Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| en:grundlagenhandbuch:aufschmelzberechnung [2026/05/24 09:30] – [Consideration of the Real Channel Geometry] neelest | en:grundlagenhandbuch:aufschmelzberechnung [2026/09/04 09:40] (aktuell) – [Temperature of the Solids at the Location of the First Melt] paal | ||
|---|---|---|---|
| Zeile 2: | Zeile 2: | ||
| ===== Phenomenology===== | ===== Phenomenology===== | ||
| - | The feet material is held in underneath the opening by the screw-elements and is transported in the screws as a solid in the direction of the screw-tips. Through contact with the heated cylinder wall, the particles in the proximity of the cylinder wall begin to sinter and melt on until finally a broad standing connected | + | The feed material is held in underneath the hopper |
| - | Through the formation of the molten film, the friction conditions in the mesh area and in the solid tailback change behind the intermeshing area. Through this process, the forced conveying of solids in a intermeshing region breaks down, and material is forced through the nip region. As a result of the 3-dimensional speed profile in the intermeshing area, an intensive mixture of the materials occurs. If the proportion of ready-fused materials is sufficient, a dispersion from the existing granules is formed from the solid phase and from the existing viscous phase polymer melt. | + | Through the formation of the molten film, the friction conditions in the mesh area and in the solid tailback change behind the intermeshing area. Through this process, the forced conveying of solids in an intermeshing region breaks down, and material is forced through the nip region. As a result of the 3-dimensional speed profile in the intermeshing area, an intensive mixture of the materials occurs. If the proportion of ready-fused materials is sufficient, a dispersion from the existing granules is formed from the solid phase and from the existing viscous phase polymer melt. |
| - | If the melted share for a completely formed dispersion is not yet sufficient an agglomeration of granule particles occurs. These are then further fused through the dissipation of the already existing melt, and through heat conduction of the surrounding walls. Also the agglomerates | + | If the melted share for a completely formed dispersion is not yet sufficient an agglomeration of granule particles occurs. These are then further fused through the dissipation of the already existing melt, and through heat conduction of the surrounding walls. Also the agglomerates |
| - | After the achievement of this condition, the further melting of the not yet molten Polymer particles can only be successful through the heat conduction from the hot melt to the solid granule particles. This is because the granule particles no longer have direct contact with the heated cylinder walls, i.e. they are longer | + | After the achievement of this condition, the further melting of the not yet molten Polymer particles can only be successful through the heat conduction from the hot melt to the solid granule particles. This is because the granule particles no longer have direct contact with the heated cylinder walls, i.e. they no longer |
| - | A condition for the described melting behavior is a sufficiently long solids-conveying section. | + | A condition for the described melting behavior is a sufficiently long solids-conveying section. |
| If the available length of conveying section to the molten film over the solid tail-back does not suffice, so the forced conveying of the solids remains up to the location where the first filling with melt took place. Principally, | If the available length of conveying section to the molten film over the solid tail-back does not suffice, so the forced conveying of the solids remains up to the location where the first filling with melt took place. Principally, | ||
| Zeile 102: | Zeile 102: | ||
| The coupling of the energy theorem and kinetics gives the differential calculus for the spherically symmetric temperature field. | The coupling of the energy theorem and kinetics gives the differential calculus for the spherically symmetric temperature field. | ||
| - | $$\frac{\partial \theta}{\partial \tau} = \frac{1 | + | $$\frac{\partial\theta}{\partial\tau} = \frac{1}{\xi^2}\frac{\partial}{\partial\xi}\left(\xi^2\frac{\partial\theta}{\partial\xi}\right)\tag{12}$$ |
| - | whereby these standardization' | + | whereby these standardizations: |
| $$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{13}$$ | $$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{13}$$ | ||
| Zeile 118: | Zeile 118: | ||
| and the boundary conditions | and the boundary conditions | ||
| - | $$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ | + | $$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ |
| as a solution of the average caloric temperature of the particle: | as a solution of the average caloric temperature of the particle: | ||
| Zeile 338: | Zeile 338: | ||
| $$C_K = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \left(\frac{\bar{h}(x_f)}{h(x)}\right)^{1+n} dx \tag{47}$$ | $$C_K = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \left(\frac{\bar{h}(x_f)}{h(x)}\right)^{1+n} dx \tag{47}$$ | ||
| - | To avoid a null division | + | To avoid a division |
| ==== Calculation of the Solid Bed Reduction Along the Melt Path ==== | ==== Calculation of the Solid Bed Reduction Along the Melt Path ==== | ||
| Zeile 344: | Zeile 344: | ||
| For the physical mathematical description of the melting of single particles in a polymer melt, according to prerequisite 2, changing effects between adjacent particles should be neglected. The energy equation in sphere coordination is reduced with the consideration of stationary relationships and natural heat at constant solid data to: | For the physical mathematical description of the melting of single particles in a polymer melt, according to prerequisite 2, changing effects between adjacent particles should be neglected. The energy equation in sphere coordination is reduced with the consideration of stationary relationships and natural heat at constant solid data to: | ||
| - | $$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \dot{q}_r) \tag{1}$$ | + | $$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \dot{q}_r) \tag{48}$$ |
| To obtain the description of heat conductivity, | To obtain the description of heat conductivity, | ||
| - | $$\dot{q}_r = -\lambda \frac{\partial T}{\partial r} \tag{2}$$ | + | $$\dot{q}_r = -\lambda \frac{\partial T}{\partial r} \tag{49}$$ |
| Put equation 2 into 1, under the assumption of constant material value to get: | Put equation 2 into 1, under the assumption of constant material value to get: | ||
| - | $$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{3}$$ | + | $$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{50}$$ |
| {{ : | {{ : | ||
| Zeile 358: | Zeile 358: | ||
| **Figure:** Sphere co-ordination on a solid particle | **Figure:** Sphere co-ordination on a solid particle | ||
| - | From the mass balance | + | The mass balance |
| Solid: | Solid: | ||
| - | $$\frac{\partial m_f}{\partial t} = -4\pi r_G^2 \rho_f \frac{\partial r_G}{\partial t} \tag{4}$$ | + | $$\frac{\partial m_f}{\partial t} = -4\pi r_G^2 \rho_f \frac{\partial r_G}{\partial t} \tag{51}$$ |
| Melt: | Melt: | ||
| - | $$\frac{\partial m_s}{\partial t} = -4\pi r^2 \rho_s \frac{\partial r}{\partial t} \tag{5}$$ | + | $$\frac{\partial m_s}{\partial t} = -4\pi r^2 \rho_s \frac{\partial r}{\partial t} \tag{52}$$ |
| - | By equating (4) and (5) one gets: | + | Setting the two equal gives: |
| - | $$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{6}$$ | + | $$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{53}$$ |
| - | Solve equation | + | Solve equation with respect to $\partial t$ and include the average flow velocity in the channel using $\left(\bar{v} = \frac{\partial z}{\partial t}\right)$ to get: |
| - | $$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \right)^{-1} \partial r \tag{7}$$ | + | $$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \right)^{-1} \partial r \tag{54}$$ |
| - | Now put equation (7) through (3) with: | + | Substituting gives: |
| - | $$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{8}$$ | + | $$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{55}$$ |
| With the definition of the constants: | With the definition of the constants: | ||
| - | $$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{9}$$ | + | $$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{56}$$ |
| - | From equation | + | From equation |
| - | $$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{10}$$ | + | $$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{57}$$ |
| The double integration of | The double integration of | ||
| - | $$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{11}$$ | + | $$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{58}$$ |
| results in the following equation: | results in the following equation: | ||
| - | $$T(r) = \frac{C_1}{A}e^{-\frac{A}{r}} + C_2 \tag{12}$$ | + | $$T(r) = \frac{C_1}{A}e^{-\frac{A}{r}} + C_2 \tag{59}$$ |
| Whereby C1 and C2 are the integration constants. With the following boundary conditions: | Whereby C1 and C2 are the integration constants. With the following boundary conditions: | ||
| - | $$T(r = \infty) = T_m \tag{13}$$ | + | $$T(r = \infty) = T_m \tag{60}$$ |
| - | $$T(r = r_G) = T_{fl} \tag{14}$$ | + | $$T(r = r_G) = T_{fl} \tag{61}$$ |
| It results in the integration constants: | It results in the integration constants: | ||
| - | $$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{15}$$ | + | $$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{62}$$ |
| - | $$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{16}$$ | + | $$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{63}$$ |
| The solution of the differential equation results in: | The solution of the differential equation results in: | ||
| - | $$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{17}$$ | + | $$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{64}$$ |
| Which is equilibrium of heat flows. | Which is equilibrium of heat flows. | ||
| Zeile 417: | Zeile 417: | ||
| On the interface of the sphere, every time period $t \geq t_0$ has to be applied: | On the interface of the sphere, every time period $t \geq t_0$ has to be applied: | ||
| - | $\dot{q}_f|r_G = \dot{q}_s|r_G \tag{17}$ | + | $\dot{q}_f|r_G = \dot{q}_s|r_G \tag{65}$ |
| Wherein $\dot{q}_s$ is the melt-side heat-flow on the interface, and $\dot{q}_f$ is the heat-flow in the sphere on the interface. | Wherein $\dot{q}_s$ is the melt-side heat-flow on the interface, and $\dot{q}_f$ is the heat-flow in the sphere on the interface. | ||
| - | For the heat-flow on the interface | + | The heat flux at the interface |
| - | $\dot{q}_s = -\lambda \frac{\partial T}{\partial r} = \frac{\lambda(T_m - T_{fl})}{1 - \exp\left(\frac{A}{r_G}\right)}\left(-\frac{A}{r^2}\exp\left(\frac{A}{r}\right)\right) \tag{18}$ | + | $\dot{q}_s = -\lambda \frac{\partial T}{\partial r} = \frac{\lambda(T_m - T_{fl})}{1 - \exp\left(\frac{A}{r_G}\right)}\left(-\frac{A}{r^2}\exp\left(\frac{A}{r}\right)\right) \tag{66}$ |
| At the position r = $r_G$, for heat-flow the following is valid: | At the position r = $r_G$, for heat-flow the following is valid: | ||
| - | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{\exp\left(\frac{A}{r_G}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{19}$ | + | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{\exp\left(\frac{A}{r_G}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{67}$ |
| - | Then extend equation (20) with $\frac{\exp(-A/ | + | Then extend equation (68) with $\frac{\exp(-A/ |
| - | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{1}{\exp\left(\frac{-A}{r_G}\right) - 1} \tag{20}$ | + | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{1}{\exp\left(\frac{-A}{r_G}\right) - 1} \tag{68}$ |
| Now substitute the constant A. | Now substitute the constant A. | ||
| - | $A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{21}$ | + | $A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{69}$ |
| With the constant A': | With the constant A': | ||
| - | $A' = \frac{1}{a_s} \frac{\rho_f}{\rho_s} \bar{v} \tag{22}$ | + | $A' = \frac{1}{a_s} \frac{\rho_f}{\rho_s} \bar{v} \tag{70}$ |
| So the heat-flow on the melt-side of the interface results in: | So the heat-flow on the melt-side of the interface results in: | ||
| - | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})A' | + | $\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})A' |
| For the heat-flow in the solid on the interface is valid: | For the heat-flow in the solid on the interface is valid: | ||
| - | $\dot{q}_{f|r=r_G} = \rho_f \bar{v}\Delta h \frac{\partial r_G}{\partial z} \tag{24}$ | + | $\dot{q}_{f|r=r_G} = \rho_f \bar{v}\Delta h \frac{\partial r_G}{\partial z} \tag{72}$ |
| - | Then put equations (23) and (24) into equation 17 to get: | + | Substituting gives: |
| - | $\rho_f \bar{v}\Delta h = \frac{\lambda(T_m - T_{fl})A' | + | $\rho_f \bar{v}\Delta h = \frac{\lambda(T_m - T_{fl})A' |
| Solve this equation with respect to $\frac{\partial r_G}{\partial z}$ to get: | Solve this equation with respect to $\frac{\partial r_G}{\partial z}$ to get: | ||
| - | $r_G \frac{\partial r_G}{\partial z} = -\frac{1}{A' | + | $r_G \frac{\partial r_G}{\partial z} = -\frac{1}{A' |
| - | For the changing of the radius $r_0$ on a length $\Delta z$, through the integration of equation (26), the following is valid: | + | For the changing of the radius $r_0$ on a length $\Delta z$, through the integration of equation (74), the following is valid: |
| - | $r_{G,i+1} = \sqrt{r_{G, | + | $r_{G,i+1} = \sqrt{r_{G, |
| - | include the constant A' in (27) to get: | + | include the constant A' in (75) to get: |
| - | $r_{G,i+1} = \sqrt{r_{G, | + | $r_{G,i+1} = \sqrt{r_{G, |
| - | $r_{0,i}$ is the radius at the beginning of the considered channel section. Equation | + | $r_{0,i}$ is the radius at the beginning of the considered channel section. Equation |
| The solid section gives: | The solid section gives: | ||
| - | $F(r_G) = \frac{N_{p, | + | $F(r_G) = \frac{N_{p, |
| {{ : | {{ : | ||
| Zeile 489: | Zeile 489: | ||
| The heat flow which exists at the interface particle/ | The heat flow which exists at the interface particle/ | ||
| - | $$f_{lh, | + | $$f_{lh, |
| The correction factor is valid in the range 0,2 < d/h <0,9 and is based on an approximation of the dimensionless temperature field. It shows a good match with the CFD results, and compares the middle temperature gradient at the particle surface with finite channel height to the middle temperature gradient with infinite expansion. | The correction factor is valid in the range 0,2 < d/h <0,9 and is based on an approximation of the dimensionless temperature field. It shows a good match with the CFD results, and compares the middle temperature gradient at the particle surface with finite channel height to the middle temperature gradient with infinite expansion. | ||
| Zeile 501: | Zeile 501: | ||
| The influence of the convection on the melt process in twin screw extruders was until here neglected. That is why a factor fk to consider the convective heat transfer is introduced in the modified melting model. This factor is also based on CFD simulations. Based on a single rotating particle surrounded by melt in an unwound twin screw channel different relations of the pellet diameter to the channel height dp/h as well as different Péclet numbers were used. The latter balances the ratio of the convection and the thermal conduction and describes the amount of the convection occurring in the flow. | The influence of the convection on the melt process in twin screw extruders was until here neglected. That is why a factor fk to consider the convective heat transfer is introduced in the modified melting model. This factor is also based on CFD simulations. Based on a single rotating particle surrounded by melt in an unwound twin screw channel different relations of the pellet diameter to the channel height dp/h as well as different Péclet numbers were used. The latter balances the ratio of the convection and the thermal conduction and describes the amount of the convection occurring in the flow. | ||
| - | $Pe = \frac{\rho_m \cdot c_{pm} \cdot v_{0z} \cdot d_p}{\lambda_m}$ | + | $$Pe = \frac{\rho_m \cdot c_{pm} \cdot v_{0z} \cdot d_p}{\lambda_m}\tag{79}$$ |
| The simulation results show that for high convective shares a considerably faster melting is to be expected. Mathematically, | The simulation results show that for high convective shares a considerably faster melting is to be expected. Mathematically, | ||
| - | $f_k = 1 + \frac{(b_1 \cdot \kappa + b_2 \cdot \kappa^2 + b_3 \cdot \kappa^3 + b_4 \cdot \kappa^4) \cdot Pe}{(b_5 \cdot \kappa + b_6 \cdot \kappa^2 + b_7 \cdot \kappa^3 + b_8 \cdot \kappa^4) \cdot \left(\frac{3}{4} \cdot \sqrt{Pe} + b_9\right)}$ | + | $$f_k = 1 + \frac{(b_1 \cdot \kappa + b_2 \cdot \kappa^2 + b_3 \cdot \kappa^3 + b_4 \cdot \kappa^4) \cdot Pe}{(b_5 \cdot \kappa + b_6 \cdot \kappa^2 + b_7 \cdot \kappa^3 + b_8 \cdot \kappa^4) \cdot \left(\frac{3}{4} \cdot \sqrt{Pe} + b_9\right)}\tag{80}$$ |
| with: $\kappa = 1 - e^{-\frac{dp}{h}}$ | with: $\kappa = 1 - e^{-\frac{dp}{h}}$ | ||
| Zeile 511: | Zeile 511: | ||
| and it compares the heat flow from the analytical calculation (without considering the convection) with the heat flow from the simulations. Therefore the following is true: | and it compares the heat flow from the analytical calculation (without considering the convection) with the heat flow from the simulations. Therefore the following is true: | ||
| - | $\dot{q}_r = \left(-\lambda_m \frac{\partial T}{\partial r}\right) \cdot f_k \cdot f_{lh}$ | + | $$\dot{q}_r = \left(-\lambda_m \frac{\partial T}{\partial r}\right) \cdot f_k \cdot f_{lh}\tag{81}$$ |
| The regression constants b1 to b9 can be taken from the following table. | The regression constants b1 to b9 can be taken from the following table. | ||
| Zeile 534: | Zeile 534: | ||
| In the modified melting model the flow in the melt-solid mixture is formed with the help of an adjustment of the viscosity. For this purpose the correction of the power law consistency with the factor $f_Φ$ is in introduced. | In the modified melting model the flow in the melt-solid mixture is formed with the help of an adjustment of the viscosity. For this purpose the correction of the power law consistency with the factor $f_Φ$ is in introduced. | ||
| - | $$K_{sm} = K \cdot f_\phi$$ | + | $$K_{sm} = K \cdot f_\phi\tag{82}$$ |
| - | $$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V, | + | $$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s}\tag{83}$$ |
| The correction factor $f_Φ$ is dependent on the volume content of the solid $Φ_{V,s}$ and compares the viscosity of the fluid-solid mixture $η_{sm}$ with the viscosity of the pure fluid $η_0$. This correction is based on Einstein but for large solid shares the viscosities calculated deviate too much. A further development according to Guth and Simah yields better results. Therefore the following is true for the correction factor: | The correction factor $f_Φ$ is dependent on the volume content of the solid $Φ_{V,s}$ and compares the viscosity of the fluid-solid mixture $η_{sm}$ with the viscosity of the pure fluid $η_0$. This correction is based on Einstein but for large solid shares the viscosities calculated deviate too much. A further development according to Guth and Simah yields better results. Therefore the following is true for the correction factor: | ||
| - | $$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s} + \frac{141}{10} \cdot \phi_{V, | + | $$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s} + \frac{141}{10} \cdot \phi_{V,s}\tag{84}$$ |
| A conversion of the volume content on the mass content yields: | A conversion of the volume content on the mass content yields: | ||
| - | $$f_\phi = 1 + \frac{5}{2}\left(\frac{\phi_{M, | + | $$f_\phi = 1 + \frac{5}{2}\left(\frac{\phi_{M, |
| If the $d_p/ | If the $d_p/ | ||
| - | $$K_{sm} = K \cdot f_\phi \cdot f_{dh}$$ | + | $$K_{sm} = K \cdot f_\phi \cdot f_{dh}\tag{86}$$ |
| - | $$f_{dh} = 1 + \frac{\frac{d_p}{h} \cdot \left(1 + \left(\frac{d_p}{h}\right)^2\right)}{\left(\frac{\rho_s}{\rho_m} - \frac{1 - \Phi_{M, | + | $$f_{dh} = 1 + \frac{\frac{d_p}{h} \cdot \left(1 + \left(\frac{d_p}{h}\right)^2\right)}{\left(\frac{\rho_s}{\rho_m} - \frac{1 - \Phi_{M, |
| Based on a solid particle in a Newtonian melt the flow conditions around this particle are considered in order to determine the correction. After this the viscosity for the flow of an equivalent volume flow of pure melt is determined. The ratio of the viscosity of this consideration and the viscosity of the solid-melt mixture leads to the correction with the factor $f_{dh}$. | Based on a solid particle in a Newtonian melt the flow conditions around this particle are considered in order to determine the correction. After this the viscosity for the flow of an equivalent volume flow of pure melt is determined. The ratio of the viscosity of this consideration and the viscosity of the solid-melt mixture leads to the correction with the factor $f_{dh}$. | ||
| Zeile 562: | Zeile 562: | ||
| * The influence of the convection on the heat flow in radial direction $f_k$. | * The influence of the convection on the heat flow in radial direction $f_k$. | ||
| - | by the assumptions made. | + | The calculation is based on the general energy equation, which, given the assumptions made, simplifies to: |
| + | |||
| + | $$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\dot{q}_r\right)\tag{88}$$ | ||
| With the influence of the correction factors flh and fk the heat flow in radial direction is given by the following equation (cf: [[en: | With the influence of the correction factors flh and fk the heat flow in radial direction is given by the following equation (cf: [[en: | ||
| - | $\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}$ | + | $$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}\tag{89}$$ |
| - | These factors | + | These factors |
| - | $r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot (T_m - T_{flow})}{\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}$ | + | $$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot \left (T_m - T_{flow}\right)} {\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}\tag{90}$$ |
| - | The factors | + | The factors |
| ===== 2D disperse melting model ===== | ===== 2D disperse melting model ===== | ||
| Zeile 590: | Zeile 592: | ||
| The filling degree of the components melt and solid is the decisive criterion for going on with the calculation according to the modified disperse melting model. In this model, the spherical particles in the screw channel after the " | The filling degree of the components melt and solid is the decisive criterion for going on with the calculation according to the modified disperse melting model. In this model, the spherical particles in the screw channel after the " | ||
| - | $$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048$$ | + | $$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048\tag{91}$$ |
| which means that the free volume occupies 25.952 %. As soon as the degree of molten material exceeds the melt rate of 25.952 %, the subsequent melting is calculated by the modified disperse melting model. This modeling takes into account the melting by a successive reduction of the radius due to convective heating in a finite channel geometry. | which means that the free volume occupies 25.952 %. As soon as the degree of molten material exceeds the melt rate of 25.952 %, the subsequent melting is calculated by the modified disperse melting model. This modeling takes into account the melting by a successive reduction of the radius due to convective heating in a finite channel geometry. | ||