mukushla 3 years ago how to choose repeating variables in Buckingham Pi Theorem...in particular

1. mukushla

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$

2. mukushla

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

3. mukushla

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

4. mukushla

@experimentX

5. mukushla

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?

6. eashmore

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

7. eashmore

It says that the repeating variables that you choose cannot be manipulated into dimensionless groups. Define the above parameters for me please.

8. mukushla

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

9. eashmore

Oh. Those are the units after each parameter. Let's see. We need 3 recurring variables. Choose d, u, and $$\rho$$ Those should work.

10. mukushla

emm...we need 4 if im not wrong

11. eashmore

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?

12. akifersoylu

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