Similitude of forced convection heat transfer of Dowtherm A and of Li2BeF4 molten salt in cylindrical tubes
By
Dajie Sun
A thesis submitted in partial satisfaction of the
requirements for the degree of
Master of Science
in
Nuclear Engineering
in the
Graduate Division
of the
University of California, Berkeley
Committee in charge:
Prof. Per Peterson, Chair Prof. Karl van Bibber Prof. Jasmina Vujic
Summer 2017
The University of California Berkeley Department of Nuclear Engineering
Content Symbols ...... ii Abstract ...... 1 Chapter 1 Introduction ...... 1 Chapter 2 Description of Experiments ...... 4 Chapter 3 Theoretical model for data reduction ...... 7 Chapter 4 Fitting Method ...... 9 Chapter 5 Results ...... 13 Chapter 6 Error and Uncertainty Analysis ...... 17 Chapter 7 Conclusions ...... 20 Acknowledgments ...... 22 References ...... 22 Appendix ...... 24 Appendix A Physical properties of Dowtherm A ...... 24 Appendix B Matlab code to calculate the Nusselt number ...... 26 Appendix C Mathematica code for Power Regression with more than 2 parameters...... 34 Appendix D OriginLab procedure for power regression with less than 3 parameters...... 34
i
The University of California Berkeley Department of Nuclear Engineering
Symbols
NuD Local Nusselt number
NuR Reduced Nusselt numer Pr Prandtl number Re Reynolds number x Position along the pipe L Length of the heating pipe D Inside diameter of the pipe Dynamic viscosity of the fluid
w Dynamic viscosity of the fluid at the wall temperature
Qm Mass flow rate
Pt Total heating power of the test section
Tin Flow temperature at the inlet
Tout Flow temperature at the outlet c p Specific heat of the fluid i Sequence Number of each node (positon of each thermocouple) hi Local heat transfer coefficient of ith section of the pipe (between node i+1 and i) kT fi, Thermal conductivity of the fluid at the temperature of Tfi, Electrical resistance of the pipe
Ti Temperature of the thermocouple at ith node
Tfi, Flow temperature at the position of ith node st li Distance between the 1 node and the ith node.
li Distance between the ith node and the (i+1)th node Uncertainty Gz The Graetz number
ii
The University of California Berkeley Department of Nuclear Engineering
Abstract
Similitude of forced convection heat transfer of Dowtherm A and of Li2BeF4 molten salt in cylindrical tubes by Dajie Sun Master of Science University of California, Berkeley Professor Per Peterson, Chair
Dowtherm A, a heat transfer oil used extensively in industry, has been found to be an attractive simulant fluid to study convective heat transfer phenomena for the high temperature molten salt flibe (Li2BeF4). This thesis analyzes extensive data for forced convection heat transfer in cylindrical channels using the heat transfer oil Dowtherm A oil, comparing this to data collected for flibe at Oak Ridge National Laboratory in the 1970’s. The use of Dowtherm enables reduced temperature, reduced scale experiments, which greatly reduces experimental costs compared to working with actual salts. Forced convection heat transfer data for flibe collected at ORNL is limited to experiments performed above 680°C, with a maximum Prandtl number of 14. Since molten salt reactors can commonly operate at lower temperatures, one is interested in also predicting convective heat transfer at Prandtl number up to 20 and above. This paper presents recent progress in the measurement and correlation of forced heat transfer coefficients of Dowtherm A with Prandtl number up to 53, including correlations for laminar, transition and turbulent flow regimes and for entrance regions.
Data was collected with Dowtherm in a 0.00546 m vertical circular tube for the following range of variables:
Reynolds number 280-200,000 Prandtl number 8.5-53 Flow temperature (°C) 30-170
Within these ranges, the Nusselt number was calculated based upon the experimental data. Correlations of the experimental data resulted in these equations:
0.0816 0.14 1.098 0.33 x x NuReD 0.0025019214 Pr , D w D with an average absolute deviation of 12.7% for 280 0.14 0.823411/3 NuRe 0.0184Pr w with an average absolute deviation of 4.3% for 3,400 1 The University of California Berkeley Department of Nuclear Engineering 0.14 1.027121/3 Nu 0.00195RePr w with an average absolute deviation of 5.2% for 50,000 Keywords: Heat transfer, Dowtherm A, forced convection, similitude 2 The University of California Berkeley Department of Nuclear Engineering Chapter 1 Introduction Because molten salts can transfer heat at low pressure and high temperature, and are chemically stable, molten salts are attractive heat transfer fluids. The molten salt flibe (Li2BeF4) has particularly attractive properties and has been studied extensively for use in molten salt cooled and fueled reactors [1][2]. A key problem for design and licensing is predicting convective heat transfer under natural circulation and forced convection conditions. Bardet and Peterson found that heat transfer oils such as Dowtherm A can serve as effective simulant fluids for the molten salt flibe and other fluoride salts [4]. By selecting appropriate length, velocity, and temperature scaling values it is possible to match the Prandtl (Pr), Froude number (Fr), Reynolds (Re) and Grashof (Gr) numbers of Flibe simultaneously in geometrically scaled experiments operating at low-temperature. This scaling also reduces length and velocity scales of the reactor flow, and thus can be applied to integral effects testing as well as separate effects testing [3]. Rohit Upadhya and Shannon Bragg-Sitton [6] provide a preliminary scaling analysis of a 300MWth FHR-HTSE’s primary loop, conducted an analysis to see whether Dowtherm A would match the Prandtl with a core temperature range of 800℃-850℃. Because results are expected to be used in reactor safety licensing analysis, it is important confirm the similitude between convective heat transport phenomena with flibe, and the simulant Dowtherm A. This masters thesis presents data analysis to compare forced convection heat transfer in cylindrical tubes, comparing data collected from recent experiments performed using Dowtherm A to earlier experiments performed at Oak Ridge National Laboratory with flibe [13]. The Dittus–Boelter (D-B) equation [10] is a popular and preferred version to calculate the heat transfer coefficient: 0.8 n NuD 0.023Re Pr (1) where NuD is the Nusselt number, Re the Reynolds number, and Pr the Prandtl number, and the exponent n takes the value 0.4 for heating (Tw> Tf) and 0.3 for cooling (Tw< Tf). The D-B equation is valid for the range of conditions 0.7Pr160 Re10000 L 10 D Considering the effect of the temperature difference between the pipe wall and the flow, which changes the viscosity, Sieder and Tate[11] recommended: 1 The University of California Berkeley Department of Nuclear Engineering 0.14 0.81/3 NuD 0.027 RePr (2) w 0.7Pr160700 Re10000 L 10 D The Dow Chemical company recommends that the Sieder and Tate equation be used for Dowtherm A [12]. In the 1970s, heat transfer was measured experimentally at Oak Ridge National Laboratory (ORNL) [13] for proposed molten salt reactors. The physical properties of molten salt fuel and Flibe are listed in Table 1 [5]: Table 1 Physical Property of Flibe and Molten Salt for Reactor Fuel Molten salt reactor fuel Flibe Density(g/cm3) 3.3754 2.012 Heat Capacity, Cp (J/(kg·K)) 1360 2386 Thermal Conductivity (W/(m℃)) 0.8 1.0 Dynamical Viscosity,μ(10-3kg/(m·s)) 27.72 17.0 Melt Point(℃) 479 459 Boiling Point(℃) Over 1400 1430 These experiments studied fuel-salt (LiF-BeF2-ThF4-UF4;67.5-20.0-12.0-0.5 mole %) flowing by forced convection through a 0.4572-cm (0.18 inch) inside diameter, horizontal, circular tube, and recommended the following heat transfer correlations: 0.33 0.14 D NuD 1.89 Re Pr, Re 1000 L w 0.14 2/31/3 NuD 0.107 Re 135 Pr, 3500 Re 12000 (3) w 0.14 0.8 1/3 NuD 0.0234 Re Pr, 12000 Re 30600 w 4 Pr 14 400 Re 30600 565CTC f 840 2 The University of California Berkeley Department of Nuclear Engineering The data collected by ORNL only extends up to a Prandtl number equal 14. Since the normal operation cold leg temperature of PB-AHTR is as high as 600°C, we are interested in Prandtl numbers up to 20 and above, shown as Figure 1. Figure 1 Prandtl number matching between flibe and Dowtherm A simulant fluid, in the flibe temperature (This figure is copied from Raluca’s doctoral thesis). Between 2008 and 2014, many experiments were done by R. Scarlat and L. Huddar with Dowtherm A in the Thermal Hydraulics Laboratory at the University of California Berkeley, to measure forced convection heat transfer inside cylindrical tubes. The next chapter describes this experimental work. 3 The University of California Berkeley Department of Nuclear Engineering Chapter 2 Description of Experiments At UC Berkeley two sets of experiments were performed to study convective heat transfer to Dowtherm A in vertical, cylindrical tubes, first using the S-HT2 facility during 2008 to 2009. Later, the original S-HT2 experiment set-up was modified to build the CIET Test Bay, the Test Bay still used the same test section with S-HT2, but replaced the S-HT2 orifice flow meter with a Coriolis meter of higher accuracy of flow rate measurement. The experimental data of laminar flow and transition flow was acquired with S-HT2 facility, while high Reynolds number turbulent flow data was acquired in the CIET Test Bay. The S-HT2 heater test section is shown schematically in Figure 2. It consists of a circular cross-section of stainless steel pipe with electrodes at each end for direct resistance heating, and 22 thermocouples (T1, T2,…T22) distributed uniformly along the pipe’s axis with an interval of 0.1063 m. The setup in Figure 2 serves as the test section of the experiment. The inlet and outlet flow temperature is given by thermocouples T0 and T23 respectively, all the thermocouples have an uncertainty of ±0.5℃ and the wall thermocouples have a location uncertainty of 0.254 cm (0.1 inch). The outlet flow goes through a flow mixer before the thermocouple, so the thermocouple measures the mixed temperature. The inside radius of the heater pipe is 5.46 mm and the outside radius is 6.35 mm. The heater is wrapped by thermal insulation fiberglass with an outside radius of 83.6 mm. The thermal conductivity of stainless steel pipe is k=17 W/(m℃) [10]and the conductivity of thermal insulation fiberglass is k=0.04 W/(m℃) [7]. In S-HT2, the Dowtherm A mass flow rate is measured with an orifice meter equipped with a pressure transducer, is not shown in these figures. The uncertainty of the flow rate is less than 0.5%, in the range of 0.04 to 0.2 kg/s. But in the CIET Test Bay, the orifice meter was replaced with a Coriolis meter, which is more accurate. The CIET Test Bay is shown in Figure 4. The Test Bay used the same heater section as S-HT2, but had a more accurate flow meter (Coriolis flow meter). A liquid mass flow accuracy of 0.10% is common for Coriolis meters, and 0.05% is readily available[9]. For more detailed description of the CIET Test Bay, please see Appendix G of Reference [8]. When electrical current flows through the pipe, heat is generated by the pipe wall and is transferred primarily to the fluid flow, with small losses through the external insulation. The total power input is calculated based on the measured oil flow rate, and the inlet and outlet temperatures T0 and T23, which also equals to the flow enthalpy change per unit time. With the wall temperature given by the thermocouples T1 to T22, the whole temperature profile along the pipe can be computed. Combining the total temperature with the total heat transferred from the pipe wall to the Dowtherm, the heat flux of each segment (the section between two neighboring thermal couples) can be calculated. The heat transfer coefficient is derived by dividing the heat flux from resistance heating, by the temperature difference calculated for the inside surface of the tube wall and for the local oil bulk temperature. The temperature difference between the outside wall and the inside wall of the pipe is calculated assuming uniform resistance heating in the wall of the tube. Also, heat losses along the axis of the thermocouple and effect of the electrical resistivity of steel are all accounted for in the data analysis. 4 The University of California Berkeley Department of Nuclear Engineering Figure 2 Diagram of the experiment Figure 3 Image of the S-HT2 experiment. 5 The University of California Berkeley Department of Nuclear Engineering Figure 4 CIET Test Bay front view (left), and side view showing natural circulation flow path (right). (SolidWorks model by A.J. Gubser [8]). 6 The University of California Berkeley Department of Nuclear Engineering Chapter 3 Theoretical model for data reduction As shown in Figure 2, the total heating power Pt is determined by the enthalpy difference between inlet and outlet flow, Tout Pt Q m c p T dT (4) T in where Qm is the mass flow rate, Tin the flow inlet temperature, Tout the outlet temperature and cp(T) the fluid heat capacity at temperature T. From the Joule–Lenz law, which states that the power of heating generated by an electrical conductor is proportional to the product of its resistance and the square of the current, the heating power of each segment (between two neighboring thermocouples) is equal to: lTii PPit L (5) Tldl 0 where Ti is the electrical resistivity of the ith segment of the pipe with temperature Ti measured by an outer wall thermocouple, Tl the electrical resistivity of the pipe at position l with temperature of Tl , and li the length of the segment. See Figure 5. Figure 5 Diagram of the Algorithm Because the total energy is calculated from the enthalpy added to the oil, it does not include heat losses from the insulation, and thus it is assumed that the power input from the heater wall is proportional to the total heat deposited by electric resistance heating (some of which is lost out of the external insulation). So, the bulk temperature of the oil of each node can be obtained from an energy balance as: 1 l TTP ii (6) f, i 1 f , iQ c T L t m p f, i T l dl 0 7 The University of California Berkeley Department of Nuclear Engineering The heat transfer coefficient hi is calculated as: Pi hi (7) D li T w,, i T f i Where Twi, is the temperature of the inside surface of the tube and D is the inside diameter of the tube. So, the Nusselt number is derived as: hDi Nui (8) kT fi, 8 The University of California Berkeley Department of Nuclear Engineering Chapter 4 Fitting Method We choose to apply the Least Square Method for fitting of the data output. Suppose that y is a function of x, with 12 ,, N as parameters, yxyx ,,,12 N (9) The target function of all data points is defined as Eq.(10) N 2 dyxy12,, Nii (10) i1 where N is the number of data points. If yx i is the expected value and the yi is the observed value, then the following assumptions are required: (1) The observed value follows a normal distribution of the expected value. The variance of the distribution is determined by the uncertainty of the measuring instruments. 2 y xyii 1 2 fydyedy 2 (11) ii2 (2) For each measurement, the uncertainty is the same and equal (3) Each measurement is independent of each other. The probability of getting yyy12,, N at the same time is then given by: N 1 2 NNN y xy 2 ii 1 2 i1 fydyedy iii (12) ii112 N 1 2 N N y xy 2 ii 1 2 i1 pfye i (13) i1 2 The most probable always maximize p in the Eq.(13), which means that the target function, d: N 2 d y xii y i1 (14) should arrive at its minimum with the most probably yx i . Because yx is a function of x with 12,, N as parameters, then d is a function of 9 The University of California Berkeley Department of Nuclear Engineering N 2 ddyxy 1212,,,,, NiNi i1 (15) The algorithm of Least Square Method with purpose of minimizing the target function of Eq.(15) is shown in Eq.(16) dfN 20 yxy ii 11i1 xi N df 20 yxy ii 22i1 (16) xi dfN 20 yxy ii N i1 2 xi From Eq.(16), the unknown parameters 12,, N are determined. Keep in mind that Eq.(16) is a nonlinear transcendental equation, and proximity initial values are needed and crucial to make iterations. 0.14 2 0.33 Nu 1 RePr w (17) By defining the Reduced Nusselt number, the expression of Nusselt number can be rewritten as below to make data fitting much easier: Nu Nu Re2 (18) R 0.33 0.14 1 Pr w Eq.(18) is used in ORNL’s report [13] for power regression for Nusselt number. From the perspective of mathematics, Eq. (18) is not right because it is a nonlinear transformation of Eq.(17) which will change the density function and alter the weight of each data point. However, this equation is kept here for the following reasons: (1) From the perspective of engineering application, the deviation is small and acceptable. (2) It has a concise form and makes computation much easier. (3) It was widely used before, there may be some confusion if we change it. There are a lot of programing languages that are capable for the computation of Eq. (18), such as C/C++, Matlab, Mathematica, etc. but Origin is highly recommended for power regression with less than 3 parameters.1 1 Origin is a proprietary computer program for interactive scientific graphing and data analysis[14]. It is produced by OriginLab Corporation and its curve fitting is performed by the nonlinear least squares fitter which is based on the Levenberg–Marquardt algorithm. 10 The University of California Berkeley Department of Nuclear Engineering For Eq. (15) with more than 2 parameters, Origin is not capable, but it can be fixed using Mathematica or Levenberg–Marquardt algorithm. For more information about the Levenberg– Marquardt algorithm, see Levenberg [15] and Marquardt [16]. As to our experimental data, it was found that Gauss–Newton algorithm sometimes is inefficient to solve Eq. (16) due to its highly dependence on initial values. So, the Levenberg–Marquardt algorithm for Eq.(15) is a better choice. If using Microsoft Excel, the power regression of Eq.(18) can be computed as the following: Plot →Add Trend line (power)→Display Equation on Chart (see Figure 6 and Figure 7). Figure 6 Procedure for power regression using Excel Figure 7 Procedure for power regression using Excel 11 The University of California Berkeley Department of Nuclear Engineering Excel works as following steps: Step 1: take a nature logarithm of Eq.(18) lnlnlnNu ReR 12 (19) Step 2: define: y N uln R (20) x l n R e and bkln ;12, then Eq.(19) can be rewritten as y k x b (21) Step 3: make a linear fitting of Eq.(21) and calculate 12, as eb 1 (22) 2 k Eq.(22) given by Excel is different from Eq.(15) or Eq.(16) because the transformation of Eq.(20) changes the probability density function. The power regression result given by Excel puts more importance on these low y-value data points, which will lead to more uncertainty. So, Excel is not recommended for our data analysis. 12 The University of California Berkeley Department of Nuclear Engineering Chapter 5 Results The equivalent data points of laminar flow (300 Figure 8 Experimental data for laminar flow The fitting line of data in Figure 8 is 0.0816 0.14 1.098 0.33 x NuDx, 0.00250 Re Pr (23) D w 13 The University of California Berkeley Department of Nuclear Engineering 45 Experimental Data Fitting Line 40 35 30 25 Nu-st 20 15 4000 6000 8000 10000 12000 14000 0.14 Re 1/3 Nust Nu Pr w Figure 9 Experimental data for transition flow The transition flow regime data (3800 0.14 0.8231/3 NuRe 0.0184Pr (24) w 14 The University of California Berkeley Department of Nuclear Engineering 600 Experiment Data 500 Fitting Line 400 Nu-st 300 200 0.14 80000 120000 160000 200000 1/3 Re NuNust Pr w Figure 10 Experimental data for high turbulent flow of high Re number The turbulent flow (50000 0.14 1.02712 1/3 Nu 0.00195Re Pr (25) w No data is available for Reynolds numbers between 14,000 and 50,000, but the extrapolation of Equation (24) is recommended for this region. 15 The University of California Berkeley Department of Nuclear Engineering Figure 11 Comparison of the empirical correlations (for laminar flow, let x/D=70). From Figure 11, we can see that in the regions of the transition flow and highly developed turbulence flow, our experiment agrees very well with the correlations developed by Oak Ridge National Laboratory from molten salt experimental data. 16 The University of California Berkeley Department of Nuclear Engineering Chapter 6 Error and Uncertainty Analysis Continued with Chapter 3, the heat transfer coefficient is lTii L Pt P T l dl h i 0 i D l TT iif i , i P L t l T l dl i 0 D lii Tdl 0 Q c T m pf i , TP it Ll i itP D TTi l dldl 00 Q c T m pf i , llii PTTtoutinmQ p c LL l i Pt Plti D liiin TTdl D liiin TT 0 LQ c T l LQ c mp mp Qm c poutin TT (26) D TTLTTiinoutin i l The uncertainty of the Nusselt number is caused by the uncertainty of the thermocouple temperature measurements, the thermocouple’s positions along the heated tube, and the flow meter flow rate. From the expression of the Nusselt number: hDi Qm cTT poutin Nui (27) kT fi, kTTLTTl iinoutini The total uncertainty can be expressed as Eq.(28): 2 2 2 2 2 NufQfTi 1 m 2 out fTfTfl 3 in 4 i 5 i (28) where 17 The University of California Berkeley Department of Nuclear Engineering 2 cTT f poutin 1 kTTLTTl iinoutini 2 cQTTL f pmiin 2 2 kTTLTTl iinoutini 2 cQTTL pmiout f3 2 (29) kTTLTTl iinoutini 2 cQTTL pmoutin f4 2 kTTLTTl iinoutini 2 cQTT 2 pmoutin f5 2 kTT LTTl iinoutini TTToutletinleti 0.5℃, linchmi 0.10.00254 For S-HT2 facilities, 2 CADd P BPCD (30) QCDPkgsmd ,/ (31) We derive 2 0.5%,QSHTm Qm (32) 0.1%,QCIETm Test Bay 3 Qkgminoutp s0.0895,47 TTcJ kgLm ,63 kW ,1.65℃ m 10,2.445℃ ,0.13438 ℃ ℃ At the inlet of the heating pipe, lTTiiin0,10.77 ℃ hDi Qm c p T out T in Nuinlet 212.54 (33) kT fi, k Ti T in L T out T in l i 2 2 2 2 2 Nuinlet fQfTfTfTfl1 m 2 out 3 in 4 i 5 i 12.42 (34) Nu 12.3742 inlet 5.8% (35) Nuinlet 212.54 At the outlet of the heating pipe, li L, T i T out 8.66℃ 18 The University of California Berkeley Department of Nuclear Engineering hDi Qmpoutin cTT Nuoutlet 269.504 (36) kT fi, kTTLTTl iinoutini 22222 NufQfTfTfTfloutletmoutinii 12345 29.8376 (37) Nu 29.8376 outlet 11.07% (38) Nuoutlet 269.504 Table 2 includes a summary of the uncertainty contributions. Table 2 uncertainty contribution of each term Inlet Outlet Uncertainty Terms S-HT2 CIET Test Bay S-HT2 CIET Test Bay 2 fQ1 m 1.12937 0.0451748 1.81581 0.0726323 2 fT2 out 44.116 44.116 575.149 575.149 2 fT3 in 10.4032 10.4032 70.93 70.93 2 fT4 i 97.3653 97.3653 242.122 242.122 2 fl5 i 0.1076 0.1076 0.267574 0.267574 Nu 212.54 212.54 269.504 269.504 Nui 12.3742 12.33 29.8376 29.8084 NuNuii 5.82% 5.80% 11.07% 11.06% From Table 2, we can see that the main source of uncertainty of the Nusselt number results from the uncertainty of temperature, while the flow rate meter uncertainty and the thermocouple position uncertainty contributed very little to the uncertainty of the Nusselt number. To make a rough estimate, the temperature difference of the wall and the flow at the outlet is 8℃, however, the uncertainty of the thermocouple temperature is ±0.5℃. The contribution of thermocouples is 0.50.522 8.84% (39) 8 In contrast, the uncertainty of the flow rate in S-HT2 facilities is only 0.5%. So, even we replaced the flow rate meter with a more accurate Coriolis flow meter in the CIET Test Bay, and reduce the flow rate uncertainty from 0.5% to 0.1%, the total improvement to the Nusselt number uncertainty is very small. 19 The University of California Berkeley Department of Nuclear Engineering Chapter 7 Conclusions For laminar flow, the pipe is divided into two regions: entrance region and fully developed region, shown in Figure 12. The Graetz number is defined as: D Gz R e Pr (40) x At the entrance region, that corresponds to Gz 1 0.05[1], the Nusselt number is a function of Reynolds number and Prandtl number. At the fully developed region, the Nusselt number is independent of Reynolds number and it is constant: 4.36constantsurface heat flux NuD (41) 3.66constantsurface temperature 20 The University of California Berkeley Department of Nuclear Engineering Figure 12 Local Nusselt numbers from entry length solutions for laminar flow in a circular pipe (This diagram is copied from reference [10]) To make a rough estimate for fully developed region, x/D must satisfy: x 0.050.05RePr 13480 674 (42) D For the laminar flow in our experimental data, the minimum of RePr is 13,500, while the maximum of x/D is 224. From Eq.(42), it is clear that the laminar flow of our data lies in the entrance region. Its Nussult number is a function of Reynolds number and Prandtl number, as shown in Eq.(23). To summarize, correlations for three ranges of Re are recommended below: 21 The University of California Berkeley Department of Nuclear Engineering 0.0816 0.14 1.098 0.33 x x 0.002501RePr , 9224, 280 340 Re 0, laminor flow D w D 0.14 0.823411/3 NuReD 0.0184Pr, 3400 50000,transitionflowRe (43) w 0.14 1.027121/3 0.00195Re Pr ,50000 Re 200000, turblent flow w Acknowledgments This thesis was done under the guidance of my advisor Dr. Per F. Peterson. The Nusselt number calculation code was written by Olga Scarlet Raluca. I made corrections and improvement of the calculation code. Also, I received some useful information about these experimental data from Connie Lee. Research funding was provided by the U.S. Department of Energy Office of Nuclear Energy, under a Nuclear Energy University Programs grant. References [1]. H. G. MacPherson, “The Molten Salt Reactor Adventure,” Nuclear Science and Engineering, Vol. 90, pp. 374-380 (1985). [2]. C.W. Forsberg, P.F. Peterson, and P. Pickard, “Molten-Salt-Cooled Advanced High-Temperature Reactor for Production of Hydrogen and Electricity,” Nuclear Technology Vol. 144, pp. 289-302 (2003). [3]. Jeffrey E. Bickel, Nicolas Zweibaum, Per F. Peterson. “Design, Fabrication and Startup Testing in the Compact Integral Effects Test (CIET 1.0) Facility in Support of Fluoride-Salt-Cooled, High-Temperature Reactor Technology”, UCBTH-14-009, 2014, p16. [4]. Philippe M. Bardet, Per F. Peterson, “Options for Scaled Experiments for High Temperature Liquid Salt and Helium Fluid Mechanics and Convective Heat Transfer”, Nuclear Technology, VOL.163, Sep. 2008, Pages 344-357. [5]. S. Cantor, “Physical Properties of Molten-Salt Reactor Fuel, Coolant, And Flush Salts”, ORNL-TM- 2316, August 1968. [6]. Rohit Upadhya, Shannon Bragg-Sitton, Piyush Sabharwall. “Preliminary Scaling and Controls Analysis of an FHR-HTSE System”, Idaho National Laboratory Summer 2013 Final Report, Aug.2013. [7]. Owens Corning Insulating Systems, LLC, Fiberglas Pipe Insulation Product Data Sheet, Pub. No. 20547-T. Printed in U.S.A. May 2017. [8]. Raluca Olga Scarlat, “Design of Complex Systems to Achieve Passive Safety: Natural Circulation Cooling of Liquid Salt Pebble Bed Reactors”, doctoral dissertation, 2012. [9]. Tom O’Banion , “Coriolis: The Direct Approach to Mass Flow Measurement”, 2013 American Institute of Chemical Engineers (AIChE), 2013. [10]. Frank P. Incropera, David P. Dewitt, “Fundamentals of Heat and Mass Transfer”, 6th 22 The University of California Berkeley Department of Nuclear Engineering Edition, p514. [11]. Sieder, E. N., and G. E. Tate, “Heat Transfer and Pressure Drop of Liquids in Tubes”, Ind. Eng. Chem., 1936, 28 (12), pp 1429–1435. [12]. Trademark of The Dow Chemical Company (1997), Dowtherm A Heat Transfer Fluid Product Technical Data, p24. [13]. J.W. Cooke, and B. Cox, ORNL-TM-4079, 1973, p1. [14]. https://en.wikipedia.org/wiki/Origin_(software). [15]. Levenberg, Kenneth (1944). "A Method for the Solution of Certain Non-Linear Problems in Least Squares". Quarterly of Applied Mathematics. 2: 164–168. [16]. Marquardt, Donald (1963). "An Algorithm for Least-Squares Estimation of Nonlinear Parameters". SIAM Journal on Applied Mathematics. 11 (2): 431–441. 23 The University of California Berkeley Department of Nuclear Engineering Appendix Appendix A Physical properties of Dowtherm A [12] 24 The University of California Berkeley Department of Nuclear Engineering 25 The University of California Berkeley Department of Nuclear Engineering Appendix B Matlab code to calculate the Nusselt number function [ out ] = Drakesol( fl ) % Raluca Scarlat, and Aaron Goswami; 16 May 2009 % Perform calculations on SHT2 data collected fall 2008. %------Constant Definitions------name=cat(2,'C:\Users\po\Desktop\Data Analysis\Downward flow\',num2str(fl),'.txt') data= load(name); s=size(data); F=s(2)-2; fQm=1; %Qm multiplier - default: set to 1 %F=15; % number of temperature measurements( including the T-100-I, which is not listed in Connie's note but I think it is the temperature of inletflow, treat T-101-I and T-103-I as one, their average as the outslet temperature) P=F; S=F-1; % number of discrete points and segments r1=.00546; r2=.00635; r3=82.6/1000; L=2.445; d=2*r1;% I changed the original L=2.445 to L=2.4511 to mathch the location of the two electrodes in line 14 Atube=pi*r1^2; Awall=2*pi*r1*L; Vhtr=L*pi*(r2^2-r1^2); Tamb=25; DPcoeff=0; DPclam=0; DPcturb=0; %z=L*[0:(1/(F-1)):1]; btm=113.26;%the height of the bottown(inlet) %z=0.0254*(btm-[113.26,104, 87, 82.5, 78.25, 61, 56.5, 52.75, 48.25, 43.75, 40, 31, 22.25, 17]);%upward flow, and the location of T-100-I is unkown z=0.0254*([17, 22.25, 31, 40, 43.75, 48.25, 52.75, 56.5, 61, 78.25, 82.5, 87, 104, 113.26]-17);%downward flow! %for Z, the first one and the last one are the location of the two electrodes %disp(z); dz=[]; Z=[]; for i=1:1:(F-1) dz=[dz,z(i+1)-z(i)]; Z=[Z,z(i)/d]; end %Cd function [out]=Cd(dP) out =(2.35E-6)*dP^2 - (2.7E-05)*dP + .00182; %temp in oC end %kIns function [out]=kIns(T) out =7.702*10^04*T+0.206; %temp in oC end 26 The University of California Berkeley Department of Nuclear Engineering %rhoOil function [out]=rhoOil(T) out=-4.3653*(10)^(-4)*T^2-0.75103*T+1074.6; %temp in oC end function [out]=betaOil(T) out=-(-2*4.3653*(10)^(-4)*T-0.75103)/(-4.3653*(10)^(-4)*T^2-0.75103*T+1074.6); %tem p in oC end %CpOil Dcoeff = 8.56*10^-07; Ccoeff = -5.70*10^-04; Bcoeff = 2.90; Acoeff = 1514.78; function [out]=CpOil(T) %temp in oC out=Dcoeff*T^3+Ccoeff*T^2+Bcoeff*T+Acoeff; end function [out]=CpOilInt(T1,T2) %temp in oC int2=Dcoeff*T2^4/4+Ccoeff*T2^3/3+Bcoeff*T2^2/2+Acoeff*T2; int1=Dcoeff*T1^4/4+Ccoeff*T1^3/3+Bcoeff*T1^2/2+Acoeff*T1; out=int2-int1; end %kOil function [out]=kOil(T) out=0.00016*T+0.1419; %temp in oC end %viscOil function [out]=viscOil(T) out=0.2208*T^(-1.18); %temp in oC end function [out]=nuOil(T) out=viscOil(T)/rhoOil(T); %temp in oC end %kHtr function [out]=kHtr(T) out=0.011*T+16.16; %temp in oC end %relHtr - stainless steel 304 resistivity as a function of temp in oC function [out]=relHtr(T) 27 The University of California Berkeley Department of Nuclear Engineering out=(0.061*T+73.10)*10^-8; end function [out]=relHtrInt(dz,Thtr) % int(relHtr dz, z=0 to L) int=0; for i=1:1:F-1 int=int+ dz(i)*(relHtr(Thtr(i))+relHtr(Thtr(i+1)))/2; end out=int; end %------Variable Definitions & Initialization------DPorf = 0; DPhtr = 0; Tin=0; Tout=0; % oil inlet and outlet temperatures Qm = 0; % oil mass flow rate in kg/s u = 0; % oil axial velocity in m/s Q = 0; % total heater power, in W q = 0; % average heater power density, in W/m3 qa = 0; % average wall heat flux, in W/m2 qi=[]; % heater power density in each segment, in W/m3 qai=[]; % wall heat flux in each segment, in W/m2 qla=0; % heat loss flux through outside insulation, in W/m2 ql=0; % heat loss as fraction of volumetric heat generation, in fraction Avg=[]; Thtr=[]; Twall=[]; Toil=[]; %assumes uniform heat flux Toil2=[]; %assumes temperature-dependent steel resistivity qlai=[]; qli=[]; dThtr=[]; %temp gradient across heater thickness h=[]; Nu=[]; Re=[]; Pr=[]; %uniform heat flux, and no temp drop across heater radius h2=[]; Nu2=[]; Re2=[]; Pr2=[]; Gr=[]; bR=[]; %buoyancy ratio Bo=[]; Gz=[]; % Gz = Re*Pr*d/l GrPrZ=[]; % = Gr*Pr*d/L %------Data Input------% extract Avg = the average of all columns in the numerical input "data" %name=cat(2,'D:\Sogou Download\SS SHT2\SHT2 Data Analysis and Write-ups_2008 to 2009\Matlab\Origin-Buoyancy\buoyancy_20090518_',num2str(fl),'.dat') %name=cat(2,'C:\Users\po\Desktop\Data Analysis\',num2str(fl),'.txt') %%move to the front of this file %data= load(name); %s=size(data); %disp(s(2)); T=[]; 28 The University of California Berkeley Department of Nuclear Engineering for i=2:1:s(2) for j=1:1:s(1) T=[T, data(j,i)]; end Avg=[Avg, sum(T)/length(T)]; T=[]; end Tin=Avg(1); Tout=Avg(F); % extract Thtr = temperature at r2; assume Thtr(0)=Thr(1)(vs. % Thtr(0)=Toil(0) ) assume Thtr(F)=Thtr(F-1) for i=2:1:(F-1) Thtr=[Thtr,Avg(i)]; end Ttemp=[]; tem1=polyfit(z(2:6),Thtr(1:5),1); tem2=polyfit(z((F-5):(F-1)),Thtr((F-6):(F-2)),1); Thtr=[z(1)*tem1(1)+tem1(2),Thtr,z(F)*tem2(1)+tem2(2)]; %disp(Thtr); % Thtr=[Tin,Thtr,Thtr(F-2)]; %also line 182 Tin=Avg(1); Tout=Avg(F); rhoIn=rhoOil(Tin); %DP=Avg(F+3); % raw pressure transducer reading, used in Cd & Qm calculations. %DPorf=Avg(F+3)*2/10*6894.75729; % in Pa %DPhtr=Avg(F+2)*5/10*6894.75729; % in Pa Qv=0.001*Avg(F+1);% put the Flow meter in the last column, F+1 %Qm=Cd(DP)* sqrt( rhoIn* DP); Qm=Qv*rhoIn; Qm=fQm*Qm; u=Qm/(rhoIn*Atube); DPcoeff=DPhtr/(rhoOil(Tin/2+Tout/2)*u^2/2/d); CpInt=CpOilInt(Tin,Tout); Q=Qm*CpInt; q=Q/Vhtr; qa=Q/Awall; %------Calculations------ 29 The University of California Berkeley Department of Nuclear Engineering Toil=Tin; for i=2:1:F Toil=[Toil,Toil(i-1)+ CpInt/CpOil(Toil(i-1))*dz(i-1)/L]; end for i=1:1:F-1 h=[h,qa/(Thtr(i)-Toil(i))]; Nu=[Nu,h(i)*d/kOil(Toil(i))]; Re=[Re,Qm*2/(pi*r1)/viscOil(Toil(i))]; Pr=[Pr,CpOil(Toil(i))*viscOil(Toil(i))/kOil(Toil(i))]; %heat flux balance: q''*2*pi*r1*dz = Qm*Cp*dT test1=Q*dz(i)/L; test2=Qm*CpOilInt(Toil(i),Toil(i+1)); test0=(test2-test1)/test1*100; end relInt=relHtrInt(dz,Thtr); Toil2=Tin; temp3=r2^2*(1+log(r1^2/r2^2))-r1^2; for i=1:1:F-1 temp= (relHtr(Thtr(i))+relHtr(Thtr(i+1)))/2 * dz(i); qi=[qi, q * temp/relInt*L/dz(i)]; temp2=CpInt/CpOil(Toil2(i)) * temp/relInt; Toil2=[Toil2,Toil2(i)+temp2]; qai=[qai,qi(i)*Vhtr/Awall]; dThtr=[dThtr,-qai(i)*temp3/(4*kHtr(Thtr(i)))]; qlai=[qlai,kIns(Thtr(i))*(Thtr(i)-Tamb)*(r3-r2)/(r2*r3*log(r2/r3))]; qli=[qli,qlai(i)*Awall/Vhtr/qi(i)]; end dThtr=[dThtr,dThtr(F-1)]; Twall=Thtr-dThtr; % Twall(1)=Tin; % also line 129 Ref=[]; Prf=[]; Grf=[]; for i=1:1:F-1 h2=[h2,qai(i)/(Twall(i)-Toil2(i))]; Nu2=[Nu2,h2(i)*d/kOil(Toil2(i))]; Re2=[Re2,Qm*2/(pi*r1)/viscOil(Toil2(i))]; 30 The University of California Berkeley Department of Nuclear Engineering Pr2=[Pr2,CpOil(Toil2(i))*viscOil(Toil2(i))/kOil(Toil2(i))]; Ref=[Ref,Qm*2/(pi*r1)/viscOil((Toil2(i)+Twall(i))/2)]; Prf=[Prf,CpOil((Toil2(i)+Twall(i))/2)*viscOil((Toil2(i)+Twall(i))/2)/kOil((Toil2(i)+Twall(i))/2)] ; Gr=[Gr,(Twall(i)-Toil2(i))*d^3*betaOil((Toil2(i)+Twall(i))/2)*9.81/nuOil((Toil2(i)+Twall(i))/2 )^2]; bR=[bR, Gr(i)/Re2(i)^2.5/Pr2(i)^(2/3)]; Bo=[Bo, 8e4*Gr(i)/(Ref(i))^3.425/(Prf(i))^0.8]; Gz=[Gz,Re(i)*Pr(i)/Z(i)]; GrPrZ=[GrPrZ,Gr(i)*Pr(i)/Z(i)]; %heat flux balance: q''*2*pi*r1*dz = Qm*Cp*dT test1=qai(i)*Awall*dz(i)/L; test2=Qm*CpOilInt(Toil2(i),Toil2(i+1)); test=(test2-test1)/test1*100; end %disp(Re2); DPclam=64/(Re2(1)/2+Re2(F-1)/2); DPcturb=1/(log(Re2(1)/2+Re2(F-1)/2)*1.8-1.5)^2; %------Comparison with Correlations------ NuDB=[]; %Dittus-Botler equation for turbulent flow, heating NuST=[]; %Siever-Tate equation for turbulent flow - recommended on Dowtherm A brochure NuE=[]; %Edwards corr for lam forced conv, with hydrodynamic + thermal entrance effects NuDBe=[]; NuSTe=[]; NuOR=[];NuORe=[];% ORNL FLiBe correlation. NuORe: the entrance effect is included. %Petukhov and Guelinski, German Heat Atlast correlations for forced convection NuHA=[]; NuHA=[]; NuHA1=[]; NuHA2=[]; NuHA3=[]; NuHAlam=[]; NuHAturb=[]; Chi=[]; Chi2=[]; NuHAlam2300=[]; NuHAturb10000=[]; EEff=[]; %entrance effect for turbulent flow (hydrodyn + thermal) viscRbw=[]; %ratio of fluid viscosity at bulk temp, to visc at wall temp for i=1:1:F-1 NuDB=[NuDB,0.023*Re2(i)^0.8*Pr2(i)^0.4]; NuST=[NuST, 0.027*Re2(i)^0.8*Pr2(i)^(1/3)*(viscOil(Toil2(i))/viscOil(Twall(i)))^0.14]; 31 The University of California Berkeley Department of Nuclear Engineering NuHA1=[NuHA1,4.364]; NuHA2=[NuHA2, 1.953*(Re2(i)*Pr2(i)/Z(i))^(1/3)]; NuHA3=[NuHA3, 0.924*Pr2(i)^(1/3)*(Re2(i)/Z(i))^0.5]; NuHAlam=[NuHAlam, ( NuHA1(i)^3+0.6^3+(NuHA2(i)-0.6)^3+NuHA3(i)^3 )^(1/3)]; NuHAlam2300=[NuHAlam2300, ( 4.364^3+0.6^3+((1.953*(2300*Pr2(i)/Z(i))^(1/3))-0.6)^3+(0.924*Pr2(i)^(1/3)*(2300/Z(i)) ^0.5)^3 )^(1/3)]; Chi=[Chi,(Re2(i)-2300)/(10000-2300)]; Chi2=[Chi2,(6.8*log(Re2(i))/log(10)-1.5)^(-2)/8]; NuHAturb=[NuHAturb,Chi2(i)*Re2(i)*Pr2(i)/(1+12.7*Chi2(i)^0.5*(Pr2(i)^(2/3)-1))*(1+Z(i)^ (-2/3))]; NuHAturb10000=[NuHAturb10000,0.00384734*10000*Pr2(i)/(1+12.7*0.00384734^0.5*( Pr2(i)^(2/3)-1))*(1+Z(i)^(-2/3))]; %if Chi(i)<2300 NuHA=[NuHA, NuHAlam(i)]; % else if Chi(i)>10000 NuHA=[NuHA, NuHAturb(i)]; if Re2(i)<2300 NuHA=[NuHA, NuHAlam(i)]; else if Re2(i)>10000 NuHA=[NuHA, NuHAturb(i)]; else NuHA=[NuHA, (1-Chi(i))*NuHAlam2300(i)+Chi(i)*NuHAturb10000(i)]; end end NuE=[NuE, 3.66 + 0.065*Re2(i)*Pr2(i)/Z(i)/(1+0.04*(Re2(i)*Pr2(i)/Z(i))^(2/3))]; EEff=[EEff, 1+(Z(i)^0.1/Pr2(i)^(1/6)*(0.68+3000/Re2(i)^0.81))/Z(i)]; %EEff=[EEff, 1];% since we have smoothed the temperature profile, meanwhile it is constant heat flux, the entrance effect can be neglected. NuDBe=[NuDBe, NuDB(i)*EEff(i)]; NuSTe=[NuSTe, NuST(i)*EEff(i)]; viscRbw=[viscRbw,viscOil(Toil2(i))/viscOil(Twall(i))]; %Below is to Caculate the ORNL FLiBe correlation if Re2(i)<1000 NuOR=[NuOR, 1.89*(Re2(i)*Pr2(i)/Z(i))^0.33*viscRbw(i)^0.14]; elseif Re2(i)<3500 NuOR=[NuOR, 1.89*(Re2(i)*Pr2(i)/Z(i))^0.33*viscRbw(i)^0.14*(3500-Re2(i))/2500+0.107*(Re2(i)^(2/3)-1 35)*Pr2(i)^(1/3)*viscRbw(i)^0.14*(Re2(i)-1000)/2500]; elseif Re2(i)<12000 NuOR=[NuOR, 0.107*(Re2(i)^(2/3)-135)*Pr2(i)^(1/3)*viscRbw(i)^0.14]; else NuOR=[NuOR, 0.0234*Re2(i)^0.8*Pr2(i)^(1/3)*viscRbw(i)^0.14]; end NuORe=[NuORe,NuOR(i)*EEff(i)]; end %disp=(size(NuORe)) %------Save to File------ 32 The University of California Berkeley Department of Nuclear Engineering %transposing... % info=[]; % info=[Qm, u, DP, dPhtr, dPorf, Q, q, qa, Tin, Tout]; info=info'; % h=[h,0]; Nu=[Nu,0]; Re=[Re,0]; Pr=[Pr,0]; % h2=[h2,0]; Nu2=[Nu2,0]; Re2=[Re2,0]; Pr2=[Pr2,0]; %z=z';Z=z/d; File=[]; for i=1:1:F-1 File(i)=fl*(1); %negative values: buoyancy files; positive values: forced conv files end File=File'; Z=Z'; Toil=Toil'; Thtr=Thtr'; Twall=Twall'; Toil2=Toil2'; h2=h2'; qai=qai'; dThtr=dThtr'; qi=qi'; h=h'; Nu=Nu'; Re=Re'; Pr=Pr'; Nu2=Nu2'; Re2=Re2'; Pr2=Pr2'; NuDB=NuDB'; NuST=NuST'; NuHA=NuHA'; NuE=NuE'; NuDBe=NuDBe'; NuSTe=NuSTe';NuOR=NuOR';NuORe=NuORe'; NuHAlam=NuHAlam'; NuHAlam2300=NuHAlam2300'; NuHAturb=NuHAturb'; NuHAturb10000=NuHAturb10000'; viscRbw=viscRbw'; EEff=EEff'; Gr=Gr'; bR=bR'; Bo=Bo'; Gz=Gz'; GrPrZ=GrPrZ'; % A2=[Re2, Pr2, Nu2]; A2=[File, Z, dThtr(1:(F-1),:),Twall(1:(F-1),:), viscRbw, EEff, GrPrZ, NuE, NuDBe, NuSTe, bR, qai/qa, Bo]; A1=[Gr, bR, Gr, Pr2, Re2, Bo, Bo.*Z, Nu2./NuHA, Nu2./NuE, Nu2./NuDBe, Nu2./NuSTe, Nu2./NuORe, Nu2./NuOR,Nu2]; %A2=[Z, Re2, Pr2, Nu2, NuDB, NuST, NuHA, NuE, NuDBe, NuSTe, NuHAlam, NuHAlam2300, NuHAturb, NuHAturb10000]; A=[A2 A1]; % headings=['z,m', 'Z', 'Thtr,oC', 'Toil,oC', 'h', 'Nu', 'Re', 'Pr', 'Twall,oC', 'Toil2,oC', 'h2', 'Nu2', 'Re2', 'Pr2'] % A=A'; headings=headings'; A=[headings A]; A=A'; name=cat(2,'D:\Sogou Download\SS SHT2\SHT2 Data Analysis and Write-ups_2008 to 2009\Matlab\Res2\calcs20080729_',num2str(fl),'.txt'); %name2=cat(2,'C:\Users\po\Desktop\Data Analysis\result\',num2str(fl),'_result.txt'); name2=cat(2,'C:\Users\po\Desktop\Data Analysis\result\','downward_result.txt'); % save (name ,'A', '-ASCII', '-tabs'); save (name2 ,'A', '-ASCII', '-tabs', '-append'); %out=(size(Z)); %out=(size(Bo)); end 33 The University of California Berkeley Department of Nuclear Engineering Appendix C Mathematica code for Power Regression with more than 2 parameters. Appendix D OriginLab procedure for power regression with less than 3 parameters. Analysis→Fitting→Nonlinear Curve Fitting→Category: Power; Function: Allometric1 34 The University of California Berkeley Department of Nuclear Engineering 35