Question: dont know if figure 1 will appear but it is...

Dont know if figure 1 will appear but it is just a representation of what is written, dont think the problem cant be solved without it...

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 either of the two pumps: OTBM450 or OTB450W.  The pump characteristics are shown in Figure 2.

Model	Watts (P2)	Current (Amps)	Discharge Head Capacity (kPa) (Liters per min)				Port Size (BSP)
			140	170	210	280	Discharge
OTB450W	650	4	70	56	47	16	1 1/4"F
OTBM450	650	4	52	43	31	0	1"M

 

Figure 2 – Pump characteristics of designated pumps

 

Selection criteria for system performance:

1. Must be one of the designated pumps
2. Minimum height of fountain 15 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.