time system of units is applied here. Let F, L, and T denote units of force, length, and
time, respectively.
By using this generalized notational system, it is convenient to write the appropriate
USCS units and then replace these with the general units. For example, with respect to ac-
celeration: USCS units, ft/s
2
; general units, LIT
2
or LT~
2
. Similarly, with respect to densi-
ty (wig): USCS units, (Ib/ft
3
)/(ft/s
2
); general units, FZrVLI
7
2
or FL-
4
T
2
.
The results are shown in the following table.
Quantity Units
V = velocity of raindrop LT~l
g = gravitational acceleration LT~^2
D = diameter of drop L
[JL 0 = air viscosity FLr^2 T
pw = water density FZ^r^2
pa = air density FL-^4 T^2- Compute the number of dimensionless parameters present
This phenomenon contains six physical quantities and three units. Therefore, as a conse-
quence of Buckingham's pi theorem, the number of dimensionless parameters is 6 - 3 =
- Select the repeating quantities
The number of repeating quantities must equal the number of units (three here). These
quantities should be independent, and they should collectively contain all the units pres-
ent. The quantities g, D 9 and /*,a satisfy both requirements and therefore are selected as the
repeating quantities. - Select the dependent variable V as the first nonrepeating
quantity
Then write Tr 1 = g^^F, Eq. a, where Tr 1 is a dimensionless parameter and Jt, y, and z are
unknown exponents that may be evaluated by experiment.
5. Transform Eq. a to a dimensional equation
Do this by replacing each quantity with the units in which it is expressed. Then perform
the necessary expansions and multiplications. Or, F^0 L^0 T^70 = (LT-^2 yLy x (FLr^1 TJLT-^1 ,
/TO 1 OjO = pzLx+y-22+\ j-lx+z-1 ^ Rq £
Every equation must be dimensionally homogeneous; i.e., the units on one side of
the equation must be consistent with those on the other side. Therefore, the exponent of
a unit on one side of Eq. b must equal the exponent of that unit on the other side.- Evaluate the exponents x, y, and z
Do this by applying the principle of dimensional homogeneity to Eq. b. Thus, O = z; O = x
- y-2z+l;0 = ~2x + z-l. Solving these simultaneous equations yields x = -^1 ^; y = -%\
z = 0.
- Substitute these values in Eq. a
Thus, Tr 1 = g-l/2D~l/2V, or Tr 1 = VI(gD)m. - Follow the same procedure for the remaining
nonrepeating quantities
Select pw and pa in turn as the nonrepeating quantities. Follow the same procedure as be-