Posts Tagged ‘Jozef Stefan’

Heat Transfer in Mechanical Engineering, Part III, Radiation

Sunday, February 28th, 2010

     Last week we talked about convective heat transfer and how hot pavement in a parking lot gives up its heat to the environment.  But how does the pavement get hot to begin with?  This week we’ll discuss radiant heat transfer to find out. 

     The sun is a huge nuclear furnace, separated from the earth by 93 million miles of space.  The space between is just a vacuum, almost completely devoid of matter.  Without contiguous solid or liquid matter between the two, heat transfer by conduction or convection can’t occur. The heat that we feel on earth is actually generated when surfaces here absorb electromagnetic energy waves emitted by the sun.  Although these waves have traveled through millions of miles of space, they have not lost their punch.  Our eyes perceive some of them as sunshine, but many others are not visible.  But even if we can’t see them, our bodies often perceive them as heat.

     But radiant heat transfer isn’t a phenomenon exclusive to the sun.  It can also occur when something is on fire.  Intense fires can transmit tremendous amounts of radiant energy across significant distances.  They can even cause combustible materials nearby to burst into flame without any direct contact being necessary.  A line of sight between the source of heat and the receiving object is all that is required, and this is because radiation moves in straight lines, it can’t bend around corners.

     In order to calculate radiant heat transfer, we still must consider the temperature difference between the bodies, as well as the area of heat transfer, just as we did when considering the cases of conductive and convective heat transfer.  But since there is no conduction or convection activity taking place, we need not concern ourselves with thermal conductivity or convection coefficients.  Instead, we have to consider something called the Stefan-Boltzmann constant, a nifty little number that looks like this:   0.000000057 Watts/m2K4.  It was discovered in 1879 by a scientist named Jozef Stefan.  It was later derived by his student, Ludwig Boltzmann, in his work on thermodynamics and quantum mechanics.  Now remember from our discussion last week the unit “K” means Kelvin (°C +273.15).

     Now, ideal radiant heat transfer problems involve calculations that need only consider the Stefan-Boltzmann constant.  By “ideal,” I mean that there is perfect emission of radiation by one object and perfect absorption of that radiation by another.  But reality is not typically so kind, and radiant heat transfer problems typically involve calculations that involve more than just the Stefan-Boltzmann constant.  They involve additional calculations of terms like emissivity factors and geometric factors.  What’s that?  Read on.

     Emissivity factors relate to how well objects actually emit and absorb radiation compared to an ideal case.  For example, a shiny object doesn’t absorb radiant energy as well as a dull, black object.  Geometric factors are included in radiant heat transfer calculations to account for the shapes and relative orientation of the objects emitting and receiving radiation.  For example, do you ever notice how the sun is hotter at noon than it is at sunset?  Well, that’s because an object with a surface that’s parallel to the surface emitting radiation will receive more radiation than one that isn’t.

     Just to give you a basic idea of how radiant heat transfer calculations work, let’s consider an ideal situation.  Suppose you own a store building with a flat roof.  The store is right on the equator and it’s the vernal equinox.  The roofing material is dull black, measures 20 meters by 10 meters, and it absorbs radiant energy like a sponge.  But today is a dark, cloudy day, and the temperature of the roof is a cool 25°C.  Now, at some point in your life I’m sure you’ve seen a documentary where scientists declared that the surface temperature of the sun is a blistering 5,400°C.  Keeping this in mind, if the sun were to suddenly pop out of the clouds directly overhead at high noon, what would be the amount of radiant heat it would transfer to the roof?

     Well, according to Jozef Stefan, the radiant heat transfer rate can be calculated to be:

     Heat Flow = (The Stefan-Boltzmann Constant) x

                               (The Area of the Roof) x

                                   ((The Sun’s Temperature)4 – (The Initial Roof Temperature)4)

Now the terms we’ll need to plug into the heat flow calculation above are found as follows:  The sun’s temperature is:  5,400°C + 273.15 = 5,673.15K.  The temperature of the roof is:  25°C + 273.15 = 298.15K.  The area of the roof is:  20 meters x 10 meters = 200 meters2.   So, the heat transfer rate is:

Heat Flow =  (0.000000057 Watts/m2K4) x (200 m2) x ((5,673.15K) 4 – (298.15K) 4)

                      = 11,808,605,250 Watts

This would be the maximum rate of heat transfer that the roof could absorb at the instant the sun popped out of the clouds.

     From our example we can conclude that even in less than ideal conditions, radiant energy from the sun has the potential to generate tremendous amounts of heat on the surface of the earth.  Much of this heat drives our weather as a result of convective heat transfer that takes place between the earth’s surface and our atmosphere.

     That wraps things up for our discussion of heat transfer in mechanical engineering.  Next week we’ll talk about the importance of vibration analysis in design.  Remember what happened to Jody Foster’s character in the movie Contact when the space/time device she was in began to shake violently?  That’s the kind of thing vibration analysis seeks to correct!