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## Sodium volumetric thermal expansion coefficient

Posted by Syeilendra Pramuditya on November 30, 2010

Read also this: Liquid Sodium Thermodynamic and Transport Properties

The volumetric thermal expansion coefficient of Sodium is around 0.0002 /K, it depends on temperature and pressure, and the following is a brief description.

Some Liquid Metal-cooled Fast Reactor (LMR) designs employ Sodium as coolant material in its primary heat transport system. And one of the important physical properties of Sodium used in thermal hydraulic analysis is the volumetric thermal expansion coefficient (usually denoted by the Greek letter Beta).

Beta is expressed as follow:

$\beta=-\frac{1}{\rho}\frac{d\rho}{dT}$

$\beta=\textrm{Volumetric thermal expansion coefficient} [K^{-1}]$

$\rho=\textrm{Density} [kg/m^3]$

$T=\textrm{Temperature} [K]$

By using the above formula, we can now evaluate $\beta$ if we know the dependency of density on temperature. This dependency is available in form of tabular data or empirical correlations. I will use the following correlation, which I obtained from a reliable source:

$\rho(P,T)=a_0+a_1T+a_2T^2+a_3T^3+a_4P$

$\frac{\partial \rho(P,T)}{\partial T}=a_1+2a_2T+3a_3T^2$

$\beta=-\frac{1}{\rho}\frac{d\rho}{dT} =-\frac{a_1+2a_2T+3a_3T^2}{a_0+a_1T+a_2T^2+a_3T^3+a_4P}$

$a_0=1011.597$

$a_1=-0.22051$

$a_2=-1.92243\times 10^{-5}$

$a_3=5.63769\times 10^{-9}$

$a_4=2.26\times 10^{-7}$

So by using the above empirical formula, now you can easily evaluate the value of $\beta$ at any “valid” temperature and pressure.

As an example, here is the plot of $\beta$ as a function of temperature, keeping the pressure constant at 101325 Pa (equal to 1 atm):

We can see that the relation between $\beta$ and temperature “looks” linear, so for pressure value of 1 atm, $\beta$ can be approximated by the following simplified equation:

$\beta(T,P = 1~atm)= 2.135\times 10^{-4} + 9.77\times 10^{-8} T$

So how accurate this formula is?

Well, Wikipedia says that the relation between volumetric and linear thermal expansion coefficient can be approximated as $\beta \simeq 3\alpha$, and this link and this link say that the linear coefficient is around $7\times 10^{-5}$, which implies that the volumetric coefficient should be around 3 times of that value, or about $0.00021~K^{-1}$. And this link says that the value is $0.000226~K^{-1}$.

So the above formula is quite good, I think.

1. ### sarisaid

nice post
repository unand
thanks….

Syeilendra said..
hello

2. ### Dedi Andriantosaid

so how accurate is it??

Syeilendra said..
Quite accurate.

3. ### nuranursaid

makasih ka atas postingan-nya.. oya kl molten steel berapa ya yg akurat? apakah sama dengan solid steel? makasih sebelumnya

Syeilendra said..
ok sama2
mohon maaf sy kurang tahu kalau molten steel, bidang saya berkaitan dgn liquid sodium
mungkin bisa dicari di tabel2 properti material