Solutions of supercritical CO2 flow through a convergent-divergent nozzle with real gas effects

International Journal of Heat and Mass Transfer 116 (2018) 127–135

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Solutions of supercritical CO2 flow through a convergent-divergent nozzle with real gas effects S.K. Raman, H.D. Kim ⇑ Andong National University, Andong, South Korea

a r t i c l e

i n f o

Article history: Received 14 November 2016 Received in revised form 4 September 2017 Accepted 7 September 2017

Keywords: Supercritical carbon dioxide Shock wave Compressible flow Real gas effect Equation of state

a b s t r a c t Supercritical CO2 (S-CO2) has a density as high as that of its liquid phase while the viscosity remains closer to its gaseous phase. S-CO2 has the potential to be used as a working fluid in compressor as it requires much less work due to its low compressibility as well as relatively small flow resistance. Besides the material properties and design calculations, the thermodynamic properties of working gases play a vital role in the design and efficiency of various machinery such as compressors, turbines, etc. Equation of state (EOS) which accounts the real gas effects is used in computational analysis to estimates the thermodynamic properties of the working fluid. Each EOS gives priority to real gas effects phenomena differently and for the analysis of the fluid dynamic machinery, as of today, there is no standard procedure for selecting a suitable equation of state (EOS). To understand the influence of real gas effects on the formation of a shock wave, S-CO2 flow through a convergent-divergent nozzle, is theoretically analyzed. The thermophysical and transport properties calculated with the six different equation of state (EOS) are used to estimate the flow characteristics. An in-house code based on the real gas approach which solves gas dynamics equations with variable gas properties is used to analytically analyze the compressible flow of S-CO2 through nozzle. The solution for the shock strength, total pressure loss, and location of the normal shock wave at different back pressure conditions is obtained. Using Becker’s solutions with varying viscosity and thermal conductivity estimated from each EOS, the properties inside the normal shock wave are calculated. Numerical results of supersonic S-CO2 flow varies with different EOS candidates and the formation of shock wave is found to be significantly influenced by the real gas effects. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Carbon dioxide above its critical point attains a supercritical state and exhibits certain peculiar properties such as higher density as a liquid but with a lower viscosity comparable to a gas. This supercritical carbon dioxide (S-CO2) has a range of engineering applications such as enhanced oil recovery, caffeine extraction, drying aerogels, textile dying, dry cleaning, etc. S-CO2 has become a subject of vital research interest for investigators all over the world mainly due to its interesting characteristics as well as its potential applications [1]. In thermodynamic power cycles, the high-density property of S-CO2 is utilized to design compact machines with less required compressor work, thereby augmenting the effective power output [2]. Application of S-CO2 as a working fluid increases the efficiency of the refrigeration cycles [3]. Naturally, CO2 has two main advantages. Firstly, easy availability ⇑ Corresponding author. E-mail address: [email protected] (H.D. Kim). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.09.019 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

of CO2 makes it cheaper to procure and secondly, it can attain supercritical state relatively close to the standard ambient temperature, without employing any powerful heating systems. Hence, the research on the power cycles based on S-CO2 as a working fluid has substantially increased in last few years [4]. The thermodynamic properties of S-CO2 such as density, local speed of sound and specific heat capacity vary significantly for slight change in temperature and pressure due to real gas effects. Equation of state (EOS) which account these real gas effects is used to calculate the thermodynamic properties of S-CO2. Till date, a number of EOSs has been developed but still researchers are interested to develop a new method to calculate the thermodynamic properties [5]. The design and efficiency of machinery mainly depends on the physical properties like specific heat capacity and density [6,7]. But only a limited attention has been devoted for the assessment of EOS and its influence of simulations [8]. Zhoa et al. [9] analyzed the significance of thermodynamic model in the S-CO2 cycle design and stated that impact of EOS on turbomachinery is important. Baltadjiev et al. [10] investigated the real gas

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Nomenclature A

j M P T

q

v u x Cp _ m R h S Z n V

x

k

l

cross sectional area, m2 isentropic exponent Mach number pressure, N/m2 temperature, K density, kg/m3 specific volume, m3/kg velocity, m/s distance coordinate along the nozzle axis from throat, mm specific heat constant at constant pressure, J/kg K mass flow rate, kg/s specific gas constant, J/kg K enthalpy, J/kg entropy, J/kg compressibility factor dimensionless distance coordinate dimensionless velocity acentric factor thermal conductivity, W/(m K) dynamic viscosity, N m/s

effects on S-CO2 centrifugal compressor and stated that the local acceleration due to negative incidence at leading edge of impeller results in supersonic flow region. In this region, the static pressure and temperature of the S-CO2 significantly decreases and influences the performance of the compressor. At these conditions, computational tools have uncertainty in accurately predicting the thermodynamic properties of S-CO2 due to lack of proven method for selecting a suitable EOS. In underground applications, S-CO2 improve the efficiency of rock breaking mechanisms as well as enhance the oil recovery. For understating these applications, researchers conducted experiments and numerical investigations on supersonic S-CO2 jet impingement, and found significant differences between both results [11,12]. Numerical results show that the structure and position of the shock wave which forms during jet impingement varies on implementing different EOSs. These differences in results directly highlights influence of various EOSs in the supersonic SCO2 flow modeling. Much attention has been devoted to the investigation of different turbulent models in S-CO2 modeling in earlier studies [13,14]. Even though variety of EOS are available, much of them were neither tested nor explored for their applicability to SCO2 flows. Hence, the assessment of the different EOSs in numerical modeling of S-CO2 is necessary. The turbine inlet temperature approximately varies from the range of 700 K to 1100 K for the S-CO2 based power cycles [1]. Practical difficulties arise while demonstrating experiments at such very high pressure and temperature with high speed. Hence, for S-CO2 is concern, numerical analysis is not only cost-effective but also safer research tool. This further demands for the assessment of different thermodynamic models on the supercritical CO2 compressible flow modeling. The present work investigates the influence of the different thermodynamic models on S-CO2 flow through a critical nozzle. The EOS developed by Aungier (ARK) [15], Peng and Robinson (PR) [16], Boston and Mathias (PRBM) [17], Plöcker (LKP) [18], Benedict-Webb-Rubin (BWR) [19] and Span and Wagner (SW) [20] are considered. The National Institute of Standards and Technology (NIST) developed a program REFPROP (REFerence fluid PROPerties) to calculate the thermodynamic and transport properties of the fluids accurately. NIST utilizes SW EOS for the calculation of thermodynamic properties of

Subscript t total state s static state 1; 2 upstream and downstream state of the shock wave respectively in nozzle inlet th throat section of nozzle e exit section of nozzle r reduced property i.e., ratio of the property value and its value at the critical point 0 stagnation condition at inlet cr ratio for the choked nozzle super entire flow in the divergent section is supersonic sub entire flow in the divergent section is subsonic Coefficients used in EOS a,b,c,d, A0, B0, C0, D0, B, C, D, a; b; c, c4

CO2, and hence throughout this work, SW EOS is represented as NIST. Detailed study on the flow properties inside shock wave using Becker’s solution [21,22] with variable properties has also been presented in this context. The real gas effects in the shock front of S-CO2 at different upstream Mach numbers are studied with varying pressure ratio. The current analytical study has been conducted using an inhouse code based on real gas approach [23,24] written in Scilab V5.5.2 programming language. Thermodynamic properties are calculated from each EOS using separate functions [25]. The solutions of S-CO2 is compared with gaseous CO2 (G-CO2) for the same pressure ratio. Since different thermodynamic models predicts real gas properties differently, the flow properties calculated with each EOS is also different. 2. Theoretical analysis 2.1. Real gas equation of state At standard atmospheric pressure and temperature, the CO2 behaves as an ideal gas, and its compressibility factor equals to unity. However, at high temperature and pressure, the compressibility factor varies from unity, due to real gas effects. In order to predict this real gas effect, six different EOSs are considered for analysis of the real gas effect in the compressible S-CO2 flow. The Table 1 shows the summary of the various EOSs that are adopted for this study, along with their critical fluid properties. 2.2. Governing equations The supersonic flow of supercritical CO2 at high pressure and temperature is analyzed theoretically based on the equation derived with the assumption of quasi-one-dimensional, inviscid, isentropic flow and ideal gas equation of state. The equations are summarized in expressions (1)–(3). S-CO2 is at high temperature and pressure and hence it is highly compressed and its compressibility factor deviates much from unity. At this point, the assumption of perfect gas behavior losses its basis and instead the qualities of real gas must be assumed. Therefore, in order to approximate the behavior of real gas, all the properties (isentropic exponent,

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S.K. Raman, H.D. Kim / International Journal of Heat and Mass Transfer 116 (2018) 127–135 Table 1 Summary of EOSs. EOS

Functional forma

ARK

P¼

PR

P¼

PRBM

P¼

LKP

Z ¼ Z þ xxR ðZ R  Z 0 Þ  Z 0 or Z R ¼ 1 þ vBr þ vC2 þ vD5 r r   c þ T 3c4v 2 þ b þ vc2 exp  v 2r r r  r   3 P ¼ qRT þ B0 RT  A0  CT 20 þ DT 20 q2 þ bRT  a  Td q3 þ a a þ Td q6 þ cTq2 ð1 þ cq2 Þ expðcq2 Þ

BWR a

RT v b RT v b RT v b 0



Critical pressure (bar)

a v ðv þbÞ

a  v ðv bÞþbð v bÞ a  v ðv bÞþbð v bÞ

Critical temperature (K)

73.773

304.13

73.830

304.21

73.830

304.21

73.830

304.21

73.830

304.21

Refer respective reference for parameters values.

viscosity, density and local speed of sound) required in the above equations are substituted after calculating with each EOS. Solutions are obtained by coupling the thermodynamic properties predicted from each EOS and the governing equations.

A 1 ¼ Ath M Pt ¼ Ps




2 jþ1




1þ

j1 2

 2ðjjþ1 1Þ M2


 j j  1 2 j1 M 1þ 2

Tt j1 2 M ¼1þ 2 Ts

ð1Þ

ð2Þ ð3Þ

The jump in values of thermodynamic properties across the normal shock can be expressed in Eqs. (5)–(7).

ht ¼ C p T t ¼ constant

ð4Þ

h j 1 2 ijj1 11 1 Ps;2 P t;2 1 þ 2 M 1 ¼ h  ijj21 Ps;1 P t;1 1 þ j221 M 22 2

ð5Þ

h j 1 2 i 1 T s;2 T t;2 1 þ 2 M 1 h i ¼  T s;1 T t;1 1 þ j2 1 M 2 2 2

ð6Þ

8 9
j2 1 = 2 < Pt;2 j2 1 M2 ¼ ; j2  1 : Ps;2

ð7Þ

Flow properties inside the normal shock wave has been analyzed by considering a region within the shock wave with a continuous change in flow properties called shock front. Profile of shockfront has been analytically captured by solving exact onedimensional gas dynamics equations [26,27]. For analysis of velocity distribution, a non-dimensional shock front, the final derivation in Ref. [26] directly written as Eq. (8). For calculations of the other properties such as pressure and entropy distribution in the shock wave the final expression of derivation from [28] are as shown in the Eqs. (9)–(11).

n¼

h u u2 i 2 ð 1 Þ ð Þ ln ðu1  uÞ u1 u2 ðu  u2 Þ u1 u2 jþ1

q u1 1 ¼ ¼ q1 u V   P 1 j1 2 1þ M1 ð1  V 2 Þ ¼ P1 V 2

ð8Þ ð9Þ

ð10Þ

S  S1 ¼ ln Cv

 1þ

j1 2

  M 21 ð1  V 2 Þ V j1

ð11Þ

The mass flow rate through the nozzle is calculated using the following equation,

rffiffiffiffiffiffiffiffi _ ¼ Ath P0 m


 jt þ1 jt  1 2 2ðj1Þ Mt Mt 1 þ 2 RT 0

j

ð12Þ

2.3. Real gas approach Real gas approach [23,24] is adopted in this present work to calculate the flow properties with real gas effects. This method assumes a semi-perfect gas in that the perfect gas governing equations are used but property values are taken from the EOS [23]. Since the method is iterative in nature, these governing equations cannot be directly used. However for clarity, Eqs. (5), (6), (7) and (12) are written with subscripts. For validation of this methodology, the supersonic flow of S-CO2 through nozzle geometry considered for numerical modeling by Kim et al. [29] is analyzed with our in-house code and compared with the two dimensional FLUENT v18 results. The SRK EOS is implemented in both the computational and analytical modeling and the pressure and Mach distributions are represented in Fig. 1. A good agreement between analytical and computational results validates the analytical modeling. From the inlet pressure and inlet temperature, the isentropic exponent, j is calculated for a particular EOS. Using the areaMach relation in Eq. (1) and inlet j value, the Mach number at the inlet is calculated. For calculation of the properties at next node, frozen flow limit concept is used, i.e., same j is initially considered between two adjacent locations. Since the nozzle is divided into large number of divisions and hence the frozen state assumption between the two nodes gives accurate results. From the area ratio at the next node and j value, the Mach number at the next node is calculated. Then pressure and temperature are calculated using the Eqs. (2) and (3). With this pressure and temperature, new j value at current node is estimated from EOS. Again same calculations are repeated at current node with this new j value until convergence is attained. Once the accurate j value at the current node is obtained, the exact Ps and Ts are calculated. Similarly, the static pressure at the nozzle exit is calculated for the case of completely supersonic flow in divergent section. If the calculated exit pressure is lower than the actual back pressure, in order to compensates this pressure difference, a normal shock is introduced at an appropriate position in the divergent section. The jump in flow properties across a normal shock is determined from Eqs. (4)–(7). Total pressure and total temperature of 500 bar and 1100 K is respectively maintained at inlet for S-CO2 modeling. While for gaseous CO2, stagnation conditions of 0.75 bar and 277.78 K were

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Fig. 1. Comparison of analytical and computational results.

maintained. The inlet conditions are carefully taken into consideration such that no phase change occurs anywhere inside the nozzle. Nozzle geometry investigated in the experiments of Berana et al. [30] is extensively studied for the CO2 compressible flow. Similar nozzle geometry as shown in Fig. 2 with shortened divergent sectional length of 8.38 mm is considered for the present work. The divergent section is reduced to avoid pressure or temperature dropping below the critical point during expansion and hence the phase change is prohibited. The convergent section is divided into 1500 nodes, and the divergent section is divided into 3500 locations to reduce the uncertainty due to the frozen limit assumption and to predict the shock wave location accurately. The nozzle is divided into large number of divisions and hence the frozen state between the two nodes gives reasonable results.

3. Results and discussion The back pressure ratio (BPR) is defined as the ratio between the back pressure (Pb) and inlet total pressure (P0) and it is a primary

Fig. 2. Dimensions considered.

of

the

two-dimensional

converging-diverging

nozzle

governing factor for an exit Mach number. The analytical results for the S-CO2 flow through a critical nozzle with various EOSs at different BPR conditions are summarized in this section. 3.1. Real gas effects in nozzle flow without shock wave All EOS candidates predicts equal mass flow rate values until near the choking condition (up to BPR = 0.87) as shown in Fig. 3. Thus, impact of variation in real gas properties different EOS models on mass flow rate result is negligible until approaching the critical pressure ratio. After a BPR of 0.87, the results start to deviates immediately indicates the sudden rise in influence of different real gas model. For the choked nozzle, maximum deviation up to 2.5% is seen between BWR EOS and ARK EOS. From Eq. (12), it can be understand that the mass flow rate is mainly depends on jt , Pt, Tth, M and Ath. For choked nozzle, except jt all other parameters are same and hence these results deviations are solely rely on j. Near the throat, the temperature and pressure significantly decreases due to local acceleration as seen in Figs. 4 and 5 respectively. This sudden decrements in P and T values, abruptly changes the j value at different rate in different EOS candidates and thus mass flow rate results deviates. For a choked nozzle, the ratio between the throat pressure (Pth) and the inlet stagnation pressure (P0) is called throat critical pressure ratio, Pth,cr. The ratio between the exit pressure (Pe) to the inlet total pressure (P0) when the flow in the divergent section is completely supersonic and subsonic flow is represented as Psuper;cr and Psub;cr respectively. Table 2, shows these pressure ratio values calculated with different thermodynamic models. In S-CO2 flow, pressure ratio values calculated with one EOS is different from another EOS mainly due to the variation in the j value. For all pressure ratio, the highest deviation is found between results of NIST and ARK EOS and these deviation gradually increases in the order of Psub;cr , Pth;cr and Psuper;cr as 0.8%, 2% and 4% respectively. It can be inferred that the impact of the EOS selection on S-CO2 modeling gradually increases as more expansion occurs. For completely supersonic flow in divergent section, the G-CO2 flow expands nearly 16% more than S-CO2 which results in higher Msuper;e for G-CO2. The expansion of GCO2 and S-CO2 along the nozzle axis is compared in Fig. 4. As the supersonic flow accelerates

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131

_ values up to BPR of 0.87 indicates that until just before the choking Fig. 3. Mass flow rate results for nozzle flow at different back pressure ratio. All EOSs estimates same m _ results. condition, the EOS choice does not alter the m

Fig. 4. Pressure distribution along nozzle axial distance. At BPR of 0.8, the distinction between the PRBM and BWR is hardly visible. From the in-set figure, the increasing gap between blue line and black line along the axial length in divergent section clearly shows the steady rise of real gas effects with acceleration of supersonic flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

along the divergent section, on set of real gas effects happens which eventually reduces the expansion of S-CO2.

3.2. Real gas effects in the nozzle flow with shock wave At constant inlet total pressure (P0), on increasing the back pressure (Pb) from Psuper;cr value, a normal shock wave forms in divergent section to compensate the rise in the back pressure. For same BPR, normal shock location in G-CO2 flow is well ahead of the S-CO2 and also, this gap is higher for BPR of 0.7 than 0.8 as

observed in Fig. 4. This increasing difference in shock location between G-CO2 and S-CO2 demonstrates the increase in real gas effect with supersonic flow acceleration in S-CO2. At both the BPR of 0.7 and 0.8, the normal shock wave location of S-CO2 flow significantly varies with different EOS. In terms of shock location, at the both BPR 0.7 and 0.8, the NIST and ARK results are two extremes and the differences between two EOS are nearly same. Thus, the influence of thermodynamic model selection on shock wave location results highly significant at all BPR. The different shock location implies distinct pre-shock Mach number as seen in Fig. 6. Subsequently, the pressure and tempera-

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Fig. 5. Temperature distribution for the back pressure ratio (BPR) of 0.7 and 0.8. Temperature rise across the normal shock significantly varies with different EOS.

Table 2 Flow conditions for the choked nozzle flow. State

EOS

Pth;cr

Psub;cr

Psuper;cr

Msuper;e

S-CO2

PR EOS NIST PRBM BWR LKP ARK

0.563 0.560 0.566 0.567 0.569 0.572

0.860 0.859 0.862 0.863 0.863 0.866

0.232 0.231 0.235 0.235 0.238 0.241

1.66 1.66 1.65 1.65 1.65 1.64

G-CO2

Ideal

0.553

0.859

0.205

1.70

Fig. 6. Mach distribution along nozzle axis for the back pressure ratio (BPR) of 0.7 and 0.8. The difference between G-CO2 and S-CO2 results gradually increases as the BPR decreases indicates that the real gas effects is prominent as more expansion occurs.

S.K. Raman, H.D. Kim / International Journal of Heat and Mass Transfer 116 (2018) 127–135

ture rise across the normal shock wave also varies with different EOS as presented in Figs. 4 and 5 respectively. Despite different EOS predicts similar distribution of pressure and Mach results after the shock, the divergence in temperature distribution is considerable. The difference in flow conditions after a normal shock directly implicates the thermodynamic properties prediction and the choice of EOS substantially impact the modeling of S-CO2 flow with shock wave. 3.3. Effect of real gas in the shock front The equations based on Becker’s solution are used for analysis of the properties inside the shock front. Variable values for physical properties such as specific heat capacity, viscosity and thermal conductivity predicted from different EOS models are used in shock front analysis. As the BPR increases, shock wave forms and moves towards the throat and the pre-shock Mach number decreases. This reduction in pre-shock Mach number is associated

133

with increase in shock thickness and decrease in shock strength as seen in Fig. 7(a) and (b) respectively. Among all the EOSs, NIST predicts highest n value for all BPR. The n value increases non-linearly with BPR and this rate of increment differs from one EOS to another. Hence, excluding NIST, the n value results from all other EOSs changes their order from BPR of 0.68 to 0.84. For instance, at BPR of 0.84, ARK and G-CO2 has respectively the minimum and maximum n value but at BPR of 0.68, the order reverses. The n and P2/P1 results are inversely proportional to each other and the proportionality constant varies form one EOS to other. So, the difference between the n results of any two EOS is not conserved in P2/P1 results. The total pressure loss and entropy increment across the normal shock linearly decreases with BPR as shown in Fig. 8(a) and (b). Except NIST EOS all other EOS have higher pressure loss compared to S-CO2 for back pressure ratio between 0.74 and 0.82 as shown in Fig. 8a. The constant difference in total pressure loss results between different EOSs at all BPR establishes the influences

Fig. 7. Variation of the shock thickness and pressure rise across shock wave for different back pressure ratio. For individual EOS, n and Ps;2 /Ps;1 are inversely related. But the differences between any two EOS are not conserved.

Fig. 8. Variation of the total pressure loss and entropy across shock wave for different back pressure ratio.

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Fig. 9. Variation of entropy inside shock front for back pressure ratio (BPR) of 0.68 and 0.82. Except for NIST, the entropy attains it peak value at the dashed vertical line.

of different EOS selection. The net entropy increase across shock calculated from various EOSs converges at higher BPR which indicates that the impact of thermodynamic models vanishes for DS calculations at higher BPR. All other properties like velocity, pressure and temperature distribution changes monotonically in the shock front, but the entropy distribution changes in a unsymmetrical bell shaped manner. The effect of EOS on shock front analysis is shown in Fig. 9. Before reaching a particular net entropy increment after the shock, the entropy reaches a peak value in the shock front where exactly the transition from supersonic to subsonic takes place. The peak entropy value calculated with NIST is higher than all other EOSs at both the BPR of 0.68 and 0.82. Each EOS predicts the net entropy and the peak entropy generation inside the shock differently at different BPR. As seen earlier, even though the net entropy increase value predicted by different EOS converges as the higher BPR, the difference in peak entropy from different EOS considerably varies. This clearly demonstrates the importance of variation real gas effects in the shock front analysis.

4. Conclusions The influence of EOS selection on S-CO2 modeling is analyzed with six different EOS candidates. Based on the real gas approach and one dimensional equations, an in-house code is developed for analytically assessing the real gas effects in S-CO2 flow through nozzle. The flow field is compared with the G-CO2 flow at same nozzle pressure ratio. By varying pressure ratio, a normal shock is introduced in divergent section and shock front analysis is also carried out. The variation of properties inside shock front of SCO2 has also been reported in terms of various EOS. The impact of EOS selection on subsonic flow solutions is negligible. The critical mass flow rate calculations considerably differs with different EOS due to the change in throat j. The real gas effect is more pronounced as more expansion happens. This increase in real gas effects with supersonic flow acceleration reduces the expansion in relative to G-CO2. All EOS predicts, the entropy distribution of S-CO2 in an unsymmetrical bell shape, but the peak and

final entropy value differs from one another. The influence of thermodynamic models selection in the calculation of net entropy change decreases with increasing back pressure ratio and eventually vanishes at higher BPR. Except net entropy change, all the results across normal shock get strongly as well as equally affected by variation in real gas effects. Irrespective of BPR, the solutions such as normal shock location, P2/P1 and T2/T1 significantly varies with choice of EOS. In numerical modeling of supersonic flow of S-CO2, solution largely varies with respect to EOS selection and hence, high attention should be devoted to EOS selection. The deviation in specific heat ratio prediction by different EOS is known to be the main reason for the change in the flow property in the supercritical CO2. Particularly at the regions were pressure and temperature change significantly due to flow acceleration or deceleration, solutions are very sensitive to the variation in real gas effects. Acknowledgments This work was supported by a grant from 2015 Research Funds of Andong National University. References [1] Y. Ahn, S.J. Bae, M. Kim, S.K. Cho, S. Baik, J.I. Lee, J.E. Cha, Review of supercritical CO2 power cycle technology and current status of research and development, Nucl. Eng. Technol. 47 (6) (2015) 647–661, http://dx.doi.org/10.1016/j. net.2015.06.009. [2] S. Jeon, Y.J. Baik, C. Byon, W. Kim, Thermal performance of heterogeneous PCHE for supercritical CO2 energy cycle, Int. J. Heat Mass Transf. 102 (2016) 867– 876, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.06.091. [3] M. Nakagawa, A.R. Marasigan, T. Matsukawa, A. Kurashina, Experimental investigation on the effect of mixing length on the performance of two phase ejector for CO2 refrigeration cycle with and without heat exchanger, Int. J. Refrig. 34 (7) (2011) 1604–1613, http://dx.doi.org/10.1016/j. ijrefrig.2010.07.021. [4] H. Li, Y. Zhang, L. Zhang, M. Yao, A. Kruizenga, M. Anderson, PDF-based modeling on the turbulent convection heat transfer of supercritical CO2 in the printed circuit heat exchangers for the supercritical CO2 Brayton cycle, Int. J. Heat Mass Transf. 98 (2016) 204–218, http://dx.doi.org/10.1016/j. ijheatmasstransfer.2016.03.001. [5] T.T. Ngo, J.H. Huang, C.C. Wang, Inverse simulation and experimental verification of temperature-dependent thermophysical properties, Int.

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