The easiest method to use to calculate objects with transient conduction is to use the lumped
capacitance method.
However, if the Biot number, which is equal to the convective heat transfer coefficient
times some length, divided by the thermal conductivity, if it's greater than 0.1, now
we need to consider that temperature gradients within the solid are no longer negligible.
So if this is the case, now our temperature is a function of our x, time, our initial
temperature, our bulk temperature, our length, our k, our alpha, thermal diffusivity, and
our h for a plane wall.
So the method used in this screencast is obtained by non-dimensionalizing the governing equations.
Let's see how to do that.
First we define a non-dimensional temperature that we call theta star.
This is equal to our temperature minus our bulk temperature divided by the initial temperature
minus the bulk temperature.
We then define a dimensionless time, which is equal to our thermal diffusivity times
our time, divided by L squared, and this is also known as the Fourier number.
And finally, we define a dimensionless length, x star, which is equal to the x where we're
looking divided by the L, and for a plane wall, the length of the plane wall is defined
as 2 times L. And now we rewrite our heat equation and initial and boundary conditions.
So now that we have our theta star as a function of all dimensionless parameters, x star, the
Fourier number, and the Biot number, we can use this to get an exact solution.
So the temperature distribution is in the form of an infinite series that looks like
this, and remember this is for a plane wall with a length of 2 times L. However, if we
have a Fourier number that's greater than 0.2, we can use what's known as the one-term
approximation, so what that means is that instead of having n equal infinity, n just
equals 1 and we just look at the first term.
So what are these?
So this term is an eigenvalue, and our constant is a function of that eigenvalue.
So let's take a look at what the one-term approximation looks like, and we'll write
it out in terms of what we're interested in.
So T minus T infinity divided by Ti minus T infinity is equal to C1 times the exponential
of this first eigenvalue squared times the Fourier number, and this is multiplied by
the cosine of this eigenvalue times x divided by L. So how do we find this C1 and this eigenvalue?
Well, we refer to a table that has these numbers as a function of the Biot number.
So what you do is you calculate your Biot number, and by the way this is just a piece
of the table, you have Biot numbers that go alot higher, and for that particular Biot
number, you find your eigenvalue, and you find your C1, and then you put it back into
this equation to solve for whatever quantity you're looking at.
Let's go through a couple of different scenarios, so let's start with here, a.
let's say you want to find some temperature at a certain depth in your solid at a certain
time.
So how do you do this?
First, you calculate the Biot number, then you look up your eigenvalue and your C, and
then you use the one-term expression and solve for T.
This is of course assuming that you have everything that you need to solve for the Fourier number
and that you have your initial and bulk temperatures.
Let's look at b.
So now what you want to do is find the centerline temperature, in other words, where x=0, and
again remember the length of this plane wall is equal to 2L, so in the center, x is going
to equal zero.
So we're going to use the same equation as we have before, however, because x=0, x star
equals zero, and therefore this cosine term is going to end up equaling one.
So to find our temperature in the center, we just calculate it like this.
In fact, we have a dimensionless form of this equation just for the centerline temperature.
So now what happens if you want to find a time where you have some temperature, you
know the depth, et.
cetera, maybe it's the centerline temperature, so where do we find time in this particular
equation?
Well that comes from the Fourier number, so if you remember, that's what's known as the
dimensionless time, so if we're going to have to calculate the time, we're going to have
to calculate the Fourier number first, and from that get time, just like this.
So I suggest that you solve for your Fourier number first, and then solve for your t, rather
than putting it all in one equation, and the reason for that is, remember to use that one-term
approximation, the Fourier number has to be greater than 0.2.
In our other situations, we are given or can calculate the Fourier number before we even
begin, so here it's important when you get to that step to check for your Fourier number.
In subsequent screencasts, I'll show you how to apply these techniques that we've just
gone over to examples.