Convection - Heat transfer in a gas or liquid by the circulation of currents from one region to...

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.

x

TkAH

Reducing heat-flow rate for better thermal insulation.

Thick (large x) cavities in house walls filled with insulating

(small k) materials;

Reducing the number of walls (small surface area A).

x

TkAH

Heat transfer by radiation. Stefan-Boltzmann law

4ATeP Total power, P [J/s=W], emitted by a hot object.

● 4th power of the absolute temperature, T.

● Surface area, A.

● Emissivity of the material, e.● Stefan-Boltzmann constant, )KW/(m107.5 428

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 NkTPV 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 = 1.3810-23 J/K

The ideal gas law NkTPV

Gas in a cylinder under a piston

Pressure, P, is given by

AmgPP atm /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.

The ideal gas law PNkTV /Doubling the temperature, number of molecules, pressure?

Keeping the volume and the number of particles constant, but doubling the temperature?

VNkTP /

N is normally very big, while k is a very small number…

AnNN NA = 6.0221023 – Avogadro number, number

of molecules in 1 mol of a substance;

n is the number of moles in the gas.

nRTkTnNPV A R = 8.31 J/molK 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.

L

reactionF

Collisions of the gas molecules with a wall..

As a result of a collision the momentum changes by

xx mvpp 22 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: dt

dp

dt

dvmmaF xx

xx

Now t is a long time interval – the time between two consecutive collisions with the wall.

For the average force on the wall it becomes:

t

mv

t

pamF xxxx

2

reactionF

change of momentum in a collision

time between collisions

Now t is – the time between two consecutive collisions with the wall.

t

mvamF xxx 2

L

xvLt /2L

mv

vL

mv

t

mvF x

x

xxx

2

/2

22

Kinetic theory of the ideal gas.

L

vmNF x

2

L

vvvmF Nxxx )...( 22

221

N

vvvv Nxxxx

)...( 222

212

Let’s try to account for all molecules of the gas:

L

Kinetic theory of the ideal gas.

L

vNmF x

2

Pressure on the wall with surface area A:

V

vmN

LA

vmN

A

FP xx

22

Velocity of a molecule: ),,( zyx vvvv 2222

zyx vvvv

The average velocity – average of a sum is equal to the sum of averages…

2222zyx vvvv

All the directions of motion (x, y, z) are equally probable!

2222

3

1vvvv zyx

V – volume of the box.

L

Pressure on the wall with surface area A:

V

vmN

V

vmN

A

FP x

22

3

1

The average kinetic energy of a molecule

The ideal gas law (experimental fact!)

KNvmNPV3

2)

2

1(

3

2 2

2

2

1vmK

kTNPV

Therefore: KNNkT3

2 kTK

2

3