In this screencast, we will look at transient conduction with an iron sphere, diameter 20
centimeters, at initial temperature 30 degrees C. It is suddenly placed, and the fact that
it says suddenly means that this is going to be transient conduction, in a bath at T
= 300 degrees C. Our initial temperature is 30 degrees C and our bulk temperature is 300
degrees C. Using the following information, the density, the heat capacity, the thermal
diffusivity, the heat transfer coefficient and the thermal conductivity, we need to find
the temperature of the sphere after 200 seconds, so our time is going to be 200 seconds.
The easiest way to solve a transient conduction problem is to use lumped capacitance, but
in order to do that, you have to have a Biot number of less than 0.1.
So let's go ahead and calculate this number.
The Biot number is the heat transfer coefficient, which is 1000 watts per meter squared K, times
a characteristic length, which for this situation is the radius, 0.1 meters, divided by 3, and
all divided by our thermal conductivity, which is 80.2 watts per meter K. And when we calculate
this, we find that this is 0.4, which is certainly greater than 0.1, so we will not be able to
use lumped capacitance.
Instead, what we will use is an exact solution which was determined, and what we will try
to do is use the one-term approximation of this exact solution.
And in order for us to be able to do that, we have to calculate the Fourier number, which
is a dimensionless time, and it has to be greater than 0.2.
So the first thing is, let's look at this governing equation.
So our dimensionless centerline temperature, which is equal to T0 minus T infinity, divided
by the initial temperature minus T infinity, this is equal to a constant times the exponential
of minus, now this is one of the transcendental roots of the exact solution, squared, times
the Fourier number.
So in order to solve for that, we have to look up in tabulated data what our C1 and
our root 1 are, and these are both a function of this Biot number, but it is not the Biot
number that we just calculated.
Here, the Biot is equal to our h times our radius of the sphere divided by k, and when
we calculate that out, we get a Biot number of 1.2.
Then, looking up in the tabulated data, we are able to find our C1, which is 1.5485,
and then our other constant which is 1.9308.
Now, let's calculate our Fourier number.
So as I mentioned before, this Fourier number is thought of as a dimensionless time, and
it's alpha, which is the thermal diffusivity, times time, divided by our radius squared.
This thermal diffusivity is actually our thermal conductivity divided by the density, divided
by the heat capacity.
So, as we are given our thermal diffusivity, that's 23.1 times 10 to the 6 meters squared
per second, and our time is 200 seconds, and we divide this all by 0.1 meters squared.
And when we calculate our Fourier number, we find that it's 0.462, which is greater
than 0.2, which means that we can go ahead and use that one term approximation.
So our T0 minus T infinity, divided by our initial temperature minus T infinity equals
1.5485 exponential of negative 1.9308 squared times 0.462.
And when we calculate that out, this equals 0.277.
So let's go ahead now and solve for this T0.
That's 0.277 times Ti minus T infinity, which is -270 degrees Celsius, plus 300 degrees
Celsius which is our T infinity, and we find that after 200 seconds, our centerline temperature
or center of the sphere is 225.3 degrees C. Now let's say instead we had used our lumped
capacitance.
Then our temperature would have been -270 degrees C times our exponential of minus h
times As, which is the surface area, divided by the density times the volume, times the
heat capacity, all multiplied by t, and this entire expression is added to our T infinity.
So here, our temperature is going to equal minus 270 degrees C times 0.18 plus 300, which
equals 251 degrees C. So you can see there is quite a difference between the two methods.
The one thing you have to remember though is using the lumped capacitance assumes that
there is no spatial differences within the solid at any given time.
So once we look for our center temperature, which we originally did, it's going to be
a lot different from the surface temperature using the exact solution, but it should be
exactly the same throughout the solid if we were using lumped capacitance.