**Appendix A**

In order to simulate the pressing/compression molding process, the flow analysis in 3D TIMON is based on the standard equations of traditional fluid dynamics: the equation of continuity, the equation of motion, and the equation of energy. These governing equations for the filling phase are given in the general and special form of the balance equations as follows:

The mass **continuity equation** (mass balance):

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\rho u\_\mathbf{x}) + \frac{\partial}{\partial y} (\rho u\_\mathbf{y}) + \frac{\partial}{\partial z} (\rho u\_\mathbf{z}) = 0,\tag{A1}$$

where ρ is the density, *t* is time and *ui* stands for the fluid velocity vector in direction x, y, and z, respectively. For incompressible flow ρ is constant and Equation (A1) can be reduced to a volume continuity equation:

$$
\frac{
\partial 
\mathbf{u}\_\mathbf{x}
}{
\partial 
\mathbf{x}
} + \frac{
\partial 
\mathbf{u}\_\mathbf{y}
}{
\partial y
} + \frac{
\partial 
\mathbf{u}\_\mathbf{z}
}{
\partial z
} = \mathbf{0}.\tag{A2}
$$

The **equation of motion** (momentum balance) in fluid flow in all three directions can be expressed in terms of deviatoric stress <sup>τ</sup>*ij* and this form of the equation is commonly called the Cauchy momentum equation:

$$
\rho \frac{\mathrm{D}u\_i}{\mathrm{D}t} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial \tau\_{ji}}{\partial \mathbf{x}\_j} + \rho \mathbf{g}\_{i\nu} \tag{A3}
$$

which in the x-direction becomes:

$$\rho \left( \frac{\partial \mathbf{u\_{x}}}{\partial t} + \mathbf{u\_{x}} \frac{\partial \mathbf{u\_{x}}}{\partial \mathbf{x}} + \mathbf{u\_{y}} \frac{\partial \mathbf{u\_{x}}}{\partial y} + \mathbf{u\_{z}} \frac{\partial \mathbf{u\_{x}}}{\partial z} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \left( \frac{\partial \tau\_{\mathbf{xx}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\mathbf{yx}}}{\partial y} + \frac{\partial \tau\_{\mathbf{xz}}}{\partial z} \right) + \rho \mathbf{g\_{x}} \tag{A4}$$

where *p*: pressure, *gi*: body force acting on the continuum, for example, gravity. In this force balance the mass is represented by the fluid density ρ and the following bracket represents the acceleration, meaning how the velocity of a particle changes with time. Therein, ∂*<sup>u</sup>*x ∂*t* stands for the change of velocity over time and the three following terms represent the speed and direction in which the fluid is moving. The right hand side of the equation shows all forces acting in the fluid, where −∂*<sup>p</sup>* ∂*x* stands for the internal pressure gradient of the fluid, the second term is representing the internal stress forces acting on the fluid (viscous e ffects are considered), and the last term represents all external forces acting on the fluid, such as gravity.

In fluid mechanics the deviatoric stress tensor <sup>τ</sup>*ij* is commonly defined as:

$$
\pi\_{i\bar{j}} = \mu \dot{\gamma}\_{i\bar{j}}.\tag{A5}
$$

.

with μ: viscosity and . γ*ij*: rate of deformation tensor reducing the Cauchy momentum equation to the Navier–Stokes equation for a simple Newtonian fluid with constant density ρ and viscosity μ:

$$
\rho \frac{\mathrm{D}u\_i}{\mathrm{D}t} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \mu \frac{\partial^2 u\_i}{\partial \mathbf{x}\_j \partial \mathbf{x}\_j} + \rho g\_{i\nu} \tag{A6}
$$

which in the x-direction looks like:

$$
\rho \left( \frac{\partial \mathbf{u}\_{\mathbf{x}}}{\partial t} + \mathbf{u}\_{\mathbf{x}} \frac{\partial \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{x}} + \mathbf{u}\_{\mathbf{y}} \frac{\partial \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{y}} + \mathbf{u}\_{\mathbf{z}} \frac{\partial \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{z}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{y}^2} + \frac{\partial^2 \mathbf{u}\_{\mathbf{x}}}{\partial \mathbf{z}^2} \right) + \rho \mathbf{g}\_{\mathbf{x}}.\tag{A7}
$$

The **equation of energy** (energy balance) for a Newtonian fluid with constant properties is given by:

$$
\rho c\_p \frac{\text{DT}}{\text{Dt}} = k \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \dot{Q}\_{\text{viscous heating}} + \dot{Q}\_{\text{\textquotedblleft}} \tag{A8}
$$

where ρ: density, *cp*: specific heat, *T*: temperature, *t*: time, *k*: thermal conductivity, *Q*viscous heating: viscous dissipation and . *Q*: arbitrary heat source (i.e., exothermic reaction).

The viscous heating for a Newtonian material is written as:

$$\dot{Q}\_{\text{viscous heating}} = \mu \dot{\nu}\_{i\dot{j}} ^2 = \mu \left( \frac{\partial u\_i}{\partial x\_{\dot{j}}} + \frac{\partial u\_{\dot{j}}}{\partial x\_{\dot{i}}} \right)^2. \tag{A9}$$

Inserting the viscous heating term into Equation (A8) gives the energy equation in Cartesian coordinates:

$$\begin{split} \rho c\_{p} \Big( \frac{\partial T}{\partial t} + \mu\_{\text{X}} \frac{\partial T}{\partial \mathbf{x}} + \mu\_{\text{Y}} \frac{\partial T}{\partial \mathbf{y}} + \mu\_{\text{Z}} \frac{\partial T}{\partial \mathbf{z}} \Big) \\ = \kappa \Big( \frac{\partial^{2}T}{\partial \mathbf{x}^{2}} + \frac{\partial^{2}T}{\partial y^{2}} + \frac{\partial^{2}T}{\partial z^{2}} \Big) + 2\mu \Big[ \left( \frac{\partial u\_{\text{x}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial u\_{\text{y}}}{\partial y} \right)^{2} + \left( \frac{\partial u\_{\text{z}}}{\partial z} \right)^{2} \Big] \\ + \mu \Big[ \left( \frac{\partial u\_{\text{x}}}{\partial \mathbf{y}} + \frac{\partial u\_{\text{y}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial u\_{\text{x}}}{\partial z} + \frac{\partial u\_{\text{z}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial u\_{\text{y}}}{\partial z} + \frac{\partial u\_{\text{z}}}{\partial y} \right)^{2} \Big] + \overset{\cdot}{Q} .\end{split} \tag{A10}$$

With the convention to denominate the viscosity of Newtonian fluids with μ and the viscosity of non-Newtonian fluids as η Equation (A9) can be rewritten as:

$$
\dot{Q}\_{\text{viscous heating}} = \eta \dot{\eta}^2. \tag{A11}
$$

In the compression molding process the polymer melt can be considered as a viscous fluid. When the material is pressed through the cavity to fill the mold, the melt is assumed to behave as a generalized Newtonian fluid (GNF). Therefore, the non-isothermal 3D flow can be described by the following simplified variant of the energy equation:

$$\rho c\_p \left( \frac{\partial T}{\partial t} + u\_\mathbf{x} \frac{\partial T}{\partial \mathbf{x}} + u\_\mathbf{y} \frac{\partial T}{\partial \mathbf{y}} + u\_\mathbf{z} \frac{\partial T}{\partial \mathbf{z}} \right) = k \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \eta \dot{\chi}^2. \tag{A12}$$

During the exothermic curing reaction of thermosets, heat generates. This effect is considered with an additional energy source terms on the right hand side in Equation (A13), where the energy balance for thermosets in a mold is shown.

$$
\rho c\_p \left( \frac{\partial T}{\partial t} + u\_\mathbf{x} \frac{\partial T}{\partial \mathbf{x}} + u\_\mathbf{y} \frac{\partial T}{\partial \mathbf{y}} + u\_\mathbf{z} \frac{\partial T}{\partial \mathbf{z}} \right) = k \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \eta \dot{\eta}^2 + \frac{\mathbf{d}Q}{\mathbf{d}t}.\tag{A13}
$$

Therein, the left terms stands for the temperature change and the convective heat transfer (more precisely thermal advection), whereas the terms on the right hand side of the equation represent the thermal conduction, shear heat generation (viscous heating), and the heat generation due to an exothermic reaction. Equation (A14) shows the relation between the heat generation velocity d*Q*d*t* and the total amount of generated heat *Q*0 in the exothermic curing reaction affected by the curing reaction rate dαd*t*:

$$\frac{\mathrm{d}Q}{\mathrm{d}t} = Q\_0 \frac{\mathrm{d}a}{\mathrm{d}t}.\tag{A14}$$

With the viscosity standardly defined as:

$$
\mu = \frac{\pi}{\dot{\nu}} \text{ or } \eta = \frac{\pi}{\dot{\nu}'} \tag{A15}
$$

and substituting μ in Equation (A9) the viscous dissipation can also be written as:

$$Q\_{\text{viscous heating}} = \tau\_{ij} \dot{\gamma}\_{ij}.\tag{A16}$$

For generalized Newtonian fluids (incompressible viscous fluids), the deviatoric stress τ can be expressed by:

$$
\pi\_{i\bar{j}} = \eta \dot{\gamma}\_{\bar{i}\bar{j}} = \eta \Big(\nabla u + \left(\nabla u\right)^{\mathsf{T}}\Big),\tag{A17}
$$

where η is the non-Newtonian viscosity that depends both on temperature *T* and the rate of deformation tensor . γ*ij*, ∇*u* is the velocity gradient tensor and (<sup>∇</sup>*u*)<sup>T</sup> is the transposed velocity gradient tensor.

Since η is a scalar it must depend only on scalar invariants of .γ. Therefore, another notation for the symmetric rate of deformation tensor .γ*ij* in the Generalized Newtonian Fluid (GNF) model in Equation (A17) is the scalar strain rate .γ:

$$\left| \dot{\mathbf{y}} = \left| \dot{\mathbf{y}}\_{\dot{\boldsymbol{\eta}}} \right| = \sqrt{\frac{1}{2} (\dot{\boldsymbol{\gamma}}\_{\dot{\boldsymbol{\eta}}} : \dot{\boldsymbol{\gamma}}\_{\dot{\boldsymbol{\eta}}})} = \sqrt{\frac{1}{2} \Pi} = \sqrt{\frac{1}{2} \sum\_{i} \sum\_{j} \dot{\boldsymbol{\gamma}}\_{\dot{\boldsymbol{\eta}}} \dot{\boldsymbol{\gamma}}\_{\ddot{\boldsymbol{\eta}}}}} \text{ with } i, \; j = \text{x, y, z} \tag{A18}$$

where .γ is the magnitude of the rate of deformation tensor .γ*ij* and II is the second invariant of the rate of deformation tensor.

The rate of deformation tensor components in Equation (A18) are defined by:

$$
\dot{\gamma}\_{ij} = \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \tag{A19}
$$

which yields:

$$\dot{\gamma} = \sqrt{2\left(\frac{\partial u\_{\mathbf{x}}}{\partial \mathbf{x}}\right)^{2} + 2\left(\frac{\partial u\_{\mathbf{y}}}{\partial \mathbf{y}}\right)^{2} + 2\left(\frac{\partial u\_{\mathbf{z}}}{\partial \mathbf{z}}\right)^{2} + \left(\frac{\partial u\_{\mathbf{x}}}{\partial \mathbf{y}} + \frac{\partial u\_{\mathbf{y}}}{\partial \mathbf{x}}\right)^{2} + \left(\frac{\partial u\_{\mathbf{y}}}{\partial \mathbf{z}} + \frac{\partial u\_{\mathbf{z}}}{\partial \mathbf{y}}\right)^{2} + \left(\frac{\partial u\_{\mathbf{z}}}{\partial \mathbf{x}} + \frac{\partial u\_{\mathbf{x}}}{\partial \mathbf{z}}\right)^{2}}\,,\tag{A20}$$

respecting the flow field in thickness direction.

The stress tensor components for GNFs in Cartesian coordinates are consequently given by:

$$\begin{aligned} \tau\_{\rm XY} &= 2\eta \frac{\partial u\_{\rm x}}{\partial x} & \tau\_{\rm XY} &= \tau\_{\rm Yx} = \eta \left(\frac{\partial u\_{\rm x}}{\partial y} + \frac{\partial u\_{\rm y}}{\partial x}\right) \\ \tau\_{\rm YY} &= 2\eta \frac{\partial u\_{\rm y}}{\partial y} & \tau\_{\rm Yz} &= \tau\_{\rm zy} = \eta \left(\frac{\partial u\_{\rm y}}{\partial z} + \frac{\partial u\_{\rm z}}{\partial y}\right) \\ \tau\_{\rm ZZ} &= 2\eta \frac{\partial u\_{\rm z}}{\partial z} & \tau\_{\rm xz} = \tau\_{\rm xz} = \eta \left(\frac{\partial u\_{\rm z}}{\partial x} + \frac{\partial u\_{\rm x}}{\partial z}\right) \end{aligned} \tag{A21}$$

$$
\tau\_{ij} = \begin{bmatrix}
\tau\_{\rm{xx}} & \tau\_{\rm{xy}} & \tau\_{\rm{xz}} \\
\tau\_{\rm{yx}} & \tau\_{\rm{yy}} & \tau\_{\rm{yz}} \\
\tau\_{\rm{xz}} & \tau\_{\rm{xy}} & \tau\_{\rm{zz}}
\end{bmatrix} = \eta \begin{bmatrix}
2\frac{\partial u\_{\rm{x}}}{\partial x} & \frac{\partial u\_{\rm{x}}}{\partial y} + \frac{\partial u\_{\rm{y}}}{\partial x} & \frac{\partial u\_{\rm{x}}}{\partial x} + \frac{\partial u\_{\rm{x}}}{\partial z} \\
\frac{\partial u\_{\rm{x}}}{\partial y} + \frac{\partial u\_{\rm{y}}}{\partial x} & 2\frac{\partial u\_{\rm{y}}}{\partial y} & \frac{\partial u\_{\rm{y}}}{\partial z} + \frac{\partial u\_{\rm{x}}}{\partial y} \\
\frac{\partial u\_{\rm{x}}}{\partial x} + \frac{\partial u\_{\rm{x}}}{\partial z} & \frac{\partial u\_{\rm{y}}}{\partial z} + \frac{\partial u\_{\rm{x}}}{\partial y} & 2\frac{\partial u\_{\rm{x}}}{\partial z}
\end{bmatrix}. \tag{A22}
$$

With Equation (A16) the stress tensor components are implemented into Equation (A10), which results in the energy equation for thermoset materials, written as:

$$\begin{split} \rho c\_{p} \Big( \frac{\partial T}{\partial t} + \boldsymbol{\mu}\_{\text{x}} \frac{\partial T}{\partial \mathbf{x}} + \boldsymbol{\mu}\_{\text{y}} \frac{\partial T}{\partial \boldsymbol{y}} + \boldsymbol{\mu}\_{\text{z}} \frac{\partial T}{\partial \boldsymbol{z}} \Big) \\ = & k \Big( \frac{\partial^{2}T}{\partial \mathbf{x}^{2}} + \frac{\partial^{2}T}{\partial y^{2}} + \frac{\partial^{2}T}{\partial z^{2}} \Big) + \tau\_{\text{xx}} \frac{\partial \boldsymbol{u}\_{\text{x}}}{\partial \mathbf{x}} + \tau\_{\text{yy}} \frac{\partial \boldsymbol{u}\_{\text{y}}}{\partial y} + \tau\_{\text{zz}} \frac{\partial \boldsymbol{u}\_{\text{z}}}{\partial z} \\ \quad + \tau\_{\text{xy}} \Big( \frac{\partial \boldsymbol{u}\_{\text{x}}}{\partial y} + \frac{\partial \boldsymbol{u}\_{\text{y}}}{\partial \mathbf{x}} \Big) + \tau\_{\text{yz}} \Big( \frac{\partial \boldsymbol{u}\_{\text{y}}}{\partial z} + \frac{\partial \boldsymbol{u}\_{\text{z}}}{\partial y} \Big) + \tau\_{\text{xz}} \Big( \frac{\partial \boldsymbol{u}\_{\text{z}}}{\partial \mathbf{x}} + \frac{\partial \boldsymbol{u}\_{\text{y}}}{\partial z} \Big) + \dot{Q}\_{\text{y}} \end{split} \tag{A23}$$

where *Q* denotes the heat generation due to the exothermic reaction in the thermoset curing process. In order to calculate the temperature distribution in the thermoset molding compound, 3D TIMON, however, assumes the heat conduction in thickness direction is dominant and therefore simplifies Equation (A23) to:

$$\rho c\_p \mathrm{d}\frac{\partial T}{\partial t} + u\_\mathrm{x} \frac{\partial T}{\partial \mathbf{x}} + u\_\mathrm{y} \frac{\partial T}{\partial y} + u\_\mathrm{z} \frac{\partial T}{\partial \mathbf{z}}\bigg) = k \frac{\partial^2 T}{\partial z^2} + \tau\_\mathrm{yz} \left(\frac{\partial u\_\mathrm{y}}{\partial z}\right) + \tau\_\mathrm{xc} \left(\frac{\partial u\_\mathrm{x}}{\partial z}\right) + Q\_0 \frac{\mathrm{d}\mathbf{z}}{\mathrm{d}t},\tag{A24}$$

where . *Q* is substituted with Equation (A14).

Using the simplification <sup>τ</sup>yz = η<sup>∂</sup>*<sup>u</sup>*y ∂*z* and τzx = η ∂*<sup>u</sup>*x ∂*z* (assumption of simple shear flow) yields to the final energy equation applied in 3D TIMON's Light 3D Heat Transfer:

$$
\rho c\_p \left( \frac{\partial T}{\partial t} + \mu\_\mathbf{x} \frac{\partial T}{\partial \mathbf{x}} + \mu\_\mathbf{y} \frac{\partial T}{\partial \mathbf{y}} + \mu\_\mathbf{z} \frac{\partial T}{\partial z} \right) = k \frac{\partial^2 T}{\partial z^2} + \eta \left( \frac{\partial \mu\_\mathbf{y}}{\partial z} \right)^2 + \eta \left( \frac{\partial \mu\_\mathbf{x}}{\partial z} \right)^2 + Q \nu \frac{\mathbf{d}\alpha}{\mathbf{d}t}.\tag{A25}
$$
