Rate of thermal conduction in various copper structures

[steveh721]

A cylinder is constructed from a super insulator. The dimensions of the cylinder are Length L: 1 M; inside diameter i.d.: 12 mm. The entire Length of the cylinder contains copper (cases described below).

Heat (T > 100C) is continuously applied to the copper on one end of the cylinder. There is a heat sink attached to the copper at the opposite end of the cylinder--it can dissipate the heat at or above the rate of heat absorption.

What will be the difference in rate of conduction, if any, for:

a) the cylinder contains solid copper (12mm dia. X L)

b) the cylinder is packed with copper powder (say, 30 microns)

c) the cylinder is packed with small diameter copper rods (say, 1mm dia. X L)

d) the cylinder is packed with small diameter copper tubes (say, 5mm o.d., 4m i.d. X L)

e) the cylinder contains a uniform copper lattice (honeycomb X L) structure surrounded by air

If there's a difference, which case results in the highest conduction rate and why? Lowest?

 

[CWatters]

The thermal conductivity will be proportional to cross sectional area and inversely proportional to length. So cases a) c) and d) should be easy enough.

Case b) and e) is tricky because instead of having a uniform cross section you have lots of voids in three dimensions. I'm not sure how you go about calculating the thermal conductivity but it will obviously be lower than for case a) which is a solid rod.

In a material with small pores the mean free path of air molecules (eg Nitrogen) might be less than the pore size. So they are more likely to collide with the walls of the pore than each other. So the thermal energy will be transferred to the conducting copper pore walls rather than travel through the air in the pores. I think this means b) is likely to have increased conductivity compared to e). Edit: and perhaps quite close to that of a).

 

[steveh721]

Thank you for your comments.

I was anticipating responses would be more toward d) and/or e) because the infrared spectrum is short wavelength/high frequency electro-magnetic energy. And, at least in the RF spectrum, high frequency energy propagates on the surface--and increased surface area has reduced impedance. Consequently, I imagined that more surface area would also result in reduced propagation time for infrared, too.

Could there be any validity to this idea?

 

[CWatters]

Pretty sure copper is opaque to infrared :-)

Conduction usually beats radaition unless the temperature difference is very high.

 

[sardar]

I would approach this question by first understanding the principles of thermal conduction. Thermal conduction refers to the transfer of heat between two objects or regions that are in contact with each other. The rate of thermal conduction is affected by factors such as the material's thermal conductivity, the temperature difference between the two objects, and the distance between them.

In this scenario, we are comparing the rate of thermal conduction for a copper cylinder with different structures: solid copper, copper powder, small diameter copper rods, small diameter copper tubes, and a uniform copper lattice structure.

a) Solid Copper Cylinder: This structure would have the highest thermal conductivity compared to the other structures. This is because solid copper has a high thermal conductivity (385 W/mK) and there are no air gaps or spaces between the particles that would impede the transfer of heat. The rate of thermal conduction would also depend on the length of the cylinder, as longer cylinders would have a higher resistance to heat flow.

b) Copper Powder: The rate of thermal conduction in a cylinder packed with copper powder would be lower than that of solid copper. This is because the powder particles are not in direct contact with each other, and there are air gaps between them. Air has a lower thermal conductivity (0.024 W/mK) compared to copper, so the presence of air in the powder would decrease the overall thermal conductivity of the cylinder.

c) Small Diameter Copper Rods: The rate of thermal conduction for a cylinder packed with small diameter copper rods would be lower than that of solid copper but higher than copper powder. This is because the rods are in direct contact with each other, but the diameter of the rods is smaller than the solid copper cylinder. The smaller diameter would create more resistance to heat flow, resulting in a lower thermal conductivity.

d) Small Diameter Copper Tubes: The rate of thermal conduction for a cylinder packed with small diameter copper tubes would be lower than that of solid copper but higher than small diameter copper rods. This is because the tubes are in contact with each other, but the thickness of the walls would create more resistance to heat flow compared to rods.

e) Uniform Copper Lattice Structure: The rate of thermal conduction for a cylinder containing a uniform copper lattice structure surrounded by air would be lower than that of solid copper. This is because the lattice structure creates small air gaps between the copper particles, which would decrease the overall thermal conductivity of the cylinder.