2019.6.26
Highlights
 Clarification of the reason why the liquid metal heat transfer coefficients in the handbook scatter several hundred percent in the low Peclet number region
 Clarification of the cause of over 100% shift to the lower side even in the high Peclet number region
 Identification of measured data which can represent correctly heat transfer coefficients over a wide range of flow conditions
Summary
When the liquid metal heat transfer coefficient of the laminar region of liquid metal is expressed by the dimensionless Nusselt number (Nu), it should be about 5 in the simplified theory, however the past measured values are incredible values of around 1 or less. Even in the turbulent region, there are several cases where the measured value shifts downward by 100% or more. These heat transfer coefficient values are illustrated in various handbooks around the world, and researchers have been often confused when calculating heat transfer. It has been clarified that this variance was caused by the measurement methods and the assumptions for the data processing by investigating the past experimental apparatuses and experimental methods and CFD analyses. In addition, it has been clarified that the proper measurement method and the data processing method to obtain correct heat transfer coefficient.
Background
Many measurements of the heat transfer coefficient of liquid metals were conducted between the early 1950s and around 1970 and the results are illustrated in various handbooks. The measured Nu number is expressed as a function of Peclet number (Pe = Re∙Pr) which is the product of the Reynolds number (Re) and the Prandtl number (Pr) as illustrated in Fig. 1. A part of data has scattered as shown in Fig. 1, and the cause has not been clarified until now. In the text book in the 1970s, the causes of the large scattering of the heat transfer coefficient were discussed on the assumption of physical effects such as wettability, oxide film and gas effects. However, all these factors could not explain the anomalous variance or shift of the experimental results, and the conclusion was that the cause was unknown. Therefore, in the reactor fuel and heat exchanger design, since the heat transfer is assumed to be poor, the design is too conservative from the actual conditions. For this reason, it has been continued to make economically expensive equipment.
Fig. 1 Dimensionless heat transfer coefficients illustrated in handbooks
Result
The motivation of the research was that the heat transfer coefficients measured in the liquid sodiumcooled reactor related experimental facilities and the intermediate heat exchanger (IHX) of the prototype reactor "Monju" were several hundred percent lower than the theoretical value in the laminar flow region. (The Pe number in Fig. 1 showed the same tendency as the result of Johnson et al.).
Research was conducted using computational fluid dynamics (CFD) to find out the cause. As a result, it has been clarified that the heat transfer in the lower plenum of IHX, which could usually be ignored, became dominant as the flow rate decreased. It has also been clarified that the heat transfer coefficient inside the heat transfer tube and in the plenum is represented by the widely used correlations.
The next question was which of the proposed correlations were reliable because the empirical correlations of the heat transfer coefficients were largely different by the researchers. In order to reproduce as realistic as possible the situation in which the experimental data were obtained, a CFD model around the test section of the apparatus was constructed on a computer as shown in Fig. 2. As a result of investigation of the apparatuses used for the experiments and the measurement data, it turned out that two causes existed. One was that the heat transfer coefficients were not true values because the assumptions of the experimental conditions were incorrect and the measured values were processed based on the wrong assumptions. For example, in the case where temperature measurement was performed using a composite pipe consisting of the test inner pipe and an outer pipe to make the heat flux uniform, the measured result was processed on the basis of the assumption that the thermal resistance at the contact surface could be neglected. However, the gap changed during the experiment, and the calculated value is composed of the heat transfer coefficient of the liquid metal and the contact conductance. It became clear by CFD calculations that the abovementioned assumption was not satisfied during the time of experiment. The result excluding these data which includes errors is illustrated in Fig. 3. Second, most studies used the linearly interpolated bulk temperature from the mixing temperatures measured at the inlet and outlet of the fluid, but this interpolation has a larger error as the Peclet number becomes smaller than 1000. In the case of liquid metal heat transfer, since the temperature difference between the wall surface and the fluid bulk is very small, thermocouple errors and the abovementioned linear interpolation which does not cause problems for fluids with the large Prandtl numbers such as water, become a major problem. This problem was also clarified by CFD calculation. In the measured heat transfer coefficients evaluated based on the temperature distribution and velocity distribution in the test section or the case where the velocity distribution was predicted by the theory and the average bulk temperature was determined by integration, good agreement with the CFD calculation results was obtained. It was found that all cases with linear interpolation were lower than the former results in the small Peclet number region. In addition, it was also found that modified Aoki's correlation of which constants are replaced by the new constants, which theoretically pursued the correlation based on Lyon's simple turbulence model, can predict the measured data with good accuracy. He was a Professor of this institute.
Fig. 2 Example of CFD simulation model
Fig. 3 Nu numbers excluding experimental data including inadequate assumptions
Fig. 4 Nu numbers deduced from the definition of the heat transfer coefficient (bulk temperatures are evaluated from the measured or estimated velocity profiles and measured temperature profiles)
As a result, the problem of the large variance in the Nu numbers has been resolved since Lubarsky and Kaufman collected and published heat transfer data in 1955.
Paper information
Journal1 : 
Nuclear Engineering and Design 
Title of original : 
Consideration on Nusselt numbers of liquid metals under low Peclet number conditions 
Author : 
Hiroyasu MOCHIZUKI 
DOI : 
Journal2 : 
Nuclear Engineering and Design 
Title of original : 
Consideration on Nusselt numbers of liquid metals flowing in tubes 
Author : 
Hiroyasu MOCHIZUKI 
DOI : 

Affiliation : 
Laboratory for Advanced Nuclear Energy, Institute of Innovative Research, Tokyo Institute of Technology 