FEAT 1997 Modelling Transient Two Dimensional Non Linear

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FEAT solves the classical equation for conduction of heat in two dimensional solids The code. can model heat conduction internal generation of heat prescribed convecfiointo a heat sink. prescribed temperatures at boundaries prescribed heat fluxes on surfaces and temperature. dependence of material properties such as thermal conductivity The code also contains pre. programmed packages for the thermal conductivity of U0 2 as a function of temperature burnup. and porosity and internal heat generation as a function of radius enrichment and bumup These. features support the use of FEAT for applications involving nuclear fuel. The finite element method makes FEAT versatile and enables it to accurately simulate complex. geometries such as the conical profile of the endcap and the dished surface of the pellet The gap. elements provide a very efficient method of simulating heat transfer between adjacent solids. The steady state version of FEAT was initially developed in 1980 and has been described. previously Tayal 1989 Since then it has been extensively used in design analyses of CANDU. fuel Currenty it is our standard tool for detailed thermal assessments of fueL The transient. version was developed in 1990 The major focus of the present paper is to describe the transient. capability of FEAT along with the associated verification and validation Ah illustrative example. dealing with transient temperature in CANDU fuel experiencing power pulse and end flux. peaking is also presented,MATHEMATICAL MODEL, The Fourier equation of heat conduction in a two dimensional thermally isotropic solid may. be written as Hsu 1986,aT xyt a r aT xyt 1 a r aT xyt t. PC k jay q x y t,in the Cartesian coordinates, ZaT r z t a r aT r z t l k a T r z t ar cT r z t 1. atC a r Bar k Jaz,in the r z coordinates and,DMATQe aT r e t l1 k aT r e t k a MTr e t 1. raL a t q r e t 3, in the r 9 coordinates In the above equations T is temperature p is density c is the specific.
heat q is heat generation rate per unit volume k is thermal conductivity x y are Cartesian. coordinates for a two dimensional plane solid r O are polar coordinates for a two dimensional. plane solid r z are cylindrical coordinates for a two dimensional axisymmetric solid All. thermal quantities and material properties vary with spatial coordinates and time. The boundary conditions may contain prescribed convection to a heat sink prescribed. temperatures at boundaries prescribed heat fluxes on surfaces and adiabatic surfaces In the. code the transient solution starts with initial conditions for the following parameters. temperatures in the sheath and the pellets heat transfer coefficient between the coolant and the. sheath outer surface heat transfer coefficient between the pellet and the sheath inner sUrface heat. generation rate and thermal properties, Because the thermal conductivity density and specific heat are non linear functions of. temperature the governing partial differential equation for temperature is solved using the finite. element method in conjunction with an incremental numerical algorithm To be consistent with. the steady state version of the FEAT code triangular finite elements are used to discretize a. two dimensional solid This permits accurate representation of complex geometries. Following the standard practice used in the finite element analysis of heat conduction energy. balance equation for each finite element may be established using Gurtin s functional form of the. Rayleigh Ritz variational principle or Galeikn method Hsu 1986 This yields the following. C jI KJ T Q 4, where CJJis the element heat capacitance matrix T is temperature T is the rate of change. of temperature K is the element thermal conductivity matri and a is the element thennal. load matrix Detailed derivations of the above equations are provided in standard text books see. for example Hsu 1986 Global energy balance equations may be formed by summing the. energy balance equations for all the individual finite elements. Three numeric algorithms the Crank Nicholson method Hsu 1986 the Galerkin method. Owen and Hinton 1980 and the backvird substitution method Huebner and Thornton 1982. were implemented in the FEAT code for integration of transient heat transfer with respect to. time Each scheme has its own specific strength regarding convergence and accuracy Writh the. three options available in the FEAT code users can choose the numeric scheme most appropriate. for the particular situation being analyzed,CONVERGENCE. Tests were conducted to study the sensitivity of transient temperatures fo Me size of the time. increment Results of one such test for CANDU fuel illustrated in Figure 1 are obtained using. different time increments Figure 2 shows that when the time step size is halved from 0 1 s to. 0 05 s the maximum temperature changes by only 0 2 5SC Hence this size of time step. provides a converged solution for this and similr situations. VERFCATION AND VALIDATION, To validate the computer code FEAT results were compared with three analytical solutions. given by Chapman 1974 Kakac and Yener 1985 and results of the ELOCA code for one. dimensional transient temperatures Sills 1979, Heat conduction in a rectangular slab of 100 mm by 100 mm receiving heat at a constant flux.
rate of 4W cm2 from the side x 0 is studied in test case 1 The initial tWriperature Is 5000 C. everywhere in the slab The material properties are p 10 000 kg m 3 c 500 l kg 0 C The. initial termal conductivity is 4 21 W m 0 C The temperature dependence ofthe thermal. conductivity is given byk 59 0 00276TW mC Results given in Figure 3 show that the. maimun difference between the FEAT and the analytical solution Kakac and Yener 1985 for. temperature atx 25 mm andy O is within 0 4 about2 0 C. Test case 2 involves a rectangular slab of side lengths of a 56 mm and b 11 2 mm Initially. the tempeture is 300oC everywhere in the slab Attime t 0 temperatures at sides x O andx. a are suddenly changed to 4000 C The material properties for the slab are p 6 490 kgt n 3 c. 350 J kg C0 and k 24 W me0 C In preparing the input for FEAT a total of 102 finite element. are used in the finite element modeL Comparison between the FEAT calculations and the. amalytical solution in the form of a series Chapman 1974 is shown in Figure 4 for temperature. t x 26 52 mm andy O The maximum difference between the two solutions is 0 07 about. Heat conduction in a long solid cylinder of radius r 10 mm and length L 50 mm is studied. test case 3 The initial temperature in the cylinder is 3009C everywhere At time t 0 the. Under is suddenly immersed in a heat sink of constant temperature 200 C The material. rties for the cylinder are p 10 000k g r 3 c 500 J kgo C and k 40 W mm C The heat. er coefficient between the heat sink and the cylinder surface is 4 000 Wm 2K In this case. total of 208 finite element nodes are used in the finite element model Results of temperature. ooc0 d oaa fcdI11M Wccodo cc tz2dc UwrC C210i32ia, at r 0 and z 0 shown in Figure 5 indicate that the maximum difference with analytical. solution is avithin 3 about 3 0C, In testcas se 4 the transient temperature in CANDU fuel experiencing power pulse but no end. flux peakin g was calculated using the two dimensional finite element code FEAT and one. dimensiona 1finite difference code ELOCA The main objective of this test case is to compare. FEAT resul ts with those of ELOCA for a one dimensional heat transfer problem Figure 6. shows for a typical power pulse the peak temperatures in the pellet calculated using ELoCA. and FEAT Xare very close for the period of power pulse The maximum difference between the. two codes i s about 2 380 C,APPITCAI IONS, The FEA T code was used to calculate the transient temperature in fuepellets and sheath near. the end cap s during a LOCA in a CAZDU 9 reactor The main objectivewas to do a sensitivity. analysis to 4establish the effect on the maximum fuel temperature and increase due to the end flux. In a typil Dal CANDU fuel temperatue assessment fuel centreline temperatures are predicted. under the as sumptlon that fuel power is axially uniform In reality the flux profile increases. abruptly ne ar the ends of fuel elements due to the reduced amount of absorbing mateiaL The. flux peaking at the ends of a fuel element result in higher fuel pellet centedine temperaes at. the ends than would be predicted by an axially symmetric simulation assuming uniform flux at. the element average, During the transient rapid coolant voiding increases the power until a few seconds later the. reactoris a toacly shut down by one of the two shutdown systems At the same time. reduced flow decreases the fuellcoolant heat transfer coefficient End flux peaking causes the. heat generation rate to increase axially towards the end of the bundle while axial heat conduction. through the endcap provides extra cooling near the endcap. Boundary conditions that must be considered to determine temperatures in the pellet sheath and. endcap as shown in Figure 1 are, No axial heat flow at z 60 mm an axial distance of approximately 3 pellet lengths beyond.
which the axial heat transfer is negligible, Convective boundary condition between the sheath outer surface and the coolant. Combined convective and radiative heat transfer between the pellet end surface and endcaP. inner surface, Heat transfer through conduction radiation or convection between the pellet outer surface. and the sheath inner surface, Heat transfer coefficient from the pellet to the endcap depends mainly on the distance between. the pellet and the endcap their temperatures and the composition and temperature of the gas in. the gap These parameters all vary during the transient and may vary spatially across the pellet. to endcap interface However in the simulation the heat transfer coefficient from the pellet to. the endcap was assumed to be uniform and constant The value chosen for this assessment was 1. kWlm2 K which was found to give reasonable agreement between FEAT steady state. simulations and temperatures inferred from post irradiation examinations of fuel bundles. During the transient period the power pulse data end flux peaking data heat trander data. between the sheath and the coolant beattransfer data between the sheath and pellet and all. other data are prepared using ELESTS Tayal 1987 and ELOCA Klein et al 1994 codes. and supplied to FEAT in the form of a data file Material properties e g density heat capacity. thermal conductivity are updated at the end of each time step using the conrelations given in. MAIRO O9 1976 for U0 2 and Zircaloy, A finite element mesh consisting of 4 663 nodes 8 316 elements 211 convective boundary. surfaces and 196 gap elements was used Time step of 0 1 s was used based on a sensitivity. study of convergence For this discretization it takes 5 3 min of central processor unit CPU. time to complete the FEAT nm on AECL s SGI computer. FIgure 7 shows typical profiles of pellet temperature along the fuel element centreline Figure. 7a shows the effect of end flux peaking when it is assumed that neighbog bundles remain in. contact after the accident whereas igure 7b shows the results when it is assumed that. neighbouring bundles separate after the accident The later case leads to higher flux peaking. factors and therefore higher temperatures, Hnd flux peaking tends to increase pellet temperature near the endcap On the other hand.
extra conduction via the endcap tends to decrease pellet temperature in that region The net. result is that the maximum pellet temperature occurs at an axial distance of 12 to15 mm from the. outside of the endplate The effects of axial heat transfer and end flux peaking diminish as the. distance from the endplate increases, FIgure 8 shows how the fuel centreline temperatures at z 14 7 mm and z 57 9 mm vary. with time The timing of the peak temperature is consistent with the power pulse profile The. peak temperature calculated using FEAT considering both end flux peaking and axial heat. transfer via endcap is about 1760C higher than that calculaed using ELOCA Nevertheless the. peak temperature occurs at the centreline and is predicted to be at about 2154C which is well. below 28400C the melting point of UO2,CONCLUSIONS, A capability for transient conduction of heat has now been added to the FEAT code Test. cases show that the calculations of FEAT agree well with independent analytical solutions. Assessment of pellet temperature during a Loss of Coolant Accident in the CANDU 9 reactor. show that the combined influences of end flux peaking and endcap cooling increase the pellet. temperature by about 1760 C compared to the situation without end flux peaking and endcap. cooling The Pealk temperature in the pellet occurs at the centreline and is predicted to be at. about 21540 C which is well below the U0 2 melting point 28400C. ACKNOWLDGEMENIS, The authors gratefully acknowledge the contributions of Z Xu for proyding independent. analytical solutions and spreadsheet calculations for test case 1 and for reviewing this. manuscript Thanks are also due to D Rattan for his review and to JILK Lau for. encouragement support and review, A J Chapman 1974 Heat Transfer MacMillan 3rd Ed New York. T R Hsu 1986 The Fini FlementMethods in Thermomechaics Allen Unwin Boston MA. KS Huebner E A Thornton 1982 The FiniteElement Methodfor Engineers John Wilby. No axial heat flow at z 60 mm an axial distance of approximately 3 pellet lengths beyond which the axial heat transfer is negligible Convective boundary condition between the sheath outer surface and the coolant Combined convective and radiative heat transfer between the pellet end surface and endcaP inner surface

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