TABLE 1. Approximate Integration of Eq. 34
Ah,fl(m) /zfe,ft(m) /zm,ft(m) /w,s/ft(s/m) At 9 S
1.0(0.30) O (0.00) 0. 5 (0.15) 2,890(9,633.3 ) 2,89 0
1.0(030) 1.0(0.30 ) 1. 5 (0.46) 3,580(11,933.3 ) 3,58 0
0.8(0.24) 2.0(0.61 ) 2. 4 (0.73) 5,160(17,308.3 ) 4,13 0
0.4(0.12) 2.8(0.85 ) 3. 0 (0.91) 7,870(26,250.0 ) 3,15 0
0.3 (0.09) 3. 2 (0.98) 3.3 5 (1.02) 11,83 0 (39,444.4) 3,55 0
0.2(0.06) 3.5(1.07 ) 3. 6 (1.10) 18,930(63,166.7 ) 3,79 0
0.1(0.03) 3.7(1.13 ) 3.75(1.14 ) 30,000(100,000 ) 3,00 0
Total 24,09 0The precision inherent in the result thus obtained depends on the judgment used in se-
lecting the increments of /*, and a clear visualization of the relationship between h and t is
essential. Let m = dtldh = AI(Q 1 — CM^1 -^5 ). The m-h curve is shown in Fig. 14a. Then, t =
Jm dh = area between the m-h curve and h axis.
This area is approximated by summing the areas of the rectangles as indicated in Fig.
14, the length of each rectangle being equal to the value of m at the center of the interval.
Note that as h increases, the increments Ah should be made progressively smaller to min-
imize the error introduced in the procedure.
Select the increments shown in Table 1, and perform the indicated calculations. The
symbols hb and hm denote the values of h at the beginning and center, respectively, of an
interval. The following calculations for the third interval illustrate the method: hm = l/2(2.0
- 2.8) = 2.4 ft (0.73 m); m = 6,000,0007(2175 - 3.4 x 80 x 2.4^15 ) = 5160 s/ft (16,929.1
s/m); At = m Ah = 5160(0.8) = 4130 s. From Table 1 the required time is t = 24,090 s =
6H41.5 min.
The t-h curve is shown in Fig. I4b. The time required for the water to reach its maxi-
mum height is difficult to evaluate with precision because m becomes infinitely large as h
approaches /*max; that is, the water level rises at an imperceptible rate as it nears its limit-
ing position.
DIMENSIONAL ANALYSIS METHODSThe velocity of a raindrop in still air is known or assumed to be a function of these quan-
tities: gravitational acceleration, drop diameter, dynamic viscosity of the air, and the den-
sity of both the water and the air. Develop the dimensionless parameters associated with
this phenomenon.Calculation Procedure:- Using a generalized notational system, record the units
in which the six quantities of this situation are expressed
Dimensional analysis is an important tool both in theoretical investigations and in experi-
mental work because it clarifies the relationships intrinsic in a given situation.
A quantity that appears in every dimensionless parameter is termed repeating; a quan-
tity that appears in only one parameter is termed nonrepeating. Since the engineer is usu-
ally more accustomed to dealing with units offeree rather than of mass, the force-length-