Thermodynamics: Thermal conductivity in a pipe

Question

You have a pipe A (inner and outer radii are r0 and r1) made from material, whose thermal conductivity coefficient K1. What thickness should different pipe B have, which has the same flow rate and heat loss, but made from material whose thermal conductivity coefficient is K2.

Note: It involves calculus (I think so).

Answer 1

In thermodynamics, we considered the amount of heat transfer as a system undergoes a

process from one equilibrium state to another. Thermodynamics gives no indication of

how long the process takes. In heat transfer, we are more concerned about the rate of

heat transfer.

The basic requirement for heat transfer is the presence of a temperature difference. The

temperature difference is the driving force for heat transfer, just as voltage difference for

electrical current. The total amount of heat transfer Q during a time interval can be

determined from:

Q Q dt kJ

t

0

The rate of heat transfer per unit area is called heat flux, and the average heat flux on a

surface is expressed as

2 W / m

A

Q

q

Steady Heat Conduction in Plane Walls

Conduction is the transfer of energy from the more energetic particles of a substance to

the adjacent less energetic ones as result of interactions between the particles.

Consider steady conduction through a large plane wall of thickness Δx = L and surface area

A. The temperature difference across the wall is ΔT = T2 – T1.  

Note that heat transfer is the only energy interaction; the energy balance for the wall can

be expressed:

dt

dE Q Q wall

in

For steady‐state operation,  

Q Q const.

It has been experimentally observed that the rate of heat conduction through a layer is

proportional to the temperature difference across the layer

The constant proportionality k is the thermal conductivity of the material. In the limiting

case where Δx→0, the equation above reduces to the differential form:

W

dx

dT Q Cond

which is called Fourier’s law of heat conduction. The term dT/dx is called the temperature

gradient, which is the slope of the temperature curve (the rate of change of temperature

T with length x).

Thermal Conductivity

Thermal conductivity k [W/mK] is a measure of a material’s ability to conduct heat. The

thermal conductivity is defined as the rate of heat transfer through a unit thickness of

material per unit area per unit temperature difference.  

Thermal conductivity changes with temperature and is determined through experiments.

The thermal conductivity of certain materials show a dramatic change at temperatures

near absolute zero, when these solids become superconductors.

An isotropic material is a material that has uniform properties in all directions.

Insulators are materials used primarily to provide resistance to heat flow. They have low

thermal conductivity