# Using a centrifuge to extract heavy water (help with the calculations please)

In summary, Cody from codys lab used electrolysis to "enrich" the water. This process is messy and centrifugation may be a better option. However, calculating the forces needed is difficult.
Hello everyone!

I have seen several DIY projects which successfully gathered heavy water
from normal water. For example Cody from codys lab used electrolysis
to "enrich" the water. This however is a messy process.

So i became curious, if this can be done easier by centrifugation.

Based on these data http://www1.lsbu.ac.uk/water/water_properties.html
around 1 in 6600 water molecules are heavy water (HDO). This would
have a density of ~1050kg/m³ compared to ~996kg/m³. This is not enough
to separate them under normal gravity but maybe this difference is enough
to be able to separate them in a centrifuge.

To get a feel for the forces needed, i want to calculate this. But here
is where i got stuck.

First, we assume a cylindrical centrifuge, spinning at a constant RPM.
We also assume that the system has reached a steady state so in the
frame of reference of the spinning liquid, nothing is moving. In that
frame, a fictitious force depending on the radius F(r) will act on the
liquid. Since the system is highly symmetric, it should be equivalent
to looking at only a 1D column along the radius.

In this steady state, the movement of particles due to the centrifugal
action and the movement due to diffusion will balance. Looking again at
the data table we find the diffusion coefficients of H2O and HDO, which
are (in SI) 2.299e-7 m²/s and 2.34e-7 m²/s respectively.

Here is my first confusion: the diffusion coefficient requires two
substances, is the second substance "normal water" (normal mixture of
H2O and HDO) ?

Second, to calculate the distribution, i am looking for a PDE i can
integrate numerically. I found the diffusion equation:
$$\frac{\partial \phi(r, t)}{\partial t} = \nabla \cdot \big[ D(r,t))\nabla \phi(r,t) \big]$$
with $\phi$ being the density and $D$ the collective diffusion coefficient.

(Can i assume the collective diffusion coefficient is the density weighted
average of the individual diffusion coefficients?)

Obviously, this equation does not include the radial force or the resultant
pressure.

I hope i can get around using the full navier-stokes equations, as they are
quite complicated. Is there a simpler form i can use? Essentially i am looking
for a 1D PDE.

It is probably easier to view this from a thermodynamic perspective. The centrifuge produces an effective potential for HDO (given by the mass difference to H2O), and your HDO molecules will be in thermodynamic equilibrium in this potential.

Rough estimate: A rim speed of 300 m/s (that is a very powerful centrifuge) leads to an effective potential of 45 kJ/kg or 0.5 meV if multiplied by the mass difference. kT at room temperature is about 25 meV. The separation will be quite weak.

That is a very quick but useful estimate. Since 300m/s rim speed is way out of reach as a hobbyist, it seems like this approach will not work.

## What is a centrifuge and how does it work?

A centrifuge is a laboratory instrument that spins samples at high speeds in order to separate components based on their density. This is achieved by using centrifugal force, which causes denser components to move towards the bottom of the sample tube while lighter components stay towards the top.

## What is heavy water and why is it important to extract it?

Heavy water, also known as deuterium oxide, is a form of water where the hydrogen atoms are replaced with deuterium, a heavier isotope of hydrogen. It is important to extract heavy water for various scientific and industrial purposes, such as in nuclear reactors and in the production of certain chemicals.

## How do I calculate the centrifugal force needed to extract heavy water?

The centrifugal force needed to extract heavy water can be calculated using the formula F = mv²/r, where F is the centrifugal force, m is the mass of the sample, v is the velocity of the centrifuge, and r is the radius of the centrifuge. It is important to consult with a centrifuge manual or an expert in order to determine the appropriate values for these variables.

## What factors can affect the extraction of heavy water using a centrifuge?

Several factors can affect the extraction of heavy water using a centrifuge, including the speed of the centrifuge, the temperature of the sample, and the type of rotor used. Additionally, the density of the heavy water relative to the other components in the sample can also impact the efficiency of extraction.

## Are there any safety precautions I should take when using a centrifuge to extract heavy water?

Yes, it is important to follow all safety guidelines provided by the manufacturer of the centrifuge. This may include wearing protective gear, properly balancing the sample tubes, and avoiding the use of glass tubes that can shatter at high speeds. It is also important to properly label and handle the heavy water sample, as it can be hazardous if mishandled.

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