The Effect of Dissipation on the Moulding of Composite Articles in Closed Moulds

Abstract

The flow and heat exchange in the process of impregnation of filler by non-Newtonian fluid during the moulding of composite articles in closed moulds are investigated. Flow is described by the modified Brinkman equation. From the rheological point of view, the medium comprises a power-law fluid. In writing the energy equation, a single-temperature model is used. Dissipative heat generation is taken into account. The problem is solved by the numerical finite difference method. The effect of dissipation (the Brinkman number) on the distribution of the mass-average temperature over the length of the cavity with different Péclet numbers is shown.

Keywords

non-Newtonian fluid impregnation heat exchange moulding

The moulding of reinforced or composite articles in closed moulds is considered. At a constant flow rate, the fluid is fed into a mould filled with porous filler material (Figure 1). Here, the temperature of the fluid T₀ and the temperature of the mould T_w are different. Processes of this type (both injection moulding and transfer moulding) are used in the production of a vast number of goods of technical, household, and special designation with different service properties. After completion of the process of mould filling and impregnation of the entire porous layer, solidification of the casting occurs as a result of processes of crystallisation or glass transition during the cooling of polymers, gelation of plastisols, or chemical curing of the polymer binder (for example, epoxy and polyester resins). In the present work it is assumed that the processes of moulding and impregnation are completed within the so-called induction period, i.e. before the onset of premature gelation or curing.

Figure 1

Diagram of the moulding process

In the literature, a fairly large number of studies are devoted to modelling of the processes of moulding of composite articles in closed moulds. A brief review of the literature on this topic is given elsewhere [1,2]. In these earlier papers [1,2], studies are mentioned that can be regarded as similar to the present formulation of the problem from the point of view of the mathematical model. Publications describing the flow of fluid through channels in which one or both of the walls are porous (permeable), and also through channels partly or entirely filled with porous material, are borne in mind.

The polymer is considered to be a incompressible non-Newtonian (power-law) fluid. The viscosity of the fluid is assumed to be fairly high, so that flow occurs at low Reynolds numbers (Re ≤ 0.01). As a result, the initial hydrodynamic section is practically absent and the velocity profile at entry into the channel can be regarded as developed [3]. Furthermore, in the equation of motion, this makes it possible to ignore inertia terms, and therefore impregnation of the porous layer will be described using the modified Brinkman equation [1,2]:

- \frac{μ}{k} u^{n} + \frac{μ}{ε^{n}} \frac{\partial}{\partial y} [{| \frac{\partial u}{\partial y} |}^{n - 1} (\frac{\partial u}{\partial y})] - \frac{d p}{dx}

(1)

where u is the axial velocity of filtration, μ is the viscosity of the fluid, k is the permeability coefficient, ∊ is the porosity, and p is pressure.

Pressure p is considered to be independent of the transverse coordinate y.

The condition of flow rate constancy is as follows:

\bar{u} = \frac{1}{h} \int_{0}^{h} u d y

(2)

Fluids with a low thermal diffusivity are considered. In this case, during channel flow the Péclet number condition Pe ≥ 100 is fulfilled. This makes it possible to ignore heat transfer by conduction along the channel axis by comparison with transfer by convection [3]. Flow occurs under boundary conditions of the first kind, when the temperature of the mould walls, T_w, is considered to be constant. It is assumed that the temperature of the fluid at entry into the mould cavity is distributed uniformly over the cross-section and is equal to T₀. The mathematical model of energy transfer is based on the so-called single-temperature model where one energy equation is used [4,5]. This approach supposes a local thermal equilibrium between the liquid and solid phases. Consequently, the energy equation has the form

ρ c_{p} u \frac{\partial T}{\partial \times} = \frac{\partial^{2} T}{\partial y^{2}} + Φ (x, y)

(3)

Here, the effective value of thermal conductivity of the heterogeneous composite medium, λ, can be calculated by different methods. In the present work, A was calculated in the following way:

λ = λ_{b} [ε^{2} ς + \frac{4 ς ε (1 - ε)}{1 + ς} + {(1 - ε)}^{2}]

(4)

The function Φ(x, y) reflects dissipative heat generation and can be written in different forms [7]. It has been noted elsewhere [8] that, with low Darcy numbers, all forms of notation give the same result. In the present work, the dissipative term is written in a form that is compatible with the limiting case of flow of a pure (free) fluid with infinitely large Darcy numbers:

Φ = \frac{η u^{n + 1}}{k} + η {| \frac{\partial u}{\partial y} |}^{n - 1} {(\frac{\partial u}{\partial y})}^{2}

(5)

The hydrodynamic and thermal boundary conditions are written in the following way:

y = 0, u = 0, T = T_{w}

(6)

y = h \frac{\partial u}{\partial y} = 0, \frac{\partial T}{\partial y} = 0

(7)

x = 0, T = T_{0}

(8)

Switching to dimensionless variables, the main equations will take the form

- \frac{U^{n}}{D a} + \frac{\partial}{\partial Y} [{| \frac{\partial U}{\partial Y} |}^{n - 1} \frac{\partial U}{\partial Y}] = \frac{dP}{dX}

(9)

U \frac{\partial θ}{\partial \times} = \frac{1}{Pe} \frac{\partial^{2} θ}{\partial Y^{2}} + \frac{B r}{Pe} \frac{1}{Da} U^{n + 1} + \frac{Br}{Pe} {| \frac{\partial U}{\partial Y} |}^{n - 1} {(\frac{\partial U}{\partial Y})}^{2}

(10)

\int_{0}^{1} U d Y = 1

(11)

where

\begin{array}{l} Y = \frac{Y}{h}, \times = \frac{\times}{h}, U = \frac{u}{\bar{u}}, D a = \frac{k}{h^{n + 1} ε^{n}}, P = \frac{p h^{n} ε^{n}}{{\bar{u}}^{n} η}, \\ Pe= \frac{\bar{u} h}{a}, Br= \frac{η {\bar{u}}^{n + 1}}{λ (T_{0} - T_{w}) h^{n - 1}}, θ = \frac{T - T_{w}}{T_{0} - T_{w}} \end{array}

The value of the mass-average temperature of the medium in dimensional and dimensionless form is defined in the following way:

T_{m} = \frac{1}{h \bar{u}} \int_{0}^{h} T u d y, θ_{m} = \int_{0}^{1} θ U d Y

(12)

The problem was solved numerically by the finite difference method using iterations and the elimination method. In the course of moulding (impregnation), the polymer is heated both by heat transfer from the hot mould walls and by dissipative heating up. In this connection, during calculations, it was of interest to assess the degree of non-uniformity of the distribution of the mass-average temperature over the length of the article at the moment of completion of the moulding (impregnation) process. Therefore, all the figures given below reflect that moment in time when the impregnation front reaches the opposite mould wall. After the completion of moulding, the stage of holding for curing (crystallisation) of the filler begins. Here, non-uniform temperature distribution over the length of the article may entail appreciable inhomogeneity in the distribution of the degree of curing of the binder throughout the article. Figures 2 and 3 show the distribution of dimensionless mass-average temperature over the length of the article at the moment of completion of the moulding process. Here, the dimensionless length of the mould amounted to X= 20. The considerable influence of the dimensionless Péclet and Brinkman numbers is shown separately. It can also be noted that calculations showed an insignificant role of the Darcy number with respect to changes in the longitudinal temperature profile. Therefore, the influence of this criterion was omitted from this study.

Figure 2

The distribution of the mass-average temperature distribution over the length of the article with different Péclet numbers: 1 – Pe = 500; 2 – Pe = 1000; 3 – Pe = 5000; 4 – Pe = 10 000. Br = −0.2; Da = 0.01; n = 0.5. Vertical axis: θ_m; Horizontal axis: X

Figure 3

The distribution of the mass-average temperature distribution over the length of the article with different Brinkman numbers: 1 – Br = 0; 2 – Be = −0.1; 3 – Br = −0.2; 4 – Br = −0.3. Pe = 1000; Da = 0.01; n = 0.5. Vertical axis: θ_m; Horizontal axis: X

In order to generalise the results presented in Figures 2 and 3 , calculations were carried out and a more universal diagram was plotted ( Figure 4 ). This links all the principal parameters of the given process and makes it possible to assess how non-uniform the temperature distribution is over the length of the article under the particular moulding conditions. The dimensionless temperature of the polymer at entry into the mould is constant and equal to 1. Greatest heating up through dissipation and heat transfer from the hot mould walls is produced by those layers of filler that reach the opposite (end) wall of the cavity at X = 20. Therefore, the mass-average temperature of the medium at X = 20 was chosen as the function in Figure 4 . Thus, the lower the value of θ_m at X = 20 (i.e. the higher the value of T_m), the more significant is the non-uniformity of the temperature distribution over the article.

Figure 4

The dependence of the mass-average temperature at X = 20 on the Brinkman number with different Péclet numbers: 1 – Pe = 1000; 2 – Pe = 2500; 3 – Pe = 5000; 4 – Pe = 7500. Da = 0.01; n = 0.5. Vertical axis: θ_m; Horizontal axis: –Br

Figure 4 makes it possible to assess the non-uniformity of temperature distribution and its dependence on practically all the processing parameters and properties of the polymer.

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