A One- Dimensional Simulation of An Electrofusion Welding Process

Halijah Osman1, Shamsuddin Ahmad2, Khairil Anuar Arshad3

Universiti Teknologi Malaysia, Skudai 81310, Johor,

MALAYSIA

Abstract A one-dimensional simulation of an electrofusion welding process based on finite difference method using control volume discretization is presented. Usually simulations of this nature are implemented with finite elements using a pack- age, such as ABAQUS, with time consuming iterative computer code. This simulation is able to predict temperature and stress histories as well as giving the gap closure time of the welding process. The results of this simple simulation compared favourably with both experimental and ABAQUS values.

1. Introduction Electrofusion (EF) welding is a thermal technique for joining polyethylene (PE) distribution pipes and has been used widely in Europe for over 20 years. The use of EF fittings is rapidly gaining preference over the older butt and socket fusion techniques because of an ability to join a wide range of PE pipe grade resins, easy assemblage prior to fusion and a low joint failure rate [1]. To assist in the creation of a strong EF joint, the pipes are scraped and cleaned to eliminate dust and contamination. The ends of two pipes are then squarely inserted into a fitting containing an embedded heating wire (a clearance (air gap) between the pipe and fitting is often present). A controlled heat flux is generated in the wire to heat the polymer around it which expands and eventually fills the air gap. Thermal contact causes the heat energy from the wire to be distributed to the PE near the contact zone and create a melt pool at the fusion interface. The fusion joint between the pipe and fitting is formed as a result of the combined action of the heat and pressure developed at the fusion interface. Pressure is a major contributing factor for generating a good bond along the EF joint ([2]-[4]). EF fittings are manufactured from polymeric materials, usually PE80 and PE100 compounds based on polyethylenes with a density no less than 930 kg/m3. Although there are several different wire-embedding technologies, almost all of EFW pipe fittings use an implanted or moulded-in resis- tance heating wire [5]. The subject of this paper is a coupler in which the shape is axially symmetric and the heating coil is wound in a monofilar manner. Usually the coupler has a constant bore along its length, larger than that of the pipe. Figure 1 ilustrates the design.

Fig. 1. Schematic diagram of a barrel-form coupler [5]. Most numerical investigations concerning the EFWprocess made use of either specially designed FE packages or FE analysis ([2]-[4],[6]-[9]) to compute temperature histories. A few studies have used FD methods ([10]-[13]) with just one having employed a control-volume formulation [12]. None of the cited ‘FD’ references have attempted to calculate the stress history which is evident in EFW. As FE simulations of the EFW process are often computationally expensive it is felt that a simpler and cheaper PC-based approach using FD-based methods may give comparable simulation results with less effort. Control-volumes provide a viable approach for the numerical modelling and simulation of thermomechanical problems [12, 14, 15], possessing the same ‘capability’ with regard to geometrical flexibility as FE methods, yet retaining simplicity and efficiency.

2. Mathematical model The mathematical model of the EFW process treats the pipe fitting system as two separate regions with an initial gap at the ‘interface’. 2.1. Temperature model The temperature distribution is governed by the transient heat conduction equation in a radial coordinate system. Allowing temperature dependent material properties, the axisymmetric heat equation is

978-1-4577-0005-7/11/$26.00 ©2011 IEEE

( )ii i i i

Tc Tρ λτ

∂ = ∇ ∇∂ (1)

( i =p-pipe,f-fitting) subject to the boundary conditions

i T h Tλ− ∇ = Δ (2)

where h is an appropriate surface film heat transfer coefficient. Across the air gap h is given by [12]

max ,2000air pg

p gair air

hg f

λ λλ λ

⎧ ⎫⎪ ⎪= ⎨ ⎬+⎪ ⎪⎩ ⎭

(3)

The flow of heat is driven by an embedded hot wire. Based on Figure 2, the wire temperature is approximated by a specified function of time,

max

max

( ) ,0 220WT TT Tτ τ

τ∞

∞−= + ≤ ≤

(3)

For this simple 1D model the wire is located on one of the nodes.

Fiq.2. Wire temperature [12] 2.2. Deformation model The deformation model is based on the equation of motion and the constitutuve equations. The equation of motion, with axial symmetry, is

2

2rr u

R Rθσ σσ ρ

τ−∂ ∂+ =

∂ ∂ (4)

where rσ and θσ are the radial and circumferential stresses, and u is the radial displacement. For a Hookean material the

stress-strain relationships are

[ ]

[ ]

[ ]

1 ( )

1 ( )

1 ( )

r z

r z

z r

u T vR Eu T vR E

T vE

θ

θ

θ

α σ σ σ

α σ σ σ

α σ σ σ

∂ − Δ = − +∂∂ − Δ = − +∂

− Δ = − + (5)

where E is the modulus of elasticity, v is Poisson’s ratio and α is the coefficient of thermal expansion. The stresses are obtained by solving equations (5) to give

1 1

1 1

1

r

z

v u u v Tv R R vu v u v TR v R vu u v TR R v

θ

σ αγ

σ αγ

σ αγ

− ∂ += + − Δ∂

∂ − += + − Δ∂∂ += + − Δ∂ (6)

where /((1 )(1 2 ))vE v vγ = + − .A combination of equation (6) with equation (4) leads to the governing equation for the dynamic displacement u

2 2

2 2 2

1 11 1

u u v u v TR R R v v R

ρ αγ τ

∂ ∂ + ∂+ − − =∂ − ∂ − ∂ (7)

Since the rate of temperature change is small, the inertia term corresponding to the motion of the particles during thermal expansion in equation (7) can be disregarded [16] reducing the problem to a pseudo-steady-state form. Initially the pipe and fitting are unstrained, at rest and at a uniform temperature,

0, 0,uu T Tτ ∞

∂= = =∂

at 0τ =

In addition we assume that all surfaces are traction free before the pipe comes into contact with the fitting, allowing immediate stress relaxation.

3. Control volume discretization To obtain difference equations a conservation principle is applied over a small control (or finite) volume that is established about each of the spatial grid points. An exhaustive and mutually exclusive set of control volumes spans the solution domain. The differential equation is integrated over each control volume (Gauss’ divergence theorem is used to convert the volume integral of the divergence to a surface integral of fluxes). To illustrate consider the linear form of equation (1) written in dimensionless form as

1 ( )U Ur

t r r r∂ ∂ ∂=∂ ∂ ∂ (8)

Fig. 3. Control volume cell surrounding grid point ir in a control volume surrounding a grid point ir (as in Fig. 3

with z = 1 and 2θ π= ). Using the CV formulation with dS rd dzθ= leads to

1 ( )

.( )

V V

V V

U UdV r dVt r r r

U dV U dVt

−

∂ ∂ ∂=∂ ∂ ∂

∂ = ∇ ∇∂

∫ ∫

∫ ∫

.

i

S

UV U ndSt

τ

−∂ = ∇∂ ∫

( )

11

1

1

2 2

2 21

12 21

11 1

1 12 21 11

( ) 2 ( )

2 ( )( )

2( )

2

ii i

i

i

i

i

i i

si s

ss

i i

i is si i

m mm m mi i i i i ii i i

i i i ii i

U Us s rt r

U Urt s s r

U Us ss s r r

U U s s s sU U Ut h h h hs s

τ

τ

π π ++

+

+

−

−

+

++

++ +

− +− −+

∂ ∂− =∂ ∂

∂ ∂=∂ − ∂

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟− ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞− = − + +⎢ ⎥⎜ ⎟Δ − ⎝ ⎠⎣ ⎦

1

12 21

2( )

i i

i is si i

U Us ss s r r

+

++

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟− ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (9)

The discretised explicit form of (9) is then

( )1

1 11 12 2

1 11

2m mm m mi i i i i ii i i

i i i ii i

U U s s s sU U Ut h h h hs s

++ +

− +− −+

⎡ ⎤⎛ ⎞− = − + +⎢ ⎥⎜ ⎟Δ − ⎝ ⎠⎣ ⎦ A similar procedure is followed for the displacement equation.

4. Results and discussion To account for a change of phase, the enthalpy method [17] may be used. Calculations for this study were performed on a PC coded using Matlab for a 125mm pipe (SDR 17.6) and

Table 1 Material properties Property Pipe Fitting

Conductivity λ(W/mK)

0.325 0.325

Specific heat c(J/kgK) 2400 2375 Density ρ 3( / )kg m 840 841

fitting with an initial air gap of 1mm. The material properties used in this paper are the average of the solid and liquid states, shown in Table 1. The coefficient of thermal expansion is 0.0005/ Kα = , Poisson’s ratio 0.45, latent heat 148600J/kg, Young’s modulus 233.9425MPa and the ambient temperature 20◦C. The temperature profile along the radial direction is shown in Fig. 4. The temperature rises more rapidly at the inner fitting surface than the outer pipe surface (consistent with experimental evidence). The gap closes at about 71.1s. The

Fig. 4. Temperature profile

history of the gap closure as well as fitting and pipe gains are shown in Fig. 5. The fitting contributes about 67% of closure and the pipe some 33% and the results compare well with those of Wood et al [12] for a 2D model using a

composite deformation (Fig. 6). Once the displacements are computed at each time step, if required the nodal stresses can be readily computed from the displacement-stress relations (6). Fig. 7 shows a representative stress profile history. The interface stress peaks just before gap closure (in agreement with [2]). There is compression around the wire which is contained in the outward direction by the relatively cold outer region of the fitting. The outer region is in small tension due to the boundary condition applied at the fitting surface.

5. Conclusion Here we have presented a very simple unrefined model of a very complex physical process that provides values of the process variables that are of the correct order of magnitude.

Fig.5. Gap closure history.

Fig.6. Gap closure history [12]

The CPU time to compute the simulation up to the gap closure is 6 min using Matlab version 4, without any attention paid to code optimization. There is a scope within Matlab to vectorize much of the calculations. This work demonstrates that a cheap and portable code can be used to simulate such a complex problem. Current work is focused on refinements of

the physical model to enhance simulation predictions.

6. Acknowledgment We would like to thank the Research Management Center of UTM for funding this project under research grant vote 78335.

Fig. 7. Radial stress history.

7. References [1] Bowman, J., Medhurst, T. and Portas, R., “Procedures for quantifying the strength of electrofusion joints”, Proceedings of Plastics Pipes VIII, Koningshof, The Netherlands, (1992), pp. B2/5.1-12. [2] Fujikake, M., Fukumura, M. and Kitao, K., “Analysis of the electrofusion joining process in polyethylene gas piping systems”, Computers and Structures, Vol. 64, (1997), pp. 939-948. [3] Kanninen, M. F., Buczala, G. S., Kuhlman, C. J., Green, S. T., Grigory, S.C., O’Donoghue, P. E. and McCarthy, M. A., “A theoretical and experimental evaluation of the long term integrity of an electrofusion joint”, Proceedings of Plastics Pipes VIII, Koningshof, Netherlands, (1992), pp. B2/3.1-10. [4] O’Donoghue, P. E., Kanninen, M. F., Green, S. T. and Grigory, S. C., “Results of a thermomechanical analysis model for electrofusion joining of PE gas pipes”, Proceedings of the 12th Plastic Fuel Gas Pipe Symposium, Boston, USA. (1991), pp. 41-342. [5] Bowman, J., “A review of the electrofusion joining process for polyethylene pipe systems”, Polymer Engineering Science, Vol. 37, (1997), pp. 674-691. [6] Nakashiba, A., Nishimura, H. and Inoue, F., “Fusion simulation of electrofusion joints for gas distribution”, Polymer Engineering Science, Vol. 33, (1993), pp. 1146-1151. [7] Nishimura, H., Inoue, F., Nakashiba, A. and Ishikawa, T., “Design of electrofusion joints and evaluation of fusion strength using fusion simulation technology”, Polymer Engineering Science, Vol. 34, (1994), pp. 1529-1534. [8] Rosala, G. F., The Process Mechanics of Polymer

Pipes Welding by Electrofusion, Ph.D. Thesis, University of Bradford, UK, (1995). [9] Rosala, G. F., Day, A. J. and Wood, A. S., “A finite element model of the electrofusion welding of thermoplastic pipes”, Proceedings of the Institution of Mechanical Engineers, Vol. 211 E, (1997). 137-146. [10] Pitman, G. L., “Electrofusion welding prediction and computer-aided design of fittings”, Proceedings of Plastics Pipes VI, University of York, UK, (1985), pp. 29.1-7. [11] Dufour, D. and Meister, E., “Polyethylene electrofusion technique: Prediction model of welding quality”, Proceedings of the International Gas Resources Conference, Japan, (1989), pp. 232-242. [12] Wood, A.S., Rosala, G. F. and Day, A. J., “Control-volume simulation of the electrofusion welding process”, IMA Journal of Mathematics Applied in Business & Industry, Vol. 9, (1998), pp. 65-88. [13] Wood, A. S., Rosala, G. F., Day, A. J., Torsun, I. and Dib, N., “Variable grid model for thermoelastic deformation”, in Advanced Computational Methods in Heat Transfer III, ( ed. Wrobel, L. C., Brebbia, C. A. and Nowak, A. J.), CMP, Southampton, UK, (1994), pp. 481-488. [14] Demirdzic, I and Muzaferija, S., “Finite volume method for stress analysis in complex domains”, International Journal for Numerical Methods in Engineering, Vol. 37, (1994), pp. 3751-3766. [15] Hattel, J. H. and Hansen, P. N., “A control volume-based finite difference method for solving the equilibrium equations in terms of displacements”, Applied Mathematical Modelling, Vol. 19, (1995), pp. 210-243. [16] Kovalenko, A. D., Thermoelasticity: Basic Theory and Applications, Wolters Noordhoff, The Netherlands, (1969). [17] Crank, J., Free and Moving Boundary Problems, Clarendon Press, Oxford, (1984).