Heat Flow Rate: Dependence on Conductivity & Temp Diff

[Jacob Wilson]

I recently read in a Khan Academy article that the rate of energy exchange through heat across a material of thickness $d$, surface area $A$, and thermal conductivity $k$ can be approximated by $$\dot{Q} = \frac{kA\Delta T}{d}$$ where $\dot{Q}$ is the heat rate and $\Delta T$ the temperature difference between the two sides of the material.

However, I am a bit confused by why the thermal conductivity of the media on either side of the material in question does not affect that heat rate across the "middle media," $\dot{Q}$. Referring to the included image, what if the thermal conductivity of the media composing the body of $T_1$ was close to zero? Would the heat rate across the middle media not be affected? According to the equation for $\dot{Q}$ above, it is not.

Is $\dot{Q}$ solely a function of temperature difference, independent of the "sandwiching" media involved?

Khan Academy article I'm talking about:
https://www.khanacademy.org/science...-heat-transfer/a/what-is-thermal-conductivity

UPDATE: I think I understand what's going on. I believe that the heat rate across any media is, in fact, independent of the surrounding media's thermal conductivity with temperature difference between the surrounding media being the only external parameter of any importance. Please correct me if I am wrong.

 

[jbriggs444]

However, say surface 1 has thermal conductivity $k_1$ and surface 2 has a thermal conductivity of $k_2$ where $k_1\neq k_2$. In that case, the heat rate of surface 1 is not equal to that of surface 2. Where is that energy going if the thermal energy leaving surface 1 (2) is not then entering surface 2 (1)?

Thermal conductivity is a property of the body of a substance, not its surface.

 

[Jacob Wilson]

Thermal conductivity is a property of the body of a substance, not its surface.

Thanks for that edit! I updated my post quite a bit to reflect that.

 

[jbriggs444]

UPDATE: I think I understand what's going on. I believe that the heat rate across any media is, in fact, independent of the surrounding media's thermal conductivity with temperature difference between the surrounding media being the only external parameter of any importance. Please correct me if I am wrong.

Yes, this is correct.

If the surrounding media is a good insulator and the media in the middle is a good conductor, a heat flow equilibrium of sorts will eventually be obtained with a high temperature difference across the insulating material and a low temperature difference across the conducting material. The result will be (once this equilibrium is obtained) an identical heat transfer rate throughout.

 

[Jacob Wilson]

Thank so much! That helps a lot. I'm new to this forum... is there a way to mark a question as resolved?

 

[sophiecentaur]

is there a way to mark a question as resolved

I am a bit confused by why the thermal conductivity of the media on either side of the material in question does not affect that heat rate across the "middle media,"

Of course it is relevant. That diagram is idealised. Heat loss or gain from the sides of the material has to be reduced as much as possible or compensated for. If you Google "Searle's Bar" and "Lee's Disc" you will see two methods for finding the thermal conductivity of, respectively, good conductors and bad conductors. The sides need to be insulated as well as possible and the apparatus can be operated in an ambient temperature that's about half way between hot and cold sinks.

is there a way to mark a question as resolved

Only occasionally, when the Mods consider a thread has totally run its course. As far as this thread is concerned, I should say the night is young.

 

[Jacob Wilson]

Heat loss or gain from the sides of the material has to be reduced as much as possible or compensated for.

Okay, yes. I was thinking that something must be idealized. Why must heat exchange from the side material be minimized though? Doesn't that just lead to a change in $\Delta T$ and thus a time dependent $\dot{Q}$? In other words, in that case we may rewrite the equation to say something like $$\dot{Q} = \frac{Ak\Delta T(t)}{d}$$

The sides need to be insulated as well as possible.

Do you just mean that there must be some sort of adiabatic boundary around the system (sandwich and middle media)? Wouldn't removing an adiabatic boundary again just lead to a time dependent $\dot{Q}$?

 

[Chestermiller]

Okay, yes. I was thinking that something must be idealized. Why must heat exchange from the side material be minimized though? Doesn't that just lead to a change in $\Delta T$ and thus a time dependent $\dot{Q}$? In other words, in that case we may rewrite the equation to say something like $$\dot{Q} = \frac{Ak\Delta T(t)}{d}$$

Do you just mean that there must be some sort of adiabatic boundary around the system (sandwich and middle media)? Wouldn't removing an adiabatic boundary again just lead to a time dependent $\dot{Q}$?

If you suddenly change one of the end temperatures while holding the other end constant at the same temperature that both ends started with, the temperature within the conductor will depend on time, and $\dot{Q}$ will be a function of both time and position. However, at long times, when the conduction reaches steady state, the temperature profile in the conductor will become linear and the rate of heat flow will become constant.

 

[sophiecentaur]

Why must heat exchange from the side material be minimized though?

If you don't eliminate / minimise it then the simple formula no longer applies - that is if you want to actually measure the conductivity. There are many ruses to eliminate the effects of heat loss in thermal experimental work.

 

[Jacob Wilson]

At long times, when the conduction reaches steady state, the temperature profile in the conductor will become linear and the rate of heat flow will become constant.

Is the steady-state condition for the conductor a fixed $\Delta T$? In other words, must the surrounding media at temperatures $T_1$ and $T_2$ be approximate heat sinks?

 

[Chestermiller]

Is the steady-state condition for the conductor a fixed $\Delta T$?

Yes.

In other words, must the surrounding media at temperatures $T_1$ and $T_2$ be approximate heat sinks?

In the scenario depicted in your figure, yes. But, more generally, whatever the temperatures are at the two very ends of a conductor, at steady state, the rate of heat flow is given by your equation and the temperature varies linearly through the conductor from T1 at one end to T2 at the other end.