1B8 :  Dimensionless Groups and Equations  5 pts 

Consider the following equation. All three of the terms in parentheses are dimensionless groups. Because k_{C} is difficult to determine directly, the other variables are measured and k_{C} is calculated from the given equation.  


What is the estimated value of k_{C} ? What are the units of k_{C} ? Show your work. The following values were measured: D = 8.0 mm, D_{AB} = 0.475 cm^{2}/s, m = 1.12 x 10^{3} Ns/m^{2}, r = 1.00 x 10^{3} g/cm^{3}, 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 righthand side of the equation to determine the value of the dimensionless group on the lefthand side. Then, we use the values and units of D and D_{AB} to determine the value and units of k_{C}.  
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  1.12E03  Ns/m^{2}  
D_{AB}  4.75E05  m^{2}/s  r  1.000  kg/m^{3}  
v  18.3  m/s  
Find:  k_{C}  ???  ??? (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.  

131 

23.579  
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 k_{g}.  



0.0193  
Answers:  k_{C}  0.0193  m/s  
1.93  cm/s  