| 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. |
|
|
|
|
|
|
|
|
|
|
|
|
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 kg. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0.0193 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Answers: |
kC |
0.0193 |
m/s |
|
|
|
|
|
|
|
|
1.93 |
cm/s |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
