4.2 Centrifugal compression system
4.2.2 Compressor model
In order to describe the dynamic behavior of the centrifugal compression system (test rig B) we used the Greitzer model as discussed in Chapter2. Because the modeling approach was similar to that for test rig A, see Chapter 3, details of the model development are omitted here. Where needed, differences with the modeling approach for test rig A are highlighted.
The structure of the applied model is identical to that depicted in Figure2.2. The resulting model equations are
4.2 CENTRIFUGAL COMPRESSION SYSTEM 69
−2 −1 0 1 2 3 4 5
0 2 4 6 8 10x 104
˙
mc (kg/s)
∆po,c(Pa)
Figure 4.2 /Data and approximation for the compressor map of test rig B; ⋄ 13,783 rpm,
∗ 15,723 rpm, ◦ 17,637 rpm, ▽ 19,512 rpm, 20,855 rpm, − = spline-polynomial approximation.
Lc
Ac
d ˙mc
dt = ∆po,c( ˙mc, N )− ∆po (4.1)
V2
c22 dpo,2
dt = m˙c− ˙mt(∆po, ut)− ˙mr(∆po, ur) (4.2) with ∆po = po,2 − po,1. The compressor characteristic ∆po,c( ˙mc, N )for the investigated test rig is shown in Figure 4.2. Note that we have again neglected the throttle duct dy-namics and omitted the relaxation equation in the model.
In contrast to the polynomial approximations used for test rig A, the stable part of the compressor curves are now approximated with spline functions, see also Appendix C.
We use spline functions because they are more suited to capture the profound choking effect—an almost vertical compressor characteristic—at high mass flows. The numerical implementation of the spline fit guarantees that the curves line up with the polynomial approximation for the unstable part at the measured surge points. However, expressing the dependency of the spline parameters on rotational speed with a quadratic polynomial in N is not straightforward and therefore omitted. Hence, compressor characteristics are only available at the speeds for which measurements were performed.
Although the test rig B is similar to test rig A, there are some important differences that we briefly address here. In summary, the main differences between the two test rigs are
Test rig B is an open system with both inlet and outlet connected to atmo-sphere, while test rig A is a recirculating system where inlet and outlet are
−0.2 −0.10 0 0.1 0.2 0.3 0.4 0.5 0.4
0.8 1.2 1.6
φc(-)
Ψc(-)
Figure 4.3 /Compressor map of test rig B in dimensionless form.
connected through a gas cooler. Hence, the compressed medium in test rig B is air (28.0134·10−3 kg/mol) that enters the system at atmospheric pressure.
Test rig B has a significantly larger plenum volume due to the long (41.9 m) discharge piping. Moreover, the length of the discharge piping gives the system a more profound distributed character and it introduces aeroacoustic phenomena that are the main topic of this chapter.
Test rig B is equipped with total pressure probes instead of the normal dynamic pressure sensors in test rig A. Hence, the model equations for test rig B contain stagnation pressures as discussed in Section2.3.1.
Finally, we recall that it is common practice to reformulate the differential equations in non-dimensional form. However, to simplify the connection of the acoustic pipeline model that will be discussed later on, we will use their full-dimensional form throughout this chapter. Hence, measurement and simulation results will also be presented in full-dimensional form. However, some important results, will be repeated in dimensionless form to enable the easy comparison with results from other chapters. The dimensionless equivalent of the compressor map is shown in Figure4.3.
4.2.3 Model identification and validation
We obtained parameter values for the developed compressor model by comparing sim-ulation results with data from actual surge measurements. Subsequently, we verified whether the model describes the surge dynamics of the investigated test rig. In Chapter6 we will discuss the results from forced-response experiments. At the start of each mea-surement the compressor was brought into surge by closing the throttle valve, after which
4.2 CENTRIFUGAL COMPRESSION SYSTEM 71
the dynamic pressure oscillations were measured for 128 s. The temperature and pres-sure data were also used to initialize the simulation model. Gas properties are obtained from gas table data provided byBaehr and Schwier(1961);Davis(1992);Mohr and Taylor (2005), using a spline interpolation when required.
To improve the agreement between measurement and simulation we selected the com-pressor duct length, the valley point of the comcom-pressor characteristic, and the throttle valve opening as tuning parameters, see also Chapter3. We used a bisection algorithm to determine the valley point that resulted in a correct prediction of the measured pressure amplitude during surge. Subsequently, we performed a series of calculations with differ-ent throttle openings and we selected the throttle opening that gave the best match with the measured surge frequency. Prior to these calculations we confirmed that the valley point and throttle opening can be determined independently from each other.
The above procedure was carried out for different values of Lc and we selected the small-est compressor duct length that gave satisfactory results. We point out that the obtained value for Lc is significantly larger than the one used in the model for test rig A. In Chap-ter6we will discuss the selection of a proper value for Lc in more detail.
The final results for two particular measurements are shown in Figures4.4and4.5, both revealing a good agreement between the measured pressure oscillations and the outcome of the tuned simulation model. However, we point out that the power spectral densities of the measurements are higher than those of the simulation results in between the har-monic peaks of the surge oscillation. This is caused by the significant flow noise (e.g.
turbulence) and some measurement noise that are present in the experimental system, which are not included in the simulation model.
More importantly, zooming in on one surge cycle reveals some differences between the measured and simulated time-series as can be seen in Figure 4.6. In the first place, we remark that the measured pressure appears to increase faster than the simulated pres-sure after reaching its minimum value. A possible explanation for this difference is the influence of rotor speed variations during surge.
Qualitative observations during the surge experiments indicated that the compressor slows down during the negative flow phase of each surge cycle. However, no quantitative data of compressor speed and drive torque is available to investigate the effect of rotor speed variations in more detail, for example by including rotor dynamics in the Greitzer model as proposed by (Fink et al.,1992).
Secondly, in Figure4.6 we observe various rapid pressure transients of decreasing am-plitude in the measured signals that are not captured by the simulation model. See Fig-ure4.7for the dimensionless equivalent of these results. From the simulated surge limit cycle in Figure 4.8 we can see that the mentioned pressure transients occur after each flow reversal.
0 10 20 30 40
(b) Auto power spectrum of ∆po
Figure 4.4 /Pressure measurement (gray) and simulation result of the tuned Greitzer model (black) during surge; N = 15,723 rpm, ˙mc0 = 2.07 kg/s, ur = 0,
(b) Auto power spectrum of ∆po
Figure 4.5 /Pressure measurement (gray) and simulation result of the tuned Greitzer model (black) during surge; N = 19,512 rpm, ˙mc0 = 2.62 kg/s, ur = 0, Lc = 10.6m.