Nusselt number correlation for turbulent heat transfer of helium–xenon gas mixtures

Biao Zhou ?Yu Ji?Jun Sun ?Yu-Liang Sun

Abstract A gas-cooled nuclear reactor combined with a Brayton cycle shows promise as a technology for highpower space nuclear power systems.Generally,a helium–xenon gas mixture with a molecular weight of 14.5–40.0 g/mol is adopted as the working fluid to reduce the mass and volume of the turbomachinery.The Prandtl number for helium–xenon mixtureswith thisrecommended mixing ratio may be as low as 0.2.As the convective heat transfer is closely related to the Prandtl number,different heat transfer correlations are often needed for fluids with various Prandtl numbers.Previous studies have established heat transfer correlations for fluids with medium–high Prandtl numbers(such asair and water)and extremely low-Prandtl fluids(such as liquid metals);however,these correlationscannot bedirectly recommended for such helium–xenon mixtures without verification.This study initially assessed the applicability of existing Nusselt number correlations,finding that the selected correlations are unsuitable for helium–xenon mixtures.To establish a more general heat transfer correlation,a theoretical derivation was conducted using the turbulent boundary layer theory.Numerical simulations of turbulent heat transfer for helium–xenon mixtures were carried out using Ansys Fluent.Based on simulated results,the parameters in the derived heat transfer correlation are determined.It is found that calculations using the new correlation were in good agreement with the experimental data,verifying its applicability to the turbulent heat transfer for helium–xenon mixtures.The effect of variable gas properties on turbulent heat transfer was also analyzed,and a modified heat transfer correlation with the temperature ratio was established.Based on the working conditions adopted in this study,the numerical error of the property-variable heat transfer correlation was almost within 10%.

Keywords Gas-cooled nuclear reactor.Space nuclear power.Helium–xenon mixtures.Convective heat transfer.Nusselt number

1 Introduction

Deep-space exploration,a landing on Mars,and the construction of planetary bases will become established activities in the near future.The development of advanced and reliable space power will become hugely important in order to achieve these ambitious milestones.Compared to typical power sources used in space,such as chemical fuel cells or solar photovoltaic arrays,the space nuclear reactor power ischaracterized by wider power coverageand longer operating duration;as such,it is regarded as a promising technology for human space exploration[1–3].Among existing technology roadmapsof space nuclear reactors,the gas-cooled nuclear reactor combined with a closed Brayton cycle is perceived as the most rational scheme for the system power exceeding 200 kW[4–6];this is because it can achieve a higher energy conversion efficiency and a smaller specific mass.Typically,pure helium isadopted as the working fluid for the terrestrial Brayton cycle because of its excellent thermal and transport properties[7].However,its small molecular weight inevitably resultsin a larger aerodynamic loading of the turbomachinery,which in turn increasesthe mass and volume of the system[8].To reduce the aerodynamic loads and obtain an acceptable heat transfer capacity,a helium–xenon mixture with a molecular weight of 14.5–40.0 g/mol is generally considered as the working fluid for Brayton cycles[9–13].The Prandtl number(Pr)of helium–xenon mixtures in this molecular weight range is approximately 0.21–0.30,which is significantly lower than that of conventional air or water(Pr＞0.70).For such low-Pr fluids,the thermal boundary layer is thicker,and the similarity of velocity and temperature will be disrupted.Hence,the convective heat transfer characteristics of low-Pr helium–xenon mixtures differ from those of fluids with medium–high Prandtl numbers.Previousstudieshave also shown that the Dittus–Bolter(DB)[14]and Colburn equations[15]obtained from experimental data of those conventional fluids overpredicted the Nusselt number of low-Pr gas mixtures[16,17].Therefore,to better describe the convective heat transfer of helium–xenon mixtures,it is necessary to establish a suitable heat transfer correlation.

The particularity of flow and heat transfer for low-Pr fluids has been widely concerned[18–20].However,most studies on this issue have mainly focused on liquid metals(Pr?0.01),where many heat transfer correlations for liquid metals were proposed[21–23].Lyon et al.[24]first derived a semi-empirical heat transfer correlation for a boundary condition with constant heat flux.Subsequently,Azer et al.[25]and Sleicher et al.[26]proposed their heat transfer correlations for liquid metals using theoretical hypotheses.Based on experimental data for liquid metals,Stomquist et al.[27],Skupinski et al.[28],and Churchill[29]presented their empirical heat transfer correlations.The Pr of liquid metals was found to be one or two orders of magnitude lower than that of the helium–xenon mixture;importantly,the flow and heat transfer characteristics of these two fluids significantly differed.To better illustrate the difference in convective heat transfer,Fig.1 presents the temperature profiles of the air,helium–xenon mixture,and liquid mercury in a heated tube[30].The temperature profile of the liquid metal was flat;however,the temperature distribution of the helium–xenon mixture wassimilar to that of air and relatively steep near the wall.These results show that the thermal resistance of liquid metal is evenly distributed on the radial section,indicating that heat transfer was dominated by molecular heat conduction.The thermal resistance of the helium–xenon mixture was concentrated in thelimited near-wall region,and turbulent heat diffusion continued to be dominant.Due to the difference in flow and heat transfer,the heat transfer correlations from liquid metals could not directly be used to calculate flow and heat transfer for helium–xenon mixtures without verification.As such,there is a need to conduct in-depth research on the heat transfer correlation for helium–xenon mixtures within the Pr range.

Fig.1 (Color online)Temperature profiles of different fluids

Thisstudy evaluatestheapplicability of existing Nusselt number correlations to helium–xenon mixtures.Subsequently,a theoretical derivation of the Nusselt number correlation was conducted to establish a more general heat transfer correlation.Numerical simulations of the turbulent heat transfer for helium–xenon mixtures were also carried out;the simulated results were used to determine related parametersin thederived correlation.Finally,theeffectsof varying gas properties on turbulent heat transfer were considered,and a new property-variable heat transfer correlation was presented.

2 Application of existing correlations

Table 1 shows that this study selected common correlations and explored their applicability to helium–xenon mixtures.Among these correlations,DB and Colburn’s correlations are empirical correlations obtained from experimental data relating to medium–high Pr fluids.Churchill’s correlation was determined by numerically fitting experimental data for different fluids with a wide range of Prandtl numbers.The uncertainty associated with the above three correlations is provided in Table 1.The Lyon’s and Stomquist’sequationswere frequently used for comparison in many studieson theconvective heat transfer of liquid metals;however,clear uncertainties have rarely been provided for these equations in published literature.Additionally,it should benoted that theapplicablerangeof most correlations in Table 1 was intentionally extrapolated to evaluate their suitability for helium–xenon mixtures.

Table 1 Existing correlations of Nusselt number

Figure 2 shows that calculations were compared with experimental data[17];the correlations from DB and Colburn had clearly overpredicted the Nusselt number of helium–xenon mixtures,while other calculations were underestimates compared with experimental data.Although predictions by Churchill’s correlation were relatively closer to the experimental data when the Reynolds number(Re)equaled 84 000,it still exceeded the error limit of 10%at Re=34 000.This indicates that the selected correlations from liquid metals are unsuitable for helium–xenon mixtures.

3 Theoretical derivation

To propose amore universal Nusselt number correlation for helium–xenon mixtures,a theoretical derivation based on the turbulent boundary layer theory and eddy viscosity hypothesis was conducted.Obtaining the temperature profile was key to deriving the theoretical heat transfer correlation[31].First,the derivation processes for the general logarithmic law of temperature was briefly reviewed.Ignoring the varying properties of gas mixtures,the wall shear stress and wall heat flux in the fully developed region may be expressed as follows[32]:

A two-layer model was used to simplify the boundary layer[33],in which the flow boundary layer was divided into linear and logarithmic regions.Similarly,the thermal boundary layer was divided into molecular conduction and turbulent conduction regions.The linear and molecular conduction regions were dominated by molecular viscosity(ν)and molecular heat diffusivity (α),respectively,neglecting the effect of turbulent momentum diffusivity(εm)and turbulent heat diffusivity(εh).Conversely,turbulent diffusion predominates in the logarithmic and turbulent conduction regions,thus ignoring molecular viscosity and molecular heat diffusivity,respectively.Using the mixing-length theory[34],the profile of dimensionless velocity may be derived by integration as follows:

Therefore,in present investigation,the derivation of Eq.(4)was sequentially conducted to obtain the dimensionless mass-average temperature()and the theoretical Nusselt number correlation.To facilitate this derivation,it was assumed that the thickness of the linear and molecular conduction regionswassmall and thereforenegligible.One study reported that the demarcation point of the flow boundary layer was almost fixed[32];that is,,where y+was defined as per Eq.(5):

In which the wall shear(τw)may be expressed as follows:

By adopting the Blasius formula to calculate the friction factor(f),the expression of y+can be simplified to Eq.(7):

At the wall surface(i.e.,y=0),the axial velocity is reduced to zero.At thecenter of tube(i.e.,y=r0),the axial velocity is the highest and can be described as follows:

By integrating the area under Eq.(3),the expression of dimensionless area-average velocity,,can be obtained:

By subtracting Eqs.(10)from(11),the following relationship can be obtained:

Combining Eqs.(12)–(14),the expression forcan be derived using Eq.(15):

However,to derive the theoretical correlation of the Nusselt number,thEstill needs to be determined.Several previous studies[39]have assumed thatas,because differences between the two parameters were＜10%for turbulent heat transfer as per Kays et al.[32].In the present study,it was assumed thatto further reduce deviation.Specifically,a correlation to calculatEwas proposed based on theoretical derivation[39],as:

Combine Eqs.(14)and(16),then,

Using the definition of Nusselt number and T+,the Nusselt number correlation may be transformed asfollows:

where f is the friction factor.Combining Eqs.(15)and(18),the following theoretical expression for the Nusselt number was obtained:

4 Numerical simulations

4.1 Thermophysical and transport properties

Accurate properties of helium–xenon mixtures are essential to numerical simulations.Typically,existing designs of space gas-cooled nuclear reactor systems have an operating pressure of 2 MPa and a temperature range of approximately 400–1300 K.Under these conditions,the theory of diluted gases based on the Chapman-Enskog approach was not found to be applicable.Tournier et al.[40]reviewed the experimental data for the properties of dense noble gas mixtures across a pressure range of 0.1–20 MPa and temperature up to 1400 K,summarizing semi-empirical property correlations.Based on the study of Tournier et al.[40],a property-calculation code for helium–xenon mixtures was developed,and its accuracy was validated in a previous study[30].Figure 3 presents the calculation of variations in gas mixture properties with temperature.

These results indicate that variations in thermal conductivity(λ)and dynamic viscosity(μ)are proportional to temperature,while the specific heat capacity(Cp)and density(ρ)decrease as temperature increases.As the value of the specific heat capacity varies very slightly with temperature,it can be set as a constant for a particular gas mixture in numerical simulations.Other property correlationswere implemented in Ansys Fluent using user-defined functions.

4.2 Numerical approach

The turbulent heat transfer of the helium–xenon mixtures in a tube was simulated using Ansys Fluent.The calculation model refers to the test section of the experiment in Taylor et al.[17],as shown in Fig.4.

Fig.2 (Color online)Predictions of Nusselt number by various correlations

Fig.3 (Color online)Variations in the properties of helium–xenon mixtures with temperature at P=2 MPa

Fig.4 (Color online)Test section of experiment for helium–xenon mixtures by Taylor et al.a Schematic diagram of experimental device;b Heated tube of test section

The experimental loop was a closed circuit driven by a gas booster pump;the flow and heat transfer of the gas mixture was investigated in the test section.Figure 4b shows the geometric parameters of the test section,which consisted of a straight tube with an inner diameter(D)of 5.87 mm.The total length(L)was 680.92 mm,of which the first part(L1)was adiabatic,while the remainder was heated by a resistance wire to obtain uniform heat flux.A two-dimensional calculation domain was adopted,as the gravitational effect was neglected.

With respect to the simulated boundary conditions,the inlet was set to ‘mass-flow-inlet’,the outlet was set to‘pressure-out’,and the wall surface adopted a non-slip boundary.The simulation parameters were set according to experimental conditions,listed in Table 2.A previous paper[41]evaluated the applicability of different turbulence models to helium–xenon mixtures;based on the findings from this paper,the SST κ–ω turbulence model wasselected for this study.To improvethe accuracy of the numerical simulation for turbulent heat transfer of low-Pr helium-xenon mixtures,a local Prtmodel(shown in Eqs.(20)and(21)),was proposed in Zhou et al.[30].Based on comparison with more experimental data,in this study the first constant in the expression of Prt,∞was adjusted slightly from the original 0.80 to 0.86.

Table 2 Simulation parameters of helium–xenon mixtures

Grid independence verification was investigated to ensure the validity of the numerical calculation.Four mesh files were tested:mesh1:20×3000,mesh2:30×4000,mesh3:40×6000,and mesh4:60×8000.The bulk temperature(Tb)of helium–xenon mixtures and the wall temperature(Tw)at x=0.45,0.55,0.60,and 0.65 m were calculated.Figure 5 shows that when the grid number exceeded 120 000,the difference in the bulk and wall temperatures as calculated by various meshes was negligible,and the maximum error was＜0.8%.Based on the calculation accuracy and efficiency,mesh3 was adopted in this study.

Figure 6 presents the grid of mesh3;the grid refinement was applied in the near-wall region,where the spacing of the first grid was 0.004 mm,spacing ratio was 1.08,and maximal aspect ratio was 28.4.Using the Run696 as an example,the corresponding y+of the first grid was y+=0.22.In fact,the wall-resolved method was enabled for the SST κ–ω model when the mesh in the near-wall region was sufficient.At that time,the y+of the first grid was generally required to be＜1.0;as such,mesh3 was able to meet thecalculation requirements.In order to further verify the simulation models adopted in the present study,additional experimental data on turbulent heat transfer for helium-xenon gas mixtures were introduced for comparison.Figure 7 shows that the calculated deviation of the wall temperature under various working conditions was essentially 5%.Therefore,the modified turbulence model may be used to study the turbulent heat transfer characteristics of helium–xenon mixtures.

4.3 Determination of parameters

Fig.5 (Color online)Simulated bulk temperature(T b)and wall temperature(T w)with various meshes

Fig.6 (Color online)Grid of mesh3

Fig.7 (Color online)Comparison of simulated wall temperature(T w)by the modified turbulent model and experimental data

To investigate the appropriate equation for the friction factor(f),four typical equations from conventional fluids[42–44](see Table 3),were selected for testing.One study[16]suggested that theexperimental friction factor of noble binary gases may be calculated using Eqs.(22)–(24).The term f istheoverall averagefriction factor,calculated using the distance from the start of the heated section to the second pressure tap(‘pt2’in Fig.4):

Table 3 Four equations used to determine the friction factor

whereΔP denotes the total pressure drop;ΔPmdenotes the acceleration pressure drop;andΔPfdenotes the friction pressure drop.Figure 9 shows that the calculated values were relatively close to each other and calculation errors were generally within 5%.Based on the good agreement between the experimental data and the concise expression,the Blasius equation was used to calculate the friction factor for helium–xenon mixtures in Eq.(19).

Notably,the Karman constant(κ)exists in the velocity distribution(see Eq.(3));as such,the simulated velocity distribution may be numerically fitted to determineκfor helium–xenon mixtures.Figure 10a shows that the fitting logarithmic law of velocity for Pr=0.30 was initially obtained by settingκ=0.42.Then,a comparison between the calculated logarithmic law withκ=0.42 and the simulation results of Pr=0.21 was conducted as shown in Fig.10b.The calculated logarithmic law withκ=0.38 was still in good agreement with the simulation results;as such,κ=0.42 was adopted in Eq.(19).

The Reynolds-averaged Navier–Stokes equations(RANS)model was developed based on the Boussinesq eddy viscosity hypothesis[45],in which eddy viscosity was calculated using mixing-length theory;as such, εm/ν was a function related only to y+.By analyzing the Prtmodel(Eqs.(20)and(21)),the Prtwas also found to be a function only related to y+for a specific helium–xenon mixture,as PEtand RElocalwere expressed as functions of y+.Therefore,the following relationship can be obtained:

Fig.8 (Color online)Variations in molecular viscosity(ν)and turbulent momentum diffusivity(εm)with y+for helium–xenon mixtures

Fig.9 (Color online)Comparison of calculations usi ng different friction factor equations with experimental data

Fig.10 (Color online)Fitted v elocity profiles of helium–xenon mixtures

It is feasible to obtain Eq.(25)by numerical fitting.Since the thermal boundary layer of low-Pr helium–xenon mixture was thicker than the flow boundary layer,the fitting range of y+should ensure that y+＞11.First,the expression was initially determined by fitting the simulation results of Pr=0.30,as shown in Fig.12(a);this expression could be described by Eq.(26).Thereafter,the cases with the Prandtl number to 0.23 and 0.21 were simulated.Theresultsareshown in Fig.12b and c,whereit is shown that the calculation of Eq.(26)was in good agreement with simulated data points.Therefore,for 0.21≤Pr≤0.30 and y+＜50,Eq.(26)was able to reasonably describe the turbulent heat diffusivity of helium-xenon mixtures.

Fig.11 (Color online)Variationsin molecular heat diffusivity(α)and turbulent heat diffusivity(εh)with the y+of helium–xenon mixtures

Fig.12 (Color online)Numerical fitting of the expression for εh/ν

The influence of the Pr on turbulent heat transfer may be quantitatively analyzed using the numerical model of the turbulent heat diffusivity.For air and water,the relationship between turbulent heat diffusivity and y+wasεh/ν∝（y+）3[36,37].Compared with Eq.(26),it can be seen that the lower Prandtl number will lead to a smaller turbulent thermal diffusivity at the same radial position,resulting in weak convective heat transfer intensity.This conclusion may beused to understand theresultsin Fig.2.The Prandtl number of liquid metals was lower than that of the binary gas mixtures.As the turbulent heat diffusion was weaker for the same Re,the Nusselt number correlations obtained from liquid metalsunderestimated theexperimental data of binary gas mixtures.Conversely,the Pr of conventional fluids was higher than that of helium–xenon mixtures,resulting in more intense turbulent heat diffusion.Therefore,predictions from the DB and Colburn correlations represented overestimations of experimental data.

For working fluids with Pr?1.0,the Prttends to be constant of 0.85.Differently,for low-Pr fluidsit tendsto be larger[46].However,it is difficult to obtain the section average Prtdirectly from the local function expressions.In this study,a constant value was used to roughly represent the section average Prt.To determine this parameter,the radial Prtdistribution of Run711 was first calculated using the Zhou Prtmodel(Eqs.(20)and(21)).As shown in Fig.13,by comparing with the prediction of the Zhou model,a Prtof 0.9 may be a candidate value;as such,this value was initially adopted as the section average Prtin Eq.(19)for the helium–xenon mixtures.

Fig.13 Radial predictions of Pr t at x=0.60 m for Run711

Once all parameters were determined,the semi-theoretical Nusselt number correlation was quantified using Eq.(28):

5 Verification and analysis

Combining Eqs.(4)and(27),the radial temperature distribution of the corresponding helium–xenon mixtures with 0.21≤Pr≤0.30,was determined as shown in Eq.(29):

To verify the correctness of the derivation,the results using Eq.(29)where the Pr was 0.20,were compared with predictions of other temperature distribution correlations(shown in Table 4).Additionally,the direct numerical simulation resultsof Kawamura et al.[47]were introduced as benchmarks.As shown in Fig.14,the calculations using Eq.(29)showed better agreement with the DNS results than those of other correlations,preliminarily proving the validity of Eq.(29)and the correctness of this derivation.

Fig.14 (Color online)Comparison of prediction by Eq.(29)with other temperature correlations

Table 4 Existing correlations for temperature distribution

To test the accuracy of the new Nusselt number correlation,predictions using Eq.(28)were compared with experimental data,as shown in Fig.15.The results show that the calculations are in good agreement with the experimental data of gas mixtures under both working conditions.This further verifies the accuracy of the derivation and its applicability to the turbulent heat transfer for helium–xenon mixtures where 0.21≤ Pr≤ 0.30.

Fig.15 (Color online)Comparison of predictions using Eq.(28)with experimental data

The ranges of the tested conditions are 18,000＜Re＜60,000,0.21≤Pr≤0.30,and Tw/Tb＜2.Figure 19 shows that the calculation deviation becomes obvious as Tw/Tbincreases.When Tw/Tb=1.8,the numerical error exceeded 30%.A seemingly exponential relationship between NuExp/Nucpand Tw/Tbwas observed when NuExp/Nucpwere presented in logarithmic coordinates.Moreover,as Tw/Tbapproaches 1.0,the constant-property correlation would become valid,and NuExp/Nucpwould approach 1.0.Therefore,by numerically fitting data points,the modified,property-variable Nusselt number correlation for the helium–xenon mixture was obtained as follows:

Fig.16 (Color online)Sensitivity analysis on the Karman constant for the derived Nusselt number correlation

The results in Fig.19 show that the error in predictions using Eq.(30)was almost within 5%and was consistently within 10%.Notably,the modified Nusselt correlation(Eq.(30))was only determined using existing and limited experimental data;to fully verify its applicability,additional experimental data are required.

Fig.17 (Color online)Sensitivity analysis on the Pr t number for derived Nusselt number correlation

Fig.18 (Color online)Sensitivity analysis on the for derived Nusselt number correlation

Fig.19 (Color online)Comparison of calculations using Eq.(28)with measured experimental data

6 Conclusion

This study evaluates existing Nusselt number correlations and derives a theoretical Nusselt number correlation for helium–xenon mixtures using turbulent boundary layer theory.The turbulent heat transfer in a tube for helium–xenon mixtures was numerically simulated using a modified Prtmodel.Based on numerical simulations,a new turbulent heat diffusivity model in the thermal boundary layer was developed,and related parameters were determined in the theoretical Nusselt number expression.A preliminary modified Nusselt number correlation was presented based on variations in gas properties.The main conclusions from this study were:

(1) Compared with the extrapolated experimental data of gas mixtures,the selected Nusselt number correlations from conventional fluids and liquid metals were unsuitable for helium–xenon mixtures with 0.21≤Pr≤0.30;

(2) The calculation error of the friction factor for helium–xenon mixtures using the Blasius equation was within 5%,and the demarcation point of U+for helium–xenon mixtures and conventional fluids was approximately equal;

(3) For 0.21≤Pr≤0.30 and y+＜40,εh/ν=0.0112(y+）1.82can reasonably describe the turbulent heat diffusivity of helium–xenon mixtures.A lower Pr led to a reduced turbulent thermal diffusivity,resulting in weak convective heat transfer intensity;

(4) Predictions by the derived semi-theoretical Nusselt number correlation were in good agreement with experimental data,verifying the accuracy of the derivation and its applicability to turbulent heat transfer for helium–xenon mixtures.

Future research should focus on obtaining additional experimental research on low-Pr helium–xenon mixtures.This additional experimental data may be used to verify correlations proposed in this study.Furthermore,the new Nusselt number correlation should be used to develop a thermal–hydraulic transient analysis code for helium–xenon cooled reactor systems.

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License,which permits use,sharing,adaptation,distribution and reproduction in any medium or format,as long as you give appropriate credit to the original author(s)and the source,provide a link to the Creative Commons licence,and indicate if changes were made.The images or other third party material in this article are included in thearticle’s Creative Commons licence,unless indicated otherwise in a credit line to the material.If material is not included in the article’s Creative Commonslicence and your intended use is not permitted by statutory regulation or exceeds the permitted use,you will need to obtain permission directly from the copyright holder.To view a copy of this licence,visit http://creativecommons.org/licenses/by/4.0/.

Author contributionsAll authors contributed to the study conception and design.Material preparation,data collection and analysis were performed by Biao Zhou,Yu Ji,Jun Sun and Yu-Liang Sun.The first draft of the manuscript was written by Biao Zhou and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.