3 Thermokinetics and Curing Behaviour Thermochemical numerical models have been developed and used for the simulation of thermosetting pultrusion processes since the 1980s in order to predict the temperature and cure development during processing. The mechanical properties and surface quality of the pultruded products are highly dependent upon the degree of cure at the end of the pultrusion process. Controlling the curing stage during the process is relatively complex as thermosetting resins are chemically reactive and sensitive to the transient temperature. Internal heat generation occurs during the curing stage due to the exothermic chemical reactions of the resin system; therefore, it is of the utmost importance to develop numerical models to predict the thermochemical behaviour of the product. In the literature, transient and steady-state simulations have been applied using numerical techniques, such as the finite difference method (FDM) with the control volume(s) (CV) approach and the finite element method (FEM) with the nodal control volume (NCV) approach. Generally, the source term in the energy equation is due to the internal heat generation in the resin system coupled with the resin cure kinetics. This is achieved in an explicit manner to obtain a fast numerical solution since the internal heat generation is highly nonlinear, i.e., it is a sequential procedure in which first the temperature is solved at the nodal points and the degree of cure is calculated and updated accordingly at the CV or NCV. The thermochemical characteristics of the pultrusion process have been analysed using different modelling approaches in terms of dimensionality, i.e., one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D). Moreover, several experimental studies have 33
Pultrusion: State-of-the-art Process Models also been carried out to measure the temperature and degree of cure profiles during the process. This chapter addresses the key points of the thermochemical models and their applications in pultrusion processes. 3.1 Numerical Investigation 3.1.1 One-dimensional Models A 1D heat transfer model of the pultrusion process for a thermosetting composite was developed in [1, 2] employing the FEM. The pultrusion of a thin rectangular profile was considered in [1]. The material was a combination of AS4-W-12K Hercules Type I graphite and Shell Epon 9310 epoxy. The fibre volume fraction (V f ) was 67%. The steady-state heat transfer equation for the material was given in a Cartesian coordinate system as [1]: 2 2 T T k + ρc u T p 0 2 2 x x x1 = (3.1) T: Temperature of the material; k: Thermal conductivity; u: Pulling speed in the x 1 -direction; ρ: Density; and C p : Specific heat. 1 2 The heat transfer through the thickness (x 2 -direction) was assumed to be constant and hence the mean temperature (thickness averaged) was used. Therefore, Equation 3.1 was simplified by integrating Equation 3.1 with respect to x 2 and is given in Equation 3.2 [1]: 34
Thermokinetics and Curing Behaviour k dt 2 2 ρc dx 1 p u dt 2 h ( Td ( x1 ) T ) 0 dx + = t (3.2) 1 T d : Die temperature varying along the length of the die; h: Heat transfer coefficient at the die part interface; and t: Thickness of the profile. A source term was also considered using the total heat of reaction of the epoxy resin during the process. The heat transfer equation was solved using the FEM. The length of the die was 91.44 cm and the thickness of the processing material was set to 1 cm. The numerical simulations were carried out for two different pulling speed values: 0.593 and 0.085 cm/s. The physical properties of the resin usually vary during the process due to the phase changes, i.e., liquid to gel and finally solid state, however in [1], constant material properties were assumed in the numerical model. Using the proposed simplified heat transfer model, the temperature evolution from the heating die was predicted. In [3, 4], a mathematical model for heat transfer and cure inside the heating die was developed utilising the FDM, in which the time increments were carried out implicitly using the Crank Nicolson method. In these studies, the assumption of no axial conduction and negligible bulk flow simplified the 2D pultrusion model into a 1D transient heat transfer model. To achieve this the transient heat transfer equation was derived in the through-thickness direction (radial direction) using a Langrangian frame of reference. The corresponding equation is given in Equation 3.3 for a pultruded rod made of fibre glass/vinyl ester using cylindrical coordinates. The advection due to the pulling speed was implemented as a convective boundary condition between the composite part and the die: 35
Pultrusion: State-of-the-art Process Models r: Radial direction; t: Time; and ρ ( ) = C T k t r r r T r p + q (3.3) q: Source term due to the exothermic reaction of the resin. The source term q was modelled using a cure kinetic model which takes into account the effect of variable initiator and inhibitor concentrations on resin reactivity. In [3], the heat transfer coefficient was defined based on the Nusselt number and altered at the die part interface in order to reflect the reduced heat transfer owing to the curing and shrinkage of the resin. Resin density, thermal conductivity and specific heat were modelled as a function of temperature and degree of cure. According to the physical material models the density initially decreases as the temperature increases, however the density increases when the composite part starts curing. The overall change in the resin density was predicted to be approximately 2%. On the other hand, it was calculated that the specific heat increases by 13% during cure and the thermal conductivity increases by approximately 90%. The net effect of these property changes was predicted as a 68% increase in thermal diffusivity and a 12% decrease in the adiabatic temperature rise. As a consequence, the temperature difference at the centreline with and without using the variable material properties was found to be approximately 15 ºC after curing occurs [3]. 3.1.2 Two-dimensional Models A 2D axisymmetric pultrusion model of a pultruded composite rod using cylindrical coordinates was developed in [5, 6] in which a CV-based finite difference (FD) approach was implemented. Two different material combinations, namely graphite/epoxy and glass/ 36
Thermokinetics and Curing Behaviour epoxy, were considered in [6] and the temperature and cure degree evolutions were compared using the developed 2D process model for the pultrusion process. The temperature at the die part interface was applied as a prescribed temperature on the outer surface of a pultruded rod with a diameter of 9.5 mm. Since graphite has a larger thermal conductivity and specific heat value, and a smaller density than fibre glass, it was found in [6] that the graphite/epoxy mixture cures earlier than the glass/epoxy mixture. On the other hand, the peak temperatures were found to be almost identical as the V f of the matrix material was the same for both graphite and glass composites and the internal heat generation is dependent on the resin density, resin volume fraction, total heat reaction of the resin and the cure kinetics of the resin. In [7], the pultrusion process of a glass fibre/vinyl ester profile was simulated in a 2D computational domain in which the solution of the thermochemical equations was carried out using the alternating direction implicit (ADI) method. In 2D, the ADI method is an unconditionally stable FD time-domain method of second-order accuracy in both time and space [8]. A rectangular pultrusion domain was employed for the composite part being processed and the heating die. The effect of the pulling speed and the length of the die on the degree of cure distribution at the die exit were investigated using the 2D ADI thermal model. Different pulling speeds were used in the simulations: 1.7, 2.04, 2.88, 3.30 and 3.73 cm/s. It was found that the through-thickness cure gradients become higher when the pulling speed increases. The degree of cure value at the die exit was calculated to be approximately 0.97 and 0.2 for the pulling speed values of 1.7 and 3.73 cm/s, respectively. Two different die configurations were also simulated in [7]. The degree of cure at the die exit was predicted to be approximately 0.97 using a die length of 1.02 m (pulling speed of 1.7 cm/s). On the other hand, the degree of cure value at the die exit was approximately 0.1 with a relatively short die length of 0.46 m (pulling speed of 1.3 cm/s). The CV/FD method was employed in [9] to perform a thermochemical simulation of the pultrusion process for the composite rod analysed 37
Pultrusion: State-of-the-art Process Models in [5, 6]. To achieve this, a cylindrical die with heaters was placed on top of the composite part to be processed in the calculation domain. The goal was to investigate the effect of the thermal contact resistance (TCR) or the heat transfer coefficient at the die part interface. A schematic view of the 2D pultrusion domain is depicted in Figure 3.1. Due to the exothermic chemical reaction of the resin and hence curing, chemical shrinkage occurs in the heated die; this leads to separation of the part from the die surface resulting in the loss of perfect thermal contact at the die part interface. In order to model this phenomena, TCR regions were defined at the interface as seen from Figure 3.1. Figure 3.1 Evolution of the centreline degree of cure in the heating die for the pultruded graphite fibre/epoxy combination with different pulling speed values. Reproduced with permission from I. Baran, C.C. Tutum and J.H. Hattel, Composites Part B: Engineering, 2013, 45, 995. 2013, Elsevier [9] The following heat transfer equations were solved using the 2D ADI method for the composite part (Equation 3.4) and the die (Equation 3.5) [9]: ( ρ T C u T 2 ) T k t z z kr ( r r r T ),c c p z,c 2 q c + = + + (3.4) r ρ C T 2 T k t z kr ( r r r T ),d d pd z,d 2 = + r (3.5) 38
Thermokinetics and Curing Behaviour Where k z and k r are the thermal conductivities in the axial direction (z) and radial direction (r), respectively; c and d denote the composite part and die, respectively. The internal heat generation (q) (W/m 3 ) due to the exothermic reaction of the epoxy resin was expressed as: q= (1 Vf) ρr HtrRr( α, T) (3.6) V f : Fibre volume fraction; ρ r : Resin density; H tr : Total heat of reaction; R r (α,t): Rate of resin reaction; α: Cure degree; and T: Temperature. R r (α,t) is generally modelled using an Arrhenius type of equation: Ea n Rr( α, T) = K0exp( )(1 α) (3.7) RT K 0 is the preexponential constant; E a is the activation energy; R is the universal gas constant; and n is the order of reaction (kinetic exponent). The degree of cure was calculated in [9] using the expression: α α α = R r (, T) u (3.8) t z 39
Pultrusion: State-of-the-art Process Models In [9], two different case studies were carried out in terms of the TCR configuration at the interface as seen from Figure 3.1. Two different optimisation case studies were performed: constant TCR (Case-1) and variable TCR (Case-2). As previously mentioned, a cylindrical die and heaters were added to the original calculation domain. The aim was to investigate the significance of the TCR for the pultrusion process while having the same centreline temperature and cure degree profiles as the original set-up. There was no heating die but a prescribed temperature was applied to the surface of the composite part in the original set-up. A curve-fitting procedure (i.e., inverse modelling) was performed using data composed of 15 centreline temperature values, measured in the validation case, in order to obtain the same centreline temperature profile of the composite within this pultrusion simulation domain (Figure 3.1). The TCR values (design variables in the curve-fitting procedure) were predicted by minimising the difference between the measured (the validation case) and the calculated (the new configuration with cylindrical die and heaters) centreline temperatures, i.e., for certain die radii. The constrained minimisation function fmincon was employed using MATLAB [10] which finds the minimum of a multivariable problem. The temperature curve-fitting procedure was repeated with 5 different die radii (r d ) selected as 10, 25, 50, 75 and 100 mm, thereby considering possible die designs for the composite rod in the validation case. The minimum error ( T T ) meas values, which give the optimum TCR values for both cases, were found to be 6,957.5, 7,979.8, 8,250.7, 297.5 and 8,275.9 in Case-1 and 6.7, 5.2, 3.7, 7.4 and 19.4 in Case-2 for the die radii of 10, 25, 50, 75 and 100 mm, respectively. The minimum error for Case-1 in which a single TCR was used is significantly higher than the error for Case-2 with respect to all die radii. This shows that the application of variable TCR (Case-2) gave much better results than the application of a single TCR (Case-1) at the interface. The centreline temperature and the cure degree profiles of the composite rod for a die radius of 10 mm are shown in Figure 3.2. It can be seen that the temperature and cure degree profiles obtained using variable TCR are almost the same as those given in the validation case. However, the results obtained when using a single TCR deviate considerably with respect cal 2 40
Thermokinetics and Curing Behaviour to the centreline temperature and cure profiles of the composite rod. In addition, it was also shown that the variable TCR includes the role of the shrinkage and also the cooling channels, which are not included in the numerical model, resulting in the TCR values being larger at the die inlet and near the die exit [9]. Figure 3.2 The centreline temperature (a) and corresponding degree of cure (b) profiles obtained using single and variable TCR at the interface, and the measured centreline temperature and cure degree profiles obtained from the validation case for a die radius of 10 mm. Reproduced with permission from I. Baran, C.C. Tutum and J.H. Hattel, Composites Part B: Engineering, 2013, 45, 995. 2013, Elsevier [9] 41