505
Chapter 6 x Viscous Flow in Ducts
Solution: For water at 20qC, take U 1.94 slug/ft3 and P 2.09E5 slug/ft˜s. Let “a”
be the small pipe and “b” the larger. For wrought iron, H| 0.00015 ft, whence H/da
0.0018 and H/db 0.0009. From the continuity relation,
Q
Va
S
4
da2
Vb
S
4
d 2b
or, since d b
2d a , we obtain Vb
1
Va
4
0.5, and a sudden
For pipe “a” there are two minor losses: a sharp entrance, K1
expansion, Fig. 6.22, Eq. (6.101), K2 [1 (1/2)2]2 | 0.56. For pipe “b” there is one
minor loss, the submerged exit, K3 | 1.0. The energy equation, with equal pressures at
(1) and (2) and near zero velocities at (1) and (2), yields
'z
h f-a ¦ h m-a h f-b ¦ h m-b
or, since Vb
Va /4, 'z
·
· Vb2 § L b
Va2 § La
f
0.5
0.56
fb
1.0 ,
a
¨
¸
¨
2g © da
¹ 2g © d b
¹̧
45 ft
Va2 ª
120
1.0 º
240fa 1.06 fb «
2(32.2) ¬
16
16 »¼
where fa and fb are separately related to different values of Re and H/d. Guess to start:
fa | fb | 0.02: then Va
Vb
21.85 ft/s, Rea | 169000, H /d a
5.46 ft/s, Re b | 84500, H /d b
Converges to: fa
Q
0.024, fb
0.0018, fa-2 | 0.0239
0.0009, fb-2 | 0.0222
0.0224, Va | 20.3 ft/s,
Va Aa | 0.111 ft 3 /s. Ans.
6.104 Reconsider the air hockey table of
Problem 3.162, but with inclusion of minor
losses. The table is 3 ft by 6 ft in area, with
1/16-in-diameter holes spaced every inch
in a rectangular grid (2592 holes total). The
required jet speed from each hole is 50 ft/s.
Your job is to select an appropriate blower
to meet the requirements.
Fig. P3.162
Hint: Assume that the air is stagnant in the manifold under the table surface, and assume
sharp-edge inlets at each hole. (a) Estimate the pressure rise (in lbf/in2) required of the
blower. (b) Compare your answer to the previous calculation in Prob. 3.162, where minor
losses were ignored. Are minor losses significant?