# Multi-Point Surrogate-Based Approach for Assessing Impacts of Geometric Variations on Centrifugal Compressor Performance

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) systems, seem to be interesting solutions for improving the efficiency of energy cycles and mitigating the effects of climate change [4]. Furthermore, an ever-growing use of centrifugal compressors in sCO2 [5], organic Rankine cycle (ORC) [6], and hydrogen [7] plants emphasizes how these turbomachines are pivotal for the current energy transition [8]. In this context, the rapid and continuous evolution of markets is forcing centrifugal compressor designers to maximize the performance of these machines in ever shorter timescales [9]. The scientific literature has recently shown an increased interest in optimization techniques for turbomachine designs [10,11]. Considering the preliminary design of centrifugal compressors, several optimization techniques were combined with low order models. To this end, Du et al. [12] used a mono-dimensional (1D) model and a genetic algorithm (GA) for the optimization of a sCO

_{2}centrifugal compressor, whereas Bicchi et al. [11] provided a design method based on artificial intelligences (AI) and a 1D single-zone model for fast development of new centrifugal compressor families. Similarly, Massoudi et al. [13] defined an approach for designing centrifugal compressors by means of the combined use of a 1D model and an artificial neural network (ANN). Another example was provided by Li et al. [14], who optimized a low-pressure centrifugal compressor by combining a simulated annealing algorithm with a 1D model. Finally, Wang et al. [15] showed the use of a 1D model and a GA for designing a sCO

_{2}centrifugal compressor. However, the scientific literature does not only report examples of preliminary design optimization. Indeed, advanced three-dimensional (3D) optimization techniques are also provided, while computational fluid dynamics (CFD) often provide a higher level of accuracy in optimizing performance. Omini et al. [16] proposed a hybrid design procedure of a new centrifugal compressor based on CFDs and GA. Ekradi and Madadi [17] presented a procedure for the 3D optimization of a transonic centrifugal compressor impeller with splitter blades by integrating GA, ANN, and a CFD solver. Finally, Ma et al. [18] developed an AI framework to achieve multi-objective optimization of the centrifugal compressor impeller.

## 2. Materials and Methods

#### 2.1. Step 1: Selection of Geometric Parameters

#### 2.2. Step 2: Parametrization of Centrifugal Compressor Stage and Parametric Analysis

#### 2.3. Step 3: Feed, Train, and Validation of Artificial Neural Network

## 3. Results and Discussion

#### 3.1. Experimental Validation

#### 3.2. Results of the Proposed Approach

_{p}/η

_{p}*) reached a maximum value of 1.01 (a relative +1% percentage increase from baseline), with a flow coefficient (ϕ/ϕ*) shifted to 1.09 (+9% from baseline). For the same perturbed geometry, the stall condition gains a η

_{p}/η

_{p}* value of 0.99 (+2.1%) with ϕ/ϕ* of 0.93 (+10.7%), whereas, for the choke condition, a value of 0.91 (+1.1%) is obtained for η

_{p}/η

_{p}* with a ϕ/ϕ* of 1.58 (+5.3%). From these results, since the CFD analyses were performed with an imposed pressure ratio, it can be stated that this perturbed geometry is suitable for working with higher mass flow rates. Indeed, the same pressure ratios are reached in this stage with higher mass flow rates compared to the baseline. Therefore, in case this perturbed geometry will work with the same mass flow rates as the baseline stage, a reduction in polytropic efficiency will be obtained. Moreover, the same perturbed stage geometry, in terms of work coefficient (τ/τ*), exhibits a −1.9%, −1.0%, and −0.1% decrease at stall, design, and choke conditions, respectively.

_{p}/η

_{p}* of about 0.99 (−1.0% from baseline) with a ϕ/ϕ* of 0.96 (−4.0% from baseline) and a τ/τ* of 1.00 (+0.0%). The same perturbed stage exhibits, at stall condition, a −2.0% reduction in η

_{p}/η

_{p}*, a gain of +1.9% for τ/τ*, and a shift to lower mass flow rates of −10.6% (ϕ/ϕ* of 0.76 instead 0.85). At choke condition, instead, the perturbed geometry compared to the baseline shows a reduction in η

_{p}/η

_{p}* of about −2.2%, the same value of τ/τ*, and a shift in ϕ/ϕ* of −2.9%. Therefore, this perturbed stage achieved the same pressure ratios of the baseline stage with lower mass flow rates.

_{p}/η

_{p}*) varies from 0.99 to 1.01 at stall (from −1.0% to +1.0% in terms of relative percentage variation respect to baseline), from 0.99 to 1.01 at design (from −1.0% to +1.0%), and from 0.88 to 0.92 at choke condition (from −2.2% to +2.2%). Moreover, the work coefficient (τ/τ*) varies from 1.04 to 1.06 at stall (from −1.0% to +1.0%), from 0.99 to 1.01 at design (from −1.0% to +1.0%), and from a value of 0.81 to 0.86 at choke (from −3.6% to +2.4%). Finally, the flow coefficient (ϕ/ϕ*) varies from 0.82 to 0.97 at stall (from −5.7% to +11.5%), from 0.96 to 1.08 at design (from −4.0% to +8.0%), and from 1.23 to 1.37 at choke condition (from −3.9% to +7.0%).

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Meridional view of a centrifugal compressor stage (

**a**) composed of the inlet duct (A), impeller (B), vaneless diffuser (C), crossover bend (D), and return vane channel (E), as well as independent/dependent geometrical parameters of the stage (

**b**).

**Figure 6.**Experimental validation of CFD predictions, in terms of polytropic efficiency (

**a**) and work coefficient (

**b**), at section 6 of the low mass flow stage.

**Figure 7.**Polytropic efficiency (

**a**) and work coefficient (

**b**) at section 6 derived from parametric analysis on the low mass flow stage.

**Figure 8.**Polytropic efficiency (

**a**) and work coefficient (

**b**) at section 6 derived from parametric analysis on the medium mass flow stage.

**Figure 9.**Polytropic efficiency (

**a**) and work coefficient (

**b**) at section 6 derived from parametric analysis on the high mass flow stage.

**Figure 10.**Low flow stage graphs describing the relative frequencies of $\mathsf{\Delta}{\eta}_{p}$ (

**a**) and $\mathsf{\Delta}\tau $ (

**b**) variations in design condition at section 6, as well as the relative frequency of choke-to-stall operating range variations $\mathsf{\Delta}\varphi $ (

**c**) with their comparable Gaussian distributions.

**Figure 11.**Medium flow stage graphs describing the relative frequencies of $\mathsf{\Delta}{\eta}_{p}$ (

**a**) and $\mathsf{\Delta}\tau $ (

**b**) variations at design condition at section 6, as well as the relative frequency of choke-to-stall operating range variations $\mathsf{\Delta}\varphi $ (

**c**) with their comparable Gaussian distributions.

**Figure 12.**High flow stage graphs describing the relative frequencies of $\mathsf{\Delta}{\eta}_{p}$ (

**a**) and $\mathsf{\Delta}\tau $ (

**b**) variations at design condition at section 6, as well as the relative frequency of choke-to-stall operating range variations $\mathsf{\Delta}\varphi $ (

**c**) with their comparable Gaussian distributions.

**Figure 13.**Comparison of ANN forecasts (cyan dots) and CFD evaluations (magenta dots) at section 6 under stall (

**a**,

**d**), design (

**b**,

**e**), and choke (

**c**,

**f**) conditions for the low flow stage.

**Figure 14.**Comparison of ANN forecasts (cyan dots) and CFD evaluations (magenta dots) at section 6 under stall (

**a**,

**d**), design (

**b**,

**e**), and choke (

**c**,

**f**) conditions for the low medium stage.

**Figure 15.**Comparison of ANN forecasts (cyan dots) and CFD evaluations (magenta dots) at section 6 under stall (

**a**,

**d**), design (

**b**,

**e**), and choke (

**c**,

**f**) conditions for the high flow stage.

Parameter | Unit | Range of Variation |
---|---|---|

Impeller inlet blade angle ${\beta}_{1}$ | deg | [−2.0; 2.0] |

Impeller outlet blade angle ${\beta}_{2}$ | deg | [−2.0; 2.0] |

Blade thickness $t$ | % | [−7.5; 7.5] |

Outlet impeller width ${b}_{2}$ | % | [−5.0; 5.0] |

Impeller throat area ${A}_{th}$ | % | [−4.0; 7.5] |

Diffusion ratio of diffuser $DR={D}_{4}/{D}_{2}$ | - | [−1.6; 1.84] |

Parameter | Reason |
---|---|

Inlet width of inlet duct ${b}_{0}$ | Fix inlet width of inlet duct |

$\mathrm{Inlet}\mathrm{width}\mathrm{of}\mathrm{impeller}{b}_{1}$ | Fix inlet width of impeller |

Inlet width of crossover bend ${b}_{4}$ | Preserve ${b}_{3}/{b}_{4}$ ratio |

Inlet width of return vane channel ${b}_{5}$ | Preserve ${b}_{4}/{b}_{5}$ ratio |

Radial position of return channel blade LE ${r}_{LE}$ | Preserve the ratio between return channel blade length and diffuser $DR$ |

Outlet width of return vane channel TE ${b}_{6}$ | Preserve TE width |

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

Value | Error with G5 [%] | Value | Error with G5 [%] | ||

G1 | 3.4 million | 0.991 | −0.9 | 1.009 | 0.9 |

G2 | 3.8 million | 0.995 | −0.5 | 1.005 | 0.5 |

G3 | 4.3 million | 0.998 | −0.2 | 1.001 | 0.1 |

G4 | 4.8 million | 1.000 | 0.0 | 1.000 | 0.0 |

G5 | 5.3 million | 1.000 | - | 1.000 | - |

Numerical Setup for CFD Computations | |
---|---|

Analysis type | RANS with adiabatic walls |

Grid type | H-type |

No. of Elements | 4.8 million |

Convective flux discretization | 2nd order TVD-MUSCL with Roe’s upwind scheme |

Viscous flux discretization | Central difference scheme |

Turbulence closure | Wilcox’s k-ω model |

Parallelization | Hybrid OpenMP/MPI architecture |

Wall treatment | Full resolution |

Near wall grid refinement | First element of 1.0 × 10^{−5} mm (y+ ≤ 1) |

**Table 5.**ANN mean absolute error in predicting polytropic efficiency ${\delta}_{{\eta}_{p}},$ work coefficient ${\delta}_{\tau}$, and operative range ${\delta}_{\Delta \varphi}$.

Stall | Design | Choke | Operative Range | ||||
---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{{\mathit{\eta}}_{\mathit{p}}}$ | ${\mathit{\delta}}_{\mathit{\tau}}$ | ${\mathit{\delta}}_{{\mathit{\eta}}_{\mathit{p}}}$ | ${\mathit{\delta}}_{\mathit{\tau}}$ | ${\mathit{\delta}}_{{\mathit{\eta}}_{\mathit{p}}}$ | ${\mathit{\delta}}_{\mathit{\tau}}$ | ${\mathit{\delta}}_{\mathbf{\Delta}\mathit{\varphi}}$ | |

Low flow stage | 0.15% | 0.30% | 0.07% | 0.08% | 0.20% | 0.14% | 0.03% |

Medium flow stage | 0.09% | 0.17% | 0.04% | 0.05% | 0.19% | 0.11% | 0.04% |

High flow stage | 0.10% | 0.08% | 0.06% | 0.06% | 0.12% | 0.12% | 0.07% |

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## Share and Cite

**MDPI and ACS Style**

Bicchi, M.; Marconcini, M.; Bellobuono, E.F.; Belardini, E.; Toni, L.; Arnone, A.
Multi-Point Surrogate-Based Approach for Assessing Impacts of Geometric Variations on Centrifugal Compressor Performance. *Energies* **2023**, *16*, 1584.
https://doi.org/10.3390/en16041584

**AMA Style**

Bicchi M, Marconcini M, Bellobuono EF, Belardini E, Toni L, Arnone A.
Multi-Point Surrogate-Based Approach for Assessing Impacts of Geometric Variations on Centrifugal Compressor Performance. *Energies*. 2023; 16(4):1584.
https://doi.org/10.3390/en16041584

**Chicago/Turabian Style**

Bicchi, Marco, Michele Marconcini, Ernani Fulvio Bellobuono, Elisabetta Belardini, Lorenzo Toni, and Andrea Arnone.
2023. "Multi-Point Surrogate-Based Approach for Assessing Impacts of Geometric Variations on Centrifugal Compressor Performance" *Energies* 16, no. 4: 1584.
https://doi.org/10.3390/en16041584