In regard to my previous post, I'll remind you of this: an object's temperature doesn't by itself determine how hot or cold it feels when you touch it. For a given temperature, when the object (or surface) is in contact with your skin, how fast how well it conducts heat is the primary property that determines how hot or cold it feels. This property is known as thermal conductivity.
One of the reasons I wanted to write the previous post on the temperature of an asphalt road in the summertime sun is to compare how hot the asphalt feels with how hot an iron manhole cover feels. The question is, which surface will burn you faster?
Another factor that comes into play in this regard is heat capacity, also called specific heat capacity, or just specific heat, which is usually described in terms of how much the temperature of a given mass of a certain substance increases when a given amount of heat is supplied to it. Turn that heat transfer process around and the question becomes how much heat (internal energy) is available to be transferred to a cooler object or surface (your skin, for instance) from the heated substance.
Now we have two questions concerning asphalt and iron: which will burn you faster, and which will give you a more serious burn due to the amount of heat it can transfer. These are related, but exactly how, I don't know. There is (or was) a spirited debate about this relationship at the physics "stack exchange" website.
We're not considering the emission of energy by radiation now. We're considering the transfer of energy by conduction, because we have two surfaces at different temperatures in contact with each other: the asphalt (or iron) and your skin. The role of the radiation is to bring the asphalt and the manhole cover to their Stefan-Boltzmann-law temperatures, as discussed in my previous post.
Are the pavement and the manhole cover even going to be at the same temperature in the direct sunlight? If we treat them both as black-body surfaces we are assuming they reach the same temp once equilibrium between absorption and emission of radiation has been established in each material.
Even when we don't treat them as black-bodies, if they are both equally good absorbers-emitters of radiation, we expect them to reach the same temperature. That's the subject of this post: studying the Stefan-Boltz law as it applies to non-black-bodies.
I mentioned in my previous post that the two problems I solved from the Milonni/Eberly textbook Lasers are elementary problems (2nd semester calculus-based freshman physics level), and in fact I found good explanations of how to use the Stefan-Boltzmann law for non-black-bodies in two freshman-level textbooks: University Physics by Sears, Zemansky and Young (SZY), and Principles of Physics by Frank Blatt.
The Stefan-Boltzmann law as Milonni/Eberly write it,
Itotal = σT4,
is applicable only to idealized surfaces or objects that are assumed to be emitting and absorbing a Planck black-body spectrum. But the fact that real objects aren't perfect emitters and absorbers can be taken into account if their emissivity, ε, is included in S-B law, which SZY's text does, using R instead of Itotal for the radiative flux:
R = εσT4
Prof. Blatt takes this a step further and explicitly shows how the
surface area of an object is used to calculate the amount of energy per unit time
it’s emitting:
P = AεσT4
where A is the surface area of the object and P is the power
(joules per second or watts in our MKS case). This is a helpful equation when it comes
to finding the radiant energy emitted by an object (the sun for instance) as opposed
to just a surface, where only the power per unit area (flux) can be calculated
(the asphalt surface or manhole surface, for instance).
Emissivity is defined as the "ratio of the energy radiated from a material's surface to that radiated from a perfect emitter, known as a blackbody, at the same temperature and wavelength and under the same viewing conditions. It is a dimensionless number between 0 (for a perfect reflector) and 1 (for a perfect emitter)".
Based on "too-much-information" I found at various websites, I'll estimate the emissivities of asphalt and of oxidized cast iron to both have a value of about 0.9, so they will both be heated by the sunlight to about the same temperature. Given the 756 watts/meter2 of solar radiation flux I used in my previous post, how much does a change from ε = 1 (black-body) to ε = 0.9 affect the temperature as calculated by the above equation?
Well, yeah, right, by a factor of the fourth root of 0.9, which is .974, which we might as well call 1, which means we might as well say the temp of both the asphalt and the manhole cover will be 153° F (67° C), as I previously calculated.
Now we need to know the thermal conductivities of asphalt and cast iron. I'm going to use 1.3 watts per meter per kelvin (1.3 W/m/K) for asphalt and 56 W/m/K for cast iron. Based on these numbers, the manhole cover is capable of burning your skin 40 times faster than asphalt.
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I crossed out the words "how fast" in the first paragraph above, but it's actually a good descriptor because time is implicitly present in the units for thermal conductivity: a watt is a joule per second. So how quickly (or slowly) something that's hot can burn you is indeed taken into account by its thermal conductivity.
If you reach into an oven where a loaf of bread has just finished baking, and accidentally let the metal oven rack touch your skin briefly, you'll get burned at least a little. You can touch the bread, however, without getting an immediate burn, and the air in the oven, which is at the same temp as the bread surface and the metal rack, would take even longer to burn you. In our asphalt and iron manhole cover exercise, the manhole cover is like the oven rack--yeah, both are metal--and the asphalt is a little like the bread. For much more about the thermodynamics of baking bread, see The Fresh Loaf for a discussion on relevant heat transfer mechanisms. Radiation is one of them.
Now what about the heat capacity of asphalt versus the heat capacity of cast iron? Heat capacity, as its name implies, is about the ability of a substance to store energy. This is the currently missing middle component in the energy conservation process we're considering here: Absorption of energy, storage of energy, and emission of energy.
First we have solar radiation as the medium for the heat absorbed by the asphalt and the iron manhole cover. Second--since we assume no radiation is being reflected or transmitted--we have energy being stored in the material as kinetic energy of the atoms, molecules, and free electrons (if there are any) that make up the material. And third we have emission of heat from the surface of the material. This was being done solely by radiation emission (so far we're ignoring the role of the heated air above the surfaces) in an equilibrium situation with absorption and storage of energy--until you came along and put one hand on the manhole cover and the other on the pavement to find out which feels hotter. Or maybe it was me who did that.
No matter who did it, that little one-second personal experiment would show the manhole cover feels hotter. You or I might not even be able to keep a hand on the manhole cover for the length of time it takes to say "Mississippi-one".
Why, then, when we use a thermometer to measure the temps of both the asphalt and the manhole cover do we not predict the manhole cover will be at a higher temp? It's a matter of perception, which is something discussed in a typical Coen-Brothers' way in A Serious Man. Our un-serious man, Prof. Gopnick, remarks after smoking pot and drinking iced tea with his next-door neighbor Mrs. Samsky, that maybe everything is just a matter of perception as the junior rabbi, Rabbi Scott, had suggested to him. But Rabbi Scott had said "perspective" not perception.
Our perception of temperature depends on the rate at which energy is conducted to or away from our skin. Thermometers register this same effect by absorbing or losing energy (from a hotter object or to a cooler object, respectively) either rapidly or slowly while coming into thermal-conductance equilibrium with the object or surface they are in contact with. How quickly or slowly this equilibrium happens depends on how well the object or surface conducts heat, and also on how well the thermometer itself conducts heat. I recall one of my favorite experiments in first-semester physics was called "The Time Constant of a Thermometer."
Now back to heat capacity. The heat capacity of asphalt is about twice that of cast iron: 900 J/kg/K for asphalt versus 460 J/kg/K cast iron. I merely leave this to the reader (and writer) as something to think about, that asphalt stores energy better than iron but conducts it about 40 times more slowly. One thing to think about in this regard is a block of iron in contact with a block of asphalt, where each is at a different temperature to begin with. Another thing is this: what is the heat capacity of a black-body? Einstein in 1907 used Planck's 1900 relation of discrete energy levels for continuously distributed frequencies to model the vibrations of solids, which are all about the heat capacity of solids, but heat capacity seems to be ignored when it comes to discussions of black-body radiation.
The "Pavements" website of the U.S. Federal Highway Administration is where I got the asphalt heat capacity number. The full name of this website is
Pavement Thermal Performance and Contribution to Urban and Global Climate
The info there is relevant to most of this blog post of mine, and also mentions "albedo" in contrast to emissivity, although the relation between the two isn't simple. Well-written, check it out. I'm outta here.
