingCalc.html