Abstract

This paper presents experimental data on a centrifugal compressor operated with CO2. Temperature and pressure at the inlet of the compressor are varied to cover the supercritical region from the liquid-like to the gas-like region. In addition, inlet conditions in the two-phase region are also included. Thus, the experimental test campaign considers thermodynamic conditions relevant for the future energy conversion with sCO2-Joule cycles. Experimental results are presented as compressor pressure ratio versus inlet mass flow rate at different rotational speeds and throttle positions. Reliable conclusions can be drawn from the experimental results since the reproducibility of the measurements has been demonstrated by conducting experiments in two different test rigs, and measurement uncertainties are reported. The entire compressor geometry is disclosed in a data repository, including computer-aided design models, and input files suitable for mean-line and grid generation programs. Thus, the experimental results are exploitable by the scientific community and pave the road for validated analysis and design tools in the context of the sCO2-Joule cycle. The presented results open the possibility to estimate uncertainties of analysis and design tools with little effort since geometry information can be quickly integrated. The experimental data are already used in this paper to obtain the accuracy of a computational fluid dynamics (CFD) code and a mean-line program for sCO2. In addition to quantifying uncertainties, the results presented can be used to identify shortcomings of existing tools. This can be an essential step in the exploration of the sCO2-Joule cycle.

Introduction

This research paper focuses on the work done in both the sCO2-HeRo and the sCO2-4-NPP projects to develop a novel heat removal system for residual decay heat in nuclear power plants, which is based on a Joule cycle using supercritical CO2 (sCO2) as working fluid [1,2]. In the first project, the Chair of Turbomachinery at University Duisburg-Essen built a compact turbomachine with an integral design as depicted in Fig. 1. It consists of a one-stage compressor and turbine on either end of the shaft with an alternator in the middle. Hybrid angular ball bearings support the rotor in a hermetic casing. Henceforward, the turbomachine is called turbine-alternator-compressor (TAC). This TAC is installed in the sCO2-HeRo cycle and described in detail, including geometry, by Hacks et al. [3]. The cycle and its components are validated regarding their performance in several experiments. The experimental analysis is meant to increase the technology readiness level (TRL) of the novel heat removal system operating with sCO2 as working fluid (sCO2-HeRo). The sCO2-HeRo project achieved TRL3 while the sCO2-4-NPP project aims for TRL5 for the full system and TRL7 for parts of the system.

Fig. 1
Cross-section of the sCO2-HeRo TAC [3]
Fig. 1
Cross-section of the sCO2-HeRo TAC [3]
Close modal

Particularly, this paper discusses pressure ratio measurements of the sCO2 compressor as a step to achieve TRL3 (experimental proof of concept) and TRL4 (technology validated in lab). Additionally, the validation of mean-line and computational fluid dynamics (CFD) performance calculations uses these measurements. Measurements benefit from being carried out in both the SUSEN cycle at Research Center Řež and the sCO2-HeRo cycle at Gesellschaft für Simulatorschulung GmbH in Essen. Measurements in both cycles are carried out with two different TAC with compressors manufactured according to the same drawings and for a wide range of compressor inlet conditions. Measurements from both cycles agree well and thus show the reproducibility of measurements. Furthermore, also the pressure ratio predictions fit well to the measurements for a wide range of inlet conditions validating the mean-line and CFD performance calculations.

Experimental and Numerical Setups

To understand the results presented in this paper, it is necessary to take a closer look at the setups of experiments and simulations using CFD and mean-line method. This paper focuses on the compressor designed for static inlet conditions of 78.3 bar, 33 °C and a mass flow rate of 0.65 kg/s at a rotational speed of 50,000 rpm and a pressure ratio of 1.5. Hence, the design operation point is close to the critical point with only a small volume flow rate. A sketch of the meridional cross section and the impeller normal section is presented in Fig. 2. The figure depicts the characteristic blade shape with 90 deg angle at the compressor trailing edge in section A-A. This particular shape results from the through flow coefficient of ϕM = 0.009 and specific speed of σyM = 0.06. Compared to compressors in other sCO2 cycles, these are very small [3]. However, they are based on a compromise in the choice of rotational speed to reduce windage losses. Furthermore, the thickness distribution of each blade (III–IV) is adjusted to minimize flow separation between two adjacent blades. Additionally, the meridional cross section in Fig. 2 shows the cover disk of the impeller and the labyrinth seals at both hub and shroud to reduce the leakage losses, which are expected to be in the range of up to 30% of the design mass flow rate. The compressor has a vaneless diffuser (V–VI), a volute with a rectangular cross section (VI–VII), and an exit cone with a circular cross section (VII–VIII). Information on the design and dimensions of the compressor are given by Hacks et al. [3]. Furthermore the whole geometry is disclosed as computer-aided design (CAD)-model and technical drawings, which can be downloaded from an online repository [4].

Fig. 2
Sketch of meridional cross section of the compressor including the axis normal section of the compressor impeller (A–A)
Fig. 2
Sketch of meridional cross section of the compressor including the axis normal section of the compressor impeller (A–A)
Close modal

Experimental Setup.

Figure 3 expands Fig. 2 by the experimental periphery of measurement points (circles) and flow path sections (squares). This piping and instrumentation diagram (P&ID) is required to explain the different experimental setups of both cycles. Adding the different specifications of the individual measurement positions and piping sections results in the experimental setup for each cycle.

Fig. 3
P&ID section around compressor with relevant flow path sections, measurement points and sensor types
Fig. 3
P&ID section around compressor with relevant flow path sections, measurement points and sensor types
Close modal

In general, pressure and mass flow measurements feature similar uncertainties. These measurements at the compressor inlet (1) and outlet (2) allow the calculation of the pressure ratio and drawing a performance map of pressure ratio as function of mass flow rate.

While the uncertainties are similar for both setups, comparison of pressure ratio measurements and resulting performance maps for different thermodynamic inlet conditions is only possible using an equivalent mass flow rate and pressure rise. This requires determining the density at compressor inlet (1). The setup in the sCO2-HeRo cycle features a direct density measurement while the setup in the SUSEN cycle only allows determining density as function of pressure and temperature. Due to the well-known large sensitivity of fluid properties of sCO2 close to the critical point to static pressure and temperature, the equivalent mass flow rate or volume flow can be determined more accurately with the setup of the sCO2-HeRo cycle. Consequently, this means that measurements taken from the sCO2-HeRo cycle are more accurate when comparing pressure ratios at different thermodynamic inlet conditions, although the uncertainties of pressure and mass flow measurements themselves are similar. For further detailed information on the measurement uncertainties, consult Appendix  A.

Furthermore, experimental setups include the different flow path sections marked by squares in Fig. 3. Flow path sections [a] and [b] are pipes connecting the measurement positions (1) and (2) with the compressor inlet and outlet (I) and (VIII) in Fig. 2, respectively. The pressure losses in these sections may distort the pressure ratio calculated based on the pressure measurements. Therefore, it is essential to understand them. The pipes in sections [a] and [b] have an internal diameter of 14 mm in the SUSEN and 15.9 mm in the sCO2-HeRo setup. Section [a] of the sCO2-HeRo setup is approximately 250 mm long and has one 90 deg bend while its 0 mm long in the SUSEN cycle. Thus, sensors at position (1) are directly in front of the compressor in the SUSEN cycle. Section [b] in the sCO2-HeRo setup is approximately 350 mm long and has one 90 deg and 45 deg bend. Section [b] in the SUSEN cycle is approximately 1700 mm long and has two 90 deg bends. Assuming smooth pipe's surface loss coefficients of 0.08 for 90 deg bends and 0.125 for 45 deg bends for both setups, one can calculate the expected pressure losses for the measurement range. The long section [b] in the SUSEN setup turns out to produce the largest pressure losses of all in both experimental setups. However, the pressure losses are limited to approximately 0.2 bar in this section. This means the pressure ratio of the compressor in the SUSEN setup calculated with pressure losses considered is less than 0.003 higher for the measurement point with the highest flow rate. It is even lower in the sCO2-HeRo setup, where the difference in pressure ratio is less than 0.001 for all measurement points. Thus, measurements tend to underpredict the pressure rise in the compressor. However, the effect is much smaller than the uncertainty of pressure measurements, and there is only marginal influence on the pressure ratio in general. Therefore, the slight underprediction of the pressure ratio due to pressure losses in the pipes is neglected and thermodynamic conditions in measurement positions (1) and (2) are considered equal to the conditions in positions (I) and (VIII) in Fig. 2 as marked in Fig. 3.

Finally, the leakage of sCO2 from the impeller exit to the impeller inlet through the shroud labyrinth seal (dashed line in Fig. 3 and shroud cavity in Fig. 2) must be considered for comparing calculation results and measurements. Due to the leakage, the mass flow at the compressor impeller inlet (II) differs from the mass flow at position (I), which is equal to the one measured at position (1) in Fig. 3. Since the leakage mass flow cannot be measured, it must be calculated by a leakage model. For an accurate calculation, the seal flow coefficient ζ is determined from measurements of the leakage over the hub seal. Since the shroud seal is of the same type (compare geometry given by Hacks et al. [3] or the drawings in the data repository [4], it is used to calculate the shroud leakage flow rate, too. The leakage flow rate via the hub seal in section [c] is characterized mainly by the pressure upstream and downstream of the stepped labyrinth seal. Unfortunately, there are no pressure measurements directly upstream and downstream of the seal. Therefore, pressure measurements in positions (2) and (4) in Fig. 3 are employed neglecting any influence of pressure rise in the diffusor, pressure distribution in the impeller cavity, or any pressure losses. Additionally, there is no direct measurement of the compressor leakage flow rate over the seal in section [c]. In the experimental setup of the sCO2-HeRo cycle, there is a second mass flow measurement at position (3) but not at position (2). Therefore, only measurement points with closed bypass valve are suitable to calculate the leakage flow rate as the difference of mass flow at positions (1) and (3). The experimental setup in the SUSEN cycle features a mass flow measurement neither at position (2) nor (3) but only at position (4). The mass flow at the latter position is always a combination of both compressor and turbine leakage flow rates. Therefore, direct measurement of the compressor leakage flow rate in the SUSEN setup is performed at standstill only with the valve after measurement position (2) being closed, meaning that the leakage flow rate is equal to the mass flow measured at position (1). To calculate the leakage rate for all measurements points, the flow coefficient ζ is determined with a cross-sectional area A = 5.65 mm2 and ϵ for the compressor hub seal with four sealing fins according to Lüdke [5] (p. 217). Then, the shroud leakage is calculated considering equal ζ and cross-sectional area A for the seal with three sealing fins.
ζ=m˙Aϵp2ρ2
(1)

For the turbomachine operated in the SUSEN cycle ζ, a value of approximately 0.5 is determined while in the sCO2-HeRo cycle, it is 0.7. The difference might relate to manufacturing tolerances of seal and impeller or the position of pressure measurements as previously described. Thus, the model includes several uncertainties, and the leakage flow rate across compressor shroud seal determined on this basis is only an estimate.

Computational Fluid Dynamics Setup.

One possibility to calculate the pressure ratio of the compressor is by CFD simulation. The validation of this approach provides the opportunity to have an insight into the three-dimensional flow field that is not available from the measurements. In addition, CFD validation will pave the way for high fidelity design and optimization techniques for sCO2 compressors. CFD simulations are conducted with the commercial code ansyscfx 19.1 using the steady RANS solver. Turbulence is accounted for using the SST k–ω turbulence model with an automatic wall function. Moreover, in all simulations, the high resolution scheme, which corresponds to a second order accuracy scheme, is selected for the advection and the turbulence modeling whenever applicable. The Rhie–Chow pressure–velocity coupling in cfx is set to second order accurate as this is found to improve the solver stability and convergence rate. Fluid properties of CO2 are implemented through real gas property table with a very fine resolution of around 0.06 bar and around 0.03 K. Properties are calculated from the Span and Wagner equation of state [6] using the NIST REFPROP database [7].

The numerical domain includes the inlet nozzle (I–II) and the radial impeller (II–IV) with the vaneless diffuser (V–VI) followed by the volute (VI–VII) and the exit cone (VII–VIII). Mesh independence study is conducted for the impeller part and the vaneless diffuser part using a single passage (II–VI) domain with rotational periodicity. Figure 4 illustrates the mesh independency results. Based on the results, a mesh with around 280,000 nodes for a single passage of the impeller is selected, which correspond to a refinement ratio of around four from the base mesh. On the other hand, mesh independency is conducted using the full geometry for the volute. Results show negligible effect of the mesh refinements. A mesh with around 2,400,000 nodes that correspond to around 2 times refinement ratio from the base mesh is selected for the volute. In comparison to the real geometry of the compressor, the hub and shroud side cavities are omitted. This is done not only to reduce computational effort but also due to the lack of sufficient information regarding the boundary condition at the hub side exit. At this side, leakage will affect mostly the flow downstream of the impeller by a reduction of the mass flow rate. In contrary, on the shroud side, the leakage flows from the impeller exit through the labyrinth seal and mixes with the flow from the compressor inlet (I). While temperature increase due to mixing is neglected, the increase in mass flow rate through the impeller is considered to shift the compressor's performance curve. The latter must be considered when comparing CFD results with measurements.

Fig. 4
Mesh independence results
Fig. 4
Mesh independence results
Close modal

Mean-Line Setup.

The mean-line analysis is incorporated into a computer program based on matlab [8], which intends to predict the performance of the sCO2-HeRo compressor and compressors with similar configuration quickly and reliably. The mean-line analysis follows a classical approach where mass and energy conservation are calculated in conjunction with appropriate loss models at different positions along the flow path corresponding to Fig. 2. Within the mean-line analysis, leakage and parasitic works are considered as well. The flow coefficient ζ calculated by Eq. (1) is also applied for the calculation of the leakage flow at the shroud and the shaft side. In the previous study of Ren et al. [8], comparison between the mean-line results and some measured results from the SUSEN loop is already performed and this study extends the validation with additional measurements at higher rotational speed.

Measurements and Simulation Results

Measurements.

Measurements with the TAC built in the sCO2-HeRo project were carried out in the SUSEN cycle first to understand whether the compressor meets the expectations of the design calculations with CFD. The measured pressure ratios showed good agreement with the pressure ratio predictions by CFD design calculations, already. However, rotational speed was limited to 30,000 rpm and inlet conditions were only well away from the critical point. They are shown in the T-s diagram in Fig. 5 for three density levels, low (diamond), intermediate (circle), and high (triangle), which lie in gas-like and liquid-like sCO2 to the right and left of the pseudo-critical line. In this paper, the original measurements in the SUSEN cycle, presented by Hacks et al. [9], are extended to include pressure ratio measurements from the sCO2-HeRo cycle at speeds up to 40,000 rpm and inlet conditions close to the critical point and even in the dual phase regime, presented by squares in Fig. 5. The rotational speed of the TAC is currently limited to 40,000 rpm by the maximum motor power and to achieve 50,000 rpm the turbine needs to be optimized to generate sufficient power. Different colors, yellow for 20,000 rpm, green for 30,000 rpm, and blue for 40,000 rpm, mark rotational speed equal to Fig. 7 in Appendix  C.

Fig. 5
Inlet conditions for measurements in SUSEN and sCO2-HeRo cycle
Fig. 5
Inlet conditions for measurements in SUSEN and sCO2-HeRo cycle
Close modal

Figure 6 compares the pressure ratio to simulation results in two diagrams. On the left the pressure ratio is depicted over mass flow measured at position (1 = I) and on the right over mass flow at position (II) according to Fig. 2. The latter is required because leakage flow rate over the shroud seal reaches values of up to 0.1 kg/s or 20% of the flow rate measured at position (1 = I) and must be considered in the comparison with CFD simulation results. Furthermore, Fig. 6 compares simulation and measurements at the reference condition of thermodynamic design inlet condition of 78.3 bar and 33 °C marked by the black cross in Fig. 5. Therefore, all pressure ratios and mass flows are converted by the help of affinity laws (summarized in Appendix  B). The equivalent pressure ratios of the experimental data found from the affinity laws were fitted to a single polynomial function for each speed line for comparison. Error propagation of the measurement uncertainties is considered and depicted by the shaded area. Since all measurements in Fig. 6 fall together on three speed lines, the measurements are conclusive and thus show reproducibility of pressure ratio measurements despite the two different experimental setups.

Fig. 6
Comparison between CFD, mean-line and experimental pressure ratios converted to the reference condition 78.3 bar and 33 °C (left: mass flow at compressor inlet (I); right: mass flow at impeller inlet (II))
Fig. 6
Comparison between CFD, mean-line and experimental pressure ratios converted to the reference condition 78.3 bar and 33 °C (left: mass flow at compressor inlet (I); right: mass flow at impeller inlet (II))
Close modal

For the raw data on measurements, pressure ratios, leakage mass flow rates, and the polynomial coefficients, please consider Annex C.

Computational Fluid Dynamics Simulation.

Computational fluid dynamics simulations are conducted at three rotational speeds of 20,000, 30,000, and 40,000 rpm. Moreover, simulation is conducted at the nominal inlet condition of 78.3 bar and 33 °C. An average deviation is calculated as
Δ¯=1ZΔ
(2)
where Z is the number of the selected measured data points and Δ is the deviation of the CFD results to the measurements expressed by
Δ=|πCFDπEXP|πEXP×100%
(3)
where πCFD and πEXP represent the pressure ratio calculated by CFD and that measured by experiments, respectively. Results show a good agreement with the experimental data with the average deviation of around 1.3% at the 20,000 rpm line, around 1.1% at the 30,000 rpm line and less than 1% at the 40,000 rpm line. This indicates a deviation of the CFD results that falls within the uncertainty of the measurements

Mean-Line Analysis.

Figure 6 also shows that the mean-line results (abbreviated ML in the legend) denoted by dashed lines are in agreement with the measurements in tendency at all rotational speeds. Additionally, the deviations of the mean-line results to the measured results are calculated by Eq. (3), where the pressure ratios of mean-line analysis πML are used instead of πCFD. The average deviation can be obtained by Eq. (2). The average deviation at 20,000 rpm is 1.2%, while it is 2.2% and 2.5% at 30,000 rpm and 40,000 rpm, respectively. This exhibits a good quality of the prediction quantitatively. The right diagram in Fig. 6 exhibits that the deviations between the mean-line and the measured results by considering the mass flow rate at the impeller inlet (II) are similar to the left one.

In summary, the mean-line analysis described in Ref. [8] is further verified by extending the rotational speed to a higher level (up to 40,000 rpm). The accuracy of the mean-line prediction regarding pressure ratio can reach a mean value of 98% by neglecting the uncertainties in the measurements. According to this comparison, the mean-line analysis is recommended to be applied for the performance prediction of sCO2 compressors, which have the same configuration as the sCO2-HeRo compressor.

Conclusions

In this paper, new experimental results regarding the sCO2-HeRo cycle conducted by University Duisburg-Essen and Gesellschaft für Simulatorschulung GmbH in Essen are presented. These experimental results are compared with those from the SUSEN cycle, with CFD and mean-line results.

The new experiments in sCO2-HeRo cycle not only reach a higher rotational speed of up to 40,000 rpm expanding the operation range of the sCO2-HeRo compressor by approximately 33%, but also contain the measurements at the same rotational speeds (20,000 rpm and 30,000 rpm) for measurements conducted in the SUSEN cycle. More inlet conditions that are closer to the critical point and even in the two-phase regime are tested in the sCO2-HeRo cycle. By identifying the different setups in both cycles, the comparison between the experimental results shows reproducibility of the pressure ratio measurements.

Computational fluid dynamics simulations are conducted as well by considering the whole compressor geometry without side cavities from the compressor inlet to the outlet. By comparing the experimental results to the CFD results, the CFD approach is validated. The comparison shows good agreement. The validated CFD approach can give an insight into the three-dimensional flow field that is difficult to obtain from the measurements and will provide high fidelity design and optimization of sCO2 compressors.

Moreover, a mean-line analysis tool is compared to both the experimental and CFD results. The mean-line results also show a good agreement with the experimental results but a larger deviation in contrast to the CFD results. On the other hand, the computing time of the mean-line analysis is much less than that of the CFD simulation. This means that this mean-line analysis providing a quick and reliable prediction of the compressor performance can be applied for sCO2 compressors with the same or the similar configuration as the sCO2-HeRo compressor.

Consequently, this study has achieved TRL4, which means that the experiments not only contribute to the development of the sCO2 TAC but also bring the development of the sCO2-HeRo system further to a higher level or, in other words, closer to the market.

Associated Contend

Supplementary Data [4] is available free of charge at following footnote link.1 The data repository contains the entire compressor geometry, including CAD models, and input files suitable for mean-line and grid generation programs.

Acknowledgment

The authors wish to thank the teams of Research Center Řež and Gesellschaft für Simulatorschulung GmbH for their support in the experiments.

Funding Data

  • Euratom Research and Training Program 2014–2018 (Grant Agreement Nos. 847606 and 662116; Funder ID: 10.13039/100018708).

Nomenclature

     
  • D =

    diameter, m

  •  
  • m˙ =

    mass flow, kg/s

  •  
  • n =

    rotational speed, 1/s

  •  
  • p =

    pressure, Pa

  •  
  • T =

    temperature, °C, K

  •  
  • y =

    specific head, J/kg

  •  
  • Δ =

    deviation

  •  
  • π =

    pressure ratio

  •  
  • ρ =

    density, kg/m³

  •  
  • σyM=2.108nV˙y3/4 =

    specific speed

  •  
  • φ=4*V˙π2*D3*n =

      flow coefficient

Subscripts

    Subscripts
     
  • EXP =

    experiment

  •  
  • ML =

    mean-line

Abbreviations

    Abbreviations
     
  • CFD =

    computational fluid dynamics

  •  
  • P&ID =

    piping and instrumentation diagram

  •  
  • sCO2 =

    carbon dioxide at supercritical state

  •  
  • sCO2-HeRo =

    supercritical CO2 heat removal system

  •  
  • sCO2-4-NPP =

    supercritical CO2 heat removal system for nuclear power plants

  •  
  • SUSEN =

    SUSEN cycle at Research Center Řež, Czech Republic

  •  
  • TAC =

    turbine, alternator, compressor system

  •  
  • TRL =

    technology readiness level

Footnotes

Appendix A: Compressor Piping and Instrumentation Diagram and Measurement Uncertainties

Table 1 shows the measurements and their uncertainties at each position together with P&ID-No. in the original publications by Hacks et al. [9] and Hofer et al. [10]. Hacks et al. [9] give the measurement uncertainties for the SUSEN cycle in Table 2 of their paper. Here, only the total error from that table is given. Hofer et al. [10] present the assessment of the measurement uncertainties in the sCO2-HeRo cycle in the sCO2-4-NPP public deliverable.

Table 1

Measurement uncertainties

SUSEN cyclesCO2-HeRo cycle
Measurement position in Fig. 3 Measured variableP&ID-No. [8]Measurement uncertaintyP&ID-No. [9]Measurement uncertainty
Compressor inlet (1)PressureTK01 P01021.10 barTK01 P01020.92 bar
TemperatureTK01 T01020.26 KTK01 T01021.80 K
Mass flow1 LKB70 CF0010.007 kg/sTK01 F01010.007 kg/s
DensityTK01 M01013.5 kg/m³
Compressor outlet (2)PressureTK01 P02011.10 barTK01 P02010.92 bar
TemperatureTK01 T02010.35 KTK01 T02011.80 K
Compressor outlet (3)Mass flowTK01 F02010.007 kg/s
TAC leakage (4)Pressure1 LKB70 CP0012.3 barTK02 P01010.92 bar
Mass flowTK02 F05010.007 kg/s
SUSEN cyclesCO2-HeRo cycle
Measurement position in Fig. 3 Measured variableP&ID-No. [8]Measurement uncertaintyP&ID-No. [9]Measurement uncertainty
Compressor inlet (1)PressureTK01 P01021.10 barTK01 P01020.92 bar
TemperatureTK01 T01020.26 KTK01 T01021.80 K
Mass flow1 LKB70 CF0010.007 kg/sTK01 F01010.007 kg/s
DensityTK01 M01013.5 kg/m³
Compressor outlet (2)PressureTK01 P02011.10 barTK01 P02010.92 bar
TemperatureTK01 T02010.35 KTK01 T02011.80 K
Compressor outlet (3)Mass flowTK01 F02010.007 kg/s
TAC leakage (4)Pressure1 LKB70 CP0012.3 barTK02 P01010.92 bar
Mass flowTK02 F05010.007 kg/s
Table 2

Affinity laws according to Bohl and Elmendorf [10]

Speed ratio
kn=nREFnRD
(4)
Size ratio
kd=dREFdRD=1
(5)
Density ratio
kρ=ρREFρRD
(6)
Mass flow
m˙REFm˙RD=kd3*kn*kρ
(7)
Pressure difference
ΔpREFΔpRD=kd2*kn2*kρ
(8)
Pressure ratio
πREF=(p1REF+ΔpREF)p1REF
(9)
Speed ratio
kn=nREFnRD
(4)
Size ratio
kd=dREFdRD=1
(5)
Density ratio
kρ=ρREFρRD
(6)
Mass flow
m˙REFm˙RD=kd3*kn*kρ
(7)
Pressure difference
ΔpREFΔpRD=kd2*kn2*kρ
(8)
Pressure ratio
πREF=(p1REF+ΔpREF)p1REF
(9)

Appendix B: Affinity Laws

Affinity laws are used to convert the raw measurement data presented in Annex C to the design inlet conditions of the compressor. The formulations according to Bohl and Elmendorf [11] that was already applied by Hacks et al. [9] are summarized in Table 2. Here, condition (RD) can be considered as the raw measurement data and condition (REF) as the reference conditions. Pressure ratio is calculated from converted pressure difference ΔpI together with the reference pressure at compressor inlet p1I.

Appendix C: Measurements—Raw Data and Leakage Calculation

Figure 7 presents the pressure ratio calculated from raw measurements as function of mass flow rate. It shows measurements in the sCO2-HeRo and SUSEN cycle at three different rotational speeds (marked by color) for different inlet conditions according to Fig. 5. The markers are the same in Figs. 5 and 7. The markers for measurements in the SUSEN cycle differentiate between three density levels, low (diamond), intermediate (circle), and high (triangle). In general, larger inlet density means larger pressure ratio (Hacks et al. [9]). Note that the measurements in the sCO2-HeRo cycle are all closer to the critical point in contrast to those in the SUSEN cycle and usually at relatively lower inlet density. Thus, the measured pressure ratio is smaller compared to measurements in the SUSEN cycle with inlet conditions left of the pseudo-critical line in Fig. 5. On the other hand, at 20,000 rpm, measurements show that those in the sCO2-HeRo lie, as anticipated, between those with inlet conditions left and right of the pseudo-critical line in the SUSEN cycle independent of the inlet conditions being closer to the critical point.

Fig. 7
Pressure ratios measured in sCO2-HeRo and SUSEN cycle over mass flow rate measured at compressor inlet (position 1 = I in Fig. 3) on the left and mass flow rate at compressor impeller inlet (position II in Fig. 2) on the right
Fig. 7
Pressure ratios measured in sCO2-HeRo and SUSEN cycle over mass flow rate measured at compressor inlet (position 1 = I in Fig. 3) on the left and mass flow rate at compressor impeller inlet (position II in Fig. 2) on the right
Close modal

Furthermore, the left diagram depicts the pressure ratio over mass flow measured at the compressor inlet (I) in Fig. 2, which is equal to the mass flow measured at position (1) in Fig. 3, and the right diagram uses the mass flow at the impeller inlet (position (II) in Fig. 2), which considers the shroud leakage, too. Note that the change of enthalpy and density from positions (I) to (II) due to the mixing with the shroud leakage is neglected. All points depicted in Fig. 7 are also tabulated in Table 3.

Table 3

Raw measurement data and calculated values for Fig. 7 

Rotation speedMeasurements according to position in Fig. 3 Compressor pressure ratioShroud leakage
CycleNp1T1ρ1m˙1p2T2p4πm˙Shroud leakage
rpmbar°Ckg/m³kg/sbar°Cbarkg/s
SUSEN cycle20,17277.533.80.2880.035.557.61.0320.02
20,17877.033.40.2879.535.262.71.0320.02
20,17378.034.00.2380.936.162.11.0370.02
20,17578.234.10.2081.136.362.21.0380.02
20,17278.434.30.1181.536.562.81.0400.02
20,17478.134.10.1681.236.362.21.0390.02
20,17478.434.30.1381.636.562.61.0400.02
19,90077.433.20.3480.435.062.01.0390.02
19,83877.333.00.3580.434.862.01.0400.02
20,19479.133.80.1583.536.064.41.0560.03
20,19178.733.50.2682.935.763.71.0540.03
20,25878.133.20.3581.935.263.01.0490.03
20,19278.933.70.2183.335.964.01.0560.03
20,25678.433.40.3082.635.563.41.0530.03
20,25577.733.00.3981.334.962.71.0460.03
20,17580.032.80.2385.434.866.41.0680.04
20,17280.132.90.2085.534.866.61.0680.04
20,17579.932.70.2785.234.666.31.0670.04
20,22178.932.30.4483.433.965.91.0580.03
20,21879.132.40.4183.934.166.11.0610.03
20,22078.732.20.4783.133.865.61.0560.03
20,21679.632.60.3284.934.466.81.0650.04
20,21779.332.40.3884.234.266.31.0620.04
30,41879.333.50.3790.338.160.41.1390.06
30,42079.233.40.3990.238.060.41.1380.06
30,41779.033.30.4389.737.860.31.1360.06
30,41878.032.80.5687.837.059.71.1260.05
30,41878.733.10.4889.137.560.11.1330.05
30,42178.533.00.5088.837.360.11.1310.05
30,42178.832.90.4590.037.261.11.1430.06
30,41979.433.10.3891.137.561.91.1470.06
30,41879.433.00.3891.137.462.81.1480.06
30,41979.132.90.4090.937.263.51.1480.06
30,41978.932.80.4390.537.163.21.1460.06
30,42379.332.90.3791.137.363.81.1490.06
30,41878.432.50.4889.736.764.61.1440.06
30,47778.232.50.5289.236.564.41.1410.06
30,47978.132.40.5489.036.464.31.1390.06
30,41878.632.60.4690.136.864.21.1460.06
30,41878.432.50.4989.736.764.61.1440.06
30,45178.432.50.5089.536.664.51.1420.06
30,47578.032.30.5788.736.264.21.1380.06
sCO2-HeRo cycle20,19275.8451.20.3079.033.763.71.0420.03
20,19276.0436.00.2779.333.964.01.0430.03
20,19375.3454.80.2878.733.563.41.0460.03
20,24876.6477.60.4179.233.863.41.0340.03
20,25177.1465.90.4279.333.963.81.0290.02
30,37873.3441.30.3382.335.462.81.1220.07
30,39274.2441.40.3682.635.763.61.1130.06
30,39074.0443.00.3782.535.563.51.1140.06
30,36774.3439.50.3283.335.963.41.1210.07
30,39473.3445.70.3582.135.362.91.1200.07
30,33972.5447.50.3181.835.062.41.1290.07
30,33874.3444.90.3183.536.163.51.1240.07
30,33773.4450.50.3282.735.463.01.1260.07
30,33974.4458.90.3083.836.163.51.1270.07
30,39674.7422.20.3482.936.063.81.1100.06
30,39674.1393.20.2982.235.863.71.1080.06
30,39673.4387.10.2981.435.463.21.1090.06
30,45173.2524.10.4882.935.061.71.1320.07
30,45273.8499.40.4683.035.462.31.1240.07
30,50773.7486.90.5181.834.962.01.1090.06
30,50973.5483.30.4981.834.961.91.1130.07
30,50973.4503.30.5681.234.461.61.1060.06
30,51073.1490.40.4981.734.861.61.1180.07
40,57273.8502.70.4492.339.363.21.2510.10
40,57773.6503.40.4992.239.263.51.2510.10
40,57773.6510.40.4692.539.263.51.2560.10
40,57874.1497.70.4792.339.463.51.2460.10
40,57973.7503.30.4692.339.363.51.2520.10
40,57973.8508.10.4492.539.363.31.2540.10
Rotation speedMeasurements according to position in Fig. 3 Compressor pressure ratioShroud leakage
CycleNp1T1ρ1m˙1p2T2p4πm˙Shroud leakage
rpmbar°Ckg/m³kg/sbar°Cbarkg/s
SUSEN cycle20,17277.533.80.2880.035.557.61.0320.02
20,17877.033.40.2879.535.262.71.0320.02
20,17378.034.00.2380.936.162.11.0370.02
20,17578.234.10.2081.136.362.21.0380.02
20,17278.434.30.1181.536.562.81.0400.02
20,17478.134.10.1681.236.362.21.0390.02
20,17478.434.30.1381.636.562.61.0400.02
19,90077.433.20.3480.435.062.01.0390.02
19,83877.333.00.3580.434.862.01.0400.02
20,19479.133.80.1583.536.064.41.0560.03
20,19178.733.50.2682.935.763.71.0540.03
20,25878.133.20.3581.935.263.01.0490.03
20,19278.933.70.2183.335.964.01.0560.03
20,25678.433.40.3082.635.563.41.0530.03
20,25577.733.00.3981.334.962.71.0460.03
20,17580.032.80.2385.434.866.41.0680.04
20,17280.132.90.2085.534.866.61.0680.04
20,17579.932.70.2785.234.666.31.0670.04
20,22178.932.30.4483.433.965.91.0580.03
20,21879.132.40.4183.934.166.11.0610.03
20,22078.732.20.4783.133.865.61.0560.03
20,21679.632.60.3284.934.466.81.0650.04
20,21779.332.40.3884.234.266.31.0620.04
30,41879.333.50.3790.338.160.41.1390.06
30,42079.233.40.3990.238.060.41.1380.06
30,41779.033.30.4389.737.860.31.1360.06
30,41878.032.80.5687.837.059.71.1260.05
30,41878.733.10.4889.137.560.11.1330.05
30,42178.533.00.5088.837.360.11.1310.05
30,42178.832.90.4590.037.261.11.1430.06
30,41979.433.10.3891.137.561.91.1470.06
30,41879.433.00.3891.137.462.81.1480.06
30,41979.132.90.4090.937.263.51.1480.06
30,41978.932.80.4390.537.163.21.1460.06
30,42379.332.90.3791.137.363.81.1490.06
30,41878.432.50.4889.736.764.61.1440.06
30,47778.232.50.5289.236.564.41.1410.06
30,47978.132.40.5489.036.464.31.1390.06
30,41878.632.60.4690.136.864.21.1460.06
30,41878.432.50.4989.736.764.61.1440.06
30,45178.432.50.5089.536.664.51.1420.06
30,47578.032.30.5788.736.264.21.1380.06
sCO2-HeRo cycle20,19275.8451.20.3079.033.763.71.0420.03
20,19276.0436.00.2779.333.964.01.0430.03
20,19375.3454.80.2878.733.563.41.0460.03
20,24876.6477.60.4179.233.863.41.0340.03
20,25177.1465.90.4279.333.963.81.0290.02
30,37873.3441.30.3382.335.462.81.1220.07
30,39274.2441.40.3682.635.763.61.1130.06
30,39074.0443.00.3782.535.563.51.1140.06
30,36774.3439.50.3283.335.963.41.1210.07
30,39473.3445.70.3582.135.362.91.1200.07
30,33972.5447.50.3181.835.062.41.1290.07
30,33874.3444.90.3183.536.163.51.1240.07
30,33773.4450.50.3282.735.463.01.1260.07
30,33974.4458.90.3083.836.163.51.1270.07
30,39674.7422.20.3482.936.063.81.1100.06
30,39674.1393.20.2982.235.863.71.1080.06
30,39673.4387.10.2981.435.463.21.1090.06
30,45173.2524.10.4882.935.061.71.1320.07
30,45273.8499.40.4683.035.462.31.1240.07
30,50773.7486.90.5181.834.962.01.1090.06
30,50973.5483.30.4981.834.961.91.1130.07
30,50973.4503.30.5681.234.461.61.1060.06
30,51073.1490.40.4981.734.861.61.1180.07
40,57273.8502.70.4492.339.363.21.2510.10
40,57773.6503.40.4992.239.263.51.2510.10
40,57773.6510.40.4692.539.263.51.2560.10
40,57874.1497.70.4792.339.463.51.2460.10
40,57973.7503.30.4692.339.363.51.2520.10
40,57973.8508.10.4492.539.363.31.2540.10

Table 4 shows the polynomial coefficients of the fitting curves of the measured pressure ratios in Fig. 6.

Table 4

Polynomial fitting of the equivalent pressure ratio at the nominal condition for the experimental measurement

Speed line rpmSecond order polynomial coefficient π = m˙2+bm˙+cValidity limits
Valueabcminmax
20,000−0.27520.10891.0490.15 kg/s0.5 kg/s
30,000−0.31430.18121.1170.34 kg/s0.64 kg/s
40,000−0.24690.22271.210.48 kg/s0.56 kg/s
Speed line rpmSecond order polynomial coefficient π = m˙2+bm˙+cValidity limits
Valueabcminmax
20,000−0.27520.10891.0490.15 kg/s0.5 kg/s
30,000−0.31430.18121.1170.34 kg/s0.64 kg/s
40,000−0.24690.22271.210.48 kg/s0.56 kg/s

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