How Does Gas Pressure Affect Heat Conduction in a Vacuum Flask?

[Fek]

Homework Statement

A vacuum flask of radius r and length l consists of two concentric cylinders separated by a narrow gap containing a gas at pressure 10^-2 Nm^-2. The liquid in the flask is at 60 degrees Celcius and the air outside is at 20 degrees celcius. Estimate the rate of heat loss by conduction.

Homework Equations

κ = 1/3 C_molecule n λ <v>

λ = ( sqrt(2) * n * σ) ^ -1

p = 1/3 nm <v²>

pV = nk_bT

J = - κ ∇T

where n is molecules per unit volume.

The Attempt at a Solution

κ is independent of pressure so the equation is still valid.

C_v per mol is 3/2 R for an ideal gas. So if I divide by avagadro's I will get heat capacity per molecule.

<v> = sqrt( 8K_BT / m pi)

The gas in the "vacuum" is air, so mainly nitrogen, so m = 14.

For the temperature of the "vacuum" gas, as it is definitely between 293K and 333K it does not matter exactly.

So now we just need the mean free path. We can estimate the collision cross section as 2a^2 where a is the atomic radius.

Then we can find n either by using the equation for pressure or the ideal gas equation
n = p / K_BT
Still left with ∇T unknown which seems like a dead end.

 

[Chestermiller]

You're trying to find J. Do you know how to determine ∇T if the gap is small and the temperature is assumed to vary linearly with radius? How does the mean free path compare with the gap between the cylinders?

Chet

 

[Fek]

Thanks for your response Chet.

∇T ≈ ΔT / s , where s is the gap between cylinders if we assume that ∇T is constant. However s is not given.

The expression for κ assumes the mean free path is much less than the size of the container, so the molecules collide with each other much more frequently than the container - I now see that this is not the case we can't use that expression.

However without using it I am struggling to see a method.

I suppose you could assume that the particles simply bounce back and forth across the gap and gain/lose internal energy each time they come into contact with a wall and transfer it that way. However it would be difficult to estimate the energy gained in each contact and the route between walls taken by the particle.

I also know Newton's Law of cooling: AH = hA ΔT , but I have not learned about what determines h.

 

[Chestermiller]

Thanks for your response Chet.

∇T ≈ ΔT / s , where s is the gap between cylinders if we assume that ∇T is constant. However s is not given.

The expression for κ assumes the mean free path is much less than the size of the container, so the molecules collide with each other much more frequently than the container - I now see that this is not the case we can't use that expression.

That's what I was afraid of. Sorry, wish I could help you, but I don't have any experience in this area.

Chet