# An AI-Based Fast Design Method for New Centrifugal Compressor Families

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Step 1: 1D Single-Zone Model Implementation and Validation

#### 2.2. Step 2: ANOVA

#### 2.3. Step 3: Sobol Sequence

#### 2.4. Step 4: Artificial Neural Network

#### 2.5. Step 5: Validation of Promising Solutions through CFD Analyses

## 3. Results and Discussion

#### 3.1. Step 1: Validation of 1D Single-Zone Model

#### 3.2. Step 2: ANOVA Results

#### 3.3. Step 3: Sobol Sequence

#### 3.4. Step 4: Artificial Neural Network

#### 3.5. Step 5: Validation of Promising Solutions through CFD Analyses

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | Area |

$a$ | Velocity of sound |

$b$ | Blade height |

$c$ | Absolute velocity |

${c}_{f}$ | Friction coefficient |

${C}_{p}$ | Specific heat ratio |

$D$ | Diameter |

${D}_{f}$ | Diffusion factor |

${f}_{inc}$ | Incidence factor |

$H$ | Entalpy |

$h$ | Specific entalpy |

$k$ | Specific heat ratio |

$L$ | Blade length |

$\dot{m}$ | Mass flow rate |

${M}_{u}$ | Peripheral Mach number |

$P$ | Pressure |

$Q$ | Volume flow rate |

$r$ | Radius |

$s$ | Specific entropy |

$T$ | Temperature |

$t$ | Thickness |

$U$ | Peripheral velocity |

$w$ | Relative velocity |

$Z$ | Effective number of blades |

$\Delta $ | Variation |

$\alpha $ | Absolute angle |

$\beta $ | Relative angle/pressure ratio |

$\gamma $ | Blade slope angle |

$\epsilon $ | Wake fraction of blade-to-blade space |

$\eta $ | Polytropic efficiency |

$\tau $ | Work coefficient |

$\varphi $ | Flow coefficient |

$\psi $ | Polytropic head |

Subscripts | |

0 | Total quantity |

1 | Impeller inlet |

2 | Impeller outlet |

3 | Diffuser outlet |

4 | Volute outlet |

bl | Blade |

bld | Blade loading |

ch | Choke |

cl | Clearance |

df | Disc friction |

e | Exit cone |

h | hub |

hs | Hub-to-shroud distortion |

hyd | Hydraulic |

in | incidence |

LE | Leading edge |

lk | leakage |

m | Meridional component |

mix | mixing |

p | Polytropic |

rc | Recirculation |

s | Shroud |

sf | Skin friction |

spl | Splitter |

TE | Trailing edg |

th | Throat |

tt | Total to total |

u | Circumferential component |

vcv | Volute circumferential velocity |

vld | Vaneless diffuser |

vmv | Volute meridional velocity |

vsf | Volute skin friction |

Superscripts | |

* | Sonic condition/scaling factor |

- | Average |

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**Figure 1.**Centrifugal compressor stage (

**a**) and its thermodynamic transformation on the h-s plane (

**b**).

**Figure 3.**Comparison between 1D model predictions, CFD, and experimental data (with error bars) in terms of ${\beta}_{tt}$ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

**Figure 4.**Comparison between 1D model predictions, CFD, and experimental data (with error bars) in terms of ${\eta}_{p}$ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

**Figure 5.**Comparison between 1D model predictions, CFD, and experimental data (with error bars) in terms of $\tau $ at the impeller outlet (

**section 2**).

**Figure 6.**Comparison between 1D model predictions, CFD, and experimental data (with error bars) in terms of $\psi $ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

**Figure 7.**Loss source breakdown evaluated with the 1D model (

**left**) and CFD (

**right**) for the low flow stage.

**Figure 8.**Loss source breakdown evaluated with the 1D model (

**left**) and CFD (

**right**) for the medium flow stage.

**Figure 9.**Loss source breakdown evaluated with the 1D model (

**left**) and CFD (

**right**) for the high flow stage.

**Figure 10.**ANOVA main effect plot of ${\eta}_{p}$ for low (

**blue**), medium (

**gray**), and high (

**black**) flow stage.

**Figure 11.**ANOVA main effect plot of $\tau $ for low (

**blue**), medium (

**gray**), and high (

**black**) flow stage.

**Figure 12.**Comparison of 1D prediction (

**blue dots**) and ANN forecast (

**red dots**) under stall, design, and choke conditions for the low flow designed impeller.

**Figure 13.**Comparison of 1D prediction (

**blue dots**) and ANN forecast (

**red dots**) under stall, design, and choke conditions for the medium flow designed impeller.

**Figure 14.**Comparison of 1D prediction (

**blue dots**) and ANN forecast (

**red dots**) under stall, design, and choke conditions for the high flow designed impeller.

**Figure 15.**New geometries investigated with ANN (

**blue dots**) in stall, design, and choke conditions and Pareto front (

**red circles**) for the low flow designed impeller.

**Figure 16.**New geometries investigated with ANN (

**blue dots**) in stall, design, and choke conditions and Pareto front (

**red circles**) for the medium flow designed impeller.

**Figure 17.**New geometries investigated with ANN (

**blue dots**) in stall, design, and choke conditions and Pareto front (

**red circles**) for the high flow designed impeller.

**Figure 18.**Comparison between old LNG and new EC geometries evaluated with the 1D model and CFD in terms of ${\beta}_{tt}$ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

**Figure 19.**Comparison between old LNG and new EC geometries evaluated with the 1D model and CFD in terms of ${\eta}_{p}$ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

**Figure 20.**Comparison between old LNG and new EC geometries evaluated with the 1D model and CFD in terms of τ at the impeller (

**section 2**) outlet.

**Figure 21.**Comparison between old LNG and new EC geometries evaluated with the 1D model and CFD in terms of $\psi $ at the impeller (

**section 2**), diffuser (

**section 3**), and volute (

**section 4**) outlet.

Component | Loss Source | Loss Correlation * | Refs. |
---|---|---|---|

Impeller(Internal losses) | Incidence | $\Delta {h}_{in}={f}_{inc}{w}_{1}^{2}{\mathrm{sin}}^{2}\left({\beta}_{1}-{\beta}_{1,bl}\right)={f}_{inc}\frac{{w}_{u1}^{2}}{2}$ $\mathrm{with}{f}_{inc}=0.5\xf70.7$ | [60] |

Blade-loading | $\Delta {h}_{bld}=0.05{D}_{f}^{2}{U}_{2}^{2}$ with ${D}_{f}=1-\frac{{w}_{2}}{{w}_{s1}}+\frac{0.75\left[\left({U}_{2}{c}_{u2}-{U}_{1}{c}_{1}\right)/{U}_{2}^{2}\right]}{\frac{{w}_{s1}}{{w}_{2}}\left[\left(\frac{Z}{\pi}\right)\left(1-\frac{{D}_{s1}}{{D}_{2}}\right)+2\frac{{D}_{s1}}{{D}_{2}}\right]}$ | [61] | |

Mixing | $\Delta {h}_{mix}=0.5\frac{{c}_{2}^{2}}{{\mathrm{cos}}^{2}{\alpha}_{2}}{\left(\frac{1-\epsilon -\frac{{b}_{3}}{{b}_{2}}}{1-\epsilon}\right)}^{2}$ | [62,63] | |

Skin-friction | $\Delta {h}_{sf}=\frac{2{c}_{f}{L}_{bl}}{{D}_{h}}{\overline{w}}^{2}$$\mathrm{with}\overline{w}=\frac{2{w}_{2}+{w}_{s1}+{w}_{h1}}{4}$ | [54] | |

Choke | $\Delta {h}_{ch}=0.5\left(0.05X+{X}^{7}\right){w}_{1}^{2}$$\mathrm{with}X=11-10{c}_{r}{A}_{th}/{A}^{*}Z$ | [34] | |

Hub-to-shroud distortion | $\Delta {h}_{hs}=\frac{1}{12}{\left(\frac{{\gamma}_{TE}-{\gamma}_{LE}}{L}\right)}^{2}{\left(\frac{{w}_{1rms}+{w}_{2}}{2}\right)}^{2}{\left(\frac{{b}_{1}+{b}_{2}}{2}\right)}^{2}$ | [34] | |

Impeller(External losses) | Disc-friction | $\Delta {h}_{df}={f}_{df}\frac{\overline{\rho}{r}_{2}^{2}{U}_{2}^{3}}{4\dot{m}}$$\mathrm{with}{f}_{df}=f\left(\frac{{U}_{2}{r}_{2}}{{v}_{2}}\right)$ | [64] |

Leakage | $\Delta {h}_{lk}=\frac{{\dot{m}}_{cl}{U}_{cl}{U}_{2}}{2\dot{m}}$ | [34] | |

Recirculation | $\Delta {h}_{rc}=8\xb7{10}^{-5}\mathrm{sinh}\left(3.5{\alpha}_{2}^{3}\right){D}_{f}^{2}{U}_{2}^{2}$ | [35] | |

Diffuser | Vaneless diffuser | $\Delta {h}_{vld}={C}_{p}{T}_{02}\left[{\left({P}_{3}/{P}_{03}\right)}^{\frac{\gamma -1}{\gamma}}-{\left({P}_{3}/{P}_{02}\right)}^{\frac{\gamma -1}{\gamma}}\right]$ | [56] |

Volute | Circumferential velocity | $\Delta {h}_{vcv}=0.25\left({\mathrm{c}}_{u3}^{2}-{\mathrm{c}}_{4}^{2}\right)$$\mathrm{if}{\mathrm{c}}_{u3}{\mathrm{r}}_{3}/{\mathrm{c}}_{4}{\mathrm{r}}_{4}\ge 1$ $\mathrm{else}\Delta {h}_{vcv}=0.5{\left({c}_{u3}-{\mathrm{c}}_{4}\right)}^{2}$ | [57,58] |

Meridional velocity | $\Delta {h}_{vmv}=0.5{c}_{m3}^{2}$ | [57,58] | |

Skin-friction | $\Delta {h}_{vsf}=0.5{c}_{f}L{\overline{{c}_{u}}}^{2}/{D}_{hyd}$ | [57,58] | |

Exit-cone | $\Delta {h}_{e}={k}_{e}{\left({c}_{4}-{c}_{5}\right)}^{2}/2g$ | [57,58] |

Grid | No. of Elements | Polytropic Efficiency | Work Coefficient | ||
---|---|---|---|---|---|

Value | Error with G5 (%) | Value | Error with G5 (%) | ||

G1 | 2.0 million | 0.989 | −1.1 | 1.008 | 0.8 |

G2 | 2.5 million | 0.994 | −0.6 | 1.005 | 0.5 |

G3 | 3.0 million | 0.997 | −0.3 | 1.002 | 0.2 |

G4 | 3.5 million | 1.000 | 0.0 | 1.000 | 0.0 |

G5 | 4.0 million | 1.000 | - | 1.000 | - |

Numerical Setup | |
---|---|

Type of analysis | RANS with adiabatic walls |

Type of grid | H-type |

No. of Elements | 3.5 million |

Discretization of convective fluxes | 2nd order TVD-MUSCL with Roe’s upwind scheme |

Discretization of viscous fluxes | Central difference scheme |

Turbulence closure | Wilcox’s k-ω model |

Parallelization | Hybrid OpenMP/MPI architecture |

Wall treatment | Wall resolution without wall functions |

Near wall grid refinement | First element of 2.8 × 10^{−5} mm (y+ ≤ 1) |

**Table 4.**Geometric parameters adopted as independent variables in the ANOVA and their range of variation.

Parameter | Range of Variation (%) | ANOVA Admissible Value (%) |
---|---|---|

$\mathrm{Impeller}\mathrm{inlet}\mathrm{hub}\mathrm{diameter}{\mathit{D}}_{1\mathit{h}}$ | Constant | - |

$\mathrm{Impeller}\mathrm{inlet}\mathrm{shroud}\mathrm{diameter}{\mathit{D}}_{1\mathit{s}}$ | (−5.0; 5.0) | −5.0; 0.0; 5.0 |

$\mathrm{Blade}\mathrm{thickness}\mathit{t}$ | (−19.0; 19.0) | −19.0; 0.0; 19.0 |

$\mathrm{Outlet}\mathrm{impeller}\mathrm{diameter}{\mathit{D}}_{2}$ | Constant | - |

$\mathrm{Outlet}\mathrm{impeller}\mathrm{width}{\mathit{b}}_{2}$ | (−16.0; 16.0) | −16.0; 0.0; 16.0 |

$\mathrm{Hub}\mathrm{inlet}\mathrm{blade}\mathrm{angle}{\mathit{\beta}}_{1\mathit{h}}$ | (−7.0; 7.0) | −7.0; 0.0; 7.0 |

$\mathrm{Shroud}\mathrm{inlet}\mathrm{blade}\mathrm{angle}{\mathit{\beta}}_{1\mathit{s}}$ | (−6.0; 6.0) | −6.0; 0.0; 6.0 |

$\mathrm{Outlet}\mathrm{blade}\mathrm{angle}{\mathit{\beta}}_{2}$ | (−8.0; 8.0) | −8.0; 0.0; 8.0 |

$\mathrm{LE}\mathrm{slope}{\mathit{\gamma}}_{\mathit{L}\mathit{E}}$ | (−14.0; 14.0) | −14.0; 0.0; 14.0 |

$\mathrm{Diffuser}\mathrm{pinch}{\mathit{b}}_{3}/{\mathit{b}}_{2}$ | (−6.0; 6.0) | −6.0; 0.0; 6.0 |

**Table 5.**ANN absolute error in prediction of polytropic efficiency ${\u03f5}_{{\eta}_{p}}$ and work coefficient ${\u03f5}_{\tau}$.

Stall | Design | Choke | ||||
---|---|---|---|---|---|---|

${\mathsf{\u03f5}}_{{\mathsf{\eta}}_{\mathit{p}}}$ | ${\mathsf{\u03f5}}_{\mathsf{\tau}}$ | ${\mathsf{\u03f5}}_{{\mathsf{\eta}}_{\mathit{p}}}$ | ${\mathsf{\u03f5}}_{\mathsf{\tau}}$ | ${\mathsf{\u03f5}}_{{\mathsf{\eta}}_{\mathit{p}}}$ | ${\mathsf{\u03f5}}_{\mathsf{\tau}}$ | |

Low flow stage | 0.1% | 0.2% | 0.2% | 0.2% | 0.8% | 0.3% |

Medium flow stage | 0.2% | 0.1% | 0.2% | 0.1% | 0.5% | 0.3% |

High flow stage | 0.2% | 0.1% | 0.2% | 0.2% | 1.0% | 0.3% |

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**MDPI and ACS Style**

Bicchi, M.; Biliotti, D.; Marconcini, M.; Toni, L.; Cangioli, F.; Arnone, A. An AI-Based Fast Design Method for New Centrifugal Compressor Families. *Machines* **2022**, *10*, 458.
https://doi.org/10.3390/machines10060458

**AMA Style**

Bicchi M, Biliotti D, Marconcini M, Toni L, Cangioli F, Arnone A. An AI-Based Fast Design Method for New Centrifugal Compressor Families. *Machines*. 2022; 10(6):458.
https://doi.org/10.3390/machines10060458

**Chicago/Turabian Style**

Bicchi, Marco, Davide Biliotti, Michele Marconcini, Lorenzo Toni, Francesco Cangioli, and Andrea Arnone. 2022. "An AI-Based Fast Design Method for New Centrifugal Compressor Families" *Machines* 10, no. 6: 458.
https://doi.org/10.3390/machines10060458