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en:grundlagenhandbuch:aufschmelzberechnung [2026/05/24 09:46] – [Particles' Influence on the Flow] neelesten: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 molten film over the solid tailback in the intermeshing area is formed. This procedure is supported through in the practice conventionally carried-out reductions of the short pitch of the screw elements. These are found in front of the geometric plastification zone, through which an extra compression of the solids occurs and thus a better contact with the heated cylinder wall can be achieved. At the same time, granule particles lying on the underside of this layer are heated through convection.+The feed material is held in underneath the hopper 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 continuous molten film over the solid tailback in the intermeshing area is formed. This process is supported in practice by reducing the pitch of the screw elements.. These are found in front of the geometric plastification zone, through which an extra compression of the solids occurs and thus a better contact with the heated cylinder wall can be achieved. At the same time, granule particles lying on the underside of this layer are heated through convection.
  
-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 are further more molten and trans-ported during their transportation through convection and through passing of further intermeshing areas in the already stated condition of dispersion.+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 continue to melt and are transported during their transportation through convection and through passing of further intermeshing areas in the already stated condition of dispersion.
  
-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 touching the heated screw surface.+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 touch the heated screw surface
  
-A condition for the described melting behavior is a sufficiently long solids-conveying section. In practice, inserted screw configurations point out that with a short conveying section, combinations of plastic elements with subsequent dust particles, results in only a partial filling in the plastification zone.+A condition for the described melting behavior is a sufficiently long solids-conveying section. However, the screw configurations used in practice feature a combination of kneading and restrictor elements after a short conveying zone. This results in the plasticising zone being partially filled.
  
 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 solid that holds a significantly larger speed component in the axial direction than the melt, reaches the melt filled area, and melt can penetrate in the cavities of the granules, then the forced conveying will collapse. In front of the location of the first filling up with melt, through a sufficiently large mass throughput, originates one with a granule filled screw area, in which, the granules rotate around the screw like in the figure. 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 solid that holds a significantly larger speed component in the axial direction than the melt, reaches the melt filled area, and melt can penetrate in the cavities of the granules, then the forced conveying will collapse. In front of the location of the first filling up with melt, through a sufficiently large mass throughput, originates one with a granule filled screw area, in which, the granules rotate around the screw like in the figure.
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.
  
-$$\frac{\partial \theta}{\partial \tau} = \frac{1 \delta}{\xi^2 \delta \xi}\left(\xi^2 \frac{\delta \theta}{\delta \xi}\right) \tag{12}$$+$$\frac{\partial\theta}{\partial\tau} = \frac{1}{\xi^2}\frac{\partial}{\partial\xi}\left(\xi^2\frac{\partial\theta}{\partial\xi}\right)\tag{12}$$
  
-whereby these standardization's:+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
  
-$$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ für } \tau \leq 0\\0 \text{ für } \tau > 0\end{cases} \tag{16}$$+$$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ for } \tau \leq 0\\0 \text{ for } \tau > 0\end{cases} \tag{16}$$
  
 as a solution of the average caloric temperature of the particle: as a solution of the average caloric temperature of the particle:
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:grundlagenhandbuch:aufschmelzberechnung:aufschmelzmodell_fuer_dispers_verteilte_fuellstoffe#Calculation_of_the_solid_bed_reduction_along_the_melt_path]]): With the influence of the correction factors flh and fk the heat flow in radial direction is given by the following equation (cf: [[en:grundlagenhandbuch:aufschmelzberechnung:aufschmelzmodell_fuer_dispers_verteilte_fuellstoffe#Calculation_of_the_solid_bed_reduction_along_the_melt_path]]):
  
-$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}$+$$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}\tag{89}$$
  
-These factors are used in the whole calculation process which was introduced in the previous chapter. Therefore the following is true for the change of the particle radius in the interval considered:+These factors govern the entire calculation process described in the previous chapter, so that the change in particle radius over the interval under consideration is given by:
  
-$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot (T_m - T_{flow})}{\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}$+$$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot \left (T_m - T_{flow}\right)} {\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}\tag{90}$$
  
-The factors fdh and fΦ, which reflect the influence of the particles on the flow, are anchored in the calculation of the middle flow velocity $\bar{v}$.+The factors $f_{dh}$ and $f_\Phi$, which reflect the influence of the particles on the flow, are incorporated into the calculation of the mean flow velocity $\bar{v}$.
  
 ===== 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 "deformation zone" are almost closely spaced packed. It is model assumption the disperse melting starts when the voids between the spheres are filled with melt. The packing density of a densest packing is  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 "deformation zone" are almost closely spaced packed. It is model assumption the disperse melting starts when the voids between the spheres are filled with melt. The packing density of a densest packing is 
  
-$$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048$$+$$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048\tag{91}$$
  
 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.