Performance prediction and loss evaluation of the carbon dioxide supersonic nozzle considering the non-equilibrium condensation

Carbon dioxide (CO2) is being considered as a promising working medium in energy conversion and refrigeration cycles due to its unique properties. When carbon dioxide flows with supersonic in turbo machinery (compressor), the non-equilibrium effect is enhanced due to the large change of fluid velocity, resulting in non-equilibrium condensation of the blade, which will seriously affect the performance of the compressor. Considering the similarities in flow characteristics between the nozzle and the compressor blade, the condensing flow of the blade can be predicted by simulating in a nozzle. The real gas model is used. The pressure and the nucleation rate are predicted based on the modified model, and the flow losses and thermal efficiency are analyzed in different states. The results show that the pressure variation in the nozzle aligns well with the experimental data. When the fluid transitions from subcritical to supercritical, the condensation interval decreases and the peak of the nucleation rate increases. The maximum supercooling decreases gradually. The flow losses are relatively large, and the thermal efficiency is low.


Introduction
As an environmentally friendly gas, carbon dioxide has been widely concerned.Carbon dioxide has a wide range of applications in air conditioning refrigeration, heat pumps and water heaters, waste heat recovery, food refrigeration and other fields due to the high density, low viscosity, non-toxicity and no damage to the ozone layer, etc.The trans-critical CO2 refrigeration cycle is widely used in automobile air conditioning with the advantages: high compressor operation efficiency, large volume cooling capacity and small system volume.In the supercritical refrigeration cycle, carbon dioxide gas undergoes the change from subcritical state to supercritical state.The properties of fluids also change over time.The velocity of the fluid suddenly increases and condensing occurs when the carbon dioxide vapor flows at high pressure in the compressor.In the non-equilibrium flow, the humidity of vapor gradually increases, the diameter of the droplet also gradually increases [1][2].The compressor blades are damaged by the condensed droplets which increase the compressor loss [3][4].Therefore, reducing or avoiding carbon dioxide vapor condensation has become the key to improving compressor performance.
Non-equilibrium condensation refers to the phenomenon that CO2 vapor expands and does not condense immediately after crossing the saturation line, but suddenly occurs when the thermodynamic state parameters reach a certain non-equilibrium degree.Wright et al. [5] described the condensation of supercritical carbon dioxide at the compressor inlet.However, the impact of fluid within the two-phase zone on compressor performance has not been verified.Gyarmathy [6] estimated that the loss due to the latent heat is released in condensing accounted for 45% of the total phase transition loss during turbine operation.Zhao et al [7] put forward a new kind of combinatorial optimization compressor structure to avoid condensation.The fluid characteristics in the nozzle and the compressor are similar.A de Laval nozzle have been extensively used to study the flow state of multiphase flows due to the complexity of directly measuring the characteristics of rotating mechanical fluids [8].Zhang [9] evaluated the condensing flow in Moses and Stein nozzles based on a modified model.The accuracy of the modified model is verified.Zhang et al [10] investigated the influence of steam inlet superheat on the nucleation process based on the modified model.The findings indicated the inlet superheat does not exert any influence on the minimum supercooling.But this research was limited to steam and lacked studies of other gases, such as CO2.The gas model has a significant impact on the accuracy of the model.By comparing the condensing flow of real gas and ideal gas under low pressure, Wen et al [11] found that ideal gas underestimated the mass flow.The calculation results by the ideal gas model are not accurate.In this study, the non-equilibrium condensation in the nozzle is studied based on the modified mathematical model in different states, and the flow losses and thermal efficiency in this process are evaluated.A reference for improving the working efficiency in the compressor is provided in this study.

Mathematical model
Condensation phenomenon exists in supersonic flow and has been widely studied.In this research, supersonic nozzle is simulated by the modified model.It is assumed that there is no relative slip between the two-phase flows.Mathematical models introduce fluid dynamics equations: continuity equation, momentum equation, and energy equation.Transport equations are used in the process of phase transition in condensation.The following are the mass equation, momentum equation and energy equation of the gas-liquid mixture.
Where , , , , , and are density, the velocity vector, temperature, time, pressure and total energy, respectively.
is the effective thermal conductivity.is the stress tensor.represents the source term in the condensing flow, defined as: Where is latent heat.The transport equations are compiled into UDFs (user defined function) [12] and the phase transition is calculated using the ANSYS Fluent platform in homogeneous condensation.In the partial differential equation, n denotes the quantity of droplets per kilogram, y is the liquid mass fraction in the homogeneous condensation.The transport equation is presented as follows: is the source term.Equation ( 5) and (6) describes the mass fraction of the droplets and the quantity of droplets respectively.
The expression of the source term is: is the nucleation rate. is the droplet radius.is the critical radius in the real gas state.The expression is as follows: is the gas constant, and is the CO2 vapour temperature; is CO2 pressure, is the supersaturation ratio, is the compressibility factor, is surface tension.Temperature is introduced in the REFPROP program to correct the surface tension model.The expression is as follows: Where , are fitting coefficients., , [13].As the droplet gradually increases, the influence of droplets on the surface tension gradually increases.Tolman [14] proposed the correction of droplet radius to surface tension.In the process of droplet growth, the influence of temperature should be considered.The expression is as follows: Where is the Tolman length, .The nucleation rate reflects the change of nucleation during condensation.Kantrowitz [15] modified the nucleation rate to more accurately calculate the nucleation process.
is the nucleation rate which is expressed as: is the specific heat ratio, is the quality of a single molecule, and are the enthalpy of CO2 vapour and liquid, respectively, is the CO2 liquid density, is the condensation coefficient, and in this work is equal to 1. is the Kantrowitz nonisothermal correction coefficient, is the Boltzmann constant, is the correction coefficient of the nucleation rate.
reflects the continuity and non-continuity of droplet growth process.The expression is as follows: is the mean free path, is the dynamic viscosity of vapour, is the gas constant.The continuous droplet growth model proposed by Gyarmathy (GY) [16] has been widely used, and its expression is as follows: is the vapour thermal conductivity.The Fuchs-Sutugin model (FS) [17] is a modification based on the Gyarmathy model, which is expressed as: By analyzing the flow in the nozzle, Young (YO) [18][19] proposed a correction based on the Gyarmathy model: Pr is the Prandtl number.
, .Gyarmathy proposed an expression to calculate the temperature of the droplet: is the vapour saturation temperature at corresponding pressure.When the droplet is small and the is larger.Molecular and macroscopic transport equations should not be ignored.The droplet growth equation in the free molecular state is given by the Hertz-Knudsen model [20], which is expressed as follows: is the condensation coefficient, which represents the number of vapour molecules on the droplet surface, indicates that vapour molecules are settling on the droplet surface, means that no vapour molecules are settling on the surface of the droplet.In the Hertz-Knudsen droplet growth model .Table 1 is the inlet boundary condition, and there is no boundary slip on the nozzle wall.Due to the velocity of the fluid is very high and there is no heat exchange with the outside, the nozzle wall can be considered adiabatic.At the same time, the turbulence model is and the double precision solver is used for calculation in ANSYS Fluent.

Mesh independence verification
The 2D axisymmetric model of the nozzle is meshed by the ICEM.The numerical calculation of the nozzle is more accurate by refining the throat mesh of the nozzle.The mesh of the nozzle is divided in Figure 2(a).(b) Pressure distribution in nozzles with different numbers of mesh cells Condensation occurs near the nozzle throat, and the vapour condenses into a large number of droplets.A significant quantity of latent heat is released during this phase transition, causing the pressure to jump.It can be found that the pressure of models with different mesh cells presents the same change trend.From the local magnification map, as the number of grids increase, the amplitude of pressure jump gradually increases, and gradually tends to be stable.Models with 9875 cells and 15314 cells showed lower pressure changes.The pressure distribution of 52258 cells and 62322 cells is basically the same.In this paper, the model with 52258 grids is used for numerical calculation for more accuracy.

Verification of gas model
The ideal gas ignores the forces between the molecules and the collision between the gas molecules and the nozzle wall.The ideal gas model simplifies the flow characteristics of the fluid and simplifies the process of calculating the condensing flow characteristics of the fluid.However, it is not conducive to evaluating the flow characteristics in the real state.It is necessary to simulate the condensation in the real gas state for evaluating the non-equilibrium condensation of vapour more accurately.
Figure 3 shows the comparison of pressure calculated using the real and the ideal gas model.Condensation occurs near the nozzle throat and a large amount of latent heat is released.It can be intuitively seen from the figure that the pressure distribution calculated based on the real gas state shows a pressure jump clearly, and it is in good agreement with the variation trend of the experimental data.The calculated pressure distribution based on the ideal gas model shows no significant pressure jump and gradually deviates from the experimental data.This further illustrates the accuracy of the calculated results based on the real gas model.

Verification of the droplet growth model
Gyarmathy proposed a droplet growth model which is adapted to fluids with different properties and droplets with larger sizes.Fuchs and Sutugin proposed interpolation formulas for studying condensation in the transition region and extended the continuity model to free molecules and supersonic flows.Young introduced a correction factor to represent the temperature difference between droplets and vapours.The Hertz-Knudsen droplet growth theory is based on the dynamic principle, considering the collision between molecules and nuclei.The Hertz-Knudsen (HK) model is suitable for condensing flows of multi-component gases.Figure 4. Pressure distributions in different droplet growth models.The pressure is calculated in Figure 4 with the four droplet growth models.Due to the phase transition in the condensation, a quantity of latent heat is released, leading to a sudden change in pressure.The figure shows that the pressure jump predicted by the four droplet growth models.The predicted results based on FS, GY and YO models are basically accord with the experimental data.The pressure calculated based on HK model in the throat of the nozzle is greater than the other three models.However, after the pressure jump reaches the peak, the pressure shows a linear decreasing trend, which is inconsistent with the change trend of the experimental data.This difference is related to the HK model overestimating the condensation rate of a single component fluid.GY model is more suitable for droplets with large diameter, such as CO2.The non-equilibrium condensation is further studied by GY model in this work.

Prediction of non-equilibrium condensation
Figure 5. Prediction of the droplet mass fraction in non-equilibrium condensation.Figure 5 shows the prediction of the droplet mass fraction in non-equilibrium condensation with different states.The vapour begins to condense into small droplets near the throat of the nozzle.As the speed of vapour flow increases, a large number of droplets appear, resulting in a gradual rise in the mass fraction of the droplets.When the fluid transitions from subcritical and nearcritical state to supercritical state, the mass fraction of the droplet increases gradually.The condensing area gradually moves towards the nozzle inlet.It indicates that vapour condensed into droplets in a supercritical state is facilitated.

Distribution of pressure
Figure 6 shows the distribution of the pressure predicted along the nozzle centre line and compared with the experimental data.As the fluid velocity increases, the vapour condenses at the nozzle throat, and a substantial quantity of droplets are produced.A quantity of latent heat is released in this process, resulting in a jump in pressure.As shown in Figure 6, the pressure distribution predicted in Case 1 and the pressure predicted at the nozzle throat in Case 2 are coincide with the experimental data.Although the pressure predicted in Case 3 is slightly higher, the overall pressure change trend is basically consistent with the experimental data.This shows the prediction results based on the modified mathematical model are accurate.

Distribution of nucleation rate and supercooling
The distribution of nucleation rate and supercooling along the nozzle centre line in subcritical, near-critical and supercritical state is presented in Figure 7.The condensation interval gradually shrinks, when the fluid gradually transitions from subcritical to supercritical state.

Prediction of the flow losses
In this paper, three loss coefficients are introduced to evaluate flow losses in supersonic nozzles.The definitions of three loss coefficients and efficiency are shown in Figure 8. Figure 9 shows the prediction of the flow losses and the thermal efficiency in the three states.The mass fraction of the droplet gradually increases, and the humidity gradually increases.Due to the condensation of the droplets, the latent heat released gradually increases, resulting in a decrease in the total pressure change at the inlet and outlet, and resulting in a gradual decrease in the potential energy loss.In the isentropic flow, the change trend of enthalpy at the inlet and outlet in the near-critical and supercritical states are smaller than that in the subcritical states.Due to the influence of the change of enthalpy at the inlet and outlet is greater than that of the velocity, the kinetic energy loss shows a gradual increase trend.In the isentropic and nonisentropic flow, the change of entropy and velocity at the outlet gradually decreases in the three states.The influence of the change of entropy is less than that of the velocity, leading to a gradual increase of the entropy loss.In the non-isentropic flow, the change of enthalpy at the inlet and outlet shows a gradual increase in these three states.In the isentropic flow, it shows a gradually decreasing trend, which makes the thermal efficiency gradually increase.

Conclusions
In this work, the nonequilibrium condensation of carbon dioxide in subcritical, near critical and supercritical states is calculated by the modified model.The conclusion is as follows: 1.The results by the real gas model align well with the experimental data.It verifies that the non-equilibrium condensation flow characteristics can be simulated accurately by the real gas model.
2. The numerical results of different droplet growth models show that the pressure calculated based on GY, FS and YO model aligns well with the trend of experimental data.In this paper, the flow characteristics of carbon dioxide in the nozzle are calculated based on GY model.
3. The pressure predicted by the modified model in different states aligns with the experimental data.It shows that the non-equilibrium condensation can be accurately simulated by the modified model.
4. When the fluid is transitioning from subcritical to supercritical state, the peak nucleation rate increases gradually, and the maximum degree of supercooling decreases gradually.In the supercritical state, the maximum supercooling is the smallest, and the vapor begins to enter the supercooling state first.
5. The potential energy loss is greatest in the subcritical state.The kinetic energy loss and entropy loss show a gradually increasing trend in subcritical, near-critical and supercritical states.And the thermal efficiency is highest in the supercritical state.

Figure 1 .
Figure 1.3D modelling of the nozzle.The 3D geometric model of the nozzle is shown in Figure 1.The nozzle geometry data obtained from Lettieri [8].The geometric length of the nozzle is 98.37 [mm], the throat area of the nozzle is 19.95 [mm 2 ].Table1.Boundary conditions.

Figure 2 .
Figure 2. Refined mesh and mesh independence verification.Figure 2(b) shows the variation trend of pressure in the model with different mesh cells.Condensation occurs near the nozzle throat, and the vapour condenses into a large number of droplets.A significant quantity of latent heat is released during this phase transition, causing the pressure to jump.It can be found that the pressure of models with different mesh cells presents the same change trend.From the local magnification map, as the number of grids increase, the amplitude of pressure jump gradually increases, and gradually tends to be stable.Models with 9875 cells and 15314 cells showed lower pressure changes.The pressure distribution of 52258 cells and 62322 cells is basically the same.In this paper, the model with 52258 grids is used for numerical calculation for more accuracy.

Figure 3 .
Figure 3.Comparison of pressure calculated using the real and the ideal gas model.

5 .
Numerical calculation Gyarmathy continuous droplet growth model is used in this study.It is assumed that no boundary slip in the fluid flows in the nozzle.The nozzle wall is considered as adiabatic.A double-precision solver and turbulent flow model are used in ANSYS Fluent.The inlet boundary conditions are presented in Table 2. Case 1, Case 2 and Case 3 are subcritical, near-critical and supercritical states, respectively.

3 Figure 6 .
Figure 6.Pressure distribution predicted along the nozzle centre line.

Figure 7 .
Figure 7. Prediction of the nucleation rate and supercooling along the nozzle centerline.

Figure 8 .Figure 9 .
Figure 8. Definitions of the loss factor and efficiency in supersonic flows.As depicted in the figure, the potential energy loss coefficient of the fluid in the adiabatic expansion flow is determined as 0, while the kinetic energy loss coefficient must be equal to 1.The entropy loss coefficient represents the increase in the specific entropy from the import to the export.Thermal efficiency is introduced to evaluate the flow efficiency in the nozzle.There are kinetic and thermodynamic losses in the non-equilibrium condensing flow of vapor in the supersonic nozzle.These two kinds of losses affect each other and exist simultaneously.
14K respectively, showing a gradually decreasing trend.It shows that when the fluid gradually reaches the supercritical state, the vapor can condense quickly without a large supercooling capacity.At the same time, in the supercritical state, the vapor first appears in the supercooled state.