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--- en:grundlagenhandbuch:massetemperatur:2d_modell:2d_ansatz [2025/06/20 08:54] – deppe2
+++ en:grundlagenhandbuch:massetemperatur:2d_modell:2d_ansatz [2026/05/28 10:58] (aktuell) – gelöscht deppe2
@@ Zeile 1: / Zeile 1: @@
-====== 2D Modell ======
-====== 2D Modell ======
-The energy equation for the calculation of the radial temperature profile in the screw flight forms the basis for the 2D model.
-$$\rho c \left(\frac{\partial T}{\partial t} + v \nabla T\right) = \lambda \nabla^2 T + \tau \nabla v$$ **Equation 1**
-A modification of the energy equation to match the existing open system, which exchanges the energy and the mass with the environment, follows.
-The cylinder moves as an ideally formed plate over the rigid screw, so that one can resort back to a pan model with kinematic reversal. The different diction for Cartesian coordinates results in the following function for the energy
- equation with the aforementioned geometry. [Sch13]
-$$\rho c \left(\frac{\partial T}{\partial t} + v_x \frac{\partial T}{\partial x} + v_y \frac{\partial T}{\partial y} + v_z \frac{\partial T}{\partial z}\right) = -\left(\frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} + \frac{\partial q_z}{\partial z}\right) - T \left(\frac{\partial \rho}{\partial T}\right)_p \left[\left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\right) - \left(\sigma_{xx} \frac{\partial v_x}{\partial x} + \sigma_{yy} \frac{\partial v_y}{\partial y} + \sigma_{zz} \frac{\partial v_z}{\partial z}\right)\right] - \left[\tau_{xy} \left(\frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x}\right) + \tau_{xz} \left(\frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x}\right) + \tau_{yz} \left(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}\right)\right]$$ **Equation 2**
-The equation consists of five terms, which have the following meanings [Ang10]:
-  * The left side equates to the change of the internal energy per unit of time and volume
-  * The first term on the right equates to the change of the supplied energy through heat conduction per unit of time and volume
-  * The second term on the right equates to the recoverable amount of work per unit of time and volume through compression
-  * The third and fourth terms on the right side equate to the non-recoverable amount of work per unit of volume and time as a result of dissipation
-As to obtain an analytic result, the equation must be simplified with further model assumptions. Apart from the viscosity, all material parameters are presumed temperature independent.
-In addition to the aforementioned ones, further assumptions are also made [Sch13]:
-  * The flow is stationary and laminarily following ($\frac{\partial T}{\partial t} = 0$)
-  * As the polymer melt is seen as incompressible ($\rho = konst.$), the second term disappears on the right side of the energy equation. Therefore the examination of the thermal capacity becomes simpler ($c_p = c_v = c$)
-  * In the direction of x and y no speed components are examined ($v_x = v_y = 0$)
-  * The screw canal is completely filled with the melt, which is wall-adhering
-  * The heat flows in the direction of the flight and those transverse to them can be neglected in comparison to the heat flow in the direction of the flight height ($q_x = q_z = 0$)
-  * Because the flight height is considerably smaller than the flight width ($h << b$), the shear stresses at the supporting flights can be neglected ($\tau_{xz} = 0$)
-  * Normal stresses are also neglectable ($\sigma_{xx} = \sigma_{yy} = \sigma_{zz} = 0$)
-With the help of the simplifying criteria the equation 4-2 can be reduced to 3 terms:
-$\rho c v_z \frac{\partial T}{\partial z} = -\frac{\partial q_y}{\partial y} - \tau_{yz} \left(\frac{\partial v_z}{\partial y}\right)$ **Equation 3**
-The sum of thermal conduction in the flight height and the energy of the dissipation results in the temperature rising lengthways in the canal, which is filled with polymer melt.
-Under consideration of the Fourier thermal conductivity approach for the heat flow in direction of the flight height and the power flow law for non-newtonian flow behavior of polymer melt with the Arrehnius-approach, the equation 4-3 can be extended to the following:
-$\rho c \bar{V}_z \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial y^2} + \left(\overline{\tau\dot{\gamma}}\right)_0 e^{-\beta(T-T_0)}$ **Equation 4**
-To simplify the energy equation further, the dimensionless operating figures as described in [Ang10], the Graetz- ($Gz$) and the Brinkmann-number ($Br$), are used. Herein, the Brinkmann-number describes the relation of disperse energy in the screw canal to the heat conduction in the direction of the flight height. The Graetz-number describes the convection in the direction of the flight length to the heat conduction in the direction of the flight height. Additionally, the dimensionless coordinates $\xi$, $\zeta$ and the dimensionless temperature $\Theta$ are placed in the energy equation.
-$Br = \frac{\left(\overline{\tau\dot{\gamma}}\right)_0 h^2}{\lambda T_Z} \approx \frac{K_{0T} v_0^{1+n} h^{1-n}}{\lambda T_Z}$ **Equation 5**
-$Gz = \frac{c \rho \bar{w}_i h^2}{\lambda \Delta z} = \frac{c \rho h}{\lambda b \Delta z} \dot{V}$ **Equation 6**
-**Equation 7**
-$\Theta = \frac{T - T_0}{T_Z}$
-**Equation 8**
-$\xi = \frac{\Delta y}{h}$
-**Equation 9**
-$\zeta = \frac{\Delta z}{l}$
-During the geometric calculation of the Graetz- $Gr$ and Brinkmann-number $Br$ some twin screw-specific adjustments are made. For the ascertainment of the Graetz-number $Gr$ the flight height $h$ and the flight width $br$ are important. These two sizes are defined using the average flight height $\bar{h}$ and the flight width $b_{max}$ of the twin screw extruder.
-For the Brinkmann-number $Br$ the viscosity $\eta$ is needed, this depends on the shear rate $\dot{\gamma}$ and is influenced by the geometry of the twin screw extruder. As there is no constant flight height $h$ in the canal, the average flight height $\bar{h}$ is used to calculate the average shear rate $\overline{\dot{\gamma}}$. Without considering the influence of the characteristics of the screw element (mixing elements, Shear elements etc.), [Kre04]
-$\overline{\dot{\gamma}} = \frac{\bar{h}}{v_0}$ **Gleichung 10**
-The function for the calculation of the temperature in the melt-filled canal, assumes the following shape:
-$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \exp\left[-\beta(T_Z \Theta)\right]$ **Equation 10**
-It is not possible yet, to solve the, as the exponential term is dependent on the temperature $T$ or rather $\Theta$. In the following equation the exponential term is partly linearized.
-$\exp\left[-\beta(T - T_0)\right] = \exp\left[-\beta(T_Z \Theta)\right] = c_1 - c_2 \beta T_Z \Theta$ **Equation 11**
-This linear equation divides the exponential term into two parts. The first part is dependent on the radial temperature $T$ or rather $\Theta$ ($c_2 \beta T_Z \Theta$) and the second is dependent on $T$ or rather $\Theta$ and $c_1$. The two unknown parameters $c_1$ and $c_2$ are calculated in two equations, which approach the exponential function with a secant which lies between the average radial temperature of the previous section $T_0$ and the cylinder wall temperature $T_Z$.
-**Image 1:** approach of exponential function using a secant [Ang10]
-In summary, the linearization with the Equation 10 produces the analytically solvable, simplified energy equation for the radial temperature (for $0 \leq \xi \leq 1$ and $0 \leq \zeta \leq 1$) [Ang10]:
-$\frac{\partial^2 \Theta}{\partial \xi^2} - Gz \frac{\partial \Theta}{\partial \xi} - c_2 \beta T_Z Br \Theta = -c_1 Br$ **Equation 12**
-The presented differential equation was successfully solved with the marginal conditions of the constant cylinder temperature and a tempered screw in the elaboration of [Sch13]. This model was adjusted according to the calculation of the temperature progression in the co-rotating twin screw extruder. As the solution contains dimensionless temperature data, these must still be converted into dimensioned data.
-This can be done using the following equation and results in the ten data of the profile of the flight height.
-$T_{S,n}(\xi) = \theta_{S,n} T_{Z,n} + T_{n-1}$ **Gleichung 14**
-Finally the arithmetic mean is formed from the radial temperature data and the average canal temperature of the section is defined. Because of the step-by-step calculation in SIGMA the axial temperature curve can be calculated.