Physics is behind how a centrifuge works

[Red_CCF]

I was wondering what the physics is behind how a centrifuge works and how to find the (radius dependent) g-force exerted along the centrifuge tubes. I look at some online sites like wiki etc. but they weren't really clear.

Thanks.

 

[Andy Resnick]

There's three main centrifuge geometries (in biochemistry)- one where the tube is vertical, one where it is held at a fixed angle, and one where it can freely swing out. This technique is used (with a density gradient sucrose gel) to separate out different populations of proteins and cellular parts.

My magic book (Wilson and Walker, Principles and Techniques of Biochemistry and Molecular Biology) has some great diagrams showing the differences, but I can't find the picture online.

 

[russ_watters]

The wiki does have the equation you are looking for, right at the top.

 

[Andy Resnick]

There's three main centrifuge geometries (in biochemistry)- one where the tube is vertical, one where it is held at a fixed angle, and one where it can freely swing out. This technique is used (with a density gradient sucrose gel) to separate out different populations of proteins and cellular parts.

My magic book (Wilson and Walker, Principles and Techniques of Biochemistry and Molecular Biology) has some great diagrams showing the differences, but I can't find the picture online.

I found a relevant image:

http://media.wiley.com/CurrentProtocols/ET/et0501/et0501-fig-0001-1-full.gif

This, combined with the RCF calculation russ_watters mentioned (a nomogram), should be sufficient to get you oriented.

 

[Red_CCF]

I'm actually more interested in the force analysis of a centrifuge and derivation of the formula you are referring to. I understand that fictitious centrifugal force is involved in making this work but I don't really know much beyond that.

 

[K^2]

Lets just consider two dimensions.

$$x = r cos(\phi)$$
$$y = r sin(\phi)$$

$$\dot{x} = \dot{r} cos(\phi) - r sin(\phi) \dot{\phi}$$
$$\dot{y} = \dot{r} sin(\phi) + r cos(\phi) \dot{\phi}$$

$$\ddot{x} = \ddot{r} cos(\phi) - 2 \dot{r} sin(\phi) \dot{\phi} - r cos(\phi) \dot{\phi}^2 - r sin(\phi) \ddot{\phi}$$
$$\ddot{y} = \ddot{r} sin(\phi) + 2 \dot{r} cos(\phi) \dot{\phi} - r sin(\phi) \dot{\phi}^2 + r cos(\phi) \ddot{\phi}$$

$$a = \ddot{r} \hat{r} + r \ddot{\phi} \hat{\phi} - r \omega^2 \hat{r} + \dot{r} \omega r \hat{\phi}$$

Please check if I made any errors, as I typed this out in haste. Should be acceleration in r direction plus acceleration in omega direction plus Centrifugal Effect plus Coriolis Effect.

Edit: just to make it perfectly clear, the few definitions I skipped in getting to the last line:
$$a = \ddot{x} \hat{x} + \ddot{y} \hat{y}$$
$$\omega = \dot{\phi}$$

$$\hat{r} = cos(\phi)\hat{x} + sin(\phi)\hat{y}$$
$$\hat{\phi} = cos(\phi)\hat{y} - sin(\phi)\hat{x}$$

 

[Cleonis]

I was wondering what the physics is behind how a centrifuge works

The most common usage of centrifuges is in biology labs where particles in suspensions are separated.
I will first discuss the necessary properties of the suspension, then I will move to the centifuge mechanics.

One particular method works as follows. In a tube a sugar solution is prepared. The solution is prepared in such a way that there is a gradient in the concentration of the solution. Near the bottom the concentration is highest, at the top the concentration is lowest. This tube must from then on be handled with care, to avoid mixing of the solution, the density gradient must remain.

Large molecules such as proteins have a particular density, that is a bit larger than water. If a suspension of protein in pure water is left to stand for a long time then eventually the proteins will settle on the bottom.
The purpose of the gradient in sugar concentration is that when you have a mix of proteins then each type will sink until it has reached a level in the solution that has the same density as the protein itself. When the descending protein has reached a level where the solution has identical density then the protein is neutrally buoyant, and it will descend no further.


Putting the tube in a centrifuge speeds up the process of separation. As the centrifuge is spinning a large centripetal force is required to force the contents of the tube along the circular trajectory.

Proteins that are on a level where they are neutrally buoyant do experience the required centripetal force, so they descend no further.

At the start of the separation process the proteins are in the topmost layer, where the suspension fluid is less dense than the protein. In that case there is not enough centripetal force. When there is not enough centripetal force the particle will move away from the central axis of rotation. As the protein molecules move away from the central axis of rotation they travel through layers of solution of increasing density. When they reach a layer with the same density they do experience sufficient centripetal force, and they remain at that level.


In general:
In all forms of centrifugation the key factor is whether there is sufficient centripetal force to sustain circular trajectory. If there is not enough centripetal force then the trajectory will spiral outward, and in the case of a spinning tube in a biology lab centrifuge that means that heavy molucules will migrate towards the bottom of the tube.


It is not helpful to cast the explanation in terms of the action of a fictitious centrifugal force. 'Centrifugal force' is just another way of saying 'not enough centripetal force'. In physics it is always better to name things by their name, rather than using roundabout expressions.