Heat Flow Rate: Dependence on Conductivity & Temp Diff

In summary: Q}##. However, if you slowly change one of the end temperatures while holding the other, then there would be no time dependent ##\dot{Q}##.If you suddenly change one of the end temperatures while holding the other... then yes, there would be a time dependent ##\dot{Q}##. However, if you slowly change one of the end temperatures while holding the other, then there would be no time dependent ##\dot{Q}##.In summary, the equation for heat rate across a material of thickness, surface area, and thermal conductivity can be approximated by $$\dot{Q} = \frac{k
  • #1
I recently read in a Khan Academy article that the rate of energy exchange through heat across a material of thickness ##d##, surface area ##A##, and thermal conductivity ##k## can be approximated by $$\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.
ac67696eb6a0451324b5a0855ce6888d34155198.jpg


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.
 

Attachments

  • ac67696eb6a0451324b5a0855ce6888d34155198.jpg
    ac67696eb6a0451324b5a0855ce6888d34155198.jpg
    19.5 KB · Views: 486
Last edited:
  • Like
Likes Chestermiller
Science news on Phys.org
  • #2
Jacob Wilson said:
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.
 
  • Like
Likes Jacob Wilson
  • #3
jbriggs444 said:
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.
 
  • #4
Jacob Wilson said:
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.
 
  • Like
Likes Jacob Wilson
  • #5
Thank so much! That helps a lot. I'm new to this forum... is there a way to mark a question as resolved?
 
  • #6
Jacob Wilson said:
is there a way to mark a question as resolved
Jacob Wilson said:
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.
Jacob Wilson said:
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.
 
  • #7
sophiecentaur said:
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 $$\dot{Q} = \frac{Ak\Delta T(t)}{d}$$
sophiecentaur said:
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}##?
 
  • #8
Jacob Wilson said:
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.
 
  • Like
Likes Jacob Wilson
  • #9
Jacob Wilson said:
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.
 
  • Like
Likes Jacob Wilson
  • #10
Chestermiller said:
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?
 
  • #11
Jacob Wilson said:
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.
 
  • Like
Likes Jacob Wilson and jbriggs444

1. How does conductivity affect heat flow rate?

Conductivity is a measure of a material's ability to conduct heat. The higher the conductivity, the faster heat can be transferred through the material. Therefore, a material with high conductivity will have a higher heat flow rate compared to a material with low conductivity.

2. What is the relationship between temperature difference and heat flow rate?

The temperature difference between two points is directly proportional to the heat flow rate. This means that as the temperature difference increases, the heat flow rate also increases. Conversely, as the temperature difference decreases, the heat flow rate decreases.

3. How does the thickness of a material affect heat flow rate?

The thickness of a material can affect heat flow rate in two ways. First, a thicker material will have a longer path for heat to travel through, resulting in a slower heat flow rate. Second, thicker materials may have a higher thermal resistance, which can decrease the heat flow rate. However, if the material has high conductivity, the effect of thickness on heat flow rate may be minimal.

4. Can heat flow rate be manipulated by changing the temperature difference?

Yes, heat flow rate can be manipulated by changing the temperature difference. This is because the temperature difference is directly proportional to the heat flow rate. By increasing or decreasing the temperature difference, the heat flow rate will also increase or decrease, respectively.

5. How does thermal conductivity vary among different materials?

Thermal conductivity varies greatly among different materials. Metals, such as copper and aluminum, have high thermal conductivity, while materials like wood and plastic have low thermal conductivity. This is because thermal conductivity is dependent on the material's molecular structure and composition.

Suggested for: Heat Flow Rate: Dependence on Conductivity & Temp Diff

Replies
6
Views
195
Replies
3
Views
303
Replies
2
Views
667
Replies
2
Views
1K
Replies
14
Views
1K
Replies
3
Views
906
Back
Top