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Heat is vibrational energy and all atoms above absoulte zero (0°K or -273°C) vibrate. The hotter the atoms are, the more they vibrate. Heat is lost (transfered) from a hot (rapidly vibrating) medium to a cold (slowly vibrating) medium by three methods. Conduction, Convection and Radiation.
Conduction of heat is where the vibrational energy of atoms is transfered directly to nieghboring atoms. As atoms rock back and forth they nudge nieghboring atoms to do likewise. The extent to which this happens depends on the atomic structure of the material, the more rigid the material is, the better it is for transmitting vibrational energy in this way. Sound will travel through the material faster and so will heat. Sometimes we want to get rid of heat. The heat sink attached to a computer's CPU is made of rigid metal to conduct heat away from the CPU at a high rate. But more commonly we want to keep heat in so we use materials that are good insulators. An insulator is is simply a poor conductor and are usually less rigid materials such as polystyrene foam.
We can calculate heat losses by conduction quite easily. For a flat surface such as a wall or a floor, the heat loss for any given material, is proportianal to the surface area and the difference in temperature between the hot and cold side, and inversly proportional to the thickness. Thermal conductivity of any given material is express as watts per meter per degree C. Builders and engineers often use U-Values which are watts per square meter per degree C. U-Values are a more convienient measure for builders as the quoted values take account of the whole anatomny of the wall or window or door, including cavities and layers of different thicknesses. The relationship between the heat transfer rate and the temperature difference is generally assumed to be linear. This does hold quite true for the narrow range of 'ordinary' temperatures considered in buildings.
The thermal conductivities of some commonly knowm materials are shown below. Note that in general, metals are good conductors as are ridid non-metals, particularly diamond.
Conductivity (U value) of multi layer wall
Thermal conductivity is the reciprocal of thermal resistance. To calculate the overall thermal conductivity (U value) of a multi layer wall we can consider it as a series of thermal resistors. For each layer we work out the thermal resitance (R value or 1/U value) and then simply add them together to arrive at the total R value. We then reciprocate this to arrive at the total conductivity or U value for the wall as a whole.
The U value of each layer is simply the watts it will conduct per unit area per degree of temperature difference. This is the thermal conductivity of the material divided by the thickness in meters. Brick has a thermal conductivity of about 0.7 W/mK so a 15 cm (0.15m) thick wall has a conductivity or U value of 0.7/0.15 = 4.667 Watts per square meter per degree. It thus has a thermal resistance or R value of 1/4.667 or 0.214. An insulator such as styrofoam has a thermal conductivity of about 0.014 W/mK (20 times better than brick). A 5 cm (0.05m) thick layer has a U value of 0.014/0.05 = 0.28 and an R value of 1/0.28 or 3.57. If we have 2 layers of 15 cm brick and a 5 cm layer of foam the total R value is 0.214 + 3.57 + 0.214 = 3.998. To convert this back to a U value we reciprocate it. 1/3.998 = 0.25 approx. This means lose 0.25 watts per square meter per degree. If instead of foam filling a cavity we had air (thermal conductoivity 0.03 W/mK), the U value of the cavity is 0.03/0.05 = 0.6, the R value is 1/0.6 = 1.667. The R value for the whole wall is 0.214 + 1.667 + 0.214 = 2.095 and the U value is 1/2.095 = 0.48 approx. This means we lose 0.48 watts per square meter per degree, about twice as much as with the foam filling.
Convection applies only where the medium is a fluid (liquid or gas). As fluid is heated it expands and becomes less dense, causing it to rise and carry heat away from a hot area while cooler fluid moves in to take its place. Convection can be enhanced by pumping when heat transfer is desired as it is in central heating systems. But when heat transfer is not desired, convection is a nuisance. Fluids, particularly gases, are among the best insulators (as indicated in the above table). This is because the mollecules in a fluid, have no rigidity and can only transfer vibrational energy when they bump into one another. But the process of convection clearly limits the effectiveness of such fluids as insulators. The trick is to trap the fluid so that it cannot move and so cannot convect. Foams exploit this technique very well as do materials such as wool.
Radiation occurs because vibrating atoms act as tiny radio stations, pumping out electromagnetic waves. The radiated energy is proportional to the forth power of the absolute temperature (in degrees Kelvin). In a dense medium this radiation is immeadiatly absorbed by neighboring atoms making radiation partly responsible for conduction. But at the surface the radiation escapes. Because the radiation rises as the forth power of the absolute temperature, doubling the temperature increases the radiation 16 times! Thus transference of heat from a hot object to its cooler surroundings can occur at a very significant rate.
Not all atoms are equal though. Some materials are highly reflective like aluminum while other are dull like coal. Only three things can happen to radiation when it encounters a medium. It can be reflected in which case it has no further effect on the medium, it can be absorbed in which case it will heat up the medium, or it is transmitted by the medium. It turns out that there is a direct correlation between how much a meterial absorbs and its emisivity , how much it radiates. Pure graphite is almost a 'perfect black body' with an emissivity of about 99% while aluminum has an emissivity of about 3%. An object blackened with graphite and radiates more than 30 times the radiation of an identical object at the same temperature coated with aluminum.
A perfect black body radiates energy according to the formula
P = sT4 where P is watts per square meter, T is the absolute temperature in degrees Kelvin (K) and s is stefan's constant (5.67 x 10-8 ).
And a real body radiates P = esT4 where e is the emissivity of the material.
Vacuum Insulation: The thermos flask
The thermos flask uses a vacuum to insulate. The vacuum is supported by means of a double wall, joined only at the neck of the flask. A vacuum lacks atoms so can neiter conduct or convect heat. Apart from the small conduction occuring at the neck, the only way heat can travel from the flask's inside wall to the outside wall, or vice versa, is by radiation. To limit this, the walls are coated with a highly reflective material such as aluminum. The performance of the flask is truly impressive, keeping the coffee hot enough to drink for several hours. The coffee may well taste awful but this does not invalidate the flask's performance as an insulator.
Taking the above equations for radiation, it can be deduced that the heat transfer rate between the hot and cold sides of the thermos is
P = es(Thi4 - Tlo4 ) where Thi and Tlo are the high and low absolute temperatures (in °K).
Note that heat loss is directly proportional to the emissivity but depends dramatically on the temperature, particularly that os the hot side. U-Values are not of much help with a thermos flask because of the forth power rule. For low temperatures, the flask walls have extremely low U-Values, unattainable by ordinary insulators of a practical thickness. For moderate temperatures such as hot coffeee, the flask performs as well as a thick insulating wall. But with higher temperatures the U-Value becomes very high and can then underperform ordinary insulation materials. However, the heat flow is independant of the gap width. It does not matter if the evecuated gap is 2mm or 0.2mm, the heat flow is the same. It is actually this fact, together with the low heat capacity of the inner wall, that is the secret of the thermos flask's success. We could with thick enough walls, match the heat transfer rates needed to keep a cup of coffee hot all day, with good insulators such as polystyrene. But to do this would require a flask the size of a large bucket.
Extreme Applications: The 'Russian doll' thermos flask
For extreme temperatures, hot or cold, multiple vacuum layers can be brought to bear. If instead of a double wall, our thermos had a tripple or a quadruple wall. What would happen when we pour in say molten aluminum at 675°C? The inner wall would quickly assume the temperature of the molten aluminum and radiate accordingly, heating up the next wall. This second wall would also rise in temperature and go on to radiate to the third wall. But it would also radiate back to the inner wall, cutting the net flow of energy between the inner wall and itself. The second wall cannot attain the same temperature as the inner wall because in order to do so it would have to radiate out more energy to the third wall than it received from the inner wall. Instead each wall moves toward an equilibrium temperature that is progressively colder as we move away from the hot interior and such that each inner wall transmits equal amounts of energy. Once equilibrium is reached, heat flow across the whole ensemble is cut by a factor of N - 1 where N is the number of walls, regardless of wall thickness. In practice such systems comprise many walls. 40 is not uncommon.
Depending on the application, such multi layered vacuum insulators can be quite easy to build, particularly if you are not restrained by space or weight. It is only the outermost wall that has to bear the crushing atmospheric pressure. The innermost wall may also need to withstand outward pressure depending on the application but the inner walls need only to contend with their own weight. A sphere is the best shape for the outer wall although a geodisic dome is a close second. All such constructions will have weak points or conduction channels. In the ordinary thermos flask this point is the neck but in a multilayer construction each layer requires support as does the inner vessel. Support can be kept minimal and would typically use ceramic beads for a very low contact area and low conductivity. The main weak point is if the inner vessel requires some sort of connection to the outside world but this would depend on the application. One way of cutting construction costs is to accept a larger gap and construct each concentric layer as a wire basket over which aluminmum foil is drapped. Very home project perhaps but it works.
Effects of imperfect vacuums
A perfect vacuum is not actually possible to achieve with pumps. Even deep space is not a perfect vacuum. A decent vacuum pump can produce pressures as low as 0.1Pa (1 millionth of an atmosphere). This is about the top end for thermos flasks. Over time, vacuums tend to degrade as gas seeps slowly through the walls of the vacuum chamber although this can be countered by an in-situ pump. So how does a pressure of 0.1Pa affect the performance?
Of crucial importance in understanding how heat is transfered by gasses in this pressure range, is a quantity known as the mean-free-path (MFP). This is basically the mean distance a gas molecule travels before it hits something, either the walls or another gas molecule. The MFP depends on what type of gas it is as this affects the size and thus the effective collision area of the molecules. The smaller they are the longer the MFP. The MFP also depends on the number of molecules per unit volume. The more there are the shorter the MFP. For a given pressure, the higher the temperature the fewer the molecules and thus the longer the MFP. For a given temperature the higher the pressure the more the molecules and thus the shorter the MFP. We need an MFP that is longer than the gaps between the walls so the gas molecules collide with the walls more often than they collide with each other. Under such conditions conduction is proportional to the pressure and convection can be ignored as the viscosity of the gas is high compared with mass density. However with a MFP shorter than the wall gap the thermal conductivity of gas is a constant. This is the point at which the thermal integrity of the vacuum breaks down and thermal losses rise dramatically. At ordinary room temperature and a pressure of 0.1Pa air has a MFP about 10cm. A typical gap would be around 1mm.