Investigation of dynamic properties of the microturbine with a maximum rotational speed of 120 krpm – predictions and experimental tests

3.1. Selection of the bearings

During the preliminary analysis of the bearing system for the microturbine rotor, the following solutions were considered: rolling element bearings, gas bearings and foil bearings. Each of these solutions has its advantages and disadvantages, some of which are briefly discussed below.

Rolling bearings are most frequently used in small fluid-flow machines as they are readily available in the market at low prices. Such bearings require lubrication, which is a disadvantage in the present case and is incompatible with the concept of the oil-free machine. The permissible rotational speed of rolling bearings is strictly related to the journal diameter (more specifically, is related to the circumferential speed of the rolling elements). It was determined that when the bearings’ journals had a diameter of 8 mm it was possible to obtain the maximum rotational speed at the level of 120 krpm [30]. Such a low diameter at the supporting points of the shaft significantly decreases its stiffness, thereby lowering the critical speed of the rotor below the nominal speed. At the resonant speed, operation of the rotor supported by rolling bearings is not safe due to large displacements [31]. Therefore, a decision was made to relinquish the use of rolling bearings.

The main advantages of foil bearings, in the discussed case, are the ability of stable operation of the rotor at very high rotational speeds and no necessity for oil lubrication. Their major disadvantages are low accuracy of the rotor alignment, high starting torque and power losses related to it [32]. The low precision alignment requires the application of higher clearance above the blades in the microturbine, which decreases the efficiency of the flow system. Due to these aspects the use of the foil bearings was abandoned. As the analysed rotor was characterised by high stiffness and good dynamic properties, there was no need to apply such an advanced solution as foil bearings.

After preliminary analyses, taking into account the advantages and disadvantages of various bearing systems, it was decided to use gas bearings fed by the microturbine’s working medium. Such bearings allow to develop machines using the oil-free technology, enable to obtain very high rotational speeds and low friction losses. Their disadvantages are low bearing load capacity and the possibility of damage at low rotational speeds (i.e. before a stable gaseous lubricating film is formed). To prevent damage to the sliding surfaces, it was decided to lubricate the bearings with a gaseous lubricating medium under pressure. The medium gets into the bearings through small diameter holes, evenly spaced over the circumference of the sleeve. In the ORC system, the fresh vapour (which evaporated in the evaporator before the microturbine) can be used to lubricate the bearings. In the discussed case, the working medium’s vapour at the evaporator outlet has a pressure greater than 10 bar, and its low consumption by the bearings should not have a significant impact on the microturbine power. The estimated vapour consumption in all bearings does not exceed 5 per cent of the total flow rate in the ORC system. In the case of a higher demand for vapour in the ORC system, the flow rate in the circulation pump can be increased without changing other parameters of the cycle.

The calculations, aimed at the selection of all dimensions of the bearings, were carried out using the MESWIR series computer programs that enable analyses of aerodynamic gas bearings. In these calculations, parameters of the low-boiling medium used for lubrication of the bearings were taken into account, but the fact that the bearings are fed was not considered. In this way, only aerodynamic phenomena stemming from the geometry of the bearings, rotational speed and viscosity of the lubricating medium were taken into account. The characteristics of the bearings were determined on the basis of an isothermal model of an aerodynamic gas bearing. Static analysis consists in determining the reaction force of a gaseous film to a given static load. During calculations the following equations are solved: Navier-Stokes momentum equation (assuming that there is constant pressure in the direction corresponding to the lubricating film thickness) and equation of continuity (with no mass transfer). The pressure distribution of the lubricating film is determined based on the Reynolds equation for compressible fluids in the following form:

where: $p$ – gas pressure, $\rho$ – gas density, $h$ – radial clearance, $R$ and $\theta$ – the radial coordinate and the angular coordinate of the polar coordinate system, $\mu$ – gas viscosity, $z$ – bearing length, $\omega$ – angular speed.

Dynamic analysis consists in calculating the stiffness and damping coefficients at the considered operating point of the gas bearing (which is taken based on the results of the static analysis). The simplified algorithm for dynamic calculations is presented in Fig. 2. Calculations are performed in the local coordinate system and then transferred to the global coordinate system of the bearing.

Fig. 2Simplified algorithm for calculating the dynamic coefficients of gas bearings

During the calculations, the focus was on assessing the operating conditions of gas bearings with different diameters, lengths and radial clearance. The diameter of the bearing was analysed in the range from 16 mm to 24 mm, its length in the range from 14 mm to 22 mm and its radial clearance in the range from 12 µm to 20 µm. Over 200 variants of the bearing’s dimensions were analysed in this way. Due to the uneven loading of the two bearings, the search area for the best solution was expanded to include cases in which each of the two bearings has different dimensions. The exemplary calculation results, in the form of characteristics showing the effect of length and radial clearance of the bearing with a diameter of 20 mm on the lift-off speed, are presented in Fig. 3. In this case, the static load of the bearing was 2.3 N. Having the lift-off speed calculated, it is known that the bearing with given dimensions, at a given speed, has the capacity to carry the static load. A lower lift-off speed at a given load is favourable as it enables stable operation of the gas bearing (even if it is not fed with the lubricating medium).

On the basis of the performed analyses, it was determined that the microturbine rotor is characterised by stable operation in a wide range of rotational speeds when it is supported on the gas bearings with a diameter of 24 mm and a length of 22 mm. The optimal value of the radial clearance is 20 µm, which is the highest value out of all the analysed ones. This choice was also influenced by practical considerations related to the manufacturing technology and manufacturing precision of the journal and of the sleeve. Due to the small benefits resulting from using two bearings with different dimensions, two identical bearings were selected. The lift-off speed of the chosen bearing is about 8,000 rpm at the load resulting from the rotor mass. It means that this bearing can be considered as aerodynamic when it operates at speeds that are higher than the lift-off speed. At lower speeds, this bearing has to be fed with the lubricating medium under pressure.

Fig. 3Lift-off speed in the gas bearing (with a diameter of 20 mm), which was loaded by a force of 2.3 N vs radial clearance; for five different values of the bearing length

3.2. Rotor dynamics analysis

The selection of the bearing system made it possible to perform the rotor dynamic analysis. The purpose of this analysis was to determine critical speeds of the rotor and to check the vibration level at different rotational speeds. For this purpose, an FEM model of the rotor was developed, taking into account the characteristics of the selected gas bearings lubricated with a low-boiling medium. A graphical representation of the model is presented in Fig. 4. It includes the generator rotor (blue colour), the rotor disc (red colour) and two keep plates of the thrust bearings. The total length of the rotor was 175 mm and both bearing journals had a diameter of 24 mm. The distance between the bearing centres was 92 mm. The following assumptions were applied to the numerical model: the shaft and keep plates of the thrust bearings are made of 1.7225 alloy steel, the microturbine’s rotor disc is made of 7075 aluminium alloy. The mass of the complete rotor was approximately 0.6 kg.

Fig. 4The FEM model of the rotor of the ORC microturbine with an electric power of 1 kW (D – discs, B – bearings, UB – unbalance)

Calculations were made for speeds up to 150,000 rpm. Their purpose was to check if there are any resonant speeds within the operating speed range or slightly higher than the maximum speed (120,000 rpm). During the rotor dynamics analysis, the characteristics of the journal bearings were taken into account. These characteristics were determined in the form of stiffness and damping coefficients for all analysed rotational speeds. Such characteristics for less loaded bearing (B1) are presented in Fig. 5. The curves representing the values of the stiffness and damping coefficients of the second bearing (B2) were very similar. Next, the coefficients were incorporated into the FEM model of the rotor (their values changed as the analysed rotational speed changed). Such an approach allowed to obtain more reliable results than for constant values of the stiffness and damping coefficients.

Fig. 5a) Stiffness and b) damping coefficients of gas bearing (B1)

a)

b)

For the purposes of the forced vibration analysis, the maximum permissible unbalance of the rotor was adopted on the basis of the ISO1940 standard. It was placed in the middle of the generator sleeve, at the node where the highest vibration level was expected. The unbalance is marked as “UB1” in Fig. 4. Such positioning of the unbalance allow to excite bending vibration of the rotor in order to calculate its first critical speed.

Fig. 6Vibration amplitudes of the microturbine rotor, calculated in the wide rotational speed range

The calculation results are presented in Fig. 6, in the form of vibration amplitudes of the bearings and discs versus the rotational speed of the rotor. The graph shows absolute vibration of the discs and relative vibration of the bearings (in other words, the vibration of the journals in relation to the bearing sleeves). It can be noticed that the vibration of the system increased as the rotational speed increased (up to a speed of approximately 60 krpm). The maximum vibration level (approx. 1.8 µm) was observed for the keep plate of the thrust bearing, located at the shaft end (D1). The bearings had the following maximum vibration amplitudes: the first bearing (B1) – 1.6 µm, the second bearing (B2) – 1 µm. At a speed of 60 krpm, the vibration amplitude calculated in the middle of the rotor (i.e. where the generator is mounted – D2) was 1.3 µm. At higher speeds, the vibration amplitude at the first bearing (B1) decreased gradually; at the second bearing (B2), it was more or less at the same level. As far as the keep plate of the thrust bearing is concerned (D1), the vibration level changed slightly as the speed increased more. For higher rotational speeds, the vibration of the generator decreased and then increased to a value of 1.7 µm. In the speed range from 60 krpm to 150 krpm, the highest increase in the vibration level can be noticed for the rotor disc (D4); at a speed of 60 krpm, the vibration amplitude was 1.1 µm and it increased to 1.8 µm at a speed of 150 krpm. Generally speaking, the level of vibration of the analysed rotor was very low as it did not exceed 2 µm in the whole speed range. Additionally, on the basis of the calculation results, it can be stated that the first critical speed of the rotor was outside the range of operating speeds; this speed was approximately 200 krpm. The overall low vibration level and such a high value of the first critical speed (i.e. beyond the range of operating speeds) indicate that the rotor–bearings system has very good dynamic properties. Other calculation results, for instance, vibration trajectories (Fig. 7) and vibration velocity spectrums, are also proof that the rotor supported on two gas bearings is characterised by very good dynamic performance. Within the entire rotor operating range analysed, only forward whirls were observed. Since the positive calculation results were obtained, it was possible to build the machine and test it.

Fig. 7Vibration trajectories of bearing journals at a) 100 krpm and b) 120; (blue line – bearing 1, red line – bearing 2)

4.1. Test rig description

The experimental research on the prototypical ORC microturbine was carried out in the laboratory at the IMP PAN. After finishing the preparatory work, the microturbine was installed on a specialised test rig (Fig. 8(a)). The test rig makes it possible to simulate the real operating conditions of the machine when coupled with an ORC system. The microturbine and its bearings were fed with the fresh vapour of the working medium sold under the trade name of HFE-7100, and the vapour that was already used was directed to a set of heat exchangers, which enabled to obtain the specified temperature and pressure. At the microturbine inlet, the working medium reached a pressure of 13.5 bar and a temperature of 180 °C. The flow rate of the working medium was controlled using a circulation pump with adjustable rotational speed. In the ORC system in question, an electric heater served as a heat source.

Because a direct vibration measurement of the rotor would have required making additional holes for sensors in the casing, which could have caused unwanted leakage of the low-boiling medium, the vibration was measured on the microturbine casing. A vibration sensor was fixed to the upper part of the casing using a magnetic pad, in the vicinity of the bearing situated between the rotor disc and the generator (Fig. 8(b)). As the microturbine casing was made of stainless steel, in order to mount the sensor it was necessary to use a ferromagnetic mounting pad screwed in the casing. The vibrations were measured using a uniaxial accelerometer (PCB HT628F01), tailored to work at an elevated temperature. A portable machine health analyser MBJ Diamond 401A (in its most extended version denoted by XT) was used to record the results. Besides the vibration measurement during operation of the microturbine, the following parameters were also measured: rotational speed of the rotor, electric power of the generator and thermodynamic parameters of the working medium. These additional measurements were carried out using a separate measurement system.

Fig. 8a) The prototypical microturbine installed on the test rig and b) its model showing where an accelerometer was mounted (1 – microturbine, 2 – turbine inlet, 3 – turbine outlet, 4 – bearing 1 inlet, 5 – bearing 2 inlet, 6 – horizontal accelerometer, 7 – vertical accelerometer, 8 – pressure sensor)

4.2. Measurement of vibrations

During the conducted research, the maximum electric power of the turbogenerator was about 1.1 kW, which was achieved when the rotor’s speed was at the level of 60 krpm. Due to limitations related to the efficiency of the circulation pump and difficulties with achieving a higher pressure and a higher flow rate of the working medium, no tests were done at speeds higher than 60 krpm. The favourable characteristics of the working medium made it possible to achieve the nominal electric power (1 kW) already at a speed of 56 krpm. If other low-boiling mediums (intended for this microturbine) were used, the nominal power could have been achieved at higher rotational speeds.

The vibration measurement results are presented in Figs. 9-13. Of all the results, several vibration velocity spectrums were chosen. These characteristics were obtained when the operating conditions of the microturbine were stable. In this case, it means that the rotor had a constant speed for at least 30 seconds and there were no changes in the pressure, temperature and flow rate of the working medium. The vibration velocity spectrums shown in Figs. 9-12 cover the frequency range from 1 Hz to 1,600 Hz whilst the vibration velocity spectrum demonstrated in Fig. 13 covers the range from 2 Hz to 3,200 Hz. The measuring range on the last graph was extended due to the necessity to register the rotational frequency component 2$X$. If the rotational speed was greater than 48 krpm, the component 2$X$ occurred at a frequency that was higher than 1,600 Hz. The measurement resolution in the frequency range from 1 Hz to 1,600 Hz was 1 Hz and after the frequency range was extended it was equal to 2 Hz. The reduction of the measurement resolution for the extended frequency range was necessary due to the capabilities of the measuring apparatus. Additional vibration measurements, performed for even more extended frequency range (up to 25,600 Hz), did not show the necessity of further extending the measuring range.

In order to facilitate making a comparison of the results presented in Figs. 9-13, all the axes of ordinates have the same scale with a maximum value of 0.5 mm/s. It should be noted that on each of the graphs one can distinguish three areas with an elevated vibration level. The first area covers the range of very low frequencies, namely the range from a minimum frequency to about 50 Hz. These vibrations are not related to operation of the microturbine and come both from the operation of other devices located in the laboratory (e.g. pumps, fans) and from the movement of people that were present in the lab during the measurements; the laboratory is located on a wooden platform supported on steel pillars. Since exciting forces coming from these sources have different frequencies, the vibration distribution is quite random in the range of low frequencies. The next area, in which an increased vibration level can be observed, is related to vibration of the rotating shaft. In this case, the vibration frequency is equal to the rotational frequency of the rotor (therefore it is denoted as 1$X$) and in Figs. 9-13 it takes the following values: 202 Hz, 355 Hz, 627 Hz, 749 Hz and 932 Hz. At these frequencies, one could observe a distinct vibration component, whose vibration velocity was high. The vibration amplitude increased as the rotational speed increased and had the following values: 0.107 mm/s, 0.124 mm/s, 0.138 mm/s, 0.265 mm/s and 0.303 mm/s. Taking into account the calculation results presented in the previous chapter, one can notice that these results are in good agreement with the experimental results. In both cases, the level of vibration originating from the rotor increased as the rotational speed increased. Bearing in mind the obtained calculation results, one can expect that at higher rotational speeds (over 60 rpm) vibration of the microturbine should not increase more but remain on the same level. On all graphs showing the vibration velocity spectrums, the third area with an increased vibration level occurs at a frequency that is two times greater than the rotational frequency of the rotor. This is vibration (denoted by 2$X$) in which the shaft gives 2 times the fundamental frequency in one revolution. In the case of the ORC microturbine, the amplitude of the 2$X$ component was very low; it was at least 3 times lower than 1$X$ component in the whole range of rotational speeds. Such vibration of a rotating system can be a result of misalignment or bending of the shaft, and may be caused by non-uniform heating and some temperature gradient in the shaft. Because in the case of the discussed machine the level of this vibration was very low, it was not analysed more widely.

Fig. 9Vibration velocity spectrum of the microturbine casing at a speed of 12,120 rpm (202 Hz)

Fig. 10Vibration velocity spectrum of the microturbine casing at a speed of 21,300 rpm (355 Hz)

Table 1 includes the measurement results presented in Figs. 9-13 as well as the ones obtained for three additional rotational speeds. These additional results were also obtained when the operating conditions were stable. The presented results are proof that the amplitude of the 1$X$ vibration component increased as the rotor speed increased in the entire investigated range of speeds. Table 1 also includes values corresponding to the overall vibration level, which were calculated as sums of all vibration components that occurred in the registered frequency ranges. In the case of this parameter, one can also notice that its value increased as the rotational speed of the rotor increased but only up to a speed of 25,020 rpm. At higher rotational speeds, the overall vibration level slightly decreased and remained in the range from 0.66 mm/s to 0.87 mm/s. It can, therefore, be concluded that the microturbine was characterised by stable operation (from the point of view of dynamic performance) and the measured absolute vibration of the casing was at a very low level.

Fig. 11Vibration velocity spectrum of the microturbine casing at a speed of 37,620 rpm (627 Hz)

Fig. 12Vibration velocity spectrum of the microturbine casing at a speed of 44,940 rpm (749 Hz)

Fig. 13Vibration velocity spectrum of the microturbine casing at a speed of 55,920 rpm (932 Hz)

During the discussed tests of the microturbine, the sound intensity level was also measured. Unfortunately, due to the noise emitted by other devices situated in the laboratory, these measurements were made difficult. In general, it was found that the noise emitted by the operating microturbine was very low. At the nominal power of 1 kW, the sound intensity level registered at a distance of 2 m from the microturbine was 63 dB (Leq A). The shut-down of the microturbine did not decrease the sound intensity level. Therefore, it can be stated that the operation of the microturbine had no influence on the sound intensity level measured in the laboratory. The noise emitted by the microturbine itself was much lower than the measured noise.