Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer

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1 Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer A.V. Baranov Gubkin Russian State Oil and Gas University, Moscow, Russia Received: 28 November 2014, Accepted: 23 December 2014 SUMMARY The non-isothermal filling of a flat mould cavity by a Newtonian fluid with simultaneous two-dimensional impregnation of the porous (reinforcing) layer is investigated. The flow in the cavity is described by inertialess Navier Stokes equations, the flow in the porous layer by Darcy s law, and the flow in the region adjacent to the liquid porous layer boundary by the Brinkman equation. Viscosity is considered to be temperature dependent. The temperature fields in the channel region and in the permeable layer are interrelated by conjugate boundary conditions of the fourth kind. Keywords: Heat exchange; Newtonian fluids; Non-isothermal flow; Polymers; Mathematical modelling; Impregnation; Moulding 1. INTRODUCTION The non-isothermal moulding of a composite product in a closed mould is considered. The liquid is forced into the mould through a feed pipe under constant pressure. It is important to stress that there is an appreciable gap between the porous impregnated layer and the mould wall. Processes of this kind are used in the production of a vast array of technical, household, and special-purpose goods with different service properties. After completion of mould filling and impregnation of the entire porous layer, solidification of the moulding occurs as a result of the processes of crystallisation or glass transition as the polymers cool down, gelation of the plastisols, or chemical curing of the polymer binder (for example, epoxy and polyester resins). If the viscosity of the binder is low, then casting is carried out at low pressures and high speed. However, this is not always the case. The viscosity of the Smithers Information Ltd., 2015 liquid may be high, and the porous layer too dense, so that impregnation requires the application of considerable pressure gradients, which may take much time. It is risky to heat the liquid itself in this case, as premature gelation of the plastisols or curing of the polymer binder may occur. In these cases, preheating of the mould itself is resorted to. Injection into a hot mould makes it possible to reduce the viscosity of the liquid and shorten the time of the subsequent stage of holding for curing. Therefore, the given processing method can be applied to materials possessing a fairly long induction period of reaction. In the literature, a fairly large number of studies have been devoted to the modelling of processes of moulding of composite products in closed moulds 1 4. However, these publications address injection into moulding cavities, the entire inner volume of which is occupied by some filler. Therefore, mould filling consists only in the impregnation of the porous material. In this case the mathematical model is generally based on Darcy s law. It is important to note that Chan and Hwang [3] and Malkin et al. 4 model the process of impregnation by a so-called rheokinetic liquid (for example, resin), the viscosity of which is dependent on the degree to which the chemical reaction has taken place. Here, Chan and Hwang 3 considered the non-isothermal case where viscosity is still also temperature dependent. The present work considers the filling of a cavity with simultaneous impregnation of the reinforcing porous layer according to the scheme shown in Figure 1. Filling of a flat moulding cavity of height 2h and length l is done via a flat-slit sprue channel under constant inlet pressure p 0. The process of mould filling is accompanied with the simultaneous impregnation of the reinforcing porous layer of thickness H. As the flow front advances in the flat cavity, two-dimensional spread of the liquid medium through the porous layer occurs with its developing front. Polymers & Polymer Composites, Vol. 23, No. 9,

2 A.V. Baranov Figure 1. Scheme of the moulding process Studies along similar lines have described the flow of liquid through channels in which one or both walls are porous (permeable) 5,6. Furthermore, there is a large number of studies that consider the flow of liquid through a channel entirely filled with porous material 7 9. Of these, the study by Chen and Hadmin 7 is of particular interest. These authors consider the problem of entirely developed stationary flow and the heat exchange of a powerlaw fluid in a flat channel filled with fibrous material. The problem is solved approximately by the integral method with involvement of boundary layer theory for the velocity profile. However, it was assumed that all properties of the fluid were independent of temperature. Khan et al. 9 solved the isothermal problem of the flow of a non-newtonian fluid described by a third-order differential model based on Rivlin Erichsen tensors. In recent years there have been a whole series of publications devoted to the modelling of the flow of liquid through a channel partially filled with porous material In the region of the flowing liquid, the flow has been described by the Navier Stokes equation, while in the porous layer Darcy s law or Forchheimer s law has generally been used. Special mention should be made of work by Kuznetsov 10 and by Xiong and Kuznetsov 11, in which the thermal problem for a Newtonian fluid is solved, as well as the hydrodynamic problem. However, the physical properties of the liquid were considered not to be dependent on temperature. Min and Kim 12 proposed a model of convective heat exchange, in accordance with which the temperature at the liquid porous layer boundary is determined from the condition of heat flow equality, while the velocity at this boundary is determined from the condition of tangential stress equality. Numerically, using the Runge Kutta method, Martins-Costa et al. 13 solved the isothermal problem of flow of an incompressible power-law fluid in a flat channel partially filled with porous material. The statements of the problems in these studies are similar to the problem considered in the present work. However, in all of these models, steady-state flow through the channel without the presence of any moving two-dimensional impregnation front in the porous body was considered. Wu and Hourng 14 used the finite element method to solve the problem of the isothermal transfer moulding of resin in a closed mould where there is a gap between the mould wall and the fibrous filler. Analysis was based only on Darcy s law. The analogous problem of moulding plastisols with impregnation of a fabric substrate was considered by Fridman et al. 15. Assuming the fabric layer to be very thin, these authors based their arguments on an approximation of rapid impregnation of the fabric. In this case, impregnation does not occur over the entire length l but only in a narrow zone adjacent to the flow front in the cavity. Fabric impregnation was considered to be one-dimensional, and the pressure profile over the depth of the fabric to be linear. As a result of these assumptions, it was established that the impregnation rate is not dependent on the axial coordinate, and the moving impregnation front is linear. Malkin et al. 16 posed and solved the problem more completely, free of the constraints of Fridman et al. 15 and covering a wide range of possible cases. The flow of the liquid in the flat cavity was described by the Navier Stokes equation, and the flow in the porous layer by Darcy s law. Solution was carried out by a numerical method, and interesting results of calculations of the distribution of the impregnation front and pressure profile in the cavity for products with a thin and thick reinforcing layer were presented. In following studies 17,18, a similar problem for an anisotropic porous layer was solved. The considerable influence of anisotropic permeability on impregnation front development was shown. This problem was further developed by Malkin et al. 19, who posed and solved the non-isothermal problem where viscosity was dependent not only on temperature but also on the degree of curing of the polymer binder. The source of non-isothermicity was the considerable difference in the initial temperatures of the material and the heated mould. Calculations showed that preheating of the mould actually leads to considerable shortening of the impregnation time, but only when the moulding process can be carried out within the induction period of the chemical reaction of curing. However, Malkin et al made the same assumption concerning the existence of tangential stress discontinuity on the permeable cavity wall. This was due to the fact that flow in the cavity occurs by shear flow with the presence of tangential stresses, which are at their maximum at the channel walls. On the other hand, the use of Darcy s law to describe the impregnation process signifies the absence of any tangential stresses in the porous layer. It would be more correct to use the Brinkman equation 3,8,13,20, which assumes the 640 Polymers & Polymer Composites, Vol. 23, No. 9, 2015

3 Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer presence in the porous layer of a certain transition zone (boundary layer) between purely shear flow in the channel and so-called potential flow in the porous layer. The thickness of this boundary layer depends on the hydrodynamic conditions of flow and on the characteristics of the porous body. However, the authors of these studies deliberately made this assumption in order to simplify the statement and solution of the problem. In the present work, in stating the problem, an attempt was made to overcome the shortcomings and assumptions noted in the above studies. Thus, in contrast to existing models, in the present case a mathematical model of non-stationary non-isothermal moulding with the use of the Brinkman equation is presented, where at the liquid porous body boundary the condition of equality not only of velocity but also of tangential stress is set. The impregnation front develops in a confined space, reaching all walls of the mould. Here, the viscosity of the liquid is temperature-dependent. Non-isothermal statement of the problem will be examined. The polymer is considered to be an incompressible Newtonian fluid. Along with ignoring inertial effects, this makes it possible to describe the impregnation of the porous layer with the aid of Darcy s law and Brinkman s law. It should be pointed out that the developed model can be used not only for the scheme of flow shown in Figure 1. With some of the simplest changes to the model, it can also describe the processes occurring by the schemes shown in Figure FLOW IN THE CAVITY: 2h y 0 Here, purely viscous flow of free liquid during the filling of an empty flat cavity will be considered. The length of the cavity is much greater than its height, and therefore, assuming the pressure Figure 2. Possible schemes of moulding in the cavity p c to be independent of the transverse coordinate y, we will confine ourselves to a single equation of motion 10,11,13,16 20 : which is supplemented by the equation of discontinuity: (2.1) (2.2) where u c and v c are the longitudinal and transverse components of velocity within the cavity, µ=µ(t c ) is the viscosity of the liquid, and T c is the temperature of the liquid in the cavity. The hydrodynamic boundary conditions for flow in the cavity are as follows: (2.3) (2.4) (2.5) (2.6) where u and v are the longitudinal and transverse components of the filtration velocity, η is the effective viscosity, p c0 =p c (x=0), = (x=0), and is the average 0 flow velocity in the cavity. For convenience of subsequent calculations, the velocity at the liquid porous layer boundary is denoted as u s. Polymers & Polymer Composites, Vol. 23, No. 9,

4 A.V. Baranov By effective viscosity, we mean the viscosity of the liquid phase in the porous medium, which is used to determine the viscous tangential stresses in the Brinkman equation. When the flow front in the cavity reaches the opposite mould wall, condition (2.6) is replaced by the following: (2.7) The notation of tangential stress equality on the porous wall in boundary conditions (2.4) was deliberately simplified, as it was assumed that: In the equation of energy, the terms reflecting dissipation and heat release from the possibly occurring chemical reaction of curing are ignored: (2.8) where λ f, ρ f, and c f are the thermal conductivity, density and specific heat of the liquid respectively. The temperature fields in the channel region and in the permeable layer are interrelated by conjugate boundary conditions of the fourth kind. Therefore, the thermal boundary conditions for flow in the flat channel are as follows: (2.9) (2.10) (2.11) where T w is the initial temperature of the mould walls and porous layer, T is the temperature within the impregnated part of the porous layer, and T 0 is the initial temperature of the liquid at entry into the cavity. Equations (2.1), (2.2), and (2.8) are supplemented by an equation describing the temperature-dependence of viscosity: (2.12) 3. FLOW IN THE BOUNDARY ZONE OF THE POROUS LAYER: 0 y d As already noted, in the porous body there exists a transition layer (a boundary layer), the flow in which cannot be described correctly by Darcy s law alone. This is due to the fact that in the region of the liquid porous layer boundary there are significant velocity gradients. In this case, different, more complex equations are resorted to. One of the classic variants is to use the Brinkman equation 3,18,13,20 : (3.1) Equation (3.1) is written for the axial component of the filtration velocity in the direction of the Ox axis. It is assumed that, in the direction of the Oy axis, for the transverse component of the filtration velocity, impregnation can be described by Darcy s equation: (3.2) The boundary conditions for the given zone on the side of the porous cavity wall are described in the form of system (2.4), where the conjugation of the velocity fields and tangential stresses on the porous wall is given. On the other side of the zone, the boundary conditions take the following form: (3.3) In the present work, as in publications by other authors 10,20, the thickness of the boundary layer δ was taken to be that distance from the porous cavity wall at which the filtration velocity u differed from the filtration velocity by less than 1%. The equation of discontinuity for the boundary layer: (3.4) To describe the temperature field within the porous layer, a one-temperature model is used. In accordance with this model, it is assumed that the processes of interphase heat exchange between the impregnating liquid and the solid skeleton of the porous layer proceed much more rapidly than impregnation itself. In accordance with Buevich and Kalinnikov 21, the equation of energy for the impregnated layer will take the following form: (3.5) where ε is the porosity, λ b, ρ b, and c b are the thermal conductivity, density and specific heat of the solid skeleton of the porous layer respectively, and λ is the effective value of the thermal conductivity of the heterogeneous composite medium. This value can be calculated in the following way 19,22 : (3.6) where ς = λ f /λ b. We stress that equation (3.5) is used to describe the temperature field through the entire thickness of the porous layer 0 y H. Here, at the liquid porous layer boundary with y = 0, just as for equation (2.8), use is made of the thermal conditions of conjugation of the temperature fields in the channel and in the porous layer (2.10). Furthermore, to solve equation (3.5), the following boundary conditions are employed: 642 Polymers & Polymer Composites, Vol. 23, No. 9, 2015

5 Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer (3.7) (3.8) As the impregnation front reaches first one and then the other opposite wall of the mould, the boundary conditions (2.10), (3.7), and (3.8) are supplemented by the following conditions: At the first stage of the process, when the moving impregnation front has not reached the opposite walls of the mould, the boundary conditions for equation (4.3) have the form: (4.4) (4.5) where, (5.1) (3.9) (3.10) 4. FLOW IN A POROUS BODY D y H In the region of the porous layer, where impregnation can be described by Darcy s law, we have: (4.1) where k is the coefficient of permeability, and and are the longitudinal and transverse components of the filtration velocity for the porous body zone where impregnation is described fully by Darcy s law. It will be convenient to give the components of the filtration velocity in the boundary zone and in the given region different notations, namely u, v and, respectively. Substituting (4.1) into the discontinuity equation for the porous body: (4.2) we will obtain an equation for the pressure distribution in the porous body: (4.3) We will emphasise once again that equation (4.3) describes the pressure field of the liquid in the region of the impregnated body that lies within the transition (boundary) layer in the porous body, and where Darcy s law can be applied. As the impregnation front reaches the opposite mould walls, the following boundary conditions are used: (4.6) (4.7) From the hydrodynamic viewpoint, the problem posed is non-stationary, as time enters all the determining relations as a parameter. The flow region boundaries in the cavity and in the porous layer, changing with time, transform the pressure and velocity fields. Here, for points of the moving impregnation front in the porous layer, we have: (4.8) The position of the flow front in the cavity is determined by means of the relation: (4.9) Once again it must be noted that the temperature field in this region is described by the same energy equation (3.5) as in the boundary zone. 5. SOLUTION OF THE PROBLEM After two integrations of the equation of motion (2.1), with account taken of the boundary conditions (2.3) and (2.4), we will obtain: Velocity component ν c is found from the equation of discontinuity (2.2). However, for subsequent solution, only its value at y = 0 will be of interest: where, (5.2) and, using the boundary condition (2.4) and equation (3.2), we will obtain: From this we obtain the equation for finding the pressure distribution p c (x) along the flat cavity: (5.3) Consideration of equation (3.1) supposes definitions of the quantity η/µ, which depends on a number of factors, including the structure of the porous layer. This is a separate, independent problem, and therefore in the present work, as in many other publications 10,11,20, it is simply assumed that η/µ = 1. Equation (3.1) will then have the form: (5.4) Polymers & Polymer Composites, Vol. 23, No. 9,

6 A.V. Baranov If we neglect the term, the general solution of equation (5.4) is known: (5.5) where C 1 and C 2 are determined from the boundary conditions (2.4) and (3.3): made of an algorithm of adaptation of the finite element grid to the changing geometry of the moving boundary (the impregnation front). At a given fixed moment in time, each ith iteration step begins with solution of equation (5.3) with the aim of finding the pressure distribution in the flat cavity p i c. For this, into the left-hand side of equation (5.3) are substituted the values p i-1 and µ i-1 found at the previous iteration step. Then, knowing the distribution p i c and departing from the boundary condition: It must be pointed out that equation (3.1) was solved on the assumption that the derivative p/ x through the thickness of the boundary layer δ in the porous body changes little. Note that a similar assumption was made in relation to the derivative dp c /dx in equation (2.1) for flow in the cavity. Furthermore, when writing the expression for C 2, it was considered that within the boundary layer 0 y δ it is possible to assume that: As a first approximation, the value of the thickness of the boundary layer δ that was obtained from solution of equation (5.4) for the isothermal case is used; this has the form 23 : where, (5.6) It must be stressed once again that, in calculations by means of equation (5.6), the thickness of the boundary layer δ was taken to be the distance from the boundary y = 0 at which the filtration velocity u differed from the filtration rate by less than 1%. The problem posed was solved by an iteration scheme using the numerical finite difference method. The main difficulty in selecting the grid is the need to calculate sufficiently accurately the derivatives p/ x and p/ y at the impregnation front. On the basis of these derivatives, the velocities of the liquid that are necessary for finding the new position of the moving boundary at the next time step are determined. With a uniform grid, striving to obtain a sufficiently accurate value of the velocity at the front would lead to a considerable increase in the number of grid nodes. Therefore, to carry out calculations, use was made of a grid that was non-uniform with respect to one of the coordinates (Figure 3). To increase the accuracy of the calculations, the grid changes after each time step, i.e. use is we set about finding the pressure distribution p i, using µ i-1 as before. The found pressure fields p i and pi c make it possible to find the velocity distribution u i, c νi, c µi, ν i, i, i. Using these velocities, we turn to solving equations of energy (2.8) and (3.5) to find the temperature distribution T i, c Ti. Thus, the ith iteration ended with finding µ i from equation (2.12). Then, for the next moment in time, using (4.8) and (4.9), we find the new positions of the flow front in the cavity and porous body. On the basis of the derived solution, calculations were carried out using the characteristic processing parameters and physical properties of the materials. Figure 4 shows the role of the non-isothermal factor, i.e. how the temperature-dependence of viscosity affects the moulding and impregnation rate of the composite article. Here, x and y 0 are in metres. It can be seen that preliminary heating of the mould actually leads to a considerable shortening of the total process time. In this case, impregnation occurs most rapidly along the hot walls of the mould, which leads to an appreciable distortion of the impregnation front by comparison with the isothermal case. 5. CONCLUSIONS Thus, in the present work a mathematical model of the nonisothermal filling of a moulding cavity 644 Polymers & Polymer Composites, Vol. 23, No. 9, 2015

7 Non-isothermal Moulding of Composite Products With Impregnation of the Porous Layer Figure 3. Calculation scheme Figure 4. The development of the impregnation front for the isothermal case (dashed lines) and for the non-isothermal case (continuous lines). Time: t = 0.5 s (1); t = 10 s (2); t = 20 s (3) having a flat gap of arbitrary height between the mould wall and the porous material has been developed. Filling of the flat cavity is accompanied with simultaneous impregnation of the porous (reinforcing) layer. The problem of non-isothermal nonsteady-state two-dimensional flow in the region of free liquid and in the porous layer of any thickness has been solved. The conditions of conjugation at the liquid porous layer boundary include velocity and tangential stress equality with the use of the Brinkman equation. The two-dimensional spread of impregnation in a confined volume is shown, where the impregnation front successively reaches all walls of the mould. Calculations indicate that the thickness of the boundary layer δ in the porous layer is commensurate with the height of the cavity 2h. Here, the impregnation rate u(y = 0) at the liquid porous body boundary, calculated by means of Brinkman s equation, may exceed the rate calculated by Darcy s law by two or more factors. Thus, models that describe the impregnation in a porous body only on the basis of Darcy s law can be used in the given problem only for cases where h << H. However, even in such a situation, the values of velocity u c (y=0) on the porous wall will be obtained with appreciable error. This in turn will influence the velocity of advance of the liquid flow front along the cavity, and accordingly the time of the entire process of moulding and impregnation. The duration of the moulding process carried out under non-isothermal conditions is of particular interest when account is taken of the possible premature gelation of plastisols or curing of the polymer binder. REFERENCES 1. Sozer E.M., Bickerton S., and Advani S.G., On-line strategic control of liquid composite mould filling process. Composites A, 31(12) (2000) Lee C.-L. and Wei K.-H., Curing kinetics and viscosity change of a two-part epoxy resin during mold filling in resin-transfer molding process. J. Appl. Polym. Sci., 77(10) (2000) Chan A.W. and Hwang S.T., Modeling non-isothermal impregnation of fibrous media with reactive polymer resin. Polym. Eng. Sci., 32(5) (1992) Malkin A. Ya., Kuznetsov V.V., Kleba I., and Michaeli W., Modeling of structural reaction injection molding process. 1. Mathematical model. Polym. Eng. Sci., 41(5) (2001) Murdoch A.I. and Soliman A., On the slip-boundary condition for liquid flow over planar porous boundaries. Proc. R. Soc. Lond. A, 455 (1999) Ariel D.P., Flow of a third grade fluid through a porous flat channel. Int. J. Eng. Sci., 41(11) (2003) Chen G. and Hadmin H.A., Forced convection of a power-law fluid in a porous channel integral solutions. J. Porous Media, 2(1) (1999) Parvazinia M., Nassehi V., and Wakeman R.J., Multi-scale finite element modeling of laminar steady flow through highly permeable porous media. Chem. Eng. Sci., 61(2) (2006) Khan A.A., Ellahi R., and Usman E.M., The effects of variable viscosity on the peristaltic flow of non-newtonian fluid through a porous medium in an inclined channel with slip boundary conditions. J. Porous Media, 16(1) (2013) Kuznetsov A.V., Fluid mechanics and heat transfer in the interface region between a porous medium and a fluid layer: a boundary layer solution. J. Porous Media, 3(3) (1999) Xiong M. and Kuznetsov A.V., Forced convection in a Couette flow in a composite duct: an analysis of Polymers & Polymer Composites, Vol. 23, No. 9,

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