Experimental Fitting of the Re-Scaled Balje Maps for Low-Reynolds Radial Turbomachinery
Abstract
:1. Introduction
2. Reynolds Number Effects
Year | Source | Inviscid loss fraction “a” | Viscosity-dependent loss fraction (1 − a) | Exponent n | Re range | Machine type |
---|---|---|---|---|---|---|
1925 | Moody | 0.25 | 0.75 | 0.33 | n.d. | Propeller turbines |
1930 | Ackeret & Muhlemann | 0.50 | 0.50 | 0.20 | n.d. | Hydraulic turbines |
1942 | Moody | 0.00 | 1.00 | 0.20 | n.d. | Pumps |
1947 | Pfleiderer | 0.00 | 1.00 | 0.10 | n.d. | Pumps |
1951 | Davis, Kottas & Moody | 0.00 | 1.00 | Variable | n.d. | All turbomachines |
1954 | Hutton | 0.30 | 0.70 | 0.20 | n.d. | Kaplan turbines |
1958 | Rotzoll | 0.00 | 1.00 | Variable | n.d. | Pumps |
1960 | Wiesner | 0.50 | 0.50 | 0.10 | 5 × 104 ÷ 5 × 105 | Radial compressors |
Fauconnet | 0.24 | 0.76 | 0.20 | |||
1961 | O’Neil & Wickli | 0.00 | 1.00 | Variable | n.d. | Radial compressors |
1965 | ASME Code PTC-10 | 0.00 | 1.00 | 0.20 | n.d. | Axial compressors |
0.00 | 1.00 | 0.10 | Radial compressors | |||
1971 | Mashimo et al. | 0.25 min | 0.75 max | 0.20 | n.d. | Radial compressors |
1974 | Mashimo et al. | 0.15–0.57 | 0.43–0.85 | 0.20–0.50 | n.d. | Radial compressors |
3. Calculation of the Correlation Coefficients
Compressors | |||||||
---|---|---|---|---|---|---|---|
Scale | D2 (mm) | b2 (mm) | ω (rad/s) | ns | ds | ηst (%) | Re/Reref |
1:10 | 9.2 | 0.55 | 68745 | 0.38 | 5.7 | 52.5 | 1/10 |
1:3 | 27.6 | 1.65 | 65217 | 0.39 | 6.2 | 57.9 | 3/10 |
1:5 | 46.0 | 2.75 | 63376 | 0.40 | 7.2 | 59.2 | 2/10 |
1:1 | 92.0 | 5.50 | 57160 | 0.42 | 8.1 | 60.6 | 1 |
Turbine | |||||||
Scale | D1 (mm) | b1 (mm) | ω (rad/s) | ns | ds | ηst (%) | Re/Reref |
1:10 | 12.7 | 1.31 | 68745 | 0.50 | 3.99 | 75.0 | 1/10 |
1:3 | 38.1 | 3.93 | 65217 | 0.47 | 4.20 | 76.1 | 3/10 |
1:5 | 63.5 | 6.55 | 63376 | 0.46 | 4.32 | 79.3 | 2/10 |
1:1 | 127 | 13.10 | 57160 | 0.41 | 4.91 | 80.0 | 1 |
- (1)
- Calculate x, the midpoint of the interval, x = 0.5·(p + q);
- (2)
- Calculate the function value at the midpoint, f(x);
- (3)
- If convergence is satisfactory (that is, p − x is sufficiently small, or f(x) is sufficiently small), return x and stop iterating;
- (4)
- Examine the sign of f(x) and replace either (p, f(p)) or (q, f(q)) with (x, f(x)) so that there is a zero crossing within the new interval.
4. Results of the Computational Fluid Dynamics Validation
5. Validation of the Correlation on Commercial Ultra-Micro Machines
- (1)
- Based on the compressor/turbine map of the selected machine, calculate Re with Equation (2);
- (2)
- With the design value for the rpm, calculate the specific speed to enter the map;
- (3)
- Draw on the map the curve for that specific speed “ns”, and extrapolate for every value of the flow rate, the compression/expansion ratio and the efficiency;
- (4)
- Calculate the corrected flow rate accordingly;
- (5)
- Use Equation (3) to calculate the reference efficiency ηref;
- (6)
- Use ηref to calculate, with Equation (5), the polytropic efficiency ηpol;
- (7)
- Calculate the polytropic work;
- (8)
- Finally, compute ns and ds.
5.1. Compressor Maps
5.2. Turbine Maps
6. The Proposed Operational Approach for the Scaling-Down of the Balje Maps
- (1)
- On the basis of the actual design data, identify an optimal value of ns/ds for radial compressors/turbines on the Balje chart;
- (2)
- Calculate the Re of the micro-scale machine;
- (3)
- Introduce an efficiency factor <1 given by Equation (5) that multiplies the total-to-static efficiency extracted from the chart;
- (4)
- Calculate the Euler work from the design data and extract the angular velocity ω and the impeller diameter D from ns and ds;
- (5)
- Check that U < Umax (the allowable material stress limit for ultra-micro components may be higher): if this constraint is not abided by, choose a different ns/ds pair and go back to Step (1);
- (6)
- If the computed Re is different from the value assumed in Step (3), go back to Step (3), substitute the old Re number with the new one, and iterate.
7. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Nomenclature
a | Constant in Equation (4) |
b | Impeller width (m) |
c, c' | Constant in Equation (4) |
D | Impeller diameter (m) |
ds | Specific diameter |
h | Enthalpy (kJ/kg) |
k | Turbulent kinetic energy |
n | Semi-empirical exponent in Equation (3) |
ns | Specific speed |
Q | Flow rate (m3/s) |
Re | Reynolds number |
U | Peripheral velocity (m/s) |
u' | slip velocity (m/s) |
V | Absolute velocity (m/s) |
Greek Symbols
η | Efficiency |
ν | Kinematic viscosity [m/s2] |
ρ | Density (kg/m3) |
ω | Angular velocity [rad/s] |
Subscripts
1 | Inlet |
2 | Outlet |
e | External |
hyd | hydraulic |
p | Polytropic |
ref | Reference |
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Capata, R.; Sciubba, E. Experimental Fitting of the Re-Scaled Balje Maps for Low-Reynolds Radial Turbomachinery. Energies 2015, 8, 7986-8000. https://doi.org/10.3390/en8087986
Capata R, Sciubba E. Experimental Fitting of the Re-Scaled Balje Maps for Low-Reynolds Radial Turbomachinery. Energies. 2015; 8(8):7986-8000. https://doi.org/10.3390/en8087986
Chicago/Turabian StyleCapata, Roberto, and Enrico Sciubba. 2015. "Experimental Fitting of the Re-Scaled Balje Maps for Low-Reynolds Radial Turbomachinery" Energies 8, no. 8: 7986-8000. https://doi.org/10.3390/en8087986