Example Problem with Complete Solution

1B-8 : Dimensionless Groups and Equations 5 pts
Consider the following equation. All three of the terms in parentheses are dimensionless groups.  Because kC is difficult to determine directly, the other variables are measured and kC is calculated from the given equation.
What is the estimated value of kC ?  What are the units of kC ?  Show your work.
The following values were measured:
D = 8.0 mm, DAB = 0.475 cm2/s, m = 1.12 x 10-3 N-s/m2, r = 1.00 x 10-3 g/cm3, v = 18.3 m/s.
Read: The key here is that the equation and the groups in parentheses are dimensionless.
This tells us that the constant 0.023 is also dimensionless.
So, we can plug numbers and units into the right-hand side of the equation to determine the value of the dimensionless group on the left-hand side.  Then, we use the values and units of D and DAB to determine the value and units of kC.
It is probably wise to begin the problem by converting all of the given values to one consistent system of units.  I chose the MKS system.
Given: D 0.008 m m 1.12E-03 N-s/m2
DAB 4.75E-05 m2/s r 1.000 kg/m3
v 18.3 m/s
Find: kC ??? ??? (units)
Assumptions: None.
Equations / Data / Solve:
Once all the given values are in a consistent set of units, we can evaluate each of the dimensionless groups in Eqn 1.
Let's double check the units on these groups using the SI units.
All the units cancel, so we can conclude that both of these groups are indeed dimensionless.
3.253 (dimensionless)
First, let's determine the units for kg.
Answers: kC 0.0193 m/s
  1.93 cm/s