Adaptive Numerical and Computational Methods for Temperature Profiling of a Cylindrical Nuclear Fuel Rod

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1 Adaptive Numerial and Computational Methods for emperature Profiling of a Cylinial Nulear Fuel Rod N. Anand, P. Aharya, V. Agrawal, S. Banerjee, N. Parmar, and R. Kothari Abstrat Aurate predition of temperature profile of the fuel rod in a nulear reator is very important due to safety fators. emperature at the enter and at the surfae is ritial for designing a nulear fuel rod. he ladding of the fuel rod is done to inresase the physial strength of the Fuel Rod. he temperature distribution in steady state operation in the nulear reator is determined by thermal balane between heat generated and heat transferred to the oolant. his paper attempts a numerial solution to the heat transfer problem of a ylinial nulear fuel rod with ladding having uniform generation throughout the fuel body, and heat transfer taking plae through onvetion and radiation at the lateral surfae of the ladding material. Solutions proposed through various omputational methods i.e. Finite Differene Method (FDM), Finite Volume Method (FVM) and Finite Element Method (FEM). hese results have been plotted and a omparative analysis has been onsidered to arrive at a desired exatness of the solution. he alulated values from eah of the methods are ompared and analyzed; appropriate reasons for the deviation of the omputed result from the exat solution have been explained. Index erms FDM, FVM, FEM, radiation, ladding. I. INRODUCION emperature profile of a nulear fuel rod an be well defined using different numerial methods with appropriate boundary onditions. Antar [] offered a simplified numerial solution of unsteady heat ondution problem in a short ylinder. In this paper, solutions to a mixed boundary heat flow problem (onvetion and radiation) in nulear rod with ladding have been proposed and ompares the result obtained in eah method [], [3] for different meshes. Nulear energy is energy released from the nulear fuel in the form of heat energy whih is finally onverted into eletriity energy. Nulear fuel is any material that an be onsumed to derive nulear energy. he most ommon fissile nulear fuels are enrihed uranium or plutonium in the form of penil-thin ylinial rods whih are staked inside tubes. Othman[] has studied the nulear fuel rod with ladding in transient ondition. he ladding prevents the esape of fission produts to the oolant. Zironium alloys have been used in a number of water ooled fission reator types due to their exellent aqueous orrosion resistane, low thermal neutron absorption ross setion and good mehanial properties. Manusript reeived August 0, 03; revised Deember 0, 03. he authors are with the S O A University, Department of Mehanial Engineering, IER, Bhubaneswar, Odisha, India ( nitish.ug@gmail.om, prats87@hotmail.om, vinit.agrawal.@gmail.om, sneha.banerjee89@gmail.om, enitishaparmar@gmail.om, reema8@gmail.om). he finite differene method (FDM) [5] is based on the differential equation of the heat ondution, whih is transformed into a differene equation. he temperature values are alulated at the nodes of the network. Using this method, onvergene and stability problem an appear. Han et al., [] used FDM method with FEM for analysis of one-dimensional fin. Dhawan et al., [6] studied the heat ondution problem in an aluminium plate using finite element method and got very good results ompared to the exat solution. he finite element method (FEM) is based on the integral equation of the heat ondution. his is obtained from the differential equation using variational alulus. In first ase the temperature values are alulated on the finite elements. hen, based on these partial solutions, the solution for the entire volume is determined. Using this method the whole surfae has been divided into elements and fields with unregulated border [7], [8]. In this paper the temperature distribution is analyzed in the radial diretion of a solid ylinder. he pratial appliation is in the eletrial wires where heat is generated in the wire and the same heat it needed to be dissipated to the surrounding so that the melting of the wire is avoided. he desired properties of the nulear fuel rod are that they must be fissionable [9], [0], should have high thermal ondutivity, orrosive resistant, good mehanial strength at high temperatures and high limiting temperature for operation. his paper presents the thermal analysis in a ylinial nulear fuel rod having isotropi physial harateristis [] []. Uniform heat generation is assumed from the entire body and with onvetion and radiation through the lateral surfae. Any kind of heat ondution in axial diretion is negleted. II. HEORY he general 3-D heat ondution problem with generation and time varying follows the equation () below: k g C x y z t In our problem we have a ylinder with radial ondution and no ondution through the axial diretion [3]. Hene, our problem is limited to -Dimesional radial ondution whih is given by equation () below: k r g C r r r t () () DOI: /IJE.0.V

In steady state the governing differential equation for ylinial oordinate system with uniform internal heat generation is given in equation (3): g r 0 r r r k (3) III.

2 In steady state the governing differential equation for ylinial oordinate system with uniform internal heat generation is given in equation (3): g r 0 r r r k (3) III. BOUNDARY CONDIION Symmetri Boundary Condition:. at r=0, 0 Robbin s Boundary Condition with Radiation: at r=r, k h B B Interfae Boundary Condition: At r = r, kf k Fig.. Error graph IV. NUMERICAL ANALYSIS FDM: FDM is based on the governing differential equation, whih is transformed into a differene equation using FD [], [5] M. he element is divided into nodes (Fig. ). Exat Solution he exat solution has been solved in two parts. First, the fuel rod part is onsidered i.e. with generation and the general equation is found. Seond, the ladding part of the fuel rod is onsidered i.e. without generation and the graph is obtained by merging both the results. he final equation followed by the temperature profile is as follows: For 0 < r < r gr gr r g ln r r For r < r < r r h k r k ln k f () gr gr B r ln r k (5) where B is the surfae temperature and an be found from the Boundary ondition K h ò (6) B B r Hene we found that the above equation an be modified into: r g E ò h h (7) B B r Here E represents the Error term. Hene when we have the error term E=0 then we have the solution for the above bi-quaati equation. he following graph gives us the value of the surfae temperature ( B ), whih is oming as 00. K with the error term as whih an be assumed to be 0. he given graph (Fig.) has been plotted using Matlab and hene to make error exatly equal to 0 is diffiult. Sine the value of error tending to 0, we assume that the value of temperature will remain unaffeted by this small error. For node i=: For node i= to m-, when r < r : Fig.. FDM domain gr i i (8) k gr i i i i i k when r = r k k i i i k k when r > r i i For node i=m, i i i (9) (0) () hr hr r k k k 3 m m () ris 3 isr k k FVM: Here the entire volume of the body is divided into 36

3 small equivalent elements (Fig. 3). Equation for i=: i i (3) Equation for i=: Equation for i= 3 to m-, when 0 < r < r : gr i i () k gr i i i (5) k i 3 i i i 3 when r = r : when r < r < r, k k i i i k k 3 Equation for i=m-: Equation for i=m: i i i (6) i i i (7) 3 i i i (8) i i i hr hr r k k k 3 m m ris 3 is r k k Fig. 3. FVM Domain (9) FEM: Finite Element Method is another method for numerial solution of the differential equation. ANSYS 3.0 well established ommerial software is used in this paper for required analysis and the details are produed in the results and disussion setion. Some of the important steps followed for FEM analysis are as follows: ) Geometry designing: he analysis was done on a ylinder of radius 0.05 meters and ladding of thikness 0.0 meters and suffiient length was provided to neglet the axial heat ondution. ) Loading: he lateral surfae was provided mixed boundary ondition i.e. onvetion and radiation simultaneously. In order to obtain a better result the end surfaes were insulated. 3) Meshing: he ylinder rod is now divided into number of hexaheal meshes as shown in Fig5b. he finest possible mesh was used so as to obtain results loser to the atual analysis. ) Solution: he meshed and the loaded geometry were solved using the ANSYS solver. In this step ANSYS solves the problem under required onditions. 5) Post-Proessing: he results obtained by the ANSYS solver are viewed in this step. he temperature and flux ontours an be seen here. A. Gauss Seidel Iteration V. SOLUION ECHNIQUES he FDM and FEM solution both results in a set of algebrai equations whih are required to be solved simultaneous. he set of equations an be solved by matrix formation however; alternative method of Gauss-Seidel iteration tehnique is applied. his tehnique requires less omputational effort and hene is preferred. Here we have a reasonable guess value for the temperature at eah node. he temperature of a node is related to the temperature around it. Hene we solve the equation is a logial order using the urrent value of the temperature of the surrounding node. he differene between the original and realulated value of eah temperature is omputed and the maximum value of this error is reorded for eah iteration. he iteration is repeated unlit the numerial error assoiated with the iteration is less than some tolerane limit. B. Soure erm Linearization In radiation heat transfer, the soure term is energy equation expressed as fourth power in temperature. Soure term A. (. in the problem (equation 0) is a bi-quaati equation and hene linearization is applied to make the equation linearized as follows: By soure term linearization S B (0) ds S S * * B B () Here Supersript * orrespond to the old value of the node or node enter. S () * *3 * B B B B * *3 herefore 3 B B B B Hene, the m th node equation in FDM and FVM solution having the term is substituted by the term * *3 3 hene linearizing the B B B bi-quaati equation and giving simpler solution. VI. RESUL AND DISCUSSION he one-dimensional radial heat ondution problem with onvetion and radiation at the surfae having infinite length has been solved. he solution of the above mentioned problem has been done by exat method, FDM, FVM and FEM. he results obtained an be seen in the Fig (a), (b) and 5. he values obtained from the different methods are 37

tabulated (able I) and the parameters used in able II. he nomenlature is mentioned in able III: finer meshes. ABLE I: RADIAL EMPERAURE DISRIBUION Radial Distane FDM FVM Exat FEM 0 58.3 60. 59.0 59.

Parameter Value Radius of fuel rod (m) 0.05 hikness of ladding material (m) 0.

4 tabulated (able I) and the parameters used in able II. he nomenlature is mentioned in able III: finer meshes. ABLE I: RADIAL EMPERAURE DISRIBUION Radial Distane FDM FVM Exat FEM ABLE II: PARAMEER AND VALUES USED Serial No. Parameter Value Radius of fuel rod (m) 0.05 hikness of ladding material (m) Generation (W/m 3 ) 0 6 hermal ondutivity of fuel rod (W/mK) 3 5 hermal Condutivity of ladding material (W/m K) 6 Convetive Heat transfer C-effiient (W/m K) 0,000 7 Emissivity Ambient emperature (K) 00 9 Stefan Boltzmann Constant (W/K ) 5.67x0-8 Following are the solution obtained by using ANSYS. 0 Fig. 5. emperature Profile It an be seen that maximum temperature is obtain at the ore of the Fuel rod and the minimum temperature in obtained at the outer surfae of the ladding material. he ladding prevents the Fuel Rod from physial damage. Hene appropriate thikness of ladding material has to be used in order to have safe operation of the Nulear Fuel Rod. VII. CONCLUSION his paper has presented various omputational tehniques involved in a thermal problem. he numerial solution with FEM, FDM and FVM represent an effiient way of obtaining temperature profile for a steady state Condutive and Radiation heat transfer proess. FDM, FEM and FVM methods an be involved for solving any omplex body by breaking the body into small domains. Also, hoie of finer grids whih requires high omputing apability an remove approximation errors to larger extent. Further optimization of the thikness of the ladding an be done onsidering the relative thermal ondutivity of the fuel to ladding. NOMENCLAURE Fig. (a). emperature ontour ABLE III: NOMENCLAURE Symb ols Parameters Unit r Radius of fuel rod M r Radius of fuel rd with ladding M h Convetive heat transfer o-effiient W/m K k hermal Condutivity of ladding material W/mK k f hermal Condutivity of fuel rod W/mK g Generatin erm W/m 3 Speifi heat Capaity kj/kgk Stefan Bltzmann Constant W/m K Fig. (b). Meshing he FDM and the FVM solutions divert from the exat solution beause st order aurate equations has been used on the boundary and nd aurate equations has been used at the entre. If equations of higher order of auray is used the FDM and FVM solutions an be loser to the exat solution. he FEM solution an be made more aurate with the use of Emissivity Unitless B Surfae emperature K i Node Number Unitless REFERENCES [] M. A. Antar, A simplified numerial solution of unsteady heat ondution problems with an appliation to a short ylinder, 38

International Journal of Mehanial Engineering Eduation, vol. 8, no. 3, pp.0-, 999. [] Y. M. Han, J. S.

ng and Y. W. Mai, ransient one dimensional heat ondution problems solved by finite element, International Journal of Mehanial Sienes, vol. 7, pp. 303-37, 005.

5 International Journal of Mehanial Engineering Eduation, vol. 8, no. 3, pp.0-, 999. [] Y. M. Han, J. S. Cho, and H. S. Kang, Analysis of a one-dimensional fin using the analyti method and the finite differene method, J. KSIAM, vol.9, no., pp. 9-98, 005. [3] B. L. Wang and Y. W. Mai, ransient one dimensional heat ondution problems solved by finite element, International Journal of Mehanial Sienes, vol. 7, pp , 005. [] Steady State and ransient Analysis of Heat Condution in Nulear Fuel Elements: ROZHGAR OHMAN, KH Numerial Analysis and Computer Siene. [5] M. N. Ozisik, Finite Differene Methods in Heat ransfer, CRC Press, 000. [6] S. Dhawan and S. Kumar, Comparative study of numerial tehniques for D transient heat ondution equation using finite element method, International Journal of Researh and Reviews in Applied Sienes, vol., pp.38-6, 009. [7] H. K. Versteeg and W. Malalasekera, An introdution to Computational Fluid Dynamis, Longman,996. [8] S. V. Patankar, Numerial Heat ransfer and Fluid Flow, Hemisphere Publishing Corporation, aylor & Franis Group, New York [9] E. A. Hoffman and W. M. Staey Nulear Design and analysis of the fusion transmutation of waste reator, Fusion Siene and ehnology, vol. 5, Jan. 00. [0]. Fukano, S. Mori, S. Akamatsu, and A. Baba, Relation between temperature flutuation of a heating surfae and generation of ypath aused by a ylinial spaer in a vertial boiling two-phase upward flow in a narrow annular hannel, Nulear Engineering and Design, vol. 7, pp [] Z. V. Stosi, hermal equivalent between nulear fuel rod model and ewperimental heater simulator, Nulear Engineering and Design, vol. 36, no. -6, pp , August 006. [] B. Cesna, Analytial model for alulation of the thermo hyauli parameters in a fuel rod assembly, Nulear Engineering and Design, vol. 0, pp , 00. [3] A. H. Malik, M. S. I. Alvi, S. Khushnood, F. M. Muhouz, M. K. K. Ghauri, and A. Shah, Experimental study of onjugate heat transfer within a bottom heated vertial onentri ylinder enlosure, International Journal of Heat and Mass ransfer, pp , 0. [] Y. M. Han, J. S. Cho, and H. S. Kang, Analysis of a one-dimentional fin using the analyti method and the finite differene method, J.KSIAM, no., pp. 9-98, 006. [5]. M. Chen and C. C. Chen, Numerial solution for the hyperboli heat ondution problems in the radial-spherial o=ordinate system using a hybrif Greens s funtion method, International Journal of hermal Sienes, vol. 9, pp , 00. N. Anand was born in Jamshedpur, India on th Otober 989. He reeived B. eh degree in mehanial engineering from Institute of ehnial Eduation and Researh, under Siksha O Anusandhan University, Odisha, India. He has keen interest in heat transfer and omputational fluid dynamis. He has undergone training at esorts onstrution equipments, ballabghar, Haryana and at National hermal Power Corporation of India, Badarpur, India. P. Aharya was born in Bhubaneswar, Odisha in 990, he reeived B. eh in mehanial engineering of 008- Bath at institute of tehnial eduation & researh, under Siksha O Anusandhan University, Bhubaneswar (Odisha), India. He has undertaken Industrial training at Vishakhapatnam Steel plant, vizag & voational training at integrated est Range, Chandipur. He has ompleted a ertified ourse on finite element Analysis using Ansys with Iter & ollaboration with IFS aademy, Pune. Apart from this he has also ompleted a Course on Pro-E, at Central Room & raining Centre, Bhubaneswar. He has keen interest in heat transfer, gas dynamis and omputational fluid dynamis. N. Parmar was born in Bhubaneswar, India in 990, who is pursuing B.eh in mehanial engineering at Institute of ehnial Eduation and Researh, under Siksha O Anusandhan University, Bhubaneswar, Odisha, India. She has undergone training on CAIA at entral room & training entre, bhubaneswar and Ansys with IFS Aademy, Pune. She has also undergone industrial training at HINDALCO, Sambalpur, Odisha, India. She has interest in Heat ransfer and ANSYS Modeling.. 39

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