### SUMMARY

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

A new finite element tool is presented, which utilises the extended FEM (XFEM) to model leaks through cracks. Heat flux and pressure boundary conditions are imposed on the crack in the form of jump terms. Enrichments are chosen to model either strong or weak discontinuities across the crack, as well as singularities at the crack tips. Excellent convergence rates are achieved for both the thermal and mechanical models, where errors are calculated relative to analytical solutions derived for this specific problem. A more general temperature approximation is also presented, which makes no assumptions about the continuity of temperature or heat flux across the crack. Results indicate that this is a robust way of modelling the temperature of a plate containing a crack with or without a leaking fluid. Thermomechanical simulations were then carried out to demonstrate the applicability of the FEM for analysing leak rates in nuclear reactor primary pipework. A two-phase flow model based on the Henry–Fauske model is chosen for the fluid aspect, and this is coupled to the structure through a convection law. Crack closure is shown to reduce the leak rate by up to 40%. © 2014 The Authors. *International Journal for Numerical Methods in Engineering* published by John Wiley & Sons, Ltd.

### INTRODUCTION

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

In nuclear power plants, the low probability of failure is justified by multilegged safety cases. There are the standard manufacturing quality assurance requirements, as well as inspection methods and defect tolerance assessments for small flaws. In addition to this, there are defence in depth arguments such as the Leak-before-Break (LbB) philosophy. It says that if there is a crack in a component, then it will be possible to detect a leak within a short enough time frame such that any corrective action may be taken before a catastrophic failure occurs [1]. Essentially, this eliminates the possibility of a guillotine type fracture, which would normally be mitigated against with pipe whip restraints and jet impingement shields. The economic benefits of not imposing these protective measures are made possible by safety arguments such as Leak-before-Break.

The full Leak-before-Break procedure requires an assessment of the critical crack size of a component. This is the size of crack at which a guillotine type fracture will occur. This can be calculated using a failure assessment diagram methodology such as R6 [2]. It is then possible to calculate the crack opening area (COA) for a crack size much smaller than the critical crack size. Analytical equations can be used to obtain COA, and if further accuracy is required, the FEM could be used. Using this COA, it is then possible to obtain the leak rate. If this leak rate is greater than a specified detection limit, a Leak-before-Break case can be made. The biggest uncertainty in a Leak-before-Break assessment is the calculation of leak rate. This paper focusses on the calculation of leak rates using a new finite element tool. This tool is designed to reduce error by incorporating crack opening and fluid calculations within elements. The fluid mechanics of the leaking fluid will be coupled to the solid via mechanical and thermal boundary conditions. In particular, the effect of heat transfer from the fluid to the crack face is investigated as it is hypothesised to reduce the COA. Therefore, the focus of this work is on the thermal exchange between the fluid and the structure, in contrast to [3], which focusses more on the mechanical aspects of a similar coupled problem.

The high inlet-to-outlet pressure ratios that occur when a through wall crack appears generally necessitate the use of critical flow models. These are either single or two-phase flow, depending on whether a gas or water coolant is used. CFD gives leak rates accurately but are impractical for LbB assessments due to the complexity of the models. Therefore, analytical or simplified 1D finite difference models are preferred as they give good accuracy relatively cheaply. The single phase mass flow rate can be derived by considering the mass, energy and momentum equations, as well as an equation of state for an ideal gas. This leads to closed-form solutions that are straightforward to solve. For the two-phase case, a model based on Henry–Fauske is employed [4]. This treats the mixture as a single fluid with properties based on the mixture quality. There is also a consideration of the thermal non-equilibrium between phases, which occurs because of the very short time the fluid is in the crack. The error is exaggerated by the fact that there is uncertainty in the crack shape [5].

It is apparent therefore that Leak-before-Break has both structural and fluid mechanics considerations. It is a present practice to analyse these two aspects independently [1]. However, modern computational techniques are making a coupled approach more feasible. The motivation for a coupled approach stems from the fact that thermal heat transfer from the fluid to the crack walls is known to affect leak rate [6]. Two-phase flow is particularly sensitive to wall heat transfer [7], this is due to the steam evolution as pressure and temperature change through the thickness. Also, the additional heating of the crack walls due to a leaking fluid causes thermal expansion, which changes the crack opening displacement (COD). So it is clear that there should be a link between fluid and structure through a thermal coupling. In addition to this, there should be a consideration for the end user, so a model that can be efficiently used in an industrial context is essential. To facilitate this requirement, a new finite element tool has been developed, which accounts for all the physics that affects leak rate. The approach outlined is founded on the extended FEM (XFEM), as this provides a framework to build an element that can capture the essential behaviour, as well as being simple to use once implemented into a finite element code.

### LITERATURE REVIEW

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

The extended FEM, first introduced by Belytschko [8], is a development of the FEM, which enables efficient modelling of problems with discontinuities or singularities. The method utilises the partition of unity concept that was presented by Melenk and Babuska [9]. With the addition of enrichment functions, which give additional information in the regions where the conventional finite element may require highly refined meshes, optimum convergence rates can be achieved. Such physical situations where this may arise are cracks in solids, where there is a discontinuity in displacement across the crack, and the singularity occurs in the strain field at the crack tip. The enrichment functions used to describe this behaviour are a Heaviside step function along the crack and a set of functions derived from the Westergaard solution at the crack tip.

Enrichment functions should capture the behaviour in specific regions using exact analytical solutions. For the mechanical model, the well-known elastic behaviour of cracks gives a solution close to the crack tip as a series expansion. The four terms used in XFEM have proven to give good accuracy and convergence rates [10]. Although these expressions were based on the traction free crack assumption, the principle of linear superposition means that they can be used for a constant pressure along the crack faces. This is shown to give good accuracy when comparing the COD from the FE approximation to analytical expressions. The thermal consideration is not as well established, although it is known that there is a singularity in heat flux at the crack tip when considering a homogeneous conducting material. Kit [11] solved the Helmholtz equation for a plate with a crack subject to various boundary conditions. These were either keeping a constant temperature or heat flux along the crack or imposing a jump condition in both quantities. This assumes a heat transfer in the out-of-plane direction, *z*, which leads to the Poisson equation. However, this work takes no account of the *z*-direction and hence, an insulated condition is assumed. Therefore, the equation to solve is simply Laplace's equation, which is carried out by using the fundamental solution and applying single layer potential theory [12] (see Appendix). This model provides a means to quantify the convergence behaviour and accuracy of the FE model.

The integration involved with XFEM is complicated by the fact that non-polynomial functions are acting within elements, thereby rendering traditional Gaussian integration techniques inaccurate. Various methods have been suggested to circumvent these problems, a commonly used technique is to decompose the elements into smaller subregions and perform the integration separately within these regions [13]. Other methods have been proposed by Bordas *et al*. who uses strain smoothing techniques [14] and Schwarz–Christoffel mapping [15] to account for the sharp discontinuity. In [14], it was found that higher levels of accuracy could be achieved with strain smoothing when the enrichment functions were polynomial; however, for non-polynomial functions, the accuracy was less than element subdivision techniques. The results in [15] are shown to be accurate. The integration scheme employed in this study is based on the method of Fries [13], where cracked elements are subdivided into quadrangles and crack tip elements are divided into six triangles each with a vertex at the crack tip. The first use of a decomposition of elements into subquadrilaterals independent of the position of the discontinuity was carried out in [16]. In [17], specific polar integration and geometrical enrichment for singular enrichment is introduced. Elguedj *et al*. [18] present the use of a fixed subintegration rule for XFEM crack growth. Laborde gives a new polar integration rule and convergence study of the XFEM in [19]. Standard Gauss quadrature in elements containing the discontinuity is used in [20].

Applications of XFEM to problems that are relevant to the one considered here are now discussed. Thermomechanical aspects of XFEM are investigated by Duflot [21]. In this work, the boundary conditions on the crack are constant temperatures using Lagrange multipliers or an insulated condition. A jump in temperature based on a thermal resistance model was considered by Yvonnet *et al.* [22], which assume a constant heat flux across the crack. This is a special case of the weak form presented in [23], which makes no assumptions about the flux across the crack. The thermal model that will be presented in the paper assumes a heat flux along the crack imposed by a Robin-type boundary condition, with change in heat flux accounted for by the relevant enrichment function.

Extended FEM is applied to coupled fluid solid problems in [24] and [3]. A microscale viscous flow model is coupled to the macroscale finite element model via pressure and displacement requirements imposed on the discontinuity. These requirements are satisfied by exploiting the partition-of-unity property of finite element shape functions. The work presented here develops this idea by imposing jump conditions across the discontinuity due to the pressure and heat flux from the leaking fluid.

### MATHEMATICAL FORMULATION

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

The mathematical derivation of the new finite element tool is presented in this section. Steady-state conditions are assumed in this analysis, as well as small deformation theory for the strains. The crack is considered to be a 1D line discontinuity in a 2D domain, see Figure 1. The crack is represented by a single contour centrally positioned between the two crack faces, see Figure 2. The use of a single contour gives rise to jump conditions, which are necessary to represent sharp changes in surface traction and heat transfer at the crack. The governing equations are the equilibrium equation and heat equation on a domain Ω bounded by *∂*Ω, respectively, and are given by

- (1a)

- (1b)

Integration of the variational terms and *δTQ* over the domain and on application of the divergence theorem provides the weak form of equation (1b) with the prescribed boundary conditions, that is,

- (4)

- (5)

where *D* is the conductivity matrix and the following solution spaces *u*_{0} and *T*_{0} of the test functions *δT* and are given by

where *H*_{1} is the Sobolev space.

The weak form for the thermal problem considered here is derived from the most general form given in [23]. In this analysis, the body contains an interface that describes a curve intersecting with the boundary. The curve is not a feature in the weak form for this application because the physics involved is limited to heat fluxes at surfaces and interfaces.

The formulation is similar to a traditional FE formulation, but in this case, the stiffness matrix has an additional contribution arising from an integral along the crack. The heat vector also has a contribution due to the bulk fluid temperature along the crack. The discretized form is as follows:

- (6)

For the temperature field, it is possible to model a jump in heat flux on either side of the crack. This requires an enrichment function that can account for a change in the derivative of temperature across the crack. Moës’ bimaterial formulation [25] is used to account for this

- (7)

where

- (8)

where *φ*_{I} is the normal distance from the interface to the node. If the crack is assumed insulated, that is, when there is no fluid in the gap, it is more appropriate to use the Heaviside function to give a jump in temperature. This does not account for different fluxes on each side of the crack however. The enrichment function in (7) is used in (10) to account for the jump in heat flux at the crack. So the temperature discretization is

- (9)

or

- (10)

where *a*_{i}, *b*_{i}, , and are additional unknowns that need to be solved for. *S*_{c} are nodes whose parent element is completely cut by the crack, and *S*_{t} are nodes whose parent element is partially cut by the crack. Equation (9) is the approximation for a discontinuous temperature across the crack, and Equation (10) is for a discontinuous heat flux across the crack. *H* is the step function which is +1 on one side of the crack and − 1 on the other. *γ*_{i} is the crack tip enrichment, and *φ* is the level set function employed to locate the crack.

For a more general formulation, the temperature can be approximated using both the weak and strong discontinuity enrichments, that is,

- (11)

The crack tip enrichments functions describe the singular behaviour at the crack tip and are based on the Westergaard solution.

- (12)

where *r* and *θ* are polar coordinates, with origin at the crack tips. For the thermal field, it is appropriate to use the following enrichment function when a constant temperature is imposed on the crack faces

- (13)

which is suitable because it is constant when switching from *π* to − *π*, which are the respective crack faces. Hence, the temperature is continuous across the crack when this enrichment function is used. Taking the derivative with respect to theta gives sinusoidal behaviour, which is discontinuous across the crack, giving a discontinuity in heat flux. When the crack faces are subject to a constant heat flux, the following enrichment function is appropriate:

- (14)

This is discontinuous from *π* to − *π*, hence, the temperature can be discontinuous across the crack.

The pressure term given in the weak form (5) is now analysed. There is a non-zero traction on each face of the crack, which is equal in magnitude but in opposite directions. The pressure term becomes

- (15)

where *δu*_{n} is a variation along the crack, and symmetry is assumed on either side of the crack.

Similarly, for the thermal case, there is a heat flux on each face of the crack, equal in magnitude but in opposite directions. This flux is defined by a convection law ; therefore,

- (16)

Considering Equation (5), *T*_{bulk} and *h* are known a priori but *T* must be obtained from the solution; therefore, it is incorporated into the left-hand side.

- (17)

With the current formulation, it is possible to model either a discontinuity in heat flux or temperature. This is because the temperature approximation can contain the Heaviside or the level set enrichment, which accounts for the strong or weak discontinuity, respectively.

Using the aforementioned convection law means that prescribing a temperature along the crack is simply a case of fixing the heat transfer coefficient to be very large. This would give a constant temperature along the crack without the need to use Lagrange multipliers. Currently, this is what is done for imposing Dirichlet conditions in XFEM [21].

### FINITE ELEMENT EQUATIONS

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

Inserting the test functions and trial functions into the weak form leads to the fully coupled mecha-nical system of equations that can be solved simultaneously with respect to the mechanical and thermal DOFs

- (18)

with the vectors *ξ* = [*U*,*A*,*B*,*T*_{u},*T*_{a},*T*_{b},*T*_{c}]^{T} and *F* = [*F*^{u},*F*^{a},*F*^{b},*Q*^{u},*Q*^{a},*Q*^{b}]^{T}. Here, the subscripts *u*, *a*, *b* and *c* denote the standard, step enriched, tip enriched and weak discontinuity enriched DOFs. The upper case letters, *U*, *A*, *B* are the DOFs for the mechanical field, *T*_{u}, *T*_{a}, *T*_{b}, *T*_{c} are the DOFs for the thermal field. The same standard linear shape function *N* is used for both the mechanical and thermal fields.

#### Submatrices

The thermomechanical and thermal element matrices are structured as follows:

- (19)

- (20)

where the subscripts *M*, *T* and *TM* correspond to the mechanical, thermal and thermomechanical components, respectively.

where *ζ*,*β* ∈ {0,1,2,3,4} denote the extra DOFs, 0 corresponds to Heaviside enriched and 1 − 4 are the tip enriched.

The second-order elasticity tensor (*C*), for plane stress, is given by

- (23)

where *E* is the elastic modulus and *ν* is the Poisson's ratio.

where *ζ* ∈ {0,1,2,3,4}and *β* ∈ {0,1}, where 0 corresponds to the weak or strong discontinuity enrichment and 1 is either a cosine or sine enrichment. If the enrichment in Equation (11) is used, *β* ∈ {0,1,2,3}.

and

- (26a)

- (26b)

where *ζ*,*β* ∈ {0,1}or {0,1,2,3} depending on the approximation used. (*D*) contains the conductivity, *k*, that is,

- (27)

The structure matrix for two dimensions reduces to

- (28)

where *K*^{T} is a 12 × 12 matrix, are 24 × 24 matrices and are 24 × 12 matrices.

In the case of a four-noded quadrilateral linear element, the standard shape functions in a reference domain are as follows:

- (29)

where *ξ*,*η* ∈ [ − 1,1] . The reference domain is mapped to the real element under a smooth map.

### FLUID MODELS

- Top of page
- SUMMARY
- INTRODUCTION
- LITERATURE REVIEW
- MATHEMATICAL FORMULATION
- FINITE ELEMENT EQUATIONS
- FLUID MODELS
- INTEGRATION
- IMPLEMENTATION
- RESULTS
- SUMMARY AND CONCLUSIONS
- FURTHER WORK
- APPENDIX: ANALYTICAL MODELS
- ACKNOWLEDGEMENTS
- REFERENCES

The crack is modelled as a 1D channel or pipe with rough walls, and the dimensions can change through the thickness of the vessel, see Figure 3. The crack opening displacement is denoted by *w*; *l* is the crack length and *t* is the length of the channel. The crack opening can either diverge or converge through the channel, current models account for a linear change in this. Choked flow must be considered here as the difference in pressure from the inside of the vessel to the outside is very high. The crack channel shown in 3 is idealised and it is known that in reality, the channel may be more complex. As the fluid flows around corners through the crack length, there are associated pressure losses. The pressure losses can be split up into three distinct categories: surface roughness, changes in direction and recirculation. These are captured in a discharge coefficient *C*_{D}, which is derived in [5]. The mass flow rate for single phase flow is given by

- (30)

where *p*_{0} and *ρ*_{0} are the inlet pressure and density and *A* is the average crack opening area. For the two-phase flow, the situation is more complicated and the SQUIRT code is used [26]. SQUIRT uses the Henry–Fauske [4] homogeneous non-equilibrium model, which is essentially the solution of two simultaneous, nonlinear equations. One of these equations accounts for the mass flux, and the other is the pressure balance due to losses along the flow path, including phase change acceleration, area change, friction, entrance and corner losses. The model accounts for the fact that there is thermal non-equilibrium between the two phases during the time the fluid is in the crack. This non-equilibrium mixture quality *X*_{c} is in the form of an empirical law, which is fitted to experimental data.

The mass flux and pressure constraints form a simultaneous nonlinear system, which can be solved by Newton–Raphson iteration, that is,

- (31a)

- (31b)

where *G*_{c} is the critical mass flux, *ν*_{gc} and *ν*_{Lc} are the specific enthalpies of the vapour and liquid, respectively, *γ* is the isentropic exponent and *X*_{c} and *X*_{E} are the non-equilibrium and equilibrium mixture quality given by

- (32a)

- (32b)

where *N* = 20 for *X*_{E} < 0.05 and *N* = 1 for . The constant *B* = 0.0523. The subscripts 0, *c*, *e*, *a*, *f*, *k* and *aa* denote the vessel pressure, critical, entrance, phase change acceleration, frictional, corners and area change acceleration pressure losses, respectively.

Properties of steam are obtained from IAPWS IF-97 steam tables [27]. The thermal non-equilibrium is accounted for by a relaxation condition, which is an expression derived from experimental data [28]. This is the most practical way of accounting for heat transfer between the liquid and the vapour. To model this process in detail would be prohibitively expensive. The heat transfer coefficient is calculated from on the flow rate, channel dimensions and fluid properties. The Dittus–Boelter correlation can be used as a good approximation

- (33)

where , *Pr* = *C*_{p}*μ* ∕ *k*_{f} and

- (34)

where *D*_{h} is the hydraulic diameter of the channel, and *C*_{p}, *k*_{f} and *μ* are the specific heat, conductivity and viscosity of the fluid respectively.