Heat-Transfer Characteristics of Liquid Sodium in a Solar Receiver Tube with a Nonuniform Heat Flux

This paper presents a numerical simulation on the heat transfer of liquid sodium in a solar receiver tube, as the liquid sodium is a promising heat-transfer candidate for the next generation solar-power-tower (SPT) system. A comparison between three mediums—solar salt, Hitec and liquid sodium—is presented under uniform and nonuniform heat-flux configurations. We studied the effects of mass flow rate (Qm), inlet temperature (Tin), and maximum heat flux (qomax), on the average heat-transfer coefficient (h) and the friction coefficient (f ) of the three mediums. The results show that the h of liquid sodium is about 2.5 to 5 times than other two molten salts when Tin is varying from 550 to 800 K, Qm is 1.0 kg/s, and qomax is 0.1 MW/m2. For maximum heat fluxes from 0.1 to 0.3 MW/m2, the h of liquid sodium is always an order of magnitude larger than that of Hitec and Solar-Salt (S-S), while maintaining a small friction coefficient.


Introduction
Compared to the parabolic trough, Fresnel and dish collectors, the solar-power-tower (SPT) plant has the remarkable advantages, such as lower electricity cost, large-scale power generation and higher efficient thermodynamic cycles [1][2][3].The SPT is equipped with a large number of heliostats on the ground, each with a tracking mechanism that accurately reflects the reflection of sunlight onto the receiver at the top of a tall tower.The concentrating magnification on the receiver can exceed 1000 times.One typical arrangement of the SPT receivers is the external tubular receiver designed for Solar Two project, in which only half of the surface of the tube is exposed to solar irradiation.This may bring about many problems, such as aggravating the plastic deformation of the receiver tube, facilitating degradation of the selective absorptive coating and decreasing the allowable solar heat flux [4,5].Since the nonuniform solar heat flux tends to cause the temperature inhomogeneity of the heat-transfer fluid (HTF) and, further, the thermal stress on the heat-transfer tubes, a much broader range of operational temperatures is required.The liquid metal as the promising candidate for the exposed cylindrical heat absorber of SPT has been proposed [6].
Nitrate salts have been used as HTFs and thermal storage mediums for decades in the concentrating SPT industry.The most commonly used HTFs are solar salt (S-S, 60% NaNO 3 and 40% KNO 3 ) and Hitec (53% KNO 3 + 40% NaNO 2 + 7% NaNO 3 ).Both the nitrate salt mixtures will decompose above 873 • K, which has seriously limited the overall efficiency in the SPT system.Furthermore, recent research efforts have shown that the nitrate salts are more suitable for use in parabolic trough systems due to their low working temperature [7].However, the next-generation SPT systems require a higher incident peak flux and operating temperature.If the liquid metal is the heat-transfer fluid, it can Energies 2019, 12, 1432 2 of 16 provide an incident peak flux above 0.6 MW/m 2 .Liquid sodium (Na), characterized by chemical stability at temperatures up to near 1173 • K has a lower melting point to 371 • K [8], as well as superior thermal conductivity and low Prandtl number.The material properties mentioned above can largely improve heat transfer when compared to conventional fluids such as oil or salt mixtures [9].In fact, during the early years of the development of central receiver systems (CRSs), liquid sodium was one of the prominent HTFs under investigation.Indeed, several test projects have been developed, and the efficiency was obtained at 88%-96% in the 1980s [10].However, the disadvantage of liquid sodium is its high combustibility when in contact with water even if no air.Fortunately, special and protective measures have also been developed from previous experiences [11].Also, this technology has continued to be investigated by several institutions around the world [12][13][14].Besides, further work has reported that liquid metals have attractive properties for CSP applications [15,16].Recently, Amy et al. [17] demonstrated how a ceramic, mechanical pump that can be used to continuously circulate liquid metal at temperatures of around 1473 • K-1746 • K.This study solves the problem that collecting, transporting, storing liquid metal above 1300 • K brings.
The prediction of heat transfer of liquid metal has been the subject of many investigations.DeAngelis et al. [18] have examined using a liquid metal as heat-transfer fluid in conjunction with a receiver.It is feasible to reach temperatures of 1623 • K at greater than 90% efficiency.Boerema et al. [15] compared liquid sodium and Hitec, and the use of liquid sodium can achieve 57% absorber area reduction and 1.1% efficiency improvement.Even if liquid sodium is an excellent heat-transfer medium, the specific application background was not mentioned in the paper.Pacio et al. [19,20] summarized the current state-of-the-art of liquid metals (LMs) as HTFs in solar power plants.
Additionally, the liquid sodium (Na) was proposed as an efficient HTF to allow extending the design ranges, and able to contribute to the development of next-generation SPT.Matsubara et al. [21] studied the spanwise heat transport in turbulent channel flow with Prandtl numbers ranging from 0.025 to 5.0.It is regrettable that they do not compare the heat-transfer property with traditional heat-transfer fluid (e.g., solar salts).Rodríguez-Sanchez et al. [22,23] focused on the thermal, mechanical and hydrodynamic analysis associated to the nonuniformity of the heat flux, and they also considered the thermal field and thermal stresses along the solid wall [24].
A higher cost, more complex system and security issues are factors that have to be considered for the experimental research on liquid metal.
Therefore, numerical simulation is an effective research method [25,26].In this work, we present a numerical simulation on the heat transfer of liquid sodium under nonuniform heat flux.A physical model of a single tube is established to investigate the heat-transfer performance of the receiver tube, and two other commonly used heat-transfer media, Hitec and solar salt, are also to be considered.First, we built solar heat-flux distributions on the whole absorber outer wall and circumferential variation of heat flux on the inner wall of the tube.Then, the heterogeneity of the temperature on the circumferential tangent plane and the solid wall is presented.Cloud images show the wall temperature distribution of the three HTFs.The temperature difference (Θ-Θ ref ) of the three HTFs along the circumferential angle is compared when q omax is 0.1 MW/m 2 , Re is ranging from 10,000 to 30,000, and inlet temperature is 550 • K. Last, the influence of three parameters, Q m (1.0 to 3.0 kg/s), q omax (0.1 to 0.3 MW/m 2 ), T in (550 • K to 800 • K) on the heat-transfer characteristics of sodium, S-S and Hitec are discussed.This research can offer technical references for the design and construction of experimental facilities.

Physical Model
The research background of this study is the heat receiver of the solar-power-tower system (Figure 1a).Since the collector is cylindrical and the fluid flows serpentinely in the collector tube, we investigate only the single collector tube (Figure 1b).We consider the conjugate heat-transfer problem of a receiver tube subject to inhomogeneous heat flux along the axial direction (z), circumferential direction (θ), and radial direction (r).The heat transport from the regional source (see Figure 2) contains three orthogonal components.The outer diameter (R) of the geometric model is 20 mm, and the tube length is 100R.The effect of different thermal boundary conditions on the heat-transfer performance of the collector tube is discussed.As shown in Figure 3, this paper deals with two heat-flux conditions of the turbulent-flow model: (a) A cosine heat flux (see Equation ( 1)) [27] is imposed on one half of the wall of the tube, while the other half is considered adiabatic.
1 2 max e cos , cos 0 0, cos 0 where the qomax is the maximum heat flux on the wall of the collector tube, θ is the circumferential angle, and z is the length along the axial direction of the receiver tube.We consider the conjugate heat-transfer problem of a receiver tube subject to inhomogeneous heat flux along the axial direction (z), circumferential direction (θ), and radial direction (r).The heat transport from the regional source (see Figure 2) contains three orthogonal components.The outer diameter (R) of the geometric model is 20 mm, and the tube length is 100R.The effect of different thermal boundary conditions on the heat-transfer performance of the collector tube is discussed.We consider the conjugate heat-transfer problem of a receiver tube subject to inhomogeneous heat flux along the axial direction (z), circumferential direction (θ), and radial direction (r).The heat transport from the regional source (see Figure 2) contains three orthogonal components.The outer diameter (R) of the geometric model is 20 mm, and the tube length is 100R.The effect of different thermal boundary conditions on the heat-transfer performance of the collector tube is discussed.As shown in Figure 3, this paper deals with two heat-flux conditions of the turbulent-flow model: (a) A cosine heat flux (see Equation ( 1)) [27] is imposed on one half of the wall of the tube, while the other half is considered adiabatic.
1 2 max e cos , cos 0 0, cos 0 where the qomax is the maximum heat flux on the wall of the collector tube, θ is the circumferential angle, and z is the length along the axial direction of the receiver tube.As shown in Figure 3, this paper deals with two heat-flux conditions of the turbulent-flow model: (a) A cosine heat flux (see Equation ( 1)) [27] is imposed on one half of the wall of the tube, while the other half is considered adiabatic.
where the q omax is the maximum heat flux on the wall of the collector tube, θ is the circumferential angle, and z is the length along the axial direction of the receiver tube.

Governing Equations
A conjugate heat-transfer model is adopted to obtain the heat flux and temperature distribution in both the fluid and the tube wall.To simplify the model, the following assumptions are made: (1) the mediums are homogeneous and isotropic; (2) the liquid phase is in a state of local thermal equilibrium; (3) the heat loss at the outer wall of the receiver tube is not considered since this paper only focuses on the heat-transfer characteristic of the HTFs in the tube.For the heat loss on the outer surface of the tube, one can refer to the references [23,28].
The governing equations for continuity, momentum, energy and standard k-ε two-equation turbulence model for the incompressible Newtonian fluid can be expressed as follows: Continuity equation: Momentum equation: Energy equation: k equation: ε equation: where the turbulent viscosity μt and the production rate of k Gk are given by where the standard constants are employed, cμ =0.09, c1 =1.44, c2 =1.92, σk =1.0, σ ɛ =1.3, and σT =0.85 [29].

Boundary Conditions
As a plurality of solar mirrors superposes the heat flux to the receiver tube wall, the radiation heat flux is highly nonuniform.The heat-flux expression is approximating the normal distribution function along the axial direction of the receiver tube and the cosine function distribution along the circumferential direction, as described in the formula (1).
It is also convenient to define the total heat flux applied to the outer surface, and note that the energy conservation in the solid implies that:

Governing Equations
A conjugate heat-transfer model is adopted to obtain the heat flux and temperature distribution in both the fluid and the tube wall.To simplify the model, the following assumptions are made: (1) the mediums are homogeneous and isotropic; (2) the liquid phase is in a state of local thermal equilibrium; (3) the heat loss at the outer wall of the receiver tube is not considered since this paper only focuses on the heat-transfer characteristic of the HTFs in the tube.For the heat loss on the outer surface of the tube, one can refer to the references [23,28].
The governing equations for continuity, momentum, energy and standard k-ε two-equation turbulence model for the incompressible Newtonian fluid can be expressed as follows: Continuity equation: Momentum equation: Energy equation: where the turbulent viscosity µ t and the production rate of k G k are given by where the standard constants are employed, c µ = 0.09, c 1 = 1.44, c 2 = 1.92, σ k = 1.0, σ ε = 1.3, and σ T = 0.85 [29].

Boundary Conditions
As a plurality of solar mirrors superposes the heat flux to the receiver tube wall, the radiation heat flux is highly nonuniform.The heat-flux expression is approximating the normal distribution function along the axial direction of the receiver tube and the cosine function distribution along the circumferential direction, as described in the Formula (1).It is also convenient to define the total heat flux applied to the outer surface, and note that the energy conservation in the solid implies that: where the total heat flux (Q) is kept constant for all the cases presented here, and R 0 = 20 mm.Considering the wall temperature of the tube varies with the time and heat flux, to improve the accuracy and practicability of the numerical calculation, we adopted the formulas shown in Table 1 to evaluate the thermophysical properties of HTFs.The collector tube material is 316 L stainless steel, and its thermal conductivity (k s ) is 18.4 W/(m•K).Based on the assumptions, the boundary conditions are expressed as follows: Table 1.Thermophysical properties of sodium [30], Solar salt and Hitec [7], where T is the fluid bulk temperature in Kelvin.

HTFs Thermophysical Properties Property Equation
Liquid sodium 371 (1) Fluid and the solid wall region When the HTF flows around a stationary solid wall in a collector tube, where the solid wall is impermeable, the normal velocity should be satisfied v n = 0.At the same time, the no-slip condition must be satisfied, and the tangential velocity v τ = 0.The heat-flux condition of the tube wall is: (2) The inlet and outlet temperature of the tube The inlet velocity, pressure, and temperature of the tube line are formulated as follows, respectively.

Numerical Methods
The governing Equations ( 2)-( 7) are discretized by the finite volume method by using O-mesh and wall-dense nonuniform mesh.Moreover, the convective terms in momentum and energy equations are discretized with the second upwind scheme.The SIMPLE algorithm is used to ensure the coupling between velocity and pressure.The discretization of momentum, turbulent kinetic energy, energy, and dissipation rate are all second upwind schemes.The turbulence model is κ-ε model.The near-wall surface flow is solved by the standard wall function method, and all the non-dimensional number of near-wall y + is controlled by 30~60.The convergence criterion for the velocities and energy is that the Energies 2019, 12, 1432 6 of 16 maximum mass residual of the cells divided by the maximum residual is less than 10 −5 and 10 −7 for the continuity, momentum, and energy equation.Based on these methods, the performance of the receiver tube for the two models with different HTFs can be rapidly predicted [31].

Parameter Definitions
To predict the thermal and hydraulic characteristics, we define the time-averaged temperature in the cross-plane among the fluid as Θ, and the average temperature of the inlet and outlet of a receiver tube is named Θ ref , also known as the qualitative temperature.
The Reynolds number and average Nusselt number in the receiver tube for the medium are given as [32].
where h is the average heat-transfer coefficient, q is the average heat flux in the tube, ∆t the difference between the average temperature of the inner wall of the tube and the qualitative temperature.
The friction coefficient is defined as [33]: where ∆p is the pressure difference between the inlet and outlet of the receiver tube.

Model Verification
We verified the calculation procedure by comparing them with empirical formulas and existing experimental data.Figure 4a exhibits the trend of Nu for S-S (Pr = 13) and Hitec (Pr = 50) as the Re ranging from 1 × 10 4 to 3 × 10 4 under the nonuniform heat flux on a receiver tube.The inlet temperature is set as the melting point for the three HTF, respectively.The result of the calculation is in good agreement with Dittus-Boetter correlation and Gnielinski correlation [7]. Figure 4b shows the variation tendency of Nu when the Pe is ranging from 65 to 203 for sodium (Pr = 0.01).The calculation results have a 2~6% difference with the Lyon-Martinelli equation correlation [34] and are consistent with the experimental data [30].These results prove that the model and its calculation procedure is suitable and reasonable.To predict the thermal and hydraulic characteristics, we define the time-averaged temperature in the cross-plane among the fluid as Θ, and the average temperature of the inlet and outlet of a receiver tube is named Θref, also known as the qualitative temperature.
The Reynolds number and average Nusselt number in the receiver tube for the medium are given as [32].
where h is the average heat-transfer coefficient, q is the average heat flux in the tube, Δt the difference between the average temperature of the inner wall of the tube and the qualitative temperature.
The friction coefficient is defined as [33]: where Δp is the pressure difference between the inlet and outlet of the receiver tube.

Model Verification
We verified the calculation procedure by comparing them with empirical formulas and existing experimental data.Figure 4(a) exhibits the trend of Nu for S-S (Pr=13) and Hitec (Pr=50) as the Re ranging from 1×10 4 to 3×10 4 under the nonuniform heat flux on a receiver tube.The inlet temperature is set as the melting point for the three HTF, respectively.The result of the calculation is in good agreement with Dittus-Boetter correlation and Gnielinski correlation [7]. Figure 4(b) shows the variation tendency of Nu when the Pe is ranging from 65 to 203 for sodium (Pr=0.01).The calculation results have a 2%~6% difference with the Lyon-Martinelli equation correlation [34] and are consistent with the experimental data [30].These results prove that the model and its calculation procedure is suitable and reasonable.

Cases Studied
The main parameters used in the calculation cases are listed in Table 2.Among them, the mass

Cases Studied
The main parameters used in the calculation cases are listed in Table 2.Among them, the mass flow rate varies from 1.0 to 3.0 kg/s, q omax is from 0.1 to 0.3 MW/m 2 , and the inlet temperature is from 550 • K to 800 • K.The temperature range is chosen between the melting point and the boiling point of the three mediums.

Heat Flux
The heat-flux distributions on the whole absorber outer wall (case 1) are consistent with the result of Equation (1) shown in Figure 5.
Energies 2019, 12, x FOR PEER REVIEW 7 of 16 550 °K to 800 °K.The temperature range is chosen between the melting point and the boiling point of the three mediums.

Heat Flux
The heat-flux distributions on the whole absorber outer wall (case 1) are consistent with the result of Equation (1) shown in Figure 5. Figure 6 shows that the solar energy flux distribution, of different mediums on a cross section when Qm is set as 1 kg/s, the inlet temperature is 550 °K, and qomax is 0.1 MW/m 2 .The distribution is unsymmetrical in the circumferential direction and is independent of the HTF in the receiver tube.It is noteworthy that the maximum heat flux through the inner wall is at the location 90°.  Figure 6 shows that the solar energy flux distribution, of different mediums on a cross section when Q m is set as 1 kg/s, the inlet temperature is 550 • K, and q omax is 0.1 MW/m 2 .The distribution is unsymmetrical in the circumferential direction and is independent of the HTF in the receiver tube.It is noteworthy that the maximum heat flux through the inner wall is at the location 90 • .
Energies 2019, 12, x FOR PEER REVIEW 7 of 16 550 °K to 800 °K.The temperature range is chosen between the melting point and the boiling point of the three mediums.

Heat Flux
The heat-flux distributions on the whole absorber outer wall (case 1) are consistent with the result of Equation (1) shown in Figure 5. Figure 6 shows that the solar energy flux distribution, of different mediums on a cross section when Qm is set as 1 kg/s, the inlet temperature is 550 °K, and qomax is 0.1 MW/m 2 .The distribution is unsymmetrical in the circumferential direction and is independent of the HTF in the receiver tube.It is noteworthy that the maximum heat flux through the inner wall is at the location 90°.

Temperature Profile
The heat-flux distribution caused by inhomogeneous radiation inevitably influences the temperature distribution on the collector tube.The heterogeneity of the temperature on the circumferential tangent plane and the solid wall is shown in Figures 7 and 8.The coupled heat transfer between solid wall and fluid has been omitted in this section.
circumferential tangent plane and the solid wall is shown in Figure ( 7)- (8).The coupled heat transfer between solid wall and fluid has been omitted in this section.
When the Re is 25,000, and the inlet temperature is 550 °K, the wall temperature distributions of the three HTFs are shown in Figure 7.It is found that the temperature distribution on the outer wall is extremely nonuniform around the tube.Compared with the other two HTFs, the temperature distribution on the solid wall of sodium is more uniform at the exit section.The outlet temperature is 561.08 °K, 550.93 °K, 551.06 °K on the area-weight-average from left to right of Figure 7.The tube wall temperature difference of sodium changes to 20 °K, solar salt changes to 24 °K and the Hitec changes to 24 °K after passing through two meters of the tube.Overall, for sodium, the temperature difference of the whole tube is the minimal, and the outlet temperature is higher than S-S or Hitec.The reasons for those are that high thermal conductivity of sodium, which enhances the turbulent flow, the heat transfer and reduces the wall temperature.Thus, in SPT projects, sodium is more suitable as a heat-transfer medium than S-S or Hitec, in terms of flow dynamics and heat transfer.Figure 8 shows the variation of the temperature difference as a function of the circumferential coordinate.We made a comparison between the present NHF model calculations (solid-lines) and the results predicted by the uniform heat flux (UHF) model (dashed lines).The temperature difference (Θ-Θref) means the difference in the temperature on the circumferential section and the qualitative temperature.The temperature difference (Θ-Θref) of the three HTFs along the circumferential angle is compared when qomax is 0.1 MW/m 2 , Re is ranging from 10,000 to 30,000, and inlet temperature is set to 550 °K.
Note that the UHF model underestimates the inner temperature of the tube on the bright side while overestimates the backlight since the UHF model imposed the same heat flux on the bright side.Energy accumulation on both sides of the tube leads to two highest temperature points shown in Figure 8(b)-(c) whereas sodium has no obvious highest point.In the UHF model, the Θ-Θref of sodium is lower than that in the NHF model, and the temperature difference is considerable as Re increasing.The results indicate that thermal conductivity still dominates even if the turbulence is significant when sodium is used as HTF.For the higher Prandtl number (S-S or Hitec), the average temperature in cross-plane of the adiabatic side must be equal to the bulk temperature.The reason is that the momentum transfer rate is higher than the heat-transfer rate, so the velocity surface is first established, and the heat-transfer delay.The changing range of Θ-Θref is larger than sodium as Re increasing for S-S or Hitec.That shows that the flow and heat transfer of S-S or Hitec are greatly influenced by turbulence.As expected, the maximum temperature difference reached at the location of 90°, which agrees with the cosine effect expressed by the Equation (1).When the Re is 25,000, and the inlet temperature is 550 • K, the wall temperature distributions of the three HTFs are shown in Figure 7.It is found that the temperature distribution on the outer wall is extremely nonuniform around the tube.Compared with the other two HTFs, the temperature distribution on the solid wall of sodium is more uniform at the exit section.The outlet temperature is 561.08 • K, 550.93 • K, 551.06 • K on the area-weight-average from left to right of Figure 7.The tube wall temperature difference of sodium changes to 20 • K, solar salt changes to 24 • K and the Hitec changes to 24 • K after passing through two meters of the tube.Overall, for sodium, the temperature difference of the whole tube is the minimal, and the outlet temperature is higher than S-S or Hitec.The reasons for those are that high thermal conductivity of sodium, which enhances the turbulent flow, the heat transfer and reduces the wall temperature.Thus, in SPT projects, sodium is more suitable as a heat-transfer medium than S-S or Hitec, in terms of flow dynamics and heat transfer.
Figure 8 shows the variation of the temperature difference as a function of the circumferential coordinate.We made a comparison between the present NHF model calculations (solid-lines) and the results predicted by the uniform heat flux (UHF) model (dashed lines).The temperature difference (Θ-Θ ref ) means the difference in the temperature on the circumferential section and the qualitative temperature.The temperature difference (Θ-Θ ref ) of the three HTFs along the circumferential angle is compared when q omax is 0.1 MW/m 2 , Re is ranging from 10,000 to 30,000, and inlet temperature is set to 550 • K.
Note that the UHF model underestimates the inner temperature of the tube on the bright side while overestimates the backlight since the UHF model imposed the same heat flux on the bright side.Energy accumulation on both sides of the tube leads to two highest temperature points shown in Figure 8b,c whereas sodium has no obvious highest point.In the UHF model, the Θ-Θ ref of sodium is lower than that in the NHF model, and the temperature difference is considerable as Re increasing.
The results indicate that thermal conductivity still dominates even if the turbulence is significant when sodium is used as HTF.For the higher Prandtl number (S-S or Hitec), the average temperature in cross-plane of the adiabatic side must be equal to the bulk temperature.The reason is that the momentum transfer rate is higher than the heat-transfer rate, so the velocity surface is first established, and the heat-transfer delay.The changing range of Θ-Θ ref is larger than sodium as Re increasing for S-S or Hitec.That shows that the flow and heat transfer of S-S or Hitec are greatly influenced by turbulence.As expected, the maximum temperature difference reached at the location of 90 • , which agrees with the cosine effect expressed by the Equation (1).The value of Θ-Θref is slightly dependent on the Re but strongly affected by the Pr.In general, the Θ-Θref is increasing as the Pr increases, decreasing as the Re increases.Comparing Figure 8(a-c), the temperature difference of S-S at the location θ = 90° is 1.93 times higher than sodium, and Hitec is 1.35 times higher that of sodium when Re is 10,000.The temperature difference of S-S is 1.29 times higher than sodium, and Hitec is 2.0 times sodium when Re is 30,000.The results reflect that sodium has a good effect on solving the uneven temperature caused by nonuniform heat flux, which may reduce the risk of hot spots and, thus, reduce pipe stresses.The value of Θ-Θ ref is slightly dependent on the Re but strongly affected by the Pr.In general, the Θ-Θ ref is increasing as the Pr increases, decreasing as the Re increases.Comparing Figure 8a-c, the temperature difference of S-S at the location θ = 90 • is 1.93 times higher than sodium, and Hitec is 1.35 times higher that of sodium when Re is 10,000.The temperature difference of S-S is 1.29 times higher than sodium, and Hitec is 2.0 times sodium when Re is 30,000.The results reflect that sodium has a good effect on solving the uneven temperature caused by nonuniform heat flux, which may reduce the risk of hot spots and, thus, reduce pipe stresses.Another concern is the inner wall temperature, also called film temperature, which could lead to the degradation of the HTFs.The inner wall temperature distribution in NHF and UHF models are shown in Figure 9a-c when Re is 25,000, and the inlet temperature is 550 K. From Figure 9a-c, the average temperature of the fluid in the tube is: 562.8 • K, 554.9 • K, 554.7 • K. Figure 9a-c show that one of the most significant advantages of sodium over S-S and Hitec is that the temperature distribution is much more uniform on both the inner and outer surface of the tube.Another concern is the inner wall temperature, also called film temperature, which could lead to the degradation of the HTFs.The inner wall temperature distribution in NHF and UHF models are shown in Figure 9(a)-(c) when Re is 25,000, and the inlet temperature is 550 K. From Figure 9(a) to (c), the average temperature of the fluid in the tube is: 562.8 °K, 554.9 °K, 554.7 °K. Figure 9(a)-(c) show that one of the most significant advantages of sodium over S-S and Hitec is that the temperature distribution is much more uniform on both the inner and outer surface of the tube.
Obviously, for the NHF model, there is a heat spot on the outer wall from 0° to 180°, especially at 90°.At the same time, the UHF model is idealistic.But, the highest heat spot temperature of sodium is the lowest, while S-S and Hitec are same.There is a considerable temperature gradient when the sodium flows in the receiver tube.Besides, not far from the entrance, the inner fluid temperature is higher and more uniform than S-S or Hitec flow in the tube.Obviously, for the NHF model, there is a heat spot on the outer wall from 0 • to 180 • , especially at 90 • .At the same time, the UHF model is idealistic.But, the highest heat spot temperature of sodium is the lowest, while S-S and Hitec are same.There is a considerable temperature gradient when the sodium flows in the receiver tube.Besides, not far from the entrance, the inner fluid temperature is higher and more uniform than S-S or Hitec flow in the tube.Corresponding velocity distribution is shown in Figure 9e,f.The velocity of sodium is the lowest, only 0.19 times the other two mediums.At a distance of 0.2 m from the inlet, the sodium flow reached fully developed, and the boundary layer is thinner, while the other boundary conditions remained unchanged.It can be explained that the molecule heat conduction of sodium dominates in fully developed turbulent heat transfer.The heat diffusion of the medium with lower Prandtl number (sodium) is much higher than the momentum transfer diffusivity.remained unchanged.It can be explained that the molecule heat conduction of sodium dominates in fully developed turbulent heat transfer.The heat diffusion of the medium with lower Prandtl number (sodium) is much higher than the momentum transfer diffusivity.The average heat-transfer coefficient of all working fluids increases with the mass flow.When the Qm increases by 0.3 kg/s, the h of S-S, Hitec and sodium increases by an average of 1303, 1355, 2775 W/m 2 ･°K, respectively.The variation of Qm has a significant influence on the heat-transfer characteristic of the HTFs, especially on sodium.For sodium, the higher the Qm, the better the heat-transfer coefficient.The reason is that the average temperature difference is getting smaller increasingly with the rise of Qm when the maximum heat flux is 0.1 MW/m 2 .However, the law of change in the average friction coefficient is reversed, all decreasing as the flow rate increases.When the Qm is higher than 0.8 kg/s, the friction coefficient decreases slowly.S-S and Hitec, with high heat-transfer rate and low-pressure drop, are exceptionally beneficial for heat transfer.The results suggest that the Qm should be higher than 0.8 kg/s to obtain greater heat-transfer performance.However, a larger mass-flow rate represents more considerable pump pressure and more substantial heat absorption capability.Besides, without increasing the solar thermal input, it is possible to cause the reduction of fluid temperature and increase the corresponding heat loss.

Effects of Inlet Temperature
The physical properties of the HTF are primarily affected by the inlet temperature (Tin).The typical operating temperatures for the three HTFs are shown in Table 3.The available temperature range is between the melting point and the boiling point.The trends of h, f for the three HTFs have been shown in Figure 11.Tin is varying from 550 °K to 800 °K, Qm is 1.0 kg/s, and qomax is 0.1 MW/m 2 .The average heat-transfer coefficient of all working fluids increases with the mass flow.When the Q m increases by 0.3 kg/s, the h of S-S, Hitec and sodium increases by an average of 1303, 1355, 2775 W/m 2 • • K, respectively.The variation of Q m has a significant influence on the heat-transfer characteristic of the HTFs, especially on sodium.For sodium, the higher the Q m , the better the heat-transfer coefficient.The reason is that the average temperature difference is getting smaller increasingly with the rise of Q m when the maximum heat flux is 0.1 MW/m 2 .However, the law of change in the average friction coefficient is reversed, all decreasing as the flow rate increases.When the Q m is higher than 0.8 kg/s, the friction coefficient decreases slowly.S-S and Hitec, with high heat-transfer rate and low-pressure drop, are exceptionally beneficial for heat transfer.The results suggest that the Q m should be higher than 0.8 kg/s to obtain greater heat-transfer performance.However, a larger mass-flow rate represents more considerable pump pressure and more substantial heat absorption capability.Besides, without increasing the solar thermal input, it is possible to cause the reduction of fluid temperature and increase the corresponding heat loss.

Effects of Inlet Temperature
The physical properties of the HTF are primarily affected by the inlet temperature (T in ).The typical operating temperatures for the three HTFs are shown in Table 3.The available temperature range is between the melting point and the boiling point.The trends of h, f for the three HTFs have been shown in Figure 11.T in is varying from 550 • K to 800 • K, Q m is 1.0 kg/s, and q omax is 0.1 MW/m 2 .
From Figure 11a, the heat-transfer coefficient of sodium is larger by about 2.5 to 5 times than the other two kinds of molten salt and decreases slowly with the temperature rising.The reason for the above unique heat-transfer feature of sodium is its high conductivity, and it decreases with increasing temperature even in turbulent flow.As shown in Figure 11c, the conductivity of sodium is 118, 154 times Hitec and S-S at 550 • K, respectively.Furthermore, the h of Hitec or S-S increases linearly with the increase of temperature whereas sodium is the opposite.The h of Hitec has not changed significantly between 750 • K and 800 • K because its boiling point is 808 • K.The greater the h, the smaller the f when the mass-flow rate is constant for the three HTFs.Besides, from Table 3, the energy-storage capacity (ρC p ) of Hitec or S-S is two times more than the sodium.11(a), the heat-transfer coefficient of sodium is larger by about 2.5 to 5 times than the other two kinds of molten salt and decreases slowly with the temperature rising.The reason for the above unique heat-transfer feature of sodium is its high conductivity, and it decreases with increasing temperature even in turbulent flow.As shown in Figure 11(c), the conductivity of sodium is 118, 154 times Hitec and S-S at 550 °K, respectively.Furthermore, the h of Hitec or S-S increases linearly with the increase of temperature whereas sodium is the opposite.The h of Hitec has not changed significantly between 750 °K and 800 °K because its boiling point is 808 °K.The greater the h, the smaller the f when the mass-flow rate is constant for the three HTFs.Besides, from Table 3, the energy-storage capacity (ρCp) of Hitec or S-S is two times more than the sodium.
Based on the above conclusions, we can select sodium as the heat-transfer medium in the collector tube on the top of the SPT, while the Hitec or S-S as the circulating medium or storage medium, respectively.

Effects of Heat Flux on the Outside Surface
Generally, the heat flux on the outside surfaces of the receiver tubes in SPT plants is quite limited to give a reasonable lifetime.The maximum heat flux allowed (allowable flux density), qomax, has progressively grown for many years, such as 0.35 MW/m 2 in the Solar One plant and 0.8 MW/m 2 in the Solar Two plant.Besides, the qomax is a crucial parameter for the receiver design, since it is directly related to the heliostat field cost, which involves the number of heliostats and the arrangement strategy [38,39].Based on the above conclusions, we can select sodium as the heat-transfer medium in the collector tube on the top of the SPT, while the Hitec or S-S as the circulating medium or storage medium, respectively.

Effects of Heat Flux on the Outside Surface
Generally, the heat flux on the outside surfaces of the receiver tubes in SPT plants is quite limited to give a reasonable lifetime.The maximum heat flux allowed (allowable flux density), q omax , has progressively grown for many years, such as 0.35 MW/m 2 in the Solar One plant and 0.8 MW/m 2 in the Solar Two plant.Besides, the q omax is a crucial parameter for the receiver design, since it is directly related to the heliostat field cost, which involves the number of heliostats and the arrangement strategy [38,39].
Figure 12a,b show the variations of h, f of the three HTFs when q omax varies from 0.1 to 0.3 MW/m 2 .The values of mass flow rate and inlet temperature are set to 1.0 kg/s and 550 • K, respectively.The increased maximum heat flux does not affect the average heat-transfer coefficient of HTFs in the tube, and the h of sodium is invariably 12.56 times that of Hitec and 13.8 times that of S-S.At the same time, the flow of liquid sodium in the tube maintains a small friction coefficient compared to the other two working fluids (Figure 12b).Therefore, liquid sodium ensures economics while maintaining efficient system operation.
Energies 2019, 12, x FOR PEER REVIEW 13 of 16 Figure 12(a) and (b) show the variations of h, f of the three HTFs when qomax varies from 0.1 to 0.3 MW/m 2 .The values of mass flow rate and inlet temperature are set to 1.0 kg/s and 550 °K, respectively.The increased maximum heat flux does not affect the average heat-transfer coefficient of HTFs in the tube, and the h of sodium is invariably 12.56 times that of Hitec and 13.8 times that of S-S.At the same time, the flow of liquid sodium in the tube maintains a small friction coefficient compared to the other two working fluids (Figure 12(b)).Therefore, liquid sodium ensures economics while maintaining efficient system operation.
As shown in Figure 12(c) and (d), when the maximum heat flux changed from 0.1 to 0.3 MW/m 2 , the temperature of the outer wall increased from 554.53 °K to 563.61 °K while the average temperature of the circumferential increased by about 2 °K.At the same time, the temperature difference between the inner and outer walls is smaller than 4 °K in both cases.As a result, the thermal stress acting on the heat-transfer tubes is greatly reduced.

Conclusions
The heat-transfer characteristic of liquid sodium is compared with solar salt and Hitec in a heat-transfer tube for the SPT system.The calculations present the effects of operational parameters (Qm, Tin, qomax) on the thermal and thermo-hydraulic performance of three mediums under a nonuniform heat flux.The following conclusions are obtained: (a) The tube wall temperature difference of sodium is 20 °K, which is 4 °K lower than other the two mediums when the Re is 25,000, and the inlet temperature is 550 °K.In either NHF model or UHF model, the temperature distribution of Sodium is more uniform and higher than S-S and Hitec, and the highest temperature on the hot spot is 4 °K lower than S-S or Hitec.
(b) Among these factors (Qm, Tin, qomax), the change of Qm has an essential influence on the heat-transfer coefficient of HTFs.However, even in high turbulence, the thermal conductivity of Sodium still plays a crucial role.Meanwhile, the change of qomax does not affect the average heat-transfer performance of the heat-transfer medium, but it has a significant influence on the Effects of q omax : (a) q omax -h, (b) q omax -f, (c,d) q omax -T.
As shown in Figure 12c,d, when the maximum heat flux changed from 0.1 to 0.3 MW/m 2 , the temperature of the outer wall increased from 554.53 • K to 563.61 • K while the average temperature of the circumferential increased by about 2 • K.At the same time, the temperature difference between the inner and outer walls is smaller than 4 • K in both cases.As a result, the thermal stress acting on the heat-transfer tubes is greatly reduced.

Conclusions
The heat-transfer characteristic of liquid sodium is compared with solar salt and Hitec in a heat-transfer tube for the SPT system.The calculations present the effects of operational parameters (Q m , T in , q omax ) on the thermal and thermo-hydraulic performance of three mediums under a nonuniform heat flux.The following conclusions are obtained: (a) The tube wall temperature difference of sodium is 20 • K, which is 4 • K lower than other the two mediums when the Re is 25,000, and the inlet temperature is 550 • K.In either NHF model or UHF model, the temperature distribution of Sodium is more uniform and higher than S-S and Hitec, and the highest temperature on the hot spot is 4 • K lower than S-S or Hitec.
(b) Among these factors (Q m , T in , q omax ), the change of Q m has an essential influence on the heat-transfer coefficient of HTFs.However, even in high turbulence, the thermal conductivity of Sodium still plays a crucial role.Meanwhile, the change of q omax does not affect the average heat-transfer performance of the heat-transfer medium, but it has a significant influence on the temperature of the tube wall.Loading excessively high heat flux on the collector tube may cause many security issues in SPT system.
(c) Under the same boundary conditions, the heat-transfer performance of Sodium is one order of magnitude higher than that of S-S and Hitec at low temperature (550 • K~700 • K), and it is twice than the two kinds of molten salts at high temperature (700 • K +).

Figure 1 .
Figure 1.(a) Solar power tower system; (b) Exposed cylindrical heat receiver and a single receiver tube

Figure 2 .
Figure 2. Schematic diagram of tower solar collector.

Figure 1 .
Figure 1.(a) Solar power tower system; (b) Exposed cylindrical heat receiver and a single receiver tube

Figure 1 .
Figure 1.(a) Solar power tower system; (b) Exposed cylindrical heat receiver and a single receiver tube

Figure 2 .
Figure 2. Schematic diagram of tower solar collector.

Figure 2 .
Figure 2. Schematic diagram of tower solar collector.

Figure 6 .
Figure 6.Circumferential variation of heat flux on the inner wall of the tube.The heat flux is non-dimensional with the maximum heat flux qomax at the same WTR and Re.

Figure 6 .
Figure 6.Circumferential variation of heat flux on the inner wall of the tube.The heat flux is non-dimensional with the maximum heat flux qomax at the same WTR and Re.

Figure 6 .
Figure 6.Circumferential variation of heat flux on the inner wall of the tube.The heat flux is non-dimensional with the maximum heat flux q omax at the same WTR and Re.

Figure 7 .
Figure 7. Temperature distribution on tube wall of sodium, S-S, and Hitec.

Figure 7 .
Figure 7. Temperature distribution on tube wall of sodium, S-S, and Hitec.

Figure 8 .
Figure 8. (a-c) Circumferential variation of the inner wall fluid temperature with the qualitative temperature for different Reynolds number at a fixed-inlet temperature (T in = 550 • K).(a) Pr = 0.001 (b) Pr = 13 (c) Pr = 50.Solid lines: the present calculations; Dashed lines: homogeneous heat-flux model (HHFM).

Figure 9 .
Figure 9. (a-c) Temperature distribution of three kinds of media under two models and (d-e) velocity distribution under NHF model.

Figure 9 .
Figure 9. (a-c) Temperature distribution of three kinds of media under two models and (d,e) velocity distribution under NHF model.

Figure 10
Figure10shows the dependence of h, f of the three HTFs on Q m , which varies from 0.2 to 1.4 kg/s.The values of maximum heat flux and inlet temperature are set to 0.1 MW/m 2 and 550 • K, respectively.

Figure 10 showsFigure 10 .
Figure10shows the dependence of h, f of the three HTFs on Qm, which varies from 0.2 to 1.4 kg/s.The values of maximum heat flux and inlet temperature are set to 0.1 MW/m 2 and 550 °K, respectively.

Figure 10 .
Figure 10.Effects of Q m on the heat-transfer performance of the HTFs in the tube: (a) Q m -h, (b) Q m -f -∆p.

Table 2 .
The calculation parameters.

Table 2 .
The calculation parameters.

Table 2 .
The calculation parameters.