=Paper=
{{Paper
|id=Vol-1852/p05
|storemode=property
|title=Dynamic design optimization of turbine-compressor
unit by Ocvirk and Dubois elastohydrodynamic
equations and Craig-Bampton approach
|pdfUrl=https://ceur-ws.org/Vol-1852/p05.pdf
|volume=Vol-1852
|authors=Diego Acerbo
}}
==Dynamic design optimization of turbine-compressor
unit by Ocvirk and Dubois elastohydrodynamic
equations and Craig-Bampton approach==
https://ceur-ws.org/Vol-1852/p05.pdf
Dynamic design optimization of turbine-compressor
unit by Ocvirk and Dubois elastohydrodynamic
equations and Craig-Bampton approach
Diego Acerbo
HSE at Bonatti S.p.A.
Via Nobel, 2 A , 43122 Parma (Italy)
email: diego.acerbo@bonatti.it
Abstract—This paper presents the results obtained analysing
the dynamic behaviour and natural vibration modes of the
components of a turbine-compressor unit, connected by toothed
coupling. The study was undertaken with the aim of improving
the dynamic behaviour of the system, identifying some critical
characteristics, mainly linked to problems of misalignment which
lead to elevated vibration levels in some bearings and to the
rupture of the lubrication film. An accurate 3D parametric
virtual model of the system was used, integrating FEM and
Multibody calculation programmes to perform a dynamic anal-
ysis of the various components considered as deformable bodies.
Information regarding some characteristics significant for the
dynamics of the system was obtained in experimental trials,
allowing the validation of the numerical models. In particular,
the contact of the teeth of the turbine and compressor hubs jibbing of the coupling. The modeling methodology followed
with the teeth of the coupling bell were simulated, as well as it is similar to that described by Cali et al. [10], [11].
the hydrodynamic effect of the lubrication in the shaft bearings.
The analysis of the results highlighted the need to use models The simplified Reynolds equations proposed by Ocvirk and
with deformable elements, and allowed the determination of the Dubois [12] for short bearings were used to simulate the
limiting conditions of misalignment. hydrodynamic reaction of the bearing on the pivot.
Index Terms—3D Flexible modeling, Computational dynamic
analysis, misalignmen, bearings, lubrication film. II. T URBO - COMPRESSOR U NIT
The P2005A turbo-compressor in the ethylene production
I. I NTRODUCTION plant of Enichem Priolo is used to compress the mixture of
Management of a modern petrochemical plant, where the hydrocarbon gases emitted at the head of the quench column.
sound functioning of the machines installed guarantees the The machine, constructed by Nuovo Pignone, consists of three
reliability and continuity of production, must include contin- main parts: the steam driven turbine, the centrifuge compressor
uous monitoring by means of dedicated systems. Checking which compresses the gas, and the toothed coupling which
for vibration is essential in verifying the correct functioning transmits the torque from the turbine to the compressor. The
of mechanical systems and will evidence possible anomalies axial turbine, delivers a maximum power of 20835 kW at 4190
at their incipient stage [1], [2], [3], [4], [5], [6], [7]. The rpm. The shaft rests on two radial hydrodynamic bearings of
present study examines a steam turbine driving a centrifuge the Michell type (tilting pad bearings) and a hydrodynamic
compressor installed in an ethylene refrigeration cycle, part axial thrust bearing of the Kingsbury type. The steam and
of the thermal cracking unit at the Enichem plant at Priolo lubrication oil seals are secured by a series of rings cut
(Sicily). Lately, different tools and methodologies have been on the rotor and mounted on the stator with a labyrinth
employed to study this typology of machinery [8], [9]. A system. The six-stages centrifuge compressor of horizontal
multibody model, composed of flexible parts and developed open case type, with three intake flanges and one delivery
with the ADAMS programme, was used to simulate the flange positioned in the lower half-casing, again rests on
dynamic behaviour of the two rotors, turbine and compressor, two segmented hydrodynamic bearings and a Kingsbury-type
linked by a toothed coupling. The model, validated by com- thrust bearing (Tab. I).
parison with experimental data obtained using the monitoring The Maag toothed coupling, Zud8 type, in AISI 8740
system, was found to be particularly useful in the analysis of steel, is 976.3 mm in length with a maximum diameter of
breakdowns, allowing the simulation of misalignment due to 383 mm. The two bells, internally toothed, rest on the hubs
and are connected together by a sleeve collar; two metal O-
Copyright © 2017 held by the authors. rings, pressure mounted in circumference slots cut into the
26
teeth of the bell, allow the bell-cylinder system an axial slip
of 6 mm. The hubs, toothed externally with 88 teeth, are
keyed on the turbine and compressor shafts; two keys and
two threaded locking rings prevent, respectively, tangential
and axial movement. The smaller longitudinal dimensions
of the hub teeth compared to those of the bells, combined
with radial and tangential gap, allow the shafts to become
misaligned. A gas balancing system, which uses a disk keyed
to the shaft, makes it possible to limit the axial thrust and
the vibrations. To prevent excessive vibrations from damaging
the seals, during operation a control system verifies that
radial movement at the bearings never exceeds the allowed
tollerances (0.24 ÷ 0.28 mm for the compressor bearings and
0.3 ÷ 0.358 mm for the turbine bearings).
III. N UMERICAL M ODEL
The numerical model was developed using: the ADAMs
calculation programme to construct the multibody model of
the turbo-compressor; and the MSC/NASTRAN calculation
programme which, through modal analysis of the components,
was used to generate transfer files simulating the flexible
behaviour of the parts in the multibody code [13], [14]. The
approach followed in order to consider the bodies flexible
was the modal approach developed by Craig and Bampton
[15], which allows the number of generalised coordinates to
be reduced to a minimum and offers greater freedom in the
definition of the constraint conditions at the boundary points.
The transfer file contains the stiffness and damping matrices Fig. 1. 3D parametric model (top); flexible multibody elements (top).
of dimensions (6Nx6N), where N is the number of points used
in modelling the flexible parts; five for the turbine, two for
the coupling and twelve for the compressor. A characteristic
aspect of this modelling is the use of kinetic reference systems,
KRF (Kinematic Reference Frame), integral with each rigid
part making up the discretized flexible body. In this approach,
each substructure of the FEM is represented by a superelement
characterised by the above stiffness and damping matrices[16].
The movements of each substructure are calculated locally
with respect to the corresponding KRF; the overall elastic
deformation of the flexible body is obtained from the set of
single movements of the rigid parts into which it is discretized. irrelevant to the dynamic behaviour. The number of hexahedral
The definition of the matrix of concentrated mass is obtained elements and nodes are reported in Table II.
through the localisation of a centre of mass for each part,
with the inertial properties referring to it. The mass of each IV. H YDRODYNAMIC B EARINGS
part is independent of the rest of the system, so that the extra-
diagonal terms are eliminated from the mass matrix. On the Particular attention was paid to the modelling of the hy-
basis of the blueprints supplied by MAAG and using digital drodynamic bearings, on which the dynamic stability of the
photogrammetry acquisition as described in [17], [18], CAD system depends.
3D geometries were developed for the following parts: the For each shaft, three-component forces were applied at the
compressor shaft; the turbine shaft; the six rotors; the thrust mid-line of each bearing: the two radial components x and z
equaliser; the Kingsbury rings of the thrust bearings; all the (Fig. 2) reproduce the hydrodynamic reaction of the bearing
parts of the diffusors of the keyed stages in the rotor; the on the journal, while the y component, using a bistop function,
threaded locking rings which axially constrain the rotors and simulates the thrust bearing [19], [20], [21].
diffusors; the hubs of the toothed coupling keyed to the stator Integrating the differential equations of Reynolds, in accor-
and the locking rings which constrain them axially (Fig. 1). dance with the Ocvirk and Dubois approximation for short
The construction of the finite element model required a bearings, the characteristics of the lubrication fluid in the
simplification of these geometries, eliminating some features bearings were obtained (eqq. 1-7) which determine, together
27
Fig. 2. Transversal cross-section of the bearing. Fig. 3. Bearing site of loading.
with the static component of the load, the elastohydrodynamic
reaction of the bearing on the pivot (eq. 8). η0 rb b3 ε0
4 sin2 Φ0
rXZ = rZX = − +
c3 (1 − ε20 )2
η0 ωb rb b3 ε0 sin2 Φ0
# (6)
kXY = + 3 π ε0 cosΦ0 sinΦ0 cos2 Φ0
c3 (1 − ε20 )2 + −
# (1) 2(1 − ε20 ) 2
5
2(1 − ε20 )2
3 π ε0 cosΦ0 sinΦ0 2(1 + ε20 ) cos2 Φ0
+ +
5
4(1 − ε20 ) 2 (1 − ε20 )3 "
η0 rb b3 π (1 + 2ε20 ) sin2 Φ0
rZZ = 5 +
" c3 2(1 − ε20 ) 2
η0 ωb rb b3 π(1 + 2ε20 ) sin2 Φ0 # (7)
kXZ = +
c3 5
4(1 − ε20 ) 2 4 ε0 cosΦ0 sinΦ0 π 2 Φ0
# (2) − 2 2
+ 3
(1 − ε0 ) 2(1 − ε20 ) 2
ε0 (1 + 3ε20 ) cosΦ0 sinΦ0 π cos2 Φ0
+ 2 3
+ 3
(1 − ε0 ) 4(1 − ε20 ) 2
FX (t) F (X , Z , 0, 0)
= X 0 0 +
" FX (t) 0
η0 ωb rb b3 π sin2 Φ0 (8)
kY Z = − 3 + k kXZ (X − X0 ) r rXZ Ẋ
c3 4(1 − ε20 ) 2 − XX − XX
# (3) kZX kZZ (Z − Z0 ) rZX rZZ Ż)
ε0 (1 + 3ε20 ) cosΦ0 sinΦ0 π (1 + 2ε20 ) cos2 Φ0 The static components of the load, determined at each
+ −
(1 − ε20 )3 5
4(1 − ε20 ) 2 equilibrium configuration (X0 , Y0 ), on varying the velocity,
are shown in Fig. 3, while the static reactions of the bearings
η0 ωb rb b3 ε0 2(1 + ε20 )sin2 Φ0
are reported in Tab. III.
kZZ = +
c3 (1 − ε20 )3
# (4) V. C ONTACT B ETWEEN THE T EETH
3 π ε0 cosΦ0 sinΦ0 cos2 Φ0
− + Three-component vector forces were used to model the
5
4(1 − ε20 ) 2 (1 − ε20 )2 forces exchanged between the teeth of the hubs and those of
" the bells. Using IMPACT and BISTOP functions, these vector
η0 rb , b3 π sin2 Φ0 forces limit the axial, radial and tangential movements of the
rXX = 3 +
c3 2(1 − ε20 ) 2 hubs inside the bells.
# (5) The axial movement of the sleeve collar-bells system
4 ε0 cosΦ0 sinΦ0 π (1 + 2ε20 ) cos2 Φ0 (±3 mm) is limited by the z components of the forces of
+ 2 2
+ 5
(1 − ε0 ) (1 − ε20 ) 2 contact between the teeth simulating the behaviour of the two
28
Fig. 5. Contact force between teeth.
Fig. 4. Detail of the coupling.
O-rings pressure mounted inside the bells. The x component
of the vector force is applied to the tooth flank in the tangential
direction, while the y component is applied in the radial
direction (Fig. 4 and Fig. 5).
VI. M ODEL VALIDATION AND A NALYSIS OF R ESULTS
The model, was validated on the basis of experimental
measurements conducted at the Enichem Priolo plant using
inductive proximity transducers, which measure the radial and Fig. 6. Frequency spectrum of the movements in the internal turbine bearing.
axial movements of the shafts in their bearings. The experi-
mental measurements of the monitoring system were filtered
and only those frequencies of major interest appear in the 40 and 90 , the maximum values admitted by the manufacturer
spectra, i.e. frequencies up to the value at the speed of rotation at speed and in transient, respectively.
and at multiples of this speed [22], [23]. For comparison, figure
6 shows the frequency spectra of the movements measured in A. Normal operation
the turbine bearing near the coupling. In correspondence with Fig. 7 shows the displacement calculated during six revo-
the first harmonic (65.3 Hz) the amplitude values are almost lutions at a speed of 3920 rpm at the two sensors mounted
coincident for both the numerical model and the experimental at ±45◦ with respect to the vertical, in the turbine bearing
measurements. The further close agreement between values located near the coupling.
for the subsequent harmonics confirms that the behaviour of The displacements, calculated under normal operating con-
the numerical model is very close to that of the real system. ditions, have amplitudes of less than 1.5 mils (0.0381 mm) and
The numerical modal analysis was performed using the in the real system can, therefore, be considered background
Lanczos method of constants. noise produced by the surface roughness of the pivot. The
Since the frequency corresponding to maximum speed coincidence of the values measured by two sensors indicates
(4000 rpm) is 66 Hz, the possibility of torsional resonance with that the pivot rotates with an almost constant eccentricity. The
the two shafts and the coupling is improbable and, therefore, values of the displacements for rigid and deformable elements
the operating anomalies are considered to be the result of shaft are compared in Fig. 8.
misalignment. Thus, as well as an analysis of the dynamic The need to use a model with deformable elements appears
behaviour of the system under normal operating conditions, a evident. Although this model presents considerable difficulty
misalignment between the shafts was simulated with values of in construction and longer calculation times, it yields results
29
Fig. 7. Displacements in the turbine bearing (coupling side). Fig. 9. Vibrations in the turbine bearing with misalignments.
Fig. 8. Comparison of displacements in the turbine bearing, rigid and Fig. 10. Comparison of displacements in the internal and external bearings.
deformable shafts.
provoke vibrations at frequencies which are multiples of the
which are comparable to those measured experimentally. In the
frequency at operating speed.
model with rigid elements, instead, the displacement values
Finally, Fig. 10 shows a comparison of the displacements
are incompatible with the normal operation of the machine,
calculated in the two bearings (internal and external with
above all considering the fact that they were calculated with
respect to the coupling) of the compressor and turbine, when
zero misalignment.
a misalignment of 40 is simulated. In the turbine, the values
B. Misalignment of the shafts of the displacements are substantially coincident and show a
fairly regular trend due to the fact that jibbing between the
Fig. 9 shows a comparison of the displacements and relative turbine itself and the coupling was provoked to simulate the
spectra of the turbine bearing at the sensor at +45◦ for misalignment. In the compressor, the internal bearing shows a
misalignments of zero, 40 and 90 . A misalignment of 40 leads trend similar to that of the turbine, while the external bearing
to displacements of about 3 mils (0.0762 mm), the limiting shows a rather irregular trend as a result of being dragged
value beyond which the critical operation alarm is activated. by the coupling and thus oscillating around the equilibrium
A misalignment of 90 . produces displacements of almost 6 position.
mils (0.1524 mm) which result in the machine blocking, given
that this exceeds the maximum allowed radial play of 0.13 C. Energy of deformation
mm. The frequency diagrams evidence that increasing the The possibility of constructing a reliable numerical model
misalignment results in a decrease in the amplitude of the with flexible bodies able to simulate the dynamic behaviour
vibrations at 3920 rpm (65.3 Hz) and conversely, a consider- of the turbo-compressor in a realistic manner also made it
able increase at double the rotation speed. Comparing these possible to obtain complete information regarding the stress
data with those of Fig. 4, it can be deduced that during the and strain states of the various machine components while
experimental measurements the machine was operating with operating. In particular is was possible to analyse the impulsive
a misalignment of less than 40 . The misalignments, therefore, interactions between the teeth of the hubs and those of the bells
30
Through the exact determination of the natural frequencies
of the three main components of the system, it was possible
to affirm that the critical operating characteristics are not
linked to phenomena of resonance produced by applied forces
originating in the bearings.
ACKNOWLEDGMENT
This study was made possible by the helpfulness of Enichem
Priolo. The authors wish to thank Engineer Antonio Rosolia,
who supplied the data and material necessary for the develop-
ment of this research.
Fig. 11. Strains in the coupling and bearings (misalignment of 90 with
jibbing). R EFERENCES
[1] W. Jiang and T. C. Lim, “Spectral-based direct multi-substructure analy-
sis of mechanical system vibration response,” Noise Control Engineering
when jibbing occurs, evaluating the stress and strain states. The Journal, vol. 63, no. 2, pp. 117–137, 2015.
simulations of misalignment with jibbing between the axis of [2] F. Fumei, L. Ping, and W. Gengshen, “Experimental research on steam-
the turbine and that of the coupling reproduce the most severe turbine rotor vibration fault diagnosis based on the fractal box counting
dimension,” Nuclear Power Engineering, vol. 27, no. 1, pp. 85–89, 2006.
operating conditions; the bells present elevated strains (Fig. 9) [3] A. Zile, D. Taradai, S. Tomashevskii, and Y. A. Shuranova, “Studying
accompanied by a displacement of the centres of rotation of the the torsional vibrations of turbine shaft trains,” Power Technology and
pivots in their bearings. The stress peaks are concentrated on Engineering, vol. 47, no. 6, pp. 470–477, 2014.
[4] M. Calı̀, G. Sequenzia, S. Oliveri, and G. Fatuzzo, “Meshing angles
the external surface of the bells and at the toothing, reaching evaluation of silent chain drive by numerical analysis and experimental
stresses close to the elastic limit (0.21 GPa). test,” Meccanica, vol. 51, no. 3, pp. 475–489, 2016.
Figure 9 shows the strain distributions in the zones of the [5] E. Pedullà, F. Lo Savio, S. Boninelli, G. Plotino, N. Grande, E. Rapis-
arda, and G. La Rosa, “Influence of cyclic torsional preloading on
bells and in the bearings where the stresses reach maximum cyclic fatigue resistance of nickel–titanium instruments,” International
values, remaining, however, below the elastic limit. From the endodontic journal, vol. 48, no. 11, pp. 1043–1050, 2015.
analysis of the dynamic behaviour of the turbo-compressor, [6] E. Pedulla, F. L. Savio, G. Plotino, N. M. Grande, S. Rapisarda,
G. Gambarini, and G. La Rosa, “Effect of cyclic torsional preloading
it emerged that in misalignments of up to 90 , the strains on cyclic fatigue resistance of protaper next and mtwo nickel–titanium
in the bells and bearings in any case allow the machine to instruments,” Giornale Italiano di Endodonzia, vol. 29, no. 1, pp. 3–8,
operate; the limits of vibration anticipated for the pivots are 2015.
[7] E. Pedullà, F. L. Savio, S. Boninelli, G. Plotino, N. M. Grande,
not exceeded and the lubrication film in the bearings is not G. La Rosa, and E. Rapisarda, “Torsional and cyclic fatigue resistance of
ruptured. During the simulations of jibbing, it was possible a new nickel-titanium instrument manufactured by electrical discharge
to distinguish two phases; in the first phase a considerable machining,” Journal of endodontics, vol. 42, no. 1, pp. 156–159, 2016.
[8] N. Bachschmid, G. Salvini, E. Tanzi, and E. Pesatori, “The influence of
amount of energy is absorbed by the bells, followed by a phase blade row dynamics on lateral and torsional shaft vibrations in steam
of settling in which the shafts find a new position of dynamic turbines,” in Proceedings of the 9th IFToMM International Conference
equilibrium on the bearings and the strain energy of the bells on Rotor Dynamics. Springer, 2015, pp. 113–127.
decreases. [9] S. Brusca, R. Lanzafame, A. Marino Cugno Garrano, and M. Messina,
“Dynamic analysis of combustion turbine running on synthesis gas,”
International Journal of Applied Engineering Research, vol. 10, no. 21,
VII. C ONCLUSIONS pp. 42 244–42 253, 2015.
The dynamic optimization of a turbo-compressor, simulated [10] G. Sequenzia, S. Oliveri, G. Fatuzzo, and M. Calı̀, “An advanced multi-
body model for evaluating riders influence on motorcycle dynamics,”
through a complete model with flexible bodies which repro- Proceedings of the Institution of Mechanical Engineers, Part K: Journal
duced all the interactions between the parts of the system was of Multi-body Dynamics, vol. 229, no. 2, pp. 193–207, 2015.
performed. The validity of the model was verified through [11] M. Calı̀, S. M. Oliveri, G. Sequenzia, and G. Fatuzzo, “An effective
model for the sliding contact forces in a multibody environment,”
the comparison of data obtained by calculation with that in Advances on Mechanics, Design Engineering and Manufacturing.
measured experimentally provided by ENICHEM. The models Springer, 2017, pp. 675–685.
usefulness is demonstrated by making it possible to study [12] G. Dubois, F. W. Ocvirk, and R. Wehe, Study of effect of a non-
newtonian oil on friction and eccentricity ratio of a plain journal
typical situations of malfunction of the real system, such as bearing. National Aeronautics and Space Administration, 1960, vol.
the jibbing of the toothed coupling, the most frequent cause of 427.
alarms signalled by the vibration monitoring system. The study [13] G. Sequenzia, S. Oliveri, M. Calabretta, G. Fatuzzo, and M. Cali, “A
allowed the simulation of the functioning of the rotors at op- new methodology for calculating and modelling non-linear springs in
the valve train of internal combustion engines,” SAE Technical Paper,
erating speed, performing an investigation into the maximum Tech. Rep., 2011.
permitted misalignment between the shafts and the coupling. [14] G. Sequenzia, S. Oliveri, and M. Calı̀, “Experimental methodology for
It was possible to verify the entity of the displacements of the tappet characterization of timing system in ice,” Meccanica, vol. 48,
no. 3, pp. 753–764, 2013.
the bearing pivots, and analyse the strains and the stress [15] M. Géradin and A. Cardona, Flexible multibody dynamics: a finite
states in the toothed bells of the turbine-compressor coupling. element approach. Wiley, 2001.
31
[16] F. Bonanno, G. Capizzi, S. Coco, A. Laudani, and G. L. Sciuto,
“A coupled design optimization methodology for li-ion batteries in
electric vehicle applications based on fem and neural networks,” in
2014 International Symposium on Power Electronics, Electrical Drives,
Automation and Motion, June 2014, pp. 146–153.
[17] M. Calı̀, D. Speranza, and M. Martorelli, “Dynamic spinnaker per-
formance through digital photogrammetry, numerical analysis and ex-
perimental tests,” in Advances on Mechanics, Design Engineering and
Manufacturing. Springer, 2017, pp. 585–595.
[18] M. Calı̀ and F. L. Savio, “Accurate 3d reconstruction of a rubber mem-
brane inflated during a bulge test to evaluate anisotropy,” in Advances on
Mechanics, Design Engineering and Manufacturing. Springer, 2017,
pp. 1221–1231.
[19] B. J. Hamrock, S. R. Schmid, and B. O. Jacobson, Fundamentals of
fluid film lubrication. CRC press, 2004.
[20] Y. Cui, Y. Wang, and J. Zhong, “Numerical analysis on the nonlinear
hysteresis phenomenon associated with instability of a steam turbine
rotor-bearing system,” in Proceedings of the 9th IFToMM International
Conference on Rotor Dynamics. Springer, 2015, pp. 2071–2081.
[21] E. Zanetti, S. Musso, and A. Audenino, “Thermoelastic stress analysis by
means of a standard thermocamera,” Experimental Techniques, vol. 31,
no. 2, pp. 42–50, 2007.
[22] F. Bonanno, G. Capizzi, G. L. Sciuto, D. Gotleyb, S. Linde, and
R. Shikler, “Extraction parameters and optimization in organic solar
cell by solving transcendental equations in circuital models combined
with a neuroprocessing-based procedure,” in 2016 International Sympo-
sium on Power Electronics, Electrical Drives, Automation and Motion
(SPEEDAM), June 2016, pp. 872–877.
[23] F. Bonanno, G. Capizzi, and G. L. Sciuto, “A neuro wavelet-based ap-
proach for short-term load forecasting in integrated generation systems,”
in 2013 International Conference on Clean Electrical Power (ICCEP),
June 2013, pp. 772–776.
32