Harald Roclawski
email: roclawsk@mv.uni-kl.de

Marc Gugau

Martin Böhle

S Y Mpo

O Tatin

CFD Analysis of a Radial Turbine during Load Step Operation of an Automotive Turbocharger

Keywords:

A method for evaluating the transient performance of a turbocharger are so called load step tests. In these test, the load of the engine is increased at constant engine speed and the time measured from the start to the end of the load step is measured. Usually these tests can be run relatively late in the development process since hardware needs to be already available. In order to judge the transient turbocharger performance at an earlier stage, engine process simulations are run using maps of compressor and turbine. For the turbine these maps usually need to be extrapolated, since only a certain range of each speed line can be measured on a standard gas stand. Furthermore, because of the exhaust gas pulsation of the engine, it is known that the turbine performance differs from the steady state case which the maps rely on. This has to be respected by additional models. Using CFD simulations, the transient performance of the turbine can be analyzed independent from steady state maps. So far these investigations have been usually performed with a constant turbine speed. In this paper a method is presented which includes the speed fluctuations of the turbocharger caused by the exhaust pulsations as well as the change in mean speed during the load step by including compressor and engine in the CFD analysis with User-Fortran models. Results for a load step from 21000 rpm to 196400 rpm are discussed.

INTRODUCTION

The unsteady operation of radial turbines for automotive turbochargers under pulsating inflow conditions has been in the focus of research for many years. In [START_REF] Baines | Turbocharger turbine pulse flow performance and modelling 25 years on[END_REF] a good summary of experimental and numerical work is presented. In case of numerical investigations, usually a single engine operating point is analyzed. The change in turbine speed due to the pulsations is usually neglected, because the change is very small. Another important unsteady operation of radial turbines are the so called load step tests which are used by the car manufacturers to evaluate the ability of the turbocharger to provide boost pressure as fast as possible [START_REF] Scheller | Potentialbewertung unterschiedlicher turbinenradgeometrien für turboaufgeladene ottomotoren[END_REF]. In these tests, the engine speed is kept constant. The test starts at steady state conditions at a defined engine torque, e.g. at 20% of the maximum torque. Then the load is increased until 90% of the maximum toque is reached. The time from 20% to 90% torque is measured and used to judge the transient performance of the turbocharger and engine. The load step tests are usually run at engine test stands. So, the availability of engine and turbocharger hardware is already required. A way of predicting the transient performance of the engine at an earlier stage in the development process is the application of 1D engine simulation software. However in this case a good model of the turbine map is necessary. Usually the turbine maps cannot be measured completely on a gas stand due to the limited operating range of the compressor which is used CFD Computational Fluid Dynamics as brake for the turbine. Therefore the measured maps have to be extrapolated, providing possibly an additional source for errors. A different approach is the usage of turbine maps obtained by CFD models in combination with a friction loss model as presented in [START_REF] Gugau | On the design and matching of turbocharger single scroll turbines for pass car gasoline engines[END_REF]. Also, the emptying and filling behavior of the turbine volute caused by the pulsating inflow is usually not considered in the 1D simulation. All these approaches have in common that no detailed analysis of the flow inside of the turbine is available, so that no detailed information about why a certain turbine performs better than another during a load step is available. In the development of turbomachinery, the application of Computational Fluid Dynamics (CFD) is the state of the art for many years to investigate the flow inside of the machines. It was already shown that with CFD simulations the performance of turbocharger turbines under pulsating flow conditions can be predicted with adequate accuracy ( [START_REF] Padzillah | Numerical and experimental investigation of pulsating flow effect on a nozzeld and nozzleless mixed flow turbine for an automotive turbocharger[END_REF]). Usually these simulations are run with constant turbine speed. It is known from experiments that due to the changing power balance between turbine and compressor which is caused by the exhaust pulsations, the turbocharge speed fluctuates with an amplitude of about maximum 5% of the mean speed ( [START_REF] Szymko | Experimental evaluation of turbocharger turbine performance under pulsating flow conditions[END_REF] and [START_REF] Capobianco | Experimental investigation into the pulsating flow performance of a turbocharger turbine in the closed and open waste-gate region[END_REF]). These changes are usually considered as small and are neglected in the CFD analysis. In contrast to the unsteady operation at pulsating inflow conditions, the change in turbine speed cannot be neglected in the load step case. In [START_REF] Sailer | Neue methoden in der turboladersimulation für die auslegung der bmw twinpower turbomotoren. 6. MTZ Fachtagung Ladungswechsel im Verbrennungsmotor[END_REF] it was already demonstrated that it is possible to run CFD simulations of turbine and compressor with variable rotor speed. In this case, the change in speed during the load step obtained from measurements was applied as boundary condition to the CFD models. A good match with experimental data was achieved. However also in this case hardware has to be already present since the change in rotor speed has to be known in advance. In [START_REF] Sailer | Neue methoden in der turboladersimulation für die auslegung der bmw twinpower turbomotoren. 6. MTZ Fachtagung Ladungswechsel im Verbrennungsmotor[END_REF] the speed fluctuations caused by the exhaust pulsations were not included in the simulations due to the insufficient temporal resolution of the speed sensor. In this paper a method for simulating a load step by means of CFD taking into account the change in mean turbine speed as well as the speed fluctuations is presented. With this method it is possible to analyze the transient performance of a turbine very early in the design process without relying on measurements. As an example, the application of the method to a 4-cylinder 1.8l Otto engine with a single stage turbocharger with a compressor wheel diameter of 47 mm is discussed.

METHODS

In order to include the change of speed during the CFD simulation, the power balance between turbine and compressor has to be solved. So, an additional model of the compressor is necessary. Also, the behavior of the engine during the load step has to be described, since the change in turbine mass flow, turbine inlet temperature and exhaust gas pulsation have to be used as boundary conditions for the CFD model. In this work, Compressor and Engine are modeled by User-Fortran. The model is described in the next section and is used for solving the power balance between compressor and turbine and for calculating the turbine mass flow and turbine inlet temperature by a simplified thermodynamic cycle analysis.

User-Fortran Model

In the User-Fortran model, the compressor map is used for calculating the operating point of the compressor. At each time step, the speed line of the compressor is extracted from the map by linear interpolation for the current speed of the turbocharger. By intersecting the compressor speed line with the engine speed line, the current operating point of the compressor (flow rate, pressure ratio and efficiency) can be calculated. At the beginning of the load step, the turbocharger operates at small speeds. Therefore the compressor map has to be extrapolated. In the current model this is solved by a linear extrapolation of the measured speed lines. It was already shown in [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF], where a load step simulation also relying on the compressor map extrapolation was compared to measurements, that the compressor map extrapolation gives reasonable results. The engine speed line is calculated according to Equation 1. Equation 1 is originally from [START_REF] Pucher | Aufladung von verbrennungsmotoren[END_REF] and is modified by multiplication with the factor C which is used to control the change of the engine speed line during the load step. The factor C can be obtained by comparing the calculated engine speed line to a measured compressor operating point. If the operating point does not lie on the speed line, the speed line has to be adjusted by modifying the factor C until the operating point matches the speed line. In this way, Equation 1 can be calibrated to a real engine.

π = C • 2 λ l • V h • T 2 T 1 • 1 n M • V (1)
Figure 1 shows an arbitrary operating point of the compressor as intersection of compressor and engine speed line. The load step is initiated by the opening of the throttle. When the throttle opens, the swallowing capacity of the engine is increased and the slope of the engine speed line becomes smaller. The opening of the throttle is simulated by decreasing the value of the factor C in Equation 1 for a defined number of timesteps. The intersection of compressor speed line and engine characteristic changes and the compressor is able to deliver more air to the engine.

With an engine model, the conditions of the exhaust gas at the turbine inlet are calculated according to the engine inlet conditions provided by the compressor. The exhaust gas mass flow is a function of the air to fuel ratio and the compressor mass flow. The air to fuel ratio is kept constant during the load step. For calculating the exhaust gas temperature, a thermodynamic cycle calculation for spark-ignition engines is performed consisting of a polytropic compression, heat addition at constant volume and a polytropic expansion. With the polytropic compression the gas temperature after compression can be calculated. The combustion is modeled by an energy balance. In this case the heat added equals the energy necessary to increase the gas temperature after polytropic compression to the combustion temperature and the heat loss into the cooling system and oil. The heat loss is modeled by empirical values. The energy balance can be solved for the combustion temperature. The turbine inlet temperature can then be calculated by the polytropic expansion. Additionally, the exhaust gas mass flow is not only increasing with load step time, but also modeled as pulsating flow as in real gasoline engines. The pulsation model is based on pulse shapes and an engine map of the pulsation number which can be calculated according to Equation 2. It is a measure for the amplitude of the exhaust pulses and a function of engine torque and speed. The frequency of the exhaust pulses depends on the engine speed.

Pulsation number = p ma x -p min p mean (2)
The map of the pulsation number was created from a calibrated GT Power model of a 1.8l Otto engine. During the load step calculation, the pulsation number can be calculated from the map as a function of engine speed and boost pressure (p mean ). At different engine speeds and a range of engine loads the calculated mass flow pulse shapes were normalized and an average pulse shape was derived. During the calculation, the pulsation number is obtained from the map based on boost pressure. Then the normalized pulse shape is scaled according to the pulsation number. Figure 2 shows as an example mass flow pulses at several moments in time during the load step. It is planned to improve this method in the future by also taking into account the depence of pulse shape on engine load.

The engine model can be calibrated to a real engine with the following parameters: engine speed line, engine compression ratio, engine heat loss, the polytropic coefficient for compression and expansion and the map for describing the exhaust gas pulsations. So far these calculations are performed by the User-Fortran model. From there, the turbine mass flow, turbine inlet temperature and actual turbocharger speed are passed as boundary conditions to the CFD model of the turbine.

After each time step, the turbine power is evaluated and passed back to the User-Fortran model. The mechanical efficiency is determined using a friction loss model where the losses are modeled as a polynomial function of turbine speed. Then the power balance of compressor and turbine is solved according 3.

n T C = dt * (P T -P C -P F ) Θ * n i * 4 * pi 2 + n i (3)
If the turbine power exceeds the compressor power, the turbocharger speed is increased. With the new speed the operating point of the compressor is moved along the engine speed line (Equation 1). Then the new operating point of the compressor can be calculated and the loop through engine model and CFD analysis starts again until a certain boost pressure is reached. At that time the load step is finished. Figure 3 shows a schematic of the User-Fortran model. The model has been already used successfully and compared to experiments for a 1D-simulation of a load step in [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF] where a map obtained from CFD simulations was used to model the turbine. In this work, the turbine side is now replaced by a CFD model.

Numerical Model

For the CFD analysis, a model of the complete turbine including volute, wheel and wheel cavities was created (Figure 4). In the future the effect of the inlet section on the filling and emptying behavior will be investigated and it will be tried to use a different boundary condition by specifying a flow profile directly at the volute inlet.

For the load step simulation, a time of 3.6s was calculated. The computational cost will greatly depend on the chosen time step. On the other hand it is necessary to resolve the exhaust gas pulsations as well as the rotation of the turbine wheel. It is possible to use a constant time step or an adaptive time step according to the turbocharger speed. In order to save computation time a constant time step of 1.11 • 10 -4 s was used which corresponds to a crankshaft angle of one degree for an engine speed of 1500 rpm. With this time step a good resolution of the exhaust gas pulsation can be achieved.

Since the time scale of the rotating turbine wheel is much smaller than the time scale of the exhaust pulsation under investigation the frozen rotor model rather than the sliding mesh model was used as frame change model which helps to reduce the computational cost of the transient simulation. The same model approach was chosen for instance by [START_REF] Aymanns | A revision of quasi-steady modelling of turbocharger turbines in the simulation of pulse charged engines[END_REF] and [START_REF] Padzillah | Numerical and experimental investigation of pulsating flow effect on a nozzeld and nozzleless mixed flow turbine for an automotive turbocharger[END_REF], confirming the general validity of this simplification. For each time step 10 iteration steps are applied. With these settings, the residuals of momentum, turbulence and energy equation are at the beginning of the load step below 1.6 • 10 -5 . Convergence becomes then worse at higher turbine speeds, but the residuals are still below 3.2•10 -4 . So the chosen time step is appropriate for this kind of simulation and is a good compromise between accuracy and computational cost. However further research in form of numerical studies is necessary to determine optimal settings. It took 6 weeks to finish the load step simulation on one compute node with 16 CPUs and 2.6 GHz CPU speed.

As boundary conditions for the fluid domains, the rotational speed of the turbine wheel was applied to the wheel domain.

At the inlet, mass flow and total temperature were specified. At the outlet, a static pressure boundary condition was used.

RESULTS

The simulation of a load step with the method proposed in this paper enables the designer to evaluate the basic turbocharger performance by tracing the operating points of turbine and compressor and other quantities as the development of the boost pressure or the turbocharger speed. In addition it is possible to analyze the flow field inside of the turbine. Another parameter of interest is the axial thrust of the turbine. One can see that it takes some time until the quasi-steady state conditions at 21000 rpm are reached. This depends on the quality of the initial boundary conditions for the compressor and for the calculation of the engine speed line. Figure 6 also shows that due to the inertia of the rotor, the mean pulse averaged speed is not increased immediately after the throttle opens. It takes 0.128s until the speed increases. Figure 7 shows the compressor map, the compressor operating points as well as the change of the engine speed line during the load step. Due to the modification of the factor C in Equation 1the change of the engine speed line is large at the beginning. Therefore the operating points of the compressor move nearly Later, the change in engine speed line is small and the operating points of the compressor are shifted along the engine speed line to higher pressure ratios. Figure 8 shows the compressor pressure ratio for the first second of the simulation. At the beginning of the load step, the compressor operates at a pressure ratio of 1.02. The time delay between the start of the load step and the time when the boost pressure is increased is 0.1645s and thus greater than the time delay for the speed increase. The reason for this the large change in engine speed line at the beginning of the load step which moves the operating point horizontally in the compressor map although the speed is already increasing. With the operating point of the compressor, the compressor power can be calculated and the power balance between turbine and compressor is solved. Figure 9 shows the power of compressor and turbine during the load step. Both turbine and compressor power are fluctuating due to the exhaust gas pulsations. The amplitude of the fluctuation of turbine power is much larger than for the compressor since the change in turbocharger speed is damped by the inertia of the rotor. It can also be seen that sometimes the compressor power exceeds the turbine power. This is the case at each minimum of the exhaust pulses and the reason for the fluctuation of the turbocharger speed (Figures 5 and6). The power to accelerate the rotor can be calculated by 4. It is plotted in Figure 10. At the beginning of the load step, the acceleration power is small and in the order of magnitude of the friction power of the bearing system. After the initiation of the load step the power increases and the rotor starts to accelerate. Until 0.6s, the compressor power is always smaller than the turbine power and the acceleration power is completely positive. Then, the acceleration power becomes occasionally negative when the compressor power is large enough to exceed the turbine power. This is the case when the turbine operates close to the pulse minimum. During the load step, the rotor is accelerated and decelerated periodically according to the exhaust gas pulsations.

Basic performance analysis

P accel = P T ur bine -P C om pr essor -P F r ict ion (4)

As it is well known in the literature, the behavior of the turbine under pulsating inflow conditions differs from steady state operation due to mass storage effects in the volute [START_REF] Baines | Turbocharger turbine pulse flow performance and modelling 25 years on[END_REF]. Therefore, the turbine operating points under pulsating flow conditions do not match the steady state turbine map. Instead, a hysteresis loop is observed when plotting the turbine flow parameter vs the pressure ratio. Figure 11 shows the steady state turbine map obtained by CFD calculations and the evolution of the hysteresis loops during the load step operation.

During the load step, the turbine operating points cover almost the complete map of the turbine in a range of pressure ratio from 1 to 3.75. One has to keep in mind that in contrast to the hysteresis loops discussed in the cited literature, the turbine speed is not constant, but is a superposition of the change in of the hysteresis loops can be discussed better using Figure 12 where only the hysteresis loops at four selected points in time are plotted. At the beginning of the load step, the pressure pulsation is small and thus, the change in turbine speed and the hysteresis loop is small, too. The operating points move approximately along a steady state speed line of the turbine at a pressure ratio between 1.2 and 1.4. During the progress of the load step, the exhaust pulsations and the mean total pressure at turbine inlet increase. The hysteresis loops are shifted along the turbine speed lines to higher pressure ratios and the deviation to the steady state map becomes greater. Also the range of pressure ratio during one hysteresis loop increases as the load step progresses. At the end of the load step, the pressure ratio changes between 1.6 and 3.7. Similar to Figures 11 and12, the operating points during the load step are plotted for the turbine wheel only in Figures 13 and14. Here the flow parameter and pressure ratio are calculated using the flow conditions at inlet and outlet of the turbine wheel. It was shown in several publications (e.g. [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF], [START_REF] Aymanns | A revision of quasi-steady modelling of turbocharger turbines in the simulation of pulse charged engines[END_REF], [START_REF] Copeland | A comparison of pulsed flow timescales within a turbine stage[END_REF] and [START_REF] Cao | Radial turbine rotor response to pulsating inlet flows[END_REF]) that the turbine wheel can be considered as quasi-steady when operating under pulsating flow conditions. In this case the hysteresis loop is very small and the operating points lie nearly on the steady state map of the turbine wheel. In Figure 13 all operating points of the turbine during the load step are plotted when evaluating flow parameter and pressure ratio at the inlet and outlet of the turbine wheel. In contrast to Figure 11, it seems that for this case a quasi-steady behavior of the turbine wheel can be observed, since all curves seem to lie on the steady state map. The operating points cover the complete map of the turbine. This shows the importance of turbine maps with a wide range for the performance analysis under transient operating conditions. A more precise assessment is possible when looking at the same selected range of operating points as in Figure 12 for the case when flow parameter and pressure ratio are calculated from the conditions at turbine wheel inlet and outlet (Figure 14). Similar to Figure 12, at the beginning of the load step, a quasi-steady state behavior of the turbine wheel can be observed since the operating points seem to lie on the steady state speed curve. During the load step, the operating points in the observed range are shifted along the steady state speed lines to higher pressure ratios. Also, a small hysteresis seems to develop as the load step progresses. This might be due to the increasing exhaust pulsations but also due to the change in mean speed of the turbine. Looking at the steady state maps, the distance in speed lines increases with increasing speed. So this might be a reason for the hysteresis. Also it seems, that at the end of the load step the hysteresis of the turbine wheel becomes smaller again, because there the load step is finished and the mean speed is constant. In [START_REF] Copeland | A comparison of pulsed flow timescales within a turbine stage[END_REF] it was shown that a rotor hysteresis occurs when the gradient of the exhaust pulsation becomes larger. Therefore the change of the pressure pulse might be another reason for the different shape of the rotor hysteresis. The influence of mean speed on Figure 14. Map of turbine wheel without volute and hysteresis loops for selected operating points rotor hysteresis might be not of great significance, but in order to get a deeper understanding, further research is necessary. It might be possible to find a better explanation by comparing these results with CFD simulations with different pulse shapes and constant speed as well as with oscillating turbine speed while the mean speed is constant.

Flow analysis

The method of simulating a load step presented in this paper enables the designer of turbochargers to judge the performance of the turbine very early in the design phase. Also the flow in the turbine can be analyzed. As an example, the flow at the turbine wheel inlet will be discussed. In this paper absolute and relative flow angles are defined against the meridional direction. Negative values indicate a flow against the direction of rotation of the turbine wheel. The absolute flow angle and incidence angle during the load step is plotted in Figure 15. As expected from the basic theory of volute design, the absolute flow angle should be nearly constant during the load step, since it is independent of mass flow. In contrast, the incidence angle β i depends on mass flow and shows therefore a great change during the load step due to the increase in turbine mass flow and due to the change in mass flow caused by the exhaust pulsations. At the beginning of the load step, the mean value for β i is at about 50 • . Optimum values for radial turbines are usually in a range of -20 • to -40 • [START_REF] Japikse | Introduction to turbomachinery[END_REF]. So at the start, the turbine operates far beyond the optimum range. During the load step, the relative flow angle is improved and changes from positive to negative incidence to a mean value of approximately -30 • . Due to the exhaust pulsation large fluctuations in incidence can be observed which are superimposed to the mean change in incidence. The amplitude of the fluctuations increases over load step time and decreases a bit at the end of the load step. As already published in several papers, the incidence during a single exhaust pulse is usually worst at the pulse maximum, when the most energy is available for the turbine and is usually in the negative range during the pulse minimum [START_REF] Cao | Radial turbine rotor response to pulsating inlet flows[END_REF] and [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF]. In Figures 16, 17 and 18, streamlines at 50% span are plotted in a blade-to-blade view of the turbine wheel at the beginning, in the middle and at the end of the load step. In each figure, the situation at pulse minimum, pulse mean and pulse maximum are plotted. The basic findings from Figure 15 are confirmed. At the beginning of the load step (Figure 16) the incidence angle is positive for all three cases and a flow separation at the suction side of the blade can be observed. Since the exhaust pulsation is small, the fluctuation of incidence is small to. For all cases β i is between 45 • and 55 • . At the middle of the load step, in (Figure 17), the level in incidence angle is improved and changes from negative to positive during the pulsation. At the pulse minimum, the negative incidence is large and flow separation at the pressure side of the leading edge can be observed. Furthermore in some rotor channels, flow separation occurs at the suction side of the blade. At the pulse mean, incidence is small and the flow field has improved. No flow separation can be observed. For the maximum pulse, positive incidence is large and the flow separates at the suction side of the leading edge. At the end of the load step (Figure 18), the incidence is completely negative. At the pulse minimum, the incidence angle is beyond the optimal range and the flow separates at the pressure side of the blade. For pulse mean and pulse maximum, the flow field is improved and only a small separation can be observed at the leading edge of the blade. The analysis of the flow field shows the challenge of designing a turbine for automotive turbocharger applications. Instead of just one or a few operating points, which is usually the case in turbomachinery design, the designer has to optimize the turbine for a wide range of operating points.

Axial thrust

The overall axial thrust of the turbocharger is an important measure for designing the bearing system and can be used for modeling the friction loss of the bearing system. The axial thrust can be measured with a high effort ( [START_REF] Smiljanovski | Tc thrust load measurements[END_REF]), can be calculated with models or by means of CFD simulations ( [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF]). In this paper, the axial thrust of the turbine wheel during the load step is calculated. Figure 19 shows the axial thrust for the Due to the exhaust pulsation the axial thrust is not constant, but fluctuating. So the axial thrust bearing is loaded periodically according to the pulse frequency. At the beginning of the load step, the axial thrust is negative. Then during the load step, the mean value increases. After 1.2s it becomes completely positive. The reason for the change in direction is, that at the beginning of the load step, the forces acting on the hub and blades are greater than the forces acting on the back of the wheel in the rear cavity. The total force acts towards the compressor side. During the load step, the turbine pressure ratio increases and the forces on the back of the wheel exceed the forces on hub and blades. Then the total force acts towards the turbine side. The pressure in the rear cavity is influenced by the exhaust pulsations and therefore the axial thrust fluctuates accordingly. The amplitude of the total axial thrust increases over time until a value of 40N. In order to judge the overall effect on the bearing system, also the axial thrust of the compressor wheel has to be considered. This will be done in the future with a model which solves for the pressure and momentum forces acting on the compressor wheel. Another way to calculate the axial thrust of the compressor is to calculate the compressor operating point by means of CFD, too, as it was already presentend in [START_REF] Sailer | Neue methoden in der turboladersimulation für die auslegung der bmw twinpower turbomotoren. 6. MTZ Fachtagung Ladungswechsel im Verbrennungsmotor[END_REF]. But, this would increase the computational cost considerably.

CONCLUSION

A method for simulating the load step of a radial turbine of an automotive turbocharger by means of CFD has been presented in this paper. It was shown how compressor and engine can be modeled using User-Fortran for calculating the power balance between compressor and turbine. According to the power balance, the turbine speed was changed dynamically during the load step calculation. The analysis of the turbine operating points during the load step reveals that also in case of the load step hysteresis loops can be observed since the time scale of the exhaust pulsations is smaller than the change in mean speed of the turbine. For the rotor, which is usually considered as being quasi-steady, it seems that there are also small hysteresis loops which might be caused by the change in mean speed and by the speed fluctuations caused by the exhaust pulsations. However, to support this hypothesis, further research with the User-Fortran model is necessary in the future. Analyzing the flow angles at the rotor inlet, it was shown that at the beginning of the load step the turbine has to operate under unfavorable positive incidence angles. During the load step, incidence is improved from positive to negative angles. Because of the exhaust pulsations incidence is fluctuating with different amplitudes during the load step. At the beginning also the fluctuations show completely positive values, then, as the mean incidence becomes better, fluctuations from positive to negative have been observed. Later, the fluctuations become completely positive. The axial thrust analysis reveals that during the load step the direction of the force changes its direction. At the beginning, the axial thrust acts towards the compressor side. As the turbine pressure ratio increases, the pressure in the rear cavity increases and the axial forces changes its direction towards the turbine side. Also, the axial thrust fluctuates according to the frequency of the exhaust pulsations. The User-Fortran model of engine and compressor in combination with the CFD analysis of the turbine enables the designer to analyze the transient performance of the turbine during the load step at an early stage in the design process without relying on map extrapolation and volute modeling for including the effect of exhaust pulsations. In the future, the method will be further investigated and improved. It is already known that the Frozen-Rotor model provides accurate results in the case of CFD simulations of radial turbines under pulsating flow conditions. In the case of the load step this still has to be investigated. Therefore additional simulations using a sliding mesh approach for modeling the rotation of the turbine wheel will be run and compared to the results presented in this paper. In order to reduce the computational cost for both frozen rotor and sliding mesh model, the optimum number of time steps and the number of iteration step per time step will be determined in numerical studies. In this context, it will also be investigated, if the usage of a constant time step or an adaptive time step based on the instantaneous turbine speed is a better choice with respect to accuracy and computational cost. Also, the model will be validated by comparison to experimental results of a load step on an engine test stand. The general ability of the model to include the speed fluctuations of the turbine caused by the exhaust gas pulsations will be validated on a turbocharger test rig with a pulse generator device. Finally, since the computational cost of the method is very high, the model will be mainly used to improve 1D methods for simulating turbine operation under pulsating flow conditions for load step simulations as well as for basic turbocharger matching tasks. In [START_REF] Roclawski | Influence of degree of reaction on turbine performance for pulsating floe conditions[END_REF] the same model was used to calculate a load step by using extrapolated turbine maps instead of the CFD model of the turbine. After calibration, the comparison to real engine measurements has shown a good agreement. The advantage of the 1D model is that it can be used investigate the turbine performance under load step conditions in a very fast and easy way. In order to further improve this model with respect to the influence of the exhaust pulsations, the CFD results will be compared to the 1D results in the future.

F a x

 x Axial thrust P C Compressor power P T Turbine power P F Friction power TC Turbocharger T F P Turbine flow parameter T 2 Temperature at compressor outlet T 1 Temperature at compressor inlet V h Engine displacement Vred Reduced Volumetric Flow Rate c Absolute velocity dt Time step w Relative velocity u Blade speed n T C Rotational speed of turbocharger n i Rotational speed at timestep i n M Engine speed Θ Moment of Inertia α Absolute flow angle β Relative flow angle λ l Lambda fresh air π Pressure Ratio

Figure 1 .

 1 Figure 1. Operating point of compressor

Figure 2 .

 2 Figure 2. Pulse shapes during the load step.

Figure 3 .

 3 Figure 3. Schematic of Load Step Simulation

Figure 4 .

 4 Figure 4. CFD model of radial turbine

Figure 5

 5 Figure 5 shows the change in speed of the turbocharger during the load step. The load step starts at quasi-steady state conditions, thus at a constant mean speed of 21000 rpm. The speed fluctuations are caused by the changing power balance between compressor and turbine and thus due to the exhaust pulsations. The change in speed is at the start of the load step below 1%. The red circle in Figure 5 indicates the opening of the throttle of the engine. Then the speed increases until a mean speed of 196400 rpm, when quasi-steady conditions are achieved again. There the change in speed is below 2%. The start of the load step is shown in Figure 6 in more detail.

Figure 5 .

 5 Figure 5. Turbocharger Speed during loadstep simulation

Figure 6 .

 6 Figure 6. Steady state conditions at start of loadstep

Figure 7 .

 7 Figure 7. Compressor Operating Points

Figure 8 .Figure 9 .

 89 Figure 8. Compressor Operating Points

Figure 10 .Figure 11 .

 1011 Figure 10. Acceleration Power

Figure 12 .

 12 Figure 12. Turbine map and hysteresis loops for selected operating points

Figure 13 .

 13 Figure 13. Map of turbine wheel without volute and unsteady operating points

Figure 15 .

 15 Figure 15. Absolute and incidence angle at turbine wheel inlet

Figure 16 .

 16 Figure 16. Streamlines at 50% span at t=0.743-0.75s.

Figure 17 .

 17 Figure 17. Streamlines at 50% span at t=1.664-1.669s.

Figure 18 .

 18 Figure 18. Streamlines at 50% span at t=3.269-3.273s.

Figure 19 .

 19 Figure 19. Axial thrust of turbine wheel during load step


Harald Roclawski
email: roclawsk@mv.uni-kl.de

Claus Oberste-Brandenburg

Martin Böhle

S Y Mpo

O Tatin

Conjugate Heat Transfer Analysis of a Centrifugal Compressor for Turbocharger Applications

Keywords:

For 2-stage turbocharging, high inlet temperatures to the high compressor stage represent an additional load for the compressor wheel which must be taken into account in the design process of the compressor. In this paper, a heat transfer analysis of a high pressure compressor for commercial vehicle applications is presented. Simulations were run with a conjugate heat transfer model. For the simulations, only the compressor side of the turbocharger is modeled. Bearing housing and turbine stage are considered by choosing appropriate thermal boundary conditions at the suitable interfaces. The simulations were validated with experiments run on a gas stand. For the experiments, temperature was measured by means of a telemetry system on 7 locations in the compressor wheel and on the compressor housing. The results are discussed for two operating points with different compressor inlet temperatures.

INTRODUCTION

For many years, turbocharging has been one of the key technologies for achieving high specific power output of internal combustion engines taking also into account more and more stringent emission limits. Additionally, car manufactures attach high importance to a good drivability, which usually is being quantified as a target torque already available at low engine speeds that is fast reached in transient response operation. High specific power output requires high boost pressures. This can be achieved by using a single stage turbocharger but with an unacceptable penalty in transient engine operation [START_REF] Guilain | Optimization of a small two-stage turbocharged diesel engine[END_REF]. Therefore, for highly downsized engines, regulated 2-stage turbocharging systems (R2S) are applied in order to overcome the conflict of high specific power and simultaneously low specific fuel consumption in agreement with the emission legislation. In case of R2S-systems, two turbochargers of different size are arranged in series. The power of both turbochargers can be regulated by bypass valves and adjusted to the requirements of the combustion engine. Heat transfer is an important issue for predicting the performance of a turbocharger during the matching process. In general, heat is transferred from the turbine through the bearing housing to the compressor side. Depending on the operating point of the turbocharger, the heat transfer process in the compressor changes. According to [START_REF] Shaaban | Impact of turbocharger nonadiabatic operation on engine volumetric efficiency and tubro lag[END_REF] at small speeds, heat is transferred to the compressor and only a small amount of heat is transferred through the compressor housing to the ambient. In this case usually the isentropic efficiency of the compressor is calculated too small because of the higher outlet temperature. At high compressor speeds, also heat is transferred to the compressor, but due to the high pressure ratio also a significant amount of heat is transferred to the casing. However the net amount of heat transfer to the compressor is small and thus the effect on compressor performance. The compressor appears to be adiabatic. Besides [START_REF] Shaaban | Impact of turbocharger nonadiabatic operation on engine volumetric efficiency and tubro lag[END_REF] many researchers investigated the effect of heat transfer on turbocharger performance numerically and experimentally (e.g. [START_REF] Heuer | Conjugate flow an heat transfer investigation of a turbo charger: Part i: Numerical results[END_REF], [START_REF] Gu | Numerical study of the heat transfer effect on a centrifugal compressor performance[END_REF]). These investigations are usually performed with single stage compressors with inlet temperatures at ambient conditions. One can expect that the heat transfer mechanism in the compressor differs for the high compressor stage due to the higher inlet temperatures. Another reason for studying heat transfer in turbochargers is the thermal load of the wheels and casings (e.g. [START_REF] Oberste-Brandenburg | On the influence of thermal boundary conditions on the thermo mechanical analysis of turbine housing of a turbocharger[END_REF] and [START_REF] Makarenko | Thermomechanical behabiour of turbocharger compressor wheels[END_REF]). Especially in case of R2S turbochargers, for the high-pressure compressor wheel, the load in terms of temperature can be significantly higher than in the single stage case. Therefore it is important to take into account the influence of high compressor wheel temperatures on durability in the design process. In the current paper, a conjugate heat transfer analysis (CHT) for predicting the temperature load on compressor wheels for turbochargers is presented. The results are compared to experimental data obtained under steady state conditions on a gas test stand where the compressor wheel temperatures are measured by means of a telemetry system. The intention of the simulation is to get a deeper understanding of the heat transfer in the high pressure compressor stage, to extend the range of operating points to higher compressor speeds which cannot be covered by the telemetry system and to learn how to create a reliable CHT-model in order to investigate other compressors in the future, where no telemetric measurements are available or possible. The latter one is especially important for passenger car applications where the thermocouples and a telemetry system cannot be installed due to the smaller compressor wheels.

METHODS

For the heat transfer analysis, an experimental and numerical approach was chosen. The experiments are necessary to validate the numerical simulations and to provide boundary conditions at the interface between compressor and bearing housing as well as on the ambient surfaces. In the following sections, the experimental setup and the numerical model are described.

Experimental Setup

For the validation of the CHT analysis, a R2S turbocharger for commercial vehicle applications was mounted on a gas stand test rig. The high pressure compressor was instrumented with thermocouples for measuring the temperatures on the volute, the compressor back plate, the bearing housing and on the compressor wheel. For the compressor wheel, temperature was measured at 7 locations (Figure 1), at the tip of the compressor wheel (CW1), at the fillet of the blades (CW2), at the back side of the wheel (CW3 and CW6), close to the bore of the compressor wheel (CW4 and CW5) and at the shaft (CW7). The position of the thermocouples is given in non-dimensional form for the axial and radial direction. The signals of the thermocouples located in the compressor wheel were transferred via a telemetry system to the data acquisition board. The transmitter of the telemetry system is placed in a sleeve at the end of the shaft at the compressor side (Figure 3). The wiring from the thermocouples to the transmitter is realized through additional holes which are drilled in the compressor wheel. For the measurements, operating points which are typical for the high pressure stage of R2S turbochargers were investigated. Of particular interest are operating points with high wheel temperatures resulting from high compressor inlet temperatures and/or high pressure ratios. The drawback of the telemetry system is that the compressor speed is limited to 140000 rpms. Therefore it is currently not possible to acquire temperature data of the compressor wheel in operating points with high speeds and high temperatures which belong to the most interesting in terms of durability. Another drawback of the telemetry system is the fact, that only 4 signals can be transmitted simultaneously. Therefore two test series were run. In the first case, the signals of the thermocouples 1, 2, 4 and 5 were transmitted, in the second case the signals of the thermocouples 1, 3, 6 and 7. So in order to get a complete test series of all seven thermocouples it s necessary to run each operating point twice. The reproducibility of the tests can be checked by analyzing the operating point (pressure ratio, volumetric flow rate, temperatures at compressor inlet T 1 and outlet T 2 ) and the signal of thermocouple CW1 which was used in both tests (see Table 1). The measurement locations on volute casing and back plate are shown in Figure 2. These were used for deriving boundary conditions for the numerical model. In the simulation, a convective boundary condition is applied by specifying the ambient temperature and a heat transfer coefficient. The temperature distribution on casing and back plate can then be calculated and compared to the measurements. In this way, the heat transfer coefficient can be adjusted iteratively until a good agreement with the experimental data is achieved. In addition, three thermocouples were mounted in the diffuser of the compressor (Figure 3). These were used for validating the calculated gas temperatures. Figure 4 shows the compressor map of the high speed compressor of the R2S-system. In addition to the R2S operation, further tests were run operating the high speed compressor as a single stage compressor. In this way, the difference in wheel temperature between single stage and R2S-system can be analyzed. The operating points where temperatures were measured for the single stage are displayed as green triangles, the R2S operating points are displayed as blue diamonds. Figure 4 also shows the operating points for which, so far, CHT simulations were run. These are displayed as red squares in the single stage case and as black dots for the R2S case. In this paper, two operating points for the R2S operation are discussed.These are labeled OP1 and OP2 in Figure 4. Table 1 shows the details of both operating points. OP1 was chosen as an operating point with small pressure ratio and volumetric flow rate, but high inlet temperature. In contrast, OP2 has a higher pressure ratio and volumetric flow rate, but a lower inlet temperature. The difference in outlet temperature for both operating points is small. In Section 2 the differences in

Numerical Model

For the CHT analysis, only the compressor side of the high pressure turbocharger including the piping between the stages was modeled. Figure 5 shows the model for the R2S case. For the single stage calculations, the 180°bend of the inlet piping was replaced by a straight pipe in order to get the same inflow conditions as on the test rig.

All solid parts of the compressor stage were meshed with tetrahedral cells. For the flow domains, hexahedral cells were In the region of interest, the compressor wheel, the y+ values are below 5. As discussed in Section 2, the comparison of the numerical results to the experimental data shows a very good agreement so that it can be assumed that the mesh is appropriate for the current investigation. The simulations were run steady state with the software AN-SYS CFX using the frozen rotor and the k-ω-SST turbulence model. For modeling the heat transfer between the solid parts in general an ideal contact was assumed and where appropriate, a thermal resistance was specified. In order to get a converged solution about 4000 iteration steps were necessary. The residuals are then below 10 -4 . To judge convergence, monitors of the compressor pressure ratio and at all locations where temperature was measured were defined. The calculation was considered as converged when the monitors give constant values.

As boundary conditions for the fluid domains, the rotational speed of the compressor wheel was applied to the wheel domain. At the inlet, total pressure and total temperature were specified. At the outlet, the mass flow rate was specified. Defining the boundary conditions for the solid domains is much more complicated, because there is usually a temperature distribution on the surfaces and not just a constant value.

Therefore, a convective boundary condition was used for the volute casing, the compressor back plate and all pipes. In this case, the heat transfer coefficient was chosen based on previous investigations and in the range of natural convection. At the shaft a constant temperature (T sha f t ) was specified according to the measured temperatures. For the bearing housing, a constant temperature (T BH ) was defined according to previous investigations of the heat transfer in turbochargers.

For the shaft, the temperature of thermocouple CW 7 (Figure 1) was used as boundary condition. Heat transfer by radiation is neglected and not included in the model. All other exterior walls of the outlet and inlet pipes were defined as adiabatic. Figure 6 gives an overview of the applied boundary conditions for the solid domains. For the compressor wheel, the compres-

RESULTS

In this section, the results of the CHT simulations for OP1 and OP2 are discussed. Table 2 shows the comparison of the calculated and measured operating points for OP1 and OP2.

A very good agreement between simulation and experiment was achieved. A comparison of the measured temperatures to the calculated temperatures of compressor wheel, volute, back plate and diffuser is shown in Table 3. The simulation provides a good match for the measurements of the material temperature. In case of OP2 the thermocouples CW3, CW6 and CW7 were broken and did not produce a result. In the case of OP1, the values for TB1 and TB2 are over-predicted by the simulation. In this case the convective boundary condition does not agree with the experience of the authors and needs to be adjusted for future calculations. OP2 The temperature distribution corresponding to the values in Table 3 on the volute is shown in Figure 7 for both operating points. The temperature increases in radial direction in the diffuser and volute due to the deceleration of the flow. A further increase can be observed in the pressure pipe. As OP2 has a larger pressure ratio and the smaller inlet temperature, the maximum temperature and the temperature increase is higher than for OP1.

∆T E X P -T C F D [K] ∆T E X P -T C F D [K] CW1 [K] -1.6 6.5 CW2 [K] -5.6 3.7 CW3 [K] -7.6 - CW4 [K] -6.5 5.2 CW5 [K] -6.3 1.6 CW6 [K] -5.2 - CW7 [K] 0 - TB1 [K] -10.29 0.3 TB2 [K] -8.55 0.92 TV1 [K] -2.35 -1.12 TV2 [K] -2.
Figure 8 shows the temperature distribution on the compressor back plate and the cover of bearing housing. At the bearing housing cover, the temperature is constant as defined by the boundary condition. For the back plate, the temperature increases with radius and is nearly constant in circumferential direction.

The temperature distribution for the compressor wheels is depicted in Figure 9. For both wheels the temperature distribution is uniform in circumferential direction, but there is a gradient in radial direction. For OP1, the temperature increases by approximately 10K, for OP2 by approximately 30K. The larger increase in temperature for OP2 is a result of the higher pressure ratio. However, the temperature level is higher in case of OP1 because of the higher inlet temperature T 1 (see Table 1). In the following, the heat transfer situation for both operating 4, the heat transfer to and from different parts of the compressor wheel is listed. The names of the different parts are defined as in Figure 10. As for the temperature distribution, differences in heat transfer can be observed for both operating points. In case of OP1, heat is added to the compressor wheel at the hub and at the blades. Heat is removed through bore, nut and wheel cavity. So, in this case the compressor wheel is basically heated due to the high inlet temperature.

For OP2, a different heat transfer situation can be observed. Here, almost all heat is added through the wheel cavity and a minor fraction through the shaft. Most of the heat is removed through the blades and also through the hub and nut of the compressor wheel. In contrast to OP1, the heat transfer to the wheel is because of the higher pressure ratio. Due to the higher temperature at compressor wheel outlet, the temperature in the wheel cavity is higher and due to the secondary flow in the cavity, the heat is transferred into the wheel. There is no additional heat transfer from the bearing housing cover (turbine side) in this case, since heat is also transferred from the cavity to the bearing housing cover. Figure 11 shows the temperature distribution in a cross section of the compressor stage and supports the observation from Table 4. However, the situation is a bit more complicated. Figures 12 and13 show the heat transfer at the compressor wheel. Red colors indicate a heat transfer to the compressor wheel, blue colors indicate a heat transfer from the compressor wheel. For both operating points heat is rejected from the wheel in the hub region for diameters smaller than about 70% of the wheel tip diameter and at the blades for a span smaller than 80%. Above these limits, heat is transferred to the wheel. Regarding Table 4, the integral values of blade and hub as well as the direction of heat transfer is different for both operating points because of the different magnitude of heat flux. A considerably different situation can be observed at the back of the compressor wheel. In case of OP1 heat is mainly rejected at the back and the bore for the shaft. In contrast, for OP2, it can be seen that heat transfer to the wheel occurs in the wheel cavity and bore. In Figure 14 the total temperature distribution in radial direction in the diffuser is shown. The temperature values are averaged in circumferential direction. The total temperature decreases with radius for both operating points. According to the first law of thermodynamics for an open system, the total temperature in the diffuser can be calculated by Equation 1 for the case that there is no work, but heat transfer.

T t = T t (r/R = 0) + q c p (1)
Equation 1 shows that the total temperature is decreased when the heat flux becomes negative, e.g. when heat is transferred from the gas to the housing. So also in this case, the simulation provides reasonable results.

In Figure 14, also a comparison with the measured total temperature from the thermocouples located in the diffuser

CONCLUSION

A heat transfer study of the high pressure compressor stage of a R2S-turbocharger has been presented. It was shown that with the applied modeling method results with a good accuracy can be achieved. Two operating points of the high pressure compressor have been discussed. It was shown that the heat transfer inside the compressor changes with higher inlet temperature. In case of OP1, the heat transfer to the compressor wheel is mainly due to the high inlet temperature, whereas in case of OP2 the heat transfer to the compressor wheel is caused by the high pressure ratio. This shows that the inlet temperature is also an important parameter especially in the case of R2S-systems and needs to be taken into account in the matching and mechanical design process. The temperature distribution in the compressor wheel was found to be uniform in circumferential direction. In radial direction, a gradient of 30 K exists for OP2. In the future, the mesh will be improved to realize smaller y+ values. Also a comparison between 2-stage and singlestage operation of the compressor will be performed. More operating points will be calculated in order to learn how to adapt the boundary conditions on the exterior walls of the compressor on dependency of the operating point. Finally, with these boundary conditions other compressors will be modeled for investigating the effect of wheel diameter and wheel type on the heat transfer in the compressor stage.

Figure 1 .

 1 Figure 1. Position of thermocouples on compressor wheel

Figure 2 .

 2 Figure 2. Position of thermocouples on volute and compressor back plate

Figure 3 .

 3 Figure 3. Position of thermocouples in radial diffusor

Figure 4 .

 4 Figure 4. Compressor Map

Figure 5 .

 5 Figure 5. CHT Model of centrifugal compressor

Figure 6 .

 6 Figure 6. Boundary conditions applied to solid domains

Figure 7 .Figure 8 .

 78 Figure 7. Temperature distribution on volute for OP1 and OP2

Figure 9 .

 9 Figure 9. Temperature distribution on compressor wheel

Figure 10 .

 10 Figure 10. Notation for compressor wheel parts

Figure 11 .

 11 Figure 11. Temperature distribution on compressor cross section

Figure 12 .Figure 13 .

 1213 Figure 12. Heatflux of compressor wheel at OP1

Figure 14 .

 14 Figure 14. Temperature distribution in radial diffuser

Table 1 .

 1 Operating points for CHT analysis

			OP1	OP2
	Vred	m 3 s	0.086	0.136
	π [-]	1.26	1.91
	u 2 [m/s]	235	371
	T 1 [K]	405.7	353.8
	T 2 [K]	435.7	442.6

T CW 1,T est1 [K] 420.07 410.68 T CW 1,T est2 [K] 420.13 410.75

Table 2 .

 2 Accuracy of operating point

			OP1	OP1	OP2	OP2
			EXP	CFD	EXP	CFD
	Vred	m 3 s	0.086 0.086 0.1369 0.1372
	π [-]	1.26	1.24	1.91	1.89
	u 2 [m/s]	235	235	371	371
	T 1 [K]	405.7 405.7 353.8	353.8
	T 2 [K]	435.7 438.0 442.6 442.91

Table 3 .

 3 Accuracy of temperatures OP1

Table 4 .

 4 Heat Transfer to compressor wheel

		OP1	OP2
	Wheel cavity -0.22 W 13.42 W
	Blades	0.6 W	-13.59 W
	Hub	3.24 W -0.178 W
	Bore	-2.92 W	2.37 W
	Nut	-0.71 W -1.96 W


Florian Thiery
email: florian.thiery@ltu.se

Rolf Gustavsson

Jan-Olov Aidanpää

S Y Mpo

O Tatin

Evaluation of Campbell diagrams for vertical hydropower machines supported by Tilting Pad Journal Bearings

Keywords:

In vertically oriented machines with tilting-pad journal bearings, there is no static load that allow to calculate the bearing properties around a determined static position. As a result, most of simulations are performed by solving Reynolds equation at each time-step which can result in long computational time. To avoid this concern, a simplified model is used that takes in account the variation of unbalance load depending on the pad configuration. This nonlinear model is used to simulate the dynamics of a hydropower turbine and compared with the Campbell diagram calculations used from hydropower industry standards. A comparison between the linear and nonlinear model is performed to evaluate how accurate the linear model is and until which limits it can be relevant. In case of qualitative discrepancies in the results in terms of natural frequencies and damping ratios, an improvement of the Campbell diagram calculation is proposed to obtain a more accurate linear model.

INTRODUCTION

In usual horizontal turbomachines, the dynamical properties of the system are calculated around the static load due to dead weight. On the contrary, for vertical machines such as pumps and hydropower turbines, the rotor is subjected to rotating unbalance loads that do not allow to calculate constant stiffness and damping coefficients for fixed operating conditions. Instead, the bearing forces depend on the relative displacements and velocities between the shaft and housing. As a result, the Reynolds equation is usually solved at each time step and the Campbell diagram is not available anymore due to the nonlinear equations of motion [START_REF] White | Rotordynamic analysis of a vertical pump with tilting-pad journal bearings[END_REF][START_REF] Ma | Numerical simulation of nonlinear oil film forces of tilting-pad guide bearing in large hydro-unit[END_REF][START_REF] Zhang | Analysis of nonlinear dynamic behaviors of rotor system supported by tilting-pad journal bearings[END_REF][4][START_REF] Cha | Nonlinear dynamic behaviour of vertical and horizontal rotors in compliant liner tilting pad journal bearings: Some design considerations[END_REF]. The main problem of solving Reynolds equation at each time-step resides in the long computational time especially when performing a design study as function of several design parameters and/or different unbalance loads. As a result, it is convenient to simplify the bearing modeling without losing the mechanical properties of the system and perform nonlinear simulations using the simpler model.

However, from an industrial point of view, it is desirable -as a first step -to obtain the Campbell diagram to evaluate critical designs for hydropower rotors. As a result, a first evaluation can be performed by assuming the bearing loads to be constant at each bearing position as it is usually done for horizontal machines. A comparison

load-between-pad

LGB lower guide bearing LOP load-on-pad TGB turbine guide bearing UGB upper guide bearing UMP unbalance magnetic pull way the natural frequencies and damping ratios of the system.

METHODS

Model

Rotor description

The model of the vertical hydropower unit used in this paper is a typical 45MW Kaplan turbine [START_REF] Synnegård | Modeling of visco-elastic supports for hydropower applications[END_REF] as seen in Fig. 2. The rotor is supported by three tilting pad journal bearings at the upper guide bearing (UGB), the lower guide bearing (LGB) and the turbine guide bearing (TGB). The shaft is described using Timoshenko beam elements, while additional mass and inertia properties are added at the exciter, generator and runner position. The Unbalance Magnetic Pull is simulated as a constant negative stiffness at the generator and exciter position. The values of these parameters are summarized in Table 2. The rotor is at first described using 15 nodes, but it is reduced to 6 nodes using the Improved Reduction System method [START_REF] Thiery | Dynamics of a misaligned kaplan turbine with blade-tostator contacts[END_REF]. The 6 master nodes kept for the simulation are the three bearing positions as well as the exciter, the generator and runner position.

Bearings modeling

The generator is set vertically and supported by tilting pad journal bearings (TPJB) that are attached to rigid bearing brackets on each side. The bearings will be considered to have a nonlinear behavior. Since the rotor is vertical, there is no static load caused by the dead weight that allows the calculation of bearing coefficients. In fact, the stiffness and bearing properties of the system will depend on the load direction. These coefficients will be determined as function of the Load on Pad (LOP) or Load between Pad (LBP) (Fig. 1), load angle ϕ and eccentricity ε. The tilting pad bearings are oriented in a way that the between-pad area is aligned with the y-direction in all three positions in Fig. 2. The calculation of bearing coefficients follows the procedure of [START_REF] Nässelqvist | Experimental and numerical simulation of unbalance response in vertical test rig with tiltingpad bearings[END_REF]. Using the parameters in Table 2, a commercial software [9] is used to calculate the bearings for both LOP and LBP as function of the eccentricity. These coefficients can be approximated using eccentricity dependent polynomial functions. For instance, the stiffness coefficient in ξ- 4 where ε is the eccentricity. The stiffness and damping coefficients in the rotating system are then assumed to be a harmonic relation of the LOP, LBP and load angle ϕ as follows:

direction is k LOP ξ = a 0 + a 1 ε + a 2 ε 2 + a 3 ε 3 + a 4 ε
k i j = Ω Ω nom k LOP i j + k LBP i j 2 + k LOP i j -k LBP i j 2 cos(N pads ϕ) (1)
c i j = c LOP i j + c LBP i j 2 + c LOP i j -c LBP i j 2 cos(N pads ϕ) (2)
where i, j = ξ, η.The stiffness coefficients are scaled with the nominal speed used for their calculation, whereas the damping coefficents are constant as function of the rotating speed. Fig. 3 shows a representation of direct stiffness and damping as function of eccentricity, calculated at the nominal speed of 166.7 RPM. In this model, the cross-coupling terms are disregarded due to their small values. Moreover, the mass parameters of the fluid are also neglected due to small forces. Since the bearing properties are given in the rotating ξη-plane following the unbalance load, they have to be transformed to the fixed coordinate system using the following transformation matrix

T = cos(ϕ) sin(ϕ) -sin(ϕ) cos(ϕ) (3)
where ϕ = arctan(y/x) + nπ is the eccentricity angle at the corresponding bearing position. As a result, the bearing matrices are transformed at each time step using K bear = T K rot T and C bear = T C rot T. The parameters for each bearing position are given in Table 1 to obtain the bearing stiffness and damping using Eq. 2. To perform rotordynamical simulations, international standards from hydropower industry have to be estimated regarding unbalance and eccentricities. First of all, the ISO 1940-1 [START_REF]Ss-iso 1940-1 : Mechanical vibration -balance quality requirements for rotors in a constant (rigid) state -part 1: Specification and verification of balance tolerances[END_REF] standard gives recommendation concerning the maximum allowed unbalance force which is dependent on the rotating mass, rotationnal speed and balancing grade. The balancing grade for hydropower rotors is G6.3, meaning that eΩ = 6.3 mm/s. The maximum allowed unbalance force is thus defined as

f unb = m × (eΩ) × Ω = m × ( 6.3 1000 ) × Ω (4)
where m represents the mass from Table 2 at the corresponding position. According to Fig. 4, the unbalance loads are reasonably assumed to be set at the generator and runner position. The distribution of the unbalance forces depend on the bearing layout in the machine. In our case, the configuration corresponds to Fig. 4(a) with a 3 bearings unit. By assuming the rotor to be rigid in comparison with the bearing stiffness, the unbalance load from the generator will be distributed between the two generator bearings, while the entire unbalance load from the runner will be on the turbine bearing.

Simulation procedure

Linear case -Campbell diagram

For calculation of the Campbell diagram, several assumptions have to be made. First of all, the stiffness and damping properties are calculated with the LOP configuration. Secondly, the load is considered static -similarly to horizontal machines -so that the bearing properties are fixed in the global coordinate system. Finally, the unbalance load is assumed to be defined by Eq. 4 at the generator and runner position. By knowing the unbalance positions, the bearing configuration and assuming the rotor to be rigid, the calculation of the stiffness and damping coefficients can be performed for each bearing by increasing the speed (and simultaneously the unbalance load) as the bearing load is known. In order to investigate more deeply the structural properties of the turbine, another evaluation of the Campbell diagram is performed at the operational speed as function of the bearing load. Since there is always uncertainties regarding the unbalance load, this diagram should help understand the global behavour of the machine.

Nonlinear case -sweep sine

In this section, the full nonlinear modeling is used to investigate the vibration properties of the system. The stiffness and damping properties are calculated for each bearing using Eq. 2 and Table 1 at each time step as function of load angle, eccentricity and speed using the following equation of motion in state space formulation:

q = 0 I -M -1 K -M -1 (ΩG + C) q + 0 -M -1 (f unb + f asyn ) ( 5
)
where M is the consistent mass matrix, C the damping matrix containing only the damping properties of the bearings, G the gyroscopic matrix and K the stiffness matrix which is updated at each time-step due to variation of eccentricity and load angle. Eq.( 5) is solved using the 4th order Runge-Kutta-Fehlberg (RKF) method with adaptive time-step. The unbalance is assumed the same as for the calculation of the Campbell diagram. To investigate the vibration response of the system, another simulation is performed where an additional asynchronous force is added at the runner position with a linear sweep-sine excitation of the form

f asyn = 0.01 f unb cos(2π f 0 t + 2π f e -f 0 t f t 2 ) (6)
The magnitude of the force is 1% of the unbalance load in order to keep the same bearing properties. f 0 and f e represents respectively the starting and ending frequency of the sweep sine, and t f is the total simulation time. To obtain the resonance frequencies from the nonlinear case, the response signal with unbalance force only is subtracted to the response signal with unbalance and sweep sine forces. Moreover, a forward and a backward sweep sine simulations are performed to distinguish between the forward and backward modes as they are close to each another for this particular turbine. The Campbell diagram as function of the rotating speed and an eccentricity e = 0.63 × 10 -3 m at the generator and runner position is displayed in Fig. 5. First of all, it should be noted that the modes of vibration with more than 90 % damping ratio are disregarded. It can be observed that the damped natural frequencies of the system are close to each other, and the damping ratios of the system are high especially for the third mode of vibration. The values of damped natural frequency and damping ratios are given in Table 3 for comparison purposes with the nonlinear case. Fig. 6 shows the Campbell diagram as function of the bearing load for a speed of 166.67 RPM by assuming that the load is equal in all the bearings. The variation of bearing load slightly influences the first mode of vibration of the entire range and the third mode of vibration for a bearing load ≤ 15 kN. The second mode of vibration stay constant compared with the two other modes have increasing frequencies. Additional information concerning the modes of vibration is available in Fig. 7. It can be observed that mode 1-2 and 5-6 correspond to a generator mode while mode 3-4 is related with the runner displacement.

RESULTS AND DISCUSSION

Linear analysis

Nonlinear analysis

The maximum displacement at the runner is displayed as function of the instantaneous frequency in Fig. 8 with the asynchronous excitation at the runner position as well. The black curve represents the response under a forward excitation, while the red curve represents backward excitation. It should be noted that the forward and backward property is specific to a node position and can be different in another node position. The response shows higher vibrations at 10.42 Hz and 10.89 Hz and lower vibrations around 9.15 Hz and 9.39 Hz. Fig. 9 shows the response at the runner position with an asynchronous force at the generator position. Similarly, higher vibrations appear at 9.31 Hz and 10.37 In the Campbell diagram displayed in Fig. 5, the 6 modes of vibration should be observed as they cross the asynchronous line. However, only 4 peaks are visible in the sweep sine excitation responses. It is reasonable to assume that mode 3 and 4 correspond to the 3rd and 4th peak response since they have the higher amplitude (lower damping ratios), close frequencies (< 10 % error) and the runner displacement is greater as compared with other nodal positions which correlates the mode of vibration shown in Fig. 7. Concerning the 1st and 2nd peak response, it can also be assumed that it corresponds to mode 1 and mode 2 even as the damping ratios are greater than mode 3 and 4. However, the Campbell diagram predicts damped natural frequencies of 6.15 Hz and 6.89 Hz that are far away from the nonlinear simulation. Another major difference with the linear case is the absence of vibration for mode 5 and 6 in the studied range. However, this may be due to the high damping ratios of the corresponding modes that suppress the vibrations.

CONCLUSION

The nonlinear simulation of a hydropower turbine has been performed in this article. The model used here has some advantages since it allows to perform fast simulations by using simplified bearing calculations based on the assumption of harmonic stiffness and damping due to variation between LOP and LBP configuration specific to vertical machines. Due to the variation of stiffness during operation, the Campbell diagram calculated in hydropower industry may not be accurate as several assumptions are done in order to calculate the eigenfrequencies and damping ratios. For instance, it was shown that the assumption of constant stiffness with LOP configuration allowed to calculate accurately the 3rd and 4th mode of vibration, but it underestimated the first two natural frequencies of the system. One of the main problem when calculating the modal properties is the assumption of equivalent load at each bearing. Even though it is true for this bearing configuration and by assuming a rigid rotor, it becomes unclear to know what is the bearing load distribution when the rotating speed is increased. One of a solution to improve the Campbell diagram is to perform the nonlinear simulation at all speeds and retrieve the average bearing load at steady state in order to use it back to calculate a more accurate Campbell diagram. However, this method is not practical since it involves a great number of simulations, but it would be a better solution to improve accuracy in the linear study. Another suggestion would be to use Floquet theory as long as the variation of stiffness and damping is periodic at steady-state.

Figure 1 .

 1 Figure 1. Tilting pad bearing configuration: (a) Load-Between-Pad (b) Load-On-Pad

Figure 2 .

 2 Figure 2. Model of a 45 MW hydropower turbine. The generator and runner are not graphically shown, but mass and inertias are set in the model in agreement with their geometry

Figure 3 .

 3 Figure 3. Direct stiffness of the bearing in the local coordinate system rotating with the load. ( ) represents the Load-Between-Pad stiffness and (•) is the Load-On-Pad stiffness

2 Figure 4 .

 24 Figure 4. Bearing configuration for different hydropower machines (a) UGB, LGB, and TGB (b) UGB and TGB (c) LGB and TGB

Figure 5 .

 5 Figure 5. Damped natural frequency and damping ratios of the first six modes of vibration as function of the rotating speed

Figure 6 .

 6 Figure 6. Damped natural frequency and damping ratios of the first six modes of vibration as function of the bearing load at the speed of 166.67 RPM

Figure 7 .Figure 8 .

 78 Figure 7. Modes of vibration of the Forsmo turbine. The bearing positions are shown in mode 6 and are the same for the other modes

Figure 9 .

 9 Figure 9. Envelope of the displacement at the runner position under a sweep sine excitation at the generator position

Table 1 .

 1 Bearing coefficients

	Bearing Properties Value	Coefficient a 0	a 1	a 2	a 3	a 4
	Position Clearance	Upper Guide 0.2 × 10 -3	k LOP ξ k LBP ξ	0.1 × 10 9 0.5 × 10 8	-0.5082 × 10 7 0.8322 × 10 6 -0.2303 × 10 7 0.7955 × 10 6	-0.2702 × 10 5 0.2939 × 10 3 -0.2546 × 10 5 0.2492 × 10 3
	Number pads	6	k LOP η	0.12 × 10 9 0.4806 × 10 5	0.4032 × 10 5	-0.8151 × 10 3 0.9483 × 10 1
	Speed	166.67 RPM	k LBP η	0.9 × 10 8	0.9406 × 10 6	0.18502 × 10 6 -0.6631 × 10 4 0.7361 × 10 2
			c LOP ξ	0.55 × 10 7 -0.4202 × 10 5 0.1995 × 10 5	-0.69 × 10 3	0.8121 × 10 1
			c LBP ξ	0.6 × 10 7	-0.2151 × 10 6 0.2896 × 10 5	-0.8074 × 10 3 0.7667 × 10 1
			c LOP η	0.69 × 10 7 -0.1545 × 10 5 0.2070 × 10 4	-0.3166 × 10 2 0.3246 × 10 0
			c LBP η	0.6 × 10 7	0.1455 × 10 5	0.6282 × 10 4	-0.2002 × 10 3 0.2250 × 10 1
	Position Clearance	Lower Guide 0.175 × 10 -3	k LOP ξ k LBP ξ	0.83 × 10 9 -0.804 × 10 8 0.6 × 10 9 -0.6689 × 10 8 0.871 × 10 7 0.786 × 10 7	-0.2150 × 10 6 0.196 × 10 4 -0.2468 × 10 6 0.2207 × 10 4
	Number pads	24	k LOP η	0.75 × 10 9 -0.486 × 10 7	0.305 × 10 6	-0.477 × 10 4	0.545 × 10 2
	Speed	166.67 RPM	k LBP η	0.67 × 10 9 0.446 × 10 7	0.269 × 10 6	-0.954 × 10 4	0.112 × 10 3
			c LOP ξ	0.19 × 10 8 -0.114 × 10 7	0.127 × 10 6	-0.387 × 10 4	0.40 × 10 2
			c LBP ξ	0.20 × 10 7 -0.6407 × 10 6 0.1727 × 10 6	-0.5473 × 10 4 0.5271 × 10 2
			c LOP η	0.17 × 10 8 -0.961 × 10 5	0.60 × 10 4	-0.943 × 10 2	0.109 × 10 1
			c LBP η	0.14 × 10 8 0.221 × 10 6	0.223 × 10 4	/0.181 × 10 3	0.248 × 10 1
	Position Clearance	Turbine Guide k LOP ξ 0.175 × 10 -3 k LBP ξ	0.3 × 10 9 0.34 × 10 9 -0.1275 × 10 8 0.1751 × 10 -0.161 × 10 8 0.249 × 10 7	-0.7955 × 10 5 0.8579 × 10 3 -0.5343 × 10 5 0.5858 × 10 3
	Number pads	8	k LOP η	0.3 × 10 9	0.506 × 10 7	0.1547 × 10 5	-0.2881 × 10 4 0.5237 × 10 2
	Speed	166.67 RPM	k LBP η	0.31 × 10 9 0.301 × 10 7	0.219 × 10 6	-0.955 × 10 4	0.135 × 10 3
			c LOP

Table 2 .

 2 Mechanical properties of the Forsmo turbine

	Symbol Item			Runner	Rotor	Exciter
	m	mass [kg]			65170	197200	2800
	J d J p	diametrical moment of inertia [kg.m 2 ] 60055 polar moment of inertia [kg.m 2 ] 104800	566500 1133000	0 0
					Turbine guide bearing Lower guide bearing Upper guide bearing
	k UMP	Magnetic stiffness [N/m]	X	-310 × 10 6	-7.6 × 10 6
		0.5	1	1.5	2
				x 10

Table 3 .

 3 Modal properties of the turbine at Ω = 166.67 RPM

	Mode	Damped natural Damping
	of vibration frequency [Hz]	ratio [%]
	Mode 1	6.15	30.22
	Mode 2	6.89	26.27
	Mode 3	9.84	18.26
	Mode 4	10.30	18.70
	Mode 5	11.89	54.78
	Mode 6	12.58	45.69

ACKNOWLEDGMENTS

The research presented was carried out as a part of "Swedish Hydropower Centre -SVC". SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu.


Erik Synnegård
email: erik.synnegard@ltu.se

Rolf Gustavsson

Jan-Olov Aidanpää

S Y Mpo

O Tatin

Forced response of a vertical rotor with tilting pad bearings

Keywords:

This paper evaluates how tilting pad bearings can affect the forced response of a vertical rotor. The stiffness and damping for a tilting pad bearing is dependent on the position in the bearing (load on pad or load between pad) and the magnitude of the load. In a vertical machine the position of the shaft in the bearing will change for each time-step and produce a periodic stiffness and damping. This periodic stiffness has shown to produce higher frequency components depending on the number of pads. For the four pad bearing considered in this paper these frequency components are 3Ω and 5Ω in the stationary coordinate system. The aim of this study is to evaluate if the periodic coefficients in the bearing could excite the system and cause resonance problems. It is found that for high loads the system can be excited due to the bearing dynamics at Ω/ω n = 0.33 and Ω/ω n = 0.2. However at low loads this effect is negligible.

INTRODUCTION

Tilting pad journal bearings (TPJB) have been used in rotating machines since the early 1900's and describing the bearing coefficients dynamically has been studied extensively in the last 60 years. TPJB's are commonly used as supports in hydropower machines and offshore pumps because of their good stability. Since hydropower machines are orientated vertically there exists no stationary operating point in the bearing due to the dead weight of the rotor. Hence, in hydropower machines and vertical pumps, the bearing coefficients need to be described dynamically at each time step. In 1964 Lund [START_REF] Lund | Spring and damping coefficients for the tiltingpad journal bearing[END_REF] published a paper where he introduced the pad assembly method and the Reynolds equation solution to solve the bearing coefficients. Over the last 50 years bearing theory has evolved, Timothy et.al [START_REF] Dimond | A review of tilting pad bearing theory[END_REF] published a review of tilting pad bearing theory. Current state of the art now include inertia effects, pad motions, thermal and mechanical deformations. However, modelling the bearing coefficients dynamically for vertical machines using e.g. Reynolds equations have been performed earlier by M.Cha et.al [START_REF] Cha | Nonlinear dynamic behaviour of vertical and horizontal rotors in compliant liner tilting pad journal bearings: Some design considerations[END_REF] and White et.al [START_REF] White | Rotordynamic analysis of a vertical pump with tilting-pad journal bearings[END_REF]. Moreover, modelling the bearing dynamically using Reynolds takes time and if rotordynamical studies are to be performed, acceptable simplified methods are preferred. For example in rotordynamics there is a need to simulate a lot of periods to reach steady state and evaluate large parameter intervals. For these cases solving Reynolds equation at each time-step would take to long time, while with the bearing model used in this paper 100 periods for this 16 DOF model takes around 4 hours on a standard laptop. Nässelqvist et.al [START_REF] Nässelqvist | Experimental and numerical simulation of unbalance response in vertical test rig with tilting-pad bearings[END_REF] proposed a method to describe the bearing coefficients dynamically as function of eccentricity and load angle. In vertical machines the bearing coefficients are dependent on shaft position and eccentricity in the bearing, this will introduce periodic coefficients, which depends on the number of pads. Earlier studies of a vertical Jeffcott rotor with two 4-pad bearings shows that higher frequency components at 3Ω and 5Ω exist in the unbalance response, see Figure 1.

In that case the shaft can be consider rigid since the aim of the study was the bearing modeling. In this paper the effects of these higher frequency components are studied where the aim is to analyse if these higher frequency components could excite the machine and cause resonance problems. The rotor in this study is similar to the one studied earlier but with a more slender shaft. In addition, experiments are also planned to see if this also can be observed experimentally.

METHODS

Rotor description

The rotor used for simulations is a vertically orientated Jeffcott rotor, see Figure 2. The rotor is modelled using 3 nodes (12 DOF's) for the shaft. At node 1 and node 3 tilting pad bearings supports the rotor, which is connected to ground through bearing brackets. Parameters for the rotor is, shaft diameter d 1 = 20.00 mm, disk diameter D = 168 mm and length L = 500 mm between the bearing centres. Due to limitations in bearing design the shaft diameter at the bearing locations are set to d 2 = 49.84. In Table 1 the rotor and bearing properties are presented.

Bearing model

For a vertical machine with tilting pad bearings the stiffness and damping varies periodically depending on the number of pads in the bearing. In Figure 3 a sketch of a 4-pad tilting pad bearing is shown. Using a commercial bearing software [6] the bearing data can be calculated as function of load angle for a fixed load in the local coordinate system (ξ, η) and this is presented in Figure 4. Here 0 • represent load on pad (LOP) and ±45 • represent load between pad (LBP). Also in Figure 4 the stiffness as function of eccentricity is presented, it can be seen that the difference between load on pad and load between pad increases for larger eccentricities. Note, only the bearing stiffness is presented in Figure 4 but the same behaviour is observed for the damping. The bearing coefficients are calculated using Nässelqvist's [START_REF] Nässelqvist | Experimental and numerical simulation of unbalance response in vertical test rig with tilting-pad bearings[END_REF] method, where a polynomial fit is made to describe the stiffness and damping for LOP and LBP as function of eccentricity . This polynom is then used to describe the bearing coefficients for a given load angle and eccentricity according to equation 1 and 2.

K i j = K LOP i j + K LBP i j 2 + K LOP i j -K LBP i j 2 cos(4α) (1)
C i j = C LOP i j + C LBP i j 2 + C LOP i j -C LBP i j 2 cos(4α) (2)
where i, j = ξ, η The bearing coefficients calculated is in the local coordinate system (ξ, η) and must be transformed using the following transformation [START_REF] Cook | Concepts and applications of finite element analysis[END_REF].

K T = T T K B T (3)
C T = T T C B T (4)
where

T = cos(α) sin(α) -sin(α) cos(α) (5)
and K B and C B is the bearing stiffness and damping matrices in the local coordinate system (ξ, η).

Numerical model

Simulations of a vertical rotor with TPJB's are performed with a rotating load

f (t) = F 0 * cos(Ω * t) + F 0 * sin(Ω * t) (6)
applied at the mass position. Here different Ω is studied to see if resonance can occur when 3Ω and 5Ω equals the first natural frequency due to the periodic coefficients in the bearing. The rotor is modelled using Timoshenko beam elements with consistent mass matrix and the bearing coefficients are evaluated at each time-step as function of eccentricity and load angle according to [START_REF] Nässelqvist | Experimental and numerical simulation of unbalance response in vertical test rig with tilting-pad bearings[END_REF]. The equation of motion can thus be written in matrix form as:

M ü + (C + C T + ΩG) u + (K + K T )u = f (t) ( 7
)
where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, K is the stiffness matrix, K T and C T is the bearing coefficients and Ω is the rotational speed. From these simulations, displacements can be compared for different rotating speeds Ω to see if resonance peaks can occur for other frequencies than the natural frequencies.

Numerical simulation

If the bearing coefficients are assumed constant, the Campbell diagram for this system can be calculated. In figure 5 the Campbell diagram is presented and the dashed black lines represent Ω, 3Ω, 5Ω. Since the 4-pad bearing setup shows frequency components at 3Ω then the region where 3Ω = ω n is of interest to study, where ω n is the natural frequencies of the system. From the fft diagram in figure 1 there is also a component around 5Ω therefore frequency ranges from 250 rpm to 500 rpm is studied to see if the first mode of vibration could be excited due to the bearing dynamics. Since the bearing coefficients are dependent on the load, two different load cases are considered. For large loads the difference between LOP and LBP is increased and the periodic excitation is increased. Therefore a large force F 0 = 90N and a smaller force F 0 = 23N is evaluated to investigate if load magnitude influence the behaviour of the system.

Rotating speed [rpm]

RESULTS AND DISCUSSION

In the numerical simulations the operating speed Ω was set in the range (250 -500) rpm corresponding to Ω/ω n = (0.18 -0.38). For each rotating speed 100 periods was recorded and from the last 20 periods the maximum absolute response was calculated for all nodes. Also the maximum and minimum whirl ratio was calculated according to equation 8.

Θ = ẏ * x -ẋ * y (x 2 + y 2 ) * Ω (8)
Where the rotational speed Ω is in rad/s. In Figure 6 displacement in the bearing is presented and it can be seen that something changes at Ω/ω n = 0.2 and Ω/ω n = 0.33. This becomes more clear if studying the disc displacement in Figure 7, here resonance peaks are clearly visible at these frequency multiples. Recall from Figure 4, here it can be seen that the difference between LOP and LBP is large at eccentricities around 45% of the bearing clearance. According to Figure 6 the displacement in the bearing is around 50% of the bearing clearance at Ω/ω n = 0.2 and 40% at Ω/ω n = 0.33. Since the difference between LOP and LBP is large in this region the frequency component at 3Ω and 5Ω is large and will excite the system. In Figure 8 the whirl ratio at the disc is shown, here the maximum and minimum whirl ratio is presented. As seen in Figure 8 the whirl interval is increased especially at the frequency ratio Ω/ω n = 0.33. This indicates some other type of dynamic behaviour of the system. In Figure 9 the displacement orbit at rotating speed 440 rpm is presented, this operating speed corresponds to Ω/ω n = 0.33 and it shows how the 4-pad bearing effect the dynamics. For low bearing loads the difference between LOP and LBP is small and the periodic behaviour in the bearing is low. This should decrease the periodic excitation and remove the resonance peaks observed in Figure 7.

Ω/ω n [-]

Ω/ω n [-]

According to Figure 10 the displacement in the bearing is around 20% of the bearing clearance at Ω/ω n = 0.2 and 15% at Ω/ω n = 0.33. Recall from Figure 4, here it can be seen that the difference between LOP and LBP is small at eccentricities smaller then 20% of the bearing clearance. Hence the excitation due to the periodic coefficients is small. Comparing the results from the disc displacement in Figure 7 to the disc displacement in Figure 11 the vibration peaks at Ω/ω n = 0.2 and Ω/ω n = 0.33 has been reduced significantly. From Figure 13 it is clear that the orbit for the smaller load is more circular then for the case of higher load. Also, the whirl ratio interval in Figure 12 is reduced to almost synchronous whirl. The difference between high and low forces is also presented in Figure 14, where the FFT of the disc displacement is presented for the different forces for the operating speed Ω = 440 rpm. Since the periodic excitation is small for low forces the frequency component at 3Ω is negligible and does not excite the first eigenfrequency. However for the large force the frequency component at 3Ω is large and has shown to influence the dynamic response of the machine. In hydropower machines the number of pads are usually larger then five hence, the periodic stiffness is smaller but with higher frequency. This should decrease the excitation of the machine due to the bearing, but for high bearing loads this can still cause increased vibrations. Most monitoring of the machines are usually in the bearings where displacement and force normally is measured. From these simulations it can be seen that when vibrations occur at the disc position there is a local minimum in the bearing displacement.

CONCLUSIONS

In a vertical machine with tilting pad bearings the bearing coefficients will change periodically due to the difference between load on pad and load between pad for a fixed load.

In comparison with a horizontal machine the operating point is strongly influenced by the dead weight, hence the bearing coefficients are relatively constant for a fixed load. This paper shows that periodic excitations from the bearing can result in increased vibrations at certain frequencies. The magnitude of the increased vibrations is strongly influenced by the initial load on the system, since the bearing coefficients are increased for higher loads the difference between on pad and load between pad will increase and this amplifies the periodic excitation. Hence for small loads in the bearing there is almost no periodic excitation and the response is almost unaffected. Interesting is the displacement in the bearing at Ω/ω n = 0.33 for high loads, the displacement is actually decreased while the displacement at the mass is increased. Since most machines usually are monitored in the bearings the increased vibrations at the disc could be missed. The shaft used for simulations in this paper has been ordered and experiments are planned to verify if the bearings dynamics can excite the system.

Figure 1 .

 1 Figure 1. Fourier transform of the bearing load from earlier simulations.

Figure 2 .

 2 Figure 2. Rotor model

Figure 3 . 4 -

 34 Figure3. 4-pad bearing geometry, with coordinate system (ξ, η) rotated α degree's from the fixed coordinate system (x, y).

Figure 4 .

 4 Figure 4. Stiffness as function of α at 30% eccentricity in the coordinate system (ξ, η) (left) and stiffness LOP and LBP as function of eccentricity (right).

Figure 5 .

 5 Figure 5. Campbell diagram for constant stiffness at 30% eccentricity assuming LOP.

Figure 6 .

 6 Figure 6. Displacement in the bearing for F 0 = 90 N.

Figure 7 .

 7 Figure 7. Displacement at the disc for F 0 = 90 N.

Figure 8 .Figure 9 . 4 Figure 10 .Figure 11 .Figure 12 .Figure 13 .FrequencyFigure 14 .

 8941011121314 Figure 8. Maximum and minimum whirl ratio at the disc for F 0 = 90 N.

Table 1 .

 1 The test rig rotor and bearing properties.

	Rotor	Symbol Item	Value Unit
		d 1	Shaft diameter	49.84 [mm]
		d 2	Shaft diameter	20.00 [mm]
		D	Disc diameter	168	[mm]
		e m	Mass unbalance radius	70	[mm]
		H	Bearing bracket stiffness	500	[MN/m]
		L	Shaft length	500	[mm]
	Bearing Symbol Item	Value Unit
		d	Shaft diameter	49.84 [mm]
		d b	Bearing diameter	50.09 [mm]
		d p	Pad diameter	50.15 [mm]
		θ	Pad angle	72	[deg]
		N	No. of pads	4	[-]
		C b	Diametrical bearing clearance 0.25	[mm]
		C p	Diametrical pad clearance	0.31	[mm]
		m	Preload factor	0.19	[-]
		x	Pad pivot offset	0.6	[-]

ηη eccentricity e/c b [%]

ACKNOWLEDGMENTS

The research presented was carried out as a part of "Swedish Hydropower Centre -SVC". SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu.