- So I have an interesting system over here.
I have two compartments, on the left compartment
I have a gas that is at a temperature of T sub a
and on the right side of this I have gas
that is a temperature of T sub b
and they are separated by a wall of depth, d,
or I guess you say of thickness, d,
and the contact area of the wall,
or the contact area of the gas onto the wall
that area is A, and I'm just drawing
a section of it, we're assuming that
these two compartments are completely separated.
Now what I am curious about, and we're
going to assume that the temperature on the left
is higher than the temperature on the right,
and so because of that you're going to have
a transfer of thermal energy from the left to the right,
and that thermal energy that gets transferred
we call that heat, and we'll denote that with the letter Q,
I'm curious about how does the rate
at which heat is transferred, or so
how much heat is transferred per unit time,
that's the rate at which heat is transferred,
how would that change depending on
how we change these different variables.
So for example, if our, if our area,
if our contact area were to go up
what would that do for Q over t?
Well then Q over t would also increase per unit time
'cause I have, I have more area for these
hot molec--these hot air particles or hot air molecules
to bump into and they'll heat that wall
and then there will be more heated wall
to heat up the colder air particles.
So in that case our rate of heat transfer would also go up.
Well what if we, what if we, and obviously if I made
my area smaller, maybe I should just write that explicitly,
if I made my contact area smaller then my rate
of heat transfer, my rate of heat transfer
would go down, and that feels like common sense.
Now what about, what about the thickness?
If I were to make it, if I were to make
the thickness larger, if I were to make this a thicker wall,
what would that do to my rate of heat transfer?
Well then I would have, I would have more
things that I would have to heat up to get it
to a certain temperature before I can,
which then can heat up the particles on the right,
and obviously this is a continuous process,
it'll always be happening, but there will just be more
stuff to heat up and it's going to take longer
and more of that, and more of that
more of that kinetic energy, that average
kinetic energy is going to get dissipated
in this wall so if this wall becomes thicker,
if the wall becomes thicker then the
rate of heat transfer is going to go down,
or if you, if the wall became thinner,
if this depth decreased then the rate of heat transfer,
then the rate of heat transfer would go up.
So you could say the rate of heat transfer
is going to be inversely proportional
to the thickness of this wall.
Now what else could we think about?
Well we could think about the temperature differential,
the temperature differential, that's T sub a minus,
minus T sub b, minus T sub b.
Well if this temperature differential, if this
temperature differential were to go up,
well what's going to happen?
Well it's common sense that well if this is super hot,
if this is super hot over here, this is way hotter
than what we have on the right,
well we're going to have more heat transferred
so the rate of heat transfer, you're going to have
more heat transferred per unit time,
your rate of heat transfer is going to go up.
And likewise, if this differential were to go down
and you could take the extreme case
if there was no differential,
if T sub a was the same as T sub b
then you would have no heat transfer
frankly in any unit of time.
So it makes sense that the rate of heat transfer
is going to be proportional to the temperature differential.
So how can we encapsulate all of this intuition
into maybe a formula for describing
thermal conductivity, for thinking about
how quickly some, how quickly this heat
will be transferred, the rate of heat transfer?
Well we could say the rate of heat transfer,
and this is really, hopefully you know comes out of
a little bit of common sense or intuition
of what would happen here, the rate of
heat transfer, I could say it's going to be proportional to
well, what are the things it's going to be proportional to?
It's going to be proportional to the area,
the more surface area we have on this wall,
more contact area, the more heat we're going
to have transferred per unit time,
so it's going to be proportional to that contact area.
It's also going to be proportional to
the temperature differential so let's multiply
this times the temperature differential,
so T sub a minus T sub b,
minus T sub b,
and it's inversely proportional to
the thickness of the wall so all of that
over the thickness of the wall.
And now another thing that you might be saying,
okay I have this proportionality constant
but wouldn't this be different for different materials?
For example, if this was a metal wall
wouldn't this conduct the heat quicker
than if this was a wood wall?
And you would be correct, a metal wall would
and so this K, this K right over here,
this is dependent, this is dependent on
the material, so what is the wall made of.
So material...
material of the wall,
and you can actually, you can actually
measure this thing and different materials
will have different thermal conductivities,
which this, which this variable
right over here would actually represent.
So going through a little bit of intuition
we were able to come up with what looks like
a fancy formula, and you will sometimes see
this formula, formula for thermal conductivity
through a solid barrier but it really comes out of
hopefully common sense, the rate of,
the amount of heat transferred per time
is going to be proportional to,
and the proportionality constant is going
to be dependent on the material.
Styrofoam for example would be very low here,
that's why coolers are made out of Styrofoam,
and it's going to be dependent on the area,
it's going to be proportional to the area,
the temperature differential, and then
inversely proportional to the thickness.
So if you wanted to really insulate something
you would want to minimize the surface area
and you would want to, and you would want to
maximize the thickness and you would want to
have something with a very low thermal conductivity,
so a thick Styrofoam wall that's maybe shaped in a sphere
might be a pretty good container for keeping
something hot or for keeping something cool.