Optimum Values of Design Variables for high Specific Speed Centrifugal Pumps SATISH KUMAR1, VIKASH GUPTA2 A.V.DOSHI3
1,2
Department of Mechanical Engineering, BRCM College of Engineering and Technology Bahal-127028, Bhiwani (Haryana) 3
Department of Mechanical Engineering, S.V.National institute of Technology Surat-395007, (Gujarat)
E-mail:
[email protected], Abstract: An optimum design computer code for high specific speed centrifugal pump has been developed to determine the fluid dynamic variables under appropriate design constraints. The optimization problem has been formulated with a non-linear objective function to minimize fluid dynamic losses and net positive suction required head of a pump stage depending on the weighting factors selected as the design compromise. The optimum solution is obtained by Hook-Jeeves direct search method. The optimized efficiency and design variables of centrifugal pump are presented in this paper as a function of specific speed. The computer program developed in this present work is best suited for specific speed between range of 60-80. Keywords: Centrifugal pump, Design optimization, Design variables, Specific speed. Symbols:
a = Constant, used in determination of the shaft power Ath= Throat area at volute in m2 B1= Runner width at inlet in m B2= Runner width at outlet in m B3=Inlet width of volute in m D= Diameter in m De= Runner eye diameter in m Dh==Hub diameter in m Dsh= Shaft diameter in m dp/r = Mean blade loading f = Weighting factor fs = Shear stress N/m2 g = Gravitational acceleration in m/s2 H = Head in m Vm = Absolute velocity in m/s hL=Total hydraulic loss in head in m Hs= Suction height in m Hd = Depression head in m Kv= Guide vanes speed ratios Km=Capacity constant Lo= Length of guide vane in m
NPSHR=Net positive suction required head in m Ns = Specific speed of pump N=Speed in rpm Po=Output power in HP Psh= Shaft power in HP Z = Number of blades Z1 = Number of guide vanes r = Radius in m Rt = Tongue radius in m Re= Reynolds number t = Blade thickness in m U= Peripheral velocity in m/s Ue= Peripheral velocity at the eye Vr= Relative velocity in m/s Vrui= Whirl component of relative velocity in m/s Vu= Whirl component of Velocity in m/s Vth= Throat velocity in m/s Q=Flow rate in m3/s
GREEK: α= Angle at which the water enter the runner β = Blade angle η = Overall efficiency θ = Volute angle θA= Maximum total angle between the side of the volute θ t= Tongue angle ν = Kinematic viscosity m2/s σ = Slip factor SUBSCRIPTS: 0= Eye of pump 2 = Outlet of the impeller 1.
σb= Blade cavitation coefficient σth=Thoma cavitation coefficient τ = Permissible torsional stress in N/m2 Φ= Flow coefficient Φ1= Flow coefficient at suction condition Ψ= Head coefficient ω= Angular velocity in rad/s
1 = Inlet of the pump x =Inlet of the volute
INTRODUCTION
Centrifugal pump has been developed as highly efficient machine owing a large number of geometric and fluid dynamic variables, and their interaction. Most of the practical design and performance analysis of centrifugal pump are essentially based on empirical or semi empirical rules as proposed by successful designers [12, 17]. When pump is designed, due concentration is focused on important factors like required head, max efficiency, a stable flow characteristics and non-cavitation performance of the pump. Further the practical turbine design process involves a number of compromises between the maximum efficiency and suppression of cavitation because cavitation reduces the life of the centrifugal pump. Present work is aimed at developing automated computer design code that can be used to find optimum configuration for the desired compromise between the maximum efficiency and minimum net positive suction head required (NPSHR).For this purpose hydraulic losses are calculated to analyze the hydrodynamic performance, and a optimization algorithm i.e. Hooke-jeeves direct search method [13], is adopted to create a generalized computer code [14, 15] for the design optimization of centrifugal pump. 2.
OPTIMIZATION PROBLEMS
This study is carried out under the following three assumptions: (i) the flow enters without any pre-swirl, (ii) the flow in the vaneless space is of free-vortex type, and (iii) the volute casing is constructed of gradually increasing circular cross-sections with a constant average velocity. The design optimization problem for centrifugal pump consists of design specifications, design constraints, design variables, and objective function. 3.
DESIGN SPECIFICATION
Design of the centrifugal pump input data: flow rate, total head, specific speed, density of liquid, operating fluid viscosity. 4.
DESIGN CONSTRAINTS
Design constraints for the centrifugal pump are given in Table 1: Table 1:-DESIGN CONSTRAINTS:
S.N.
Geometric parameter
Hydraulic parameter
1. 2.
0.75≤Dsh/Dh≥0.95 0.8≤De/D1≥1
0.3≤σf ≥.45 0.1≤Φ≥0.2
3.
0.3≤D1/D2≥0.8
0.5≤Ψ ≥0.7
5.
4. 5.
1.1≤D3/D2≥1.3 1.4≤B3/B2≥2
0.8≤ Vm / Vm1≥1 1.2 ≤ Vm0/Vm2.≥1.5
6. 7. 8. 9. 10. 11.
.006≤t/D2≥0.016 0.04≤b2/D2≥0.2 5≤Z≥12 20≤β1≥40 20≤β2≥40 25≤ α2≥45
0.2≤ Vu / U2 ≥0.6 Vrui/ Vr≥1.4 ∆h≥0
DESIGN VARIABLES
(i) Geometric parameter Vane angle, Number of vanes, Runner discharge width, Hub/Tip ratio, Inclination of the mean stream line with axial direction and Blade cavitations factor (ii) Hydraulic parameter Flow coefficient, Head coefficient, Blade velocity, Relative velocity and other hydraulic parameters needed to describe the flow direction and magnitudes become direct function of geometry. 6.
OBJECTIVE FUNCTION
Objective function is constructed by a combination of the loss of efficiency (1-ηp) and positive suction head required (NPSHR). Design optimization can state in the format of a non-linear constrained optimization equation given below:
NPSH R Z = f1 (1 − η P ) + f 2 ………………………………….. H
(1)
where coefficients f1, f2 are the weighting factor for the loss of efficiency. The outlet performance a compromise optimal design can be obtained by defining the weighting factor f1 & f2 between 0 & 1. 6.1 Evaluation of efficiency Calculated internal hydraulic losses [11, 12, and 16], in this paper, depends upon the geometrical and hydraulic parameters, are more accurate. 6.2 Evaluation of the NPSHR NPSHR is expressed under the no-pre whirl condition as given below:
NPSH R = (1 + σ f )
Vm22 v2 + σ f 2 …………………………………….. (2) 2g 2g
where the blade cavitation coefficient σf is the parameter needed to define the inlet performance of pump. The average value of σf is taken in range of 0.3 to 0.45[10]. For present work σf is assumed as 0.4. 6.3 Optimization Program A simple and efficient optimization technique called the Hooke-Jeevas direct search method is adopted to create a generalized computer code for the design optimization of centrifugal pump as shown in Fig. 1. A systematic search has to be initiated for the solution within the constrained domain of variables.
Optimization requires the sorting values of design variables, geometric and hydraulic constrained specified as input to the optimization procedure. Calculation of the objective function at the shorting point search procedure is then carried out to locate a new feasible point with the improved value of the objective function. The search is continued until a point is searched where no further improvement in the objective function is possible. START INITIALISE X=XC and ∆i NO
i=1, i+=1
i
