Preamble

Scaling method for Lead-Bismuth natural circulation under marine rolling conditions Jianing Xu, Zhen Wang, Shichao Zhang, Chao Chen, Jie Yu, and Gang Pei 1 Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei, 230031, China University of Science and Technology of China, Hefei 230026, China Lead-bismuth fast reactors have broad application prospects in the field of marine power. The thermal- hydraulic characteristics of Lead-Bismuth Eutectic (LBE) under marine conditions have a significant effect on the reactor safety. A new scaling analysis method for the LBE natural circulation under rolling conditions was derived to understand the corresponding mechanism. The design criterion for the scaled-down facility was obtained and the influence of varying rolling processes with different amplitudes and periods was discussed. The numerical models were simulated using Fluent code to verify the scaling criterion. The results show that the time-averaged parameters of the LBE prototype under different rolling processes can be accurately simulated by the scaled-down model, with a maximum error of less than 9.08%. The scaled-down cases reflected the periodic changes of the flow characteristics in the prototype. The low-flow phases caused by the instability during the rolling motion led to the periodic brief scaling distortion of the mass flow rate. In addition, the transient deviation of the temperature difference decreases with the reduction of the rolling amplitude and period. These conclusions have important application value for the design of the LBE experimental device under the marine motion.

Keywords

Marine reactor, Rolling condition, Lead-bismuth eutectic, Natural circulation system, Scaling analysis method

Nomenclature List of Symbols A Cross-section area d Pipe diameter L Total circulation length u Fluid velocity a M Modified acceleration g Gravity acceleration T h , T c Fluid temperature of hot and cold section ρ Density β s Thermal expansion coefficient a c , a t Centripetal and Tangential acceleration H Height between cooling heating section center f Friction loss coefficient ∆ T Temperature difference η Dynamic viscosity e t Dimensionless time correction factor ω Angular velocity ε Angular acceleration θ Rotation angle T M Rolling period k Form loss coefficient C P Specific heat capacity at constant pressure r Rolling radius q Heat flux z Axial length of heating and cooling section k s Thermal conductivity of solid wall t Time λ Thermal conductivity

Supported by the National Key R & D Program of China under the Grant No. 2022YFB1902503 and the Youth Innovation Promotion Association of Chinese Academy of Sciences (CAS) under the Grant No.2023466.

Zhen Wang, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei, 230031, China, 15656920911, Chao Chen, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei, 230031, China, 15215518857,

Subscripts R Ratio between the scaled-down model and prototype T Theoretical value 0 Reference constant Abbreviations Ri Richardson number Fri Friction number Qs Heat source number St Stanton number Re Reynolds number LBE Lead-Bismuth Eutectic TALL Thermal-hydraulic ADS Lead-bismuth Loop CHEOPE CHEmical OPErational transient NCL − SJTU Natural Circulation Loop-Shanghai Jiaotong University LECO Lead-bismuth eutectic Experimental Circulation facility under Ocean conditions CMT Core Makeup Tank HLM Heavy Liquid Metal UDF User-Defined Function SST Shear Stress Transport

1. INTRODUCTION

The lead-based fast reactor is one of the six types of Generation-IV advanced reactors.

Lead-Bismuth Eutectic (LBE) has a low melting point, a high boiling point, outstand- ing heat transfer and neutron characteristics, and stable chem- ical properties, and it is a promising candidate coolant for the advanced reactor systems[ ]. For small-scale and special- ized applications such as ocean and deep-sea exploration, the lead-bismuth fast reactor represents a viable option[ The additional forces imposed by the ocean conditions on the fluids lead to the complex spatial dynamics, such as incli- nation, heaving, and particularly rolling motions[ ]. As the ocean and the floating nuclear power continue to develop[ ], the changes in flow and heat transfer characteristics under

the marine conditions have attracted increasing attention[ Zhang et al.[ ] analyzed the pressure drop of the single- phase flow in the horizontal pipe sections under rolling mo- tion and developed a frictional factor correlation.

Yan et ] fully considered the channel structure and adopted a 3 rod bundle test section to study the pressure drop charac- teristics of the forced air-water two-phase flow under rolling conditions. The study pointed out that the flow rates of the gas and liquid were important influencing factors for the dy- namic frictional resistance in the rod bundle. Meanwhile, the

rolling amplitude had a significant influence on the two-phase 25

frictional pressure gradient, while that of the period was rel- atively small. Yan and Gu et al.[ ] adopted the CFD code to analyze the influence of rolling motion on the flow and heat transfer characteristics of the turbulent flow in typical rod bundles. For the short-period and the large-amplitude cases,

the additional force has a significant impact on the flow char- 31

acteristics, and the correlation of the frictional resistance and heat transfer coefficient under the steady state is no longer re- liable. Bai et al.[ ] studied the flow characteristics of the he- lical coil once-through steam generator in the floating nuclear power plants. The improved Relap5 code was used to inves- tigate the influence of the rolling motion. In addition to the

local phenomena, the significant attention has also been paid 38

to the thermal-hydraulic characteristics of the natural circula- tion within the integral facilities under the ocean conditions.

Tan et al.[ ] pointed out that the rolling inertial force leads to an increase in the resistance coefficient of the natural cir- culation loop, thus reducing the average flow rate. Cong et ] discussed the influence of the rolling motion on the stability of the natural circulation system. The study indicated that the swing period, the maximum swing angle, the swing

phase difference, and the heat power have significant impacts 47

on the system, while the swing radius has little impact. Wang et al.[ ] classified the flow instabilities of the system under ocean conditions and discussed the characteristics of the den- sity wave oscillations under the natural circulation. The reso- nance phenomena should be avoided to enhance the stability of the system.

To study the thermal-hydraulic characteristics of the LBE, many experimental facilities have been built such as the KYLIN-II experimental loop[ ], the Thermal-hydraulic ADS Lead-bismuth Loop (TALL) experimental facility[ the Chemical Operational transient (CHEOPE) facility[

and the Natural Circulation Loop-Shanghai Jiaotong Univer- 59

sity (NCL-SJTU) facility[ ], among others. Researchers

at Xi’an Jiaotong University built the Lead-bismuth Eutec- 61

tic experimental Circulation facility under Ocean conditions (LECO)[ ]. The influence of marine conditions on the forced circulation flow and resistance characteristics of LBE was studied based on the rolling bench and CFD codes[ ]. The majority of existing facilities are designed for the stationary and land-based conditions. Therefore, further research on the thermal-hydraulic characteristics of LBE flow under the mo-

tion conditions is crucial for the development of the oceanic 69

LBE reactors. Considering factors such as site, cost, and safety, the ex- perimental facilities cannot reproduce the full-scale reactor and its associated ships and ocean environment. It is typi- cally used in engineering to obtain a set of scaling methods and establish the experimental facilities based on these meth- ods. Over the years, many researchers have conducted exten- sive studies on the scaling theory. Theories such as the linear analysis[ ], the power-volume analysis[ ], the Hierarchical Two-Tiered Scaling (H2TS)[ ], and the Dynamical System Scaling (DSS)[ ] have been proposed. A wealth of scaling methods has been gradually developed and widely applied to the analysis of the land nuclear reactors based on the above theories. In recent years, Li et al.[ ] derived the cor- responding scaling methods respectively for the single-phase natural circulation in a rectangular loop and the gravity-driven drainage phenomenon in the core makeup tank (CMT) based on the H2TS and DSS theories. The Relap5 code was used to establish the system models[ ]. Xu et al.[ ] conducted the scaling analysis on the transient process of the natural cir- culation in the pressurized water reactor system, and the study compared the differences among the several methods.

However, the research on the scaling analysis of the small heavy liquid metal (HLM) reactors under marine conditions is relatively lacking. In recent years, Zhao et al.[ ] explored the rationality of using the scaled loops to study the LBE nat- ural circulation fast reactors, but the research was based on the steady-state parameters. Xu et al.[ ] developed the scal- ing methods for the transient process of the LBE circulation.

Similar to the previous studies, their study limited the consid- eration of the dynamic process to the power varying with time under land condition. In this study, scaling research on the LBE natural circulation under the marine rolling motions was conducted. Considering the influence of additional forces, a scaling method was developed for dynamic LBE natural cir- culation under rolling motion based on the H2TS theory. Nu- merical simulations of the rolling processes were conducted using Fluent to verify the accuracy of the scaling method.

The influence of the different rolling characteristics was dis- cussed, and the dynamic scaling deviation was subsequently evaluated.

2. SCALING METHOD UNDER ROLLING CONDITIONS

Several motions of the ships and other floating devices in the ocean are shown in Figure a. In this study, the rolling

motion among them was analyzed. For the convenience of 114

analysis, an LBE natural circulation rectangular loop model was built as shown in Figure It is assumed that the

flow area in the loop remains uniform, and the Boussinesq 117

hypothesis[ ] is applied to the LBE fluid. Additionally, it is considered that the loop has good thermal insulation per- formance except for the heating and cooling sections[ Moreover, the specific numerical model will be introduced in Section 3.1.

The mass conservation equation of the LBE natural circu- lation can be expressed as:

ρu i A i = ρu 0 A 0 (1) 125

Sketch of the ship motion, Sketch of the simplified natural circulation loop.

Subscript i denotes any position in the loop, and 0 denotes the reference value. Similarly, the momentum conservation equation under the rolling condition is as follows:

= β s a M ρ ( T h − T c ) H

Herein, denotes the corrected acceleration under the dynamic conditions. For the rolling conditions, we have:

a M = g + a c + a t (3) 132

denote the centripetal acceleration and the tan- gential acceleration respectively during the rolling motion.

They give:

a c = ω 2 r (4) 136

a t = εr (5) 137

In this study, the rolling case was simplified to the sinu- soidal excitation and the variation of the rolling angle with time can be expressed as[

θ ( t ) = θ max sin � 2 π T M t � (6) 141

represents the rolling amplitude while represents the rolling period. Then, the angular velocity and angular acceleration can be expressed respectively as:

d t = θ max 2 π T M cos � 2 π T M t � (7) 145

ω ( t ) = dθ

ε ( t ) = dω

d t = − θ max

We defined the dimensionless parameters: u + = u/u 0 , 147

t + = t/t 0 and ∆ T + = ∆ T/ ∆ T 0 . Thus, Eq. ( 2 ) can be 148

expressed as:

d t + = β s a M H ∆ T 0 u 2 0 ∆ T + − 1 2

To ensure the similarity between the scaled-down and the prototype loops, the following criterion can be derived from the continuity and momentum equations:

Flow area criterion: Richardson number criterion:

Π Ri,R = � β s gH ∆ T 0 u 2 0

R = 1 (11) 157

Friction number criterion:

Π F ri,R = � fL

R = 1 (12) 159

Where, the friction loss coefficient can be expressed as f = 160

Rolling acceleration criterion:

Π a,R = 1 (13) 163

Eq. ( ) can be described in detail as follows: Rolling amplitude criterion:

θ max,R = 1 (14) 166

Centripetal acceleration and tangential acceleration crite- rion:

R = � ω 2 r g

R = 1 (15) 169

R = � εr

R = 1 (16) 170

Where, ω = θ 2 π T M , ε = θ � 2 π T M

Then, the local phenomena of the natural circulation are discussed. The energy balance equation for the heating sec- tion is:

� = 4 q d (17) 175

By defining z + = z/l 0 , Eq. ( 17 ) can be written as: 176

∂z + = 4 ql ρC P du 0 ∆ T 0 (18) 177

The heat source number criterion can be given:

Π Qs,R = � 4 qL ρC P du 0 ∆ T 0

R = 1 (19) 179

The same analysis is applied to the cooling section and the Stanton number is obtained.

� = − k s ∂T s ∂y (20) 182

Π st,R = � 4 Lk s ρC P du 0 δ 0

R = 1 (21) 183

Finally, the similarity criteria of the LBE natural circula- tion under the rolling condition can be summarized as fol- lows:

3.1. CFD model and boundary conditions

The prototype LBE loop shown in Fig. a was investi- gated using the Fluent. The simple rectangular loop is com- posed of a heat source, a heat exchanger, and the adiabatic pipe sections. The rotating axis of the rolling motion was set at the midpoint of the bottom horizontal pipe. The spe- cific design parameters of the rolling loop are presented in performed to investigate the effects of the grid number on the LBE flow rate and the temperature difference between the hot and cold sections in the loop. As shown in Fig. b, when the number of the grids is around one million, the circulation parameters tend to stabilize. To obtain a balance between the computational accuracy and time, the number of grids was set Parameters Value Height of the loop (m) Width of the loop (m) Heating section (m) Cooling section (m) Pipe Diameter (mm) Height between cooling and heating sections (m) Power (kW) In the model, the constant heat flux boundary condition is used for the heat source section, while the wall temperature boundary condition is used for the cooling section. The den- sity variations resulting from the temperature difference drive the LBE natural circulation. The thermal properties of liquid ies, Reference [ ] compared the common turbulence models to analyze their influence on the LBE flow and heat trans- fer characteristics. The results showed that the predictions of the SST k-omega model are in good agreement with the ex- perimental correlations, especially in the low flow rate region corresponding to the natural circulation. Therefore, the SST k-omega model was adopted in this study.

3.2. Validation of computational method

Currently, the Fluent does not include the computational model for the marine conditions. Thus, after establishing the lead-bismuth natural circulation model, it is necessary to use the User-Defined Function (UDF) to add the model of addi- tional force for simulating the rolling motion into the source term. The reliability of the prototype loop and the rolling UDF code was verified by comparing the results of two stud-

ies: one by Li et al.[ 19 ] from the University of Science and 225

Technology of China, based on the KYLIN-II natural circu- lation loop, and the other by Yuan et al.[ ] from Xi’an Jiao-

Sketch of the CFD geometric model, Relationship between the LBE circulation parameters and the number of mesh.

Parameters Expressions

tong University, based on the LECO facility. Figures 3 a and 228

1. Static circulation model validation

The KYLIN-II natural circulation loop is structurally sim- ilar to the present model but differs in geometric parame- ters. The same modeling method was applied to simulate the KYLIN-II loop, and the results were then compared with those from previous research[ ] to verify the reliability of the modeling methodology.

The steady conditions of the natural circulation were cal- culated. By varying the power of the heating section, multiple sets of corresponding data on the temperature difference and the mass flow rate were obtained. A comparison of the cal- culated values from the reference, the experimental data[ that the simulation results are very close to the calculated re- sults from the previous research, and both sets of results fol- low the same trend as the experimental data.

The calculated values of the flow rate and the tempera- ture difference show good agreement with the experimental results at the high input powers. However, when the power is below 5 kW, the calculated mass flow rate is higher than the experimental value, while the temperature difference is lower.

This is mainly due to the relatively significant heat dissipation 252

show that the numerical model method for the LBE natural circulation loop is reasonable.

2. Rolling code validation

Then, the UDF code of the rolling additional force is

verified by comparing with the Xi’an Jiaotong University’s 258

research[ ]. As shown in Fig. b, the LECO loop differs from the loop used in this study only in the location of the cooling section and the pipe diameter of the heating section (test section). Since the differences are minor, the comparison remains relevant. It should be noted that there is an electro- magnetic pump in the LECO loop. However, the validation was only based on the results of the natural circulation. The experimental and preheating sections were subjected to heat flux boundary conditions, while a constant wall temperature boundary condition was applied to the cooling section. The

remaining parts of the loop were regarded as the adiabatic 269

KYLIN-II natural circulation facility, Lead-bismuth eutectic Experimental Circulation facility under Ocean conditions Power-Temperature difference cauve, Power-Flow rate cauve pipes[ The dynamic and time-averaged values of the LBE flow rate were calculated. As shown in Fig. a-d, the UDF code was validated by comparing the simulation results with the reference data. It can be seen that the curves of dynamic and time-averaged values of the mass flow rate are both in good agreement with the previous study[ ] at varying amplitudes and periods.

In summary, the LBE natural circulation rolling model de- veloped in this study is valuable for the further analysis ac- cording to the verification presented in Sections 3.2.

3.3. Dynamic motion cases of prototype circulation

To comprehensively analyze the impact of the characteris- tics of rolling period and amplitude on the scaling method, four rolling motion processes as shown in Table were set up. For the prototype LBE circulation loop, the rolling angles were 20° and 10°, and the rolling periods were 12 s and 8 s respectively. The curves of the flow rate variations with time under the steady-state and motion conditions in the prototype loop were calculated separately and the results in Fig. obtained.

It can be seen that the LBE natural circulation exhibits reg- ular periodic fluctuations under the sinusoidal rolling exci- tation. It should be noted that the cycles presented in Fig. 6 [FIGURE:6] are the several cycles after the rolling motion stabilized.

The amplitude of flow rate fluctuation increases with the in- crease of rolling amplitude and with the decrease of rolling period. This is consistent with the conclusions of the previ- ous research[ ]. Furthermore, the backflow of the LBE occurred in the loop when the fluctuation was large. This in-

dicates that rolling conditions have a significant impact on the 300

stability of the natural circulation.

Comparison of different rolling conditions.

4. RESULTS AND DISCUSSION

The scaling method was applied to the rolling processes of the prototype loop established in Section 3.3. Two com-

mon scaled-down cases with the length ratios L R = 0 . 5 and 305

L R = 0 . 25 were selected to evaluate the scaling distortion. 306

The scaling criteria of the scaled-down models are presented in Table The steady state and time-averaged deviations of the nat- ural circulation flow rate and the temperature difference be- tween the hot and cold sections for the scaled-down models are shown in Table . The flow rates and the temperature dif- ferences of the LBE corresponding to all scaled-down cases are consistent with the theoretical values, with a maximum deviation of 9.08%. This shows that the criteria based on the above scaling method can reflect the time-averaged character- istics of the prototype natural circulation loop.

Then, the dynamic scaling distortions under the rolling pro- cesses were discussed.

To better compare the circulation characteristics under the different cases, the parameters of the scaled-down loop were normalized and the dimensionless curve of the LBE mass flow rate was obtained. The vertical ordinate is defined as follows:

W ∗ = W − W 1 W 2 − W 1 (23) 324

Where represents the transient flow rate variation and

W 1 and W 2 represent the maximum and minimum values dur- 326

ing the rolling processes, respectively. Moreover, to compare the relationship of the circulation characteristics between the scaled-down cases and the prototype within each period more clearly, the horizontal ordinate is defined as follows:

t ∗ = t e t t 0 (24) 331

Where is the time required for the LBE to complete a cycle in the loop, which depends on the time-averaged flow

max,R rate and the total length of the loop. To eliminate the overall distortion caused by the accumulation of time-averaged flow rate errors, the dimensionless time correction factor is defined

as e t = u t /u R,T . Where u R and u R,T represent the ratio 337

of the time-averaged flow velocity between the scaled-down cases and the prototype cases, and the ratio of the theoretical flow velocity, respectively. totype and the scaled-down cases under the different rolling processes. The dimensionless time range of one circulation cycle was selected for the analysis. It is noted that all curves are around 0.5 at the start of the dynamic processes. This in- dicates that the transient values of the flow rate at this mo- ment are close to the time-averaged values throughout the whole cycle, and all cases start a new circulation cycle si- multaneously. The normalized flow rates of all prototypes and scaled-down models exhibit the standard sinusoidal fluc- tuations, which correspond to the applied rolling additional force. Meanwhile, the curves corresponding to the different length ratios are almost identical and highly consistent with the prototype within each period. These results demonstrate that the scaling method can accurately simulate the fluctu- ation characteristics of the flow rate under the rolling pro- cesses. difference which is defined as , where

and ∆ T 2 are the maximum and minimum temperature dif- 360

ferences between the hot and cold sections. Obviously, the temperature difference of the loop does not follow the ideal sinusoidal periodic fluctuations. The curves exhibit different fluctuations with the changes in the rolling amplitude and pe- riod. As shown in Figs. a and b, the temperature differ-

ence exhibits the significant fluctuations around the minimum 366

value within the period. The curve rapidly reaches the mini- 367

mum value and subsequently increases by approximately 0.3.

Then, the curve gradually decreases until it reaches the next period. Comparing Figs. a and c with Figs. b and as the rolling amplitude decreases, the oscillations of the tem- perature difference become increasingly slight, and the curves gradually approach the sinusoidal curve.

Error (%) The differences between the scaled-down models and the prototypes have been compared. In general, the scaled-down models agree with the dynamic characteristics of the tem- perature difference of the prototypes under all motion pro- cesses. However, the scaling cases of the temperature differ- ences show some deviations, which are different from those of the flow rate. As shown in Fig. a, the temperature differ- ence curves of the two scaled-down cases both exhibit hys- teresis compared with the prototype. This is not obvious in the process where the dimensionless curves decrease from

their maximum value, but it is more significant in the oscil- 384

shows that the hysteresis of the curves gradually diminishes 386

as the rolling period decreases. The same conclusion can be obtained by comparing Figs. c and d. In addition, the influence of the length ratios on the scaling deviations is not

significant. 390

scaling errors, with Fig. illustrating the errors of the nor- malized mass flow rate and Figs. illustrating the er- rors of the normalized temperature difference. The ordinate is calculated as , where represents any parameter, and the subscript represents the theoreti- cal values. Fig. presents the results under fixed rolling processes, which compares the cases with different length ra- tios. Although the normalized flow rate curves are consis- tent, the scaled-down cases under different rolling conditions

show different dynamic deviations significantly. The curves 401

in Figs. a and b show such a process within one motion pe- riod. The deviations first maintain a low level for a relatively long time, then increase suddenly, and subsequently enter a short transitional stage. Following this, the deviations fluc- tuate again momentarily and eventually enter a new period.

The scaling deviations are within 10% and 20% for most of the time within one period, and the dashed lines mark this range clearly. This indicates the good agreement between the scaled-down models and the prototypes for most of the time.

The conclusion is similar to that for the dimensionless curve.

However, the flow rate shows significant distortion during 412

some brief moments, and the density of data points confirms

Rolling process 1, Rolling process 2, Rolling process 3, Rolling process 4 this. It can be easily explained from Fig. that the backflow can be observed in the rolling processes 1 and 2. The flow rate shows two extremely low regions near zero during its “forward-backward-forward” circulation within a rolling pe- riod. The low flow rate in the prototype loop causes the tran- sient distortion since the scaled-down cases cannot accurately and synchronously reflect the moments of the extremely low flow rates.

In addition, the backward flow rate in process 1 remains consistently small throughout the period. Consequently, the errors observed in the region between the two distortion peaks Rolling pro- cess 1, Rolling process 2, Rolling process 3, Rolling process 4 are larger for the scaled-down cases of process 1 than for those of process 2. Processes 3 and 4 exhibit distinct error curves, as shown in Figs. c and d. The deviations in Pro- cess 3 are smaller than those in Process 4 because its flow rate fluctuations are slight relative to the time-averaged value.

Furthermore, the curves only exhibit the unidirectional fluc- 430

tuations, as the natural circulation in these processes did not experience backflow.

In summary, the periodic distortion in the flow rate of the scaled-down model is due to the low flow rate regions dur- ing the rolling motion. The backflow is a characteristic of

Rolling process 1, Rolling process 2, Rolling pro- cess 3, Rolling process 4 the rolling motion and may cause the instability within the circulation[ ]. When the natural circulation during the mo- tion processes exhibits backflow, the scaling deviations dur- ing the fluctuation show no obvious relationship with indi- vidual parameters such as rolling amplitude and rolling pe- riod. Excluding the brief periods of the fluctuation, there is good agreement between the models with different length ra- tios and the prototype in all rolling processes. paring the scaled-down cases corresponding to the tempera- Rolling process 1, Rolling process 2, Rolling process 3, Rolling process 4 ture difference between the hot and cold sections, while Fig. shows the results under the different rolling processes for a given length ratio. Overall, the temperature difference dis- tortion exhibits different profiles from those of the mass flow

rate distortion and exhibits no significant deviations. In Fig. 450 [FIGURE:450]

a, the transient error of the temperature difference in rolling process 1 is slightly larger. As shown in Fig. a, the distor- tion is attributed to the hysteresis of the scaled-down curves and their sustained fluctuations within the range of the low temperature difference. Correspondingly, the dynamic dis-

L R = 0 . 5 , b L R = 0 . 25

tortion of rolling processes 2-4 is relatively small. The results are consistent with the previous analysis of Fig. . Moreover, similar to the analysis of the steady cases, Fig. shows that

the length ratio has no significant influence on the dynamic 459

deviations. As shown in Fig. , for a given length ratio, the temperature errors of the LBE natural circulation decrease with the decreasing rolling amplitude and the rolling period.

Based on the time-averaged parameters, the transient er- rors of the similarity criterion numbers for the various scaled-

down cases are listed in Table 6 [TABLE:6] . For convenience of com- 465

parison, the distortion analysis for the Friction number is neglected here since the criterion can be easily satisfied by adjusting the loop resistance coefficient in the experimental facilities[ The deviation of the Richardson number is within 18.56%, while the deviations of the heat source num- ber and Stanton number are less than 9.78%. The deviation of the Richardson number is slightly larger, which can be re- vealed from the analysis of multiple groups of independent variables. Generally, it is clear that the method is applicable to the scaling analysis of the marine rolling motion.

The transient Richardson number distortion, in which the steady and time-averaged values are substituted by the tran- sient parameters was calculated for the comprehensive com- parison, as shown in Fig. a-d. The relative error shows periodic fluctuations with the rolling motion, and in Process 1, the errors of approximately 150% occur at the brief mo- ments. It is easy to understand from Eq. ( ) that the error sources of the Richardson number are the scaling distortion of the flow rate and the temperature difference. Periodic distor- tion caused by the flow instability also induces the dynamic errors of criterion numbers. The same analysis was conducted for the other criterion numbers, and the similar conclusions were obtained. The corresponding figures have been omitted due to the space limitations.

Rolling process 1, Rolling process 2, Rolling process 3, Rolling process

5. CONCLUSIONS

To study the dynamic natural circulation of the LBE under marine rolling conditions, a scaling analysis method was es- tablished and the array of similarity criteria was derived. The

Length ratio Rolling dynamic processes Relative Error of Ri (%) Relative Error of Qs (%) Relative Error of St (%) models were simulated with the Fluent code under the dif- ferent rolling processes. The dynamic scaling distortion for the natural circulation parameters was calculated and the ra- tionality of the scaling method was evaluated. The following (1) The rolling motion exerts the additional forces on the circulation flow. A scaling analysis method for the dynamic rolling process of LBE natural circulation was established by introducing the acceleration criterion. The scaled-down mod- els can be used to effectively simulate the periodic processes of the prototype. The time-averaged values of the LBE flow rate and temperature difference are accurately reproduced, and the errors in all are within 9.08%. Furthermore, the

length ratio has no significant effect on the scaling deviation. 507

(2) The phenomenon of backflow may be induced by the rolling motion, and causes the flow rate to be in an extremely low range at the certain moments. This leads to the periodic and instantaneous scaling distortion. The transient deviation of the flow rate is determined by the specific motion processes when there is backflow in the rolling loop, and its correlation with any individual parameter is not obvious. There is good agreement between the model and the prototype in all rolling processes excluding the brief periods of the oscillation. (3) The temperature difference between the hot and cold sections of the natural circulation does not follow the ideal Z. J. Deng, S. B. Cheng, H. Cheng, Experimental investigation on pressure-buildup characteristics of a water lump immerged in a molten lead pool. Nucl. Sci. Tech. , 1-15 (2023). 10.1007/s41365-023-01188-1 Y. Lu, J. He, Z. Q. Zhu et al., Preliminary calibration test and analysis of electromagnetic flow-meter in liquid lead-bismuth. Nucl. Tech. , 77-82 (2014). j.0253- (in Chinese) J. He, Q. Y. Huang, Z. Q. Zhu et al., Experiment and analysis on flow rate and temperature of liquid PbBi in PREKY. Nucl.

Sci. Tech. 26(1) , (2015). sinusoidal fluctuations. Instead, it produces irregular fluctua- tions as it varies with the rolling parameters. Despite a slight hysteresis, the scaled-down models reflect the fluctuations of the prototype accurately. When the length ratio is fixed, the temperature difference errors of the LBE natural circulation decrease with the decreasing rolling amplitude and the rolling (4) The relative deviations of the Richardson number, Stan- ton number, and heat source number calculated from the time- averaged system parameters are within an acceptable range.

Moreover, their dynamic errors show the periodic fluctuations with the rolling motion, which correspond to the distortion of the circulation characteristics.

In this study, only the rolling condition of the marine mo- tion is analyzed. For the future work, other types of marine

motions will be further considered and the uniform similarity 534

criteria of the complex dynamic processes will be compre- hensively evaluated.

6. ACKNOWLEDGEMENTS

The authors highly regard the efforts of group members and the numerical computations were performed on Hefei ad- vanced computing center.

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