Buckingham Pi / Dimensional analysis

[beth92]

Homework Statement

A capillary filled with water is placed in a container filled with a chemical of concentration [itex] C_{0} [/itex], measured in number of molecules per unit volume. The chemical diffuses into the capillary of water according to the following relation (where x is distance along capillary):

[itex] \frac{\partial C}{\partial t} = D \frac{\partial^{2}C}{\partial x^{2}} ~~~~ C(0,t) = C_{0} ~,~ C(x,0)=0 [/itex]

a) Find dimensions of diffusion coefficient D
b) For capillary with cross sectional area A, the number of molecules entering the capillary N in a fixed time T is measured. We can assume that there is a law relating the 5 quantities [itex] D, C_{0}, N, T, A [/itex]. Use dimensional analysis to find the general form of this law.
c) If experiments show that [itex] N [/itex] is proportional to [itex] \sqrt{T} [/itex], then give the simplest law which expresses N as a function of the 4 other quantities.

Homework Equations

Buckingham Pi theorem says that the law will have the form [itex] F(\pi_{1},\pi_{2},...) = 0 [/itex] where the [itex] \pi_{i} [/itex] are dimensionless quantities created using the 5 given physical quantities.

The Attempt at a Solution

a) I calculate the dimensions of D to be L²/T

b) I calculate my dimensionless quantities as:

[itex]

\pi_{1} = N \\

\pi_{2} = A^{3}C_{0}^{2} \\

\pi_{3} = \frac{T D}{A}

[/itex]

c) This is the part I'm not sure of. What does it mean by the 'simplest' law? From the buckingham Pi theorem I get that:

[itex] N = g( A^{3}C_{0}^{2},\frac{TD}{A} ) [/itex]

Where g is some unknown function.

But I'm not sure what to do with the fact that N varies with the square root of T. Can I take that term out and say:

[itex] N = \sqrt{\frac{TD}{A}} f(A^{3}C_{0}^{2}) [/itex]

This doesn't seem right/complete to me but I can't think of anything else.

 

[Orodruin]

But I'm not sure what to do with the fact that N varies with the square root of T. Can I take that term out and say:

N=TDA‾‾‾√f(A3C20) N = \sqrt{\frac{TD}{A}} f(A^{3}C_{0}^{2})

This doesn't seem right/complete to me but I can't think of anything else.

Yes, you can do this. If ##f## would depend on ##\pi_3##, then ##N## would not vary as ##\sqrt{T}##.