Solution

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Example Solution:
Heat flow is defined by

$$J =\frac{ΔE}{A*t}$$

We know that the change of the energy is given by

$$ΔE = \frac{nRΔT}{\gamma-1}$$

Where (n) is the number of moles

We know that Heat flow then

$$J = \frac{nRΔT}{(\gamma-1)A*t}$$

We know that the time is given by because (the free path of atoms is much longer than δ) furthermore λ>>δ therefore the average collision between the molecules will be δ:

$$t = \frac{δ}{v}$$

Likewise:

$$J = \frac{nRΔTv}{(\gamma-1)A*δ}$$

Now we have to:

$$A*δ=V$$

So from the previous equations we have to:

$$J = \frac{nRΔTv}{(\gamma-1)V}$$

Another way to write this equation is:

$$J = \frac{n'KΔTv}{(\gamma-1)}$$

Where (n') is the number of the atoms per unit volume:

$$v=\sqrt{\frac{3R(T+Δ)}{\mu}}$$

How ΔT<<T is much less than this

$$v\approx\sqrt{\frac{3RT}{\mu}}$$

Introducing this velocity into the equation for the flow we have to

$$J = \frac{n'KΔT}{(\gamma-1)}\sqrt{\frac{3RT}{\mu}}$$