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Heat Transfer
We all know from text-book Physics that heat can be transferred by conduction, convection and radiation. New books add evaporation and this does, indeed, involve heat change as the state of a substance moves from solid to liquid or liquid to gas. These heat changes are called Latent Heat of fusion and Latent Heat of vaporisation and are used in refrigeration cycles and many other applications.
If we start with radiation first we can introduce a simple formula given by Josef Stefan (1835 - 1893).
Rate of loss of energy = sigma x surface area x (Temperature)4
where sigma is Stefan's constant with value 5.76x10-8 Wm-2K-4
If we think of the earth heated by the sun then an equilibrium temperature is reached when the incoming radiation is balanced by the loss of heat from the earth.
The only way that earth can lose heat is by radiation and so we can calculate the loss of heat using the above equation. Assuming a temperature of 300 degree Kelvin for the earth's surface.
The rate of loss of heat is 458 watts per square metre.
Now the surface area of the earth is 5.1x1014 square metres
So rate of loss of heat is 2.3 x 1017 watts
On page 1 we have listed a value of 1.1 x 1017 so we see that our calculations are in reasonable accord. The added corrections that need to be made are twofold :
a . the earth is not a perfect emitter (ie the emissivity is less than 1 ) and so corrections must be made.
b . there are interactions between the outgoing radiation and the atmosphere and these can be very complex. They give rise to the Greenhouse Effect, amongst other things, and we know that clear skies at night gives rapid cooling of the earth's surface.
( another aspect of radiation theory is that surrounding objects will radiate back towards the object under investigation. Our earth calculation was much simplified since we could neglect interactions from deep space as it has a temperature close to zero degree Kelvin ).
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With convection we need a material to be present. The fluid (gas or liquid) adjacent to a hot object gets heated and, therefore, becomes less dense. This causes the heated fluid to rise and new fluid comes in to take its place . A circulation of fluid is thus set up and these circulation patterns are called convection currents.
Although convection is a very complex theoretical problem one usually uses a very simplified calculation proceedure.
the rate of heat loss = heat transfer coefficient x surface area of object x ( Tbody - Tambient)
The symbol used for the heat transfer coefficient is h and the unit is Wm-2K-1
An experiment to determine h for air cooling has been given at www.picotech.com/experiments/heat_transfer_coefficieny/heat.html. This experiment monitors the cooling of a slab of aluminium and the value of h is approximately 10 Wm-2K-1 .
In modern electronics heat has to be moved quite rapidly from components such as IC processors. Here a combination of convection and thermoelectric cooling is used. Melcor , page 16, has provided a software package to evaluate thermoelectric coolers and the approriate Heat Sinks . the summary is below:
About 100 watts of heat will be removed from any device with the given Peltier cooling device and a heat sink with resistance 0.068 is required to extract the heat from the Peltier device. This thermal resistance is the reciprocal of h with an area included
so Temp. Difference (K) == resistance (K/W) x rate of heat transfer (W).
Thus device and upper surface of Peltier will have a temperature of 50 degree C
Lower surface of Peltier will be 30 degree C
and resistance x power gives approximately 7 degreeC between ambient and the Peltier temperature.
Solar heating, page 13, stresses the importance of keeping convection losses to a minimum by having a cover glass over the heat exchange panel. If this were not present then the hot panel would be cooled by the air currents and very little heat would be transferred to the circulating water.
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Conductive losses also require a material to be present but here we are concerned with solid materials so there is no possibility of convection currents being established. The formula for the rate of heat conducted through a slab of material of thickness t is
Watts transmitted == Kappa x A x ( Thot - Tambient ) / t
Kappa is the thermal conductivity of the material and a few values are given below
Material Kappa (Wm-1K-1)
Aluminium 230
Copper 380
Stainless Steel 25
Brick 0.5
Wood 0.15
Glass 1.0
Cork 0.05
Glass Wool 0.04
So, a bar of copper ( and most other metals) will transmit heat very readily whereas a good thermal insulator would be made from brick, glass, cork etc.
For the building trade many materials are standardised in their sizes and, although scientists still use Kappa, it is conventional to replace the thermal conductivity by a combination of Kappa and sample thickness. This gives us the so-called U-value.
For instance the length of a brick is 225 mm and therefore the heat loss for a brick in a length-wise direction is Watts transmitted == (0.5/.225) x A x Temp. Difference
Watts transmitted == U x A x Temp. Difference
and the U - values for brick is about 2 Wm-2K-1
It is now much easier to calculate heat loss as only the area and temperature difference are required.
Suppose we have a house : with wall area 140 m2 and U- value 2 Wm-2K-1 : window area 20 m2 and U- value 7 Wm-2K-1 floor area 60 m2 and U- value 1 Wm-2K-1 : roof area 60 m2 and U- value 0.5Wm-2K-1 . The rate of transmitted heat is evaluated for each section in turn and then added together to yield the total heat lost. For a temperature difference of 10 degreeC this would give a rate of loss of heat as 2800 + 1400 + 600 + 300 = 4200 Watts. The numbers have been chosen purely for illustration and would not be considered accepable today!
As in many cases features in a building will be a combination of two materials as below.
We know the temperatures of the inside and outside but not the interfacial temperature. By setting up two equations for each of the materials it is possible to eliminate the interfacial temperature and the combined U - value, Utotal , is given by
1/Utotal = 1/Uinsulation + 1/Ubrick
Now if Ubrick is 2 and, say, Uinsulation is 0.2 then Utotal is 0.18 i.e. the U - value is lower than either of the component parts.
Building Regulations are now demanding much better insulation in homes to cut down fuel wastage. This means low U-values for walls, floors, roofs and windows in new built property ;
external walls 0.3 ( this demands insulated cavities as the brick alone has a U-value of 2) windows 1.8 ( this demands a double-glazed unit with a gap of 20 mm and low emissivity ) roof 0.5 ( 3 inch insulation layer ) floor 1.1 (not defined)
Advancing to year 2010 we see projections to even higher standards of insulation as shown www.odpm.gov.uk
For building structures which are capable of giving these U-values consult www.tek.kingspan.com/uk/thermal_performance.htm .
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