Convection ; good conductor & bad conductor difference?

[dreamLord]

Hi.

Suppose I have 2 cylindrical bodies of the same radius and length - one is a good thermal conductor, while the other is a bad conductor. The upper end in each case is kept at a constant temperature T1, while the lower end is in contact with a steel disk of same radius, which is in free contact with air.

Now, my question is, at any instant of time, which cylinder will be losing more energy to the air through its curved surface area? What difference does a bad conductor and a good one make in heat transportation by convection?

Thanks!

 

[256bits]

Which one would you think would lose more heat to the air
- a cylinder made up of a material with low thermal resistance such as copper, or,
- a cylinder made up of a material with high thermal resistance such as styrofoam.

 

[dreamLord]

That's what I'm having trouble figuring out. A good thermal conductor should allow heat to flow through it easily, and hence heat losses should be low in that case. But, at the same time, it's temperature will rise faster, and hence it will lose heat to the air faster. Which logic is flawed out of the two?

 

[mikeph]

The rate of change of an object's temperature (for fixed heat inflow) is measured by the specific heat capacity, not the thermal conductivity. If you're talking about a transient heat problem then you need to know this material property as well.

If not then your temperatures will not change, and all you need to know is that a more thermally conductive material will conduct through it more heat per unit area per kelvin temperature gradient.

 

[dreamLord]

So are you saying that the amount of heat lost to air is independent of the nature of the conductor, but rather depends on the specific heat capacity?

 

[mikeph]

Not at all. In a steady state (no changes over time) it will only depend on the thermal conductivity. In a transient case (changing over time, e.g. when you switch the heat source on at t=0 and measure the results) you need thermal conductivity and the specific heat capacity, because the conductor has to first heat up before it can conduct heat to the ambient air; the rate of this heating is governed by the heat capacity.

 

[dreamLord]

Okay, so I'll ask about when steady state has not yet been acquired.

When I switch on the heat source at t=0, the conductor heats up with time. In this case, which conductor loses more heat energy to the surroundings? How do I go about calculating heat lost by both conductors to the surroundings at any instant of time(or a small interval of time)? I know the temperature at the top surface, the specific heat capacity of the conductor as well as it's thermal conductivity, and room temperature.

 

[mikeph]

You need to also specify the heat capacities.

If the really good thermal conductor also has a HUGE specific heat capacity, then it will absorb a lot of heat before it increases its temperature by 1 degree. Since the heat flow is proportional to the temperature difference, you might have a curious case where a poor thermal conductor (with a low heat capacity) transports more heat initially because its temperature rises much faster.

The overall heat flowing through the objects would then be some complicated function of time that depends on the heat capacity and the thermal conductivity. It's found by solving the heat equation.

 

[dreamLord]

Do all good conductors have a higher heat capacity as compared to poor conductors?

Assuming the above is true, please find the flaw in my reasoning.

Initially, both bodies have the same temperature difference at the ends (1 end maintained at T1, other at room temperature). Thus, in any time interval, the good conductor will allow more heat to pass through it. This means that it will have a higher temperature at a time 't' than that of the poor conductor.

Now, convection depends on the temperature difference and the specific heat capacity. Assuming my first question is true, the good conductor has a higher heat capacity as well as a higher temperature than air. Thus, it will lose more heat.

 

[mikeph]

Do all good conductors have a higher heat capacity as compared to poor conductors?

Not at all! Gases have very low thermal conductivities (due to large mean free paths) and very high specific heat capacities because they have extra molecular degrees of freedom (energy can be stored in rotation and translations that don't occur in solids).

I'm not sure where convection is coming into your 3rd paragraph, is that a typo?

 

[dreamLord]

Ah okay, I see, thanks.

No, it's not a typo. I was trying to talk about conduction and convection in the initial stages - conduction for when heat flows from the top surface to the lower one through the conductor, and convection when heat is lost to the air. Am I wrong?