Convection - Heat transfer in a gas or liquid by the
circulation of currents from one region to another.
Can be forced or spontaneous (natural).
Hot and cold liquid is brought in a thermal contact;
it reduces the distance across which the
conduction occurs and increases the contact area.
!T
H = " kA
!x
Reducing heat-flow rate for better
thermal insulation.
!T
H = "kA
!x
Thick (large Δx) cavities in house walls filled with insulating
(small k) materials;
Reducing the number of walls (small surface area A).
Heat transfer by radiation. Stefan-Boltzmann law
Total power, P [J/s=W], emitted by a hot object.
P = e!AT
4
● 4th power of the absolute temperature, T.
● Surface area, A.
● Emissivity of the material, e.
#8
2
4
● Stefan-Boltzmann constant,$ = 5.7 " 10 W/(m ! K )
Emissivity, e, varies between 1 (black body) and 0 (reflecting surface).
A good emitter of radiation (e ≈ 1) is also a good absorber.
A radiator and a solar heater should be black.
A thermos bottle should be silver-coated.
Emissivity, e, depends on the wavelength of the radiation.
For a solar heater, what values of e are the best for sunlight and for
thermal waves radiated near 100 °C?
Gases
Gas is matter in a rarefied state.
The molecules are moving freely most of the time, and only once
in a while undergo short-term collisions.
The macroscopic state of a gas in thermodynamic equilibrium is
determined completely by its temperature, pressure, and volume.
The ideal gas law
PV = NkT
P is the pressure, V is the volume, T is the absolute temperature…
N is the total number of molecules in the gas and
k is Boltzmann’s constant, k = 1.38×10-23 J/K
The ideal gas law
PV = NkT
Gas in a cylinder under a piston
Pressure, P, is given by
P = Patm + mg / A
Where m is the total mass of the
piston and the lead and A is the
area of the piston.
We can:
• add or remove the lead shots to
change the pressure of the gas;
• tune the temperature of the
thermal reservoir.
V = NkT / P
The ideal gas law
Doubling the temperature, number of
molecules, pressure?
P = NkT / V
Keeping the volume and the number of particles
constant, but doubling the temperature?
N is normally very big, while k is a very
small number…
N = nN A
NA = 6.022×1023 – Avogadro number, number
of molecules in 1 mol of a substance;
n is the number of moles in the gas.
PV = nN A kT = nRT
R = 8.31 J/mol⋅K universal gas constant
Kinetic theory of the ideal gas
Kinetic energy is the only form of
molecular energy that is important and it
is preserved in the collision events.
Collisions of the gas
molecules with a wall..
L
As a result of a collision the momentum changes by
!p = 2 p x = 2mv x
Force due to one molecule as a function of time
Collisions of the gas molecules with a wall (cont.)
Newton’s second law for
an instantaneous force:
For the average force
on the wall it becomes:
dv x dp x
Fx = ma x = m
=
dt
dt
!p x 2mv x
Fx = ma x =
=
!t
!t
change of
momentum in
a collision
time between
collisions
Now Δt is a long time interval –
the time between two consecutive
collisions with the wall.
2mv x
Fx = ma x =
!t
Now Δt is – the time between two consecutive collisions with the wall.
!t = 2 L / v x
L
2mv x 2mv x
mv
Fx =
=
=
!t
2L / vx
L
2
x
Kinetic theory of the ideal gas.
Let’s try to account for all molecules of the gas:
2
m(v12x + v22x + ... + v Nx
)
F=
L
2
1x
2
2x
2
Nx
(v + v + ... + v )
v =
N
2
x
mN v
F=
L
2
x
Kinetic theory of the ideal gas.
Nmv
F=
L
2
x
Pressure on the wall with surface area A:
L
2
x
N mv
F N mv
P= =
=
A
LA
V
2
x
V – volume of the box.
Velocity of a molecule:
r
v (v x , v y , v z )
The average velocity – average of a sum is
equal to the sum of averages…
All the directions of motion (x, y, z)
are equally probable!
2
2
2
v = vx + v y + vz
2
2
2
v = vx + v y + vz
2
vx = v y
2
2
2
1 2
= vz = v
3
2
Pressure on the wall with surface area A:
2
N
m
v
F N mv
1
P= =
=
A
V
3 V
2
x
2 1 2
2
PV = N ( mv ) = N K
3 2
3
L
The average kinetic energy of a molecule
The ideal gas law (experimental fact!)
Therefore:
2
NkT = N K
3
1 2
K = mv
2
PV = N kT
3
K = kT
2