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I've recently read some chapters on heat transfer, and I was confused by an equation that described the thermal conductivity for ideal gases.

$$k=\frac{9\gamma-5}{4}\frac{c_{\nu}}{\pi d^{2}}\sqrt{\frac{Mk_{B}T}{N\pi}}$$

I've only learned the basics regarding heat transfer, and I couldn't see the steps needed to derive the above equations. What I know is:

(The thermal conductivity for gases) $$k=\frac{1}{3}c_{\nu}\rho\bar{c}\lambda_{mfp}$$ (Average molecular speed) $$\bar{c}=\sqrt{\frac{8RT}{\pi M}}$$ (The speed of sound in fluid) $$c=\sqrt{\gamma RT}$$ (The density of an ideal gas) $$\rho=\frac{p}{RT}$$ (The ratio of specific heats) $$\gamma=\frac{c_{p}}{c_{\nu}}$$

Edited:

(The mean free path of a gas molecule) $$\lambda_{mfp}=\frac{k_{B}T}{\sqrt{2}\pi d^{2}p}$$

So my question is:

  1. How do I derive the mean free path for gas molecules?
  2. How do I derive the thermal conductivity equation (the first one)?

Edit:

The textbook I am using is: Fundamentals of Heat and Mass Transfer (Eighth edition) by Theodore L. Bergman and Adrienne S. Lavine (pg. 67).

Textbook link

Added symbols meaning.

I forgot to add the equation for the mean free path of gas molecules. I guess it is just too general of a question to ask how to derive it. Please ignore my first question.

To be more specific, I need confirmation that I am using the correct formulas for the derivation (equations after the first one) and algebraic steps to reach the first formula.

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  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Jun 26, 2022 at 18:55
  • $\begingroup$ So that the context and assumptions are clear, please cite the source of the first equation. Nobody knows what you've been reading except you! $\endgroup$ Jun 26, 2022 at 20:48
  • $\begingroup$ It might be worth reading what Feynman say on the subject and then information about the Maxwell-Boltzmann distribution. $\endgroup$
    – Farcher
    Jun 26, 2022 at 23:02

1 Answer 1

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If all else fails, check the citations.

On the same page in Bergman et al.'s Fundamentals of Heat and Mass Transfer as your first equation, the authors point to Zhang's Nano/Microscale Heat Transfer, McGraw–Hill, New York, 2007.

In turn, Zhang refers (on pg. 41) to the so-called Eucken formula (E. Eucken, Physik. Zeitschr 14, 324–332, 1913) that relates the Prandtl number $\text{Pr}=\frac{c_p\mu}{k}$, where $\mu$ is the dynamic viscosity, to the heat capacity ratio $\gamma=\frac{c_p}{c_v}$ as

$$\text{Pr}=\frac{4\gamma}{9\gamma-5}.$$

In addition, Tien & Lienhard's Statistical Thermodynamics, Hemisphere, New York, 1985, which Zhang cites, obtains the relationship between the dynamic viscosity and certain other parameters as

$$\mu\approx\frac{1}{2}\rho \bar c \lambda_{\text{mfp}}\left(=\frac{1}{2}\frac{m\bar c}{\sqrt{2}\pi d^2}=\frac{1}{\pi^2d^2}\sqrt{\pi mk_B T}\right),$$

where I've given some equivalent relations to compare with other results in the literature; here, $m$ is the molecular mass.

Thus, \begin{align} k&=\frac{c_p\mu}{\text{Pr}};\\ &=\frac{9\gamma-5}{4\gamma}\cdot c_p\cdot\frac{1}{2}\rho\bar c\lambda_\text{mfp};\\ &=\frac{9\gamma-5}{4}\cdot c_v\cdot\frac{1}{2}\frac{Mp}{RT}\sqrt{\frac{8RT}{\pi M}}\frac{k_BT}{\sqrt{2}\pi d^2p};\\ &=\frac{9\gamma-5}{4}\frac{c_v}{\pi d^2}\sqrt{\frac{Mk_BT}{N\pi}},\end{align}

where I've simply plugged in the relations you're already familiar with (note that your density formula is missing the system mass $M$) to obtain your first equation (Bergman et al.'s Eq. 2.12).

A parenthetical note: I emphasize that I'm reviewing only the steps in the derivation of Eq. 2.12 as referenced by the authors of that textbook. Many of the terms in this answer appear with slightly different values in other derivations (e.g., a fixed Prandtl number of 1, or different prefactors; for instance, Bird et al.'s Transport Phenomena has not $\mu=\frac{1}{2}\rho \bar c \lambda_{\text{mfp}}$ but $\mu=\frac{1}{3}\rho \bar c \lambda_{\text{mfp}}$ while noting that the prefactor of $\frac{1}{3}$ is "very roughly" approximated). In fact, dozens of (sometimes incompatible) relations of a similar type exist in the literature on the kinetic theory of gases, as a result of more or less hand-wavy arguments, estimates, and assumptions; this answer does not assert that this particular derivation or result is the best one. It merely identifies the links that you reported lacking.

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  • $\begingroup$ Oh, wow, this was much more complicated than I originally expected. Thanks for the detailed explanation! This is very helpful. $\endgroup$
    – Yun
    Jun 27, 2022 at 1:15
  • $\begingroup$ You’re welcome! I never thought that the mystery coefficient would date back 110 years! They had a good handle on gases from the start. $\endgroup$ Jun 27, 2022 at 4:16

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