Natural Convection Heat Transfer Experiments in Fluoride Salt

Effective high-temperature thermal energy exchange and delivery at temperatures above 600 °C have the potential for significant national impact by reducing both capital and operating costs of energy conversion and transport systems. Today, there are no standard commercially available high-performance heat transfer fluids effective above 600 °C. High pressures associated with water and gaseous coolants (such as helium) at elevated temperatures impose limiting design conditions for the materials in most energy systems, while alkali metal coolants are very reactive when exposed to air or water. Liquid salts offer high-temperature capabilities at low vapor pressures, good heat transport properties, low reactivity with air and water, and reasonable costs. They are, therefore, leading candidate fluids for next-generation energy production and transport.

Day [5] reviews available heat transfer correlations for laminar flow over isothermal vertical cylinders and uses computational fluid dynamics to model a vertical cylinder in a fluid with Pr = 0.7. Day parametrically varied cylinder aspect ratios (L/D) from 1 to 10 and found that classical heat transfer solutions do not always agree with his numerical results. He also notes that conventional criteria used to determine when a flat plate solution could be used for estimating natural convection heat transfer over a vertical cylinder were not always sufficient, observing that as cylinder aspect ratios increase, the critical Rayleigh number above which a flat plate solution can be used also increases.

Godaux and Gebhart [8] experimentally studied the transition between laminar and turbulent natural convection over a heated vertical stainless steel foil in water. They identified a flow or velocity transition that influenced the thermal transition, although it occurred before the transition. They determined that the thermal transition is not well correlated with the Grashof number for water where Pr ∼7, and they suggest that the total amount of convected thermal energy would be a better predictor of thermal transition. However, no specific methodology for predicting thermal transition was recommended.

Vliet and Liu [6] performed experiments under uniform surface heat flux conditions using water and suggested that transition between laminar and turbulent flow started between $Raz*$ = 3 × 10¹² and 4 × 10¹³, with fully developed turbulent flow occurring at $Raz*$ = 10¹⁴.

Experiments using water, spindle oil, and Mobiltherm oil conducted by Fujii et al. [9] showed transition for uniform surface heat flux experiments starting at $Raz*$ = 0.2 × 10¹³ to 2.5 × 10¹³ and ending at $Raz*$= 1 × 10¹³ to 5 × 10¹³. This observation is very similar to the ranges noted by Vliet.

For turbulent flow, 10⁹ < Gr_L Pr < 10¹² they also recommend the correlation of McAdams given in Eq. (14).

For turbulent flow, Fujii et al. suggest several potential correlations with the same viscosity corrections [9].

A variety of other correlations have also been proposed for isothermal and constant heat flux conditions [1–4].

Only a limited amount of heat transfer data is available for liquid fluoride salt systems. Less than ten sets of experiments are available that characterize fluoride salt heat transfer, and the quality of the data and reported experimental detail vary greatly. The data were taken in the 1950 s through the 1970 s. Ambrosek et al. [16] present a summary of available FLiNaK forced convection heat transfer data. Ambrosek identified four data sources and made an effort to assimilate the data to develop generalized conclusions on FLiNaK heat transfer. A more recent assessment of heat transfer was made with a variety of fluoride salts and is presented by Yoder [17], who shows that in general, fluoride salt heat transfer could be predicted using conventional heat transfer correlations, but the uncertainties are relatively large. A more general review of the state of technology for molten salt coolants, including heat transfer, is presented by Holcomb and Cetiner [18]. Natural convection heat transfer data for fluoride salts are lacking and will be needed to validate thermal designs.

The experiment uses approximately 3.5 L of FLiNaK salt (46.5% LiF, 11.5% NaF, 42% KF). A three-dimensional schematic of the experimental cell is shown in Fig. 1, which depicts the cell inserted into an electrically heated furnace used to maintain cell temperatures. The cell consists of a nickel crucible that holds the liquid salt. Tabs located at the top and bottom of the crucible's outer surface are included to center the crucible inside a stainless steel vessel, which maintains the correct inert atmosphere over the salt pool. An argon cover gas is used over the salt, and a small argon flow rate is maintained to continuously sweep that area. A 1.27 cm diameter heater located in the center of the cell is used to induce natural circulation in the liquid salt. Four thermocouples located inside the heater system are used to measure heater surface temperature during operation. A three-junction thermocouple probe provides salt temperature measurements at three axial locations within the cell. The heater and thermocouple probe were fabricated by Delta-M Corporation (Oak Ridge, TN) [19] using nickel as the outer sheath for both the thermocouple probe and the heater. Temperature measurement errors for the heater thermocouples and the salt temperature probe were estimated from the thermocouple calibration curves to be 0.55 °C (a 1σ value). The heater is powered using a BK Precision VSP-12010, 0-120V, 0-10A DC power supply. Error in the measured power was less than 0.25%.

The upper flange is made of stainless steel. The two large ports visible in the rendering (Fig. 1) allow viewing of the salt during operation through sapphire windows. Argon over pressure in the cell is maintained at a few pounds per square inch to ensure that no air could ever enter the cell. A picture showing the assembly and the insulated furnace is provided in Fig. 2.

The dimensions of the cell are shown in Fig. 3. The nickel crucible has a 12.4 cm inside diameter and is 30.5 cm in height. During an experiment, the salt level is approximately 2.5 cm below the top of the crucible. The furnace surrounding the outer vessel was manufactured by Watlow [20] and has three vertical heating zones that allow higher heating rates at the top of the crucible to prevent salt from deforming the crucible during the melting process. The furnace heaters are controlled using Omega proportional–integral–derivative power controllers. The outer surface of the furnace is insulated with Fiberfrax insulation. Approximately 5 cm of insulation is extended to cover the top flange of the vessel. The vessel and the furnace sit on firebrick to provide insulation and to position them vertically. A more detailed discussion of the test cell and a description of other testing that was performed in the cell can be found in Ref. [21].

Tests were run with heater power and salt temperatures parametrically varied to determine the natural circulation heat transfer coefficients at the surface of the heater. The temperature range was chosen (∼560 °C to 640 °C) to maintain sufficient margin above the freeze point of FLiNaK salt (454 °C) and remain below operating limits of existing materials compatible with the salt. Because there were heater failures during testing, three heaters were used for this experiment. Failures occurred at weld joints due to incorrect weld filler material being used on the first two heaters. Typical thermocouple locations for the thermocouple probe and the three heaters used in testing are shown in Figs. 4 and 5(a)–5(c). The thermocouple probe (see Fig. 4) was installed midway between the outside diameter of the heater element and the nickel liner inner wall (∼3.1 cm from the heater centerline), and it remained in the same location for the duration of testing.

To perform these tests, a steady overall salt temperature was established using the furnace system. The heater element power was then set at the desired level. The cell was allowed to stabilize until steady salt and heater temperatures were achieved. Steady-state operation was assumed to have been achieved when thermocouple temperatures showed less than a 1/2 degree variation over approximately 1/2 h. Both heater and salt temperature data were then taken, and the heater voltage and current were recorded.

Temperature and heater power data are shown in Table 1 for 11 separate tests, which represent data from all three heaters. Salt temperatures were taken using the thermocouple array, with thermocouple locations as indicated in Fig. 4. Heater thermocouple locations depend on the specific heater used. A thermocouple was also placed in the argon cover gas above the salt melt, and another thermocouple was placed in the argon-filled space between the nickel crucible and stainless steel outer vessel (vertical centerline of the melt). These measurements are also shown in Table 1.

A radiation correction q″_rad was calculated assuming gray body radiation between the heater element and crucible. The crucible was assumed to be at the measured salt temperature. Ni emissivity was assumed to be 0.1 [22], and the FLiNaK was assumed to be transparent. The transparent assumption is an open question with fluoride salts, as very little absorptivity measurements have been taken using these salts. The radiation correction was less than 5% for all data points. In calculating the uncertainty in measured heat transfer coefficients, a 100% uncertainty was assumed in the radiation correction term.

In calculating T_b as used in the heat transfer coefficient, the local measured salt temperature using the thermocouple probe (Fig. 4) was corrected to account for the elevation differences between the thermocouple probe and heater thermocouple locations. Measured probe temperatures were interpolated linearly between measurement elevations and the elevation of the heater thermocouples (for the bottom heater thermocouple, the salt thermocouple probe temperatures were linearly extrapolated). The maximum salt temperature correction using this procedure was 2.1 °C, and the average was 0.35 °C. The heater thermocouples were located directly inside the nickel heater sheath. To determine the temperature at the heater outer surface, a conduction calculation was performed to determine the temperature drop across the heater sheath using the measured inner sheath temperature and assuming a uniform heat flux over the heated length. The calculated temperature drop across the sheath for all heater powers was less than 1.1 °C.

In examining Tables 1 and 2, stratification in the cell salt temperatures can be seen in some of the tests. Test 6 has the highest heater power of all of the tests, and it shows a salt temperature increase of about 17 °C from the bottom to the top of the cell as measured by the salt thermocouple probe. For the other tests, the maximum salt temperature difference from bottom to top was less than 10 °C, with the salt temperature difference generally increasing with heater power. Although not always explicitly stated, many natural convection correlations assume a uniform fluid temperature, T_b. Since the salt in these experiments is not always uniform from the top to the bottom of the cell, the experiment may not exactly match the assumptions built into the development of the correlations. This may cause additional discrepancy when comparing the correlations to the data, but in practical application of the correlations, this condition is often not exactly met anyway, so comparisons shown here should indicate how well these correlations might be expected to perform in a fluoride salt engineering application.

Using the criteria proposed by Popiel [1] in Eq. (1), the L/D of the experimental heater design is such that heater curvature is important and needs to be taken into consideration. Because only one heater diameter was used in these tests, however, no conclusions regarding the impact of heater curvature could be made based on the data presented here. Boundary layer thicknesses in these experiments should be less than 5 mm in all tests based on laminar flat plate assumptions, and the cell walls should therefore have no influence on boundary layer development.

Several correlations discussed previously are compared to the present data in Fig. 7 (local formulations). For these comparisons, salt properties were calculated from the correlations given by Richard et al. [23] (the correlation for β, the thermal coefficient of volume expansion was taken from Davis [24]). Typical properties for FLiNaK at 700 °C are given in Table 3 as well as those for water, sodium, and helium for comparison. Although the heater in the experiment should provide a near-uniform heat flux surface, the correlations shown in Fig. 7 are predominantly developed for isothermal surfaces and are therefore plotted as a function of Ra_z. This comparison was made because in many cases, the exact surface condition being analyzed is either unknown or is somewhere between isothermal and uniform heat flux conditions; understanding how well these correlations perform under either set of conditions is important to the designer.

and is one indication of how well the correlations fit the experimental data (note that R² can become negative with this definition).

The turbulent flow correlation of McAdams, which is recommended by Popiel, as well as Rohsenow and Choi, is also plotted. As shown in the figure, this correlation tends to overpredict the data above Ra = 10⁹, while the Fujii et al. correlation tends to underpredict measured Nusselt numbers for Ra_z > 10⁹. The data point that appears to be an outlier at Ra_z ∼2.5 × 10⁹ is data from the top thermocouple of heater number 1. Because no obvious experimental reason could be identified that would indicate this thermocouple was giving an erroneous reading, this data point was included in the correlation evaluations.

Figure 8 compares the data to correlations developed for uniform heat flux conditions. Since Ra^*_z was less than 1.1 × 10¹² for all of the data, only laminar flow correlations are compared in Fig. 8. The correlations of Vliet and Liu [6] and Fujii et al. [9] also do a good job of predicting the FLiNaK data. As indicated in Table 4, the average percent errors for these correlations are 10.1% for the Vliet correlation and 7.4% for the Fugi et al. correlation. The correlation of Jarall and Campo [15] developed from experiments in air shows an average error of 17.4% and tends to consistently overpredict experimental Nusselt numbers. The fluoride salts have significantly higher Pr than air. In the testing discussed here, the Pr ranged from ∼9 to 13 (compared to Pr < 1 for air) and would have a significant impact on the boundary layer behavior and the applicability of the Jarall and Campo correlation to the present data.

In comparing Figs. 7 and 8, as well as the statistics in Table 4, choosing a correlation developed for isothermal surfaces versus a correlation developed for uniform heat flux surfaces only has a minor impact on the fluoride salt data predictions. Prediction variations between correlations developed from the same data type (isothermal or uniform heat flux) are larger than the difference between choosing among “isothermal” or “uniform heat flux” correlations. For instance, the isothermal correlations examined here have a variation of from 9% to 23% average error. The correlations developed for uniform heat flux surfaces have a variation of 7% to 18%, while the best existing correlation developed for isothermal conditions is 9%, and the best existing correlation developed for a uniform heat flux is 7%. For the data presented here, either the “isothermal” or “uniform heat flux” form of the correlations can be used with similar performance expected.

All three of the best estimate correlation fits do a good job of characterizing the experimental data. For all cases, the parameter b lies below the value of 1/3 expected for fully developed turbulent flow. This may indicate that an Ra_z value higher than 10⁹ may be appropriate for turbulent transition in this data, and that a laminar flow treatment is sufficient for the entire data set.

As Figs. 7–9 indicate, the liquid salt data can be predicted well using conventional natural convection correlations developed using other fluids. The near-uniform heat flux data presented here seem to be predicted well using correlations developed for either isothermal or uniform heat flux surfaces. Correlations developed explicitly for the geometry of interest using conventional formulations can improve the errors significantly.

Natural convection heat transfer testing was conducted using a 1.27 cm diameter instrumented heater element in FLiNaK salt. These data were acquired for Rayleigh numbers ranging from 3 × 10⁵ to 8 × 10⁹ and salt temperatures from 560 °C to 640 °C. Experimental heat transfer coefficients could be well predicted using conventional formulations for vertical cylinders developed using other fluids. Errors using existing correlations at higher Ra (>10⁹ - turbulent flow) were significantly larger than those in the laminar region. Average correlation errors as high as ∼20% were seen in the turbulent region, and correlation errors as low as ∼10% were seen in the laminar region. This is likely due to the lack of data for turbulent flow natural convection rather than the fact that the fluid tested here is a fluoride salt. When correlations were fitted to data from this experiment, average errors were reduced by a factor of about two.

Conventional natural convection correlations appear to be sufficient for predicting fluoride salt heat transfer, but users should expect errors of up to ∼20% when they are used.

• U.S. Department of Energy (Contract No. DE-AC05-00OR22725).

The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.²

A_s =

heater surface area (based on heated length of heater)

C_p =

salt specific heat

D =

heater outer diameter

g =

acceleration of gravity (9.8 m/s²)

Gr_L =

Grashof number (isothermal surface) (gρ²βL³(T_w−T_b)/μ²)

Gr_z =

Grashof number (isothermal surface) (gρ²βz³(T_w−T_b)/μ²)

$Grz*$ =

Grashof number (constant heat flux surface) (gρ²βz⁴q″/ μ²k = Gr_z × Nu_z)

h =

heat transfer coefficient

i =

current

k =

salt thermal conductivity

l =

liquid

L =

heater length

Nu_L =

average Nusselt number (h_nc L/k)

Nu_z =

local Nusselt number (h_nc z/k)

P =

heater power= V × i

Pr =

Prandtl number (μ C_p/k)

$q″$ =

average heater heat flux: P/A_s

$qrad″$ =

radiation correction

R² =

coefficient of determination

Ra_L =

Rayleigh number (Gr_L Pr)

Ra_z =

Rayleigh number (Gr_z Pr)

$Raz*$ =

Rayleigh number (⁠$Grz*$ Pr)

SSE =

sum of squares of error (Σ [Nu_ex–Nu_pr]²)

$SSTc$ =

sum of squares total referenced to mean (Σ[Nu_ex –Nu_m]²)

T =

temperature

v =

vapor

V =

voltage

z =

distance from bottom of heater element

β =

salt thermal coefficient of volume expansion

μ =

salt dynamic viscosity

ρ =

salt density

σ =

error

Subscripts

b =

at bulk salt temperature

cr =

critical

ex =

experimental

FP =

flat plate

L =

average over length L

L_h =

heated length

m =

mean of experimental

nc =

natural circulation

pr =

predicted

w =

at wall temperature

z =

local