Optimization of the fuel rod's arrangement cooled by turbulent nano fluids flow in pressurized water reactor(PWR)

M.Hatami*,M.J.Z.Ganji,I.Sohrabiasl,D.Jing *

1 International Research Center for Renewable Energy,State Key Laboratory of Multiphase Flow in Power Engineering,Xi'an Jiaotong University,Xi'an 710049,China

2 Department of Mechanical Engineering,Esfarayen University of Technology,Esfarayen,North Khorasan,Iran

3 Department of Renewable Energies Engineering,Faculty of New Sciences&Technologies,University of Tehran,Tehran,Iran

4 Department of Mechanical Engineering,Babol University of Technology,Babol,Iran

1.Introduction

One of the most important applications of nano fluids is cooling process.Although using nano fluids as working fluids in the primary cooling loop of light water reactor(LWR)has a number of limitations because any change in the reactor core materials affects the criticality and hence the effective neutron multiplication factor,but some researchers worked on numerically[1,2].Anyway,many other applications of nano fluids for cooling processes are presented in the literature such as modeling TiO2nano fluid for cooling the fuel rods which is presentedviaFig.1 by Zarifi and Jahanfarnia[1].

Nano fluids application is not limited for rod's cooling.For example,Aziz[3]considered the outcome of time-dependent chemical reaction on the flow of a viscous fluid past an unsteady stretching sheet.Also,magneto hydrodynamic squeezing flow of a viscous fluid between parallel disks was analyzed by Domairry and Aziz[4].Mustafaet al.[5]solved the problem of the fluid flow between parallel plates by Homotopy Analysis Method(HAM).Turkyilmazoglu[6]solved momentum and energy equations of nano fluids analytically to deduce the flow and heat transport phenomena in two theoretical cases,single phase and multi-phase.When the nanoparticles are uniformly distributed across the condensate boundary layer called it single phase and when the concentration of nanoparticles through the film is allowed to vary from the wall to the outer edge of the condensate film in the light of modified Buongiorno's nano fluid model named multi-phase.Solution of particle's motion in different fluids media has been considered by the authors widely[7-10].

Comparison of the single and two-phase modeling forthe nano fluids has been considered by the researchers.For instance,Haghshenas Fardet al.[11]compared the results of the single phase and two phase numerical methods for nano fluids in a circular tube.They reported that for Cu-water the average relative error between experimental data and CFD results based on single-phase model was 16%while for two-phase model was 8%.In another numerical study,G?ktepeet al.[12]compared these two models for nano fluid convection at the entrance of a uniformly heated tube which found the same results and confirm the accuracy of two-phase modeling.Mohyud-Dinet al.[13]in an analytical study,considered the three dimensional heat and mass transfer with magnetic effects for the flow of a nano fluid between two parallel plates in a rotating system.As one of their main outcomes,thermophoresis and Brownian motion parameters are directly related to heat transfer but are inversely related to concentration pro file.Also they found that higher Coriolis forces decrease the temperature boundary layer thickness.Three-dimensional flow of nano fluids under the radiation(due to solar oretc.)has been analyzed by Hayatet al.[14]and Khanet al.[15].They also computed and examined the effects of different parameters on the velocity,temperature,skin friction coefficient and Nusselt number of nano fluid flow.Other works in the field of nano fluids flow and heat transfer analysis can be found in[16-22].

Fig.1.Control volume between fuel rods filled by nano fluids[1].

There are some simple and accurate approximation techniques for solving nonlinear differential equations called the Weighted Residuals Methods(WRMs).Collocation,Galerkin and Least Square Method(LSM)are examples of the WRMs which are introduced by Ozisik[23]for using in heat transfer problems.Stern and Rasmussen[24]used collocation method for solving a third order linear differential equation.Vaferiet al.[25]have studied the feasibility of applying Orthogonal Collocation method to solve diffusivity equation in the radial transient flow system.Recently Hatami and Ganji[26]used LSM for heat transfer study through porous fins also they used this accurate method for fully wet circular porous fin[27],semispherical porous fins[28]and straight solid and porous fins[29].Shaoqin and Huoyuan[30]developed and analyzed least-squares approximations for the incompressible magneto-hydrodynamic equations.Ghasemiet al.[31]found that LSM is more appropriate than other analytical methods for solving the nonlinear heat transfer equations.

Based on the above short review,all the researches in this field focused on the application of nano fluids in a case study of cooling rods in pressurized water reactors(PWRs),so the lack of numerical optimization[32,33]study on the rods geometry cooled by nano fluids flow is observed to find the best conditions of cooling or heat transfer by Nusselt number analysis.In the present study,the authors aim to optimize the heat transfer mechanism for fuels rod in a typical PWR cooling by Al2O3nano fluids.

2.Problem Description

As described above nano fluids have many applications in industry in which one of them is cooling process in PWR.In this study as shown in Fig.2,an arrangement offuelrods for a typical PWR is considered which is cooled by Al2O3-water nano fluids.The main aim of this study is to find the best values for rod's diameters and distance from each other to reach the best cooling performance or maximum heat transfer by Nusselt number.

3.Numerical Analysis

The numerical simulation is performed with a three dimensional turbulent flow system.Dimensional governing equations are[2]:

Continuity equation:

Fig.2.Control volume of fuel rods considered in this study.

Table 1Thermal propertied of Al2O3-water nano fluid in different volume of fraction[34]

Table 2Different mesh and grid numbers for mesh independency study

Momentum equations:

Energy equation:

where Γ=μ/Prand Γt=μt/Prtand

where turbulent viscosity can be calculated by

In this paper,RNGk-ε model is selected for turbulence model due to its efficiency in boundary layer modeling near walls.Transport equations for RNGk-ε are:

and

whereGkrepresents the generation of turbulence kinetic energy due to mean velocity gradients andGbrepresents the generation of kinetic turbulent energy due to buoyancy forces.In this method constants areC1ε=1.42 andC2ε=1.68[2].

Thermal properties of Al2O3-water nano fluid can be used from Table 1 or by the following equations

These types of dynamic viscosity(Eq.(10))and thermal conductivity(Eq.(11))are proposed by Wanget al.(1999)and Hamilton-Crosser(1962),respectively which is introduced in Ref.[2].For heat transfer analysis,heat transfer coefficient can be calculated by,

Fig.3.Results of different generated meshes(Table 2)a)Nusselt number and b)Y+.

Fig.4.Validation of the results by experimental work by Pak and Cho[34].

Table 3Different geometries proposed by CCD for rod's diameter and distance

And then the non-dimensional average Nusselt number(Nu)over the wall can be found by

Also,Nazififardet al.[2]showed thatNucan be calculated by

In this study,ANSYS-FLEUNT commercial software package is used to solve the set of Eqs.(1)-(7).As mentioned above,RNGk-ε turbulence model is used while the SIMPLEC algorithm is utilized to deal with the pressure-velocity coupling.The second order upwind scheme is performed to discretize the convection terms because it offers a better accuracy.The convergence criteria are set such that the residual errors for continuity,momentum,energy,kand ε reduce to less than 10-5.

Fig.5.a and r parameters for geometry and 9 different geometries proposed by CCD.

4.Optimization Analysis

Generally,the structure of the relationship between the response and the independent variables is unknown.The first step in response surface methodology(RSM)is to find a suitable approximation to the true relationship.The most common forms are low order polynomials( first or second-order).Second order model can significantly improve the optimization process when a firstorder model suffers lack of fit due to interaction between variables and surface curvatures.A general second-order model is defined as[33]:

Fig.6.Different temperature pro files for nano fluids cooling the fuel rods.

wherexiandxjare the design variables andAare the tuning parameters.CCD or central composite design is one of modules in RSM to obtain the points of each factor according to their levels.CCD contains an imbedded factorial or fractional factorial design with center points that is augmented with a group of‘star points’that allow estimation of curvature.If the distance from the center of the design space to a factorial point is±1 unit for each factor,the distance from the center of the design space to a star point is±α with|α|＞ 1.In CCD technique,optimization is based on a parameter called “desirability”.Desirability is an objective function that ranges from zero outside of the limits to one at the goal.The numerical optimization finds a point that maximizes the desirability function.The characteristics of a goal may be altered by adjusting the weight or importance.For several responses and factors,all goals get combined into one desirability function.The goal of optimization is to find a good set of conditions that will meet all the goals,not to get to a desirability value of 1.0.Desirability reflects the desirable ranges for each response(di).The simultaneous objective function is a geometric mean of all transformed responses:

Fig.7.Different velocity pro files for nano fluids cooling the fuel rods.

Fig.8.Effect of different geometries on Nusselt number for 4.33%Al2O3 nano fluid.

wherenis the number of responses in the measure.If any of the responses or factors falls outside their desirability range,the overall function becomes zero.For simultaneous optimization each response must have a low and high value assigned to each goal.On the CCD worksheet,the “Goal” field for responses must be one of the five choices: “none”, “maximum”, “minimum”, “target”,or “in range”,according their definition:

Maximum:

Fig.9.Effect of different nanoparticles volume fraction on Nusselt number for geometry number 8.

Minimum:

whereYiis theith response value andwtis the weight of that response.Weights give added emphasis to the goal.Weights greater than 1(maximum weight is 10),give more emphasis to the goal and less than 1(minimum weight is 0.1),give less emphasis to the goal.

5.Results and Discussions

As mentioned above,the main aim of this study is to optimize the fuel rod's arrangement in a pressurized water reactor(PWR),numerically.A sample case geometry is modeled by Solid Works software as shown in Fig.2 for mesh generation.After mesh generation and defining the boundary condition according to Fig.2,it is necessary to examine the mesh independency andY+ranges to be in acceptable ranges.Fig.3 shows the mesh independency study for six different mesh numbers detailed in Table 2.Meshes consist of QUADS and HEXAS cells as described in detailviaTable 2.As seen in this figure,mesh accuracy is suitable for about 600000 grid number,alsoY+is in acceptable range(below 10.0)for this mesh numbers for RNGk-ε turbulent model.To validate the results,a comparison of the optioned results(Nusselt number)with the previous experimental data by Pak and Cho[34]is performed which can be found in Fig.4.This figure also confirms the high accuracy of the applied numerical modeling compared to experimental results.After this validation,CCD is applied to find the most important cases for theaandrparameters(half of rod's distance and diameter)as shown in Fig.5.CCD proposed 9 critical designs for rod's arrangement as shown in Fig.5 and their details are presentedviaTable 3.All these nine geometries are designed and their results are excluded from 3D numerical modeling in various Reynolds number and nanoparticles concentration.Since this study focuses more on the optimization arrangement of fuel's rod,just Al2O3nanoparticles are considered due to its wide application in this purpose,while,as Turkyilmazoglu[35]discussed,many other types of nanoparticles can enhance the heat transfer or cooling process and can be examined in the optimized geometry.A short investigation of TiO2nanoparticles on the cooling performance also is done in the last section of current paper.

Figs.6 and 7 present the temperature and velocity of the nano fluid between the rods in these geometries.This is completely evident that when the rod's diameter is larger or distance between the rods is smaller,thermal/velocity pro files around the neighborhood rods affect on each other.As seen in Fig.8,case 6 has the maximum Nusselt number among the tested cases,but it is necessary to check the other possible cases which are not designed.This fact is possible by response surface methodology(RSM)analysis which is introduced in the following.Fig.9 confirms that increasing the nanoparticle volume fraction makes an increase in heat transfer and Nusselt number consequently.

RSM analysis results are presentedviaFigs.10-13 for low(around 10000)and high(around 40000)Reynolds numbers.As these figures confirm when distance parameter(a)is in the minimum level and diameter parameter(r)is in the maximum possible level,cooling the rods will be better due to higher Nusselt number in this situation.This result is independent from Reynolds number value(high or low)because in high and low Reynolds numbers the same results were observed.By calculating the desirability functions,the most efficient cases are presentedviaTables 4,5 for this problem.As seen in both tables,whena=6 andr=5 the best efficiency of cooling process will occur.Finally to show the effect of nanoparticles type on theNuvalues,Table 6 is presented when φ=4.33%.As seen,in all Reynolds numbers Al2O3has largerNunumber compared to TiO2due to its higher thermal conductivity.Also,this table shows the percentage improvement of nanoparticles on the cooling process which confirms that Al2O3averagely 17%and TiO210%improve the Nusselt numbers.

6.Conclusions

In this paper,a numerical optimization of the fuelrod's diameter and distance is performed for a typical pressurized water reactor(PWR)cooling by Al2O3-water nano fluid.Response surface methodology(RSM)by using computational fluid dynamic(CFD)outcomes is applied to find the best performance or maximum heat transfer for the cooling process.The tests are performed for three nano particles concentration and Reynolds numbers.Finally,a central composite design or CCD analysis is applied on the results to optimize the geometries.Results confirm that the optimum case is when the distance of the rods is minimum and diameter in the maximum values.

Fig.11.Desirability function for 4.33%Al2O3-water nano fluid in low Reynolds flow(10000).

Acknowledgments

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China(No.51422604,21276206)and the National 863 Program of China(No.2013AA050402).This work was also supported by the China Fundamental Research Funds for the Central Universities.

Fig.10.Effect of geometry parameters on Nusselt number for 4.33%Al2O3-water nano fluid in low Reynolds flow(10000).

Fig.12.Effect of geometry parameters on Nusselt number for 4.33%Al2O3-water nano fluid in high Reynolds flow(40000).

Fig.13.Desirability function for 4.33%Al2O3-water nano fluid in high Reynolds flow(40000).

Table 4Geometry optimization for 4.33%Al2O3-water nano fluid for turbulant and low Reynolds flow

Table 5Geometry optimization for 4.33%Al2O3-water nano fluid for turbulant and high Reynolds flow

Table 6Comparison of different nanoparticles on Nusselt improvement when φ=4.33%,a=7.1 and r=4.25

[1]E.Zarifi,G.Jahanfarnia,Subchannel analysis of TiO2nano fluid as the coolant in VVER-1000 reactor,Prog.Nucl.Energy73(2014)140-152.

[2]M.Nazi fifard,M.Nematollahi,K.Jafarpur,K.Y.Suh,Numerical simulation of water based alumina nano fluid in subchannel geometry,Sci.Technol.Nucl.Installations2012(2014),928406,http://dx.doi.org/10.1155/2012/928406.

[3]M.Abd-El Aziz,Unsteady fluid and heat flow induced by a stretching sheet with mass transfer and chemical reaction,Chem.Eng.Commun.197(2010)1261-1272.

[4]G.Domairry,A.Aziz,Approximate analysis of MHD squeeze flow between two parallel disks with suction or injection by homotopy perturbation method,Math.Probl.Eng.2009(2009)603916.

[5]M.Mustafa,T.Hayat,S.Obaidat,On heat and mass transfer in the unsteady squeezing flow between parallel plates,Meccanica47(2012)1581-1589.

[6]M.Turkyilmazoglu,Analytical solutions of single and multi-phase models for the condensation of nano fluid film flow and heat transfer,Eur.J.Mech.B.Fluids53(2015)272-277.

[7]M.Hatami,G.Domairry,Transient vertically motion of a soluble particle in a Newtonian fluid media,Powder Technol.253(February 2014)481-485.

[8]M.Hatami,D.D.Ganji,Motion of a spherical particle on a rotating parabola using Lagrangian and high accuracy Multi-step Differential Transformation Method,Powder Technol.258(May 2014)94-98.

[9]M.Hatami,D.D.Ganji,Motion of a spherical particle in a fluid forced vortex by DQM and DTM,Particuology16(October 2014)206-212.

[10]A.S.Dogonchi,M.Hatami,G.Domairry,Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow,Powder Technol.274(April 2015)186-192.

[11]M.Haghshenas Fard,M.Nasr Esfahany,M.R.Talaie,Numerical study of convective heat transfer of nano fluids in a circular tube two-phase model versus single-phase model,Int.Commun.Heat Mass Transfer37(2010)91-97.

[12]S.G?ktepe,K.Atal?k,H.Ertürk,Comparison of single and two-phase models for nano fluid convection at the entrance of a uniformly heated tube,Int.J.Therm.Sci.80(2014)83-92.

[13]S.T.Mohyud-Din,Z.A.Zaidi,U.Khan,N.Ahmed,On heat and mass transfer analysis for the flow of a nano fluid between rotating parallel plates,Aerosp.Sci.Technol.(2015),http://dx.doi.org/10.1016/j.ast.2015.07.020.

[14]T.Hayat,M.Imtiaz,A.Alsaedi,M.A.Kutbi,MHD three-dimensional flow of nano fluid with velocity slip and nonlinear thermal radiation,J.Magn.Magn.Mater.396(2015)31-37.

[15]J.A.Khan,M.Mustafa,T.Hayat,A.Alsaedi,Three-dimensional flow of nano fluid over a non-linearly stretching sheet:An application to solar energy,Int.J.Heat Mass Transf.86(2015)158-164.

[16]M.Hatami,D.D.Ganji,Natural convection of sodium alginate(SA)non-Newtonian nano fluid flow between two vertical flat plates by analytical and numerical methods,Case Stud.Therm.Eng.2(March 2014)14-22.

[17]M.Fakour,A.Vahabzadeh,D.D.Ganji,M.Hatami,Analytical study of micropolar fluid flow and heat transfer in a channel with permeable walls,J.Mol.Liq.204(April 2015)198-204.

[18]S.E.Ghasemi,M.Hatami,A.Kalani Sarokolaie,D.D.Ganji,Study on blood flow containing nanoparticles through porous arteries in presence of magnetic field using analytical methods,Physica E:Low-dimensional Syst.Nanostruct.70(June 2015)146-156.

[19]S.E.Ghasemi,M.Hatami,G.H.R.Mehdizadeh Ahangar,D.D.Ganji,Electrohydrodynamic flow analysis in a circular cylindrical conduit using least square method,J.Electrost.72(1)(February 2014)47-52.

[20]M.Rahimi-Gorji,O.Pourmehran,M.Hatami,D.D.Ganji,Statistical optimization of microchannel heat sink(MCHS)geometry cooled by different nano fluids using RSM analysis,Eur.Phys.J.Plus130(February 2015)22.

[21]G.Domairry,M.Hatami,Squeezing Cu-water nano fluid flow analysis between parallel plates by DTM-Padé Method,J.Mol.Liq.193(May 2014)37-44.

[22]A.R.Ahmadi,A.Zahmatkesh,M.Hatami,D.D.Ganji,A comprehensive analysis of the flow and heat transfer for a nano fluid over an unsteady stretching flat plate,Powder Technol.258(May 2014)125-133.

[23]M.N.Ozisik,Heat Conduction,second ed.John Wiley&Sons Inc,USA,1993.

[24]R.H.Stern,H.Rasmussen,Left ventricular ejection:Model solution by collocation,an approximate analytical method,Comput.Boil.Med.26(1996)255-261.

[25]B.Vaferi,V.Salimi,D.Dehghan Baniani,A.Jahanmiri,S.Khedri,Prediction of transient pressure response in the petroleum reservoirs using orthogonal collocation,J.Pet.Sci.Eng.98-99(2012)156-163.

[26]M.Hatami,D.D.Ganji,Thermal behavior of longitudinal convective-radiative porous fins with different section shapes and ceramic materials(SiC and Si3N4),Ceram.Int.40(5)(June 2014)6765-6775.

[27]M.Hatami,D.D.Ganji,Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis,Int.J.Refrig.40(April 2014)140-151.

[28]M.Hatami,G.H.R.Mehdizadeh Ahangar,D.D.Ganji,K.Boubaker,Refrigeration efficiency analysis for fully wet semi-spherical porous fins,Energy Convers.Manag.84(August 2014)533-540.

[29]S.E.Ghasemi,P.Valipour,M.Hatami,D.D.Ganji,Heat transfer study on solid and porous convective fins with temperature-dependent heat generation using efficient analytical method,J.Cent.South Univ.21(12)(December 2014)4592-4598.

[30]G.Shaoqin,D.Huoyuan,Negative norm least-squares methods for the incompressible magneto-hydrodynamic equations,Acta Math.Sci.28 B(3)(2008)675-684.

[31]S.E.Ghasemi,M.Hatami,D.D.Ganji,Thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation,Case Stud.Therm.Eng.4(November 2014)1-8.

[32]A.Aziz,Heat Conduction With Maple,R.T.Edwards,Philadelphia(PA),2006.

[33]M.Hatami,Experimental optimization of the vanes geometry for a variable geometry turbocharger(VGT)using a Design of Experiment(DoE)approach,Energy Convers.Manag.106(2015)1057-1070.

[34]B.Pak,Y.Cho,Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particle,Exp.Heat Transfer11(2)(1998).

[35]M.Turkyilmazoglu,A note on the correspondence between certain nano fluid flows and standard fluid flows,J.Heat Transf.137(2)(2015)024501.