# A Numerical Study on Blade Design and Optimization of a Helium Expander for a Hydrogen Liquefaction Plant

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## Abstract

**:**

## 1. Introduction

## 2. Operation Conditions

## 3. Expander Design

#### 3.1. Preliminary Design

#### 3.2. Meanline Design

#### 3.3. 3D Shape Generation

#### 3.4. Numerical Analysis

## 4. Optimization

#### 4.1. Optimization Methodology

_{adj}(correction coefficient of determination) of the response surface function. That is, if R2

_{adj}> 0.9, the response surface function is very accurate, and if 0.9 > R2

_{adj}> 0.7, the response surface function is relatively accurate [15,16].

#### 4.2. Optimization of the Expander

#### 4.3. Optimization Results at Design Point

#### 4.4. Comparison of Performance Curve

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DOE | Design of experiments |

DP | Design point |

PS | Pressure side |

RSM | Response surface method |

RPM | Revolutions per minute |

SS | Suction side |

TPD | Tons per day |

Nomenclature | |

α | Absolute flow angle [°] |

β | Blade beta angle, relative flow angle [°] |

C | Absolute flow velocity [m/s] |

d | Rotor diameter [mm] |

D_tip | Rotor tip diameter [mm] |

Ds | Specific diameter |

Δ | Difference |

η | Efficiency |

h | Enthalpy [J/kg] |

Hx | Heat exchanger |

In | Inlet |

Ns | Specific speed |

Out | Outlet |

Ω | Revolution speed [rad/sec] |

Pr | Pressure ratio |

Q | Flow rate [㎥/sec] |

R | Radial |

S | Entropy [J/kg-K] |

T | Temperature [K] |

U | Rotor tip velocity [m/s] |

W | Relative velocity [m/s] |

Z | Axial |

Subscript | |

0 | Volute |

1 | Nozzle inlet |

2 | Nozzle throat |

3 | Nozzle outlet |

4 | Rotor inlet |

5 | Rotor throat |

6 | Rotor outlet |

Isen | Isentropic |

T | Total |

TT | Total to total |

## References

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**Figure 1.**The 0.5-TPD class hydrogen liquefaction plant: (

**a**) schematic of hydrogen liquefaction plant and (

**b**) T–s diagram of hydrogen liquefaction plant.

**Figure 3.**Velocity triangles at design points: (

**a**) rotor inlet (incidence angle −26.74°) and (

**b**) rotor outlet (deviation angle 22.8°).

**Figure 4.**Performance prediction curves from meanline analysis: (

**a**) expansion ratio vs. mass flow rate; (

**b**) expansion ratio vs. efficiency; (

**c**) expansion ratio vs. power; (

**d**) expansion ratio vs. expander outlet temperature.

**Figure 7.**Performance prediction curves from numerical analysis: (

**a**) expansion ratio vs. mass; (

**b**) expansion ratio vs. efficiency; (

**c**) expansion ratio vs. power; (

**d**) expansion ratio vs. expander outlet temperature.

**Figure 11.**Isentropic total-to-total efficiency of expander by shape optimization without assuming the extra mechanical losses.

**Figure 15.**Comparison of entropy distribution at mid-span of the rotor (Plane 1): (

**a**) original model and (

**b**) optimized model.

**Figure 16.**Comparison of entropy distribution at Plane 2: (

**a**) original model and (

**b**) optimized model.

**Figure 17.**Comparison of relative velocity distribution at Plane 1: (

**a**) original model and (

**b**) optimized model.

**Figure 18.**Comparison of relative velocity distribution at Plane 2: (

**a**) original model and (

**b**) optimized model.

**Figure 19.**Performance comparisons between the original model and optimized model: (

**a**) expansion ratio vs. efficiency; (

**b**) expansion ratio vs. expander outlet temperature; (

**c**) expansion ratio vs. expander power.

Process @ T–s Diagram | Component |
---|---|

①-② | Heat exchanger 1 |

②-③ | Heat exchanger 2 |

③-④ | Expander 1 |

④-⑤ | Heat exchanger 3 |

⑤-⑥ | Expander 2 |

⑥-⑦ | Heat exchanger 4 |

⑦-⑧ | Heat exchanger 3 |

⑧-⑨ | Heat exchanger 2 |

⑨-⑩ | Heat exchanger 1 |

⑩-① | Compressor with cooler |

Parameter | Unit | Expander 1 (③–④) | Expander 2 (⑤–⑥) |
---|---|---|---|

Pt_in | bar | 9.76 | 5.98 |

Tt_in | K | 49.1 | 24.8 |

Pt_out | bar | 6.1 | 1.62 |

Tt_out | K | 42.8 | 17 |

Pr | - | 1.6 | 3.69 |

Efficiency | % | 74.7 | 76.3 |

Power | kW | 5.32 | 6.2 |

Ns | Unit | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 0.392 |
---|---|---|---|---|---|---|---|---|---|---|---|

Efficiency | - | 0.76 | 0.81 | 0.85 | 0.87 | 0.87 | 0.84 | 0.8 | 0.72 | 0.61 | 0.85 |

RPM | - | 38,191 | 57,286 | 76,382 | 95,477 | 114,573 | 133,668 | 152,764 | 171,859 | 190,955 | 75,000 |

Power | kW | 5.4 | 5.7 | 6.0 | 6.1 | 6.1 | 6.0 | 5.6 | 5.1 | 4.3 | 6.0 |

Ds | - | 10.0 | 6.67 | 5.0 | 4.0 | 3.33 | 2.86 | 2.5 | 2.22 | 2.0 | 5.09 |

D_tip | mm | 105.59 | 70.39 | 52.8 | 42.24 | 35.2 | 30.17 | 26.4 | 23.46 | 21.12 | 53.77 |

U_tip | m/s | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 | 211.15 |

Parameter | Unit | Dimension |
---|---|---|

RPM | - | 75,000 |

Number of nozzles | - | 17 |

Nozzle inlet radius (N1) | mm | 36.5 |

Nozzle outlet radius (N2) | mm | 29.5 |

Nozzle height | mm | 3.3 |

Nozzle outlet blade angle | ° | 76.21 |

Number of rotors | - | 14 |

Rotor inlet radius (R1) | mm | 26 |

Rotor outlet shroud radius (R2) | mm | 13.2 + 0.3 |

Rotor outlet hub radius (R3) | mm | 8.4 |

Rotor outlet blade angle | ° | −60 |

Cycle | Design Parameters | Metamodel |
---|---|---|

1 | blade $\beta $ angle | $U=0.872630-0.000602{\beta}_{H3}-0.001696{\beta}_{H4}+0.000415{\beta}_{S3}+0.000553{\beta}_{S4}\phantom{\rule{0ex}{0ex}}+0.000022{\beta}_{H3}{\beta}_{H3}+0.000134{\beta}_{H4}{\beta}_{H4}+0.000015{\beta}_{S3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}+0.000084{\beta}_{S4}{\beta}_{S4}-0.000050{\beta}_{H3}{\beta}_{H4}-0.000017{\beta}_{H3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}-0.000003{\beta}_{H3}{\beta}_{S4}-0.000040{\beta}_{H4}{\beta}_{S3}-0.000188{\beta}_{H4}{\beta}_{S4}\phantom{\rule{0ex}{0ex}}+0.000112{\beta}_{S3}{\beta}_{S4}$ |

2 | meridional contours | $U=0.875270-0.003063{Z}_{H2}-0.001480{R}_{H3}+0.003210{Z}_{S2}+0.001197{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000112{Z}_{H2}{Z}_{H2}-0.000010{R}_{H3}{R}_{H3}-0.000145{Z}_{S2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}-0.000180{R}_{S3}{R}_{S3}+0.000207{Z}_{H2}{R}_{H3}+0.000153{Z}_{H2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}-0.000100{Z}_{H2}{R}_{S3}-0.000222{R}_{H3}{Z}_{S2}-0.000020{R}_{H3}{R}_{S3}\phantom{\rule{0ex}{0ex}}+0.000170{Z}_{S2}{R}_{S3}$ |

3 | blade $\beta $ angle | $U=0.884060-0.000268{\beta}_{H3}-0.001175{\beta}_{H4}+0.000102{\beta}_{S3}+0.000020{\beta}_{S4}\phantom{\rule{0ex}{0ex}}-0.000064{\beta}_{H3}{\beta}_{H3}-0.000192{\beta}_{H4}{\beta}_{H4}-0.000044{\beta}_{S3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}-0.000162{\beta}_{S4}{\beta}_{S4}-0.000140{\beta}_{H3}{\beta}_{H4}+0.000080{\beta}_{H3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}+0.000095{\beta}_{H3}{\beta}_{S4}+0.000055{\beta}_{H4}{\beta}_{S3}+0.000190{\beta}_{H4}{\beta}_{S4}\phantom{\rule{0ex}{0ex}}-0.000060{\beta}_{S3}{\beta}_{S4}$ |

4 | meridional contours | $U=0.885210-0.002624{Z}_{H2}-0.001767{R}_{H3}+0.002823{Z}_{S2}+0.001001{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000214{Z}_{H2}{Z}_{H2}+0.000050{R}_{H3}{R}_{H3}-0.000225{Z}_{S2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}-0.000099{R}_{S3}{R}_{S3}+0.000005{Z}_{H2}{R}_{H3}+0.000242{Z}_{H2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}+0.000095{Z}_{H2}{R}_{S3}-0.000023{R}_{H3}{Z}_{S2}-0.000022{R}_{H3}{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000050{Z}_{S2}{R}_{S3}$ |

5 | blade $\beta $ angle | $U=0.892050+0.000054{\beta}_{H3}-0.000543{\beta}_{H4}-0.000093{\beta}_{S3}-0.000231{\beta}_{S4}\phantom{\rule{0ex}{0ex}}-0.000071{\beta}_{H3}{\beta}_{H3}-0.000583{\beta}_{H4}{\beta}_{H4}+0.000030{\beta}_{S3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}-0.000016{\beta}_{S4}{\beta}_{S4}-0.000507{\beta}_{H3}{\beta}_{H4}+0.000073{\beta}_{H3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}+0.000067{\beta}_{H3}{\beta}_{S4}+0.000132{\beta}_{H4}{\beta}_{S3}+0.000340{\beta}_{H4}{\beta}_{S4}\phantom{\rule{0ex}{0ex}}+0.000005{\beta}_{S3}{\beta}_{S4}$ |

6 | meridional contours | $U=0.892940-0.001352{Z}_{H2}-0.001308{R}_{H3}+0.001674{Z}_{S2}+0.000508{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000555{Z}_{H2}{Z}_{H2}-0.000188{R}_{H3}{R}_{H3}-0.000321{Z}_{S2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}-0.000192{R}_{S3}{R}_{S3}-0.000210{Z}_{H2}{R}_{H3}+0.000367{Z}_{H2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}+0.000183{Z}_{H2}{R}_{S3}+0.000200{R}_{H3}{Z}_{S2}+0.000153{R}_{H3}{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000165{Z}_{S2}{R}_{S3}$ |

7 | blade $\beta $ angle | $U=0.895790+0.000402{\beta}_{H3}+0.000905{\beta}_{H4}-0.000169{\beta}_{S3}-0.000332{\beta}_{S4}\phantom{\rule{0ex}{0ex}}-0.000215{\beta}_{H3}{\beta}_{H3}-0.001522{\beta}_{H4}{\beta}_{H4}-0.000081{\beta}_{S3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}-0.000123{\beta}_{S4}{\beta}_{S4}-0.001088{\beta}_{H3}{\beta}_{H4}+0.000100{\beta}_{H3}{\beta}_{S3}\phantom{\rule{0ex}{0ex}}+0.000218{\beta}_{H3}{\beta}_{S4}+0.000233{\beta}_{H4}{\beta}_{S3}+0.000660{\beta}_{H4}{\beta}_{S4}\phantom{\rule{0ex}{0ex}}-0.000065{\beta}_{S3}{\beta}_{S4}$ |

8 | meridional contours | $U=0.895840-0.000051{Z}_{H2}-0.000798{R}_{H3}+0.000898{Z}_{S2}+0.000019{R}_{S3}\phantom{\rule{0ex}{0ex}}-0.000257{Z}_{H2}{Z}_{H2}-0.000155{R}_{H3}{R}_{H3}-0.000118{Z}_{S2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}+0.000011{R}_{S3}{R}_{S3}-0.000070{Z}_{H2}{R}_{H3}+0.000135{Z}_{H2}{Z}_{S2}\phantom{\rule{0ex}{0ex}}-0.000023{Z}_{H2}{R}_{S3}+0.000120{R}_{H3}{Z}_{S2}+0.000060{R}_{H3}{R}_{S3}\phantom{\rule{0ex}{0ex}}+0.000030{Z}_{S2}{R}_{S3}$ |

Expansion Ratio | Efficiency | ΔEfficiency [%] * | Outlet Temperature [K] | ΔTemperature [%] * | ||
---|---|---|---|---|---|---|

Original | Optimized | Original | Optimized | |||

1.74 | 0.750 | 0.764 | 1.9 | 41.306 | 41.167 | 0.34 |

1.67 | 0.752 | 0.767 | 1.94 | 41.813 | 41.674 | 0.33 |

1.59 (DP) | 0.754 | 0.769 | 1.98 | 42.415 | 42.281 | 0.32 |

1.49 | 0.758 | 0.768 | 1.32 | 43.252 | 43.161 | 0.21 |

1.40 | 0.752 | 0.752 | 0.08 | 44.164 | 44.144 | 0.05 |

1.35 | 0.733 | 0.734 | 0.1 | 44.73 | 44.718 | 0.03 |

1.34 | 0.723 | 0.725 | 0.15 | 44.973 | 44.948 | 0.06 |

1.30 | 0.690 | 0.685 | −0.73 | 45.528 | 45.526 | 0.00 |

1.24 | 0.586 | 0.555 | −5.21 | 46.564 | 46.66 | −0.21 |

1.21 | 0.466 | 0.419 | −10.07 | 47.319 | 47.488 | −0.36 |

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## Share and Cite

**MDPI and ACS Style**

Lim, H.; Seo, J.; Park, M.; Choi, B.; Park, J.; Bang, J.; Lee, D.; Kim, B.; Kim, S.; Lim, Y.; Alford, A. A Numerical Study on Blade Design and Optimization of a Helium Expander for a Hydrogen Liquefaction Plant. *Appl. Sci.* **2022**, *12*, 1411.
https://doi.org/10.3390/app12031411

**AMA Style**

Lim H, Seo J, Park M, Choi B, Park J, Bang J, Lee D, Kim B, Kim S, Lim Y, Alford A. A Numerical Study on Blade Design and Optimization of a Helium Expander for a Hydrogen Liquefaction Plant. *Applied Sciences*. 2022; 12(3):1411.
https://doi.org/10.3390/app12031411

**Chicago/Turabian Style**

Lim, Hyungsoo, Jeongmin Seo, Mooryong Park, Bumseog Choi, Junyoung Park, Jesung Bang, Donghyun Lee, Byungock Kim, Soowon Kim, Youngchul Lim, and Adrian Alford. 2022. "A Numerical Study on Blade Design and Optimization of a Helium Expander for a Hydrogen Liquefaction Plant" *Applied Sciences* 12, no. 3: 1411.
https://doi.org/10.3390/app12031411