| ### Statement | | ### Statement |
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| $5.4.16*.$ [Insert problem description here] | | $5.4.16*.$ [Insert problem description here] |
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| __Example Statement__: | | __Example Statement__: |
| $1.1.1.$ Between two flat parallel plates located at a distance δ from each other, there | | $1.1.1.$ Between two flat parallel plates located at a distance δ from each other, there |
| is a monatomic gas (the free path of atoms is much longer than δ). Estimate | | is a monatomic gas (the free path of atoms is much longer than δ). Estimate |
| the heat flux density if the temperature of the plates is maintained at T and | | the heat flux density if the temperature of the plates is maintained at T and |
| T + ∆T, respectively, and the unit volume of gas contains n' atoms; µ is the | | T + ∆T, respectively, and the unit volume of gas contains n' atoms; µ is the |
| mass of the atom | | mass of the atom |
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| ### Solution | | ### Solution |
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| [Your solution should be placed here] | | [Your solution should be placed here] |
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| __Example Solution__: | | __Example Solution__: |
| Heat flow is defined by | | Heat flow is defined by |
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| $$J =\frac{ΔE}{A*t} $$ | | $$J =\frac{ΔE}{A*t} $$ |
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| We know that the change of the energy is given by | | We know that the change of the energy is given by |
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| $$ΔE = \frac{nRΔT}{\gamma-1}$$ | | $$ΔE = \frac{nRΔT}{\gamma-1}$$ |
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| Where (n) is the number of moles | | Where (n) is the number of moles |
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| We know that Heat flow then | | We know that Heat flow then |
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| $$J = \frac{nRΔT}{(\gamma-1)A*t}$$ | | $$J = \frac{nRΔT}{(\gamma-1)A*t}$$ |
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| 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: | | 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: |
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| $$t = \frac{δ}{v}$$ | | $$t = \frac{δ}{v}$$ |
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| Likewise: | | Likewise: |
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| $$J = \frac{nRΔTv}{(\gamma-1)A*δ}$$ | | $$J = \frac{nRΔTv}{(\gamma-1)A*δ}$$ |
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| Now we have to: | | Now we have to: |
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| $$A*δ=V$$ | | $$A*δ=V$$ |
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| So from the previous equations we have to: | | So from the previous equations we have to: |
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| $$J = \frac{nRΔTv}{(\gamma-1)V}$$ | | $$J = \frac{nRΔTv}{(\gamma-1)V}$$ |
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| Another way to write this equation is: | | Another way to write this equation is: |
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| $$J = \frac{n'KΔTv}{(\gamma-1)}$$ | | $$J = \frac{n'KΔTv}{(\gamma-1)}$$ |
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| Where (n') is the number of the atoms per unit volume: | | Where (n') is the number of the atoms per unit volume: |
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| $$v=\sqrt{\frac{3R(T+Δ)}{\mu}}$$ | | $$v=\sqrt{\frac{3R(T+Δ)}{\mu}}$$ |
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| How ΔT<<T is much less than this | | How ΔT<<T is much less than this |
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| $$v\approx\sqrt{\frac{3RT}{\mu}}$$ | | $$v\approx\sqrt{\frac{3RT}{\mu}}$$ |
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| Introducing this velocity into the equation for the flow we have to | | Introducing this velocity into the equation for the flow we have to |
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| $$J = \frac{n'KΔT}{(\gamma-1)}\sqrt{\frac{3RT}{\mu}}$$ | | $$J = \frac{n'KΔT}{(\gamma-1)}\sqrt{\frac{3RT}{\mu}}$$ |
