Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| en:grundlagenhandbuch:scale-up [2026/01/28 10:11] – [Constant Heat Flow] deppe2 | en:grundlagenhandbuch:scale-up [2026/09/04 11:04] (aktuell) – [Model Theory] paal | ||
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| Zeile 3: | Zeile 3: | ||
| ===== Model Theory ==== | ===== Model Theory ==== | ||
| - | For reasons of economy the development of new recipes in the area of polymer processing is usually carried out on laboratory machines. Only once an appropriate product quality has been attained the process transfer to the production machine is initiated. By reason of this two-tiered approach the transfer of processes and operating points in polymer processing assumes central importance. To minimize the risk of quality loss during the scale-up, the adoption of model laws appears to be expedient. These model laws rest upon the foundations of the similarity theory [1]. | + | For reasons of economy the development of new recipes in the area of polymer processing is usually carried out on laboratory machines. Only once an appropriate product quality has been attained the process transfer to the production machine is initiated. By reason of this two-tiered approach the transfer of processes and operating points in polymer processing assumes central importance. To minimize the risk of quality loss during the scale-up, the adoption of model laws appears to be expedient. These model laws rest upon the foundations of the similarity theory [[en: |
| - | The first thermodynamic theorem | + | The first law of thermodynamics |
| Zeile 12: | Zeile 12: | ||
| $$P + \dot{Q} = \dot{m} c_v \Delta T + p \dot{V} \tag{1}$$ | $$P + \dot{Q} = \dot{m} c_v \Delta T + p \dot{V} \tag{1}$$ | ||
| - | By transformation | + | By rearranging |
| $$\frac{P}{\dot{m} c_v \Delta T} + \frac{\dot{Q}}{\dot{m} c_v \Delta T} = 1 + \frac{p \dot{V}}{\dot{m} c_v \Delta T} \tag{2}$$ | $$\frac{P}{\dot{m} c_v \Delta T} + \frac{\dot{Q}}{\dot{m} c_v \Delta T} = 1 + \frac{p \dot{V}}{\dot{m} c_v \Delta T} \tag{2}$$ | ||
| Zeile 26: | Zeile 26: | ||
| With the equations 3 to 5 the basic characteristic values for the setup of the model laws are determined. | With the equations 3 to 5 the basic characteristic values for the setup of the model laws are determined. | ||
| - | For the calculation of the pressure-throughput behavior and the machine performance in the case of co-rotating twin screw extruders it is furthermore crucial to have knowledge of the velocity profile of the polymer melt in the screw channel. To obtain identical flow conditions in both the model design and the main design, it is therefore obvious to postulate invariance of the standardized velocity profile in the channel. This postulation corresponds, | + | For the calculation of the pressure-throughput behavior and the machine performance in the case of co-rotating twin screw extruders it is furthermore crucial to have knowledge of the velocity profile of the polymer melt in the screw channel. To obtain identical flow conditions in both the model design and the main design, it is therefore obvious to postulate invariance of the standardized velocity profile in the channel. This postulation corresponds, |
| $$\frac{\varphi_s}{\varphi_{s, | $$\frac{\varphi_s}{\varphi_{s, | ||
| - | By considering a metering section of a co-rotating twin screw extruder, which is completely loaded with melt, the total volume flow rate thus consists of a fraction of drag flow and a fraction of pressure flow, in case the leakage flow is neglected [4]. In doing so, for Newtonian liquids the following equation applies: | + | By considering a metering section of a co-rotating twin screw extruder, which is completely loaded with melt, the total volume flow rate thus consists of a fraction of drag flow and a fraction of pressure flow, in case the leakage flow is neglected [[en: |
| $$\dot{V} = \dot{V}_z - \dot{V}_p \tag{7}$$ | $$\dot{V} = \dot{V}_z - \dot{V}_p \tag{7}$$ | ||
| Zeile 89: | Zeile 89: | ||
| $$\frac{T}{T_0} = \left(\frac{\dot{\gamma}}{\dot{\gamma}_0}\right)^{-\xi} \tag{14}$$ | $$\frac{T}{T_0} = \left(\frac{\dot{\gamma}}{\dot{\gamma}_0}\right)^{-\xi} \tag{14}$$ | ||
| - | In reference | + | With regard |
| **Table:** Screw speed and channel depth exponents | **Table:** Screw speed and channel depth exponents | ||
| Zeile 107: | Zeile 107: | ||
| $$h_{max} = D - a \tag{18}$$ | $$h_{max} = D - a \tag{18}$$ | ||
| - | Because of the invariance of the flight depth exponent at a given geometry, the principle of total energetic similarity generally cannot be realized for co-rotating twin screw extruders. By choosing appropriate boundary conditions, however, it is possible to obtain partial energetic similarity. According to Christiano [5] it appears expedient to differentiate three cases in doing so. | + | Because of the invariance of the flight depth exponent at a given geometry, the principle of total energetic similarity generally cannot be realized for co-rotating twin screw extruders. By choosing appropriate boundary conditions, however, it is possible to obtain partial energetic similarity. According to Christiano [[en: |
| ===== Model transfer===== | ===== Model transfer===== | ||
| Zeile 115: | Zeile 115: | ||
| The starting point for this approach is the characteristic value Π5. of the melt conveying zone (see table). By assuming a constant melt temperature at the screw tip and in conjunction with the requirements specified beforehand (see table), the viscosity ratio of model design and main design results: | The starting point for this approach is the characteristic value Π5. of the melt conveying zone (see table). By assuming a constant melt temperature at the screw tip and in conjunction with the requirements specified beforehand (see table), the viscosity ratio of model design and main design results: | ||
| - | $$\frac{\eta}{\eta_0} = \left(\frac{D}{D_0}\right)^{2\psi+\chi-2-\omega} \tag{1}$$ | + | $$\frac{\eta}{\eta_0} = \left(\frac{D}{D_0}\right)^{2\psi+\chi-2-\omega} \tag{19}$$ |
| Furthermore, | Furthermore, | ||
| - | $$\frac{\dot{\gamma}}{\dot{\gamma}_0} = \left(\frac{D}{D_0}\right)^{1-\psi-\chi} \tag{2}$$ | + | $$\frac{\dot{\gamma}}{\dot{\gamma}_0} = \left(\frac{D}{D_0}\right)^{1-\psi-\chi} \tag{20}$$ |
| If both terms are logarithmized and the diameter ratio is eliminated, one receives: | If both terms are logarithmized and the diameter ratio is eliminated, one receives: | ||
| - | $$\lg\frac{\eta}{\eta_0} = \frac{2\psi + \chi - 2 - \omega}{1 - \psi - \chi} \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{3}$$ | + | $$\lg\frac{\eta}{\eta_0} = \frac{2\psi + \chi - 2 - \omega}{1 - \psi - \chi} \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{21}$$ |
| According to Equation 1, at a constant temperature for the viscosity function it also applies: | According to Equation 1, at a constant temperature for the viscosity function it also applies: | ||
| - | $$\lg\frac{\eta}{\eta_0} = (n - 1) \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{4}$$ | + | $$\lg\frac{\eta}{\eta_0} = (n - 1) \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{22}$$ |
| By that method the screw speed exponent for a constant specific energy input derives: | By that method the screw speed exponent for a constant specific energy input derives: | ||
| - | $$\chi = \frac{n - n\psi + 1 + \omega - \psi}{n} \tag{5}$$ | + | $$\chi = \frac{n - n\psi + 1 + \omega - \psi}{n} \tag{23}$$ |
| By inserting the screw speed exponent into Eq. $\frac{\dot{m}}{\dot{m}_0} = \frac{\dot{V}}{\dot{V}_0} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ the respective throughput for this operating point results. | By inserting the screw speed exponent into Eq. $\frac{\dot{m}}{\dot{m}_0} = \frac{\dot{V}}{\dot{V}_0} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ the respective throughput for this operating point results. | ||
| Zeile 139: | Zeile 139: | ||
| Based on the energy balance of the melt zone for the heating capacity results | Based on the energy balance of the melt zone for the heating capacity results | ||
| - | $$\Pi_6 = \frac{\dot{q} L_{sz} \left(1 - \frac{\beta}{\pi}\right)}{h v \rho c_v \Delta T_{sz}} \tag{1}$$ | + | $$\Pi_6 = \frac{\dot{q} L_{sz} \left(1 - \frac{\beta}{\pi}\right)}{h v \rho c_v \Delta T_{sz}} \tag{24}$$ |
| - | The expression in brackets in Equation 6 can be considered as an approximately constant value [3]. Are, furthermore, | + | The expression in brackets in Equation 6 can be considered as an approximately constant value [[en: |
| - | $$\frac{\dot{q}}{\dot{q}_0} = \frac{L_{sz, | + | $$\frac{\dot{q}}{\dot{q}_0} = \frac{L_{sz, |
| By assuming a constant melt temperature at the screw tip for both the model design and the main design, it consequently has to apply | By assuming a constant melt temperature at the screw tip for both the model design and the main design, it consequently has to apply | ||
| - | $$\chi = \psi - \omega \tag{3}$$ | + | $$\chi = \psi - \omega \tag{26}$$ |
| to obtain identical heat flow densities. The melt flow, which is to be metered, derives as before by inserting the exponent into the throughput equation $\frac{\dot{m}}{\dot{m}_0} = \frac{\dot{V}}{\dot{V}_0} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi}$. | to obtain identical heat flow densities. The melt flow, which is to be metered, derives as before by inserting the exponent into the throughput equation $\frac{\dot{m}}{\dot{m}_0} = \frac{\dot{V}}{\dot{V}_0} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi}$. | ||
| - | ===== Constant Residence Time ===== | + | ==== Constant Residence Time ==== |
| Zeile 157: | Zeile 157: | ||
| The medium residence time can be described as a quotient of the free channel volume and the volume throughput. | The medium residence time can be described as a quotient of the free channel volume and the volume throughput. | ||
| - | $$\bar{t} = \frac{V_{free}}{\dot{V}} \tag{1}$$ | + | $$\bar{t} = \frac{V_{free}}{\dot{V}} \tag{27}$$ |
| The substitution of both values by | The substitution of both values by | ||
| - | $$\dot{V} \sim b h D N \text{ and } V_{free} \sim b h L \tag{2}$$ | + | $$\dot{V} \sim b h D N \text{ and } V_{free} \sim b h L \tag{28}$$ |
| - | and the formation of the quotient for the model design and for the main design leads to Equation | + | and the formation of the quotient for the model design and for the main design leads to Equation |
| - | $$\frac{\bar{t}}{\bar{t}_0} = \frac{L D_0 N_0}{L_0 D N} = \left(\frac{D}{D_0}\right)^{\psi+\chi} \tag{3}$$ | + | $$\frac{\bar{t}}{\bar{t}_0} = \frac{L D_0 N_0}{L_0 D N} = \left(\frac{D}{D_0}\right)^{\psi+\chi} \tag{29}$$ |
| To be able to comply with the requirement of a constant residence time of the material in the model design as well as in the main design, the equation | To be able to comply with the requirement of a constant residence time of the material in the model design as well as in the main design, the equation | ||
| - | $$\chi = -\omega \tag{4}$$ | + | $$\chi = -\omega \tag{30}$$ |
| - | has to apply. This means that at a constant screw length to diameter ratio (ω = 0) both machines are operated with an identical screw speed. | + | has to apply. This means that at a constant screw length to diameter ratio $(ω = 0)$ both machines are operated with an identical screw speed. |
| - | ====== Nomenclature Scale-up | + | ===== Nomenclature Scale-up ===== |
| - | ===== Latin Symbols | + | ==== Latin Symbols ==== |
| ^ Symbol ^ Meaning ^ | ^ Symbol ^ Meaning ^ | ||
| Zeile 204: | Zeile 204: | ||
| | $V_{free}$ | Free channel volume | | | $V_{free}$ | Free channel volume | | ||
| | $\dot{V}$ | Volume flow rate | | | $\dot{V}$ | Volume flow rate | | ||
| - | | $\dot{V}_z$ | Volume flow rate resulting from drag flow | | + | | $\dot{V}_s$ | Volume flow rate resulting from drag flow | |
| | $\dot{V}_p$ | Volume flow rate resulting from pressure flow | | | $\dot{V}_p$ | Volume flow rate resulting from pressure flow | | ||
| - | ===== Greek Symbols | + | ====Greek Symbols ==== |
| ^ Symbol ^ Meaning ^ | ^ Symbol ^ Meaning ^ | ||
| Zeile 225: | Zeile 225: | ||
| | $\psi$ | Flight depth exponent | | | $\psi$ | Flight depth exponent | | ||
| - | ===== Indices | + | ==== Indices ==== |
| ^ Index ^ Meaning ^ | ^ Index ^ Meaning ^ | ||
| Zeile 233: | Zeile 233: | ||
| | $\dot{q}$ | For constant heat flow density | | | $\dot{q}$ | For constant heat flow density | | ||
| - | ====== References | + | ===== References ===== |
| - | [1] Pawlowski, I.; Die Ähnlichkeitstheorie in der physikalisch-technischen Forschung, Springer-Verlag 1971 | + | [Chr94] Christiano, J. P.: Scale-up study of co- rotating fully intermeshing twin screw extruders using 47mm, 69mm, and 96.5mm diameters, Antec (Tagung), S.239-247, 1994 |
| - | [2] Mayer, A.; Extruderbaureihen - Ein Beitrag zur Auslegung und Optimierung von Einschneckenextrudern , Dissertation RWTH Aachen 1984 | + | [Erd69] Erdmenger, R.: Mehrwellenschnecken der Verfahrenstechnik, |
| - | + | ||
| - | [3] Potente, H.; Auslegung von Schneckenmaschinen-Baureihen, | + | |
| - | + | ||
| - | [4] Erdmenger, R.; Mehrwellenschnecken der Verfahrenstechnik, | + | |
| - | + | ||
| - | [5] Christiano, J. P.; Scale-up study of co-rotating fully intermeshing twin screw extruders using 47mm, 69mm, and 96.5mm diameters, Antec (Tagung), S.239-247, 1994 | + | |
| + | [May84] Mayer, A.: Extruderbaureihen - Ein Beitrag zur Auslegung und Optimierung von Einschneckenextrudern, | ||
| + | [Paw71] Pawlowski, I.: Die Ähnlichkeitstheorie in der physikalisch-technischen Forschung, Springer-Verlag 1971 | ||
| + | [Pot81] Potente, H.: Auslegung von Schneckenmaschinen-Baureihen, | ||