TRANSURANUS: A Fuel Rod Analysis Code Ready for Use

Transcription

1 BG TRANSURANUS: A Fuel Rod Analysis Code Ready for Use K. Lassmann C. O'Carroll J. van de Laar C. Ott 2 _. 1 EC JRC, Institute for Transuranium Elements, Karlsruhe, Germany 2 Paul Scherrer Institute, Villigen, Switzerland 1 Introduction The basic concepts of fuel rod performance codes are discussed in a contribution to this conference [2]. In the following paper the TRANSURANUS code [3], a so-called 1i/2-dimensional code is presented. The TRANSURANUS code was developed at the Institute for Transuranium elements and is now in use in various institutions through Europe. It is based on a clearly defined mechanical/mathematical framework into which the detailed models describing the physical behaviour can easily be incorporated. The code is well structured and therefore easy to understand. In this paper emphasis is put on the description of this mechanical/mathematical framework rather than the physical models or available options. The latest developments concerning high burn-up models are outlined. 2 General Concept The mechanical/mathematical concept of the TRANSURANUS code consists of a superposition of a one-dimensional radial and axial description (the so-called quasi two-dimensional or 11/2-D models). The fuel rod is divided into axial slices and at a given time t the rod is analysed slice per slice. After all slices have been analysed, the slices need to be coupled together which means that quantities such as the inner pin pressure or the axial friction forces between fuel and cladding are determined (axial coupling). The structure of the TRANSURANUS code reflects the structure of the theoretical model: Level 1 is the main driver of the code in which the time integration is organised. The new time v 1 = t n + At is determined, where the time step At is the minimum of rifely different time step criteria given by stability criteria or criteria to control the accuracy. This time step control refined over many years of code development is a very important feature of the code. Level2 controls the axial loop over all slices, the axial coupling and its convergence. A special technique is applied to minimize computer memory. Level 3 controls the analysis of a slice, i.e. at this code level the thermal and mechanical analysis is performed for which all physical models are needed. A clear distinction is made between explicit and implicit models. For implicit (or mixed explicitimplicit) model special procedures for obtaining convergence are necessary. One of the main advantages of this clear structure is that the user can easily incorporate new models. The user is more or less only concerned with the decision whether a model is explicit or implicit and he needs only a limited knowledge how to interface this model with the TRANSURANUS variables. 3 Basic Equations In the following sections the basic equations are briefly outlined in order to make the underlying theoretical concepts better understandable. 3.1 Thermal Analysis Thermal analysis of the whole fuel rod is obtained by a superposition of one-dimensional radial and axial energy conservation equations [4]. Here, only simplified equations are given. The energy equation (heat conduction equation) for fuel, cladding and structure is c p lil dt (1) r dr dr whereas the energy equation for the coolant is given by <7*/,c c p + c vp w (2) dt dz A where: c ~ specific heat, p=density, 9 = temperature, r - radius, - thermal conductivity, q" = heat flux density, q'" power density, w = coolant velocity, A = area, cl,o - cladding-outer, cl.c - cladding-coolant and z = axial coordinate. The main features of the thermal analysis are summarized as follows: 1. The solution method includes the well-known Finite Difference Method and the Finite Element Method as special cases. The standard usage is an optimum combination of both which makes the solution extremely accurate. 2. The methods includes explicit, implicit or Crank- Nicholson integration procedures. Standard usage for transient conditions is the Crank- Nicholson scheme. 3. Phase changes (melting, boiling) are considered. 4. An important aspect of the thermal analysis is the heat transfer between fuel and cladding. The TRANSURANUS code includes a detailed model for this gap conductance which is fully described in Reference [5]. The URGAP model is available on request. VVER REACTOR FUEL PERFORMANCE, MODELLING AND EXPERIMENTAL SUPPORT 95

2 3.2 Mechanical Analysis The mechanical analysis consists of the calculation of stresses, strains and the corresponding deformations. Dynamic forces are in general not treated and the solution is therefore obtained by applying the principal conditions of equilibrium and compatibility together with constitutive relations. In the following a brief overview is given which should allow the reader to understand the basic mechanical concepts underlying the TRANS- URANUS code. More details are given in References [3] and [6]. In order to derive an adequate solution, the following assumptions are made: 1. The geometric problem is confined to onedimensional, plane and axisymmetric idealization, i.e. the axial deformation is constant across the radius (modified plane strain condition). 2. The elastic constants are isotropic and constant within a cylindrical ring, a so-called coarse zone. 3. The constitutive equations are given by e to ' =z et +e ex (3) where tot = total, el = elastic, ex = sum of all nonelastic strains, =1 r = radial, / = tangential, a - axial. e«one important theoretical concept of the TRANSURANUS code is that all volume changes due to different processes such as densification and swelling, cracking, etc. are expressed via strains. The assumptions, together with the compatibility equations e '' = ~JR' e ' = 7' e " = constant = C 3 and the equation of equilibrium d<5r (51 -<5r dr R lead to the classical semianalytic solution of the problem. The radial deformation is obtained by u{ R) = C -v R (4) (5) where /?=/ +*/, R = radius of the deformed geometry and r - radius of the reference geometry. Similar expressions result for the stresses and strains. Eq. (6) shows that equilibrium is always taken at the deformed geometry. In addition, geometrical second order effects are taken into account approximately (not shown here). The constants C h C 2 and d are determined from the wellknown boundary conditions. The integrals included in the solutions (e.g. Eq. 6) must be evaluated numerically. This is the reason why the solution is called a semianalytic solution. From the theoretical assumptions it follows that fuel and cladding can be divided into an arbitrary number of rings (coarse zones) which are further subdivided in fine zones in order to allow for the numerical integration. Since any discretization can be chosen this is called a variable multizone concept. Creep of fuel and cladding shows a highly nonlinear behaviour with regard to stress where stress exponents of 5 and more must be anticipated. Therefore the calculation of creep is not an easy task and the corresponding numerical procedures must be designed very carefully. In the TRANS- URANUS code both principal techniques, explicit and implicit treatments, are used. The treatment of plasticity and cracking is given in Reference [3]. In the case of a whole fuel rod axial friction forces between fuel and cladding are generated which are calculated by a one-dimensional axial model called URFRIC (TRANSURANUS FRLCtion model [7]). 4 Flexibility of the TRANSURANUS Code Up to now no distinction was made between the different types of fuel rods and in fact this was not necessary as can be seen from the basic equations. The general concept of the model is that the basic equations apply to all type of fuel rods and reactor conditions. However, specific models are needed for specific problems. The complete set of models and options available is fully described in Table 1 of Reference [3]. The basic equations of material conservation are applied to plutonium redistribution (model of Bober et al. [8]), to pore migration and redistribution of oxygen (OXIRED model [9]). Different models are optional for densification, gas release and swelling, relocation, for predicting the fuel structure, cladding failure, the radial power density, the formation and closure of the cenral void (FBR), the waterside corrosion etc. In order to be as flexible as possible the relevant material properties such as the elastic constants, the thermal conductivity, the specific heat, the density etc. are formulated in specific subroutines which allow for the incorporation of different correlations. At present up to 30 different correlations for fuel and cladding are possible and the TRANS- URANUS code includes material data for all reactor types. Possible fuel materials are oxide, mixed oxide, carbide and nitride, the cladding materials are Zircaloy, steel and niobium and the coolant may be water, sodium, sodium-potassium and helium. 96

3 Similar to the formulation of the basic material properties, the general boundary conditions were formulated as flexible as possible and different geometries ranging from an analysis of the fuel alone to the analysis of Fuel, Cladding, Coolant and Structure. The TRANSURANUS code is written in standard FORTRAN 77 which does not allow for variable dimensioning. However, the radial and axial discretization is very flexible through the usage of pseudo-variable dimensioning. JUUU Exact Solution 5 Verification of the Basic Equations and Numerical Aspects The TRANSURANUS code has been verified extensively by three steps: 1. Verification of the numerical techniques by comparison with analytic solutions or by comparison with other techniques. 2. Verification of specific models with experiment. 3. Verification of the TRANSURANUS code by comparison with irradiations which is an ongoing activity. In the following examples of the three verification steps are given Figure 1 FDM * FEM Present Method i i i i i Radius [mm] Radial temperature distribution in a fuel rod (test case). Compared are the analytic, exact solution with different options of the TRANSURANUS code (FDM = finite difference method, FEM = finite element method, Present solution = optimum of accuracy) 5.1 Verification of the Numerical Techniques During the development of the TRANSURANUS code all possibilities were used to test the basic equations regarding accuracy, computer time consumption and stability. In the following a few examples shall be given in order demonstrate the capabilities of the TRANSURANUS code. Thermal analysis: The numerical techniques of the thermal analysis have been carefully designed to be fast reliable and accurate: examples are shown in Figure 1. Special emphasis was put on the problem of obtaining convergence and the total amount of numerical effort [10]. This is the prerequisite for the analysis of complicated power histories in which the thermal analysis has to be done up to several thousand times for a single analysis. analytic solution TRANSURANUS. explicit TRANSURANUS, implic.t n= 5» analytic solution V A TRANSURANUS. explicit V TRANSURANUS. implicit rate - V g radius ~ 02-K) 3 - avio 3 - Figure 2 rn radius Radial distributions of normalized stresses and tangential creep rate for a Norton exponent of n = 5. Compared are the analytical solutions with the explicit and the implicit numerical solution 97

4 Mechanical analysis: The mechanical analysis was also tested extensively. In particular, the highly nonlinear creep was analysed using explicit and implicit techniques. The agreement between both techniques and the analytic solution was confirmed for a thick-walled cylinder. Figure 2 may serve as an example. It is worthwile to note that the explicit technique cannot be applied for standard fuel conditions because the high rates in the centre of the pin, which determine the stability criterion, would lead to extremely small time steps. In fact, the difference in computer time between an explicit and an implicit technique is up to a factor of 5000 for a standard irradiation history. cc o T]«100 M Mass Transfer: Plutonium redistribution and pore migration are solved by Finite Difference techniques where the balance equation for each zone is written in conservative form. Fully implicit formulations are used and the resulting systems of equations are penta- and bidiagonal which can be solved effectively. The numerical schemes of radial plutonium redistribution and pore migration can be Figure RELflT VE RADIUS Comparison between the analytic, asymptotic solution of Clement (solid line) and the numerical solution (open circles). Inner and outer fuel radii are 0.75 and 3 mm; centre temperature 2550K, fuel surface temperature 1100 K, initial plutonium concentration MWd/t WWd/t TUBfiN 5 a MWd/t MWd/t MWd/t LUIT4 B 3 3. o 1 G 4 * * (a) (b) MWd/t MWd/t MWd/t TUBRNP MWd/t MWd/t TUBRNP Figure 4 (C) (d) Comparison of the radial total Pu concentration measured with that calculated by the TUBRNP model: (a), (b) STRO fuels with enrichment 235 U = 2.9%, calculations with TUBRNP for and MWd/t; (c), (d) EPRI fuels with enrichments 235 U=5.75% and 8.25%, calculations with TUBRNP for and MWd/t. 98

5 compared with asymptotic solutions given by Clement [11] which are valid as long as the change from the initial conditions is relatively small. Comparisons between numerical and the analytic solutions of plutonium redistribution are shown in Fig. 3 for two different time steps: the agreement is excellent. The slight deviation between the solutions at the inner boundary at 500 h is due to the fact that the asymptotic solution is already of limited use under these conditions. 5.2 Verification of Specific Models The TRANSURANUS code includes many physical models which cannot be discussed here. One of the recently developed model is the TRANS- URANUS burn-up model TUBRNP [12]. This model calculates the local concentration of several isotopes by the following burn-up equations: ^235 (' dn 242 (r) a.242^242 (7a) (7b) (7c) (7d) (7e) (7f) The local concentrations of 238 U and 240 Pu, and iv 2 4o(/-)> are written as /V238/1OO ^240/2 ('') where /(r), /=1,2 are radial shape functions with normalisation factors defined by, ""' f,{r)rdr -r = 1, /=1,2 (8) where /, and r ml are the inner and outer fuel radii. The shape functions take into account the resonance absorption in 238 U and 240 Pu that lead to the formation of 239 Pu and 241 Pu, respectively. These distribution functions can be interpreted as the combination of a constant production of Pu from thermal neutron capture plus a highly nonlinear term for production due to resonance absorption: an example is given in Figure 4. The TUBRNP model has been used to analyse the radial distribution of Pu in MOX fuels [13], an example is shown in Figure 5. It has also be used to determine the thickness of the so-called "rim-zone" [14], see Figure Verification by Comparison with Experiments LWR conditions have been analysed by experiments from the following projects: Halden, Riso, Studsvik, Tribulation and others, FBR conditions were mainly tested within the CABRI project. Here, only an example from the FUMEX exercise organised by IAEA [15] shall be given. Blind predictions were made for 10 complicated irradiations performed in the Halden reactor. Figure 7 shows the comparison between predicted and measured temperatures for different burnup levels, gas compositions etc. The agreement is good although the tendency to overpredict high temperatures needs further clarification. 6 Conclusions The following conclusions can be drawn: 1. The TRANSURANUS code is a quasi twodimensional (I1/2-D) code designed for the treatment of a whole fuel rod for any type of reactor and any situation. The fuel rods found in the majority of test or power reactors can be analysed for very different situations, as given for instance in an experiment, under normal, offnormal and under accident conditions. The time scale of the problems to be treated may range from milliseconds to years. 2. The TRANSURANUS code consists of a clearly defined mechanical/mathematical framework into which physical models can easily be incorporated. This framework has been extensively tested and the programming very clearly reflects this structure. The code is well structured and therefore easy to understand. 3. The code has a comprehensive material data bank for oxide, mixed oxide, carbide and nitride fuels, Zircaloy and steel claddings and different coolants. The subroutines which include the material properties are of identical structure and the incorporation of new data is straightforward. 4. The code can be employed in different versions, as a deterministic and a statistical code. It is evident that the relevance of statistical analysis is increasing: this is an area for future growth. 5. For the user the following practical aspects are of relevance: The implementation on any computer is simple since the TRANSURANUS code is written in standard FORTRAN

6 V - TUBRNP IIK)(1( 1 242IM 241I ) u 239I! NORMALISED RADIUS Figure 5 The isotopic plutonium composition plotted as a function of fuel radius as determined by SIMS (PWR) and predicted by TUBRNP. The curves labelled 1 to 4 correspond to the predictions for isotopes 239 to Depth of the restructured zone (n m) =70 GWd/t) Enrichment(% 60 Burnup (GWd/t) 70 Figure 6 The depth of the restructured zone as a function of enrichment and average burnup. The surface is the prediction (8 ff ) of the TUBRNPmode with a threshold value of i b =70 GWd/t. The points are the values of the depth of this zone derived from the Xe profiles (5 EPMA ). 100

7 Figure 7 Selected Temperatures from FUMEX 1, , 3.3, 4a and 4b Blind Prediction (TRANSURANUS, CEC) Comparison between blind temperature predictions employing the TRANSURANUS code (CEC version) with measured data of the FUMEX exercise [15] The TRANSURANUS code system consists of 2 preprocessor programms (MAKROH and AXORDER) which largely helps in setting up new data cases. A new Windowsbased interactive interface is under development. One postprocessor, the very detailed plot program URPLOT, enables the user to plot all important quantities as function of the radius, the axial coordinate or the time. The postprocessor URSTAT may be used to evaluate statistical analyses. The TRANSURANUS code exhibits short running times. For example a simple LWR analysis of a whole fuel rod takes about 20 s CPU time on a modern workstation. A data case with a very complicated power history and detailed discretization may take approximately 2 minutes. It is important to note that all techniques were chosen in such a way that the computer costs depend more or less linearly on the discretization. The TRANSURANUS code is now in use in various institutions through Europe and is now available to all interested parties. References [1] K. Lassmann, TRANSURANUS: a fuel rod analysis code ready for use, J. Nucl. Mat. 188 (1992) [2] K. Lassmann et al., Main Concepts and Objectives of Fuel Performance Modelling and Code Development, Seminar on VVER Reactors Fuel Performance, Modelling and Experimental Support, Varna, St. Constantine, Bulgaria, 7-11 November 1994, paper 4.1 [3] K. Lassmann, H.Blank, Modelling of Fuel Rod Behaviour and Recent Advances of the TRANSURANUS Code, Nucl. Eng. Design., 106(1988) [4] K. Lassmann, T. Preufier, An Advanced Method for Transient Temperature Calculation In Fuel Element Structural Analysis, Nucl. Techn., 60 (March 1983), [5] K. Lassmann, F. Hohlefeld, The Revised URGAP-Model to Describe the GAP Conductance Between Fuel and Cladding, Nucl. Eng. Design, 103 (1987) [6] K. Lassmann, URANUS - A Computer Programme for the Thermal and Mechanical Analysis of the Fuel Rods in a Nuclear Reactor, Nucl. Eng. Design, 45 (1978) [7] K. Lassmann, Treatment of Axial Friction Forces in the TRANSURANUS Code, Nuclear science and technology, Trans, of two int. seminars on mathematical/mechanical modelling of reactor fuel elements, CEC Report EUR EN (1991), p. 185 [8] M. Bober, G. Schumacher, D. Geithoff, Plutonium redistribution in fast reactor mixed oxide fuel pins, J. Nucl. Materials, 47 (1973) [9] K. Lassmann, The OXIRED Model for Redistribution of Oxydes in Nonstoichiometric Oxides, J. Nucl. Materials, 150 (1987) [10] K. Lassmann, A Fast and Simple Iteration Scheme for the Temperature Calculation in a Fuel Rod, Nucl. Eng. Design, 103 (1987) [11] C.F. Clement, Analytic solutions to mass transport equations for cylindrical nuclear fuel elements, J. Nucl. Materials, 68 (1977) [12] K. Lassmann, C. O'Carroll, J. van de Laar, C.T. Walker, The radial distribution of Plutonium in high burn-up UO 2, J. Nucl. Materials, 208 (1994) [13] C. O'Carroll J. van de Laar, C.T. Walker, C. Ott, R. Restani, Validation of the TUBRNP model with the radial distribution of Plutonium in MOX fuel measured by SIMS AND EPMA, IAEA TCM on Water Reactor Fuel Element Modelling at High Burnup and Experimental Support, Bowness-on-Windermere, UK, September 1994, Paper 2.9 [14] C. O'Carroll, C.T. Walker, K. Lassmann and J. van de Laar, First steps towards modelling high burnup effects in UO 2 fuel, IAEA TCM on Water Reactor Fuel Element Modelling at High Burnup and Experimental Support, Bownesson-Windermere, UK, September 1994, Paper 2.4 [15] C. O'Carroll, J. van de Laar and K. Lassmann, Report on ITU FUMEX Results, Paper pres. at IAEA RCM on Fuel Element Modelling at Extended Burnup (FUMEX), Bowness-on- Windermere, UK, September 1994 S8EXT PAGSfS) 101