SIMILARITY AND DIMENSIONLESS GROUPS A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 14 CHAPTER THREE SIMILARITY AND DIMENSIONLESS GROUPS 3-1 Similarity The Performance of a turbomachine like pumps, water turbines, fans or blowers for incompressible flow can be expressed as a function of: (i) density of the fluid ρ (ii) Speed of the rotor N (iii) Characteristic diameter D (iv) Discharge Q (v) Gravity head (gH) (vi) Power developed P and (vii) Viscosity μ. Using Buckingham pi theorem Turbomachine Performance = f [ρ, N, D, Q, gH, P, μ] Taking N, D as repeating variables and grouping with other variables as non-dimensional groups π1 = [ρa1 Nb1 Dc1 Q] (3.1) π2 = [ρa2 Nb2 Dc2 gH] (3.2) π3 = [ρa3 Nb3 Dc3 P] (3.3) π4 = [ρa4 Nb4 Dc4 μ] (3.4)

π1 = Q/ND3 = Φ, Flow coefficient (3.5) A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 15 Equating powers of mass, length and time in the LHS and RHS of the π terms, we obtain: π1 = Q/ND3 = Φ, Flow coefficient (3.5) π2 = gH/N2D2 = Ψ Head coefficient (3.6) π3 = P /ρN3D5 = Θ Power coefficient (3.7) π4 = μ /ρND2 = Re Reynolds number (3.8) For model studies for similar turbomachines, we can use: (3.9) (3.10) (3.11) (3.12) Suffix 1 Model ; Suffix 2 Prototype

A course in Turbomachinery…………………………………. …. …Lecturer: Dr A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 16 3.2 Net Positive-Suction Head The net positive-suction head (NPSH) is the head required at the pump inlet to keep the liquid from cavitating or boiling. The pump inlet or suction side is the low-pressure point where cavitation will first occur. Fig. 3.1 Cavitation bubbles forming and collapsing on the suction side of an impeller blade. The NPSH is defined as: 2 2 3.13 where pi and Vi are the pressure and velocity at the pump inlet and pv is the vapour pressure of the liquid. Given the left-hand side, NPSH, from the pump performance curve, we must ensure that the right-hand side is equal or greater in the actual system to avoid cavitation. If the pump inlet is placed at a height Zi above a reservoir whose free surface is at pressure pa, we can use Bernoulli’s equation to rewrite NPSH as: p pv where hfi is the friction head loss between the reservoir and the pump inlet. Knowing pa and hfi, we can set the pump at a height Zi that will keep the right-hand side greater than the “required” NPSH. pi vi pv i i 3.14

3.3 The Specific Speed Ns Fig. 3.2 A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 17 Fig. 3.2 If cavitation does occur, there will be pump noise and vibration, pitting damage to the impeller, and a sharp drop off in pump head and discharge. In some liquids this deterioration starts before actual boiling, as dissolved gases and light hydrocarbons are liberated. 3.3 The Specific Speed Ns The pump or hydraulic turbine designer is often faced with basic problem of deciding what type of turbo machine will be the best choice for a given duty. Usually the designer will be provided with some preliminary design data such as the head H, the volume flow rate Q and rotational speed N when a pump design under a consideration. A non-dimensional parameter called specific speed referred to and conceptualised as the shape number, is often used to facilitate the choice of most appropriate machine. Addison (1955) defines a pump as being of standardized 'size' when it delivers energy at the rate of one horsepower when generating a head of one foot. This imaginary wheel has a speed termed the specific speed. He shows that the specific speed Ns for pumps is given by:

European units (NSp, Eur). A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 18 N The equivalent expression relating to hydraulic turbines takes the form: N√ Fig. 3.3 Maximum efficiency as a function of pump specific speed for the three main types of dynamic pump. The horizontal scales show non-dimensional pump specific speed (NSp), pump specific speed in customary U.S. units (NSp, US), and pump specific speed in customary European units (NSp, Eur). Ns = H3/4 (3.15) Ns = HS/4 (3.16)

3.4 Characteristic Number Ks: A course in Turbomachinery…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 19 3.4 Characteristic Number Ks: Characteristic numbers are dimensionless numbers used in fluid mechanics to describe a character of the flow. To compare a real situation with a small-scale model. For pump:- ( where ω is in radians per second, Q in cubic metres per second and (gH) is as usual. This gives a non-dimensional result. For turbine:- (√ (gH)S/4 EX.1 An axial flow pump with an impeller rotor diameter of 30 cm handles water at a rate of 2.7 m3/minute running at 1500 rpm. The energy input is 125 J/kg and Total to Total efficiency is 0.75. If a geometrically similar pump has a diameter of 20 cm running at 3000 rpm, find its: i) Flow rate ii) input power. Ks (pump) = (gH)3/4 (3.17) Ks (turbine) = (3.18)