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--- en:grundlagenhandbuch:massetemperatur [2026/05/24 17:28] – [Radial Clearance and Groove Flow] neelest
+++ en:grundlagenhandbuch:massetemperatur [2026/06/05 20:18] (aktuell) – [Determination of the Average Temperature] deppe2
@@ Zeile 267: / Zeile 267: @@
 The coupling of the temperature calculations in the channel area and the clearance area [[en:grundlagenhandbuch:massetemperatur#references |[Ste92]]] can be performed by means of a balance of the enthalpy being fed and the enthalpy streaming off.
-{{ :grundlagenhandbuch:massetemperatur:modifizierter_ansatz:de_sigma150_dlg_grundlagenhandbuch_massetemperatur_004.png?nolink |}}
+{{ :en:grundlagenhandbuch:de_sigma150_dlg_grundlagenhandbuch_massetemperatur_006.svg?nolink&600 |}}
 **Figure:** Control room for the enthalpy balance to calculate the medium temperature.
@@ Zeile 273: / Zeile 273: @@
 The enthalpy change in the control room corresponds with the difference between the inflowing and the outflowing enthalpy:
-$$\Delta \dot{H}_{z_0,1} = \dot{H}_{z_1} - \dot{H}_{z_0} \tag{1}$$
+$$\Delta \dot{H}_{z_0,1} = \dot{H}_{z_1} - \dot{H}_{z_0} \tag{46}$$
 The alteration results from the temperature increase in the channel plus the enthalpy change of the flows in the radial clearance and in the grooves.
-$$\Delta \dot{H}_{z_0,1} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \Delta \dot{H}_{gap} + \Delta \dot{H}_{groove} \tag{2}$$
+$$\Delta \dot{H}_{z_0,1} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \Delta \dot{H}_{gap} + \Delta \dot{H}_{groove} \tag{47}$$
 with:
-$$\Delta \dot{H}_{zgap} = \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{gap}) \tag{3}$$
+$$\Delta \dot{H}_{zgap} = \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{gap}) \tag{48}$$
-$$\Delta \dot{H}_{zgroove} = \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{4}$$
+$$\Delta \dot{H}_{zgroove} = \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{49}$$
 $\overline{T_{z01,i}}$ is the integrally averaged temperature in the control room:
-$$\overline{T_{z01,i}} = \frac{1}{\Delta z} \cdot \int_{z_0}^{z_1} T_{channel}(z) \cdot dz \tag{5}$$
+$$\overline{T_{z01,i}} = \frac{1}{\Delta z} \cdot \int_{z_0}^{z_1} T_{channel}(z) \cdot dz \tag{50}$$
 Is it simplifyingly assumed that the isotherms run at right angle to the screw axis, the medium temperature in the neighboring channel $\overline{T_{z01,i+1}}$ can be determined:
-$$\overline{T_{z01,i+1}} = \frac{1}{\Delta z} \cdot \int_{z_0+\frac{t}{\sin(\varphi_s)}}^{z_1+\frac{t}{\sin(\varphi_s)}} T_{channel}(z) \cdot dz \tag{6}$$
+$$\overline{T_{z01,i+1}} = \frac{1}{\Delta z} \cdot \int_{z_0+\frac{t}{\sin(\varphi_s)}}^{z_1+\frac{t}{\sin(\varphi_s)}} T_{channel}(z) \cdot dz \tag{51}$$
 The overall temperature increase in the interval $[z_0, z_1]$ thus calculates to:
-$$T_{z_1} = \frac{\dot{H}_{z_i}}{c \cdot \rho \cdot \dot{V}_z} \tag{7}$$
+$$T_{z_1} = \frac{\dot{H}_{z_i}}{c \cdot \rho \cdot \dot{V}_z} \tag{52}$$
 with
-$$\dot{H}_{z_i} = \Delta \dot{H}_{z_{0,1}} + \dot{H}_{z_0} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{thread}) + \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{8}$$
+$$\dot{H}_{z_i} = \Delta \dot{H}_{z_{0,1}} + \dot{H}_{z_0} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{thread}) + \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{53}$$
 one eventually receives the following equation for the calculation of the temperature development:
-$$T_{z_1} = T_{z1,channel} + \frac{\dot{V}_x}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) + \frac{\dot{V}_{groove}}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{9}$$
+$$T_{z_1} = T_{z1,channel} + \frac{\dot{V}_x}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) + \frac{\dot{V}_{groove}}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{54}$$
 First experiments for the verification of the temperature model were performed with a twin screw extruder of the ZSK 30 type. A polypropylene (PP 1100H) served as the experimental medium. The polymer was plasticated in a subsidiary extruder and laterally conveyed into the twin screw extruder (see figure).
@@ Zeile 310: / Zeile 310: @@
 It can be recognized that the model is in a position to describe the experimental results with sufficient accuracy.
+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_massetemperatur_007.svg?nolink&700 |}}
 **Figure:** Comparison of measured and calculated melt temperatures in thread elements.
@@ Zeile 317: / Zeile 319: @@
 The energy equation for the calculation of the radial temperature profile in the screw flight forms the basis for the 2D model.
-$$\rho c \left(\frac{\partial T}{\partial t} + v \nabla T\right) = \lambda \nabla^2 T + \tau \nabla v \tag{Equation 1}$$
+$$\rho c \left(\frac{\partial T}{\partial t} + v \nabla T\right) = \lambda \nabla^2 T + \tau \nabla v \tag{55}$$
 A modification of the energy equation to match the existing open system, which exchanges the energy and the mass with the environment, follows.
@@ Zeile 327: / Zeile 329: @@
 $$-T \left(\frac{\partial p}{\partial T}\right)_V \left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\right) - \left(\sigma_{xx} \frac{\partial v_x}{\partial x} + \sigma_{yy} \frac{\partial v_y}{\partial y} + \sigma_{zz} \frac{\partial v_z}{\partial z}\right) $$
-$$-\left[\tau_{xy} \left(\frac{\partial v_y}{\partial x} + \frac{\partial v_x}{\partial y}\right) + \tau_{xz} \left(\frac{\partial v_z}{\partial x} + \frac{\partial v_x}{\partial z}\right) + \tau_{yz} \left(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}\right)\right]\tag{Equation 2}$$
+$$-\left[\tau_{xy} \left(\frac{\partial v_y}{\partial x} + \frac{\partial v_x}{\partial y}\right) + \tau_{xz} \left(\frac{\partial v_z}{\partial x} + \frac{\partial v_x}{\partial z}\right) + \tau_{yz} \left(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}\right)\right]\tag{56}$$
 The equation consists of five terms, which have the following meanings [[en:grundlagenhandbuch:massetemperatur#references |[Ang10]]]:
@@ Zeile 348: / Zeile 350: @@
   * 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:
+With the help of the simplifying criteria the equation 56 can be reduced:
-$$\rho c v_z \frac{\partial T}{\partial z} = -\frac{\partial \dot{q}_y}{\partial y} - \tau_{yz} \left(\frac{\partial v_z}{\partial y}\right)\tag{Equation 3}$$
+$$\rho c v_z \frac{\partial T}{\partial z} = -\frac{\partial \dot{q}_y}{\partial y} - \tau_{yz} \left(\frac{\partial v_z}{\partial y}\right)\tag{57}$$
 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:
+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 57 can be extended to the following:
-$$\rho c \bar{v}_z \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial y^2} + \left(\overline{\tau\dot{\gamma}}\right)_0 e^{-\beta(T-T_0)}\tag{Equation 4}$$
+$$\rho c \bar{v}_z \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial y^2} + \left(\overline{\tau\dot{\gamma}}\right)_0 e^{-\beta(T-T_0)}\tag{58}$$
 To simplify the energy equation further, the dimensionless operating figures as described in [[en:grundlagenhandbuch:massetemperatur#references |[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.
-$$Br = \frac{\left(\overline{\tau\dot{\gamma}}\right)_0 h^2}{\lambda T_Z} \approx \frac{K_{0T} v_0^{1+n} h^{1-n}}{\lambda T_Z}\tag{Equation 5}$$
+$$Br = \frac{\left(\overline{\tau\dot{\gamma}}\right)_0 h^2}{\lambda T_Z} \approx \frac{K_{0T} v_0^{1+n} h^{1-n}}{\lambda T_Z}\tag{59}$$
-$$Gz = \frac{c \rho \bar{v}_z h^2}{\lambda \Delta z} = \frac{c \rho h}{\lambda b \Delta z} \dot{V}\tag{Equation 6}$$
+$$Gz = \frac{c \rho \bar{v}_z h^2}{\lambda \Delta z} = \frac{c \rho h}{\lambda b \Delta z} \dot{V}\tag{60}$$
-$$\Theta = \frac{T - T_0}{T_Z}\tag{Equation 7}$$
+$$\Theta = \frac{T - T_0}{T_Z}\tag{61}$$
-$$\xi = \frac{\Delta y}{h}\tag{Equation 8}$$
+$$\xi = \frac{\Delta y}{h}\tag{62}$$
-$$\zeta = \frac{\Delta z}{l}\tag{Equation 9}$$
+$$\zeta = \frac{\Delta z}{l}\tag{63}$$
 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.
@@ Zeile 376: / Zeile 378: @@
 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.) [[en:grundlagenhandbuch:massetemperatur#references |[Kre04]]].
-$$\overline{\dot{\gamma}} = \frac{\bar{h}}{v_0}\tag{Equation 10}$$
+$$\overline{\dot{\gamma}} = \frac{\bar{h}}{v_0}\tag{64}$$
 The function for the calculation of the temperature in the melt-filled canal, assumes the following shape:
-$$\frac{\partial \Theta}{\partial \xi} = \frac{1}{Gz} \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \exp\left[-\beta(T_Z \Theta)\right]\tag{Equation 11}$$
+$$\frac{\partial \Theta}{\partial \xi} = \frac{1}{Gz} \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \exp\left[-\beta(T_Z \Theta)\right]\tag{65}$$
 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.
-$$\exp\left[-\beta(T - T_0)\right] = \exp\left[-\beta(T_Z \Theta)\right] = c_1 - c_2 \beta T_Z \Theta\tag{Equation 12}$$
+$$\exp\left[-\beta(T - T_0)\right] = \exp\left[-\beta(T_Z \Theta)\right] = c_1 - c_2 \beta T_Z \Theta\tag{66}$$
 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 $T_Z$.
@@ Zeile 394: / Zeile 396: @@
 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 $0 \leq \zeta \leq 1$) [[en:grundlagenhandbuch:massetemperatur#references |[Ang10]]]:
-$$\frac{\partial^2 \Theta}{\partial \xi^2} - Gz \frac{\partial \Theta}{\partial \xi} - c_2 \beta T_Z Br \Theta = -c_1 Br\tag{Equation 13}$$
+$$\frac{\partial^2 \Theta}{\partial \xi^2} - Gz \frac{\partial \Theta}{\partial \xi} - c_2 \beta T_Z Br \Theta = -c_1 Br\tag{67}$$
 The presented differential equation was successfully solved with the marginal conditions of the constant cylinder temperature and a tempered screw in the elaboration of [[en:grundlagenhandbuch:massetemperatur#references |[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.
@@ Zeile 400: / Zeile 402: @@
 This can be done using the following equation and results in the ten data of the profile of the flight height.
-$$T_{0,n}(\xi) = \theta_{\xi,n} T_{Z,n} + T_{n-1}\tag{Equation 14}$$
+$$T_{0,n}(\xi) = \theta_{\xi,n} T_{Z,n} + T_{n-1}\tag{68}$$
 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.