Dimensional Analysis

[mattman]

Experimentation can suck. Take some experiment: like dropping a rubber ball. You want to figure out what changes, how high the ball bounces back. That might sound easy, but it is not. Consider all the variables in that experiment: variables such as the balls’ velocity, the air viscosity, the elasticity of the rubber and the hardness of the floor. Now imagine you wanted to test dropping the ball, you could vary each variable on this list separately; like turning seven nobs, independently. To change volume you could drop 15 balls, each a different size and graph how high they bounce back. This is silly. Finding what controls bounce height would take forever. The experiment would need to be simplified. It was recognized then, that these variables were related. These variables could be combined – into dimensionless groups. In the case of the rubber ball, one the dimensionless group of interest was the drag coefficient.

Drag Coefficent = Drag Force/(0.5*Ball Aera * Velocity^2 * Density)

These dimensionless groups simplify the variable space. Instead of seven knobs to turn, you could now turn only one or two. Many systems have dimensionless numbers: from plasma systems to fluid systems to heat conduction. The polywell may have its own dimensionless numbers. The man who first devised the method of finding dimensionless numbers was Lord Rayleigh, in England, in the 19th century.

But, finding dimensionless groups in a new system can be tricky. First, you need to gather all the independent variables. Pick the relevant ones; the ones that will likely affect the outcome of the experiment. Next you need to identify what you are trying to measure; name that variable as a dependent. Once that is done there is a nice formula which predicts how many dimensionless groups you expect to find. This is the Buckingham-pi theorem [11]. Edgar Buckingham created this theorem to predict how many groups there were in a given experiment. He was a physicist working for the US government during the first half of the 20th century.

Dimensionless Groups = # of Variables - # of dimensions in problem

This method has been used in the past to simplify complex problems. Applying this method to a complex system like the polywell should help us simplify experimentation. This is not an easy task – and people are welcome to critique how well we did. The goal is to reduce the variables you need to explore when operating this machine. We hope to also find established numbers that may apply to this system.

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Example: Apply Rayleigh's method to Just the magnetic confinement field

For example, let’s apply this method to just the magnetic confinement field. Here is a list of all the independent variables I expect will affect the actual magnetic containment field, the machine makes.

1. The number of rings [dimensionless]
2. The ring width - [meters]
3. The circumference of the rings - [meters]
4. The minimum spacing between each ring - [meters]
5. The magnetic field each ring creates per ring section - [tesla/meter – (kilogram / meter * amp * second^2)]
6. The magnetic field of the electron cloud pressing back against the ring field - [tesla – (kilogram / amp * second^2)]

That is a list of all the independent variables; we would expect to be important. Variables 1 through 4 deal with the geometry of the system. Variables 5 and 6 deal with the magnetic field strength. There is also a 7th variable; the dependent one, the actual magnetic containment field we generate. Now according to Rayleigh's method if one variable depends on 6 other independent variables, then a function can be written to relate these, this is shown below.

Resulting Field = Function ( Width, Number, Diameter, Spacing, R_Field, E_Field)

...where this can be written as...

Resulting = Const*Width^a*Number^b*Diameter^c*Spacing^d*R_Field^e* E_Field^f

This function is essentially, a mathematical representation of our experiment. We are testing, to figure out how changing six of these factors, affects the 7th factor. This is a really important concept to grasp. Think about it. If, in reality, these six factors are the independent reasons for the 7th factor being what it is, then they are related in reality. If they are related in reality, then they can be related mathematically. This is an important and profound concept to understand.

Applying Buckingham-Pi:

We do not know what this function is, nor do we need to know, for our current purposes. Let us assume this function exists. Now, let us apply the Buckingham-pi theorem to this function. You should include the variable you seeking; so there are 7 variables. There are also 4 dimensions: meters, amps, kilograms and seconds. There are therefore 3 dimensionless groups.

This theorem predicts how many dimensionless groups you should find in Rayleigh's relationship. It cannot tell you which of these groups are physically meaningful. In more complex problems, you would now use matrix mathematics to find these groups. In fact, it was assuming that these factors could be entered into a matrix which led Buckingham to create the formula in the first place [12]. The wikipedia article steps through two examples of applying matrix algebra to find the dimensionless numbers. This is unneeded here, as it is easy to see all the dimensionless groups one could make from our variables. These are included below.

Dimless group 1 = Magnetic Field Generated by electron cloud/ (ring no*Circumfrence * Magnetic Field per ring section)

Dimless Group 2 = Width/Circumfrence

Dimless Group 3 = Spacing/Circumfrence

Dimless Group 4 = Width/Spacing

Unfortunately for us, there are 4 listed here, where the formula predicts only 3. If a more experienced expert could weigh in on this unexpected result, we would appreciate it. There is also the possibility of not including the number of rings in group 1, since this variable has no dimensions. However if a machine, for example, had 48 rings this number would clearly contain needed information – so it was included. Some of these dimensionless groups are not physically meaningful. For example, what use would the ratio of the ring width over the ring spacing, have? It is clear that there is a dimensionless group to describe the ratio of the externally applied magnetic field to the magnetic field generated by the electron cloud. You could call it the Whiffleball Number.

[happyjack27]

in my Mon Dec 26, 2011 3:35 am post here:

viewtopic.php?t=3468&postdays=0&postorder=asc&start=15

i used the method of images to reduce the "degrees of freedom" of a polywell down to 3+1=4:

*target wiffleball radius
*amp turns on the magrid (mag. field strength)
*charge density on the magrid (elec. field strenth)
*(+1)magrid coils center-to-midplane radius

the target plasma density and net charge could then be determined by the method of images.

[happyjack27]

i believe in non-dimensionalizing you also do a linear translation to zero-center and unit scale it. that is you, you drop off any additive constants at the end that do nothing but shift a number line (e.g. x = 2y - 1 becomes x = 2y), then set 1 of the 4 parameters to a constant value: "1". then the remaining 3 parameters are your 3 dimensions.


also, in your last 3 groups, you have essentially 3 equations of 3 variables (width, circumference, and spacing):

Dimless Group 2 = Width/Circumfrence

Dimless Group 3 = Spacing/Circumfrence

Dimless Group 4 = Width/Spacing


knowing any 2 out of 3 of these ratios is sufficient to determine the 3rd. so there you go. you can drop either 2,3, or 4. your choice.

[mattman]

Your Right. That is a really good point.


There are 3 groups, but if you know two of the dimensionless groups, the 3rd is irrelevant.


That is what the Buckingham-pi theorem predicted...