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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.

Related links:

3 Responses to “Sodium volumetric thermal expansion coefficient”

  1. sari said

    nice post
    please visit this
    repository unand
    thanks….

    Syeilendra said..
    hello
    I have visited your link, nice

  2. so how accurate is it??

    Syeilendra said..
    Quite accurate.

  3. nuranur said

    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

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