CHPT 10 EXTENSION
THERMAL CONDUCTIVITY
MATHEMATICAL MODELS
Mathematical models are a powerful way to describe and understand the relationships in nature.
Scientists and engineers use the following mathematical model to describe and understand how
thermal energy is conducted through materials:
Q AT

t
L
Where
Q
t
is the amount of thermal energy [Q] transferred in a certain amount of time [t], typically

The Greek letter kappa, , represents the thermal conductivity constant of a particular
substance, measured in Joules per second meter degrees Celsius,
J . It is frequently reported as Watts per meter per Kelvin, W / m K.
s m C
measured in Joules per second, J / s which can be abbreviated as Watts, W.
Material
Thermal
conductivity
Material
J / (s mC)
Diamond
Silver
Copper
Brass
Aluminum
Iron
Steel
Lead
Mercury
Glass, ordinary
Concrete
Fiberglass
Asbestos
Brick, red
1000
406.0
385.0
109.0
205.0
79.5
50.2
34.7
8.3
0.8
0.8
0.04
0.08
0.6
Thermal
conductivity
J / (s mC)
Cork board
Wool felt
Polystyrene (Styrofoam)
Polyurethane
Wood
Air (0° C)
Snow(dry)
Ice
Water (20° C)
Helium (20°C)
Hydrogen (20°C)
Nitrogen (20°C)
Oxygen(20°C)
Silica aerogel
0.04
0.04
0.033
0.02
0.12-0.04
0.024
0.05
1.6
0.6
0.138
0.172
0.0234
0.0238
0.003
A
is the area exposed to both the warmer and cooler areas, measured in square meters, m2
T
is the temperature gradient, i.e., the temperature difference between the warmer and cooler
areas, typically measured in degrees Celsius, C
L
is the thickness of the material, measured in meters, m
Use this mathematical model to make predictions about thermal conductivity. For each prediction,
use the GUESS method, i.e., identify what is Given, Unknown, and the Equation. Then Substitute
and Solve.
Example 1:
Lglass = 0.01 m
Predict how much thermal energy is lost
through a window on a cold day (0C).
Assume that the window is made from
ordinary glass [ = 0.8 J ] that is
Tin = 20C
Aglass = 1.0 m2
s m C
m 2,
0.01 m thick and covers1.0
and that
the temperature inside is 20C.
G
glass = 0.8
Given
 = 0.8
U
Unknown
A = 1 m2
T =Thot side – Tcold side = 20C – 0C = 20C
L = 0.01 m
Q=?
E
Equation
S
Substitute
Solve
J
s m C
t
Q AT

t
L
Q

t
S
Tout = 0C
0.8
J
s m C
 1.0 m 2  20  C
0.01 m
Q = 1,600 J
t
s
Lair = 0.01 m
Example 2:
Suppose you were able to prevent a layer of
air from moving. Predict how much thermal
energy would be lost through that layer of air
on a cold day (0C). Assume that the air is
the same size and thickness as the glass from
the previous example: 0.01 m thick
covering1.0 m2. Assume the temperature
inside is 20C.
G
Given
  0.024
Tcool = 20C
Aair = 1.0 m2
air = 0.024
[from chart on reverse]
J
s m C
U
Unknown
E
Equation
S
Substitute
A = 1 m2
T =Thot side – Tcold side = 20C – 0C = 20C
L = 0.01 m
Q=?
t
Q AT

t
L
Q

t
S
Solve
0.024
J
 1.0 m 2  20  C
s m C
0.01 m
Q
= 48 J / s
t
Thot = 0C
Example 3:
Lfoil = 0.0003 m
Predict how quickly a 0.1 m x 0.1 m
sheet of aluminum foil will conduct
thermal energy when it is exposed
to 425C on one side and 25C on
the other. Assume the sheet is
0.0003 m thick.
Tcool = 25C
Afoil = 0.01 m2
Thot = 425C
foil = 205
G
Given
 = 205.0
U
Unknown
A = 0.1 m x 0.1 m = 0.01 m2
T =Thot side – Tcold side = 425C – 25C = 400C
L = 0.0003 m
Q=?
E
Equation
S
Substitute
S
Solve
[from chart]
J
s m C
t
Q AT

t
L
Q
= _________________
t
Q
= 2,730,000 J / s
t
A note about units:
Understanding the units involved in a mathematical model is helpful in understanding the relationship
between parts of that model. Examine the units in this relationship.
Q AT

t
L
J
 m 2  C
J s m C

s
m
In English: the rate at which energy is transferred through an object by conduction is proportional to
its area (A), its thickness (L), the temperature difference (T), and the type of material from which
the object is made ().
Hopefully, this statement matches your previous experience and intuition.
Name _____________________________________ Period _____ Date _______________
CHPT 10 PRACTICE
THERMAL CONDUCTIVITY
MATHEMATICAL MODELS
1) Predict how quickly thermal energy will travel through
a cylinder of Plaster of Paris if the cylinder is 0.05 m
thick and covers 0.007 m2. Assume temperature on
the exposed side is 425C and the protected side is
25C. Assume the thermal conductivity constant of
Plaster of Paris is about the same as a brick.
LPoP = 0.0003 m
Tcool = 25C
APoP =
m2
Thot = 425C
PoP =
G
Given
U
Unknown
E
Equation
S
Substitute
S
Solve
2) Predict how quickly thermal energy will travel through a layer of cardboard if the cardboard is 0.01 m
thick and covers 0.01 m2. Assume temperature on the exposed side is 425C and the protected side
is 25C. Assume the thermal conductivity constant of cardboard is 0.04 J / (sC m)
G
Given
U
Unknown
E
Equation
S
Substitute
S
Solve
3) Predict how quickly thermal energy will leave a
human body (T = 34C, A = 2 m2) at room
temperature (T = 20C). For the sake of
calculation, assume air is 0.024 J / (s C m) and
that the air is 0.05 m ‘thick’.
Lair = 0.0003 m
Tbody = 25C
Abody =
m2
Tair = 425C
air =
G
Given
U
Unknown
E
Equation
S
Substitute
S
Solve
4) Predict how quickly thermal energy will travel through a tile of silica aerogel if the tile is 0.1 m thick and
covers 0.0225 m2. Assume temperature on the exposed side is 3,000C and the protected side is 25C.
G
Given
U
Unknown
E
Equation
S
Substitute
S
Solve