Some questions on heat transfer

[MexChemE]

Hello, forum! I'm just starting a new course on heat transfer and we're using Incropera's book. Last time I studied heat transfer was in my transport phenomena course, using BSL, so it was kind of a culture shock using the new book, because the methods used are kind of different in some cases. However, I am enjoying Incropera's book, the problems are very entertaining, and interesting for both MechE and ChemE majors. Anyway, some doubts have popped up when solving some problems, and I'm trying to be a different student that when I took my TP course, when I always went for understanding the math first and leaving the physics in second place. Now I want to do things differently.

First question: Incropera's book introduces energy balances on a surface. The authors mention accumulation and generation terms are not accounted for in a surface balance, because they are volumetric effects; fair enough. But then they say surface balances apply for both steady and transient state, but transient state implies accumulation, negating the last statement. I know the temperature of a surface may change in time if we have a transient state process, so I don't get why they say accumulation terms won't be taken into account.

Next: When using the heat equation to model heat conduction in a solid rod with volumetric heat generation, BSL used the following boundary condition for heat flux at the center of the rod:
q_r |_{r=0} = q_0
"The radial heat flux at the center of the rod is equal to some finite quantity." This was done in order to avoid getting an infinite heat flux at the center, because of the mathematics of radial heat conduction in cylindrical coordinates:
q_r = \frac{S_e r}{2} + \frac{C_1}{r}
Now, in a problem from Incropera's, I had to define boundary conditions for the exact same physical situation (including electric heat generation). Naturally, taking the methods from BSL as dogmas, I set my boundary condition for r = 0 as the heat flux being finite at that point, however, the solutions manual states the heat flux at that point is zero. My thoughts on this are: BSL just stablished the condition it used in order to avoid a mathematical error (division by zero), after all, a heat flux equal to zero is also a finite heat flux, is it not? The way I'm interpretting the physics is that for both cases, the heat flux is zero at the center of the rod. It should be, anyway, as the temperature is at a maximum at the center of the rod.

Last: In that same problem, the conducting rod is covered by a non-conducting cladding. Again, I had to define a boundary condition for the point where both materials meet. I was going for the boundary condition where the heat flux at the point where both materials meet is equal for both regions. However, the solutions manual used a boundary condition where the temperature of the rod and the temperature of the cladding are equal where they meet, argumenting thermal equilibrium. It is a logical answer, but I know temperature may not always be continuous among two different contacting materials. So, apart from the case when there are no air spaces at the interface, in which situations can we say temperature is continous?

That would be all for now. Thanks in advance for any input!

 

[Chestermiller]

Hello, forum! I'm just starting a new course on heat transfer and we're using Incropera's book. Last time I studied heat transfer was in my transport phenomena course, using BSL, so it was kind of a culture shock using the new book, because the methods used are kind of different in some cases. However, I am enjoying Incropera's book, the problems are very entertaining, and interesting for both MechE and ChemE majors. Anyway, some doubts have popped up when solving some problems, and I'm trying to be a different student that when I took my TP course, when I always went for understanding the math first and leaving the physics in second place. Now I want to do things differently.

First question: Incropera's book introduces energy balances on a surface. The authors mention accumulation and generation terms are not accounted for in a surface balance, because they are volumetric effects; fair enough. But then they say surface balances apply for both steady and transient state, but transient state implies accumulation, negating the last statement. I know the temperature of a surface may change in time if we have a transient state process, so I don't get why they say accumulation terms won't be taken into account.

What is the mass of the interface? Given the answer to this question, how much heat can accumulate at the interface, even under transient conditions?

Next: When using the heat equation to model heat conduction in a solid rod with volumetric heat generation, BSL used the following boundary condition for heat flux at the center of the rod:
q_r |_{r=0} = q_0
"The radial heat flux at the center of the rod is equal to some finite quantity." This was done in order to avoid getting an infinite heat flux at the center, because of the mathematics of radial heat conduction in cylindrical coordinates:
q_r = \frac{S_e r}{2} + \frac{C_1}{r}
Now, in a problem from Incropera's, I had to define boundary conditions for the exact same physical situation (including electric heat generation). Naturally, taking the methods from BSL as dogmas, I set my boundary condition for r = 0 as the heat flux being finite at that point, however, the solutions manual states the heat flux at that point is zero. My thoughts on this are: BSL just stablished the condition it used in order to avoid a mathematical error (division by zero), after all, a heat flux equal to zero is also a finite heat flux, is it not? The way I'm interpretting the physics is that for both cases, the heat flux is zero at the center of the rod. It should be, anyway, as the temperature is at a maximum at the center of the rod.

I don't know what either book says, but, for the problem you describe: $2\pi r q=\pi r^2S$, or, $q=S\frac{r}{2}$. The boundary condition at r = 0 is T=T₀, and the value of T₀ is determined by making good on the boundary condition at the other boundary.

Last: In that same problem, the conducting rod is covered by a non-conducting cladding. Again, I had to define a boundary condition for the point where both materials meet. I was going for the boundary condition where the heat flux at the point where both materials meet is equal for both regions. However, the solutions manual used a boundary condition where the temperature of the rod and the temperature of the cladding are equal where they meet, argumenting thermal equilibrium. It is a logical answer, but I know temperature may not always be continuous among two different contacting materials. So, apart from the case when there are no air spaces at the interface, in which situations can we say temperature is continous?

That would be all for now. Thanks in advance for any input!

Aside from air spaces or some other resistance, the temperature is continuous. Also, if there is no phase change at the interface, the heat flux is continuous. There is, of course, nothing wrong with both the heat flux and the temperature being continuous at a boundary.

 

[MexChemE]

What is the mass of the interface? Given the answer to this question, how much heat can accumulate at the interface, even under transient conditions?

Well, if the surface has an infinitesimal thickness, its mass has to be zero, so it makes sense saying it can't accumulate energy. But then, how to account for temperature changes in the surface using this kind of balances? Do we have to do a shell balance in the whole control volume?

I don't know what either book says, but, for the problem you describe: $2\pi r q=\pi r^2S$, or, $q=S\frac{r}{2}$. The boundary condition at r = 0 is T=T₀, and the value of T₀ is determined by making good on the boundary condition at the other boundary.

The expression you wrote for heat flux is equivalent to the expression I wrote after getting rid of C₁ applying the boundary condition as in BSL, this is done in order to avoid dividing C₁ by zero when evaluating the heat flux at r = 0. Even if we apply the $T |_{r=0} = T_0$ boundary condition, the heat flux at that point has to be zero because T₀ is the maximum temperature of the system, is it not?

Aside from air spaces or some other resistance, the temperature is continuous. Also, if there is no phase change at the interface, the heat flux is continuous. There is, of course, nothing wrong with both the heat flux and the temperature being continuous at a boundary.

Got it.

Thanks so far!

 

[Chestermiller]

Well, if the surface has an infinitesimal thickness, its mass has to be zero, so it makes sense saying it can't accumulate energy. But then, how to account for temperature changes in the surface using this kind of balances? Do we have to do a shell balance in the whole control volume?

I don't understand this question.

The expression you wrote for heat flux is equivalent to the expression I wrote after getting rid of C₁ applying the boundary condition as in BSL, this is done in order to avoid dividing C₁ by zero when evaluating the heat flux at r = 0. Even if we apply the $T |_{r=0} = T_0$ boundary condition, the heat flux at that point has to be zero because T₀ is the maximum temperature of the system, is it not?

You typically will not know To yet when you set up this problem. Usually, the only temperature you will know is air temperature or the surface temperature. You will be solving for To. Since you already know the flux at all radial positions, you only need one boundary condition. This will be enough to determine To from the solution.

 

[MexChemE]

I don't understand this question.

What I meant to ask was if it was possible to account for temperature changes on a surface using the surface energy balance, but judging from your other answer, I guess not.

You typically will not know To yet when you set up this problem. Usually, the only temperature you will know is air temperature or the surface temperature. You will be solving for To. Since you already know the flux at all radial positions, you only need one boundary condition. This will be enough to determine To from the solution.

This is much clearer now, thank you very much!

 

[Chestermiller]

What I meant to ask was if it was possible to account for temperature changes on a surface using the surface energy balance, but judging from your other answer, I guess not.

You can do this as long as you don't include any accumulation at the surface.