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en:berechnungen:faserlaengenabbau [2025/01/20 10:55] – cschallen:berechnungen:faserlaengenabbau [2025/09/03 12:03] (aktuell) – [Calculation modelling ‘Fiber length distribution’] neelest
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 ======Fiber length degradation====== ======Fiber length degradation======
  
-===== Simplified calculation approach ‘fibre length reduction’ =====+REX/PSI offers two calculation models for fiber degradation.
  
-see also [[en:berechnungen:faserlaengenabbau|]]+===== Simplified calculation approach ‘fiber length reduction’ =====
  
-==== Original calculation approach  ====+->  [[..:grafische_darstellung_der_ergebnisse:faserlaengenabbau|For graphical representation of the fiber length reduction]]
  
-The calculation takes into account the temporal regressive decrease in the average fibre length during shearing processes. This is described by the following equation:+The calculation takes into account the temporal regressive decrease in the average fiber length during shearing processes. This is described by the following equation:
  
 $$\frac{dl}{dt_v}=-c \cdot l^2$$ $$\frac{dl}{dt_v}=-c \cdot l^2$$
Zeile 13: Zeile 13:
 with $$l = \frac{L-L_∞}{L_0-L_∞}, 0 < l ≤ 1$$ with $$l = \frac{L-L_∞}{L_0-L_∞}, 0 < l ≤ 1$$
  
-The solution of Equation 1 and the insertion of the boundary conditions produces the+The solution of the Equation and the insertion of the boundary conditions produces the
 following descriptive equation: following descriptive equation:
  
 $$l(t_V) = \frac{1}{\frac{t_V}{t_0}+1}$$  $$l(t_V) = \frac{1}{\frac{t_V}{t_0}+1}$$ 
  
-where $t_0$ is the time constant at which a halving of the fiber length occurs. The time+Where $t_0$ is the time constant at which a halving of the fiber length occurs. The time
 constant is determined by an energy-related view of the fiber breakage, taking into constant is determined by an energy-related view of the fiber breakage, taking into
 account the flow processes during plasticization. Energy is used to break a fiber. This account the flow processes during plasticization. Energy is used to break a fiber. This
Zeile 29: Zeile 29:
 $$E_{diss} = η \cdot \dot γ^2 \cdot V \cdot t_V$$ $$E_{diss} = η \cdot \dot γ^2 \cdot V \cdot t_V$$
  
-Where $V$ is the melt volume of the area being studied, $η$ the viscosity of the plastic, $\dot γ$ +Where $V$ is the melt volume of the area being studied, $η$ the viscosity of the plastic, $\dot γ$ the shear rate and $t_V$ the dwell time.
-the shear rate and $t_V$ the dwell time.+
  
 The given melt volume can be expressed for Euler's second buckling case as follows, The given melt volume can be expressed for Euler's second buckling case as follows,
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 $$l(t_V) = \frac{κ \cdot Φ \cdot E \cdot ε_B }{t_V \cdot ζ \cdot 2 \cdot η \cdot \dot γ^2 + κ \cdot Φ \cdot E \cdot ε_B}$$  $$l(t_V) = \frac{κ \cdot Φ \cdot E \cdot ε_B }{t_V \cdot ζ \cdot 2 \cdot η \cdot \dot γ^2 + κ \cdot Φ \cdot E \cdot ε_B}$$ 
  
-===== Calculation modelling ‘Fibre length distribution’ =====+===== Calculation modelling ‘Fiber length distribution’ =====
  
-see also +-> [[..:grafische_darstellung_der_ergebnisse:faserlaengenverteilung|For graphical representation of the fiber length distribution]]
  
-===  Required properties of the reinforcing fibres or compound  ===+===  Required properties of the reinforcing fibers or compound  ===
  
   * Tensile strength   * Tensile strength
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   * Modulus of elasticity   * Modulus of elasticity
   * Density   * Density
-  * Fibre weight fraction+  * Fiber weight fraction
  
-=== Results of the fibre breakage calculation ===+=== Results of the fiber breakage calculation ===
  
-  * Number of weighted fibre lengths:+  * Number of weighted fiber lengths:
  
 $$L_n = \frac{∑_in_i \cdot l_i}{∑_i n_i}$$ $$L_n = \frac{∑_in_i \cdot l_i}{∑_i n_i}$$
  
-  * Volume-weighted fibre length: +  * Volume-weighted fiber length: 
  
 $$L_v = \frac{∑_i n_i \cdot {l_i}^2}{∑_i n_i \cdot l_i}$$ $$L_v = \frac{∑_i n_i \cdot {l_i}^2}{∑_i n_i \cdot l_i}$$
  
-  * Frequency distribution of fibre lengths: $l_i$+  * Frequency distribution of fiber lengths: $l_i$
  
 === Procedure for calculating the fraction === === Procedure for calculating the fraction ===
  
-When calculating the fibre length reduction in the plasticising process, a distinction is made between different zones and the damage mechanisms that occur there. +When calculating the fiber length reduction in the plasticising process, a distinction is made between different zones and the damage mechanisms that occur there. 
  
-=== Calculation of fibre breakage at the solid bed - melt film interface ===+=== Calculation of fiber breakage at the solid bed - melt film interface ===
  
-When processing short fibre-reinforced granulates, fibres are partially exposed from the granulate grain at the interface between the solid bed and the melt film. The melt flows around these fibres, which are anchored on one side, and can fail due to this stress. This failure does not occur with long fibre-reinforced materials. Due to the rod shape of the LGF granules and the high l/d ratio, vertically oriented fibres do not occur in the melt film.+When processing short fiber-reinforced granulates, fibers are partially exposed from the granulate grain at the interface between the solid bed and the melt film. The melt flows around these fibers, which are anchored on one side, and can fail due to this stress. This failure does not occur with long fiber-reinforced materials. Due to the rod shape of the LGF granules and the high l/d ratio, vertically oriented fibers do not occur in the melt film.
  
 {{ :berechnungen:abbildung_9_2_10_1_en.svg?600&nolink |}} {{ :berechnungen:abbildung_9_2_10_1_en.svg?600&nolink |}}
  
-A fracture criterion is defined to check whether the fibres clamped at the interface break. This sets the bending stress exerted by the melt on the individual fibre in relation to the tensile strength of the fibre. If the bending stress of the fibre is higher than the tensile strength of the fibre, the fibre is assumed to have failed. +A fracture criterion is defined to check whether the fibers clamped at the interface break. This sets the bending stress exerted by the melt on the individual fiber in relation to the tensile strength of the fiber. If the bending stress of the fiber is higher than the tensile strength of the fiber, the fiber is assumed to have failed. 
  
 Breakage condition: Breakage condition:
Zeile 83: Zeile 82:
 $$\frac{σ_{hydro}}{R_m} > 1$$ $$\frac{σ_{hydro}}{R_m} > 1$$
  
-=== Calculation of fibre breakage in a melt flow ===+=== Calculation of fiber breakage in a melt flow ===
  
-In addition to fibre breakage at the interface, damage occurs in the melt film and in the melt vortex to fibres that move freely in the melt. The fibre breakage that occurs here is calculated as follows:+In addition to fiber breakage at the interface, damage occurs in the melt film and in the melt vortex to fibers that move freely in the melt. The fiber breakage that occurs here is calculated as follows:
  
-It is fundamentally assumed that the failure occurs due to buckling of the fibres under compressive load. The fracture criterion thus results from the critical buckling force calculated according to the second Euler buckling case and the hydrodynamic force $F_i$ exerted by the melt on the fibre. +It is fundamentally assumed that the failure occurs due to buckling of the fibers under compressive load. The fracture criterion thus results from the critical buckling force calculated according to the second Euler buckling case and the hydrodynamic force $F_i$ exerted by the melt on the fiber. 
  
 $$\frac{F_i}{F_{critical buckling force}} = \frac{8 \cdot ζ \cdot η_m \cdot {l_i}^4}{π^3 \cdot E_f \cdot {d_f}^4} (-D:A) > 1$$ $$\frac{F_i}{F_{critical buckling force}} = \frac{8 \cdot ζ \cdot η_m \cdot {l_i}^4}{π^3 \cdot E_f \cdot {d_f}^4} (-D:A) > 1$$
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 $ζ$ = drag coefficient $ζ$ = drag coefficient
  
-The probability of breakage $P_i$ is defined depending on whether the breakage criterion is fulfilled. The fibre interaction coefficient $C_{FB}$ takes into account the influence of the fibre-fibre interaction or the fibre volume fraction on the fibre breakage. In addition to the fibre volume fraction, the fibre interaction coefficient is also dependent on the flow velocity in the channel and the fibre length.+The probability of breakage $P_i$ is defined depending on whether the breakage criterion is fulfilled. The fiber interaction coefficient $C_{FB}$ takes into account the influence of the fiber-fiber interaction or the fiber volume fraction on the fiber breakage. In addition to the fiber volume fraction, the fiber interaction coefficient is also dependent on the flow velocity in the channel and the fiber length.
  
 $$ $$
Zeile 110: Zeile 109:
 $$ $$
  
-The break position of the fibre along the length $l_i$ is described with the assumption of a normal distribution and the definition of a break transition matrix.+The break position of the fiber along the length $l_i$ is described with the assumption of a normal distribution and the definition of a break transition matrix.
  
 $$R_{ji} = normpdf (l_j, \frac{l_i}{2}, Sl_i)$$ $$R_{ji} = normpdf (l_j, \frac{l_i}{2}, Sl_i)$$
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 $$∑_j R_{ji} = 2 P_i$$ $$∑_j R_{ji} = 2 P_i$$
  
-The breakage model used here is based on Phelps' model from 2009 for calculating fibre breakage in the injection mould. However, the model was extended to include the fibre interaction coefficient by modifying the calculation of the probability of breakage. +The breakage model used here is based on Phelps' model from 2009 for calculating fiber breakage in the injection mould. However, the model was extended to include the fiber interaction coefficient by modifying the calculation of the probability of breakage. 
  
-=== Calculation of fibre breakage in a screw channel segment ===+=== Calculation of fiber breakage in a screw channel segment ===
  
-For the various sections of the screw channel, the fibre breakage that occurs and the resulting fibre lengths in the individual sections are calculated and then added using volumetric weighting. +For the various sections of the screw channel, the fiber breakage that occurs and the resulting fiber lengths in the individual sections are calculated and then added using volumetric weighting. 
  
 The following is an overview of the different variants that occur and which breakage calculations are taken into account.  The following is an overview of the different variants that occur and which breakage calculations are taken into account. 
  
-**Calculation of fibre breakage in the mixed solid bed + melt film**+**Calculation of fiber breakage in the mixed solid bed + melt film**
  
-//KGF: Breakage model for calculating fibre breakage at the interface + breakage model for melt flow in the melt film//+//KGF: Breakage model for calculating fiber breakage at the interface + breakage model for melt flow in the melt film//
  
 {{ :berechnungen:abbildung_9_2_10_2_en.svg?600&nolink |}} {{ :berechnungen:abbildung_9_2_10_2_en.svg?600&nolink |}}
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 {{ :berechnungen:abbildung_9_2_10_3.svg?600 |}} {{ :berechnungen:abbildung_9_2_10_3.svg?600 |}}
  
-**Calculation of fibre breakage in the mixed area of solid bed + melt film + melt vortex**+**Calculation of fiber breakage in the mixed area of solid bed + melt film + melt vortex**
  
-//KGF: Fracture model for calculating fibre breakage at the interface + fracture model for melt flow in the melt film + fracture model for melt flow in the melt vortex//+//KGF: Fracture model for calculating fiber breakage at the interface + fracture model for melt flow in the melt film + fracture model for melt flow in the melt vortex//
  
 {{ :berechnungen:abbildung_9_2_10_4_en.svg?600&nolink |}} {{ :berechnungen:abbildung_9_2_10_4_en.svg?600&nolink |}}
Zeile 146: Zeile 145:
 {{ :berechnungen:abbildung_9_2_10_5_en.svg?600&nolink |}} {{ :berechnungen:abbildung_9_2_10_5_en.svg?600&nolink |}}
  
-**Fibre breakage calculation in the pure melt range**+**Fiber breakage calculation in the pure melt range**
    
 //Fracture model for melt flow// //Fracture model for melt flow//
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 ===Further topics=== ===Further topics===
   * [[en:berechnungen:einfache_berechnung|]]   * [[en:berechnungen:einfache_berechnung|]]
 +  * [[en:berechnungen:prozess_iterieren]]
   * [[en:berechnungen:durchsatz|]]   * [[en:berechnungen:durchsatz|]]
   * [[en:berechnungen:druckverlauf|]]   * [[en:berechnungen:druckverlauf|]]
Zeile 159: Zeile 159:
   * [[en:berechnungen:temperaturverlauf|]]   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]   * [[en:berechnungen:leistung_und_schubspannungen|]]
 +  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]   * [[en:berechnungen:verweilzeitverteilung|]]