A Reference Correlation for the Viscosity of Krypton From Entropy Scaling

Abstract

We present a new wide-ranging correlation for the viscosity of krypton based on critically evaluated experimental data. For the first time, such a correlation has as its basis the entropy scaling approach. We base the residual contribution on the Lennard-Jones fluid, resulting in one adjustable parameter for the entire phase diagram away from the dilute-gas limit. The estimated uncertainty is less than 2.0 % (at the 95 % confidence level) over the entire phase diagram, except in the extended critical region. The correlation is valid from 70 K to 5000 K for the dilute gas, and from 115.775 K to 750 K in the fluid phase, with a pressure limit equal to that of the melting curve

Appendix A: Derivation of High-Temperature Limit

The second virial coefficient is defined by

$$\begin{aligned} B_2 = -2\pi \int _{0}^{\infty } [(\exp (-\beta V(r))-1)r^2] \mathrm{d} r \end{aligned}$$

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where V(r) is the potential, and $\beta =1/(k_\mathrm{B}T)$. The product $\beta V$ is dimensionless. Any potential that is finitely valued in the entire domain of integration has the infinite temperature limit

$$\begin{aligned} \boxed {\lim _{\beta \rightarrow 0}n_\mathrm{eff}= \frac{3}{2}} \end{aligned}$$

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where

$$\begin{aligned} n_\mathrm{eff}= -3\frac{B_2-\beta \dfrac{\mathrm{d} B_2}{\mathrm{d} \beta } }{\beta ^2 \dfrac{\mathrm{d} ^2B_2}{\mathrm{d} \beta ^2} } \end{aligned}$$

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The derivation begins with substitution for $B_2$, yielding

$$\begin{aligned} n_\mathrm{eff}= -3\frac{-2\pi \int [(\exp (-\beta V)-1)r^2] \mathrm{d} r-2\pi \int \beta V\exp (-\beta V)r^2 \mathrm{d} r}{-2\pi \int (\beta V)^2\exp (-\beta V)r^2 \mathrm{d} r} \end{aligned}$$

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and after joining terms

$$\begin{aligned} n_\mathrm{eff}= -3\frac{\int [(1+\beta V)\exp (-\beta V)-1]r^2 \mathrm{d} r}{\int (\beta V)^2\exp (-\beta V)r^2 \mathrm{d} r} \end{aligned}$$

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which has an indefinite form for $\beta \rightarrow 0$. In this case, two applications of de l’Hôpital’s rule are required, such that

$$\begin{aligned} \lim _{\beta \rightarrow 0}n_\mathrm{eff}= -3\lim _{\beta \rightarrow 0}\frac{\frac{ \mathrm{d} ^2}{ \mathrm{d} ^2 \beta ^2}\int [(1+\beta V)\exp (-\beta V)-1]r^2 \mathrm{d} r}{\frac{ \mathrm{d} ^2}{ \mathrm{d} ^2 \beta ^2}\int (\beta V)^2\exp (-\beta V)r^2 \mathrm{d} r} \end{aligned}$$

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or

$$\begin{aligned} \lim _{\beta \rightarrow 0}n_\mathrm{eff}= -3\lim _{\beta \rightarrow 0}\frac{\int [V^2(\beta V - 1)\exp (-\beta V)]r^2 \mathrm{d} r}{\int V^2(V^2\beta ^2 - 4V\beta + 2)\exp (-V \beta )r^2 \mathrm{d} r}, \end{aligned}$$

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and finally yielding

$$\begin{aligned} \lim _{\beta \rightarrow 0}n_\mathrm{eff}= -3\lim _{\beta \rightarrow 0}\frac{\int -V^2r^2 \mathrm{d} r}{\int 2V^2r^2 \mathrm{d} r} = \frac{3}{2} \end{aligned}$$

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