Sequence of insulating layers of different conductivities for pipe

[biplab93]

Homework Statement

Two insulating materials of thermal conductivity K and 2K respectively are to be used as two layers of insulation for lagging a pipe carrying hot fluid. If the radial thickness both the layers is to be same, then -

i) the first material (thermal conductivity K) should be the inner layer, and the second material should be the outer layer

ii) the second material should be the inner layer

iii) the order of the materials is irrelevant

iv) numerical values required to give a specific answer

Homework Equations

The heat transfer in this case would be given by the following equation, I suppose:

q=$\frac{2*pi*ΔT}{1/h1r1 +ln(r2/r1)/K1 + ln(r3/r2)/K2 +1/r3h3}$

q= radial heat transfer per unit time per unit length of pipe
ΔT= temperature drop across the pipe thickness i.e. hot fluid temperature inside the pipe - ambient temperature outside the pipe
h1= heat transfer coefficient of the hot fluid
h3=heat transfer coefficient of the ambient air outside the pipe
r1=inside radius of first layer
r2=outside radius of the first layer=inside radius of the second layer=r1+t, t=radial thickness of one layer of insulation
r3=outside radius of the second layer=r2+t=r1+2t
K₁=thermal conductivity of the first layer
K₂=thermal conductivity of the second layer

The Attempt at a Solution

In the book where I found the question, (i) is the given answer, but I don't understand how that is. Looking at the equation, I guess ln(r2/r1)/K₁+ln(r3/r2)/K₂ should be maximum, but I don't know how to reach the conclusion that K₁=K and K₂=2K is the optimum option.

 

[mfb]

Just use r2=r1+t, r3=r2+t

Then you can write those logarithms as ln(1+t/r1) and ln(1+t/r2), where r2>r1. Which one is larger? How can you maximize the sum with this knowledge?

 

[Chestermiller]

Just use r2=r1+t, r3=r2+t

Then you can write those logarithms as ln(1+t/r1) and ln(1+t/r2), where r2>r1. Which one is larger? How can you maximize the sum with this knowledge?

So the logarithms are ln(1 + x) and ln(1+x/(1+x)), where x = t/r1. Now, expand each log term in a taylor series in x, and retain terms only up to x².

 

[mfb]

It is sufficient to know that ln() is monotonically increasing.

 

[rezeew]

I would like to clarify that the order of the insulating materials does not affect the overall heat transfer in this scenario. The equation provided is correct, and it shows that the heat transfer is dependent on the thermal conductivity of the materials and their respective thicknesses. Therefore, the order of the materials is irrelevant and any combination of K and 2K can be used as long as the total thickness of the insulation is the same.

However, if we want to optimize the insulation for maximum heat transfer reduction, we can consider the equation and see that the term ln(r2/r1)/K1+ln(r3/r2)/K2 should be minimized. This means that the first material with a thermal conductivity of K should have a smaller thickness compared to the second material with a thermal conductivity of 2K. This can be achieved by using K as the inner layer and 2K as the outer layer. But this is not the only option, as long as the thicknesses of the materials are adjusted accordingly, the order can be reversed.

In conclusion, the order of the insulating materials can be chosen based on practical considerations such as cost, availability, and ease of installation. As long as the total thickness of the insulation is the same, the heat transfer will not be affected.