Development and testing of the FAST fuel performance code: Normal operating conditions (Part 1)

Nuclear Engineering and Design 282 (2015) 158–168

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Development and testing of the FAST fuel performance code: Normal operating conditions (Part 1) A. Prudil a,1 , B.J. Lewis b,2 , P.K. Chan a,∗ , J.J. Baschuk c,3 a b c

Royal Military College of Canada, Chemistry and Chemical Engineering, 13 General Crerar Crescent, Kingston, Ontario K7K 7B4, Canada University of Ontario Institute of Technology (UOIT), Energy Systems and Nuclear Science, 2000 Simcoe Street North, Oshawa, Ontario L1H 7K4, Canada Atomic Energy of Canada Limited (AECL), Fuel and Fuel Channel Safety, 1 Plant Road, Chalk River, Ontario K0J 1J0, Canada

h i g h l i g h t s • • • • •

FAST is a general purpose nuclear fuel model developed using Comsol Multiphysics. Presents the development of the FAST code for normal operating conditions. Multiphysics, multidimensional approach using commercial finite-element platform. Presents proof-of-concept comparison to experimental data and other CANDU fuel codes. Demonstrated improved agreement with the end-of-life sheath strain measurements.

a r t i c l e

i n f o

Article history: Received 24 February 2014 Received in revised form 15 September 2014 Accepted 17 September 2014

a b s t r a c t The Fuel And Sheath modeling Tool (FAST) is a general purpose nuclear fuel performance code. FAST includes models for heat generation and transport, thermal expansion, elastic strain, densification, fission product swelling, cracked pellet, contact, grain growth, fission gas release, gas and coolant pressure and sheath creep. The equations are solved on a two-dimensional (radial-axial) geometry of a fuel pellet and sheath using the Comsol Multiphysics finite-element platform. This paper presents the FAST code for normal operating conditions and results of the proof-of-concept testing against the ELESIM and ELESTRESIST fuel codes as well as experimental data from seven irradiated fuel elements. In these seven cases, all of the codes were found to under-predict the measured average sheath strains. However, the FAST code was found to under-predict the mid-pellet sheath strains by a smaller margin than the other two codes. A larger data set is required to assess relative accuracy of the codes. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Nuclear fuel is an important consideration for the design and operation of all nuclear reactors. Fuel design changes can be used to improve performance of existing reactors (e.g., mitigating reactor aging phenomena or power flattening), address environmental concerns (e.g., recycled fuel or actinide burning), address reactor safety concerns (e.g., hydrogen generation from cladding oxidation

∗ Corresponding author at: Department of Chemistry and Chemical Engineering, PO Box 17000, Station Forces, Kingston, Ontario K7K 7B4, Canada. Tel.: +1 613 541 6000x6145. E-mail addresses: [email protected] (A. Prudil), [email protected] (P.K. Chan). 1 Tel.: +1 613 541 6000x7867. 2 Tel.: +1 905 721 3142. 3 Tel.: +1 613 584 3311x46286. http://dx.doi.org/10.1016/j.nucengdes.2014.09.036 0029-5493/© 2014 Elsevier B.V. All rights reserved.

or higher melting temperature materials) or address operational issues (such as refueling interval). Additionally, the fuel matrix and fuel cladding are the first (of multiple) barriers to the release of fission products to the environment, making understanding the impact of fuel design particularly important. Computer modeling tools with predictive capability can be used to assess new designs to support fuel qualification. Computer models are necessary because of the high cost and difficulty associated with performing in-reactor measurements. These models, in effect, act as advanced interpolation (and in some cases extrapolation) tools to help bridge the gaps between the application (power reactors) and the experimental results (in- and out-reactor experiments). Like all other computer programs, nuclear-fuel modeling codes must always be designed to accommodate the finite computing resources available to run them. In order to accommodate these limits, phenomena must be modeled using less computationally expensive approximations to obtain a model with

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acceptable fidelity within the constraints. This has historically favored the development of fuel modeling codes employing a fixed one-dimensional or quasi-two-dimensional representation of the fuel-element geometry. These codes typically also utilized many empirical correlations with limited coupling between phenomena. This is representative of many early codes such as GAPCON (Hann et al., 1973), FRAP-T (Thompson et al., 1975), and ELESIM (Notley, 1970) and to a lesser degree their successors FRAPCON (Berna et al., 1978), FRAPTRAN (Cunningham et al., 2001), ELESTRES (Tayal, 1987) and ELOCA (Sills, 1979). In the time since these models were first developed, there have been many advancements in both computer hardware and software that have expanded modeling capabilities. This includes the development of higher frequency and higher capacity devices, the popularization of parallel computing, and the development of more efficient algorithms for solving systems of linear equations. These advancements have made feasible much more computationally expensive modeling codes which require fewer simplifying assumptions. These advanced codes have the potential of greater predictive capabilities and more diverse feature sets than those previously available. This has led to the development of a new fuel modeling paradigm employing features such as fully coupled multidimensional, multiphysics techniques and unification of normal and transient modeling domains into a single code. There are numerous examples of codes with one or more of these features, such as FALCON (Electric Power Research Institute (EPRI), 2004), TRANSURANUS (Lassmann, 1992; Lassmann et al., 1998), FEMAXI (Nuclear Energy Agency, 2011), and BISON (Hasen et al., 2009; Williamson et al., 2012). A common trait of most fuel modeling codes is that they have been developed as purpose-written, standalone, computer programs in which the physical models are included along with the numerical methods directly in the source code. This architecture offers some advantages, particularly in terms of computational efficiency, protection of intellectual property, and guarding against accidental modification. The main disadvantage of this architecture is that significant modifications are difficult, and generally require editing the source code. Thus, in order to model non-standard fuels or geometries the end-user would have to have intimate knowledge of the internal structure of the code, and access to the source code. An example of this would be modeling a pellet with a blindhole using the ELESTRES code. This would require modifying the source code to accept new inputs, utilize a new mesh, and apply additional boundary conditions. This can make most existing fuel codes difficult to use for research-oriented applications, in which it is desirable to simulate the performance of novel fuel designs and experimental configurations. An alternate architecture has also emerged which provides greater separation of the modeling tasks from the numerical solution tasks. The two main advantages of this architecture are a reduction in the difficulty associated with modifying the model and the ability to use an existing numerical solution infrastructure. This is the methodology employed by the FAST code (the subject of this work) and the BISION code (Hasen et al., 2009; Williamson et al., 2012). The FAST model has been developed on the Comsol Multiphysics (v.4.4) finite-element platform. A significant reduction in development time and cost was achieved by utilizing the builtin pre-and-post-processing tools for various tasks such as building model geometry and finite-element meshes, solving linear systems and graphing results, rather than developing custom tools for the same task. The Comsol platform is extremely flexible, allowing the solution to a wide range of ordinary, and partial differential equations with arbitrary coupling of the dependent variables. These equations are represented in either strong or weak form and are automatically discretized as part of the solution process, making

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them easy to modify as needed. A drawback of this implementation is that the FAST code requires a Comsol installation and a license to run. The FAST code is comprised of mechanistic and empirical separate effects models (also called phenomena or physics models) coupled together to obtain a simultaneous solution. This code has evolved from fully coupled two-dimensional (radial-axial) models developed previously (Morgan, 2007; Shaheen, 2011). The FAST code has four broad motivations, which have guided the design decisions: • Support modeling of CANDU fuel (i.e. solid UO2 , collapsible zircaloy-4 sheathing). • Improve prediction of sheath strain including circumferential ridging effects to support predictions of Stress Corrosion Cracking (SCC). • Serve as a flexible research tool which can be adapted for fuel design optimization, experimental design/analysis, and prototyping new material/phenomena models. • Utilize non-proprietary models where available to limit intellectual property issues. Section 2 of this paper summarizes some of the key theory employed in the FAST code for modeling CANDU fuel under normal operating conditions (NOC). In order to apply the code to other fuel types, modification would be necessary to account for differences in fuel designs and irradiation conditions. For example, LWR fuel would require models to account for the difference in flux depression, length of fuel elements/pins and cladding corrosion. Section 3 presents a proof-of-concept validation of the FAST model with comparisons to experimental measurements and Canadian industry standard codes. Section 4 presents the ongoing and future development of the code as well as some potential applications.

2. Model development The behavior of nuclear fuel during irradiation is a complicated multiphysics problem involving many branches of science and engineering. The geometry employed in the FAST code is described in Section 2.1. Phenomena models are summarized in the following Sections 2.2–2.5. The references to the material property models used in the FAST code are provided in Section 2.6. A complete description of the FAST code, including implementation details, is available in reference (Prudil, 2013).

2.1. Model geometry The model geometry consists of one half-pellet in the radialaxial plane (axisymmetric) with an accompanying sheath (see Fig. 1). This includes options for a central hole as well as dishing and chamfering of one or both ends of the pellet. The model assumes that the single pellet is representative of all pellets within an element. This is equivalent to assuming no strong axial dependence of the irradiation conditions over the length of an element (∼0.5 m in length). This allows a periodic boundary condition to be applied which bounds the model in the axial direction. The boundary conditions required to implement this periodicity are described in details in Section 2.3.1. A sample mesh has also been included in the right-hand side of Fig. 1. It is worth noting that the default geometry and mesh can be modified as needed using the Comsol Graphical User Interface (GUI) or by importing CAD or mesh files generated by other software packages.

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where the heat transfer coefficients, hgap , are associated with gaseous conduction, solid-to-solid surface conduction, and radiative heat transfer respectively. The gaseous and the solid–solid coefficients are obtained from the model of Campbell et al. (Campbell et al., 1977). The gaseous term is: hgap,gas =

kg . 1.5(Rf + Rs ) + dgap + g

(4)

The variables kg , kf and ks are the thermal conductivity of the gas, the fuel, and the sheath, respectively (units of W m−1 K−1 ). The average local gap distance is denoted by dgap (m), which is effectively increased by the surface roughness of the fuel and sheath materials, Rf , Rs (m), and the temperature jump distances at the surfaces, g (m). The temperature jump distance is dependent on the gas temperature (T), pressure (P) and composition. It is calculated by P0 g= P

 

g0,i (T/T0 )

i

Fig. 1. Fuel element cross-section with symmetry planes identified (left) and the corresponding geometry in FAST (right).

2.2. Heat generation and transport The primary requirement of any fuel modeling code is to determine the temperature, T, throughout the fuel element because most material properties are temperature dependent and many phenomena are thermally driven. Heat transport in solid components is modeled by the heat-conduction equation: ∂T = ∇ · (k∇ T ) + Qprod Cp ∂t

(1)

where , Cp and k are the material properties of density, specific heat capacity and thermal conductivity, respectively. These are not constants; they are dependent on many factors such as temperature, porosity, burnup, radiation damage and/or manufacturing conditions. These dependencies are accounted for using empirical and semi-empirical correlations. The volumetric heat production rate, Qprod , accounts for heat produced in the fuel. In FAST this is assumed to be proportional to the thermal neutron flux in the fuel. The flux model employed in the code was taken from ELESTRES-IST for heavy water reactors (Tayal, 1987). This model is the analytic solution to 1D neutron diffusion plus an additional exponential correction term obtained by curve-fitting the flux profile predictions from reactor physics simulations. This term accounts for the uneven distribution of neutron absorbing species in the fuel. The thermal neutron flux profile in this model is given by

−1 yi s+0.5

.

(5)

Here, y is the mole fraction, i indicates the gas component, g0 is the jump distance at a reference temperature (T0 ) and pressure (P0 ), and s is an empirically derived value for the temperature exponent of viscosity. The solid–solid conduction term is given by:

 hgap,solid =

2kf ks



kf + ks

1.16 × 105 H



Pi dgap

(6)

where Pi is the local average contact pressure at the interface (Pa) and H is the Meyer hardness of the zircaloy (Pa). In the case of an open gap, the contact pressure is zero and the solid conduction term does not contribute. The radiative component of the gap conductivity is calculated assuming gray body radiation between infinite parallel surfaces (i.e., unity view factor). This yields a radiative heat transfer coefficient of: hgap,rad =

SB 1 εe,f

+

1 εe,s

−1

2 2 + Tsheath )(Tfuel + Tsheath ) (Tfuel

(7)

Here fmag is a proportionality coefficient to achieve the required linear power, In and Kn are the nth order modified-Bessel functions of the first and second kind, r is the radial coordinate, Pr is the pellet radius, and the parameters flux , ˇflux , and flux are the flux depression parameters tabulated from the reactor physics codes as a function of initial pellet radius, enrichment and average burnup. Note that these parameters are dependent on the neutron spectrum of the reactor and therefore may need to be defined for other reactor types. Heat transport across the pellet-to-sheath gap is modeled assuming one-dimensional steady-state heat transfer due to the high aspect ratio. The radial heat flux is given by

where is  SB the Stefan–Boltzmann constant and εe is the effective emissivity of the fuel and sheath. This heat transfer mechanism only becomes significant in fuel with a large fuel-to-sheath gap, which is usually a result of sheath lift off. This occurs when the internal gas pressure exceeds the external coolant pressure (as a result of excessive fission gas release or a loss of coolant pressure). This leads to reduced solid–solid and gaseous heat conduction, further increasing the temperate and internal gas pressure. The heat conduction from the fuel sheath to the bulk coolant occurs through thin-film heat transfer. The effective heat transfer coefficient, hfilm , is determined by a number of parameters including the fluid properties (viscosity, temperature, pressure), the direction and speed of the fluid flow as well as the temperature and surface morphology of the solid. Generally, it is calculated by thermohydralic codes (e.g. CATHENA) to analyze coolant subchannels. For the sake of this model, the heat transfer coefficient is considered an input parameter. Typical values for CANDU fuel during normal operating conditions range from 38 to 50 kW m−2 K−1 . During accident conditions, the heat transfer coefficient can be significantly reduced by the loss of turbulent flow and/or coolant density. The net radial heat flux from the sheath to the bulk coolant, Qr , is given by

Qr = (hgap,gas + hgap,solid + hgap,rad )(Tfuel − Tsheath )

Qr = hflim (Tsheath − Tcoolant ).



Qprod = fmag  = fmag

I0 (flux r) +

I1 (flux Hr ) K0 (flux r) + ˇflux eflux (r−Pr ) K1 (flux Hr )



. (2)

(3)

(8)

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2.3. Deformation mechanics In the reactor, the geometry of the fuel element deforms as a result of a number of processes including: mechanical loading, thermal expansion, material creep, fuel densification and fission product swelling. The FAST code calculates the net deformation as the sum of the individual strains. 2.3.1. Elastic deformation The deformation due to mechanical loading of the fuel is calculated assuming a pseudo-steady-state which assumes the internal stress, , is in equilibrium with the body load, Fv , and neglecting inertial effects −∇ ·  = Fv .

(9)

The stress in the material is calculated according to the standard isotropic linearly-elastic (Hookean) model  = [Celastic ][εelastic ]

(10)

where Celastic is the stiffness matrix and εelastic is elastic strain vector. In order to account for the presence of inelastic strains, each component of the elastic strain is calculated as the total strain minus the inelastic strain, εelastic = εtotal − εinelastic .

(11)

The total strain is then written in terms of the spatial derivatives of the displacements. In the 2D axisymmetric case, as used in this work, the components of the strain are given by εr =

∂u , ∂r

ε =

εrz =

1 2

u , R



εz =

∂u ∂v + ∂r ∂z

∂v , ∂z



,

εr = ε z = 0.

(12)

Together, Eqs. (9) through (12) form the basis of the elastic material model in FAST. A second, modified version of this Hookian mechanics model has also been included in order to account for the presence of circumferential cracks in the pellet. These cracks result in a larger pellet radius by preventing the periphery of the fuel from constraining the thermal expansion of the hot center. The limiting case is when there are infinitely many cracks in the radial-axial plane so that there is no tensile stiffness in the circumferential direction. This situation can be modeled by adding a term, εcracks , to the circumferential component of the elastic strain in Eq. (11) which forces the elastic strains to be less than or equal to zero. This effectively allows the pellet to resist compressive elastic strains but not tensile strains. The ‘cracked’ pellet model can be written as: εelastic, = εtotal, − εinelastic, − εcrack,

(13)

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boundaries come into contact and then grows linearly with the penetration distance (dpen ) at a rate determined by the penalty factor, Pf .

  ⎧ ⎨ Pest exp Pf dpen dpen < 0 Pest Pi = ⎩ Pest + Pf dpen

(15)

dpen ≥0

Ideal contact is given in the case of Pest = 0 and Pf = infinite, however, finite values can be used to obtain an approximation to the contact. To provide physically meaningful results, the penalty factor must be selected sufficiently large that the contact pressure is approximately independent of the penalty factor and penetration depth is sufficiently small that it can be neglected. This formulation helps reduce the non-linearity which would be caused by a sharp change in the penalty function thus improving model stability. For the results presented in this work values of Pest = 15 MPa and Pf = 1 × 1013 Pa m−1 were used for the pellet-to-sheath contact, while Pest = 100 MPa and Pf = 1 × 1014 Pa m−1 were used for the axial contact. These values were found to maintain the penetration depths around ∼1–3 ␮m which is approximately the surface roughness of the material. Friction between the contact surfaces has also been neglected in this model because it significantly increases the computation complexity of the model. The expected contribution from friction is small due to assumption of circumferential and periodic symmetry as well as a distributed axial gap. The periodic boundary condition as described in Section 2.1 requires the radial displacement of the top and bottom surfaces of the sheath to be equal. The axial displacement on the bottom edge of the sheath and bottom land point (on the pellet) is set to zero to constrain the problem in the axial direction. The pellet-to-sheath contact force is applied normal to the inside edge of the sheath. The pellet-to-pellet contact force is applied in the axial direction to the top edge of the sheath. For the pelletto-pellet contact, the penetration distance is assumed equal to the length of the pellet in the axial direction minus the length of the sheath (accounting for the initial pellet to pellet gap, thermal expansion and creep).

2.3.2. Thermal strain The thermal expansion strains, εthm , were approximated using empirical correlations (Prudil, 2013). An isotropic MATPRO correlation for stoichiometric UO2 was used for the pellet. To account for the anisotropic thermal expansion of the sheath, different correlations for the thermal strain were used for the axial direction and the radial/circumferential directions.

or equivalently εelastic, = min(εtotal, − εinelastic, , 0).

(14)

The ‘cracked’ model is intended to provide an estimation of the upper-bound effect of cracking in the radial-axial plane while providing a calculation of the circumferential ridging. Although it is known that the elastic properties of the fuel sheath are not isotropic, this is not included because the total elastic strains are small compared to the plastic deformations which result from the effect of creep (Sills, 1979). This makes the elastic anisotropy negligible. The pellet-to-sheath and pellet-to-pellet contact is modeled using a penalty method to apply a force normal to the sheath. The penalty function was selected such that the interface pressure, Pi , grows exponentially to the estimated contact pressure (Pest ) as the

2.3.3. Pellet densification and fission product swelling When fresh fuel is inserted in a reactor, the high temperatures and radiation leads to a reduction in fuel porosity and an increase in the density. The volumetric strain resulting from the fuel densification is given by εvol,dens =

Vdens 1 − p0 = −1 V0 1 − p0 (1 − FPoreR )

(16)

where p0 is the initial porosity and FPoreR is the fraction of initial porosity which has been removed from the fuel. The fraction of porosity removed from the fuel was calculated using a modified version of an empirical correlation developed by Hastings (Hastings and Evans, 1979) for CANDU fuel. The original Hastings model was

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converted to a differential form for modeling non-constant temperatures leading to dFPoreR dBu = 0.02867(0.6 − FPoreR ) max(ln(1 − 1.6667FPoreR ) dt dt + 8.67 × 10−10 T 3 , 0)

∂ε ,ir (17)

where T is the temperature in K and Bu is the burnup of the fuel in MW h k U−1 . According to this model, the fraction of initial porosity which can be removed from the fuel saturates at 60%. The fission product swelling effect is divided into two sources with different mechanisms: solid fission product swelling and gaseous fission product swelling. Solid-fission product swelling occurs when the space occupied by two fission product atoms in the fuel matrix is greater than the space occupied by a single UO2 atom. The volumetric strain due to solid fission product swelling is assumed to be linearly proportional to the fuel burnup. Olander suggests a volumetric strain of εvol,SFP = 0.0032

Bu 225

(18)

where Bu is the burnup in units of MWh kg U−1 (Olander, 1976; SCDAP/RELAP5 Development Team, 1997). The gaseous fission product swelling is caused by the formation of fission gas bubbles on the grain boundaries. The MATPRO (SCDAP/RELAP5 Development Team, 1997) correlation for the volumetric strain rate of the gaseous fission products, εvol ,GFP , in units of s−1 is calculated by d(εvol,GFP ) = 8.8 × 10−56 (2800 − T )11.73 exp(−0.0162(2800 − T ) dt − 8.0 × 10−27 Buf)

dBuf . dt

(19)

Here Buf is the fuel burnup measured by number of fissions per cubic meter. As suggested in MATPRO, this equation is applied in the range of 1000–2000 K. Below this temperature the rate of fission gas swelling is assumed to be zero due to a slow rate of diffusion of fission gas from the fuel matrix. Similarly, it is assumed zero above 2000 K because grain boundary saturation is assumed to have occurred. A limitation of this model is that it does not account for the detailed power history or grain size. This could be accounted for by calculating the fission gas swelling from the grain boundary fission gas concentration calculated as part of the fission gas release model. However, the MATPRO model is used since it is easier to implement as it does not require a coupling to the fission gas release model. 2.3.4. Sheath creep The creep rate was taken from the MATPRO 11 correlation (EG&G Idaho, Inc., 1979). This correlation calculates both the thermal and irradiation creep in the circumferential direction. The thermal creep rate, ε ,c , is given by ∂ε ,c ∂t

 = 5 × 10−23  2

3.47 × 10−23

 3 | |

 exp

−U T



 − ε ,c

 exp

−U T

The creep is accelerated by fast-neutron irradiation because it increases the point-defect concentration. The strain rate due to irradiation creep is given by



∂t

=

2.4. Fission gas release calculation The release of fission gas from irradiated UO2 fuel to the element free volume is a very complicated phenomenon. The model used in this work is based on that employed in reference (Morgan, 2007). The release process is modeled in two steps. In the first step, fission gas is produced in the fuel grains and then diffuses to the grain boundary where it accumulates forming intergranular bubbles. The second step occurs when the intergranular bubbles grow large enough to interconnect and release gas to the element free volume.

2.4.1. Release to the grain boundaries The fission gas release to the fuel grains can be modeled by a Booth diffusion process (Beck, 1960; White and Tucker, 1983; Morgan, 2007; El-Jaby, 2009). In this model, the fuel grains are treated as idealized homogenous spheres in which the fission gas atoms are produced uniformly and exhibit Fickian diffusion. The grain surface is assumed to be a perfect sink with atoms diffusing across the grain surface entering the intergranular bubbles. The model for fission gas diffusion was implemented as a separate two-dimensional Cartesian geometry to represent the fuel grains which is coupled to the pellet model. In this geometry, the xcoordinate corresponds to the radial coordinate of the pellet model and the y-coordinate corresponds to the normalized radial coordinate within each fuel grain. This numerical implementation was validated against an analytical solution published by both Kidson (1980) and Rim et al. (1981) for a step change in model parameters. The release rate to the grain boundary, Rgb , is Rgb (t) =

(21)

12



D 2

gd

∂C  ∂y 

(23) y=1

where, gd is the local fuel grain diameter and D is the diffusion coefficient for fission gas in the UO2 crystal matrix. This diffusion coefficient was obtained from Morgan (2007) who followed the work of Turnbull et al. (Friskney et al., 1977; Turnbull et al., 1982, 1977) and White and Tucker (1983). The average local UO2 grain size was determined by solving the grain growth relationship provided by Khoruzhii et al. (1999). The rate of grain growth in m s−1 is given as dgd = 1.46 × 10−8 exp dt



×

U = 212.7 − 0.5324T + 1.17889 × 10−4 T 2 + 3.3486 × 10−7 T 3 .

(22)

T7

where, Fflux is the flux of neutrons (m−2 s−1 ) with kinetic energy greater than 1 MeV.

(20)

where, T is the temperature in Kelvin,  is the circumferential component of the stress and U is the apparent activation energy divided by the ideal gas constant R given by



0.65 2.2 × 10−7  exp −5000/T Fflux

 −32, 100  T

exp(7620/T ) Frate T exp(5620/T ) 1 − − gd 2.23 × 10−3 6.71 × 1018

 .

(24)

Here T is the temperature in K and Frate is the fission rate density (m−3 ). Note that this model does not consider the distribution of grain sizes within a region; it only considers the average grain size. It was shown that this simplification produces accurate results for the fission gas release calculation despite the potentially wide variations in the grain-size distribution (Shaheen, 2011).

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2.4.2. Gas release to the fuel element Once fission gas has been released to the grain surface, it becomes trapped in intergranular bubbles between fuel grains. The amount of gas accumulated on grain boundaries is described in terms of the volume average of concentration, Gb . The amount of gas required at the grain surface to achieve inter-linkage is the grainboundary saturation, Gbsat . When there is not sufficient fission gas to maintain the interlinked network, the bubbles become isolated and cannot release gas to the free volume. This effectively contains a portion of the fission gas on the grain boundary as intergranular bubbles even after the bubbles have been interlinked. Beginning with the model proposed by White and Tucker (1983) and utilizing values appropriate for UO2 , the grain boundary saturation can be approximated by Gbsat =

4.1622 × 1016 (Pext + 2.504 × 106 ) Tgd

(25)

where Pext is the external pressure acting on the fission gas bubbles. Olander argues the external pressure should be equal to the local hydrostatic stress in the solid, PHS . This can calculated from the results of the solid-mechanics model as PHS = −

 +  +   r z

(26)

3

where sigma is the linear stress components of the stress from Eq. (10). According to this formulation, the hydrostatic stress is positive for compressive stress, and negative for tensile stress. Thus, compressive loads on the fuel reduce fission gas release by increasing the concentration required to achieve grain boundary saturation and vice versa. Physically, the tensile stress on the fuel is limited by the presence of cracks which occur along the three primary directions. To account for this, a second modified version of the hydrostatic stress, PHSC , can also be utilized which limits the components of the stress to compressive values PHSC = −

 min( , 0) + min( , 0) + min( , 0)  r z 3

.

(27)

Note that the user can select to use either model for the hydrostatic stress independent of the solid-mechanics model used. The kinetics of the gas release from the grain boundary is poorly understood. Traditionally, NOC fuel modeling codes usually assume that fission gas release from the grain boundary occurs instantly

163

relative to NOC timescales. However, transient fuel models usually assume the fission gas release is too slow to be significant on the timescale of fuel transients (roughly seconds to several minutes). This suggests that fission gas release kinetics occurs over a moderate timescale of tens of minutes or hours. For simplicity, a first-order kinetic model has been used. According to this theory, the release rate from the grain boundary from a small fuel volume, ∂V, is ∂Re = max ∂V





Gb − Gbsat ,0 fg

(28)

where  fg is the time constant of fission gas release. The release rate of gas atoms to the element, Re , can then be calculated by integrating over the volume. A default value for the time constant of 6 h has been used in the code, although user input may override this value as appropriate. Tests indicated very little change in total fission gas released during NOC for time constant values between 30 min and two days. 2.5. Gas pressure calculation The internal gas pressure is calculated using a non-homogenous temperature form of the ideal gas law. This is given by P=

nRgas



(29)

1 dV V T

where n is the number of moles of gas within the element, Rgas is the ideal gas constant and V is the volume occupied by the gas. In this model, the gas volume is divided into sub-volumes which are calculated individually and added together. Since the gaseous regions are not meshed, the temperature in these regions has been approximated using the temperatures on the boundaries of the gas volumes. This approach converts the volume integrals into boundary integrals. 2.6. Material properties models References for the material property models used in the FAST code are listed in the Table 1. These have been taken from the MATerial PROperties for light-water reactor analysis (MATPRO) compilation, the theory manuals of fuel performance codes employed at Atomic Energy of Canada Limited (AECL), and open

Table 1 List of material property models used in FAST code. Property

UO2 models

Zircaloy-4 models

Mixed gas models

Thermal conductivity

Lucuta (Lucuta et al., 1996), AECL-ELESTRES (Williams, 2010) or MATPRO (SCDAP/RELAP5 Development Team, 1997) MATPRO (SCDAP/RELAP5 Development Team, 1997) Fink (Fink, 1982)

MATPRO (SCDAP/RELAP5 Development Team, 1997)

Sills and Peggs (AECL, 1975) or Semi-empirical model (Prudil, 2013)

Emissivity Heat capacity Thermal expansion strains Young’s modulus Poisson’s ratio Density Fracture stress Meyer’s hardness Temperature jump distances

MATPRO (SCDAP/RELAP5 Development Team, 1997) MATPRO (SCDAP/RELAP5 Development Team, 1997) MATPRO (SCDAP/RELAP5 Development Team, 1997) Olander (Olander, 1976) MATPRO (SCDAP/RELAP5 Development Team, 1997)

MATPRO (SCDAP/RELAP5 Development Team, 1997) Brooks and Stansbury (Brooks and Stansbury, 1966) MATPRO-9 (EG&G Idaho, Inc., 1976) and Sills (Sills, 1979) MATPRO (SCDAP/RELAP5 Development Team, 1997) and Sills (Sills, 1979) Sills (Sills, 1979) Sills (Sills, 1979)

Ideal gas model

Ideal gas model

MATPRO (SCDAP/RELAP5 Development Team, 1997) Sills and Peggs (AECL, 1975) or Semi-empirical model (Prudil, 2013)

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literature publications. These models are also summarized in (Prudil, 2013). For some parameters, such as thermal conductivity, multiple models have been implemented. In these cases, the user is able to select the model they would like to use for any simulation.

3. Benchmark/validation The FAST code has undergone a proof-of-concept validation against both predictions from the ELESIM and ELESTRES fuel performance codes and experimental data. The code-to-code benchmark is a comparison of pellets temperatures as a function of time/burnup (see Section 3.1). The second part of the NOC validation compares the end-of-life predictions of fission gas release and sheath deformation to experimental data and the predictions of the ELESIM and ELESTRES codes (see Section 3.2). In order to directly compare the codes, the ELESTRES correlation for thermal conductivity of UO2 was used in the FAST code.

3.1. Code-to-code benchmarking A code-to-code comparison of the predicted fuel surface and centreline (maximum) temperatures as a function of average element burnup was conducted for constant linear element powers of 25, 40, 55 kW m−1 up to an average element burnup of 200 MWh kg U−1 similar to other analyses (Morgan, 2007; Shaheen, 2011). The purpose of the code-to-code comparison is twofold: firstly it provides a simple check that the FAST code produces reasonable agreement with an established model, and secondly it is to help provide insight into the differences between the codes. This can help interpret the results of the post-irradiation predictions in the next section. The code-to-code comparison was performed using the element geometry and coolant conditions specified by Atomic Energy of Canada Limited (AECL) Chalk River Laboratories (CRL). The results of this comparison are shown in Fig. 2. Three configurations of the FAST code were tested: FAST-Pext = 0 is no external pressure acting on the fission gas bubbles, FAST-UC is the uncracked configuration (i.e., no cracking in solid-mechanics and hydrostatic pressure calculations) and FAST-C is the fully cracked configuration (i.e., using the cracked solid-mechanics and hydrostatic pressure models). At the start of irradiation, all three configurations of the FAST code predict pellet surface temperatures which are ∼10 K greater than the ELESIM and ELESTRES codes. This discrepancy arises from inherent differences in the implementations of the models in the various codes such as the axial dependence of the contact pressure and the transient creep down of the sheath. As the burnup increases, the predictions of the uncracked configuration of the FAST code begin to diverge from the others. In each of the three linear powers this initially begins with a small dip in predicted surface temperature followed by a sustained increasing trend. These features are a result of the onset of fission gas release in the element. According to Eq. (5), a small amount of fission gas will initially decrease the temperature jump distance thereby reducing the surface temperature. This effect is not seen in the ELESIM and ELESTRES results despite using the same gaseous conduction models for the gap. This may be due to the relatively large (10 MWh kg U−1 ) minimum burnup/time-step required in these codes, or greater predicted solid–solid conduction (therefore less dependence on the gaseous conduction). As the fission gas volume increases, the decrease in the temperature jump distance is more than offset by the drop in thermal conductivity of the gas mixture and the reduction in the solid-to-solid contact pressure (because of increased gas pressure). These effects collectively cause the gap

Fig. 2. Code-to-code comparison of predicted pellet surface temperature (a) and centerline temperature (b) as a function of burnup.

heat-transfer coefficient to decrease, and the surface temperatures to rise. The three versions of the FAST code predict the onset of fission gas release at very different burnups. The uncracked model predicts fission gas release almost at the start of irradiation because the tension leads to very low grain boundary saturation near the fuel surface. This is in contrast to the cracked model which only predicts significant fission gas for the 55 kW m−1 case at around 80 MWh kgU−1 . As expected, the fission gas release from the zero external bubble pressure test is somewhere between the uncracked models. In the 25 and 40 kW m−1 cases, the early fission gas release of the uncracked-FAST model causes the temperatures predictions to diverge significantly from the other models. At 55 kW m−1 , the furthest outlier is the cracked-FAST model due to the late prediction of fission gas release. The centreline temperature predictions (Fig. 2b) show some similarities to the pellet surface temperature predictions. Again, the uncracked-FAST model for 25 and 40 kW m−1 cases predicts steadily rising temperatures due to fission gas release. At these powers the centreline temperature predictions from the zero external pressure and cracked FAST models are also in agreement with each other. The three versions of FAST initially predict slightly larger centreline temperatures than the other codes. A notable difference is the change in the order of the predicted temperatures. The ELESTRES code predicted the greatest surface temperatures at 55 kW m−1 , however, it predicts centreline temperatures lower than both the Pext = 0 and cracked case. Similarly, the ∼30 K difference in fuel surface temperatures between the cracked-FAST and ELESIM results at 55 kW m−1 and 200 MWh kg U−1 is also reversed. Since all of these codes use the same flux depression model, the change in order of these temperatures must be the result of differences in the calculated thermal conductivity of the fuel. The

Fig. 3. Sample power history from Case 1026 (highest power case).

7.5 7.58 11.6 9.0 0.405 0.405 0.405 0.405 7.5 7.58 11.6 9.0 20.87 15.54 18.52 15.52 0.45 0.45 0.827 0.85 0.45 0.445 0.827 0.85 0.047 0.047 0.176 0.181 0.584 0.26 0.236 0.215 12.141 12.16 14.328 12.236 0.1 0.085 0.08 0.08

Axial chamfer (mm) Dish depth (mm)

23 31 26 31

Table 3 Summary of fuel element geometry for the 7 cases provided by AECL-CRL.

The validation exercise is a comparison of the predicted end-oflife condition of seven irradiated fuel elements which underwent post-irradiation examination (PIE). The data for this validation was provided by AECL-CRL. The cases were selected to cover a range of power, burnup and geometries, where a complete description of the cases is available in reference (Prudil, 2013). The maximum and average linear powers and discharge burnup of these cases is included in Table 2 along with a sample power history in Fig. 3. A summary of the important fuel element manufacturing parameters is included in Table 3. The PIE yielded measurements of the fission gas release volume, grain size, sheath strain, and circumferential ridge heights of each element. The results of this comparison exercise have been summarized using the case number on the horizontal axis as illustrated in Fig. 4. Results are presented from the same three versions of the FAST code used in the code-to-code comparison (Pext = 0, uncracked, and cracked). The error bars on the average experimental value indicate the maximum and minimum measured values for each element. This provides a sense of scattering in the experimental results. From the comparison of end-of-life fission gas volume, it is apparent that the uncracked-FAST model greatly over predicts the fission gas in all cases. By comparison, the cracked-FAST and Pext = 0 models predict far less gas release. This is consistent with

Radial chamfer (mm)

3.2. Post-irradiation prediction

2.0 1.93 2.29 2.87

Land width (mm)

Pellet length (mm)

Initial grain size (␮m)

FAST and ELESTRES codes were run using the same thermal conductivity correlation. Thus these differences must be the result of input parameters to the thermal conductivity correlation, like local porosity and burnup which are treated differently in each code. For example, the FAST code calculates the burnup locally as the time integral of Eq. (2) (with scaling factor for unit consistency), thus it depends on time as well as the radial and axial position within the pellet. In contrast, the ELESTRES code calculates an average burnup which depends only on time. Due to flux depression, the local burnup at the surface can be 2–3 times the burnup at the center axis for cases with a high average burnup, leading to significant differences in the predicted material properties.

389 1018 1026 1281–1284

20.2 30.4 31.1 32.1 26.4 23.7 22.6

Initial grain size (␮m)

24.5 37.3 53.4 41.2 33.7 30.1 28.8

Sheath thickness (mm)

422.6 132 301.8 552.2 454.6 404.8 385.3

Outer diameter (mm)

389 1018 1026 1281 1282 1283 1284

Diametral clearance (mm)

Time average linear power (kW m−1 )

Stack axial clearance (mm)

Max Power (kW m−1 )

Pellets per element

Max burnup (MWh kg U−1 )

AECL element number

AECL case number

Pellet density (kg m−3 )

Table 2 Summary of the seven fuel element power histories provided by AECL-CRL.

165

10 592 10 758 10 690 10 650

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Fig. 4. FAST proof-of-concept validation for normal operating conditions showing (a) the fission gas release, (b) mid-pellet sheath strain, (c) circumferential ridge strain and (d) circumferential ridge height benchmarked against average and maximum measurements as well as the ELESTRES and ELESIM codes. Error bars indicate the approximate scatter in measurements from multiple elements based on the average and maximum values.

the trends observed in the code-to-code comparison, where the uncracked-FAST model began accumulating fission gas at very low burnups, followed by the Pext = 0 case and then the uncracked case. The measured fission gas volumes are found to be between the cracked and uncracked models except in case 1282, in which the cracked model over predicted the fission gas by a small amount (∼4 ml). This supports the assumption that the physical behavior of a partially cracked fuel element should be between the extremes of the fully elastic and fully cracked models. However, it is also observed that the mean range (i.e., difference between the upper and lower bounds) is 26.7 ml, which is very large compared to the gas release volumes. In fact, the mean error in the predicted fisson gas volumes is 21.0, 6.9 and 4.6 ml for the uncracked, cracked and Pext = 0 cases, respectively. The Pext = 0 configuration was found to predict larger fission gas release volumes than both the ELESTRES and ELESIM codes. This is an improvement for cases 1026 and 1281 (both of which showed significant gas release); however, it is an over prediction in case 1282 (and cases 389 and 1283 to a much smaller degree). Thus, the FAST code appears to overpredict fission gas release at low burnups (low fission gas release), but shows an improved predictive capability at higher burnup (and higher fission gas release). The FAST predictions of circumferential sheath strain at the midpellet show improved agreement with the experimental measurements compared to the other codes. In all cases, all of the codes were found to consistently under-predict the measured strains. Since the

FAST code predicted the largest circumferential strains at the midpellet this led to an improved agreement. The three variations of the FAST code showed far less variation in the predicted strain than in the fission gas predictions. The most notable exception is case 389 where the uncracked model predicted a strain significantly larger than the other two models. In four of the seven cases, the uncracked model predicted the largest strains. This was initially unexpected as cracking allows for greater expansion of the pellet leading to a larger sheath strain. However, the uncracked models also predicted a much greater quantity of fission gas, leading to higher temperatures (as seen in the code-to-code comparison). Higher temperatures drive larger thermal strains in the pellet which also leads to larger sheath strains. For case 389, it was found that the larger thermal strains exceeded the effect of the cracks. However, case 1026 shows the opposite relationship because the cracked model did not over predict the fission gas by a significant a margin. The strain predictions at the pellet-to-pellet interface (also known as the circumferential ridge) followed many of the same trends as the mid-pellet strains. Again, both ELESTRES and ELESIM codes were found to under-predict the measured values, with the FAST code closer to the measured values. However, the difference between FAST and ELESTRES predictions has narrowed slightly. This is because the ELESTRES code predicted much larger ridge heights than the FAST code. ELESTRES over predicted the mean ridge height in five of the seven cases while FAST underpredicted six of those seven cases.

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4. Ongoing and future work Other developments of the FAST code include: • An extension of the code for modeling the high-temperature transient behavior the fuel (Prudil et al., 2012, 2013). This work involves the addition of high-temperature creep phenomena, sheath oxidation, time-dependent irradiation conditions, multipellet models to account for axial dependence, and support for large variations in time-steps as well as additional validation. This work will be documented in a second paper. • An iodine-stress corrosion cracking model is currently being integrated into the FAST code to predict fuel failures during power ramps (Oussoren et al., 2013). This requires modeling the production and transport of iodine within the element and integration with a sheath crack propagation model (Lewis et al., 2011). • The code is also being used to model advanced reactor fuels comprised of mixtures of thorium, uranium and plutonium oxides for a Super-Critical–Water Reactor (SCWR) GEN-IV concept (Bell et al., 2013). This work requires replacing the material property correlations and a number of phenomenological models. The aim of this project is to use the revised model to optimize the fuel design. The Comsol platform on which the FAST code is constructed offers a large and highly flexible feature set. This allows modifications to the FAST code to be made easily compared to existing fuel codes, and without modification of any source code or numerical methods. All of the material/phenomena models, geometry, mesh, initial conditions and boundary conditions to be edited through the graphical user interface. This makes it well suited to be adapted for many tasks not supported by traditional codes such as modeling fuel experiments (e.g., electrically heated elements), developing novel designs (e.g., new geometries), or prototyping new material/phenomena models (e.g., optimizing fitting parameters for a new material property correlation). However, it should be noted that the FAST code is also limited to the features available in Comsol. 5. Summary and conclusions An overview of the FAST fuel performance code was presented including a description of the key models employed in the code. Proof-of-concept demonstration tests were conducted to demonstrate the feasibility of the code as a general purpose fuel performance model. This included code-to-code comparison of fuel surface and centreline temperatures for constant power histories as well as a comparison between code predictions and post-irradiation measurements of irradiated fuels. In the seven test cases, the FAST code more closely predicted end-of-life sheath strains compared to the ELESIM and ELESTRES codes. A larger data set is required to determine if this improvement is consistent for a wide range of conditions or a statistical anomaly due to the small sample size. Finally, various ongoing improvements to the code were briefly discussed along with its potential for experimental design and analysis. Acknowledgements The authors would like to acknowledge the support of our colleagues at RMC, as well as B. Leitch, A. Williams, S. Yatabe and M.R. Floyd (AECL-CRL) for assistance and providing validation data. Funding for this work was provided by NSERC (IRCPJ 355249 – 2006), UNENE (749500), COG (WP 22303, 22326 and 22324) and Ontario Research Fund grant (753180).

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