1 Introduction
2 Methodology
2.1 Numerical Methodology
2.1.1 Computational Model
Parameter Name | Value |
---|---|
Pump inlet diameter Ds (mm) | 80 |
Pump outlet diameter Dd (mm) | 80 |
Impeller inlet diameter D1 (mm) | 72 |
Impeller outlet diameter D2 (mm) | 136 |
Impeller inlet width b1 (mm) | 34.5 |
Impeller outlet width b2 (mm) | 17.8 |
Impeller blade number z | 6 |
Impeller blade angle at the leading edge β1 (°) | 38 |
Impeller blade angle at the trailing edge β2 (°) | 23 |
2.1.2 Grid Distribution
Scheme index | Node number (×106) | Head (m) |
---|---|---|
A | 0.976 | 20.122 |
B | 1.126 | 19.698 |
C | 1.434 | 19.776 |
D | 1.817 | 20.185 |
E | 2.387 | 20.196 |
F | 3.482 | 20.245 |
G | 4.290 | 20.275 |
H | 6.603 | 20.281 |
I | 8.390 | 20.284 |
Component | Suction Pipe | Inlet Pipe | Impeller | Volute | Discharge Pipe |
---|---|---|---|---|---|
Nodes number | 779544 | 581578 | 933510 | 1216305 | 779544 |
2.1.3 Numerical Setup
2.2 Optimization Procedure
2.2.1 Objective Function
2.2.2 Decision Variables
Parameter | Range |
---|---|
Blade angle at trailing edge β2(°) | [20, 40] |
Impeller outlet diameter D2(mm) | [128, 134] |
Impeller outlet width b2(mm) | [15, 20] |
Blade wrapping angle θ(°) | [80, 120] |
2.2.3 Automatic Simulation Technique
2.2.4 Design of Experiment
NO | β2(°) | D2(mm) | b2(mm) | θ(°) | ηy(%) | H (m) | Cp* |
---|---|---|---|---|---|---|---|
1 | 32 | 131 | 19 | 80 | 79.48 | 22.11 | 0.1829 |
2 | 36 | 133 | 16 | 110 | 82.76 | 22.11 | 0.0946 |
3 | 24 | 129 | 20 | 115 | 83.44 | 20.68 | 0.1419 |
4 | 36 | 128 | 18 | 105 | 84.41 | 21.12 | 0.0531 |
5 | 28 | 132 | 16 | 105 | 82.36 | 21.00 | 0.1030 |
… | … | … | … | … | … | … | … |
36 | 40 | 133 | 17 | 90 | 81.48 | 22.71 | 0.1171 |
37 | 28 | 129 | 19 | 100 | 81.88 | 20.91 | 0.1115 |
38 | 28 | 134 | 15 | 110 | 82.53 | 21.55 | 0.0828 |
39 | 32 | 130 | 15 | 85 | 80.83 | 20.43 | 0.1652 |
40 | 40 | 128 | 17 | 95 | 83.59 | 21.11 | 0.1445 |
2.3 Surrogate Model
2.3.1 Response Surface Model
2.3.2 Artificial Neural Network
2.4 Modified Discrete Genetic Algorithm
2.4.1 Modifications
2.4.2 Benchmark
Name | Formula | Dim | Decision Domain | Optimum Variable | Global Optimum |
---|---|---|---|---|---|
Ackley | \(f(x) = - a\exp \left( { - b\sqrt {\frac{1}{d}\sum\limits_{i = 1}^{d} {x_{i}^{2} } } } \right) - \exp \left( {\frac{1}{d}\sum\limits_{i = 1}^{d} {\cos (cx_{i} )} } \right) + a + \exp (1)\) | 6 | [−32.768, 2.768] | [0, ……, 0] | 0 |
Bulkin N.6 | \(f(x) = 100\sqrt {\left| { \, x_{2} - 0.01x_{1}^{2} \, } \right|} + 0.01\left| { \, x_{1} + 10 \, } \right|\) | 2 | x1∈[−15, −5], x2∈[−3, 3] | [−10, 1] | 0 |
Drop- Wave | \(f(x) = { - }\frac{{1 + \cos \, \left( {12\sqrt {x_{1}^{2} + x_{2}^{2} } } \right)}}{{0.5 \, (x_{1}^{2} + x_{2}^{2} ) + 2}}\) | 2 | [−5.12, 5.12] | [0, 0] | −1 |
Griewank | \(f(x) = \sum\limits_{i = 1}^{d} {\frac{{x_{i}^{2} }}{4000}} - \prod\limits_{i = 1}^{d} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right)} + 1\) | 6 | [−600, 600] | [0, ……, 0] | 0 |
Function | Algorithm | CR | SSavg | SSmin | SSmax | SSmedian | SSstd |
---|---|---|---|---|---|---|---|
Ackley | GA | 0 | - | - | - | - | |
MDGA | 100% | 90.2 | 29 | 311 | 64.5 | 69.86 | |
Bulkin N.6 | GA | 0 | - | - | - | - | |
MDGA | 100% | 6.9 | 2 | 52 | 3 | 11.37 | |
Drop-Wave | GA | 25% | 229 | 130 | 538 | 151 | 174.23 |
MDGA | 100% | 7.45 | 2 | 42 | 4 | 9.65 | |
Griewank | GA | 0 | - | - | - | - | |
MDGA | 100% | 72.35 | 23 | 129 | 73 | 32.76 |
3 Experimental Validation
3.1 Test Rig Setup
Parameter | Sensor | Model | Range | Uncertainty |
---|---|---|---|---|
Inlet pressure | Pressure sensor | HM90 | 0-0.2MPa | ±0.1% |
Pressure at P2 | Pressure sensor | HM90 | 0-0.7MPa | ±0.1% |
Flowrate | Turbine flowmeter | LW-80 | 12-120m3/h | ±0.5% |
Outlet pressure | Pressure sensor | WIKA | 0-1MPa | ±0.1% |
3.2 Result Comparison
4 Results and Discussions
4.1 Performance Comparison
Parameter | β2 (°) | D2 (mm) | b2 (mm) | θ (°) |
---|---|---|---|---|
Original case | 23 | 136 | 17.8 | 110 |
Optimized case | 20 | 133 | 15 | 120 |
Parameter | \(\eta_{{0.6Q_{{\text{d}}} }}\) | \(\eta_{{1.0Q_{{\text{d}}} }}\) | \(\eta_{{1.4Q_{{\text{d}}} }}\) | \(H_{{0.6Q_{{\text{d}}} }}\) | \(H_{{1.0Q_{{\text{d}}} }}\) | \(H_{{1.4Q_{{\text{d}}} }}\) | \(C_{{0.6Q_{{\text{d}}} }}^{{{\text{V1}}}}\) | \(C_{{1.0Q_{{\text{d}}} }}^{{{\text{V1}}}}\) | \(C_{{1.4Q_{{\text{d}}} }}^{{{\text{V1}}}}\) |
---|---|---|---|---|---|---|---|---|---|
Original case | 67.34 | 79.86 | 77.84 | 21.51 | 19.42 | 15.00 | 0.1620 | 0.0887 | 0.0896 |
Optimized case | 69.25 | 82.23 | 79.55 | 21.29 | 18.94 | 14.24 | 0.1649 | 0.0533 | 0.0716 |
Change | 2.84% | 2.97% | 2.20% | −1.02% | −2.47% | −5.07% | 1.79% | −39.91% | −20.09% |