Cars, trailers, footpaths, people's heads, and attics get hot. A lot of things get hot in the sun. How can we predict exactly how hot something is going to get? What can we do to make it cooler?
Thank goodness for "Simplified Solar Heating Model" by Ephraim M. Sparrow, University of Minnesota. He kindly gives us the constants and most everything we need to do the calculations. This little project took about three weekends to complete, maybe two days of 'work'.
I could test his model because I had a Christmas present Radio Shack temperature sensor (Catalog No. 22-325, about $50). Here are the results. The measured temperatures around my house are awfully close to the predicted ones. The most error was for my two cars where I just guessed on the reflectivity and emissivity values (they are metal flake finishes). For the others I used published values for reflectivity and emissivity. Not bad for a simplified model Dr. Sparrow!
Clearly, everybody who wants to have cool cars, and other cool things in the Sun, should be happiest with dull-finish bright white.
What I did was to take Sparrow's paper and translate his equations and constants into an Excel spreadsheet. Download the Excel Spreadsheet here. It will save you a couple of hours of getting his equations into a form for Excel:
Now for a bit more explanation on my part.
Most things in the Sun do not continue to get hotter and hotter. This is because (1) they emit heat (they radiate it..those darn little infrared photons), and (2) the air also conducts heat away as well (those darn little atoms vibrating). At some temperature, the rate of heat gain from the Sun and the heat loss from radiation and air conduction balance out, and that is how hot your car got, out there in the parking lot!
The reflectivity of an object is the inverse of its ability to absorb radiant heat. Reflectivity is easy to see. Shiny metals are highly reflective (shiny aluminum is ~.71). White paint is pretty reflective (see the table above). It is important to be reflective in the invisible infrared part of the spectrum, but most materials that are reflective in visible are also strongly reflective in infrared. Dark dull things are not reflective. Reflectivity is easy to see.
Emissivity is related to the ability of something to give off heat. It is not easy to see. You can feel the heat of it, if you move near. But, it turns out emissivity is also a surface property. Interestingly, shiny metals which are highly reflective, also have very low emissivity. This is why shiny aluminum or chrome gets really hot in the Sun. While it is reflecting most of the heat away, it can emit heat only very slowly and so gets very hot. Perfect emissivity is called 'black body.' A black body basically radiates its heat as fast as possible (given things like external temperature). So, because there is perfect emissivity like perfect reflectivity, emissivity goes as a ratio from 0 to 1, just like reflectivity.
Human skin, and a lot of nature's things, stay cool in the sun because their emissivity is very close to 1. Man made things can come close to nature's things, but most have emissivity much lower than 1. You can paint shiny metal with white paint, or pretty much anything that isn't another shiny metal, and radically increase it's emissivity -- its ability to give off heat and stay cooler in the Sun.
It is interesting that conduction of heat into the air is strongly dependent on airflow. This is why driving a car for just a few minutes is enough to cool it down quickly. The airflow of driving continues to keep the homeostatic temperature low because the moving air is continuously cooling the car's surface.
So with the little Excel file above, Dr. Sparrow's paper, and a little Excel knowledge, anybody can design things to keep cool in the sun. The equation, Argh! above, that has to be solved to compute the homeostasis temperature, gives the rate of BTU change. A British Thermal Unit is the amount of energy needed to raise a cubic foot of water one degree Fahrenheit. Below the homeostatic temperature, the object gets hotter. Above it, it gets cooler. So we have to solve Dr. Sparrow's equations for a BTU rate of change of zero BTUs. That gives us the temperature that the object will reach balance in gaining and losing heat. That's when the temperature of the object, car, your head, or attic, stops changing.
So far so good. Unfortunately, though, Dr. Sparrow's equations requires us to solve for the roots of a fourth-order polynomial (it's not his fault, a little joke from God), and, while there exist closed form solutions for fourth order, they are pretty horrendous. I spent most of my time on this one figuring out which computer language I was going to code up my own polynomial solver to give away, or trying to find a solver that I knew just about everybody already has for free on their PC. I wanted you to have everything you need to look like an incredible expert, like Dr. Sparrow.
It took some real digging to find a really easy solution, and I found, well-hidden, that good old Microsoft Excel has a standard "add-in" called "Solver" that will do the trick. You have to go to "Tools Add-ins" and add in Solver. It's a free Microsoft add-in. Then when you bring it up, you only need to fill in three fields: Where the BTU gain/loss equation is (the "target cell"), the fact that you want this field to be solve for the value of 0 (gain or loss), and what variable you want to run free until solved, this is the surface temperature (cell) you are predicting. Here is the Excel Solver Window used in the above Excel download solving for shiny Aluminum (predicting a hot 160 Degrees Fahrenheit homeostasis).
You just click "Solve" and it goes off and solves the fourth order polynomial, and predicts the temperature. And, as my little experiment showed, the prediction ain't too bad.
So now all you serious architects, design engineers, and even you science project kids, can build better stuff, and also predict just how hot your car is going to get in the Sun (and the shade, in still air, and in the wind, painted, and made to shine, etc.) And we have to thank Dr. Sparrow for giving us a simplified model, of what is otherwise some pretty daunting physics, that seems to work!
Google for "material reflectance emissivity tables" and you'll find all kinds of published data for common materials that you can use in your industrial design projects. Just imagine all those golfers with colorful beanie hats you invent with electric motors running their head-cooling beanie props!
I would be interested to hear of any other experimental results with Dr. Sparrow's simplified model.
As a follow up, you can also probably predict how long it takes your car to get as hot as it is going to get. My theory is just to use these same equations and then treat your car as a mass equivalent to the water. A cubic foot of water weighs 62.42796 pounds (Thank you John Walker). BTUs translate pretty much directly. A car that weighs a ton should heat up like about 16 cubic feet of water.
Similarly, you should be able to predict just how fast your car cools down to the wind of driving, or your (properly capped) head to the wind of running or "being beanie cooled."
Addendum: The next day.
Just how hot could the Sun really get you, if it weren't 100 million miles away? Well, here is its cool 'surface' (~125,000 degrees F, thanks) with Jupiter and Earth to ~scale (864,327, 88,411, and 7,926 miles in diameter, respectively). Above this 'surface' it gets about five million degrees Fahrenheit. An interesting rule: the earth is about 1/10 the diameter of Jupiter which is about 1/10 the diameter of the Sun, and the Sun diameter is about 1/100 the distance from the Earth to the Sun (~91,600,000 to ~94,800,000 miles). Another exercise for the reader is how close the Sun needs to get to vaporize your car (iron boils at 5,182 degrees F, aluminum 4,472 F, or about 1/1000th the radiant temperature of the Sun). The Sun could be much harsher.