Heat transfer question - Graphing temperature versus radius

[theBEAST]

Here is the solution to the problem, all you need to look at is the diagram, the conductivity values and the graph.

What I don't get is why the temperature is not continuous when the material changes. Shouldn't there be equilibrium at the boundaries between the different materials?

 

[Simon Bridge]

For much the same reason that electric fields are not continuous across the boundaries either.

Think of it in terms of what temperature is measuring.
Model the solid as a lattice of masses joined by springs - different materials have different strength springs.

 

[theBEAST]

For much the same reason that electric fields are not continuous across the boundaries either.

Think of it in terms of what temperature is measuring.
Model the solid as a lattice of masses joined by springs - different materials have different strength springs.

Hmmm, the thing is, in my class notes, the professor drew this out:
http://i.imgur.com/f0or1ND.jpg

Which shows that with two different materials, at the boundary the temperature is the same... Did my professor make a mistake?

 

[Simon Bridge]

Um - yes and no. Same goes for the simple hint I gave too.

Your prof is presenting a vastly simpler system with idealized layers, so he's using a very simple model to describe what happens.

Notice, for instance, that the first pic has curved lines through a single material while your prof's pic has striaght lines?

If it helps - think of the boundary between two layers having much lower thermal conductivity, but being very thin. Thinner than the thickness of the vertical dashed lines. i.e. if you put a sheet of lead onto a sheet of concrete, there will probably be small pockets of air between them.

 

[Chestermiller]

Here is the solution to the problem, all you need to look at is the diagram, the conductivity values and the graph.

What I don't get is why the temperature is not continuous when the material changes. Shouldn't there be equilibrium at the boundaries between the different materials?

You are correct. At the interface between two materials, the temperature is continuous, and the heat flux is continuous (unless there is a change of phase at the interface). The heat flux being continuous at the interface means that the temperature gradient times the thermal conductivity is continuous across the interface. The continuity of temperature and heat flux applies irrespective of whether the heat transfer is steady state or transient, or whether the boundary is flat or curved. In the case of a curved boundary, the normal component of the heat flux is continuous.

 

[Simon Bridge]

You are correct. At the interface between two materials, the temperature is continuous, and the heat flux is continuous (unless there is a change of phase at the interface). The heat flux being continuous at the interface means that the temperature gradient times the thermal conductivity is continuous across the interface. The continuity of temperature and heat flux applies irrespective of whether the heat transfer is steady state or transient, or whether the boundary is flat or curved. In the case of a curved boundary, the normal component of the heat flux is continuous.

... which leaves only to explain the discontinuities in the graph ;)

 

[Chestermiller]

... which leaves only to explain the discontinuities in the graph ;)

Dear Simon,
I really don't know what to say. I haven't a clue as to why the person who prepared that graph screwed up. My personal opinion is that he simply does know his way around heat transfer.

One thing I do know, based on my significant formal training in heat transfer as well as over 35 years of practical experience modeling physical problems in industry including many, many, heat transfer systems much more complex than this one, is that the graph is incorrect (for the reasons that I cited in my previous response). I am totally confident about this.

Incidentally, the graph is not the only error in the problem statement. The statement indicates that, within the Pb, the stainless steel and the concrete, the temperature will be varying in inverse proportion to the radius. This is incorrect. The temperature will be varying linearly with 1/r, with a constant term present in the linear relationship.

I would be glad to provide pointers on how to solve this problem once the OP has had more of an opportunity to try it on his own (given the additional information that I have provided him).

Chet

 

[Simon Bridge]

I don't think the person drawing the chart all that screwed up ... you'll get similar stuff yourself by embedding thermocouples in a material or whatever - depending on the situation.

... it can be like the way electric potential changes at a junction.
The junction is not usually made of either the first or the second material alone ... but some irregular mixture of the two - and maybe another substance as well. The material properties can change sharply in the small distance of the junction.

The design is for a nuclear waste container.
It goes Waste (20cm) -> lead (4cm) -> steel (1cm) -> concrete (3cm)

The simple models assume perfect contact between materials.
On the scale of the diagram, drops across as much as 1mm (where the contact is not perfect) will look like a discontinuous step.

The simple model also assumes that thermal conductivity is a constant with temperature. Is it?

 

[Chestermiller]

I don't think the person drawing the chart all that screwed up ... you'll get similar stuff yourself by embedding thermocouples in a material or whatever - depending on the situation.

... it can be like the way electric potential changes at a junction.
The junction is not usually made of either the first or the second material alone ... but some irregular mixture of the two - and maybe another substance as well. The material properties can change sharply in the small distance of the junction.

The design is for a nuclear waste container.
It goes Waste (20cm) -> lead (4cm) -> steel (1cm) -> concrete (3cm)

The simple models assume perfect contact between materials.
On the scale of the diagram, drops across as much as 1mm (where the contact is not perfect) will look like a discontinuous step.

The simple model also assumes that thermal conductivity is a constant with temperature. Is it?

I guess we have a disagreement between heat transfer "experts" here. See Bird, Stewart, and Lightfoot, "Transport Phenomena", Section 10.6, Heat Conduction Through Composite Walls. This is a book that has been a mainstay in Chemical Engineering curricula for over 50 years, and was recently updated with a wonderful second edition in 2002. Everything I said in my post agrees with what they say there: the temperature at the interfaces between the materials is continuous, and the heat fluxes at the interfaces between the materials is continuous. They show two worked problems, one for a flat geometry and the other for a cylindrical geometry. These worked problems include graphs of the temperature profiles. The temperature profiles are continuous across the interfaces. In the after-chapter problems, they pose a nuclear fuel rod problem with heat generation within the rod (with a radial functionality to the heat generation rate which closely resembles the functionality in our problem), and they give the answer to the problem (which satisfies the same boundary conditions at the interfaces). This problem is also accompanied by a temperature profile graph with the temperature profile continuous across the material interfaces.

Incidentally, in terms of your question regarding the temperature dependence of thermal conductivity, the problem statement calls for using constant values for the thermal conductivities.

Chet

 

[nschaefe]

I think you are both correct. You cannot have a true discontinuity between surfaces, as the heatflux and temperature at the interface must be constant. However, consider thermal contact resistance between the two surfaces, which is thin but also has a very, very low thermal conductivity. This would be a steep curve / large temperature drop connecting the two surfaces, but is probably not shown on the graph above because of its scale.

Granted I think the graph is a poor illustration of this concept, and the magnitude of the drop seems to be much, much higher than you would expect. Additionally, if the graph includes contact resistance, its thermal conductivity and equivalent thickness should be noted in the figure. That would be my best guess for why the graph is drawn this way.

 

[Chestermiller]

I think you are both correct. You cannot have a true discontinuity between surfaces, as the heatflux and temperature at the interface must be constant. However, consider thermal contact resistance between the two surfaces, which is thin but also has a very, very low thermal conductivity. This would be a steep curve / large temperature drop connecting the two surfaces, but is probably not shown on the graph above because of its scale.

Granted I think the graph is a poor illustration of this concept, and the magnitude of the drop seems to be much, much higher than you would expect. Additionally, if the graph includes contact resistance, its thermal conductivity and equivalent thickness should be noted in the figure. That would be my best guess for why the graph is drawn this way.

OMG. I was not able to see those heat transfer coefficients for the interfaces until I blew the figure up to 175%. But, sure enough, they are there: 2000, 2000, and 1000. Even at 175% zoom, it's still not easy to see the subscript (indicating which interface each refers to). I think we are now all on the same page. That certainly explains the temperature jumps.

Chet

 

[Simon Bridge]

I guess we have a disagreement between heat transfer "experts" here.

I don't believe I have said anything that disagrees with Bird, Stewart, and Lightfoot, "Transport Phenomena", Section 10.6, Heat Conduction Through Composite Walls. I'm quite sure those worthies would confirm that, IRL, you do get these kinds of jumps.
(checking: the text cited actually does not seem to talk about interface effects.)

OMG. I was not able to see those heat transfer coefficients for the interfaces until I blew the figure up to 175%.

... well I completely missed them so ... well spotted.
The values square well with the jumps of around 5degC, on the resolution of the r-axis.

I suspect the coefficients for the interfaces are more obvious in the original problem statement.