Variable grid model for thermo-elastic deformation A.S. Wood, G.F. Rosala, A.J. Day, I. Torsun & N. Dib University of Bradford, West Yorkshire,

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1 Variable grid model for thermo-elastic deformation A.S. Wood, G.F. Rosala, A.J. Day, I. Torsun & N. Dib University of Bradford, West Yorkshire, ABSTRACT A preliminary variable space grid formulation is described for a thermally loaded and deforming elastic body undergoing a change of phase. The model is numerically simulated using finite difference techniques and some results are compared with measured data. INTRODUCTION Electro-fusion welding (EFW) is a process for joining lengths of plastic pipe, typically used for the transmission of gas or water. Essentially the procedure involvesfirstinserting two pipe ends to be joined into a electro-fusion fitting containing an embedded heating coil. An interference fit between pipes and fitting is not expected and so, at least initially, an air gap will be present. A current is then passed through the coil, the temperature of which increases due to Joule heating. The surrounding plastic material of the pipes and fitting subsequently melts and the air gap closes due to thermal expansion of the material, causing fusion of the molten plastic. Once the current is switched off the fused unit is allowed to cool and solidify. The joint created is along the circumference of each pipe in contact with the fitting, shown in Figure 1. The modelling aspect of the process involves Fitting Pipe 1 Pipe 2 Figure 1: Schematic diagram of an EFW fusion region.

2 482 Heat Transfer the coupled problems of heat transfer with a change of phase and t her moelastic deformation. Most successful commercial modelling packages, such as ABAQUS, use sophisticated finite element codes to evolve the t her momechanical problem, which can take hours of processing time to simulate minutes of real-time evolution. This note is concerned with generating a computationally cheap (PC compatible), yet adequate, simulation package for certain highly topical thermo-mechanical problems. The period to 'gap closure' is treated in this note. MODELLING In this study we shall assume circular symmetry which will facilitate the development of a one-dimensional radial model. Heat Transfer The configuration of a radial section through the pipe and fitting is shown in Figure 2. There are four distinct regions - the pipe bore, the pipe wall, the air gap and the fitting wall. r&, r^, Tg and r/ represent the external radii of each of the four regions. Bore Pipe Gap Fitting r, r/ ' Figure 2: Radial configuration of model. Bore The bore simply contains air, with (assumed) constant thermal properties &a, pa, c%, which is heated from an initial temperature Ua by free convection from the inner surface of the pipe wall. The usual linear heat equation is applicable subject to the boundary conditions ^=, r = ; k = h*(un-u), r = r,. (2) r Anticipating the fact that the pipe will deform as it is heated so r& r&(t). To facilitate a numerical solution process it is convenient to introduce the idea of a variable space grid that moves with the deforming body. The rationale is to track the temperature on an identifiable moving point and in place of estimating J7(r»,t), < r+ < r&, for?% fixedwe now monitor

3 U(ri(t),t). By the chain rule Heat Transfer 483 du_ dt du_ dr and Equation 1 can then be written as dt du < ^ = ^(^ < >, (3) in which ' = - and ' = -. Pipe Wall In the pipe we consider temperature dependent thermal properties. The pipe is initially at a temperature UQ and is, in the first instance, heated by free convection across the air gap. It loses heat to the bore by free convection from its inner surface. The governing non-linear equation is The variable space grid approach restructures the equation to subject to the boundary conditions ^) +^r ([/')', r& < r = n(^) < r, r J du r = r&,, (4) (5) The second of these boundary conditions will change once the air gap has closed and the pipe and fitting have fused. Air Gap The simplest way to treat heat transfer across the air gap is to use an 'overall' heat transfer coefficient hpf which is a function of the gap size g = rg rp. Typically hpf oc - with an upper bound being imposed on hpf as g ->. Fitting Wall The variable grid heat transfer equation (and boundary conditions) are essentially the same as those for the pipe wall except that the fitting may have different temperature dependent thermal properties. We have pcu = pcu'r subject to +- (U'f T o, (6) dr (7)

4 484 Heat Transfer in which Ua is the temperature of the surrounding air. The temperature of the pipe/fitting unit is increased by the heating element through free convection, described by in which U^ is the wire temperature. The variable grid model requires a driving force for the movement of the nodes?% ( ). Typically, in Stefan problems, the motion of the phase-change boundary can be used to drive internal nodes. However, this needs to be balanced with the problem of tracking the phase-change boundary itself. In this paper we adopt an enthalpy approach for temperature description in which the enthalpy E is defined by UR U f pc du -f pl, U > Um, (8) Um is the (assumed unique) melting temperature of the pipe and/or fitting material. At U = Um I pcdu < E < I pc du -f pl, where UR is a reference temperature. In this case E = pcu and Equations 4 and 6 can be easily modified. The bore simply has pure heat conduction and so working solely in terms of temperature is sufficient. The enthalpy approach dispenses with the need to explicitly track any melt fronts. Thermo-elastic Deformation The most natural way to describe the motion of selected points in the pipe and fitting walls is to utilize the thermo-mechanical deformation. This will automatically define changes in the volume occupied by the pipe and fitting. It is assumed that material deformation is due solely to temperature changes in the body. Theory tells us that an element of material of length i subject to a uniform temperature change AC/ undergoes an extension/contraction At to a length f given by f = 1(1 -f aac/). a. is the coefficient of linear thermal expansion. Equivalently we can write Af = aifcjj and if U is treated as a continuously varying function of time t then we can deduce the differential equation

5 Heat Transfer 485 In practice we have volumetric expansion. Consider a solid wedge of the pipe wall in which the natural thermal expansion in the axial and circumferential directions is restricted. It can be argued that the radial deformation should be multiplied by a factor of 6?r to account for this restricted movement and is now modelled by -«- f- Since the temperature rise across the pipe wall will not be uniform we need a composite deformation model. This can be accomplished by dividing the length i into N contiguous cells of size Ar,, i = 1,..., TV, bounded by nodes?%, i =,..., TV. Assuming that the temperature is uniform in cell i and equal to the temperature at node r, then, from Equation 9, cell i expands or contracts according to This gives the motion of node i relative to node i 1. If we take r\>(i) as the origin of this moving grid system then the absolute motion of each node (relative to a fixed origin) is n = h + Ar, = r,-_i + r,, i = 1,..., TV, (1) j=i in which TO = n> and rjy = r^. To move the inner pipe surface, r&, we assume circumferential expansion which leads to the equation TO = 67raro(/o This formulation recognises that the inner surface of the pipe will be the last to suffer temperature changes and is, in essence, a rigid surface. The governing heat equation at the node r,- can then be written where U E/(pc). At the innermost node " T^ In the fitting wall, it is the outer surface that is last to feel the effect of temperature changes and, in consequence, we measure motion relative to r/(t). Thus, TJ 67r#r/C/y and, for internal nodes, N Ar, = r,-+i - Ar,-+i, i =,..., TV - 1. (12) i where r# = r/ and TO = Tg. In thefittingwall, the temperature in cell i + 1 is taken to be uniform and equal to the temperature at node 2, so

6 486 Heat Transfer An+i = 67raA;+if/; and the heat transfer equation at internal nodes can now be written as *»» + * The gap <? is then naturally determined from g Tg ry NUMERICAL SOLUTION The construction of the numerical solution is quite straightforward. Finite differences offer the simplest solution. Each region (bore, pipe and fitting) is divided into TV cells (those defined by the moving grids). In the pipe and fitting the distribution of nodes will evolve non-uniformly by virtue of Equations 1 and 12. In the bore, given r& at any time, the (uniform) internal node distribution is described by 2 n = jjn, i =,..., N. The heating element, which is embedded at a depth &%,, is located on the first internal node of the fitting wall. If d^ =, the wire is located on node zero. For the purposes of this note an empirical formula for the wire temperature is imposed, based on the experimental evidence, with the aim of identifying whether the correct trends of thermo- mechanical behaviour can be established by the model. The heat equation in each region is replaced by the usual central spatial differences and backward time differences [1]. Care needs to be taken to correctly incorporate the convective boundary conditions at the pipe and fitting wall surfaces. Stability of the explicit numerical scheme is governed by heuristic arguments which follow the line of reasoning that a virtual increase in the value of u should lead to a virtual increase in the subsequent value u ~*~*, where u ~ U(ri,tm)- This statement is satisfied if the coefficient of u on the righthand side of the difference equations is positive, and it leads to restrictions on the size of the time step. PRELIMINARY RESULTS AND DISCUSSION Figure (a) shows various crucial temperature profiles for one particular laboratory experiment involving EFW. The characteristic peak in the interface temperature (as the heating stage is completed) is clearly visible. Figure (b) is a blow up of the heating phase. Table 1 compares several measured interface temperatures with those predicted by the simulation model described in this note. These preliminary results are encouraging in that the correct trends are established. The predicted fitting interface (inner surface) temperatures are

7 Heat Transfer 487 Table 1: Measured and simulated interface temperatures and gap size. Time s Temperature, C Simulated Fitting Measured Fitting Pipe Gap mm initially higher than the experimental values, and ultimately lower. This can be explained by the fact that, in practice, the heating wire cross-sections along an axial slice of the fitting are not uniformly distant from the fusion interface at which thermo-couples are located. Further, the imposed wire temperature formula (a + b\/t) provides an initially larger heating rate than that used in the experiment. The pipe interface (outer surface) temperatures show a small rate of increase initially, as would be expected since the air gap is acting much like an insulating layer. Once the gap reduces to something like.5mm the pipe surface begins to heat quite rapidly and very quickly approaches thefittingtemperature once the gap has closed, at 71s. Once the gap has closed, temperature flow between the pipe andfittingis still modelled by a heat transfer coefficient (albeit quite large) since both materials are still solid. The gap closure time of 71s agrees reasonably well with the empirically deduced value of about 75s. This second value is determined by observing the time at which the interface temperatures converge on Figure (b). The efficiency of the present model simulation is demonstrated by the low CPU time of just 8.3s (the model was coded using Prospero Fortran and was compiled for a 33MHz 486DX processor). It must be stressed that the model presented is still at a development state and there is clearly scope for improvements, such as tweeking the heat transfer coefficient h^. However, it has been demonstrated that fast, and adequate, thermo-mechanical code can be generated.

8 488 Heat Transfer References 1. Smith, G.D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, 2nd edn., Oxford, Acknowledgments The authors are indebted to a supportive consortium including Glyned Plastics, Fusion Group, Uponor-Aldyl, Wavin, Stewarts and Lloyds, FINA Research, BP Chemicals, TWI, WRc, British Gas and the SERC (Grant GR/H 4563). Fitting outer surface temperature Figure 3: Experimental EFW temperature evolution (a) for the entire fusion process and (b) for the heating phase.