Science Facts - Temperature Conversions

I eventually got tired of converting temperature from C to F to K. Therefore I wrote a program to do this automatically - if any temperature is entered, then the other 2 are automatically computed.

In addition, the program also uses Stefan's law to determining the blackbody temperature of an object at equilibrium.

P * (1-Albedo) = s * T^4

Javascript Calculator

To Use - just type a value into any cell and the others will be automatically computed. Or use the mouse wheel with Alt, Ctrl, & Shift and their combinations.

		Compute this when changed
Power	W/m2	Power Albedo Temp
Albedo	% Reflected	Power Albedo Temp
Temperature	°K °C °F	Power Albedo Temp
Set Power

Stefan's Law

Stefan's law is used to determine the blackbody temperature of an object at equilibrium.

P * (1-Albedo) = s * T⁴

P  Power - W/m²
s  Stefan's Constant = 5.67 x 10^-8 W m^-2 K^-4
T  Temperature in degrees Kelvin

However, there is an important limitation

There is no information on how fast a body reaches equilibrium

This is very important because the Sun's brightness varies with clouds, day and night, the seasons, and so forth.

For thin (low mass) items in a vacuum, equilibrium may be reached in a few minutes.

On the Moon (no atmosphere and 28 day rotation), the maximum temperature is within a few degrees of what is predicted. However, on the dark side the minimum predicted temperature is never reached ... it is not even close. This indicates that objects heat up faster than they cool down. This asymmetry means that Stefan's law can be used to predict maximum temperatures, but is fairly worthless at computing the minimum temperature of a rotating body. It also means that using this equation to compute the expected temperature from the average energy will always give the wrong results.

Peak Power

On a typical day, the TOA (Top of Atmosphere) solar output is 1,361 W/m2, enough energy to boil water. Yet, in the north, in winter, it is comfortable to stand in the sun. This is because

Some of the TOA energy is absorbed in the stratosphere
Most objects reflect a lot of light

Based on the IPCC radiation diagram, during the middle of the day, 268 W/m2 (67*4) are absorbed by the stratosphere. Subtracting that from the TOA value yields the amount of energy available at the surface (assuming a clear day).

1,361 W/m2 at TOA - 67*4 W/m2 absorbed by the stratosphere = 1,093 W/m2 at the surface
Which can produce a temperature of 211°F (albedo = 0%)

For items on the Earth's surface (think roads), the predicted temperature (211°F) is never reached because of convection - the air gets hot and carries some heat away. Working with black asphalt samples, I was able to measure a maximum temperature of about 135°F. Assuming that the peak power at the surface is 1,093 W/m2 and an albedo of 4%, the expected temperature is 204°F. This difference is due to convection.

For a closed car (i.e. no convection), on a hot summer's day, the peak temperature (inside the car) is also about 135°F. (Yes, I have measured this.) Using this value (in the calculator above) allows computation of the effective albedo.

135°F -> 674 W/m2
(1093 - 674)/1093 = 0.38    (Albedo = 38%)

Thus, assuming an albedo of 38%, a car can not get hotter than 135°F and a passive solar hot water heater (albedo ~ 4%) can get hot enough to almost boil water ... which is close to what is observed.

Note: This is why solar water heaters work in Canada.

To simplify using the calculator above, buttons are provided for TOA (1,361 W/m2) and TOA minus the energy absorbed in the stratosphere (1,093 W/m2).

TOA/4

The average energy received at the surface is 1/4 of the peak (because the Earth rotates)

        TOA/4 * (1 - Albedo) = 340.25 * 0.70 = 238 W/m2  -> -18°C,  -1°F
(TOA-Strat)/4 * (1 - Albedo) = 273.25 * 0.70 = 191 W/m2  -> -32°C, -25°F

However, these are just averages. For instance, the number of hours of daylight various through out the year ... from zero to 24 near the poles.

In addition, the albedo of sand, ocean, deserts, and people varies considerably.

At equilibrium, the rate of absorbing energy is equal to the rate of emitting energy. However, in a system where the energy input is pulse width modulated (the Earth spins), the difference in these rates will cause the average temperature to be different than the value computed via Stefan's law.

Author: Robert Clemenzi