Edited by: Kaiyi Shi, Liupanshui Normal University, China
Reviewed by: Claudio Tenreiro, University of Talca, Chile; Khalil Ur Rahman, Pakistan Nuclear Regulatory Authority (PNRA), Pakistan
This article was submitted to Nuclear Energy, a section of the journal Frontiers in Energy Research
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Numerical simulation has been widely used in nuclear reactor safety analyses to gain insight into key phenomena. The Critical Heat Flux (CHF) is one of the limiting criteria in the design and operation of nuclear reactors. It is a twophase flow phenomenon, which rapidly decreases the heat transfer performance at the rod surface. This paper presents a numerical simulation of a steady state flow in a vertical pipe to predict the CHF phenomena. The detailed Computational Fluid Dynamic (CFD) modeling methodology was developed using FLUENT. Eulerian twophase flow model is used to model the flow and heat transfer phenomena. In order to gain the peak wall temperature accurately and stably, the effect of different turbulence models and wall functions are investigated based on different grids. Results show that O type grid should be used for the simulation of CHF phenomenon. Grids with Y+ larger than 70 are recommended for the CHF simulation because of the acceptable results of all the turbulence models while Grids with Y+ lower than 50 should be avoided. To predict the dryout position accurately in a fine grid, Realizable kε model with standard wall function is recommended. These conclusions have some reference significance to better predict the CHF phenomena of vertical pipe. It can also be expanded to rod bundle of Boiling Water Reactor (BWR) by using same pressure condition.
Critical Heat Flux (CHF) is a twophase flow phenomenon that is characterized by a heat transfer mechanism change which rapidly decreases the efficiency of the heat transfer performance at the heater surface. When it occurs, heated surfaces are no longer wetted by boiling liquid and the vapor phase start to occupy the heat surface. As a result, the energy is directly transferred from the heat surface to vapor. It results in rapid reduction of the heat removal ability and sharp rise of the vapor temperature, as well as the rod surface temperatures which are important for the nuclear safety. During the design and operation of the nuclear reactors, CHF value should be calculated in advance. But experience and thousands of data points have shown the complexity of CHF phenomenon for different conditions. In the past decades, both experiment and numerical simulation are widely used to predict the CHF value during the design and operation process. During the experimental process, the wall temperature is monitored by use of thermocouples. Along with the increasing of heat power, CHF phenomenon is detected when the temperature of one thermocouple has a sharp rise. Similar to the experiment, this sharp rise of the wall temperature is also a signal of CHF in the numerical simulation. So it is of great significance to predict the wall temperature accurately and stably.
Nowadays, experimental method is widely used in the CHF prediction. But most of them are focused on the vertical pipe. Large scale experiments of rod bundle are few reported or openaccess around the world. In the reactor design area, CHF lookup table is the main method to determine the CHF value. But both these two methods will give a large safety margin, and then reduce the power level. Compared to experimental measurements and CHF lookup table, numerical simulation has its own advantages. Thermalhydraulics system or subchannel codes can efficiently characterize bulk flow behavior, while computational fluid dynamics (CFD) analysis may provide relatively accurate results for the velocity and temperature profiles around the rods. Although the subchannel analysis is more widely used for CHF prediction, CFD method can work as a supplement to provide more accurate results. Furthermore, numerical simulation can reduce the cost significantly when we use different geometries, materials, and test conditions. More accurate simulation techniques using these combined analysis tools can be used to optimize experimental designs and improve the testanalysis design cycle for advanced fuel rods.
Grids and turbulence models are found to be very important in the Computational Fluid Dynamic (CFD) simulation. At the moment, turbulence models used in two phase flow are still same with that of single phase flow. Especially in Eulerian two phase model, the liquid and vapor phases are applied with same turbulence model separately. In single phase flow, the results of Chen et al. (
Although different turbulence models are used for the calculation of two phase flow, few researchers has ever compared the effect of different grids and turbulence models. In general, a grid dependency study should always be performed in high quality CFD simulation. Even though it is hard to obtain fully grid independent results in some cases, it is necessary to discuss the effect of key grid parameters. Thakrar et al. (
However, as we can see above, few researchers have ever shown the sensitivity analysis on turbulence models and wall functions for CHF phenomenon. The grid dependency study is also ignored in the CFD simulation of boiling twophase flow, especially on the CHF phenomenon. It is still not clear that if the turbulence models have the same performance on CHF phenomena as that of subcooled boiling flow, especially on the value and position of highest wall temperature. So based on Eulerian twophase model, this paper presents numerical simulations of CHF phenomenon by the use of CFD method to investigate the effect of different grids, turbulence models and wall functions. Lack of open CHF experimental data of rod bundle, especially the wall temperature distribution, we use vertical pipe instead. Considering both the flows in rod bundle and vertical pipe are vertical upward flow, the flow and heat characteristics are similar to each other as long as we use same pressure condition. The type of boiling crisis is dryout which can be identified by the distribution of wall temperature and void fraction. Results are focused on the wall temperature distribution to give a reference for the detector of CHF phenomena.
Eulerian twophase models are used as the basic models to simulate the two phase flow in a vertical pipe. All the interfacial mass, momentum and energy transfer models are based on the interfacial area density model. Furthermore, CHF model will also be shown below.
The conservation equation of Eulerian two phase model includes six equations which are mass, momentum and energy equations for two phases separately. These six equations will be solved for six parameters which include pressure, velocities of liquid and vapor, temperatures of liquid and vapor, volume fraction.
(1) Mass equations
(2) Momentum equations
(3) Energy equations
Where α,ρ,
The interfacial momentum transfers between liquid and vapor phases are decided by the five forces which are the drag force, lift force, wall lubrication force, virtual mass force and turbulence dispersion force.
For fluidfluid flow, each secondary phase is assumed to form droplets or bubbles. This assumption gives a method to derivate the drag force which is the most important force of these five parameters. Drag force refers to the force that droplets or bubbles suffer because of the different velocities. As we know, because of viscosity, the general form of the drag force is:
When it is used for a particle in the fluid, this form will be changed to
Where
Associated with Re number whose form is
We can gain a new form of drag force of the droplets or bubbles which is
So in per unit mixture volume, the drag force can be expressed as
Where
Lift force act on a particle mainly due to velocity gradients in the continuous phase flow field. It is calculated by the function from Drew and Lahey (
In this equation,
Wall lubrication force which tends to push the bubbles away from the wall can be expressed as
Where
Virtual mass effect should be included when the secondary phase accelerates relative to the primary phase. It is defined as
The turbulent dispersion force acts as a turbulent diffusion in dispersed flows. It plays an important role in driving the vapor away from the vicinity of the wall to the center of the channel. Bertodano (
Where
The interfacial energy transfers includes two parts which are the heat from liquid to vapor phase at the near wall region and the heat transfer between vapor and liquid phases in the subcooled bulk. As the bubbles depart from the wall and move to the subcooled region, there is heat transfer from the bubble to liquid that is defined as
The heat transfer coefficient can be computed using Ranz and Marshall (
The interface to vapor heat transfer is calculated using the constant time scale return to saturation method which was provided by Lavieville et al. (
In this equation, δt is the time scale.
Depends on the theory, interfacial mass transfer can also be divided in to two parts: liquid evaporation near the wall and liquid evaporation or vapor condensation in the main stream. The former is calculated on the basis of evaporation heat flux which will be introduced below.
The latter can be calculated directly based on the interfacial energy transfer and latent heat
Different from the heat transfer of single phase flow on the near wall region, the heat transfer of twophase flow includes three different types heat transfer. As we can see from the subcooled boiling flow, the energy is transferred directly from the wall to the liquid. Part of this energy named convective heat flux will cause the temperature of the liquid to increase and part which is called evaporative heat flux will generate vapor. Interphase heat transfer will also cause the average liquid temperature to increase; however, the saturated vapor will condense. In addition, there is a quenching heat flux which model the cyclic averaged transient energy transfer related to liquid filling the wall vicinity after bubble detachment. These basic mechanisms are the foundations of the Rensselaer Polytechnic Institute (RPI) models developed by Kurul and Podowski (
To model boiling departing from the nucleate boiling regime, or to model it up to the critical heat flux and post dryout condition, it is necessary to include the vapor temperature in the solution process. In such cases, some of the energy may be transferred directly from the wall to the vapor. So in total, the wall heat partition will be expressed as
The four heat fluxes
Where
The nucleate site density
The empirical parameters
The bubble departure diameter
The area of influence
The coefficient
The last parameter is the frequency of bubble departure
Associated with
In the model, the critical heat flux is defined to occur when the volume fraction of vapor reach given values which is provided by Tentner et al. (
Where α_{v, 1} = 0.9 and α_{v, 2} = 0.95.
The numerical analysis is performed using the commercial CFD code FLUENT 16.0 (Ansys Fluent,
CHF experiment data in an upward flow vertical pipe from Becker (
Geometry and boundary condition.
It has been tested that the grid type has limit influence on the temperature distribution in the singlephase flow (Chen et al.,
O type grid, arbitrary Hexaprism grid and Triprism grid.
All the turbulence models can be divided into high Reynolds and low Reynolds turbulence models according to its application. The standard kε model (Launder and Spalding,
The turbulence model should be used with suitable wall function or resolve the boundary layer with a fineenough mesh without any wall functions. As we know, the nearwall region can be subdivided into three sublayers, i.e., viscous sublayer, buffer layer and fully turbulent region (Pope,
In order to validate the correlation of Y+, grids and turbulence models, five sets of grids are used under different turbulence models and wall functions. The detailed information about grid is listed in Table
Grids information.
ICEM1  1056000  0.5  182.2–393.9  7.12–633.5 
ICEM2  1152000  0.25  99.7–192.9  5.04–320.3 
ICEM3  1536000  0.187  68.9–129.0  4.05–215.8 
ICEM4  1904000  0.125  53.1–105.2  3.32–152.2 
ICEM5  1632000  0.1  12.7–78.7  3.73–133.3 
Computational matrixes of grids, turbulence model and wall function.
Standard kε  EnhWF  EnhWF  EnhWF/StdWF 

RNG kε  EnhWF  EnhWF  EnhWF/StdWF 
EnhWF/StdWF  
Realizable kε  EnhWF  EnhWF  EnhWF/StdWF 
EnhWF/StdWF  EnhWF 
Standard kω  EnhWF  EnhWF  EnhWF  
SST kω  EnhWF  EnhWF 
Wall functions.
Standard wall function  The momentum and energy are connected using empirical formula in the nearwall region and fully developed turbulence region 
The first node should be set in the log law region where 

High numerical stability  
Scalable wall function  Limiter: Y + = max(Y +, 11.225) 
It is more suitable for low Y+ region  
Nonequilibrium wall function  Separation and adhesion of boundary layer, high gradient of turbulence 
It is not suitable for kω turbulence models  
Enhanced wall function  It can be used in Coarse grid or grid with Y+ around 1. So it is suitable for complex models and flow 
Not all the possible combinations are calculated in this paper for the numerous computational time and resource. Because of the same basic mechanism for the three kε models, the results of each can be used to the others. So the simulation of ICEM1 to ICEM5 on Realizable kε models and enhanced wall function are calculated to investigate the effect of different grids. This work is also done for the standard kω models on ICEM3~ICEM5. The effect of turbulence models are compared based on ICEM3 and ICEM4. We believe the conclusion can be extended to the other grids. For the wall function part, different wall functions can only be used on different kε models. So in the kω models, no wall functions are calculated except the enhanced wall function.
The effect of grid type, grids number, turbulence and wall functions are discussed below. Results are compared among the highest temperature values and dryout location of the inner surface of the pipe. The temperature distribution is the key saferelative parameter in the CHF experiment and should be paid more attention to discuss.
As has been validated, the grid type has limit influence on the temperature distribution in single phase flow. But for two phase flow, a little difference is found among three grid types. The grid types are shown in Figure
Temperature distribution of the inner surface of the pipe.
Radius distribution of temperature at the 4.5 m height.
Grid independent calculation is always an important requirement in the numerical simulation. It is necessary to estimate the quality of gird through the calculations of different grids' level. However, it is not easy to gain fully gridindependent results for all the simulation, especially for two phase flow. But It has been tested that all the kε models and kω models with enhanced wall function can be used for a large scale of Y+ values from one to hundreds in single phase flow. Zhang et al. (
Figure
Surface temperatures under different grids on standard kε model and enhanced wall function.
The surface temperature results including ICEM4 are shown in Figure
Surface temperatures under different grids on RNG kε model and enhanced wall function.
For a further validation, the comparison results including the ICEM5 are shown in Figure
Surface temperatures under different grids on Realizable kε model and enhanced wall function.
Surface temperatures under different grids on Standard kω model.
Y+ value of ICEM5 on Realizable kε model and enhanced wall function.
Different turbulence models are applied to the CHF calculation with enhanced wall function based on ICEM3 and ICEM4. The results are shown in Figures
Surface temperatures under different turbulence models on ICEM3.
Surface temperatures under different turbulence models on ICEM4.
Figure
When ICEM4 is employed to the CHF calculation, large differences are found among the different turbulence models. The results are shown in Figure
Heat transfer and flow characters in near wall regions are important to the calculation of temperature and velocity in twophase flow. The wall functions and wall boiling model are also influenced by each other. Because the wall functions use different empirical correlations to solve the equations in viscous sublayer and buffer layer, different wall functions will have different performance. In FLUENT, kω models are low Reynolds models and set to be used with wall functions unless the grid Y+ doesn't meet the requirements of the low Reynolds number modeling. If the requirements are met, then FLUENT doesn't apply wall function but resolves the near wall region. So there is no need to discuss the effect of different wall functions when use kω models.
Based on ICEM3, the CHF calculation results of kε models under different wall functions are shown in Figure
Surface temperatures under different wall functions on
The purpose of scalable wall functions is to force the usage of the log law in conjunction with the standard wall functions approach. This is achieved by adding a limiter in the Y+ calculations Y + is calculated by
So when the Y+ is larger than 11.25, Y+ is the same with original value, and then the scalable wall function is the same as standard wall function. Otherwise, Y+ is equal to 11.25 which bring different results. For the cases in this paper, all the Y+ values are larger than 11.25, so there is no difference for the results between these two wall functions. When compared with experimental data, the standard wall function and scalable wall function show the best performance while results of the other two are also acceptable.
Based on ICEM4, the comparison is only investigated between standard wall function and enhanced wall function on two kε models. The results are shown in Figure
Surface temperatures under different wall functions on ICEM4.
CHF phenomenon in a vertical pipe is investigated by use of CFD simulation. Simulations are calculated among different grids type, grids number, different turbulence models and wall functions to detect the effect of these parameters. Several conclusions are obtained after comparing the simulation results and experimental data. The conclusion about the effect of grid and turbulence models can help someone reduce their workload on mesh validation and set up solve algorithm quickly.
type or axissymmetric grid is recommended for the CFD simulation of vertical pipe or the other geometries.
Grids with Y+ larger than 70 are recommended for the CHF simulation while Grids with Y+ lower than 50 should be avoided for the simulation of CHF phenomenon in the vertical pipe.
The temperature distributions of kω models are 20K higher than that of kε models in all the cases. But they are all acceptable when comparing with experimental data.
The predicted dryout position of enhanced wall function is always behind that of the others if used with a fine grid. To predict the dryout position accurately in a refine grid, Realizable kε model with standard wall function is recommended.
XD is the main author of the paper, he finished the research and write. ZZ is the supervisor and give the guide for the other authors. DL is the coauthor who works for the comparison of the experiment and simulation. ZT is the associated supervisor and give the reference too. GC works with the other authors for the discussion part.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work is supported by the Research on Key Technology of Numerical Reactor Engineering (J121217001), and the authors also appreciate deeply the suggestions from Dr. Tenglong Cong.