Nowadays the main use of energy is to convert heat energy into mechanical energy. Even though the heat source may come from nuclear reactor, solar energy, geothermal energy, industrial waste heat, chemical fuel combustion and so on, the conversion system is mainly the closed Brayton cycle with special working fluids. Among the numerous alternative cycling mediums, the supercritical carbon dioxide (SCO2) is becoming more and more popular due to its high critical temperature which is close to normal temperature and high density which considerably reduced the machine size (Feher, 1968; Dostál et al., 2011; Deng et al., 2017; Yao and Zou, 2020).

It is the compressor that works near the critical point of CO2 in the closed Brayton cycle system, so the application advantage of CO2 as the working medium in Brayton cycle system is mainly reflected in CO2 compressor (Wright et al., 2010; Lettieri et al., 2015; Lee et al., 2016). However, there are few studies which theoretically analyze how the physical properties of CO2 near the critical point affect the compressor performance. The main difficulty lies in the lack of a set of specifications to restrict other variables so that only the real gas effect is considered. Baltadjiev (Baltadjiev, 2012) avoided the above problem by comparing the one-dimensional nozzle flow of real gas and ideal gas. The results indicated a reduction of 9% in the choke margin compared to its performance at ideal gas conditions. Although it is difficult to directly extend the conclusion of one-dimensional nozzle flow to three-dimensional rotating compressor flow, lessons can be drawn from the application of the isentropic pressure exponent ns (Schultz, 1962) which is of great significance for the flow analysis of real gases. In addition, some researches (Chen et al., 2019a) simply changed the inflow condition of SCO2 compressors to study the variation of compressor performance. However, due to the inconsistency of the flow field under different conditions, it is hard to ensure that the performance is not affected by other factors.

One possible method is to obtain a series of dimensionless parameters that affect the compressor performance through dimensional analysis (Simon et al., 2017). By remaining these dimensionless parameters unchanged, the performance comparison will be meaningful. This solution, which is called similitude method, is widely adopted in the field of air compressor experiments. In fact, for two similar air compressors, the performance of the two compressors is identical because the physical properties of the working fluids remain unchanged (Dixon and Hall, 2014). In condition that the fluid properties are changed, some researchers have studied the similitude method between different ideal gas compressors. There are several ways to deduced the similarity criteria, including dimensional analysis (Roberts and Sjolander, 2005; Chen et al., 2019b), kinematic similarity derivation (Zou and Ding, 2018) and differential equation method (Zhu et al., 2008). However, when it comes to the near-critical CO2, the ideal gas assumption can not be satisfied, so the above methods are no longer applicable. Pham (Pham et al., 2016) proposed a set of similitude parameters by introducing the isentropic pressure exponent ns and compressibility factor Z into the similar parameters of air compressor. However, there may be some deviations due to the lack of strict physical derivation. Jeong (Jeong et al., 2020) used Pham’s similitude parameter, but made some numerical correction to the efficiency similitude parameter. Good agreement had been achieved. However, this correction was obtained by trials. Xu (Xu et al., 2021) theoretically deduced a unified performance conversion method for turbomachines working with all kind of fluids, which could be the basis of this work.

In this paper, a set of nondimensional parameters which determine the compressor performance are firstly deduced by dimensional analysis. Since physical properties of CO2 vary with different compressor inlet conditions, it is hardly possible to keep all the physical property parameters unchanged. Thus, this paper proposed a set of new similitude parameters which may ensure approximate similarity between real gas compressors by combining the physical property parameters and the deduced nondimensional parameters. Then keeping these similitude parameters unchanged, the variation of a SCO2 compressor performance is studied when its inlet condition changes along an isotherm line and an isobar line near the critical point. Numerical simulations are also adopted for validation. Finally, using this similitude method, the performance distribution of a SCO2 compressor in a certain range of inlet pressure and temperature is given. The influence of the real-gas effect on the compressor performance is summarized, and some suggestions on the designed inlet flow condition of the SCO2 compressor are given.


Physical properties of carbon dioxide

The specific heat Cp, compressibility factor Z and enthalpy H of CO2 in a certain range are displayed in Figure 1. This range (T=310K360K,P=7.4MPa8.2MPa) is selected because the incoming flow for most SCO2 compressors lies in it. A sharp tip of the Cp surface could be found near the critical point, indicating that in SCO2 compressors the specific heat Cp of the working fluid could no longer be regarded as constant. It can also be found that near the critical temperature, the compressibility factor Z deviates a lot from 1, which indicates that the ideal gas hypothesis is no longer applicable. In addition, the enthalpy of CO2 is not only related to temperature, but also affected by the pressure. Obviously there is a coupling effect between pressure rise and enthalpy increase. Overall, the real gas CO2 behaves totally differently from the ideal gas near the critical point so it is necessary to study the real-gas effect of CO2 on the performance of compressors.

Figure 1.

Physical properties of carbon dioxide.

Dimensional analysis

The performance of a compressor can be expressed in terms of the control variables, geometric variables, and fluid properties. The geometric variables are characterized by the impeller diameter D, while others are gathered as L=(l1,l2,l3,). The control variables include the inlet total temperature T01, the inlet total pressure P01, the mass flow rate m˙ and the rotational speed n. The fluid properties consist of the gas constant Rg, the specific heat ratio γ and kinematic viscosity υ. The compressor performance is reflected in terms of enthalpy rise ΔH, pressure ratio π and efficiency η. Since there is a certain conversion relationship between these three performance parameters, only the enthalpy rise ΔH and efficiency η are selected to represent the compressor performance.

Due to the strong nonlinear characteristics of the physical properties near the critical point for CO2, the state equation of ideal gas is no longer applicable. In addition, the specific heat ratio cannot reflect the isentropic change process of working fluid alone. Hence, the compressibility factor Z and the isentropic pressure exponent ns are introduced to describe real gas properties. They are defined as follows:


where αT is the isothermal compressibility and is defined as (V/P)T/V. For ideal gases, αT=1/P, so ns=γ.

Now it is convenient to write the two functional relationships as


Choosing the parameters (T01,P01,Rg,D) as dimensions, these two relationships can be reduced to the following forms based on Buckingham’s π-theory (Buckingham, 1914):


where (m˙RgT01)/(P01D2) is nondimensional mass flow, (nD)/RgT01 is nondimensional blade tip speed, Z0 and ns are fluid properties which are already nondimensional, (nD2)/υ is the Reynolds number and L/D is nondimensional shape parameter.

Obviously, full similarity can only be achieved when the above six nondimensional parameters are all identical. However, no matter for the similarity between the real gas and the ideal gas compressor, or for the similarity between the real gas compressors with different inlet conditions, it is impossible to ensure that all physical parameters are equal. Therefore, refer to Chen’s work (Chen et al., 2019b), we derived a set of new similitude parameters to achieve approximate similarity by combining those nondimensional parameters and fluid property parameters. The forms are as follows:


In the Equations 7 and 8, the bracket represents a new parameter composed of its internal parameters. Its specific expression will be derived in the next section.

Similitude parameters

In this section, the specific expressions for the mass flow similitude parameter, the compressibility similitude parameter, the pressure ratio similitude parameter and the efficiency similitude parameter are deduced based on some similarity assumptions.

(m˙RgT01)/(P01D2),Z0,ns is the similitude parameter for the mass flow. In this paper, it is assumed that similar compressors shares similar shapes of characteristic curves, and the relative positions of two similar points on their respective characteristic curves are the same. Consequently, the similitude parameter of mass flow, which means the swallowing capacity of the compressor, can be defined as


According to the real gas thermodynamics, the mass flow and critical mass flow are


Therefore, the similitude parameter for the mass flow is


(nD)/RgT01,Z0,ns is the similitude parameter for the blade tip speed, which should be nondimensionalized by the sound speed. According to the real-gas thermodynamics, the sound speed is a=nsZ0RgT0. Then the similitude parameter for the blade tip speed could be written as:


η,Z0,ns is the similitude parameter for the isentropic efficiency. The author’s previous work (Xu et al., 2021) has theoretically derivated the similitude parameter for efficiency. It it proved that the polytropic efficiency ηp is much more suitable for performance conversion between similar compressors working with different gases. This is because the polytropic efficiency ηp is independent of the pressure ratio, enthalpy increase, inlet flow conditions, and fluid physical properties. The influences of all these factors are actually manifested in the conversion process from ηp to ηs. So here we choose the polytropic efficiency ηp as the similitude parameter, which is


Since it is much more popular to use isentropic efficiency ηs rather than polytropic efficiency ηp in engineering, conversion between these two kinds of efficiency is necessary. Detailed conversion method has been presented in the author’s previous work (Xu et al., 2021).

ΔH/(RgT01),Z0,ns is the similitude parameter for the enthalpy rise, it can be deduced that


For similar velocity triangles, there is


Above all, the final similitude parameters are as follows


Approximate similarity

While the fluid properties are not completely identical, only approximate similarity can be achieved using the above similarity criterion. Thus, it is necessary to evaluate the degree of similarity. The similarity of compressor can be evaluated from two aspects, namely, kinematics similarity and dynamic similarity. The kinematics similarity means that the inlet and outlet velocity triangles of the compressor are proportional respectively, which can be quantified by cm1/u and cm1/cm2. The dynamic similarity means that the magnitude and direction of the same force on the corresponding point in the flow field are proportional, which specifically include the axial Mach number Ma1a, the rotational Mach number Mu and the Reynold number Re. Since Mu and Re are the similitude parameters, only cm1/u, cm1/cm2 and Ma1a are considered.

For two compressors with the identical geometry but different working media, assuming that the working fluids are A and B respectively, the following two equations can be derived when Πm˙, ΠU and Re are identical.


Setting nsA to a constant value of 1.29, Figure 2 shows the variation of (c1/u)B/(c1/u)A with axial Mach number Ma1a when nsB changes from 1.3 to 1.9. It can be seen that greater difference in ns will lead to larger deviation in kinematics similarity. If the ns difference is small, the variation of the incoming Mach number Ma1a has little effect on the similarity of the compressor. However, when the ns difference is large, the degree of kinematics similarity decreases with a growing Ma1a while the degree of dynamic similarity increases. In general, for different inlet conditions, if the isentropic pressure exponent ns changes a lot, there will be assignable errors exist no matter for the kinematics similarity or for the dynamic similarity. Therefore, further validation is necessary for the similitude method.

Figure 2.

Evaluation of approximate similarity for different working fluid B when nsA=1.29.

Results and discussion

Validation of the similitude method

A typical SCO2 centrifugal compressor with near-critical designed inlet condition is selected for further validation. Detailed dimensions and the operation conditions of this compressor at the design point are listed in Table 1. With identical geometry, numerical simulations are performed with ideal CO2 gas and real CO2 gas. The commercial software ANSYS CFX 19.0 is used to solve steady viscous Reynolds Averaged Navier-Stokes equations with a time marching finite volume method. The spatial discretization adopts a second-order upwind scheme and the temporal discretization adopts a second-order backward Euler scheme. The shear stress transport (SST) turbulence model is used. In the numerical calculation, only one passage is simulated so that surfaces at circumferential side are set as periodic boundary conditions. Total temperature, total pressure, and inflow angle are used as the inlet boundary conditions, and static pressure is used as the outlet boundary condition. All the wall faces are set as adiabatic with no slip.

Table 1.

Informations of the simulated compressor.

320 K7,600 kPa30 mm68 mm152 mm−33°

Fully structured grids were generated using Autogrid5. An H-type grid topology is used in the mainstream passage, and an O-type topology is adopted near the blade wall and inside the tip gap. The mesh around the leading edge and trailing edge is locally refined. Mesh dependency test was done by computing the total to total isentropic efficiency and total to total pressure ratio changes for different grid sizes. It was found that grids with around 1.1 million nodes are dense enough to provide grid-independent results. Sufficiently fine cells near the walls were generated to ensure the values of y+ close to unity. Figure 3 demonstrates the studied centrifugal compressor and the structured grid.

Figure 3.

Sketch of computational mesh.

The CFX bulit-in ideal gas model for CO2 is selected for the ideal CO2 compressor simulation. As for the real gas CO2 compressor, an external real gas properties (RGP) look-up table is used to define the properties of CO2 such as enthalpy, entropy, pressure, temperature, density, sound speed, specific heat in constant pressure and viscosity. The abovementioned properties are taken from the widely used NIST REFPROP 9.0 database (Lemmon et al., 2002). RGP table resolution is gradually increased when getting closer to the critical point and the minimum interval of the pressure and the temperature are 10 kPa and 0.1 K respectively. The table covers sufficient wide ranges of temperature and pressure to prevent clipping or extrapolating of the properties. To validate the accuracy of the fluid thermodynamic property interpolation, the simulated properties at a point near the critical point and a point away from the critical point are compared with the REFPROP database. Results are showed in Table 2 and it is found that the error is less than 0.2%, indicating that the look-up table is suitable for CFD simulations.

Table 2.

Validation of the RGP look-up table.

T = 307.93 K, P = 6.48 MPaT = 368.85 K, P = 13.62 MPa

A real gas SCO2 compressor and an ideal gas CO2 compressor with identical (Πm˙,ΠU,Re) are simulated. The characteristic curves of the ideal gas CO2 compressor can be easily transformed to the similar real gas SCO2 compressor characteristic curves. Results are displayed in Figure 4. The simulated real gas SCO2 compressor characteristic curves are presented for comparison. The original ideal gas CO2 compressor characteristic curves are also showed. It can be seen that without transformation, performance maps for ideal gas and real gas are totally different. But if the similitude method is imposed, fair good agreement can be achieved for the pressure ratio. As for the efficiency curves, the proposed similitude method is able to predict the efficiency with errors less than 1% except for the choking region. The maximum error which occurs at the choke point is nearly 2%.

Figure 4.

Comparison of the characteristic curves.

The performance parameters at design point are listed in Table 3 for each compressor. It can be observed that the relative deviation of performance prediction via the similitude method is no more than 1%, which demonstrates the high accuracy of the similitude method at the design point.

Table 3.

Comparison of the performance at design point.

Real gas CO214,000131.512.2370.9472
Ideal gas CO217,52299.952.2130.9463

Velocity triangles at 10%, 50%, 90% span are given in Figure 5, in which the red arrows refer to real gas and the blue arrows refer to ideal gas. It can be observed that the velocity triangles of the ideal gas compressor are highly similar to those of the real gas compressor at identical locations except for the shroud region. Due to the existence of the low speed region near the impeller shroud, the mass flow through this region accounts for only a small proportion, so the deviation of the velocity triangle in this region has little effect on the prediction of the pressure ratio. Generally speaking, the assumption that the velocity triangle is similar in deriving the performance similitude parameters is reasonable.

Figure 5.

Comparison of the velocity triangles at design point.

Comparison of relative Mach number contours at inlet and outlet is presented in Figure 6. It is showed that the distribution of relative Mach number at these planes in the ideal gas compressor is in good accordance with that in real gas compressor. There are only some differences in numerical values, which further proves that the proposed similitude method can guarantee the dynamic approximate similarity.

Figure 6.

Comparison of the relative Mach number at design point.

Real-gas effect on the SCO2 compressor performance

In this section, the ideal gas CO2 compressor above is transformed to real gas SCO2 compressor with different inlet conditions, and the real-gas effect of CO2 on compressor performance is studied on that basis.

Figure 7 shows the variation of compressor performance with the inlet total temperature along the isobar line P01=7.6MPa. The curve represents the transformed data from the ideal gas CO2 compressor, and the scatter represents the CFD results under the similar conditions. It can be seen that the similitude method can accurately predict the change of the compressor pressure ratio and temperature ratio. The efficiency may be a little different but the change trend is correctly predicted, which is enough for the discussion. Along the isobar line, with the increase of inlet temperature, the compressibility factor Z0 gradually increases, while the isentropic component ns gradually decreases, which means that the real-gas effect is weakened gradually. Correspondingly, compressor pressure ratio gradually decreases, the efficiency gradually decreases, while the temperature ratio gradually increase. That is to say, along an isobar line, if the inlet condition approaches the critical point from the gas side, the real-gas effect becomes stronger, and higher the pressure ratio of the compressor can be obtained through lower temperature rise, with a little efficiency increment.

Figure 7.

Performance variation with the inlet total temperature at the isobar line P01 = 7.6 MPa.

Figure 8 shows the variation of compressor performance with the inlet total pressure along the isotherm line T01=320K. The curve represents the transformed data from the ideal gas CO2 compressor, and the scatter represents the CFD results under the similar conditions. It can be seen that along the isotherm line, with the increase of inlet pressure, the compressibility factor Z0 gradually decreases, while the isentropic component ns gradually increases, which means that the real-gas effect is enhanced gradually. Correspondingly, compressor pressure ratio and efficiency gradually increases, while the temperature ratio gradually decrease. This lead to the same conclusion about the advantage of real-gas effect as the previous paragraph.

Figure 8.

Performance variation with the inlet total pressure at the isotherm line T01 = 320 K.

In the same way, the contour of compressor performance parameters in a larger temperature and pressure range can be drawn, as shown in Figure 9. The pseudo-critical line is also drawn as a reference. The pseudo-critical line in this paper is defined as the path where the gradient of Cp is minimum in the TP map. It can be described by the following equation:

Figure 9.

Performance variation with the couple effect of inlet total temperature and pressure.

where a=4.29,b=4.27,c=1.48.

It can be seen that for similar SCO2 compressors, with the increase of inlet pressure or decrease of inlet temperature, the compressor’s pressure ratio gradually increases. When the inlet condition moves across the pseudo-critical line, the change rate of pressure ratio increases sharply. As for the temperature ratio and the efficiency, there is an optimal T-P relationship which makes the temperature ratio lowest and the efficiency highest. The optimal pressure is about 150 kPa higher than the pseudo-critical pressure near the critical point. In sum, the real-gas effect enables the compressor to obtain higher pressure ratio with a lower temperature rise and a bit higher efficiency, which is the advantage of the SCO2 as the working fluid. This advantage will be enhanced when the inlet temperature gets close to the critical temperature and when the inlet pressure gets close to the optimal pressure. Therefore, it is recommended that the designed inlet condition of SCO2 compressors should be taken near the critical temperature and about 150 kPa higher than the pseudo-critical line.


Several conclusions are drawn as follow:

  • Based on the dimensional analysis method and the real gas thermodynamics, a set of approximate similarity criteria for real gases is established, which consists of similitude parameters for mass flow ((m˙RgT01/P01D2)/(ns/Z0)((2/(ns+1)))((ns+1)/(ns1))), for compressibility nD/nsZ0RgT0, for viscosity nD2/υ, for enthalpy rise ΔH/(nsZ0RgT0) and for efficiency ηp.

  • When the similitude parameters remain unchanged, the pressure ratio of SCO2 compressor decreases with the increase of inlet temperature and increases with the increase of inlet pressure. As for the temperature ratio and the efficiency, at a specific inlet temperature, there is always an optimal inlet pressure which can minimize the compressor temperature ratio and maximize the efficiency. The optimal pressure is about 150 kPa higher than the pseudo-critical pressure.

  • The real-gas effect enables the SCO2 compressor to obtain higher pressure ratio with lower temperature rise, which is the advantage of SCO2 as a working fluid. It is recommended that the inlet temperature of SCO2 compressor should be as close to the critical temperature as possible, and inlet pressure should be about 150 kPa higher than the pseudo-critical pressure.


Latin letters


Mass flow (kg/s)


Absolute velocity (m/s)


Specific heat at constant pressure [J/(kg·K)]


Impeller diameter (m)


Enthalpy (J/kg)


Mach number (−)


Impeller tip-speed Mach number (−)


Rotational speed (rpm)


Isentropic pressure exponent (−)


Pressure (kPa)


Radius (mm)


Gas constant (J/(kg·K))


Temperature (K)


Impeller blade tip speed (m/s)


Compressibility factor (−)

Greek letters


Isothermal compressibility (−)


Polytropic efficiency (−)


Isentropic efficiency (−)


Specific heat ratio (−)


Similitude parameter (−)


Pressure ratio (−)


Density (kg/m3)


Temperature ratio (−)


Kinematic viscosity (m2/s)



Total thermodynamic condition


Impeller inlet




Total to total