Question: a high flying fountain is required for decoration and air...

A high flying fountain is required for decoration and air conditioning purpose for a high building.  The fountain consists of a concrete tank in which a submersible pump is installed.  Water is then pumped to a height L₂, and diverted in another pipe section L₁ at an angle $\alpha$ outwards through a nozzle as shown in Figure 1.  For easier design decision, the angle $\alpha$ is fixed at 30 degrees.

Figure 1 – Configuration of the high flying fountain

You are required to design the system using one of the three pumps: 75/35, 75/56, 115/576.  The pump characteristics are shown in Figure 2.

Figure 2 – Pump characteristics of designated pumps

 

Selection criteria for system performance:

1. Must be one of the designated pumps
2. Height of fountain above ground to be maintained between 15 m to 20 m
3. Larger water flowrate preferred. More water gives a better impression of the fountain

Which set of system design parameters would you suggest?

Formulae applicable to this question:

Pressure drop in a pipe is given by:

$\mathrm{\Delta }P=P_1-P_2=\frac{128\mu LQ}{\pi D^4}$

where

• P₁ = upstream fluid pressure (N/m²)
• P₂ = downstream fluid pressure (N/m²)
• L = length of pipe section (m)
• Q = fluid mass flowrate (kg/s)
• $\pi$ = circle constant = 3.14159
• $\mu$ = dynamic viscosity of water at operating temperature = 0.29
• D = internal diameter of the pipe

Pressure drop through pipe bend is given by:

$\mathrm{\Delta }P=P_1-P_2=\frac{1}{2}{f}_s\rho v^2\frac{\pi R_b}{D}\frac{\theta }{{180}^{{}^o}}+\frac{1}{2}{k}_b\rho v^2$

where P₁, P₂, D and $\pi$  have the same meaning as before.

• R_b = bend radius
• v = velocity of fluid flow at the centre line of the bend.
• $\rho$ = the density of water (very heavily salted mineral water) (= 1,029 kg/m³)
• $\theta$ = the angle of circular segment of the bend sustained from the centre of bend in degrees

The pipe flow factor k_b depends on the angle of bend. The value of k_b can be interpolated from Table 1 and 2:

Table 1:  k_b values for 90 degrees bends

Rb/D	0.5	0.6	0.7	0.8	1.0	2.0	3.0	4.0	5.0	6.0	8.0	10.0
kb	0.85	0.68	0.56	0.48	0.39	0.24	0.18	0.16	0.15	0.14	0.13	0.13

Note: For R_b/D > 10.0, k_b = constant = 0.13.

Table 2:  k_b values for 120 degrees bends

Rb/D	0.5	0.6	0.7	0.8	1.0	2.0	3.0	4.0	5.0	6.0	8.0	10.0
kb	0.95	0.74	0.60	0.53	0.44	0.26	0.26	0.18	0.16	0.15	0.15	0.15

Note: For R_b/D > 10.0, k_b = constant = 0.13.

The coefficient of pipe bend friction $f_s$ depends on Reynolds number Re.  Reynolds number is given by:

$Re=\frac{\rho vD}{\mu }$

If Re < 2300, then it is laminar flow,

$f_s=\frac{64}{Re}$

If Re > 4000, then it is turbulent flow,

$\frac{1}{\sqrt{f_s}}=-2{\log }_{10}(\frac{ϵ}{3.7D}+\frac{2.51}{Re\sqrt{f_s}})$

Stainless steel will be used throughout the system.  The pipe material roughness factor $ϵ$ = 0.0015.

If 2300 < Re < 4000, then it is critical flow.  Interpolate between laminar and turbulent flows using Re value.