Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- en:grundlagenhandbuch:werkzeug [2026/02/08 20:47] – [Distributor] neelest
+++ en:grundlagenhandbuch:werkzeug [2026/07/09 11:04] (aktuell) – [Die Module] neelest
@@ Zeile 4: / Zeile 4: @@
 The die module is divided into two considerations of dies:
-  * die-tool
+  * [[en:grundlagenhandbuch:werkzeug:grossflaechenschmelzefilter#die|die-tool]]
-  * large-area melt filter
+  * [[.:werkzeug:grossflaechenschmelzefilter|large-area melt filter]]
 Both tools are used to calculate the decrease in pressure and both can indicate the pressure at the screw tip during a SIGMA-simulation. The die-tool and the large-area melt filter can be intercoupled as well as used separately during a simulation.
@@ Zeile 11: / Zeile 11: @@
 The basis of calculation in the case of the die-tool is based on analytical methods and in the case of the large-area melt filter on the network theory.
-===== Large-area Melt Filter =====
-While producing high-quality extrusion products like fibers and films serious demands are made on the plastic filtration. Here the filter has to ensure that even minute impurities are completely removed from the melt. In storing and handling of the granule a contamination of the melt with small particles like metal abrasion or dust etc. can not be ruled out. These result in the later product showing unacceptable losses of quality or even torn fibers or films.
-With filtering of delicate plastics or with high quality requirements of the filter rate the residence time and the residence time distribution have to be kept as low as possible. In addition, only filter arrangements which use large filter areas in small housing volume can be considered [Hen82]. These requirements can only be fulfilled by a concentric arranging of the filter elements in form of a discus (Disc-Filter). The filter elements are here stretched on perforated supports. Those allow for the filtered melt to flow to the filter housing's exit.
-==== Geometry and Designations of the Melt Filters ====
-The designations shown in the following illustration are used to subdivide the large-area melt filter into sections.
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_werkzeug_001.png?nolink |}}
-^ No. ^ Designation ^ Short term ^
-| 1 | Melt inlet, feed pipe | Inlet |
-| 2 | Distributor | Distributor |
-| 3 | Pressure disc under the filter disc stack | Pressure disc |
-| 4 | Torpedo cone in the central pipe | Torpedo |
-| 5 | Collecting pipe or mandrel | Central pipe |
-| 6 | Cross-hole into the central pipe | Cross-hole |
-| 7 | Melt outlet, outlet pipe | Outlet |
-Inside the single sections the geometrical sizes (diameter and lengths) have to be indicated separately.
-The combination of melt inlet and distributor is summed up as inlet in the input dialog. The pressure disc, the torpedo, the central pipe and the cross-hole are combined as the middle section. The melt outlet remains the outlet.
-==== Calculation Model ====
-The analytical calculation of the large-area melt filter is realized with the help of the network method. At first the large-area melt filter has to be replaced by a suitable rheological model. This is achieved by dissection into rheologically single segments. Afterwards these analytically recordable flow resistances are linked.
-In this chapter the simplified assumptions are summed up and the boundary conditions are defined. This is required for the calculation.
-=== Conditions for Calculation ===
-First of all the following basic assumptions are true:
-  * stationary flow (even and independent of time)
-  * laminar flow (no turbulent effects)
-  * neglect of elastic effects
-  * no thixotropic and rheopectic effects (no time-dependent viscosity)
-=== Boundary Conditions at the Melt Filter (general) ===
-In addition, the following assumptions are made for the whole filter:
-  * Viewing of the filter disc stack as a porous, thick-walled pipe
-  * No inlet and outlet pressure losses
-  * Constant density, independent of pressure and time
-  * Viscosity independent of pressure
-=== Simplification of the Single Element of the Filter ===
-The following simplifications are made inside the single resistance element in order to allow for an analytical calculation of the elements:
-  * constant geometry, for example conical segments are replaced by cylindrical ones
-  * constant mass flow; drain and inflow only at the element nodes
-  * constant viscosity $\eta = f(\dot{\gamma}_i) = const.$
-As neither geometry nor mass flow change within one segment the shear rate remains constant. Because of a likewise nearly constant temperature changes in viscosity can be neglected locally.
-=== Dissection into Elements with the Same Boundary Condition ===
-The modelling of the filter calls for a dissection into analytically describable elements. The interlinking of these elements is described below.
-For this purpose the filter is divided into a number of "i" segments. This means that the housing gap elements and central pipe elements have the same length and the filter elements have the same height (cf. figure).
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_werkzeug_002.png?nolink |}}
-==== Feed Pipe ====
-The feed pipe is a single cylindrical pipe; the pressure loss is calculated with the formula:
-$$\Delta p = \frac{8 \cdot \eta \cdot \dot{V} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi} = \frac{8 \cdot \eta \cdot \dot{m} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi \cdot \rho_m}$$
-For the representative shear rate (cf. calculation of the die) the following is true:
-$$\overline{\dot{\gamma}}_{pseudoplastic} = \overline{\dot{\gamma}}_{newtonian} \cdot 0,815 = \frac{4 \cdot \dot{m}}{R^3 \cdot \pi \cdot \rho_m} \cdot 0,815$$
-==== Distributor ====
-The distributor consists of three segments (cone frustum, pipe and annular slit) that are connected in series. The pressure loss in the distributor results from adding up the pressure losses in the three segments.
-For the pressure losses and the shear rates the following relations are true:
-**Pipe:**
-$$\Delta p = \frac{8 \cdot \eta \cdot \dot{V} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \eta} = \frac{8 \cdot \eta \cdot \dot{m} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \eta \cdot \rho_m}$$
-$$\overline{\dot{\gamma}}_{pseudoplastic} = \overline{\dot{\gamma}}_{newtonian} \cdot 0,815 = \frac{4 \cdot \dot{m}}{R^3 \cdot \pi \cdot \rho_m} \cdot 0,815$$
-**Annular slit:**
-$$\Delta p = \frac{12 \cdot \eta \cdot \dot{V} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3} = \frac{12 \cdot \eta \cdot \dot{m} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3 \cdot \rho_m}$$
-$$\overline{\dot{\gamma}}_{pseudoplastic} = \frac{\dot{V}}{(R_A^2 - R_I^2) \cdot \bar{R}} ; \text{with } \bar{R} = R_A \left[1 + k^2 + \frac{1 - k^2}{\ln(k)}\right]^{1/2} , k = \frac{R_I}{R_A}$$
-**Cone frustum:**
-$$\Delta p = \frac{8 \cdot \eta \cdot \dot{V} \cdot L}{R^4 \cdot \pi} \cdot \frac{1 - \left(\frac{R_2}{R_1}\right)}{3 \cdot \left(\frac{R_1}{R_2} - 1\right)} = \frac{\eta \cdot \dot{m} \cdot L}{\left(\frac{\bar{D}}{2}\right)^4 \cdot \pi \cdot \rho_m} \cdot \frac{1 - \left(\frac{D_2}{D_1}\right)}{3 \cdot \left(\frac{D_1}{D_2} - 1\right)}$$
-$$\overline{\dot{\gamma}}_{pseudoplastic} = \frac{4 \cdot \dot{m}}{\left(\frac{\bar{D}}{2}\right)^3 \cdot \pi \cdot \rho_m} \cdot \frac{\left[1 - \left(\frac{D_2}{D_1}\right)\right]^{3/4}}{\left[3 \cdot \left(\frac{D_1}{D_2} - 1\right)\right]} \cdot 0,815$$
-Index 1 refers to the input cross-section and index 2 refers to the output cross-section.
-==== Housing Gap ====
-The gap between housing and outer surface of the filter element can in sections be viewed as an annular slit segment. In this connection it is important that the mass flow decreases gradually around each filter flow. The following relations for the pressure loss are true:
-$$\Delta p_i = \frac{12 \cdot \eta \cdot \dot{V} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3} = \frac{12 \cdot \eta \cdot \dot{m} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3 \cdot \rho_m}$$
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \frac{\dot{V}}{(R_A^2 - R_I^2) \cdot \bar{R}} ; \text{mit } \bar{R} = R_A \left[1 + k^2 + \frac{1 - k^2}{\ln(k)}\right]^{1/2} , k = \frac{R_I}{R_A}$$
-$$\dot{V}_i \neq \dot{V}_{i-1} ; \dot{m}_i \neq \dot{m}_{i-1}$$
-==== Filter Element ====
-The pressure loss in the filter element itself is calculated with the help of a standardized filter parameter, the specific pressure loss.
-$$\Delta p_i^F = dp^F \cdot \frac{\dot{V} \cdot \eta \cdot n_i}{nd} , \text{mit } [dp^F] = \frac{bar \cdot h}{kg \cdot Pas}$$
-The calculation of this filter parameter is carried out according to the manufacturers' instructions. Pall Corporation indicates pressure factors (K) in tabular form. The pressure loss of a filter stack (not of the whole melt filter) is calculated in accordance with the following equation:
-$$\Delta p_{Pall}[psi] = K \cdot \frac{\eta[poise]}{1000} \cdot \frac{\dot{V}[lbs/h]}{nd} \cdot \frac{1}{A[ft^2]} \cdot \frac{1}{\rho[g/cm^3]}$$
-A similar procedure is recommended by Fairey Industrial Ceramics Ltd. In this case, the pressure loss is given by:
-$$\Delta p_{Fairy}[bar] = K \cdot \frac{\eta[poise]}{3000} \cdot \frac{\dot{V}[kg/h]}{1724,4} \cdot \frac{1}{\rho[g/cm^3]} \cdot \frac{60}{nd}$$
-The pressure loss for a specific filter stack calculated in accordance with these recommendations can now be standardized for the specific pressure loss per disc independent of the material:
-$$dp^F = \frac{\Delta p_{manufacturer}}{\dot{V} \cdot \eta} \cdot nd$$
-After calculating the pressure loss according to the instructions of another manufacturer the standardization according to this latter equation is then carried out analogously.
-In order to calculate the viscosity in accordance with [Pal07] a constant shear rate of $\dot{\gamma} = 20s^{-1}$ is assumed.
-==== Cross-Hole ====
-The cross-holes are pipe sections in which the melt flows simultaneously through every filter section. In this process the filter flow is divided into nq/i cross-holes. The pressure loss of the number of cross-holes in segment "i" is therefore given by:
-$$\Delta p = \frac{8 \cdot \eta \cdot \frac{i}{nq} \cdot \dot{V} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi} = \frac{8 \cdot \eta \cdot \frac{i}{nq} \cdot \dot{m} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi \cdot \rho_m}$$
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \tilde{\dot{\gamma}}_{newtonian} \cdot 0,815 = \frac{4 \cdot \frac{i}{nq} \cdot \dot{m}}{R^3 \cdot \pi \cdot \rho_m} \cdot 0,815$$
-==== Central Pipe ====
-Above the torpedo the central pipe can in sections be calculated as a single pipe flow in accordance with:
-$$\Delta p = \frac{8 \cdot \eta \cdot \dot{V} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi} = \frac{8 \cdot \eta \cdot \dot{m} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi \cdot \rho_m}$$
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \tilde{\dot{\gamma}}_{newtonian} \cdot 0,815 = \frac{4 \cdot \dot{m}}{R^3 \cdot \pi \cdot \rho_m} \cdot 0,815$$
-In the lower part an annular slit flow around the torpedo cone is calculated in accordance with the formula
-$$\Delta p = \frac{12 \cdot \eta \cdot \dot{V} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3} = \frac{12 \cdot \eta \cdot \dot{m} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3 \cdot \rho_m}$$
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \frac{\dot{V}}{(R_A^2 - R_I^2) \cdot \bar{R}} ; \text{with } \bar{R} = R_A \left[1 + k^2 + \frac{1 - k^2}{\ln(k)}\right]^{1/2} , k = \frac{R_I}{R_A}$$
-In this case a constant gap height within each segment is assumed. In both cases the calculation is each time carried out with a gradually increasing mass flow.
-==== Outlet Pipe ====
-The calculation of the outlet pipe is carried out analogously to that of the feed pipe.
-===== Assessment of the Entire Melt Filter =====
-The pressure loss of the entire melt filter results from adding up the single pressure losses along a current path (cf. network method). In accordance with Kirchoff's loop law this total pressure loss has to be constant and not depended on the path examined. The system of equations of the entire melt filter that has to be solved can be described as follows:
-$$\Delta p^{ges}(i) = \Delta p^E + \Delta p^V + \sum_{j=0}^{i-1} \Delta p_j^G + \Delta p_i^F + \Delta p_i^Q + \sum_{j=1}^{n_i} \Delta p_j^R + \Delta p^A = const \forall i$$
-Here the variable "i" indicates the height at which the path examined branches away from the longitudinal housing gap flow into the radial filter element flow. The terms can here be related to the geometrical resistances of the rheological model as follows:
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_werkzeug_003.png?nolink |}}
-This system of equations for the entire melt filter is only depended on the single filter currents.
-  * By means of dividing the filter into a number of segments "i" the same number of possible current paths – a system of "i" equations - arises.
-  * These equations contain as the unknown the number of "i" filter capacities as well as the total pressure loss. This means that the total number of the unknowns in the system of equations is "i+1".
-  * This system of equations is under-determined and therefore approximately solvable.
-  * In the numeric calculation the total pressure loss is iteratively minimized. For this purpose the programme determines the total pressure loss and changes the filter currents marginally. Subsequently, it again calculates the pressure loss and compares the results. In this way the "optimum" flow distribution with the least pressure loss is determined step by step.
-===== Die =====
-For the calculation of the die the decrease in pressure is calculated in dependence of throughput and geometry. The following basic geometries a die can be composed of are available:
-  * a pipe flow,
-  * a gap flow and
-  * an annular slit flow.
-For a point balance these basic geometries are calculated separately and subsequently superimposed.
-With conical pipe, gap and annular slit geometries the single elements are subdivided lengthwise into intervals. The geometrical sizes for each interval are averaged. Therefore analytical equations for constant geometries can be used for the calculation.
-Because of the viscosities, geometries, and velocities existing in plastics processing very low Reynolds numbers always arise as a result. This means that always a laminar flow is formed. Therefore turbulent effects are not considered.
-==== Single Pipe Flow ====
-Consequently, for the pressure loss of a laminar flow through a pipe with the diameter D and the length L the following is true:
-$$\Delta p = \frac{\eta \cdot \dot{V} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi} = \frac{\eta \cdot \dot{m} \cdot L}{\left(\frac{D}{2}\right)^4 \cdot \pi \cdot \rho_m}$$
-==== General Gap Flow ====
-For the pressure loss of a laminar flow through a rectangular channel with the width B, the height H and the length L the following is true:
-$$\Delta p = \frac{12 \cdot \eta \cdot \dot{V} \cdot L}{B \cdot H^3} = \frac{12 \cdot \eta \cdot \dot{m} \cdot L}{B \cdot H^3 \cdot \rho_m}$$
-==== Specific Gap Flow, Uncoiled Annular Slit ====
-For the pressure loss of a laminar flow through an annular slit with the external diameter DA, the internal diameter DI and the length L the following is true:
-$$\Delta p = \frac{12 \cdot \eta \cdot \dot{V} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3} = \frac{12 \cdot \eta \cdot \dot{m} \cdot L}{\frac{D_A + D_I}{2} \cdot \pi \cdot \left(\frac{D_A - D_I}{2}\right)^3 \cdot \rho_m}$$
-==== Representative Sizes ====
-As the above mentioned methods of calculation are only valid for Newtonian liquids, the method of representative sizes (viscosity and shear rate) is used. This procedure is based on the finding that there is a position in the flow channel where the shear rate of a Newtonian liquid is equal to that of a non-Newtonian liquid. When the ratio of the shear rate in the Newtonian as well as in the non-Newtonian case is known the representative viscosity can be determined.
-For the determination of the representative shear rates the following relations apply:
-==== Single Pipe Flow ====
-The representative distance from the middle of the channel in the circular tube amounts to a constant 0,815 according to [Mic91]. The range of the flow index which is important in practice shows minimal deviations (<1,8%).
-Therefore the representative shear rate is calculated as follows
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \tilde{\dot{\gamma}}_{newtonian} \cdot 0,815 = \frac{4 \cdot \dot{m}}{R^3 \cdot \pi \cdot \rho_m} \cdot 0,815$$
-==== General Gap Flow ====
-With a gap flow with the width B, the height H and the length L the following is true for the relation of the shear rates:
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \tilde{\dot{\gamma}}_{newtonian} \cdot 0,772 = \frac{6 \cdot \dot{m}}{B \cdot H^3 \cdot \rho_m} \cdot 0,772$$
-==== Consideration of the Annular Slit ====
-On examining the shear rate in annular slits with the outside radius RA, the inside radius RI and the length L, the ratio of the shear rates is approximated with the help of a function of the radius. The following is true:
-$$\tilde{\dot{\gamma}}_{pseudoplastic} = \frac{\dot{V}}{(R_A^2 - R_I^2) \cdot \bar{R}} ; \text{mit } \bar{R} = R_A \left[1 + k^2 + \frac{1 - k^2}{\ln(k)}\right]^{1/2} , k = \frac{R_I}{R_A}$$
-==== Representative Viscosity Calculation ====
-With the knowledge of the shear rates the viscosity for the pressure calculation can be determined. In this case the power law is chosen.
-In calculating the viscosity it is assumed that there are isothermal relations in the die. The starting melt temperature is determined through a precalculation of the screw. In this way the melt temperature at the screw tip is used in the calculation of the die.
-===== References =====
-[Mic91] Michaeli, W.: "Extrusionswerkzeuge für Kunststoffe und Kautschuk" Carl Hanser Verlag, München, 1991