# A Review of Turbine and Compressor Aerodynamic Forces in Turbomachinery

## Abstract

**:**

## 1. Summary

_{CFD}= (k

_{seal}+ k

_{shroud}) = (40% + 60%), i.e., the shroud cavity produces 50% more cross-stiffness than the stage seal.

_{shroud}+ k

_{seal}). Hence, the accepted methods for calculating the cross-coupled (destabilizing) stiffness are way off the experimental results. Prediction from full three-dimensional CFD models will become commonplace in the near future.

## 2. Introduction

#### A Summary of Seal and Stage Reaction Forces and Dynamic Force Coefficients

_{(t)}, Y

_{(t)}) around the center of the seal housing or a stage casing. The rotor motion generates a dynamic pressure field, which produces a reaction force with components (F

_{X}, F

_{Y}). The generally accepted force–rotor displacement model is

^{2}fluid compressibility, characteristic of gases and supercritical fluids, leads to force coefficients that are functions of the whirl frequency (ω), whilst being modestly affected by the rotor motion amplitude [18].

_{(t)}= r cos(ωt) and Y

_{(t)}= r sin(ωt); and Equation (1) becomes

_{eff}> 0 produces a centering force as the radial force F

_{r}< 0, whereas C

_{eff}> 0 promotes dynamic stability, since the tangential force F

_{t}< 0 opposes the rotor forward precession whirl. Seal or stage configurations that produce a significant effective damping coefficient are desirable, and those that do not generate a cross-coupled stiffness (k = 0) are best. Note that c

_{(ω)}> 0 increases the effective stiffness of the seal or the stage.

_{eff}decreases. Furthermore, at a certain cross-over frequency

## 3. A Brief Review of Aerodynamic Forces in Turbines and Compressors

_{XY}= −K

_{YX}= k are the cross-coupled stiffness coefficients. K

_{XY}> 0 to induce forward whirl, while K

_{YX}< 0 for backward whirl, i.e., opposite to the sense of the shaft rotation.

#### 3.1. Aerodynamic Forces in (Unshrouded) Axial Turbines and Axial Compressors

_{t}) is generated that tends to cause forward (destabilizing) whirl; i.e., in the direction of rotor spinning and with F

_{t}= (k × e). The Thomas “clearance–excitation” force is proportional to the mass flow rate ($\dot{m}$) through the turbine and the enthalpy drop (ΔH

_{D}) across a stage, and inversely proportional to shaft speed (Ω). Thus, this force is load-dependent, a function of the applied torque (T

_{0}). Recall that mechanical power equals (T

_{0}× Ω)~$\left(\dot{m}\text{\hspace{0.17em}}\Delta {H}_{D}\right)$.

_{t}~(k × e). Alford’s cross-coupled stiffness (k) equals

_{o}is the compressor stage torque, and D and h

_{B}are the mean tip diameter and blade length, respectively. Above, β is “the change of thermodynamic efficiency per unit of rotor displacement, expressed as fraction of blade height.” Alford recommended β ranging from 1.0 to 1.50. Most importantly, as argued by Alford, the aerodynamic force is forward (direction of shaft rotation) for both compressors and turbines. Hence, in Alford’s theory, there is no distinction between the physical operation of both types of machines.

_{spool}), known as the blade loading indicator, that denotes the effect of a non-axisymmetric pressure distribution on the rotor spool. This effect, first identified by Song and Martinez-Sanchez [32] (1997), induces an additional tangential force in turbines and is suspected to play a role in compressors as well. Under a steady shaft offset displacement (eccentricity), Spakovszky predicted a strong tendency towards backward whirl for operation at low flow coefficients, whereas forward whirl appears for high flow coefficients. These findings are in agreement with Ehrich’s measurements [6] (1993). Spakovszky found β

_{spool}> 0 over most of the compressor operating range, i.e., inducing forward whirl, though its magnitude is typically ~50% of Alford’s β parameter. However, under forced whirling conditions, β

_{spool}is comparable in magnitude to Alford’s β, but acting in the direction opposite of the whirl. The blade loading indicator (β

_{spool}), a function of the stage geometry and mean flow, determines the direction of whirl due to tangential blade loading forces.

#### 3.2. Destabilizing Forces in (Shrouded and Unshrouded) Centrifugal Compressors

_{ref}as a reference fluid mass (density x volume of solid disk with tip diameter D

_{o}and thickness b

_{2}). For m~0, the whirl frequency ratio, WFR = k/(CΩ), relates the whirl frequency of unstable motions to shaft speed. A WFR = 0.5 means that the compressor cannot stably operate at a shaft speed above two times (=1/WFR) the system first natural frequency.

_{r}= 0.067 mm to 0.081 mm, representative of actual applications.

- Impellers have negative direct stiffness and large direct and cross-coupled inertia coefficients.
- The cross coupled stiffnesses are much larger for tight clearance impellers, thus generating large destabilizing forces at rotor speeds below the rotor-bearing system critical speed, i.e., WFRs > 1.
- The shroud generates most of the destabilizing forces in a pump impeller.
- Enlarged clearances in neck ring seals aggravate the generation of cross-coupled forces, since the swirl flow into the seal increases. Enlarged clearances are the result of transient rubbing events in the operation of a pump.
- At reduced flow rates (<Q
_{BEP}), the impeller cross-coupled stiffnesses are smaller. - Impeller–volute interaction forces are small and benign.
- In a radial flow impeller, the cross-coupled stiffnesses are small, i.e., the destabilizing force is negligible, since the projected axial area of the shroud is quite small.

_{C}/ΩC

_{D}) > 1 for low shaft speeds (<2 krpm), and WFR→0.5 at the highest speed (5 krpm) for the closed impeller, whilst for the open impeller WFR > 4 at low speed and WFR→1 at the highest speed.

_{pump}) and direct damping $\overline{C}$ (=3.14

_{pump}) and a nearly identical whirl frequency ratio (WFR~0.35

_{pump}). The pump, on the other hand, evidenced large cross-coupled damping $\overline{c}$~7.91

_{pump}and cross-coupled inertia $\overline{m}$~−0.58

_{pump}. The most important difference relates to the compressor developing a positive direct stiffness $\overline{K}$ = 1.25, while = −2.5

_{pump}.

#### 3.3. Wachel’s Equation and Its Effect on Compressor Stability

_{o}is the torque = (compressor power/shaft angular speed), D is the impeller tip diameter, and h

_{W}is the diffuser width. Above MW is the gas molecular weight, and (ρ

_{out}/ρ

_{in}) is the ratio of gas densities discharged over the inlet. The parameter β ranges from 2 to 5 for most high performance compressors. Note that Wachel’s equation, derived from correlation studies of field experiences with unstable compressors, is in actuality a modified form of Alford’s force model, as explained by Evans and Fulton [7]. The ratio (MW/10) appears arbitrary, and API [48] sets MW = 30 for hydrocarbon gas processing compressors. The API formula is known as the PACC #, i.e., the predicted aerodynamic cross-coupling number.

_{A}or until the log-dec δ

_{A}= 0 obtained for an overall cross-coupled coefficient with magnitude Q

_{o}.

_{A}< 0.1, or Q

_{o}/Q

_{A}< 2, or Q

_{o}/Q

_{A}< 10, and the location in region B of the point is defined by the intersection of the critical speed ratio CSR (operating speed/undamped natural frequency) and the average density at the normal operating condition, see Figure 9. The Level II analysis accounts for forces from all stages, including their labyrinth seals, damper seals, impeller/blade flow aerodynamic effects, and internal friction.

_{shroud}is the axial length of the shroud, and (Q/Q

_{design}) is the flow relative to the design condition. C

_{mr}, a constant for a given impeller design, ranges from 4.0 to 7.5 (when using SI units). If Equation (13) is to be applied to a turbine or a turbo expander, take the inlet plane as the reference for dynamic head.

_{Alford}= 6355 N/m; whereas Wachel’s Equation (12) produces k

_{Wachel}= 11.13 k

_{Alford}, and using API’s formula with MW = 30 predicts k

_{API}= 3.27 k

_{Alford}. Similarly, using Moore and Ransom’s Equation (13) gives k

_{Moore-Ransom}= 1.71 k

_{Alford}, while the prediction from an unsteady CFD study of the forces produced by the expander delivers k

_{CFD}= 2.32 k

_{Alford}.

_{CFD}= (k

_{LS}+ k

_{shroud}) = (40% + 60%), i.e., the shroud cavity produced 50% more cross-stiffness than the eye seal. Predictions showed that the total k (>K) grew with the power two of the shaft speed. Later, Hoopes et al. [58] reported, at one operating condition, k

_{CFD}= 2.99 MN/m = (k

_{LS}+ k

_{shroud}) = (34 % + 66%) = 1.53 k

_{Wachel}= 2.35 k

_{Moore-Ransom}. The authors noted the generation of a non-uniform circumferential pressure field regulated by the LS varying clearance as the impeller whirled.

_{Wachel}= 439 kN/m, k

_{API}= 1.07 k

_{Wachel}, k

_{Moore-Ransom}= 0.935 k

_{Watchel}, and k

_{CFD}= 1.245 k

_{Watchel}. Similarly, for the 5th stage, k

_{Wachel}= 424 kN/m, and k

_{API}= 1.07 k

_{Wachel}, k

_{Moore-Ransom}= 1.127 k

_{Watchel}, whereas k

_{CFD}= 1.917 k

_{Watchel}. The CFD cross-coupled stiffnesses (k’s) were larger, since they accounted for the asymmetric pressure field in the shroud that also exerted a force on the impeller. Note again the wide variation in predictions depending on the model used. The same reference produced a table (see below) that separated the individual contributions of the eye seal and shroud to the force coefficients, k and C. The tabulated results made evident the benefits of the SB upstream of the LS. Note that, for the baseline condition (one without SBs), the eye seal contributed 57% of the total k. On the other hand, for the stage with the installed SB, the total k magnitude was much lower (~38% of baseline), with the shroud cavity contributing 134% of the total k; i.e., k

_{LS}< 0. With the SB, the eye seal produced a k < 0.

_{A}) based on Alford’s Eq., Wachel’s Eq., API 617 formula, their own MPACC, and a Q derived (somehow) from a CFD study of one open stage. Table 4 summarizes the results, which reveal the MPACC number and CFD based predictions produced the best correlation with the measured log-dec for full power conditions. The paper also calls to attention the lack of experimental results related to unshrouded stages.

_{shroud}+ k

_{LS}).

_{model}~1.14 k

_{exp}.

_{CFD}~k

_{exp}for the shroud contribution. In addition, the authors noted k

_{Alford}~0.25 k

_{exp}(shroud), k

_{Wachel}~0.59 k

_{exp}, and k

_{Moore-Ransom}~0.28 k

_{exp}. Clearly, the conventional methods for predicting the cross-coupled (destabilizing) coefficients produce very different results from experimental results or a prediction from a full CFD model.

_{2}turbomachinery, with the added caveat of likely fluid condensation in turbines handing the supercritical fluid.

## 4. Closure

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

C_{eff} | (C−k/ω). Seal effective damping coefficient [N-s/m] |

C, c | Direct and cross-coupled damping coefficients [N/m] |

C_{r} | Seal nominal clearance [m] |

D | Rotor diameter, tip diameter of blade [m] |

F_{X}, F_{Y} | Seal reaction forces in Cartesian coordinates [N] |

h_{B}, h_{W} | Blade tip length, diffuser width [m] |

K, k | Direct and cross-coupled stiffnesses [N/m] |

K_{eff} | (K + c ω). Seal effective stiffness coefficient [N/m] |

L | Seal axial length [m] |

M, m | Direct and cross-coupled mass coefficients [kg] |

MW | Gas molecular weight [kg] |

$\dot{m}$ | Seal mass flow rate [kg/s] |

P | Pressure [Pa] |

Q | Estimated cross-coupled stiffness coefficient for whole compressor [N/m] |

T_{o} | Compressor (turbomachine) stage torque [Nm] |

U_{s} | ½ R D. Rotor surface speed [m/s] |

WFR | k/((Cω). Whirl frequency ratio |

β | Blade efficiency parameter (Alford’s equation) |

δ | Logarithmic decrement (a measure of damping ratio) |

ρ | Gas density [kg/m^{3}] |

ω | Excitation (whirl) frequency [rad/s] |

Ω | Shaft speed [rad/s] |

## Abbreviations

CSR | Operating speed/first bending critical speed of rotor on rigid supports |

LS | Labyrinth seal |

PDS | Pocket damper seal |

PACC | Predicted Aerodynamic Cross-Coupling number |

MPACC | Modified PACC |

SB | Swirl brake |

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**Figure 1.**Seals in a Multistage Centrifugal Pump or Compressor. Adapted from Childs [11].

**Figure 2.**Straight-through and Back-to-back Compressor Configurations and 1st Mode Shapes. Adapted from Childs [11].

**Figure 4.**Knives on a rotor (TO) impeller eye seal: baseline and modified with a swirl brake. Reproduced with permission from Ref. [20].

**Figure 6.**Diagram of tip-clearance-induced blade forces in an axial turbine. Reproduced from Ehrich [21].

**Figure 7.**Typical pump impeller geometry, copied with permission from Ref. [11].

**Figure 9.**Test open and closed face impellers in Ref [43].

**Figure 10.**Stability Experience Plot as per API 8th edition and Fulton–Sood Map [7]. Region A: stable, Region B: unstable.

$\overline{\mathit{K}}$ | $\overline{\mathit{k}}$ | $\overline{\mathit{C}}$ | $\overline{\mathit{c}}$ | $\overline{\mathit{M}}$ | $\overline{\mathit{m}}$ | WFR | Note | |
---|---|---|---|---|---|---|---|---|

CT-volute | −2.5 | 1.10 | 3.14 | 7.91 | 6.51 | −0.58 | 0.35 | Φ = 0.092 |

CT-diffuser | −2.65 | 1.04 | 3.80 | 8.96 | 6.60 | −0.90 | 0.27 | Φ = 0.092 |

Radial flow impeller | −0.42 | −0.09 | 1.08 | 1.88 | 1.86 | −0.27 | - | BEP, vaneless diffuser |

S-Diffuser (2) | −5.0 | 4.4 | 4.2 | 17.0 | 12.0 | 3.5 | 1.05 | 2 krpm |

S-Diffuser (4) | −2.0 | 7.5 | 4.2 | 8.5 | 7.5 | 2.0 | 1.78 | 4 krpm |

S-with swirl brake | −2.2 | 7.7 | 3.4 | 8.6 | 6.7 | 3.1 | 2.26 | 4 krpm, BEP, Type D |

S-with face seal | −4.2 | 5.1 | 4.6 | 13.5 | 11.0 | 4.0 | 1.11 | BEP, Type A |

Yoshida et al. [45] | $\overline{\mathit{K}}$ | $\overline{\mathit{k}}$ | $\overline{\mathit{C}}$ | $\overline{\mathit{c}}$ | $\overline{\mathit{M}}$ | $\overline{\mathit{m}}$ | WFR | Note |
---|---|---|---|---|---|---|---|---|

1.82 | 1.48 | 4.30 | −0.081 | 3.04 | −0.053 | 0.34 | Φ = 0.424 |

**Table 3.**CFD predicted cross-coupled stiffness (k) and direct damping (C) for the 2nd stage of a compressor. Baseline Model: without swirl brake (SB), and model with swirl brake. Results from Kim and Venkataraman [59].

Baseline No SB | With SB | ||||||
---|---|---|---|---|---|---|---|

Total | Eye Seal | Shroud | Total | Eye Seal | Shroud | ||

k [MN/m] | 1.261 | 0.714 | 0.548 | k | 0.470 | −0.159 | 0.629 |

100% | 57% | 43% | 100% | −34% | 134% | ||

C [N-s/m] | 764 | 657 | 107 | C | 410 | 317 | 93 |

100% | 86% | 14% | 100% | 77% | 23% | ||

WFR = k/(Ω C) | 0.77 | 0.51 | 2.39 | 0.53 | −0.23 | 3.15 |

**Table 4.**Log-dec measured, and predicted and estimated cross-coupled stiffness for the compressor of Wu and Kuzdal [61].

Test | API 617 PACC | MPACC | Wachel’s Eq. | Alford’s Eq. | CFD Open Stages + Laby Seals + MPACC for Shrouded Stages | |
---|---|---|---|---|---|---|

Log-dec, δ | 0.45 | 0.39 | 0.44 | 0.17 | 0.19 | 0.46 |

Estimated Q | 0.599 × Q_{Wachel} | 0.473 × Q_{Wachel} | 32,914 lbf/in = Q_{Wachel} | 0.957 × Q_{Wachel} | 0.424 × Q_{Wachel} |

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**MDPI and ACS Style**

San Andrés, L.
A Review of Turbine and Compressor Aerodynamic Forces in Turbomachinery. *Lubricants* **2023**, *11*, 26.
https://doi.org/10.3390/lubricants11010026

**AMA Style**

San Andrés L.
A Review of Turbine and Compressor Aerodynamic Forces in Turbomachinery. *Lubricants*. 2023; 11(1):26.
https://doi.org/10.3390/lubricants11010026

**Chicago/Turabian Style**

San Andrés, Luis.
2023. "A Review of Turbine and Compressor Aerodynamic Forces in Turbomachinery" *Lubricants* 11, no. 1: 26.
https://doi.org/10.3390/lubricants11010026