Question 3.8: Water in a tank flows through an outlet 25 m below the water......
Water in a tank flows through an outlet 25 m below the water level into a 0.15 m diameter horizontal pipe 30 m long, with a 90° elbow at the end leading to a vertical pipe of the same diameter 15 m long. This is connected to a second 90° elbow which leads to a horizontal pipe of the same diameter, 60 m long, containing a fully open globe valve and discharging to atmosphere 10 m below the level of the water in the tank. Taking e/d = 0.01 and the viscosity of water as 1 mN s/m², what is the initial rate of discharge?
From equation 3.20, the head lost due to friction is given by:
h_{f}=\frac{-\Delta P_{f}}{\rho g}=4\phi\frac{l}{d}\frac{u^{2}}{g}=8\phi\frac{l}{d}\frac{u^{2}}{2g} (3.20)
h_{f}\,=\,4\phi\,{\frac{l}{d}}\,{\frac{u^{2}}{g}} m water
The total head loss is: h={\frac{u^{2}}{2g}}+h_{f} + losses in fittings
From Table 3.2., the losses in the fittings are:
={\frac{2\times0.8u^{2}}{2g}} (for the elbows) + \frac{5.0u^{2}}{2g} (for the valve)
={\frac{6.6u^{2}}{2g}} m water
Taking Φ as 0.0045, then:
10={\frac{(6.6+1)u^{2}}{2g}}+4\times0.0045\left[{\frac{30+15+60}{0.15}}\right]{\frac{u^{2}}{2g}}={\frac{(7.6+12.6)u^{2}}{2g}}
from which u² = 9.71
and: u = 3.12 m/s
The assumed value of Φ may now be checked.
R e={\frac{d u\rho}{\mu}}={\frac{(0.15\times3.12\times1000)}{(1\times10^{-3})}}=4.68\times10^{5}For Re = 4.68 x 10^{5} and e/d = 0.01, Φ = 0.0046 (from Figure 3.7) which approximates to the assumed value. Thus the rate of discharge = 3.12 x (π/4)0. 15² = 0.055 m³/s or (0.055 x 1000) = \underline{\underline{55\ kg/s}}.
| Table 3.2. Friction losses in pipe fittings | ||
| Number of pipe diameters | Number of velocity heads (u²/2g) | |
| 45° elbows (a)^{*} | 15 | 0.3 |
| 90° elbows (standard radius) (b) | 30-40 | 0.6-0.8 |
| 90° square elbows (c) | 60 | 1.2 |
| Entry from leg of T-piece (d) | 60 | 1.2 |
| Entry into leg of T-piece (d) | 90 | 1.8 |
| Unions and couplings (e) | Very small | Very small |
| Globe valves fully open | 60-300 | 1.2-6.0 |
| Gate valves: fully open | 7 | 0.15 |
| \frac{3}{4} open | 40 | 1 |
| \frac{1}{2} open | 200 | 4 |
| \frac{1}{4} open | 800 | 16 |
*See Figure 3.17.
