Recent Advances in Computational Convective Heat Transfer Study in

a Sub-Channel for Nuclear Power Reactor and Future Directions

A. S. MOLLAH

Department of Nuclear Science and Engineering,

Military Institute of Science and Technology,

Mirpur Cantonment, Mirpur, Dhaka-1216,

BANGLADESH

Abstract: - A nuclear power reactor's primary use is to generate thermal energy, which in turn produces

electricity. The primary heat source is a nuclear fission event occurring inside the fuel rod. The convection heat

is transmitted through the coolant by the heat energy generated at the fuel rod wall boundary. Better heat

transfer is produced in the flow area by turbulence and irregularity. As a result, turbulent flow heat transfer may

present a significant challenge when predicting and assessing the thermal performance of nuclear power

reactors. Computational techniques in convective heat transfer have become indispensable for solving

challenging issues in the fields of science and engineering thanks to the development of current sophisticated

numerical methods and high-performance computer hardware. The development of novel computational

techniques and models for complicated transport and multi-physical phenomena is constantly in demand

throughout applicable disciplines. This chapter's objective is to provide some recent developments in

computational techniques for convective heat transfer, taking into account research interests in the community

of mass and heat transfer, and to showcase relevant applications in nuclear power plant engineering domains

including future directions. This study describes the most recent advancements in nuclear reactor convective

heat transfer research utilizing the computational fluid dynamics (CFD) method, particularly at Ansys Fluent.

This work examines the convective heat transfer and fluid dynamics fluid dynamics for turbulent flows across

three rod bundle sub-channels that are typical of those employed in the PWR-based VVER type reactor. In this

paper, CFD analysis is carried out using the software tool Ansys Fluent. Temperature distribution profile,

velocity profile, pressure drop, and turbulence properties were investigated in this study. Boundary conditions

i.e. temperature, velocity, pressure, heat flux, and heat generation rate were applied in the sub-channel domain.

The main obstacles and bright spots for the CFD methods in nuclear reactor engineering are discussed, which

helps to further its further uses. We intend to research a full-length fuel bundle model for VVER-1200 in the

future to gather specific fluid characteristic data and use the findings to analyze safety and operate nuclear

power facilities in Bangladesh. This paper presents a thorough analysis of the sub-channel thermal hydraulic

codes used in nuclear reactor core analysis. This review discusses several facets of previous experimental,

analytical, and computational work on rod bundles and identifies potential future directions based on those

earlier studies.

Key-Words: - Computational fluid dynamics (CFD), Rod bundle, Sub-channel, VVER-1200, Convective heat

transfer, Nuclear reactor, NPP, Thermal hydraulic, Test facility.

Received: April 18, 2023. Revised: September 23, 2023. Accepted: November 19, 2023. Published: December 31, 2023.

1 Introduction

Nuclear power plants (NPP) facilities in operation

today are categorized as Generation III or older.

Because of the relatively low operating

temperatures, these NPPs are not as energy-efficient

as they may be, [1]. To create the Generation IV of

nuclear reactors, several nations have started an

international cooperative effort. The long-term

objective of creating such nuclear reactors is to raise

the thermal efficiency from the existing range of

30–35% to 45–50%, [1]. Nuclear power reactors

come in a variety of forms, [1]. They all share the

ability to generate thermal energy, which can then

be used directly, transformed into mechanical

energy, and finally, in the great majority of

instances, into electrical energy. Given the impact of

greenhouse gases on our environment, nuclear

power will contribute more to the world's energy

needs in the future, [2]. A recent International

Energy Outlook report, [3], claims that the need for

clean energy is expected to expand by 49% between

2007 and 2035 (Figure 1). Around 440 nuclear

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power stations are currently in operation around the

world, while another 60 are in various stages of

construction as of April 2023, [4], [5]. One hundred

reactors are in a shutdown status, and several of the

reactors have been operating for quite some time

previously. As the globe works to reduce the

damaging consequences that fossil fuels have on the

environment. The existing NPPs have relatively low

operating temperatures, but the future generation of

nuclear power plants will have even higher thermal

efficiency, increasing the production of secure

electricity, [1]. Along with improved safety features,

nuclear power plants' capacity is growing to lower

their capital costs. More emphasis is placed on

passive core cooling characteristics as a result of the

nuclear accidents caused by Three Mile Island

(1979), Chernobyl (1986), and Fukushima (2011).

The new design of the reactors, particularly the

reactor core and its accompanying systems, must

take into account the lessons discovered through the

studies, tests, and accidents of nuclear systems in

the past. The thermal hydraulic analysis can

primarily guarantee the safety of the nuclear reactor

system under all core conditions, [6], [7], [8], [9].

To prevent core damage, heat must be removed

from the reactor core structure at the same rate that

it is generated. The reactor core is often cooled by

pumping a working fluid, or "coolant," through it.

Depending on the purpose of the nuclear reactor, the

heat that has accumulated in the coolant is then put

to use for a variety of purposes. Nuclear power

reactors use the common steam cycles to convert

heat into electricity. Predicting the temperature and

velocity distributions in the reactor's various

sections is one of the main goals of reactor thermal-

hydraulics (TH), [10], [11], [12], [13], [14], [15],

[16], [17], [18], [19], [20], [21], [22].

Fig. 1: Usage of energy on a global scale, 2007-

2035 (quadrillion Btu), [3].

The reactor core, which generates heat and has

the greatest temperatures, is the most significant

component of the nuclear reactor. For varied reactor

operation conditions, such temperatures must be

estimated. Temperatures must be kept below

prescribed safety limit values for fuel materials to

ensure safe reactor operation. Predicting the stresses

that the flowing coolant will impose on the reactor's

interior structures is a key goal of reactor thermal

hydraulics. Mechanical failures of the structures

may result from excessively high or persistent

oscillatory forces. As a result, the structure-

mechanics analysis uses the mechanical loads in the

nuclear reactor information from the thermal-

hydraulic analysis to examine the integrity of the

system. The impact of temperature distributions

must frequently be considered in such analyses,

particularly when thermal stresses are large. The

reactor-physics analysis supplies information about

the distribution of heat sources, whereas the former

provides input data (the moderator density and other

nuclear statistics not specifically mentioned here) to

the thermal-hydraulic analysis. Thus, if one wishes

to understand how a nuclear reactor works, it is

essential to grasp thermal hydraulics (TH)

characteristics. A quick change in temperature or

pressure, for instance, might have serious effects on

the entire system in one area of the reactor’s safety

analysis. TH needs to be looked at to guarantee a

reactor's safe and reliable power production.

However, the TH-related phenomena can sometimes

be very complicated, necessitating the use of

simulation techniques.

Multiphysics simulations are frequently

employed because a reactor is subject to numerous

different physical phenomena. These can be

achieved by combining various computer codes.

Researchers' diverse experimental and analytical

studies of various sub-channel geometries are

identified. The past research also provides

perspectives for future directions. In the early

phases of sub-channel thermal hydraulic analysis,

the focus of research articles was mostly on

analytical and experimental work done for the

creation of sub-channel analysis codes, [23], [24],

[25], [26], [27], [28], [29], [30], [31], [32], [33],

[34], [35], [36], [37], [38], [39], [40], [41], [42].

These attempts were made to increase the accuracy

of CFD predictions and to derive models from CFD

for novel fuel geometries, decreasing the time

needed for fuel assembly design and optimization,

improving safety, and running the fewest number of

tests possible to conserve resources and time. The

use of the CFD technique to study rod bundle flow

distributions and heat transfer is also presented in

the open literature and is briefly covered in each

subsection with the main conclusions and the areas

that still need to be filled. This article's goal is to

provide a concise review of key ideas in the subject

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of nuclear thermal hydraulics as well as the various

computer programs available for simulating TH-

related events with the System codes (SYS-TH),

sub-channel codes, and CFD codes. The final goal

of this work is to use the CFD code Ansys Fluent to

assess convective heat transfer in sub-channels of

rod bundles and triangle tubes of VVER-type

nuclear reactors, [43], [44], [45], [46], [47], [48],

[49], [50]. Additionally, CFD approaches assist in

obtaining the precise temperature and flow

distributions within assemblies of rod bundles in

square and triangle sub-channels.

2 Need for Thermal-Hydraulic

Analysis in Nuclear Reactor

A nuclear power plant's central core is made up of

several hundred fuel assemblies, each of which

contains several fuel rods. Coolants circulate these

rods or pins, picking up the heat produced by

nuclear fission reactions. The safety authorities

require safety assessments before granting a nuclear

power plant a license, assuring the avoidance of

nuclear disasters like core meltdowns and other

similar events. All of the components must be

designed to meet safety requirements due to this

necessity. The pressure drop and heat transfer

effectiveness under standard, sporadic, and

accidental conditions are two technical issues in the

core. Regarding the safety concern, the clad

temperature is restricted. The study of thermal

hydraulics and mechanics focuses on the physics

and mechanics of liquid flow, energy transmission,

and interactions with surrounding structures in big,

intricate systems like nuclear reactors. The study of

fluid flow, energy (such as heat) transfer, and

interactions between fluid flow, energy, and

supporting structures is known as thermal

hydraulics, [6], [7], [41]. Material interactions with

radioactive radiation are occasionally another

crucial factor in nuclear systems. Consequently, the

goals of thermal-hydraulic analysis can frequently

be divided into two major groups:

a. Design: boundaries established to ensure

that the plant's sturdy design can operate

economically with few shutdowns and low

operational expenses;

b. Safety: the bounds set forth by legal or

regulatory regulations to safeguard the

public's health or way of life.

3 Computer Codes for TH

Thermal hydraulics-related phenomena related to

computer codes have been reviewed in this initial

study, [51]. These can be classified into two primary

groups, one of which is concerned with the flow's

characteristics and the other with heat transfer. The

most often used method in the past has been using

SYS-TH codes. These rely primarily on

experimental validation and make use of coarse

meshes, huge control volumes, and spatial and

temporal simplifications. Sub-channels are used to

separate the core into these smaller, interconnected

control volumes and main. Sub-channel codes,

Computational Fluid Dynamics (CFD) programs,

and System Thermal Hydraulics (SYS-TH) codes

are the three main types of codes used for thermal-

hydraulic analyses of nuclear reactors. In nuclear

power facilities, large-scale phenomena are

addressed with SYS-TH codes, [52]. The CFD

method is most frequently used to examine

particular components, and in some situations, to

verify the output of SYS-TH codes. Numerous

codes, including TRACE [53], CATHARE2 [54],

VIPRE [55], RETRAN-3D [56], ATHLET [57],

[58], COBRA [58], [59], and RELAP [60], [61]

have been developed in response to the significance

of thermal-hydraulic computations. A sub-channel

is a flow passage established between a few or a lot

of rods and the wall of the channel or shroud tube.

As shown in Figure 2, either coolant-centered sub-

channels or rod-centered sub-channels may be used

to construct the sub-channels. The output of certain

CFD simulations can then be combined with SYS-

TH codes to get more precise approximations. Flow

3D [62], PorFlow [63], Flica-4 [64], Sub-chanflow

[65]. Ansys Fluent/CFX [66], [67], OpenFoam [68],

Comsol [69], and STAR-CCM+ [70] are a few

examples of CFD programs. One of the most

promising developments for reactor applications is a

novel technique called computational multifluid

dynamics (CMFD), which focuses on simulating

two-phase flow on the mesoscale or microscale.

Recent work has improved and validated CFD

algorithms for two-phase/multiphase flow issues

including coupling and spacer grid related to reactor

safety, [71], [72], [73], [74], [75], [76], [77], [78].

4 Sub-channel Analysis for VVER

Type Reactor

4.1 Reactor Description

The primary sources of nuclear electricity are light

water reactors (LWRs). The pressurized water

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reactor (PWR) family includes the VVER reactors,

[1]. The pressurized water reactor (PWR) is by far

the most prevalent nuclear system. In Figure 2, a

generic PWR system is shown. A brief description

of Figure 2 is given below:

 Heat is produced by the reactor vessel's core.

 Heat is transferred to the steam generator via

pressurized water in the primary coolant loop

(Loop 1)

 Steam is produced inside the steam generator

when heat from the main coolant loop evaporates

the water in a secondary loop (Loop 2). By

directing the steam into the main turbine, the

steam line generates energy by turning the

turbine generator.

 The condenser is where the leftover steam is

vented and condensed into water (Loop 3). Using

several pumps, the resultant water is pumped out of

the condenser, warmed up, and then fed back to the

steam generator.

Electrically powered pumps circulate water to

cool the fuel assemblies located in the reactor's core.

The electrical grid provides power to these pumps

and other facility-running systems. If off-site power

is interrupted, alternative pumps that run on on-site

diesel generators can provide emergency cooling

water.

Fig. 2: Pressurized water reactor schematic

In terms of plant performance and safety, the

VVER-1200/AES-2006 surpasses the VVER-1000,

[79]. The Generation 3+ V-392M reactor plant

design for AES-2006 is a development of prior

designs employing the VVER-1000 water-cooled

and water-moderated reactor, which has been

successfully operated for many years. The

architecture of the fuel assembly and the core shape

are what separate a Western PWR from a VVER.

The thermal power was increased to 3200 MWt and

extra passive safety mechanisms were included to

control accidents that exceed the parameters of the

design foundation. There will be 1200 MWe of

power produced by the VVER 1200. Both the

Novovoronezh-2 location in Russia and the Rooppur

site in Bangladesh are actively working on the

building of a VVER-1200, [79]. The pressurized

water-moderated VVER-1200/AES-2006 reactor

has 163 fuel assemblies. In addition to the 312 fuel

rods, each fuel assembly has 18 directing channels

for control rods or burnable poisons. A single rod is

used to represent the typical behavior of all the fuel

rods in each channel when modeling the fuel rods,

and the individual sub-channels are pooled to

produce an equivalent flow area in the core analysis.

The reactor coolant system circulates coolant in a

closed circuit to transmit heat from the reactor core

to the secondary side of the reactor. Figure 3 depicts

a schematic illustration of the nuclear reactor's flow

routes, [1].

Fig. 3: Flow path diagram for PWR, [1]

Coolant is transported from intake connections

to the perforated reactor bottom via a double-walled

downcomer, where it rises to the reactor vessel.

After flowing through the lower lattice and

perforated bottom (1), the stream is divided into

control bundle channels and 312 operational fuel

channels of the active core (2). After leaving the

working fuel bundle channels, the heated stream

enters the mixing chamber (3). It then moves via

apertures in the guide tubes' walls to the mixing

chamber through the control bundle channels and

guide tubes (4). The reactor coolant system is made

up of a reactor, a pressurizer, and four circulation

loops. Each of these loops has a steam generator, a

reactor coolant pump set, and major coolant

pipelines that connect the loop equipment to the

reactor. Between the primary and secondary sides is

a steam generator. Figure 4 depicts the fuel

assembly model, [1]. Table 1 presents the primary

side's major design and thermal-hydraulic

performance during the reactor plant normal

operation.

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Fig. 4: VVER-1200 assembly configuration, [48]

The FAs are made to generate heat and transmit

it from the fuel rod surface to coolant over the

course of their intended service life while staying

within the permitted design parameters for fuel rod

damage. The FAs have a nominal height of 4570

mm. The height of the fuel rod's power-generating

portion in the hot state of the reactor is 3750 mm. A

tube made of zirconium alloy covers the fuel rod.

Inside the cladding are stacked sintered UO2 pellets

with a 5% (4.95%) maximum enrichment. Ceramic

uranium dioxide (UO2) pellets with a melting point

of 2800 °C serve as the fuel. The VVER-1200 core

is constructed using four distinct enriched fuel types

(1.6%, 2.4%, 3.6%, and 4.95%). The cylindrical

pellets are then placed into helium-filled tubes made

of a corrosion-resistant zirconium metal alloy

containing 1% Nb to help with heat conduction. An

average fuel rod produces 167.8 W/cm of linear

heat.

Table 1. Thermal hydraulic parameters, [48]

Parameter

Value

Reactor nominal thermal power,

3300

Coolant inventory in the reactor

coolant system , m3

290

Coolant inventory in PRZ at

nominal power operation, m3

Primary pressure at the core outlet,

absolute, MPa

16.2

Coolant temperature at reactor inlet,

298.2

Coolant temperature at reactor

outlet,oC

328.9

Coolant flow rate through rector, m3/h

86000

Primary side design parameters:

-gauge pressure, MPa

-temperature, oC

17.64

350

Pressure of the primary side hydraulic

tests, MPa

-for tightness;

-for strength

17.64

24.5

A sub-channel in a nuclear reactor is a section

of the fuel assembly that is enclosed by fuel rods.

The long, cylindrical tubes that contain the nuclear

fuel pellets are called fuel rods. The heat from the

fuel rods is transferred to the coolant as it passes

through the sub-channels. The size and shape of the

sub-channels, which are distributed in a regular

pattern throughout the reactor core, are determined

by the reactor's design and operational conditions.

The thermal-hydraulic behavior of sub-channels is

an important consideration in the design and

operation of nuclear reactors. Therefore, reactor

operators continuously monitor the sub-channel

temperatures and coolant flow rates to ensure the

safe and efficient operation of the nuclear reactor. In

cross-section, reactor core fuel assemblies can be

square, hexagonal, or circular as depicted in Figure

Fig. 5: An example of a sub-channel, [10]

Rod diameter, rod-to-rod pitch (p), type, and

position of rod gaps, as well as, for arrays within

shrouds, the distance between the rods and the

shroud as well as the geometry of the shroud, are the

key elements that determine the thermal-hydraulic

performance of rod bundles. Radial and axial

variations in the fuel rod power output also cause

significant variations in the array's coolant flow rate

and thermal coolant conditions. A 1200 MWe

nuclear power reactor's hexagonal fuel sub-

assembly underwent a 3D CFD analysis for

turbulent flow in the interior, edge, and corner sub-

channels. Since the three sub-channels geometrical

forms and boundary conditions varied, so did the

effects of the operational parameters. The findings

may be used to support the following observations,

[48]:

1) When compared to the other two sub-channels,

the corner sub-channel has much higher turbulent

attributes such as turbulent kinetic energy,

turbulence dissipation rate, and turbulent eddy

viscosity over the axial length. This eddy viscosity

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causes the flow velocity in the corner sub-channel to

drop.

2) The fluid temperature is lower at the edge sub-

channel region than in the other two sub-channels

due to the larger Nusselt Number there. However,

the corner sub-channel has a higher chance of DNB

and the creation of a hot spot in the fuel rod next to

it than the other two sub-channels due to a falling

Nusselt number and increasing temperature and

fluid enthalpy along its axial length.

3) The length of the fuel sub-assembly should be

such that the Nusselt number can achieve its fully

developed state to ensure proper heat evacuation

from the fuel rod surface. In this simulation, the

edge and interior sub-channels both reach the fully

developed Nusselt number. Here, the algorithm

should more than take into account the length of the

fuel subassembly to find a fully developed Nusselt

number in the corner sub-channel.

4) The increase in friction factor slope is greater at

the corner sub-channel along the direction of fluid

flow because there is higher turbulent viscosity

there. Because the coolant flow area is larger near

the edge, the friction factor is higher.

5) In all three sub-channels, the relative pressure

decrease is virtually constant for different Reynolds

numbers.

4.2 Thermal Hydraulic Analysis of the Sub-

channel

The behavior of the coolant and fuel rods in the sub-

channel under various operating conditions is

investigated as part of the thermal-hydraulic

analysis of a sub-channel of a VVER-1200 reactor

fuel assembly. The geometry of the VVER sub-

assembly results in three different types of sub-

channel meshes: triangular (interior sub-channel)

over the bulk of the bundle, edge sub-channel, and

corner sub-channel, as shown in Figure 6, [48].

Fig. 6: Sub-channels of the fuel sub-assembly of

VVER-1200 reactor

Calculating several parameters, such as the

coolant's flow rate, temperature, pressure, and heat

transfer coefficient as it travels through the fuel

assembly sub-channel, is covered by the study.

Nuclear processes inside the fuel rods produce heat,

which is delivered to the coolant via the cladding

and coolant gap. The heat transfer coefficient

determines how effectively heat is transferred

between the coolant and the fuel rods. A thermal

hydraulic analysis of a sub-channel in the fuel

assembly of a VVER-1200 reactor is required to

guarantee the safe and efficient operation of the

reactor. It aids in the detection of potential issues

and the optimization of the reactor's design and

operational parameters to improve both its usability

and safety.

Turbulent flow is a type of fluid flow in which

the fluid moves in an irregular and chaotic manner,

with fluctuations in velocity and pressure occurring

randomly in both time and space. This type of flow

occurs when the fluid is moving at high velocities or

when the flow encounters obstructions or

irregularities in the channel or pipe. Turbulent flow

is characterized by high Reynolds numbers

(Re>4000), which indicate that the inertial forces in

the fluid are much larger than the viscous forces. In

turbulent flow, the fluid particles move in a

complex, three-dimensional pattern, with eddies and

vortices forming and breaking down randomly

Table 2 contains a listing of the VVER-1200

reactor’s boundary conditions as well as its

operating temperature and pressure, [48].

Table 2. Boundary conditions for sub-channel

model

The general CFD methodology employed in the

current investigation is presented in Figure 7 in the

form of a self-explanatory flow chart.

The Ansys Fluent code is used to solve each of

these equations. Two numerical approaches have

been used in Ansys Fluent:

• solver depending on pressure

• solver based on density

The density-based technique was primarily

employed for high-speed compressible flows,

whereas the pressure-based approach was created

for low-speed incompressible flows. Recently, both

approaches have been improved and recast to

address problems and work for a variety of flow

situations outside of their original or conventional

meaning.

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Fig. 7: A flow chart describing the analysis's overall

methodology

The momentum equations are used in both

techniques to determine the velocity field. The

density field is obtained using the continuity

equation in the density-based approach, whereas the

pressure field is derived using the equation of state.

According to the assumptions of a steady,

incompressible flow, the differential equations

governing flow, turbulence, and heat transfer are as

follows, [47]:

Continuity Equation:

󰇛

󰇍



󰇜  (1)

Conservation of momentum:

󰇛󰇜  󰇟  󰇛  󰇛󰇜󰇜󰇛µ 

µ󰇜

󰇛󰇜󰇛µ󰇜

󰇠   (2)

Conservation of energy:

󰇟

󰇍



󰇛  󰇜󰇠  󰇟  

󰇍



 󰇛

.

󰇍



)

(3)

Where ke is the effective conductivity and ke is the

turbulent thermal conductivity defined according to

the turbulence model being used. The first three

terms on the right hand side of Eq. (3) represent

energy transfer due to conduction, species diffusion,

and viscous dissipation, respectively, [47].

Heat Transfer in Fluid:

  󰇛󰇜    

(4)

Turbulent kinetic energy equation:

󰇛󰇜  󰇣󰇡   

󰇢󰇤    

(5)

Specific dissipation rate equation:

󰇛󰇜  󰇟󰇛  󰇜󰇠 





 (6)

Turbulent Eddy viscosity:

  󰇛󰇜 (7)

Where values of constants in the k-ε turbulence

model are:

Cμ= 0.09; Ce1= 1.44; Ce2=1.92; σk=1.0; σe = 1.3.

Ansys 2021R1 is the numerical simulation tool

utilized in this work. Establishing governing

equations, boundary conditions, and initial

conditions are typically steps in the process of

creating a mathematical model. The standard

principles of conservation in physics, such as mass

conservation, momentum conservation, and energy

conservation, apply to generic fluid movement.

These conservation laws are expressed

mathematically in the governing equation. The

continuity equation and the momentum equation can

be utilized directly as the governing equations if

there is no heat exchange in the fluid flow inside the

hydraulic turbine. The differential equation related

to the numerical solution is as follows:

Governing equation

Continuity equation:

∂/∂+∂/∂(f)=0 (8)

Since the density of an incompressible fluid, ,

is constant across time (∂/∂=0), the continuity

equation for that fluid can be reduced to the

following differential form.

∂/∂=0 (9)

Universal momentum equation:

∂/∂+(∂/∂)=+1/(∂/∂) (10)

The constitutive relation

=(∂/∂+∂/∂)- −(2/3)  (∂/∂) 

of Newtonian fluid is substituted into the above

equation.

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∂/∂+∂/∂()=−(1/)(∂/∂)+(/)(∂2/∂∂

)+(1/3)(/)(∂/∂)(∂/∂) (11)

The momentum equation may be further

reduced for incompressible fluids, where the

continuity equation necessitates ∂/∂=0.

/∂+∂/∂()=−(1/)(∂/∂)+(/)(∂2/∂∂)

(12)

where  is the mass force on the unit mass fluid,

 is the stress tensor,  is the second-order

unit tensor,  is the fluid density, is the

pressure, and  is the time.

The turbulence equation must be included since

the flow in a hydraulic turbine is typically turbulent.

The standard k-ε model, RNG k-ε model, and

Realizable k-ε model are often used CFD turbulence

models in Fluent fluid analysis. To varied degrees,

the three k-ε models can reproduce the features of

fluid flow; nevertheless, when powerful swirl flow

and curved streamline flow are simulated, the

Standard k-ε model exhibits distortion. Like the

RNG k-ε model, the Realizable k-ε model can be

used for numerical calculations of different kinds of

fluid flows and offers advantages in swirl flow

prediction. The water flow in the nuclear reactor's

sub-channel is simulated using the Realizable k-ε

model. Here are the k-ε equations displayed.

∂/∂()+∂/∂()=∂/∂[(+/)(∂/∂]

+− (13)

∂/∂()+∂/∂()=∂/∂[(+/)(∂/∂)]+1

−2{2/(+√)} (14)

Where=(2/), =2, 1=max(0.43,/(+5)), =

(/), =√(2), =1/2(∂/∂+∂/∂)

The formula mentioned above has the following

values: t = fluid density (d), k = turbulent kinetic

energy, d = dissipation rate, d = dynamic viscosity,

and v = coefficient of kinematic viscosity. The

coordinate components are  and ; the speed in

the direction of i and j, respectively, is denoted by

 and  (i, j = 1, 2, 3); the word responsible for

producing turbulent kinetic energy due to average

velocity is . S represents the tensor coefficient for

average strain rate; k and ε's Prandtl numbers are 

and , respectively; The turbulent viscosity

coefficient is denoted by Vt; The turbulent field (k

and ), the rotation rate, average strain, and the

system's angular rate of rotation all influence ;

the model coefficient is C1, and the constants are

 = 1.0,  = 1.2, and 2 = 1.9.

The two conveyed variables in the k ˵

turbulence model are the rate of dissipation of the

turbulent kinetic energy, ε, and the energy itself; k.

Fluent employed the following equations in the

conventional k- ϵ model, [66]:

where k and ε are the inverse effective Prandtl

numbers for k and ε, respectively, which are

computed variables that Fluent uses to modifyeff,

eff is the effective dynamic fluid viscosity.

It is anticipated that the SST k- model will

yield more precise and dependable predictions for a

larger class of flows by utilizing the k- model in

the near wall region and the k-ε model for the free

stream zone [66]. The SST k- model makes use of

the following transport equations, [66]:

where Yk and Y represent the turbulence-induced

dissipation of k and , D is the cross-diffusion

term, G is the production of , and Gk and G are

the effective diffusivity of k and .

Boundary conditions, such as wall boundary

conditions, import and export boundary conditions,

etc., are necessary for the control equations to have

a conclusive solution. The correctness of the

computation results is directly impacted by the

choice of boundary conditions.

Fluent discretizes the governing equations using

the finite volume method. Its main concept is as

follows: a non-repetitive control volume surrounds

each grid point in the calculating region, which is

separated into grids. A set of discrete equations is

obtained by integrating the governing equations

over each control volume. The discrete equations in

this paper were interpolated using the second-order

upwind scheme, closed using the Realizable k-ε

turbulence model, and solved with pressure and

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velocity coupling using the Semi-Implicit Method

for Pressure Linked Equations (SIMPLE) algorithm.

For the purpose of studying the sub-channels with

coolant as a water channel within, which removes

heat from the fuel rods and transfers it to the steam

generator, the flow pattern is turbulent in nature. In

the fluid domain, steady-state heat transfer is

maintained in addition to forced convection flow

analysis. Construction of the computational domain

is done using ANSYS 21. The ANSYS 2021 R1

geometry module is used to create the center, edge,

and corner sub-channel geometries. Figure 8

represents the center, edge, and corner sub-channels

geometry that is used in this study, [48].

4.2.1 Subchannel Geometry

Depending on the core geometry, the sub-channel

can have either a square or a triangular shape. A

group of connected sub-channels that are part of a

simulated rod bundle are thought to contain a one-

dimensional flow. Through cross-flow mixing, the

channels are connected, [48]. Compared to SYS-TH

codes, the meshing utilized in sub-channel codes is

more precise. We need to take geometrical

simplification into consideration in many nuclear

thermal-hydraulic problems. The geometry is highly

essential since it affects the boundary condition in

the CFD calculation, unlike the system code where

geometry-related information must be provided by

the code user in an averaged fashion. The mesh has

a significant role in the precision of the calculated

findings. However, because it has limitless degrees

of freedom, it is difficult to directly compare to

other mesh configurations and represents

geometry quantitatively. The user-supplied

nodalization utilized in system codes is replaced by

grid generation techniques in the CFD approach.

Geometrical data, such as the pressure drop

coefficient, is frequently provided as an average in

the system codes. This strategy may be troublesome

since the geometry has an impact on the boundary

conditions, which are crucial to CFD computations.

4.2.2 Generation of Mesh

Mesh generation comes next, after geometry

creation. The Ansys fluent mesh tool was used to

generate mesh for each of the three different

geometrical models (different sub-channels).

Similar to meshing in finite element simulations, the

development of a mesh in CFD simulations affects

the simulation's accuracy and processing time. The

mesh generated by the problem's grid generation

approach will attempt to fit the geometry of the

system under simulation. Ansys's meshing

capability reduces the time and effort needed to get

precise results.

a. Center Sub-channel b. Edge Sub-channel

c. Corner Sub-channel

Fig. 8: Geometry of 3 sub-channels

Ansys contributes by developing more

automated and improved meshing tools because

meshing frequently consumes a significant portion

of the time required to achieve simulation results.

The elements take inflation at the model's borders

and are orientated during the mesh production of

tetrahedral structures. A properly defined turbulence

velocity profile requires a domain length of 3.75 m,

which is 10% of the total heat generated by the fuel

rod and long enough to produce the desired pattern.

The geometry and mesh were created via the

ANSYS 2021R1 program. Figure 9, Figure 10 and

Figure 11 shows outlet mesh, center sub-channel

whole domain mesh, edge sub-channel whole

domain mesh, and corner sub-channel whole

domain mesh receptively.

A cell's stretching is gauged by its aspect ratio.

The distances between the cell centroid and face

centroids and nodes are two examples of the

distances that are used to compute it. The ratio of

the greatest value to the minimum value of any of

these distances is used to calculate it. Below are

some tetrahedral mesh statistics, including

skewness, aspect ratio, orthogonal quality, and y+:

Average Skewness: 7.89x10-2 (Range: 3.786 x 10-6-

0.7)

Orthogonal quality: 0.9654 (Range: 0.783-1.000)

Aspect ratio: 19.876 (Range: 3.783-68.238)

Aspect ratios of other elements are computed

from the aspect ratio of a perfect tetrahedral

element. When normalized concerning a perfect

tetrahedral, the aspect ratio of an element is the ratio

of its longest edge to the shortest normal dropped

from a vertex to the opposite face. The aspect ratio

of a perfect tetrahedral element is 1.0 by definition.

To assess the mesh quality, the Ansys software

software computes the aspect ratio. The majority of

the parts in a high-quality mesh have an aspect ratio

of fewer than five. This study's aspect value ranges

from 3.783-68.238), with an average value of

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19.876. This value agrees well with the values seen

in the literature, [25].

Fig. 9: Demonstration of computational geometry,

[43]

Fig. 10: Tetrahedral grid generation over the

geometry (Centre sub-channel), [43]

(a)

(b)

Fig. 11: a) Edge sub-channel mesh and b) Corner

sub-channel mesh, [48]

For the convergent solutions, the residual value

of the energy equation is set to be 10-8, the residual

value of the momentum equation, k, and  equations

to be 10-6, and the residual value of the continuous

equation to be 10-4. The number of grids that meet

the convergence requirement is about 4.8x106.

Several mesh sizes were investigated for this

purpose, and the convergence criterion was very

well satisfied at 4.8x106. The residuals of every

equation were tracked, and the behavior graph for

each is displayed in Figure 12. After 3000 iterations,

every chosen truncation criterion was met, with the

continuity equation criterion being the first. With a

quick beginning decrease and a less steep rate in the

final iterations, the residual reduction reported by

other authors, [43], [44] is obtained for all

equations.

Fig. 12a: Residual values vs iteration with 4.8x106

meshes

It is also appropriate to monitor the drag

coefficient to examine convergence, as stated in the

fluent user handbook [66]. The value of this

coefficient has not changed much since the thirtieth

iteration, as seen in Figure 12b, indicating solution

stability [66].

Fig. 12b: Drag coefficient for every cycle

4.2.3 Boundary Conditions

The solid surface walls of the fuel rod claddings do

not slip or have a smooth boundary condition for the

flow computations in the domains. The constraint

was put on the symmetry planes in the symmetries.

The sub-channels lower portion and top surfaces

were configured as periodic interfaces, which aid in

simulating the flow in situations with fully

developed turbulent flow. That is shown in Figure

13, [48].

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Fig. 13: Boundary conditions of center sub-channel,

[48]

The mass flow rate utilized as the entrance

velocity for the sub-channels is 5.66 ms-1, and the

Reynolds number Re = 3.75×104 is determined from

Equation (8).

Re = (ρuDh)/µ (8)

Where, ρ = Density of the fluid (g/cc); u = Velocity

(m/sec); Dh = Hydraulic diameter (m); µ = Viscosity

(Pa.s).

The non-dimensional Nusselt number, Nu, is

utilized to directly assess the heat transmission

characteristics. Equation (9) gives Nu's definition.

Nu=

 (9)

Where, h is a convective heat transfer coefficient

(W/m2.K); and where k is a thermal conductivity of

the fluid [W/m.K].

The following formula is used to get the

convective heat transfer coefficient, [1].

h= 

󰇛󰇜 (10)

where, q is the heat flow rate of the fuel (W), A

is the heat transfer area(m2); and temperature

difference ΔT(K).

4.2.4 Set the Border and the Starting Conditions

On the borders of the flow domain, flow, thermal,

and turbulence variables should be provided. All

solids and liquids in the simulation should have their

material qualities stated. If the boundary and

beginning conditions at the CFD level are not

provided in the validation problem or if they must

be imposed in the application problem, they should

be appropriate and their sensitivity should be

assessed later. The commercial CFD code Ansys

Fluent chooses a few physical models that are

coherent with the meshing: It is decided to use the

Reynolds-Averaged Navier-Stokes (RANS) model,

specifically the k-turbulent model, which entails

adding a further turbulent viscosity to the

momentum and energy equations. The momentum

equation and the energy equation are not connected.

The turbulent viscosity is calculated using the

turbulent energy per unit mass, k-, and the

dissipation per unit mass. Each of these two terms is

the answer to a transport equation. The wall law

model is applied, which is a two-layer all y+

treatment (Figure 14). The non-dimensional wall-

adjacent grid height, which depends on the fluid

characteristics and the skin friction coefficient, is

simply the y+ value (equation 11).

y+=

 (11)

Where, y+ is a non-dimensional measurement of

distance from a wall,density of the fluid, ux is the

velocity of the fluid in x-direction and y is the

distance of the first node from the wall and is fluid

viscosity.

For coarser meshes (y+ ≥30), ANSYS Fluent

offers the option to use the normal wall treatment

approach; for low wall y+ meshes (y+≤1), it offers an

improved wall treatment option. The value of y+

must fall between 1 and 30 to be employed in the

current simulation's k- model with an enhanced

wall function technique. The y+ distribution of the

mesh created for the computation using the

enhanced wall function method and the k- model

is displayed in Figure 14. It can be observed that y+

values range from <1 to ~5. As a result, the values

of y+ satisfy the conditions of the turbulent model

used in this investigation.

Fig. 14: y+ value over wall fuel rod for polyhedral

mesh, [43]

Fig. 15: Turbulence kinetic energy along flow

channel in 3 sub-channels

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Fig. 16: Specific dissipation rate along flow channel

in 3 sub-channels

Fig. 17: Eddy viscosity along flow channel in 3 sub-

channels

Fig. 18: Static enthalpy along flow channel in 3 sub-

channels

5 Results and Discussion

5.1 Turbulence Kinetics

Due to the high Reynolds number, coolant water

flow that is moving axially is turbulent. The flow

characteristics and temperature distribution of the

fluid are significantly influenced by turbulent

features such turbulence kinetic energy, specific

dissipation rate, eddy viscosity and static enthalpy.

According to Figure 15 and Figure 16 the corner

sub-channel has a greater concentration of all

features than the inner or edge sub-channels.

Additionally, compared to the center and edge sub-

channels, the corner sub-channel has a somewhat

higher rate of specific dissipation. And according to

Figure 17 the corner sub-channel has a lower

concentration of all features than the center or edge

sub-channels. Additionally, compared to the center

and edge sub-channels, the corner sub-channel has a

somewhat lower eddy viscosity. Figure 18 depicts

that the static enthalpy of the corner sub-channel is

higher than edge sub-channel as well as lower than

the center sub-channel.

5.2 Velocity Distribution

The velocity distribution along three sub-channels is

depicted in Figure 19 based on flow channel. 5.66

ms-1 is the constant inlet velocity. All of the sub-

channel velocities sharply increase to 10.4% of the

overall length. Then, a progressive increase in

velocity was seen along the center and edge sub-

channels. However, as seen in Figure 15 and Figure

16 the higher turbulence kinetic energy and specific

dissipation rate causes the velocity of the corner

sub-channel to drop. According to Figure 17, lower

eddy viscosity is another cause of the velocity of the

corner sub-channel drop. This results from the

formation of huge eddies and their subsequent

splitting into smaller eddies. Internal fluid friction

increases in this area more than in other sub-

channels throughout these processes, which causes

flow velocity in the corner to drop.

Fig. 19: Velocity distribution along flow channel in

3 sub-channels

5.3 Temperature Distribution

Figure 20 displays the temperature distribution

along the flow channel in 3 sub-channels. The

temperature distribution was measured using the

reference line that ran along the axial direction of

the graph, which showed that the temperature in the

inlet was maintained at 571.2 K throughout the

process. The temperature down the path went up

because the uniform heat flux of the fuel rod was

distributed evenly across the sub-channel. The 2.3 K

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rise in temperature represents the 15% increase in

height of the active fuel rod. A comparison was

made between the analytical solution to the

temperature problem and the data that was

generated numerically. The error percentage was

found to be almost 0.18 %.

Fig. 20: Temperature distribution along flow

channel in 3 sub-channels

5.4 Pressure Drop Variation

The average pressure at the entrance is 16.2 MPa,

and the outflow pressure is constant at 16.19 MPa.

Here, the use of water as a coolant result in a total

pressure drop of 12.9 kPa, 13.7 kPa, and 54.2 kPa in

the center, edge, and corner sub-channels,

respectively. Figure 21 represents the variation of

pressure drop along flow channels in 3 sub-

channels. Viscosity shear stresses in the fluid and

turbulence along the inner walls of the flow channel

prevent fluid from flowing through the flow

channel. The formula is as follows, [1], [66]:

Where, ∆ = pressure loss (Pa) f =friction

coefficient, = length of the flow channel (m), D =

hydraulic diameter (m)  = density (kg/m3 ), and Vm

= velocity (m/s).

There is a drop in pressure in the sub-channel

along the axial direction of the channel. Pressure

decreases as a result of eddies or vortices forming

within the turbulent flow. These eddies are produced

by the mixing of fluid particles traveling at different

speeds and directions, which can result in pressure

pockets inside the flow. More turbulence from these

eddies increases the overall pressure drop in the

axial direction. The subchannel’s pressure

distribution is described in Figure 21.

Fig. 21: Pressure drop along the axial distance of the

sub-channel in 3 sub-channels

5.5 Reynolds Number Variation

According to equation (12), velocity and Reynolds

number are related. The hydraulic diameter, Dh,

which varies in the three sub-channels, determines

the Reynolds number. Dh of the interior sub-channel

was found to be 6 mm, for the edge sub-channel to

be 10 mm, and for the corner sub-channel to be 4

mm using equation (12). The change in Reynolds

number is depicted in Figure 22. Greater Reynolds

numbers in the edge sub-channel indicate that

convective heat transfer occurs here more frequently

than in the other two sub-channels. The length of the

sub-channel must be fixed for adequate heat

removal while taking into account the bare

minimum length required to get a fully developed

Reynolds number. Reynolds number peaks in the

corner sub-channel in 5Dh and steadily drops

throughout the flow channel. This implies that the

convective heat transfer in this sub-channel is

decreasing, but the static fluid enthalpy in the corner

sub-channel is growing, as illustrated in Figure 18.

Therefore, there is a chance that nucleate boiling

(DNB) will develop at the corner sub-channel for

low Reynolds number and high static enthalpy of

fluid, [1]. A two-phase flow might develop as a

result of nucleate boiling, which would reduce the

convective heat transfer coefficient.

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Fig. 22: Reynolds number (Re) variation along flow

channel in 3 sub-channels

5.6 Friction Factor

The velocity in various sub-channels is used to

compute the friction factor. Figure 23 illustrates

how the friction factor varies along three sub-

channels of the flow channel. Due to a larger

coolant flow area, the friction factor is higher in the

edge sub-channel. Interestingly, the additional eddy

viscosity in the flow area of the coolant sub-channel

causes its friction factor gradient along the flow

channel to be higher than that of the interior sub-

channel while having a smaller flow area. The

following formula can be used to determine the

friction factor (f):

f=(τw / (0.5 * ρ * U2))*L/D

Where, τw is the wall shear stress, ρ is the fluid

density, L is the length and D is the hydraulic

diameter. The velocity of the free stream is U.

Fig. 23: Variation of friction factor along flow

channel in 3 sub-channels

5.7 Nusselt Number and Convective Heat

Transfer Coefficient Variation

The Nusselt number variation along flow channels

in 3 sub-channels is depicted in Figure 24. And the

convective heat transfer coefficient variation of the

center sub-channel along the flow channel is

depicted in Figure 25.

Fig. 24: Nusselt number variation along flow

channel in 3 sub-channels

Fig. 25: Heat transfer coefficient variation of center

sub-channel

5.8 Validation

Validation is very important for any simulation

model. For this purpose, in this study validation is

done with another model. In this study k-omega

SST model is done for all simulating working

processes and validation is done by the k-epsilon

model. That is shown in Figure 26, Figure 27 and

Figure 28. The velocity, temperature, and pressure

drop variation between k-omega SST and k-epsilon

model is shown in Figure 26, Figure 27 and Figure

28 respectively. In addition, there is a slight change

in the value of velocity, temperature, and pressure

drop along the flow channel but the nature of the

graph is almost the same between the two models.

Fig. 26: Validation of velocity

Fig. 27: Validation of temperature

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Fig. 28: Validation of pressure drop

6 Future Directions

According to the literature review on the topic, it is

also possible to model numerous experimental

studies on turbulent mixing under various fluid

conditions by using sub-channel and CFD analysis.

As a result, trustworthy CFD mixing forecasts can

be made. Thus, to reliably estimate the safety

margins of the nuclear reactor core, the study will

focus on the following sub-channel analysis gap

areas:

a. better heat transfer and mixing correlation thermal

hydraulic sub-channel analysis tools;

b. enhance code to handle numerous sub-channels

and fuel rod configurations;

c. CFD code to incorporate validation and

verification of previous research studies; and

d. Create a thermal hydraulic test facility (Figure

29) on a laboratory scale, [78], [80] to validate CFD

models and offer technical assistance to NPP

operators.

Fig. 29: General view of the integrated thermal

hydraulic test facility, [80].

7 Conclusions

Sub-channel analysis algorithms are frequently

employed in nuclear reactor core thermal-hydraulic

analyses to calculate various thermal-hydraulic

safety margins. The safety margins and operating

power limits of the nuclear reactor core under

various key system variables, such as system

pressure, coolant input temperature, coolant flow

rate, and thermal power and its distributions, are the

primary parameters for sub-channel analysis. Due to

the intricacy of the rod bundle shape, the range of

turbulent scales, and the resource limitations, a full-

scale computational fluid dynamic (CFD)

simulation of a nuclear reactor core is time- and

labor-intensive to complete. Knowledge of forced

convective heat transfer within sub-channels formed

between multiple nuclear fuel rods or heat

exchanger tubes, not only in the fully developed

regime but also in the developing regime or laminar

flow regime, is necessary for the development of

new procedures in nuclear reactor safety aspects and

optimization of modern nuclear reactors, [66], [67],

[68].

With the use of ANSYS's turbulence model, the

three sub-channels such as center, corner, and edge

using three fuel rods have been analyzed. The

effects of turbulent flow on temperature distribution,

velocity variance, pressure drop, friction factor,

Reynolds number, etc. have been examined for

different sub-channels of VVER-1200 via a rod

bundle with three sub-channels that are a good

representation of those employed in the VVER-

1200. A 311 rod bundle is projected to require

around 500 times as many computing cells as a 3

rod bundle with 700 mm lengths. As a result, there

will be a total of one billion cells, or around 500

times the number of three rod bundles. The

necessary computer memory will be 500 times

larger, or 1000 GB RAM. The development of

parallel CFD software and structural mesh-

generating tools for rod bundles is the focus of

research activities to address these issues. The study

provides valuable information for future research

and practical applications in the field of nuclear

power plants.

In terms of the model's application, the

suggested models can be duplicated and used in

other fields, such as air conditioning, heating,

cooling, and particularly hydraulic turbines. This

study adds to the body of knowledge regarding the

performance simulation of thermal hydraulics for

different kinds of nuclear reactors.

As it is a new type of power plant (VVER-

1200), more research is needed to understand its

thermal hydraulics behavior and for upgrading its

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operation capacity. In conclusion, despite significant

developments in CFD over the past couple of

decades that have made it possible to perform

thermal hydraulics analysis along with neutronics,

further study is needed to develop suitable coupling

codes such as SuperMC and SUBCHANFLOW,

[28], [65], [70], [71], [72], [73]. [74], [78]. Future

research on the full-dimensional fuel bundle model

is intended to give in-depth information on the

characteristics of liquids and apply the findings to

the operation and safety analyses of VVER-1200

nuclear power plants that will be constructed in

Bangladesh.

Acknowledgments:

The author would like to thank everyone who

provided assistance, inspiration, and support. As

some of them might not be alive to witness the

outcome of their encouragement, we have chosen

not to name them all. But, we do hope that they can

at least read this acknowledgment, wherever they

may be.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

A. S. Mollah contributed towards the problem

selection, the simulations, the analysis, wrote the

paper and finally approved the final manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This research received no external funding.

Conflict of Interest

The author declares that no known competing

financial interests or personal relationships could

have appeared to influence the work reported in this

paper.

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