Conceptual Questions Regarding the Dynamics of Heat Transfer

[zak.hja]

Homework Statement

Three building materials, plasterboard [k = 0.30 J/(s m Co)], brick [k = 0.60 J/(s m Co)], and wood [k = 0.10 J/(s m Co)], are sandwiched together (as I've tried to show below). The temperatures at the inside and outside surfaces are 28.9°C and 0°C, respectively. Each material has the same thickness and cross-sectional area. Find the temperature (a) at the plasterboard-brick interface and (b) at the brick-wood interface.

Outside (0 C) --- Wood | Brick | Plasterboard --- Inside (28.9 C)

2. Homework Equations
(Q/t)=(kAΔT)/L

The Attempt at a Solution

I was able to solve for the correct temperatures of 21°C and 18°C for the plaster-brick and brick-wood interfaces, respectively; however, this solution was based on the assumption that the rate of heat transfer would be the same for all three materials. This seemed proper at the time. As I think about it in retrospect, however, wouldn't the rates of heat transfer differ among the materials due to their differing conductivity (k)? As in, certain materials are more "receptive" of heat and will transfer it more readily -- how can that be, if all three rates are assumed to be the same?

EDIT: I may have mixed up my terminology. The fact that heat itself (Q) is the same across all three materials -- that makes more sense to me than the rates being the same (as I figure t would differ). The same amount of heat will progress from one end to another, if I've understood correctly. How is the heat received differently by the materials -- as in, do any of the materials absorb some of the heat, preventing it from being transferred onward to the next interface? I'm just confusing myself more and more!

If someone could straighten out this issue for me, I would deeply appreciate it! Thank you!

 

[Chestermiller]

EDIT: I may have mixed up my terminology. The fact that heat itself (Q) is the same across all three materials -- that makes more sense to me than the rates being the same (as I figure t would differ). The same amount of heat will progress from one end to another, if I've understood correctly. How is the heat received differently by the materials -- as in, do any of the materials absorb some of the heat, preventing it from being transferred onward to the next interface?

The heat transfer is assumed to be occurring at steady state, without the temperatures anywhere within the sandwich changing with time. So, all the heat that enters one side of the sandwich must exit the other side. Where else can it go? If the materials absorb and retain heat, then that would correspond to a non steady state (transient) situation. The temperatures at various locations within the sandwich would be changing with time. But that is not the case in the problem described here.

Chet

 

[zak.hja]

The heat transfer is assumed to be occurring at steady state, without the temperatures anywhere within the sandwich changing with time. So, all the heat that enters one side of the sandwich must exit the other side. Where else can it go? If the materials absorb and retain heat, then that would correspond to a non steady state (transient) situation. The temperatures at various locations within the sandwich would be changing with time. But that is not the case in the problem described here.

Chet

Thank you for responding!
Hmm, I think I follow: are we assuming that the heat has already been "conducted through", as opposed to viewing the transfer as a gradual process (wherein the temperatures aren't the same until equilibrium is reached)?

 

[Chestermiller]

Thank you for responding!
Hmm, I think I follow: are we assuming that the heat has already been "conducted through", as opposed to viewing the transfer as a gradual process (wherein the temperatures aren't the same until equilibrium is reached)?

The terminology you are using here is very imprecise and difficult to relate to.

The heat is being conducted through all three slabs at a constant rate. So the rate it enters each slab is the same as the rate at which it exits that slab and flows into the next slab. At steady state, none of the heat is being retained by the slabs any longer. It's like an electric current flowing through three resistors in series. The voltage is like the temperature.

The temperatures of the three slabs will never be equal to one another unless the heat flow is shut off. As long as the heat is flowing, the slabs will never attain equilibrium with one another (if that is what you mean by equilibrium). Or, by equilibrium, do you mean steady state?

Chet

 

[zak.hja]

The terminology you are using here is very imprecise and difficult to relate to.

The heat is being conducted through all three slabs at a constant rate. So the rate it enters each slab is the same as the rate at which it exits that slab and flows into the next slab. At steady state, none of the heat is being retained by the slabs any longer. It's like an electric current flowing through three resistors in series. The voltage is like the temperature.

The temperatures of the three slabs will never be equal to one another unless the heat flow is shut off. As long as the heat is flowing, the slabs will never attain equilibrium with one another (if that is what you mean by equilibrium). Or, by equilibrium, do you mean steady state?

Chet

Yes, I meant the latter -- I apologize if my terminology is imprecise, as I'm still a bit unfamiliar with the topic. I understand why the rates are constant now; thank you! However, there's one thing that's still a bit unclear to me: is the steady state independent of the thermal conductivity and thickness of the materials? For instance, would the same principle that you've described (heat conducted at a steady state) apply to any situation where we have heat conducted from one end of a material and out to another, regardless of the variables described in the equation (Q/t)=(kAΔT)/L? For instance, if we looked at heat flowing from the inside of a house, through a fixture of insulation and plywood to the outside, the rate of heat transfer would remain the same -- correct?

 

[Chestermiller]

Yes, I meant the latter -- I apologize if my terminology is imprecise, as I'm still a bit unfamiliar with the topic. I understand why the rates are constant now; thank you! However, there's one thing that's still a bit unclear to me: is the steady state independent of the thermal conductivity and thickness of the materials? For instance, would the same principle that you've described (heat conducted at a steady state) apply to any situation where we have heat conducted from one end of a material and out to another, regardless of the variables described in the equation (Q/t)=(kAΔT)/L?

Yes.

For instance, if we looked at heat flowing from the inside of a house, through a fixture of insulation and plywood to the outside, the rate of heat transfer would remain the same -- correct?

Yes. It would be the same through the insulation as through the plywood. This is the same as the electrical example I gave in which you have three different resistors in series. Even though the resistances can be very different, the current through each of the resistances is the same. It is the voltage drop across each of the resistors that can be very different.

Chet

 

[zak.hja]

Yes.

Yes. It would be the same through the insulation as through the plywood. This is the same as the electrical example I gave in which you have three different resistors in series. Even though the resistances can be very different, the current through each of the resistances is the same. It is the voltage drop across each of the resistors that can be very different.

Chet

The assistance is very much appreciated -- thank you, Chet!

 

[morrobay]

The heat transfer is assumed to be occurring at steady state, without the temperatures anywhere within the sandwich changing with time. So, all the heat that enters one side of the sandwich must exit the other side. Where else can it go? If the materials absorb and retain heat, then that would correspond to a non steady state (transient) situation. The temperatures at various locations within the sandwich would be changing with time. But that is not the case in the problem described here.

Chet

On a related subject ; Can you give the function for Q(t) in non steady state heat transfer ? I cannot find any reference for this situation :
At t₀ T₁ < T₂ . For two cubes of a metal in contact with a conduction bar.(T₁ increasing, T₂ decreasing ) Of course densities, thermal conductivity and specific heat are given What is Q(t) ? thank you

 

[Chestermiller]

On a related subject ; Can you give the function for Q(t) in non steady state heat transfer ? I cannot find any reference for this situation :
At t₀ T₁ < T₂ . For two cubes of a metal in contact with a conduction bar.(T₁ increasing, T₂ decreasing ) Of course densities, thermal conductivity and specific heat are given What is Q(t) ? thank you

Please start a new thread on this question. Also, if it is a homework problem, please use the appropriate homework forum and the homework template.

Chet

 

[morrobay]

Please start a new thread on this question. Also, if it is a homework problem, please use the appropriate homework forum and the homework template.

Chet

Sure, I have a homework question on this in this section. See; dQ/dt = kA(dT/dx) Last post yesterday.
(correction area should be .0042m²)
Its not so much the numerical answer but the function Q(t) that I am looking for so maybe I should just ask
in the Classical Physics ? Note the homework question evolved and post # 7 contains most information,
but is not in proper template. If its ok Ill be glad to restate this question.