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Research Papers: Natural and Mixed Convection

Natural Convection Heat Transfer Experiments in Fluoride Salt PUBLIC ACCESS

[+] Author and Article Information
Graydon L. Yoder, Jr.

Oak Ridge National Laboratory,
Oak Ridge, TN 37831
e-mail: yodergljr@ornl.gov

Dennis Heatherly

Oak Ridge National Laboratory,
Oak Ridge, TN 37831

Mario Caja

Electrochemical Systems, Inc.,
Oak Ridge, TN 37830

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 1, 2016; final manuscript received August 3, 2017; published online December 12, 2017. Assoc. Editor: Ali Khounsary.

J. Heat Transfer 140(4), 042501 (Dec 12, 2017) (9 pages) Paper No: HT-16-1549; doi: 10.1115/1.4038132 History: Received September 01, 2016; Revised August 03, 2017

A liquid fluoride salt experiment has been constructed and used to acquire natural convection heat transfer data. The experiment used FLiNaK salt in a small cell that included a cylindrical electrical heater, 1.27 cm in diameter, oriented vertically in a FLiNaK bath. Thermocouples internal to the heater were used, along with FLiNaK temperature measurements and heater power measurements, to determine natural circulation heat transfer coefficients. These data were acquired for Rayleigh numbers ranging from 3 × 105 to 8 × 109 and salt temperatures from 560 °C to 640 °C. Test results show that measured heat transfer coefficients are consistent with conventional natural convection heat transfer coefficients for cylinders, but the average error can be as high as 20% using these correlations. Correlations developed by fitting the data for this experiment show much lower errors (<10%).

FIGURES IN THIS ARTICLE
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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.

An extensive amount of work has been conducted to examine natural convection heat transfer and fluid dynamics around a heated vertical cylinder. Experimental and analytical methods have been applied to characterize natural convection behavior around slender vertical cylinders (boundary layer thicknesses are significant compared to the cylinder's diameter) and large vertical cylinders (boundary layer thicknesses are small compared to the cylinder's diameter). For the large vertical cylinders, vertical flat plate correlations are often used. Several review articles focused on natural convection over vertical cylinders have been published over the years [14]. Popiel [1] presented a review of natural convection heat transfer over slender cylinders and proposes a criterion [2] for the conditions under which the presence of cylinder curvature cannot be neglected Display Formula

(1)GrL0.25D/L11.474+48.92/Pr0.006085/Pr2

Others propose alternate criteria, such as that of Sparrow and Gregg [3]: Display Formula

(2)GrL0.25D/L35

Cebeci [4] proposes this same criterion (Eq. (2)) in calculating average heat transfer for fluids with Pr < 0.72, and Display Formula

(3)GrL0.25D/L25.1
for fluids with Pr < 6.

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.

A critical Rayleigh number is also traditionally used as the criterion for transition from laminar flow over a vertical cylinder to turbulent flow. For example, Vliet and Liu [6] suggest that the transition from laminar to turbulent flows for constant wall temperature conditions starts at Raz between 2.8 × 108 and 4 × 109 and end at Raz = 1010. Some sources such as Rohsenow and Choi [7] have suggested the use of Raz = 109 as a transition point. Popiel [1] recommends a critical Grashof criterion to determine this transition for vertical flat plates Display Formula

(4)Grcr109

Popiel's study investigating multiple data sets with a wide Pr range indicated that the critical Ra increased with increasing Pr. Based on this and other data, he also recommends using Display Formula

(5)Grcr=4.×109
for vertical cylinders.

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.

For uniform wall heat flux conditions, a modified Grashof number is typically defined Display Formula

(6)Grz*=gρ2βz4q/μ2k=GrzNuz

and Display Formula

(7)Raz*=Grz*Pr

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.

Multiple correlations have been developed for predicting the heat transfer coefficient under natural circulation conditions over vertical cylinders. Many start with correlations for vertical flat plates and make corrections for cylinder curvature. Simplified analytical treatments for natural circulation over vertical flat plates with uniform temperature show that Display Formula

(8)NuzRaz1/4
for laminar flow, and Display Formula
(9)NuzRaz1/3
for turbulent flow [10,11]. More detailed analysis and evaluation of experimental data generally confirm these dependencies.

For an isothermal vertical flat plate in laminar flow, Popiel [1] recommends using the correlation of Le Fevre [12] for the local Nusselt number Display Formula

(10)Nuz-FP=0.75Grz1/4Pr1/2[4(0.609+1.221Pr1/2+1.238Pr)]1/4

and a correlation by Churchill and Chu [13] for average Nusselt number Display Formula

(11)NuL-FP=[0.825+0.387RaL1/6/(1+(0.492/Pr)9/16)8/27]2

For vertical isothermal cylinders in laminar flow, he recommends the following correlation for local Nu with Pr = 6: Display Formula

(12)Nuz/Nuz-FP=1+1.043[Grz0.25(z/D)]0.914

and Display Formula

(13)NuL/NuL-FP=1+A[1024GrL0.25(L/D)]B
for 0.01 < Pr < 100, where
A=0.0571322+0.20305Pr0.43

and

B=0.91650.0043Pr0.5+0.01333ln(Pr)+0.0004809/Pr

There are less data available for turbulent flow natural circulation than there are for laminar flow, and less analytical evaluation has been done. Popiel [1] recommends the following simple correlation by McAdams [14] for the average Nusselt number for Display Formula

(14)4×109RaL2.5×1010,NuL=0.13RaL1/3

The local Nuz would then also be Display Formula

(15)Nuz=0.13Raz1/3

Rohsenow and Choi [7] recommend the following for laminar flow: for 104 < GrL Pr < 109Display Formula

(16)NuL=0.56(GrLPr)1/4
the local Nuz would be Display Formula
(17)Nuz=0.42(GrzPr)1/4.

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

Fujii et al. [9] performed a comprehensive series of experiments over a large diameter vertical cylinder cooled by three liquids: water and two oils, each with differing Pr. For an isothermal cylinder, Fujii et al. considered the upper limits for laminar flow to be GrzPr ≤ 1–5 × 1010 and suggest the following correlations for laminar flow over a vertical isothermal cylinder Display Formula

(18)Nuz(μw/μb)0.21=0.45(Raz)1/4
for water, and Display Formula
(19)Nuz(μw/μb)0.21=0.49(Raz)1/4
for the oils.

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

For constant surface heat flux conditions, based on experimental data taken using water on a vertical plate, Vliet and Liu [6] proposed the correlation Display Formula

(20)Nuz=0.568Raz*0.22
for turbulent flow (2 × 1012 < Raz* < 1016), and Display Formula
(21)Nuz=0.6Raz*0.2
for laminar flow (Raz* < 1012).

Fujii et al. [9] also developed correlations appropriate for constant heat flux surfaces using the same apparatus and fluid set as discussed previously. For both spindle oil and Mobiltherm oil under laminar flow conditions, he found Display Formula

(22)Nuz(μw/μb)0.17=0.62Raz*0.2

and Display Formula

(23)Nuz(μw/μb)0.17=0.055Raz*2/7
for turbulent flow. He also observed that at least in the turbulent and transition regions, the equations developed for isothermal wall conditions can generally be applied to the case of uniform heat flux without introducing significant error.

Jarall and Campo [15] performed natural circulation tests using electrically heated vertical tubes in air. Three tube diameters ranging from 1.6 cm to 4.8 cm were tested. These tests provided near uniform heat flux at the tube surface and covered Raz* < 2 × 1012. They were able to correlate all of their data using Display Formula

(24)Nuz=1.285(Raz*z/D)0.165

A variety of other correlations have also been proposed for isothermal and constant heat flux conditions [14].

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.

Experiment Design.

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].

Experimental Results.

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.

Heat Transfer Data.

Table 2 shows heat transfer coefficients on the heater surface calculated from the data. Heater power was calculated from current and voltage readings from the DC power supply. The local natural circulation heat transfer coefficient, hnc, was calculated as follows: Display Formula

(25)hnc=(qqrad)/(TwTb)

The surface heat flux, q, was assumed to be uniform along the length of the heated region and was calculated using the heater power, P, the heater length, Lh, and the heater diameter, D Display Formula

(26)q=P/πDLh

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 Tb 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.

A plot of the heat transfer coefficient versus (Tw-Tb) is shown in Fig. 6. The data are plotted as a function of the heater thermocouple location. For instance, the green squares are from the bottom heater thermocouple and tend to have higher heat transfer coefficients than the remainder of the data. Because the thermal boundary layer grows as z1/4 (for laminar flow) with the boundary layer thickness equal to zero at the beginning of the heated length, the thermal resistance tends to be relatively lower (and heat transfer coefficients relatively higher) at the bottom of the heater. The bottom heater thermocouple data seem to show higher scatter in measured Nuz than the other measurement locations. However, as will be seen in later discussion and data assessment, much of this apparent scatter is removed using conventional correlating parameters. Error bars shown in Fig. 6 represent measurement errors based on thermocouple error, the error in measured heater power, and the error assumed in the radiation term. The error bars are calculated by propagating these errors through Eq. (25) giving Display Formula

(27)[σh/h]2=[σP/P]2+[σqrad/(qqrad)]2+[σTw/(TwTb)]2+[σTb/(TwTb)]2

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, Tb. 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.

Data Assessment and Correlation Evaluation.

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 Raz. 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.

In Fig. 7, for some cases (lower Nu values), the error bars of the data are smaller than the symbol size and cannot be seen in the figure. The laminar and turbulent equations are indicated in the figure as solid lines. In the laminar region—the Popiel correlation for a flat plate, Eq. (10) modified by Eq. (12), as well as the Rohsenow and Choi recommended correlation, Eq. (17), also modified for heater curvature by Eq. (12)—do a reasonable job of predicting the data below Raz = 109 (assumed here to be the turbulent transition). The laminar flow correlation of Fujii et al., Eq. (19), is also shown in the figure. Since Fujii et al. recommend using this equation below Raz ≅ 1010, it is shown for all Raz values plotted. The salt data compare very favorably to these traditional laminar flow correlations. This is confirmed by the high coefficient of determination, R2, for these equations shown in Table 4. R2 is defined as Display Formula

(28)R2=1SSE/SSTc

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 Raz > 109. The data point that appears to be an outlier at Raz ∼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.

Best estimate fits to the data of the form Display Formula

(29)NuzARazb
are shown in Fig. 9. Best estimate correlations fitting all of the data, fitting data with Raz < 109 (here assumed to be laminar flow) and fitting data with Raz>109 (assumed to be turbulent flow) are plotted in the figure. The Rohsenow and Choi recommended correlation and the McAdams correlation are shown again in Fig. 9 for reference. The last three rows in Table 4 show fit parameters for the best estimate equations and provide the R2 value for each, as well as the average percent error. The best estimate correlation fits show significantly lower average error than existing correlations. The best estimate laminar correlation gives an average error of 4.9% compared to the Popiel correlation with 9.6%. Similarly, the best estimate turbulent correlation fit has an average error of 4.7% compared to the average error in the McAdams correlation of 22.7%. The larger errors associated with the use of conventional correlations cannot be completely explained. However, the knowledge of thermophysical properties of salts is not complete, and other authors have noted discrepancies between salt heat transfer data and conventional forced convection salt heat transfer correlations that they have attributed at least in part to the lack of consistent physical property data [16,17]. The salt has a relatively high Prandtl number compared to water and air that may not be captured well in some of the correlations, as evidenced by Ref. [9], who developed separate correlations for oils and water while using the same test apparatus to collect data for both. Although there is some discrepancy between the FLiNaK data taken here and conventional Nusselt number correlations, these correlations should serve as a good first estimate to predict natural convection heat transfer in fluoride salts.

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 Raz 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. 79 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

  • As =

    heater surface area (based on heated length of heater)

  • Cp =

    salt specific heat

  • D =

    heater outer diameter

  • g =

    acceleration of gravity (9.8 m/s2)

  • GrL =

    Grashof number (isothermal surface) (gρ2βL3(Tw−Tb)/μ2)

  • Grz =

    Grashof number (isothermal surface) (gρ2βz3(Tw−Tb)/μ2)

  • Grz* =

    Grashof number (constant heat flux surface) (gρ2βz4q″/ μ2k = Grz × Nuz)

  • h =

    heat transfer coefficient

  • i =

    current

  • k =

    salt thermal conductivity

  • l =

    liquid

  • L =

    heater length

  • NuL =

    average Nusselt number (hnc L/k)

  • Nuz =

    local Nusselt number (hnc z/k)

  • P =

    heater power= V × i

  • Pr =

    Prandtl number (μ Cp/k)

  • q =

    average heater heat flux: P/As

  • qrad =

    radiation correction

  • R2 =

    coefficient of determination

  • RaL =

    Rayleigh number (GrL Pr)

  • Raz =

    Rayleigh number (Grz Pr)

  • Raz* =

    Rayleigh number (Grz* Pr)

  • SSE =

    sum of squares of error (Σ [Nuex–Nupr]2)

  • SSTc =

    sum of squares total referenced to mean (Σ[Nuex –Num]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

  • Lh =

    heated length

  • m =

    mean of experimental

  • nc =

    natural circulation

  • pr =

    predicted

  • w =

    at wall temperature

  • z =

    local

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Rohsenow, W. M. , and Choi, H. , 1961, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, NJ.
Godaux, F. , and Gebhart, B. , 1974, “ An Experimental Study of the Transition of Natural Convection Flow Adjacent to a Vertical Surface,” Int. J. Heat Mass Transfer, 17(1), pp. 93–107. [CrossRef]
Fujii, T. , Takeuchi, M. , Fujii, M. , Suzaki, K. , and Uehara, H. , 1970, “ Experiments on Natural-Convection Heat Transfer From the Outer Surface of a Vertical Cylinder to Liquids,” Int. J. Heat Mass Transfer, 13(5), pp. 753–787. [CrossRef]
Bejan, A. , 1984, Convection Heat Transfer, Wiley, New York
White, F. M. , 1991, Viscous Fluid Flow, 2nd ed., McGraw-Hill, New York.
Le Fevre, E. J. , 1956, “ Laminar Free Convection From a Vertical Plane Surface,” Ninth International Congress of Applied Mechanics, Brussels, Belgium, Sept. 5–13, pp. 168–174.
Churchill, S. W. , and Chu, H. H. S. , 1956, “ Correlating Equations for Laminar and Turbulent Free Convection From Vertical Flat Plate,” Int. J. Heat Mass Transfer, 18, pp. 175–183.
McAdams, W. H. , 1954, Heat Transmission, 3rd ed., McGraw-Hill, New York.
Jarall, S. , and Campo, A. , 2005, “ Experimental Study of Natural Convection From Electrically Heated Vertical Cylinders Immersed in Air,” Exp. Heat Transfer, 18(3), pp. 127–134. [CrossRef]
Ambrosek, J. , Anderson, M. , Sridharan , K. , and Allen, T. , 2009, “ Current Status of Knowledge of the Fluoride Salt (FLiNaK) Heat Transfer,” Nucl. Technol., 165(2), pp. 166–173. [CrossRef]
Yoder, G. L. , Jr., 2014, “ Examination of Liquid Fluoride Salt Heat Transfer,” International Congress on Advances in Nuclear Power Plants (ICAPP), Charlotte, NC, Apr. 6–9, Paper No. 14332. https://www.researchgate.net/publication/261759411_Examination_of_Liquid_Fluoride_Salt_Heat_Transfer
Holcomb, D. , and Cetiner, M. , 2010, “ An Overview of Liquid-Fluoride-Salt Heat Transport Systems,” Oak Ridge National Laboratory, Oak Ridge, TN, Report No. ORNL/TM-2010/156. https://info.ornl.gov/sites/publications/Files/Pub25407.pdf
Delta-M Corporation, 2017, “Process Instrumentation for Demanding Applications,” Delta M Corp., Oak Ridge, TN, accessed Nov. 2, 2017, http://www.deltamcorp.com/
Watlow, 2017, “Product Selector,” Watlow, St. Lewis, MO, accessed Nov. 2, 2017, http://www.watlow.com
Yoder, G. L. , Heatherly, D. , Williams, D. , Caja, J. , Caja, M. , Elkassabgi, Y. , Jordan, J. , and Salinas, R. , 2014, “ Liquid Fluoride Salt Experiment Using a Small Natural Circulation Cell,” Oak Ridge National Laboratory, Oak Ridge, TN, Report No. ORNL/TM-2014/56. http://info.ornl.gov/sites/publications/files/Pub40598.pdf
Monarch Instruments, 2008, “ Table of Total Emissivity,” Monarch Instruments, Amherst, NH, accessed Nov. 2, 2017, http://www.monarchinstrument.com/pdfs/TableofEmissivity.pdf
Richard, J. , Wang, D. , Yoder, G. L. , Carbajo, J. , Williams, D. , Forget, B. , and Forsberg, C. , 2014, “ Implementation of Liquid Salt Working Fluids Into TRACE,” International Congress on Advances in Nuclear Power Plants (ICAPP), Charlotte, NC, Apr. 6–9, Paper No. 14214. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.706.975&rep=rep1&type=pdf
Davis, C. B. , 2005, “ Implementation of Molten Salt Properties Into RELAP5-3D/ATHENA,” Idaho National Laboratory, Idaho Falls, ID, Report No. INEEL/EXT-05-02658. https://inldigitallibrary.inl.gov/sites/sti/sti/3028316.pdf
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References

Popiel, C. O. , 2008, “ Free Convection Heat Transfer From Vertical Slender Cylinders: A Review,” Heat Transfer Eng., 29(6), pp. 521–536. [CrossRef]
Popiel, C. O. , Wojtkowiak, J. , and Bober, K. , 2007, “ Laminar Free Convective Heat Transfer From Isothermal Vertical Slender Cylinders,” Exp. Therm. Fluid Sci., 32(2), pp. 607–613. [CrossRef]
Sparrow, E. M. , and Gregg, J. L. , 1956, “ Laminar-Free-Convection Heat Transfer From the Outer Surface of a Vertical Circular Cylinder,” Trans. ASME, 78, pp. 1823–1829.
Cebeci, T. , 1974, “ Laminar-Free-Convective-Heat Transfer From the Outer Surface of a Vertical Slender Circular Cylinder,” Fifth International Heat Transfer Conference, Tokyo, Japan, Sept. 3–7, pp. 15–19. http://adsabs.harvard.edu/abs/1974hetr....3...15C
Day, J. , 2012, “ Laminar Natural Convection From Isotheral Vertical Cylinders,” Master's thesis, University of North Texas, Denton, TX. https://digital.library.unt.edu/ark:/67531/metadc177190/
Vliet, G. C. , and Liu, C. K. , 1969, “ An Experimental Study of Turbulent Natural Convection Boundary Layers,” ASME J. Heat Transfer, 91(4), pp. 517–531. [CrossRef]
Rohsenow, W. M. , and Choi, H. , 1961, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, NJ.
Godaux, F. , and Gebhart, B. , 1974, “ An Experimental Study of the Transition of Natural Convection Flow Adjacent to a Vertical Surface,” Int. J. Heat Mass Transfer, 17(1), pp. 93–107. [CrossRef]
Fujii, T. , Takeuchi, M. , Fujii, M. , Suzaki, K. , and Uehara, H. , 1970, “ Experiments on Natural-Convection Heat Transfer From the Outer Surface of a Vertical Cylinder to Liquids,” Int. J. Heat Mass Transfer, 13(5), pp. 753–787. [CrossRef]
Bejan, A. , 1984, Convection Heat Transfer, Wiley, New York
White, F. M. , 1991, Viscous Fluid Flow, 2nd ed., McGraw-Hill, New York.
Le Fevre, E. J. , 1956, “ Laminar Free Convection From a Vertical Plane Surface,” Ninth International Congress of Applied Mechanics, Brussels, Belgium, Sept. 5–13, pp. 168–174.
Churchill, S. W. , and Chu, H. H. S. , 1956, “ Correlating Equations for Laminar and Turbulent Free Convection From Vertical Flat Plate,” Int. J. Heat Mass Transfer, 18, pp. 175–183.
McAdams, W. H. , 1954, Heat Transmission, 3rd ed., McGraw-Hill, New York.
Jarall, S. , and Campo, A. , 2005, “ Experimental Study of Natural Convection From Electrically Heated Vertical Cylinders Immersed in Air,” Exp. Heat Transfer, 18(3), pp. 127–134. [CrossRef]
Ambrosek, J. , Anderson, M. , Sridharan , K. , and Allen, T. , 2009, “ Current Status of Knowledge of the Fluoride Salt (FLiNaK) Heat Transfer,” Nucl. Technol., 165(2), pp. 166–173. [CrossRef]
Yoder, G. L. , Jr., 2014, “ Examination of Liquid Fluoride Salt Heat Transfer,” International Congress on Advances in Nuclear Power Plants (ICAPP), Charlotte, NC, Apr. 6–9, Paper No. 14332. https://www.researchgate.net/publication/261759411_Examination_of_Liquid_Fluoride_Salt_Heat_Transfer
Holcomb, D. , and Cetiner, M. , 2010, “ An Overview of Liquid-Fluoride-Salt Heat Transport Systems,” Oak Ridge National Laboratory, Oak Ridge, TN, Report No. ORNL/TM-2010/156. https://info.ornl.gov/sites/publications/Files/Pub25407.pdf
Delta-M Corporation, 2017, “Process Instrumentation for Demanding Applications,” Delta M Corp., Oak Ridge, TN, accessed Nov. 2, 2017, http://www.deltamcorp.com/
Watlow, 2017, “Product Selector,” Watlow, St. Lewis, MO, accessed Nov. 2, 2017, http://www.watlow.com
Yoder, G. L. , Heatherly, D. , Williams, D. , Caja, J. , Caja, M. , Elkassabgi, Y. , Jordan, J. , and Salinas, R. , 2014, “ Liquid Fluoride Salt Experiment Using a Small Natural Circulation Cell,” Oak Ridge National Laboratory, Oak Ridge, TN, Report No. ORNL/TM-2014/56. http://info.ornl.gov/sites/publications/files/Pub40598.pdf
Monarch Instruments, 2008, “ Table of Total Emissivity,” Monarch Instruments, Amherst, NH, accessed Nov. 2, 2017, http://www.monarchinstrument.com/pdfs/TableofEmissivity.pdf
Richard, J. , Wang, D. , Yoder, G. L. , Carbajo, J. , Williams, D. , Forget, B. , and Forsberg, C. , 2014, “ Implementation of Liquid Salt Working Fluids Into TRACE,” International Congress on Advances in Nuclear Power Plants (ICAPP), Charlotte, NC, Apr. 6–9, Paper No. 14214. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.706.975&rep=rep1&type=pdf
Davis, C. B. , 2005, “ Implementation of Molten Salt Properties Into RELAP5-3D/ATHENA,” Idaho National Laboratory, Idaho Falls, ID, Report No. INEEL/EXT-05-02658. https://inldigitallibrary.inl.gov/sites/sti/sti/3028316.pdf

Figures

Grahic Jump Location
Fig. 4

Thermocouple probe—thermocouple locations

Grahic Jump Location
Fig. 5

(a) Heater 1 thermocouple locations, (b) heater 2 thermocouple locations, and (c) heater 3 thermocouple locations

Grahic Jump Location
Fig. 1

Schematic of liquid salt cell

Grahic Jump Location
Fig. 2

Assembly inserted in furnace

Grahic Jump Location
Fig. 3

Salt cell dimensions (mm)

Grahic Jump Location
Fig. 6

Heat transfer coefficient data

Grahic Jump Location
Fig. 7

Comparison of FLiNaK data to correlations developed for isothermal surfaces

Grahic Jump Location
Fig. 8

Comparison of FLiNaK data to correlations developed for uniform heat flux surfaces

Grahic Jump Location
Fig. 9

Best estimate data fit

Tables

Table Grahic Jump Location
Table 2 Calculated values of heat transfer coefficients
Table Grahic Jump Location
Table 1 Temperature data from natural circulation testing in a liquid salt cell
Table Footer NoteHeater #1 had a 23.83 cm heated length, Lh.
Table Footer NoteHeater #2 had a 27.64 cm heated length, Lh.
Table Footer NoteHeater #3 had a 21.59 cm heated length, Lh.
Table Footer NoteaHeater #3 had no bottom thermocouple.
Table Grahic Jump Location
Table 3 FLiNaK property comparison
Table Grahic Jump Location
Table 4 Comparison of Nu equations of the form Nuz = A(Raz)b or Nuz= A(Raz*)b
Table Footer NoteaCorrelations developed for uniform heat flux conditions.
Table Footer NoteNote: The McAdams correlation had errors sufficiently large that the R2 value defined using Eq. (27) was negative.

Errata

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