Thermal conduction between difference surfaces

[sgstudent]

Homework Statement

from this video: It showed that a better conductor of heat will transfer heat as well as absorb heat more quickly than a lousy conductor. However, during the transfer the heat is passed from eg metal at 30 degrees to a solid at 20 degrees . So why would the metal being a better conductor allow more heat to be transferred than plastic to ice?

Homework Equations

none

The Attempt at a Solution

I'm thinking since the heat is being transferred to the solid the metal/plastic particles would vibrate transferring energy to the solid. So for the metal, being a better conductor of heat more energy can be transferred to the ice per unit time than the plastic which is a bad conductor of heat. Since heat can only be transferred from a region of higher temperature to lower temperature, so only a fixed amount of energy can be transferred to the first few molecules (as the solid may be a bad conductor) before that heat dissipates slightly and more heat is transferred into it. Hence, even if the first layer of the solid's temperature reaches equilibrium, the metal would still be able to transfer more heat than the plastic.

 

[apelling]

Essentially I agree with what you have said. To add some more detail it might be useful to consider this equation

$$\frac{ΔQ}{Δt}=-kA\frac{Δθ}{Δx}$$

This basically says that the rate of flow of heat through a material depends on the conductivity times the contact area times the temperature gradient.

To simplify matters assume the metal, ice and plastic blocks are all cubes of the same size so A and Δx are the same for all. When two materials are in contact the rate of flow of heat through both must be equal by conservation of energy and assuming that all heat flow stays in the materials ie perfect insulation.

This leads to the conclusion that a material of high thermal conductivity will have a small temperature difference along its length while a material of low thermal conductivity will need a bigger temperature difference to maintain equal heat flow.

So when a metal is in contact with the ice the ice has a large temperature difference across it while the metal has a small change in temperature between front and back surface. This means that where the metal meets the ice the temperature will be close to 19°C (this ambient temperature I got from the video), the temperature difference across the ice is nearly the largest it can be and heat flows quickly. But with the polymer the situation is reversed because ice is more conductive than a polymer. So the temperature difference across the ice will now be smaller, the temperature where ice meets polymer will be about 3°C because ice is about 6 times more conductive than the average polymer and heat flow is less.

Another way of looking at this is that the metal stays warm when in contact with ice because it can absorb heat from its surroundings whereas the polymer cools down on contact with ice because it cannot absorb heat well.

 

[sgstudent]

Essentially I agree with what you have said. To add some more detail it might be useful to consider this equation

$$\frac{ΔQ}{Δt}=-kA\frac{Δθ}{Δx}$$

This basically says that the rate of flow of heat through a material depends on the conductivity times the contact area times the temperature gradient.

To simplify matters assume the metal, ice and plastic blocks are all cubes of the same size so A and Δx are the same for all. When two materials are in contact the rate of flow of heat through both must be equal by conservation of energy and assuming that all heat flow stays in the materials ie perfect insulation.

This leads to the conclusion that a material of high thermal conductivity will have a small temperature difference along its length while a material of low thermal conductivity will need a bigger temperature difference to maintain equal heat flow.

So when a metal is in contact with the ice the ice has a large temperature difference across it while the metal has a small change in temperature between front and back surface. This means that where the metal meets the ice the temperature will be close to 19°C (this ambient temperature I got from the video), the temperature difference across the ice is nearly the largest it can be and heat flows quickly. But with the polymer the situation is reversed because ice is more conductive than a polymer. So the temperature difference across the ice will now be smaller, the temperature where ice meets polymer will be about 3°C because ice is about 6 times more conductive than the average polymer and heat flow is less.

Another way of looking at this is that the metal stays warm when in contact with ice because it can absorb heat from its surroundings whereas the polymer cools down on contact with ice because it cannot absorb heat well.

Hi thanks for the reply. Could you elaborate on this: But with the polymer the situation is reversed because ice is more conductive than a polymer. So the temperature difference across the ice will now be smaller, the temperature where ice meets polymer will be about 3°C because ice is about 6 times more conductive than the average polymer and heat flow is less.

why would the temperature difference be smaller? Thanks

 

[haruspex]

A graph might help. On the x axis, distance, starting somewhere inside the source of heat, passing through the boundary, finishing inside the ice. On the y axis, temperature. Take the y values at the endpoints to be fixed, left hand end much higher than right.
When it's metal on the left, the slope of the line through that part will be shallow, so the line arrives at the boundary still high up, then must descend steeply through the ice. That means the heat transfer through the ice will be rapid. Change the left hand to be plastic and that side of the line will slope more steeply, so the right half will be shallower.

 

[sgstudent]

Essentially I agree with what you have said. To add some more detail it might be useful to consider this equation

$$\frac{ΔQ}{Δt}=-kA\frac{Δθ}{Δx}$$

This basically says that the rate of flow of heat through a material depends on the conductivity times the contact area times the temperature gradient.

To simplify matters assume the metal, ice and plastic blocks are all cubes of the same size so A and Δx are the same for all. When two materials are in contact the rate of flow of heat through both must be equal by conservation of energy and assuming that all heat flow stays in the materials ie perfect insulation.

This leads to the conclusion that a material of high thermal conductivity will have a small temperature difference along its length while a material of low thermal conductivity will need a bigger temperature difference to maintain equal heat flow.

So when a metal is in contact with the ice the ice has a large temperature difference across it while the metal has a small change in temperature between front and back surface. This means that where the metal meets the ice the temperature will be close to 19°C (this ambient temperature I got from the video), the temperature difference across the ice is nearly the largest it can be and heat flows quickly. But with the polymer the situation is reversed because ice is more conductive than a polymer. So the temperature difference across the ice will now be smaller, the temperature where ice meets polymer will be about 3°C because ice is about 6 times more conductive than the average polymer and heat flow is less.

Another way of looking at this is that the metal stays warm when in contact with ice because it can absorb heat from its surroundings whereas the polymer cools down on contact with ice because it cannot absorb heat well.

What do you mean by "When two materials are in contact the rate of flow of heat through both must be equal by conservation of energy and assuming that all heat flow stays in the materials ie perfect insulation."? Won't the heat be transferred from one of the materials to the other material? But if one material is more conductive than the other, then the heat flow from the end of one material to the start of the other material will be faster than the heat from from the end of the first material to the end of the second material? Something like: http://postimage.org/image/7nssd85gx/full/

So there might be accumulation of heat at the less conductive area making that part hotter than the other parts for a while?

 

[apelling]

Sorry I should have stated that I was assuming that the situation had settled to a steady equilibrium situation. Indeed initially heat will flow faster through the more conductive material. This will change the temperature at the interface until the temperature gradient in the less conductive material makes the heat flow equal through both.
If you understand electrical potential dividers it is a similar situation. Current is the same through two resistors in series so a higher potential difference is across the less conductive/more resistive resistor.

 

[sgstudent]

Oh. But actually if I placed 2 cuboids of the sane mass and shape and specific heat capacity, however one of them have is a very bad conductor while one is a very good conductor. If apply 100J of heat on the more conductive end the final product would be that both cuboids are at the same temperature. However, how will the heating process go? Will it go like this: Initially the edge of the conductive end will have 100J of heat but as it gets conducted it will decrease in temperature. However, due to the other cuboid which is a bad conductor of heat, once some heat flows there it would accumulate that heat (but only to the same temperature as the heat supplier aka the safe of the good conductor). At this point of time, the temperature of the good conductor will appear to stabilize. But as the bad conductor slowly conducts the heat to the rest of its surrounding, more heat is drawn from the good conductor slowly. As a result despite continuously appearing to have stabilized in temperature, the good good conductor slowly decreases in temperature (decreases starting from the edge of the cuboid touching the bad conductor). After a while all the heat will be evenly distributed with 50J of energy on each cuboid.

Is this correct? Thanks for the help :)