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en:grundlagenhandbuch:drehmoment_und_antriebsleistung [2026/05/24 17:49] – [Torque calculation] neelesten:grundlagenhandbuch:drehmoment_und_antriebsleistung [2026/06/16 11:11] (aktuell) – [Enthalpy model with heat flow] neelest
Zeile 48: Zeile 48:
 The estimation of the power consumption must differ in the 3 function sections in the figure on the barrel wall and in the melting section. The estimation of the power consumption must differ in the 3 function sections in the figure on the barrel wall and in the melting section.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_001.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_001.svg?nolink&600 |}}
  
 **Figure:** Model for the power calculation. **Figure:** Model for the power calculation.
Zeile 124: Zeile 124:
 The starting point for the energetic examination is an energy balance at an extruder (see figure). The starting point for the energetic examination is an energy balance at an extruder (see figure).
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_002.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_002.svg?nolink&600 |}}
  
 **Figure:** Energy balance at a co-rotating twin screw extruder. **Figure:** Energy balance at a co-rotating twin screw extruder.
Zeile 136: Zeile 136:
 $$P = P_{Diss} + P_{Vol} = P_{Diss} + \Delta p \cdot \dot{V} \tag{24}$$ $$P = P_{Diss} + P_{Vol} = P_{Diss} + \Delta p \cdot \dot{V} \tag{24}$$
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_003.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_003.svg?nolink&600 |}}
  
 **Figure:** Power fractions in dependence on the dimensionless volume throughput. **Figure:** Power fractions in dependence on the dimensionless volume throughput.
Zeile 177: Zeile 177:
 The calculation is performed on the basis of the models introduced in chapter 3 of [[en:grundlagenhandbuch:drehmoment_und_antriebsleistung#references |[Kre04]]]. In the figure the variations predicted by means of the model are contrasted with the results of the flow simulation. A sufficient description of the volumetric strain energy can be recognized for the whole area. The calculation is performed on the basis of the models introduced in chapter 3 of [[en:grundlagenhandbuch:drehmoment_und_antriebsleistung#references |[Kre04]]]. In the figure the variations predicted by means of the model are contrasted with the results of the flow simulation. A sufficient description of the volumetric strain energy can be recognized for the whole area.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_004.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_004.svg?nolink&600 |}}
  
 **Figure:** Dimensionless pumping capacity in dependence of the dimensionless volume throughput. **Figure:** Dimensionless pumping capacity in dependence of the dimensionless volume throughput.
Zeile 220: Zeile 220:
 The figure displays the comparative relation of the dissipated energies calculated with the Polyflow and the model predictions for a twin screw channel. The figure displays the comparative relation of the dissipated energies calculated with the Polyflow and the model predictions for a twin screw channel.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_005.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_005.svg?nolink&600 |}}
  
 **Figure:** Comparison of the model predictions and the dissipated energies calculated with Polyflow for a twin screw channel. **Figure:** Comparison of the model predictions and the dissipated energies calculated with Polyflow for a twin screw channel.
Zeile 240: Zeile 240:
 For the verification of the model experiments were performed with a laboratory extruder. Silicon oil of the Baysilone 50.000 type was utilized as experimental medium. For the measurement of the power the drive unit of the extruder was suspended in an oscillating way and the section modulus was measured by means of a spring-balance, which was attached to a lever arm. Deriving from the measurement of the section modulus and the screw speed as well as the knowledge of the transmission gear ratio the drive power could thus be determined. The figure displays a comparison of the measured dimensionless drive power and the dimensionless drive power calculated with the model for closely intermeshing thread-mixing elements. The accordance is good, particularly by considering the simplifications made in the modeling. For the verification of the model experiments were performed with a laboratory extruder. Silicon oil of the Baysilone 50.000 type was utilized as experimental medium. For the measurement of the power the drive unit of the extruder was suspended in an oscillating way and the section modulus was measured by means of a spring-balance, which was attached to a lever arm. Deriving from the measurement of the section modulus and the screw speed as well as the knowledge of the transmission gear ratio the drive power could thus be determined. The figure displays a comparison of the measured dimensionless drive power and the dimensionless drive power calculated with the model for closely intermeshing thread-mixing elements. The accordance is good, particularly by considering the simplifications made in the modeling.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_006.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_006.svg?nolink&600 |}}
  
 **Figure:** Comparison of the measured and the calculated dimensionless drive power for closely intermeshing thread-mixing elements. **Figure:** Comparison of the measured and the calculated dimensionless drive power for closely intermeshing thread-mixing elements.
Zeile 258: Zeile 258:
 $$P_{Diss.channel.tf} = f \cdot P_{Diss.channel.vf} \tag{37}$$ $$P_{Diss.channel.tf} = f \cdot P_{Diss.channel.vf} \tag{37}$$
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_007.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_007.svg?nolink&600 |}}
  
 **Figure:** Boundary conditions for the simplified simulation of a partly filled screw section. **Figure:** Boundary conditions for the simplified simulation of a partly filled screw section.
Zeile 264: Zeile 264:
 For the verification of this equation exemplary finite element simulations were performed with the boundary conditions as illustrated in the figure. The starting point of the simulations was a fully loaded rectangular channel with a ratio of channel height to channel width of b/h=40. For the exemplary simulations of party filled channels the depth of the reference channel is reduced. Additionally, the boundary conditions are then also changing for the simulation at the rear face of the flight, which represents the open area. In contrast to the reference computation at this area non-negligible velocities are defined; it is predetermined that at this area tangential forces do not emerge. In all cases the dissipated energy and the shear rate were calculated. For the verification of this equation exemplary finite element simulations were performed with the boundary conditions as illustrated in the figure. The starting point of the simulations was a fully loaded rectangular channel with a ratio of channel height to channel width of b/h=40. For the exemplary simulations of party filled channels the depth of the reference channel is reduced. Additionally, the boundary conditions are then also changing for the simulation at the rear face of the flight, which represents the open area. In contrast to the reference computation at this area non-negligible velocities are defined; it is predetermined that at this area tangential forces do not emerge. In all cases the dissipated energy and the shear rate were calculated.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_008.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_008.svg?nolink&600 |}}
  
 **Figure:** Dependence of the relative dissipated energy and the relative shear rate on the loading content. **Figure:** Dependence of the relative dissipated energy and the relative shear rate on the loading content.
Zeile 317: Zeile 317:
 First of all, a specific enthalpy curve of a semi-crystalline plastic is considered (Picture 1). At the crystallite melting temperature, the plastic has a corresponding specific enthalpy depending on the type. This enthalpy is divided in PAM into $\Delta h_f$ solid enthalpy (enthalpy change to increase the solid temperature) and $\Delta h_a$ melting enthalpy (enthalpy required to dissolve the crystalline regions). First of all, a specific enthalpy curve of a semi-crystalline plastic is considered (Picture 1). At the crystallite melting temperature, the plastic has a corresponding specific enthalpy depending on the type. This enthalpy is divided in PAM into $\Delta h_f$ solid enthalpy (enthalpy change to increase the solid temperature) and $\Delta h_a$ melting enthalpy (enthalpy required to dissolve the crystalline regions).
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_009.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_009.png?nolink&600 |}}
  
 **Picture 1:** Specific enthalpy curve of a partially crystalline polymer **Picture 1:** Specific enthalpy curve of a partially crystalline polymer
  
-The specific enthalpy change at a supporting point compared to the filling point of the material results from the multiplication of the solid enthalpy by a percentage factor which is dependent on the filling temperature, the current solid temperature and the crystallite melting temperature. The factor indicates to what percentage of the crystallite melting temperature the solid was heated, starting from the starting temperature of the material. The melting enthalpy is initially not taken into account in the solids conveying zone. The reason is shown in figure 1. For the determination of the enthalpy increase must be referred to the black imaginary straight lines. If the melting enthalpy is taken into account, the straight line would have a too large gradient at low temperatures, resulting in too large a deviation. In the solids conveying range, however, the temperature increase lies precisely in this small value range. Formally expressed, the specific enthalpy change at a support point in the solids conveying range compared to the filling point is given by equation 0-1+The specific enthalpy change at a supporting point compared to the filling point of the material results from the multiplication of the solid enthalpy by a percentage factor which is dependent on the filling temperature, the current solid temperature and the crystallite melting temperature. The factor indicates to what percentage of the crystallite melting temperature the solid was heated, starting from the starting temperature of the material. The melting enthalpy is initially not taken into account in the solids conveying zone. The reason is shown in figure 1. For the determination of the enthalpy increase must be referred to the black imaginary straight lines. If the melting enthalpy is taken into account, the straight line would have a too large gradient at low temperatures, resulting in too large a deviation. In the solids conveying range, however, the temperature increase lies precisely in this small value range. Formally expressed, the specific enthalpy change at a support point in the solids conveying range compared to the filling point is given by equation 44:
  
-$$\Delta h_{FF} = \Delta h_f \cdot \frac{T_{FS} - T_E}{T_K - 0}\tag{Equation 0-1}$$ +$$\Delta h_{FF} = \Delta h_f \cdot \frac{T_{FS} - T_E}{T_K - 0}\tag{44}$$ 
  
-For the driving power of an element area the difference of the specific enthalpy change between two supporting points is calculated with equation 0-2.+For the driving power of an element area the difference of the specific enthalpy change between two supporting points is calculated with equation 45.
  
-$$\Delta \Delta h_{FF} = \Delta h_{FF_n} - \Delta h_{FF_{n-1}}\tag{Equation 0-2}$$ +$$\Delta \Delta h_{FF} = \Delta h_{FF_n} - \Delta h_{FF_{n-1}}\tag{45}$$ 
  
-Using the calculated specific enthalpy difference, the pressure difference and the introduced heat flow, the required drive power required between two support points in the solids transport zone can be calculated from this (equation 0-3).+Using the calculated specific enthalpy difference, the pressure difference and the introduced heat flow, the required drive power required between two support points in the solids transport zone can be calculated from this (46).
  
-$$P_{D/FF} = \dot{m} \ast (\Delta \Delta h_{FF} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{Equation 0-1}$$ +$$P_{D/FF} = \dot{m} \ast (\Delta \Delta h_{FF} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{46}$$ 
  
 ==== Melting zone and melt conveying zone ==== ==== Melting zone and melt conveying zone ====
Zeile 337: Zeile 337:
 The melting zone and the melt conveying zone differ from the solids conveying zone in that the material is two-phase. Consequently, a separate consideration of the specific enthalpy change must be carried out for the solids bed and the melt zone. The melting zone and the melt conveying zone differ from the solids conveying zone in that the material is two-phase. Consequently, a separate consideration of the specific enthalpy change must be carried out for the solids bed and the melt zone.
  
-For the specific enthalpy change of a pure heating of the melt, the following applies in relation to the crystallite melting temperature through the integration of the specific heat capacity via the temperature equation 0-4.+For the specific enthalpy change of a pure heating of the melt, the following applies in relation to the crystallite melting temperature through the integration of the specific heat capacity via the temperature equation 47.
  
-$$\Delta h_{RS} = cp_0 \ast (T_M - T_K) + \frac{m_{cp}}{2} \ast (T_M^2 - T_K^2)\tag{Equation 0-4}$$ +$$\Delta h_{RS} = cp_0 \ast (T_M - T_K) + \frac{m_{cp}}{2} \ast (T_M^2 - T_K^2)\tag{47}$$ 
  
-In addition to the enthalpy, energy was already introduced into the molten material by heating the melt in order to heat the solid up to the crystallite melting temperature. Formally, the specific enthalpy introduced up to the melting point can be determined according to equation 0-1 with equation 0-5.+In addition to the enthalpy, energy was already introduced into the molten material by heating the melt in order to heat the solid up to the crystallite melting temperature. Formally, the specific enthalpy introduced up to the melting point can be determined according to equation 44 with equation 48.
  
-$$\Delta h_{FA} = \Delta h_a + \Delta h_f \ast \frac{T_K - T_E}{T_K - 0}\tag{Equation 0-5}$$ +$$\Delta h_{FA} = \Delta h_a + \Delta h_f \ast \frac{T_K - T_E}{T_K - 0}\tag{48}$$ 
  
-From the addition of the equations 0-4 and 0-5, equation 0-6 results for the specific enthalpy increase from the starting point of the filling temperature to the current melt temperature for the melt range:+From the addition of the equations 47 and 48, equation 49 results for the specific enthalpy increase from the starting point of the filling temperature to the current melt temperature for the melt range:
  
-$$\Delta h_S = \Delta h_a + \Delta h_f \ast \frac{T_K - T_E}{T_K - 0} + cp_0 \ast (T_M - T_K) + \frac{cp_m}{2} \ast (T_M^2 - T_K^2)\tag{Equation 0-6}$$ +$$\Delta h_S = \Delta h_a + \Delta h_f \ast \frac{T_K - T_E}{T_K - 0} + cp_0 \ast (T_M - T_K) + \frac{cp_m}{2} \ast (T_M^2 - T_K^2)\tag{49}$$ 
  
-The solid bed in the melting zone, on the other hand, undergoes a different specific enthalpy increase. This can be determined with equation 0-7, following equation 0-1 in the solids conveying area.+The solid bed in the melting zone, on the other hand, undergoes a different specific enthalpy increase. This can be determined with equation 44, following equation 50 in the solids conveying area.
  
-$$\Delta h_{FA} = \Delta h_f \ast \frac{T_{FS} - T_E}{T_K - 0}\tag{Equation 0-7}$$ +$$\Delta h_{FA} = \Delta h_f \ast \frac{T_{FS} - T_E}{T_K - 0}\tag{50}$$ 
  
 The enthalpy increases of the solid bed and the melt zone must now be weighted and added up on the basis of the degree of melting in order to obtain the total enthalpy change at a supporting point in the melting zone and the melt conveying zone. For weighting, the total mass flow is multiplied by the respective percentage of the solid or melt. By adding the two specific enthalpy changes, the total enthalpy increase at one support point can be calculated. The differentiation between melting zone and melt area is made by the degree of melting. This is calculated in advance in SIGMA and can also be displayed in visual form. The enthalpy increases of the solid bed and the melt zone must now be weighted and added up on the basis of the degree of melting in order to obtain the total enthalpy change at a supporting point in the melting zone and the melt conveying zone. For weighting, the total mass flow is multiplied by the respective percentage of the solid or melt. By adding the two specific enthalpy changes, the total enthalpy increase at one support point can be calculated. The differentiation between melting zone and melt area is made by the degree of melting. This is calculated in advance in SIGMA and can also be displayed in visual form.
  
-In the pure melt conveying zone, for example, the melting degree is equal to one, so that the second term of equation 0-8 is omitted.+In the pure melt conveying zone, for example, the melting degree is equal to one, so that the second term of equation 51 is omitted.
  
-$$\Delta h_{AS/S} = m_{asv} \ast \Delta h_S + (1 - m_{asv}) \ast \Delta h_{FA}\tag{Equation 0-8}$$ +$$\Delta h_{AS/S} = m_{asv} \ast \Delta h_S + (1 - m_{asv}) \ast \Delta h_{FA}\tag{51}$$ 
  
-The required driving power of an element area can be determined with the available results via the difference of the specific enthalpy increase of two supporting points (equation 0-8) via equation 0-9.+The required driving power of an element area can be determined with the available results via the difference of the specific enthalpy increase of two supporting points (equation 51) via equation 52.
  
-$$\Delta \Delta h_{AS/S} = \Delta h_{AS/S_n} - \Delta h_{AS/S_{n-1}}\tag{Equation 0-9}$$ +$$\Delta \Delta h_{AS/S} = \Delta h_{AS/S_n} - \Delta h_{AS/S_{n-1}}\tag{52}$$ 
  
-$$P_{D_{AS/S}} = \dot{m} \ast (\Delta \Delta h_{AS/S} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{Equation 0-10}$$ +$$P_{D_{AS/S}} = \dot{m} \ast (\Delta \Delta h_{AS/S} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{53}$$ 
  
 ==== Power calculation of a single-stage compounding process ==== ==== Power calculation of a single-stage compounding process ====
Zeile 369: Zeile 369:
 In the single-stage compounding process, two polymers are metered into the hopper and then compounded. This means that the components are dosed in a certain ratio in the hopper and simultaneously plasticized. As a result, both materials have the same temperature at one point along the extrusion process. However, the polymers differ in their melting behaviour. At the same temperature, both materials also have different enthalpy levels. The consequence is that the components mixed in the hopper cannot be seen as a unit, but that the enthalpy levels must be considered separately and then added, weighted, to the corresponding mass flow. The advantage of this method is that the model is not only applicable for two polymers, but for any number of them. In addition, the basic model already modelled can be used and only a weighting and addition of both enthalpy levels is required. In the single-stage compounding process, two polymers are metered into the hopper and then compounded. This means that the components are dosed in a certain ratio in the hopper and simultaneously plasticized. As a result, both materials have the same temperature at one point along the extrusion process. However, the polymers differ in their melting behaviour. At the same temperature, both materials also have different enthalpy levels. The consequence is that the components mixed in the hopper cannot be seen as a unit, but that the enthalpy levels must be considered separately and then added, weighted, to the corresponding mass flow. The advantage of this method is that the model is not only applicable for two polymers, but for any number of them. In addition, the basic model already modelled can be used and only a weighting and addition of both enthalpy levels is required.
  
-In the solids transport zone, the enthalpy level of the individual polymers can be calculated using equation 0-1. These are then weighted with equation 0-11 and added together to obtain the enthalpy level at a supporting point.+In the solids transport zone, the enthalpy level of the individual polymers can be calculated using equation 44. These are then weighted with equation 54 and added together to obtain the enthalpy level at a supporting point.
  
-$$\Delta h_{FF_{C1}} = \frac{\dot{m}_1}{\dot{m}_{SS}} \ast \Delta h_{FF1} + \frac{\dot{m}_2}{\dot{m}_{SS}} \ast \Delta h_{FF2} + \cdots + \frac{\dot{m}_n}{\dot{m}_{SS}} \ast \Delta h_{FFn}\tag{Equation 0-11}$$ +$$\Delta h_{FF_{C1}} = \frac{\dot{m}_1}{\dot{m}_{SS}} \ast \Delta h_{FF1} + \frac{\dot{m}_2}{\dot{m}_{SS}} \ast \Delta h_{FF2} + \cdots + \frac{\dot{m}_n}{\dot{m}_{SS}} \ast \Delta h_{FFn}\tag{54}$$ 
  
-Subsequently, identical to equation 0-2 and equation 0-3, the enthalpy change and from this the required drive power between two adjacent support points can be calculated.+Subsequently, identical to equation 45 and equation 46, the enthalpy change and from this the required drive power between two adjacent support points can be calculated.
  
-The same method is used in the melting zone and the melt conveying zone. The enthalpy levels of the polymers are considered separately in the first step and then added together. Equations 0-6 and 0-7 of the basic model provide the enthalpy levels of the corresponding melt and solid content for the individual polymers. In equation 0-8 the proportions are weighted with the enthalpy level. For a process with several polymers that melt simultaneously, the weighting of the different mass flows must also be taken into account. Equation 0-12 applies to the specific change in the enthalpy when compounding polymers when mixed in the hopper:+The same method is used in the melting zone and the melt conveying zone. The enthalpy levels of the polymers are considered separately in the first step and then added together. Equations 49 and 50 of the basic model provide the enthalpy levels of the corresponding melt and solid content for the individual polymers. In equation 51 the proportions are weighted with the enthalpy level. For a process with several polymers that melt simultaneously, the weighting of the different mass flows must also be taken into account. Equation 55 applies to the specific change in the enthalpy when compounding polymers when mixed in the hopper:
  
-$$\Delta h_{AS/S_{C1}} = \frac{\dot{m}_1}{\dot{m}_{SS}} \left(\dot{m}_1 \ast m_{asv_1} \ast \Delta h_{S_1} + \dot{m}_1 \ast (1 - m_{asv_1}) \ast \Delta h_{FA_1}\right) + \frac{\dot{m}_2}{\dot{m}_{SS}} \left(\dot{m}_2 \ast m_{asv_2} \ast \Delta h_{S_2} + \dot{m}_2 \ast (1 - m_{asv_2}) \ast \Delta h_{FA_2}\right)$$ $$ + \cdots + \frac{\dot{m}_n}{\dot{m}_{SS}} \left(\dot{m}_n \ast m_{asv_n} \ast \Delta h_{S_n} + \dot{m}_n \ast (1 - m_{asv_n}) \ast \Delta h_{FA_n}\right)\tag{Equation 0-12}$$ +$$\Delta h_{AS/S_{C1}} = \frac{\dot{m}_1}{\dot{m}_{SS}} \left(\dot{m}_1 \ast m_{asv_1} \ast \Delta h_{S_1} + \dot{m}_1 \ast (1 - m_{asv_1}) \ast \Delta h_{FA_1}\right) + \frac{\dot{m}_2}{\dot{m}_{SS}} \left(\dot{m}_2 \ast m_{asv_2} \ast \Delta h_{S_2} + \dot{m}_2 \ast (1 - m_{asv_2}) \ast \Delta h_{FA_2}\right)$$ $$ + \cdots + \frac{\dot{m}_n}{\dot{m}_{SS}} \left(\dot{m}_n \ast m_{asv_n} \ast \Delta h_{S_n} + \dot{m}_n \ast (1 - m_{asv_n}) \ast \Delta h_{FA_n}\right)\tag{55}$$ 
  
-Subsequently, identical to equation 0-9 and equation 0-10, the specific enthalpy difference and the required drive power between two supporting points can be calculated.+Subsequently, identical to equation 52 and equation 53, the specific enthalpy difference and the required drive power between two supporting points can be calculated.
  
 ==== Power calculation for a two-stage compounding process ==== ==== Power calculation for a two-stage compounding process ====
Zeile 392: Zeile 392:
 ==== Power model for a melt extruder ==== ==== Power model for a melt extruder ====
  
-In the melt extruder, the specific increase in enthalpy due to melting of the solid is eliminated. The melt conveyed in the extruder only undergoes a change in enthalpy due to a temperature variation. For this reason, the calculation of the enthalpy increase up to the melting point can be neglected and the reference point for the enthalpy change is not the filling temperature but the crystallite melting temperature. With later differentiation of the enthalpy changes this portion would be shortened out again anyway. The specific enthalpy change of a melt at a support point in the melt extruder in relation to the crystallite melting temperature can be calculated using Equation 0-6 with Equation 0-13.+In the melt extruder, the specific increase in enthalpy due to melting of the solid is eliminated. The melt conveyed in the extruder only undergoes a change in enthalpy due to a temperature variation. For this reason, the calculation of the enthalpy increase up to the melting point can be neglected and the reference point for the enthalpy change is not the filling temperature but the crystallite melting temperature. With later differentiation of the enthalpy changes this portion would be shortened out again anyway. The specific enthalpy change of a melt at a support point in the melt extruder in relation to the crystallite melting temperature can be calculated using Equation 49 with Equation 56.
  
-$$\Delta h_{SE} = cp_0 \ast (T_M - T_K) + \frac{cp_m}{2} \ast (T_M^2 - T_K^2)\tag{Equation 0-13}$$ +$$\Delta h_{SE} = cp_0 \ast (T_M - T_K) + \frac{cp_m}{2} \ast (T_M^2 - T_K^2)\tag{56}$$ 
  
-For the driving power, the difference of the specific enthalpies is calculated again (equation 0-14):+For the driving power, the difference of the specific enthalpies is calculated again (57):
  
-$$\Delta \Delta h_{SE} = \Delta h_{SE_n} - \Delta h_{SE_{n-1}}\tag{Equation 0-14}$$ +$$\Delta \Delta h_{SE} = \Delta h_{SE_n} - \Delta h_{SE_{n-1}}\tag{57}$$ 
  
-Equation 0-15 applies to the drive power:+Equation 58 applies to the drive power:
  
-$$P_{D_{SE}} = \dot{m} \ast (\Delta \Delta h_{SE} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{Equation 0-15}$$ +$$P_{D_{SE}} = \dot{m} \ast (\Delta \Delta h_{SE} + \frac{\Delta p}{\rho}) - \dot{Q}\tag{58}$$ 
  
  
 ==== Power model for a melt extruder with several polymers ==== ==== Power model for a melt extruder with several polymers ====
  
-In the case of a twin-screw extruder, which functions as a melt extruder and is equipped with a melt consisting of several polymers, the model must be modified for a melt extruder. The procedure is similar to that for compounding. The individual enthalpy levels of the components are determined in a first step and then weighted with the aid of the mass flows. Formally, equation 0-16 is obtained.+In the case of a twin-screw extruder, which functions as a melt extruder and is equipped with a melt consisting of several polymers, the model must be modified for a melt extruder. The procedure is similar to that for compounding. The individual enthalpy levels of the components are determined in a first step and then weighted with the aid of the mass flows. Formally, equation 59 is obtained.
  
 $$\Delta h_{SE_M} = \frac{\dot{m}_1}{\dot{m}_{ges}} \left(cp_{01} \ast (T_M - T_K) + \frac{cp_{m1}}{2} \ast (T_M^2 - T_K^2)\right) + \frac{\dot{m}_2}{\dot{m}_{ges}} \left(cp_{02} \ast (T_M - T_K) + \frac{cp_{m2}}{2} \ast (T_M^2 - T_K^2)\right) $$ $$\Delta h_{SE_M} = \frac{\dot{m}_1}{\dot{m}_{ges}} \left(cp_{01} \ast (T_M - T_K) + \frac{cp_{m1}}{2} \ast (T_M^2 - T_K^2)\right) + \frac{\dot{m}_2}{\dot{m}_{ges}} \left(cp_{02} \ast (T_M - T_K) + \frac{cp_{m2}}{2} \ast (T_M^2 - T_K^2)\right) $$
-$$+ \cdots + \frac{\dot{m}_n}{\dot{m}_{ges}} \left(cp_{0n} \ast (T_M - T_K) + \frac{cp_{mn}}{2} \ast (T_M^2 - T_K^2)\right)\tag{Equation 0-16}$$ +$$+ \cdots + \frac{\dot{m}_n}{\dot{m}_{ges}} \left(cp_{0n} \ast (T_M - T_K) + \frac{cp_{mn}}{2} \ast (T_M^2 - T_K^2)\right)\tag{59}$$ 
  
 Subsequently, as in the case of a melt extruder, which is loaded with a polymer, the difference of the enthalpy change can be formed and from this the drive power between two support points. Subsequently, as in the case of a melt extruder, which is loaded with a polymer, the difference of the enthalpy change can be formed and from this the drive power between two support points.
Zeile 418: Zeile 418:
 With the relationships shown, the individual required drive powers between two support points or the individual elements can be determined and visualized in SIGMA. Finally, the individual calculated element powers must be added up to obtain the total power. The sole consideration or difference of the initial and final enthalpy to determine the power leads to errors in various cases. This is the case, for example, if the melt temperature drops once during the extrusion process due to fillers and the melt is then heated again. The subsequent heating of the melt must be done by new energy from outside. This requires a further input of power, which would be neglected if the initial and final enthalpy were simply compared. With the relationships shown, the individual required drive powers between two support points or the individual elements can be determined and visualized in SIGMA. Finally, the individual calculated element powers must be added up to obtain the total power. The sole consideration or difference of the initial and final enthalpy to determine the power leads to errors in various cases. This is the case, for example, if the melt temperature drops once during the extrusion process due to fillers and the melt is then heated again. The subsequent heating of the melt must be done by new energy from outside. This requires a further input of power, which would be neglected if the initial and final enthalpy were simply compared.
  
 +====Enthalpy model with heat flow====
 +The enthalpy model neglects the heat flow between the polymer and the barrel wall. While in larger industrial extruders the proportion of heating power to total power is small compared to the drive power, neglecting the heat flow can lead to significant errors, particularly in smaller laboratory extruders or processes with a large temperature difference between the melt and the barrel.
 +
 +To account for the heat flow, the heat transfer coefficient is calculated according to [[en:grundlagenhandbuch:drehmoment_und_antriebsleistung#references |[TM00]]]. First, the Brinkmann number is calculated and then the Nusselt number is regressed from it.
 +
 +$$Br=\frac{K*v_{0}^{n+1}}{\lambda*\Delta T*h^{n-1}}\tag{60}$$
 +
 +$$Nu=2,9*Br^{0,8}\tag{61}$$
 +
 +The heat transfer coefficient can then be calculated using the thermal conductivity of the melt and the channel height:
 +
 +$$\alpha=\frac{Nu*\lambda}{h}\tag{62}$$
 +
 +The heat flow is calculated as follows:
 +
 +$$\dot Q=\alpha*l_{Node}*U_{Barrel}*\frac{b_{eff}}{b_{max}}*\Delta T\tag{63}$$
 +
 +The heat flow is subtracted from the calculated specific enthalpy difference according to equation 46 in order to determine the corrected drive power. This calculation is performed when the 'Enthalpy w. heat flow' power model is selected.
 ==== Validation ==== ==== Validation ====
  
 The new power model was implemented and verified in SIGMA. Identical to other models, the new power model was compared with the values of the experimental investigations to validate the model. The deviations of different process points and material combinations are shown in the following picture. The new power model was implemented and verified in SIGMA. Identical to other models, the new power model was compared with the values of the experimental investigations to validate the model. The deviations of different process points and material combinations are shown in the following picture.
  
-{{ :en:grundlagenhandbuch:drehmoment_und_antriebsleistung:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_0010.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_drehmoment_und_antriebsleistung_0010.png?nolink&600 |}}
  
 **Figure:** Comparison of experimental investigations and simulations **Figure:** Comparison of experimental investigations and simulations
Zeile 441: Zeile 459:
  
 [TK78] Tadmor, Z.; Klein, I: Engineering Principles of Plasticating Extrusion, Robert E. Krieger Publishing Company, Huntington, New York, 1978 [TK78] Tadmor, Z.; Klein, I: Engineering Principles of Plasticating Extrusion, Robert E. Krieger Publishing Company, Huntington, New York, 1978
 +
 +[TM00] Tenge, S.; Mewes, D.: “Experimental investigation of the energy balance for the metering zone of a twin screw extruder”; Polymer Engineering & Science; Volume 40; 2000; S. 277–289