Thermal expansion

1
Thermal expansion

• Linear expansion
• Most materials expand when heated. As long as the
  temperature change isn't too large, each
  dimension of an object experiences a change in
  length that is proportional to the change in
  temperature.
• or, equivalently,
• where L0 is the original length, and is the
  coefficient of linear expansion, which depends on
  the material.

Material ( 10-6/C) Material ( 10-6/C)
Aluminum 23 Glass 8.5
Copper 17 Iron 12
2
Thermal expansion

• Volume expansion
• For small temperature changes, we can find the
  new volume using
• or, equivalently,
• where V0 is the original volume.

3
Bimetallic strip

• A bimetallic strip is made from two different
  metals that are bonded together. The strip is
  straight at room temperature, but it curves when
  it is heated. How does it work?
• What is a common application of a bimetallic
  strip?

4
Bimetallic strip

• A bimetallic strip is made from two different
  metals that are bonded together. The strip is
  straight at room temperature, but it curves when
  it is heated. How does it work?
• The metals have equal lengths at
• room temperature but different
• expansion coefficients, so they have
• different lengths when heated.
• What is a common application of a bimetallic
  strip?
• A bimetallic strip can be used as a switch in a
  thermostat. When the room is too cool the strip
  completes a circuit, turning on the furnace. The
  furnace goes off when the room (and the strip)
  warms up.

5
What happens to holes?
When an object is heated and expands, what
happens to any holes in the object? Do they get
larger or smaller? 1. The holes get smaller
2. The holes stay the same size 3. The holes
get larger
6
Holes expand, too

• Holes expand as if they were filled with the
  surrounding material.
• If you draw a circle on a disk and then heat the
  disk, the whole circle expands.
• Removing the material inside the circle before
  heating produces the same result the hole
  expands.

7
Holes expand, too
8
Thermal Stress

• If an object is heated or cooled and it is not
  free to expand or contract, the thermal stresses
  can be large enough to cause damage. This is why
  bridges have expansion joints (check this out
  where the BU bridge meets Comm. Ave.). Even
  sidewalks are built accounting for thermal
  expansion.
• Materials that are subjected to thermal stress
  can age prematurely. For instance, over the life
  of a airplane the metal is subjected to thousands
  of hot/cold cycles that weaken the airplane's
  structure.
• Another common example occurs with water, which
  expands by 10 when it freezes. If the water is
  in a container when it freezes, the ice can exert
  a lot of pressure on the container.

9
A black can and a white can

• Two cans, one black and one white, are at room
  temperature. They are then exposed to a heat
  lamp. Which one heats up fastest? The cans are
  identical except for their surfaces.
• 1. the black can
• 2. the white can
• 3. they heat up at the same rate

10
A black can and a white can

• We've probably all noticed, by leaving black
  objects out in the sun, that they heat up
  fastest. The black can absorbs radiation more
  efficiently than does the white can, which
  reflects more of the radiation away.

11
A black can and a white can

• The same two cans are then filled with hot water.
  Which cools down fastest?
• 1. the black can
• 2. the white can
• 3. they cool down at the same rate

12
Heat transfer

• Heat naturally flows from higher-temperature
  regions to lower-temperature regions.
• The three basic mechanisms by which heat is
  transferred are conduction, convection, and
  radiation. We'll look at each of these
  separately, but in a given situation more than
  one mechanism might be important.

13
Conduction

• Thermal conduction involves energy in the form of
  heat being transferred from a hot region to a
  cooler region through a material. At the hotter
  end, the atoms, molecules, and electrons vibrate
  with more energy than they do at the cooler end.
  The atoms, molecules, and electrons don't flow
  from one place to the other - the energy flows
  through the material, passed along by the
  vibrations.

14
Conduction

• The rate at which heat is conducted along a bar
  of length L depends on the length, the
  cross-sectional area A, the temperature
  difference between the hot and cold ends, TH -
  TC, and the thermal conductivity k of the
  material.
• The rate of energy transfer is power, so

15
Thermal conductivity

• Metals generally have high thermal conductivities
  because of the free electrons that move around
  randomly. These are very efficient at
  transferring energy through the metal. Copper,
  for instance, has a thermal conductivity of 400
  W/(m K), compared to 0.024 W/(m K) for foam
  insulation.

16
R values

• Insulating materials are rated in terms of their
  R values, which measures their resistance to
  conduction. The higher the R, the lower the
  conductivity.
• In terms of the thickness, L

17
A conduction sandwich

• A typical conduction problem involves creating a
  sandwich of two (or more) layers and determining
  the temperature at the interface(s) between the
  layers.
• Consider a two-layer problem where one layer has
  twice the thickness and six times the thermal
  conductivity as the other layer, but the layers
  have the same area.

18
A conduction sandwich

• To find the temperature at the interface between
  the layers (after thermal equilibrium has been
  reached) you should
• 1. find that the unknown temperature is halfway
  between the temperature on one side and the
  temperature on the other side (T ?)
• 2. set up a ratio where the change in temperature
  across a layer is proportional to the thickness
  of the layer (T ?)
• 3. set up a ratio where the change in temperature
  across a layer is inversely proportional to the
  thickness of the layer (T ?)
• 4. set the rate of heat flow through one layer
  equal to the rate of heat flow through the other
  layer (T ?)

19
A conduction sandwich

• To find the temperature at the interface between
  the layers (after thermal equilibrium has been
  reached) you should
• 1. find that the unknown temperature is halfway
  between the temperature on one side and the
  temperature on the other side (T 12 C)
• 2. set up a ratio where the change in temperature
  across a layer is proportional to the thickness
  of the layer (T 8 C )
• 3. set up a ratio where the change in temperature
  across a layer is inversely proportional to the
  thickness of the layer (T 16 C)
• 4. set the rate of heat flow through one layer
  equal to the rate of heat flow through the other
  layer (T 18 C )

20
Convection

• Heat transfer in fluids generally takes place via
  convection, in which flowing fluid carries heat
  from one place to another. Convection currents
  are produced by temperature differences. Hotter
  (less dense) parts of the fluid rise, while
  cooler (more dense) areas sink. Birds and gliders
  make use of upward convection currents to rise,
  and we also rely on convection to remove
  ground-level pollution.
• Forced convection, where the fluid does not flow
  of its own accord but is pushed, is often used
  for heating (e.g., forced-air furnaces) or
  cooling (e.g., fans, automobile cooling systems).

21
Thermal radiation

• Thermal radiation involves energy transferred via
  electromagnetic waves. Often this is infrared
  radiation, but it can also be visible light or
  radiation of higher energy.
• Thermal radiation is relatively safe, and is not
  the dangerous nuclear radiation associated with
  nuclear bombs, etc.
• All objects continually absorb thermal energy and
  radiate it away again. When everything is at the
  same temperature, the amount of energy received
  is equal to the amount given off and no changes
  in temperature occur. If an object emits more
  than it absorbs, though, it cools down.

22
Thermal radiation

• For an object with a temperature T (in Kelvin)
  and a surface area A, the net rate of radiated
  energy depends strongly on temperature
• where Tenv is the temperature of the surrounding
  environment, and the Stefan-Boltzmann constant is
  
• s 5.67 x 10-8 W/m2
• e is the emissivity. It is a measure of how
  efficiently an object absorbs and emits radiated
  energy. Highly reflective objects have
  emissivities close to zero. Black objects have
  emissivities close to 1. An object with e 1 is
  called a perfect blackbody.

23
Thermal radiation

• The best absorbers are also the best emitters.
  Black objects heat up faster than shiny ones, but
  they cool down faster too.
• This is exactly what we observe with our black
  can and white can, as the cans cool.

24
Newtons Law of Cooling

• In many situations (and our cans are an example
  of this), the temperature of a hot object that is
  cooling follows an exponential decay. What does
  this tell us about what the rate at which the
  object loses energy? How does this rate of energy
  loss depend on temperature?

25
Newtons Law of Cooling

• In many situations (and our cans are an example
  of this), the temperature of a hot object that is
  cooling follows an exponential decay. What does
  this tell us about what the rate at which the
  object loses energy? How does this rate of energy
  loss depend on temperature?
• Exponential decay is characteristic of a rate
  that is proportional to a quantity, in this case
  the temperature difference between the object and
  the surroundings.
• h is the heat transfer coefficient

26
Newtons Law of Cooling

• With the rate of energy loss being proportional
  to the temperature difference, the exponential
  equation for the temperature of the object as a
  function of time is
• ? is the decay rate, which depends on the surface
  area and emissivity.

27
Heat

• What is heat?

28
Heat

• Heat is energy transferred between a system and
  its surroundings because of a temperature
  difference between them.

29
Specific heat

• The specific heat of a material is the amount of
  heat required to raise the temperature of 1 kg of
  the material by 1C.
• The symbol for specific heat is c.
• Heat lost or gained by an object is given by

Material c (J/(kg C)) Material c (J/(kg C))
Aluminum 900 Water (gas) 1850
Copper 385 Water (liquid) 4186
Gold 128 Water (ice) 2060
30
A change of state

• Changes of state occur at particular
  temperatures, so the heat associated with the
  process is given by
• Freezing or melting
• where Lf is the latent heat of fusion
• Boiling or condensing
• where Lv is the latent heat of vaporization
• For water the values are
• Lf 333 kJ/kg
• Lv 2256 kJ/kg
• c 4.186 kJ/(kg C)

31
Which graph?

• Simulation
• Heat is being added to a sample of water at a
  constant rate. The water is initially solid,
  starts at -10C, and takes 10 seconds to reach
  0C.
• You may find the following data helpful when
  deciding which graph is correct
• Specific heats for water cliquid 1.0 cal/g C
  and
• cice csteam 0.5 cal/g C Latent heats for
  water heat of fusion Lf 80 cal/g and heat of
  vaporization Lv 540 cal/g
• Which graph shows correctly the temperature as a
  function of time for the first 120 seconds?

32
Which graph?
Which graph shows correctly the temperature as a
function of time for the first 120 seconds? 1.
Graph 1 2. Graph 2 3. Graph 3 4. Graph 4
5. Graph 5 6. None of the above
33
Ice water

• 100 grams of ice, with a temperature of -10C, is
  added to a styrofoam cup of water. The water is
  initially at 10C, and has an unknown mass m. If
  the final temperature of the mixture is 0C, what
  is the unknown mass m? Assume that no heat is
  exchanged with the cup or with the surroundings.
• Use these approximate values to determine your
  answer
• Specific heat of liquid water is about 4000 J/(kg
  C) Specific heat of ice is about 2000 J/(kg C)
  Latent heat of fusion of water is about 3 x 105
  J/kg

34
Ice water

• One possible starting point is to determine what
  happens if nothing changes phase. How much water
  at 10C does it take to bring 100 g of ice at
  -10C to 0C? (The water also ends up at 0C.)
• You can do heat lost heat gained or the
  equivalent method
• Plugging in numbers gives
• Lot's of things cancel and we're left with
• 100 g 2m, so m 50 g. So, that's one possible
  answer.

35
Ice water

• Challenge for next time find the range of
  possible answers for m, the mass of the water.

36
Whiteboard