Question

how to choose repeating variables in Buckingham Pi Theorem...in particular

Answer 1

nondimensional groups for heat transfer in a pipe...effective parameters :
\[h : Mt^{-3}T^{-1}\]\[u : Lt^{-1}\]\[k : MLt^{-3}T^{-1}\]\[c_p : L^2t^{-2}T^{-1}\]\[\mu : ML^{-1}T^{-1}\]\[\rho : ML^{-3}\]\[d : L\]

Answer 2

we have 7-4=3 nondimensional group and 4 repeating variables..how to choose them considering the dimensions i gave u

Answer 3

and sorry \[\mu : ML^{-1}t^{-1}\]

Answer 4

@experimentX

Answer 5

we have a rule that No repeating parameter should have dimensions that are a power of the dimensions of another repeating parameter. why ...and how apply it here?

Answer 6

I seem to have forgotten this from fluids (or maybe I never really learned it).

You might find this helpful: http://www.eng.wayne.edu/legacy/forms/4/Buckinghamforlect1.pdf

Answer 7

It says that the repeating variables that you choose cannot be manipulated into dimensionless groups.

Define the above parameters for me please.

Answer 8

sorry i was out

h: heat transfer coefficient
u: flow mean velocity
k: heat conductivity coefficient
cp: specific heat capacity
\(\mu\): viscousity
\(\rho\) : density
d: diameter of pipe

Answer 9

Oh. Those are the units after each parameter.

Let's see. We need 3 recurring variables. Choose d, u, and \(\rho\) Those should work.

Answer 10

emm...we need 4 if im not wrong

Answer 11

I'm working out of that document I linked and it's ringing some bells.

It looks like we have 7 variables\[f(h, u, k, c_p, \mu, \rho, d) \]

Between these 7 variables, we have 4 different dimensions. Mass (M), time (t), temperature (T), and length (L).

n = 7; m = 4

n-m gives 3 dimensionless groups. We then need to choose 3 variables (these three cannot be formed into dimensionless groups themselves. That is to say that we cannot manipulate these three independently to create a dimensionless group. For example, if we had length and diameter, we couldn't choose these are two of our three. )

Once we choose three variables, we rewrite the four dimensions in terms of the three variables we chose.

Does this follow?

Answer 12

ı think 3 repeating variables occur; u, d, ρ,