Dies ist eine alte Version des Dokuments!
Scale-up
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 [Paw71].
The first law of thermodynamics for a stationary flow process is the starting point of this approach.
Figure: Energy balance of a twin screw extruder.
$$P + \dot{Q} = \dot{m} c_v \Delta T + p \dot{V} \tag{1}$$
By rearranging the equation, the following dimensionless form is obtained:
$$\frac{P}{\dot{m} c_v \Delta T} + \frac{\dot{Q}}{\dot{m} c_v \Delta T} = 1 + \frac{p \dot{V}}{\dot{m} c_v \Delta T} \tag{2}$$
In this form it is composed of three dimensionless characteristic values, which clearly define the operating point of an extruder.
$$\Pi_P = \frac{P}{\dot{m} c_v \Delta T} \tag{3}$$
$$\Pi_{\dot{Q}} = \frac{\dot{Q}}{\dot{m} c_v \Delta T} \tag{4}$$
$$\Pi_{p,T} = \frac{p \dot{V}}{\dot{m} c_v \Delta T} \tag{5}$$
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, as shown by Mayer [May84], with the postulation made by Potente [Pot81] for a constant dimensionless pressure gradient for both machines. As furthermore demonstrated by Potente [Pot81], a constant dimensionless pressure gradient can only be realized under the condition of a constant helix angle $\varphi_s$.
$$\frac{\varphi_s}{\varphi_{s,0}} = \left(\frac{D}{D_0}\right)^0 \tag{6}$$
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 [Erd69]. In doing so, for Newtonian liquids the following equation applies:
$$\dot{V} = \dot{V}_z - \dot{V}_p \tag{7}$$
By transformation Equation 7 can be turned into Equation 8:
$$\dot{V} = \left(1 - \frac{\dot{V}_p}{\dot{V}_z}\right) \dot{V}_z \tag{8}$$
The fraction in this equation is denominated as pressure flow-drag flow ratio or as dimensionless pressure gradient respectively. Among other things it influences the homogeneity of the melt and has to be constant also for power law fluids. Under this condition it holds that:
$$\dot{V} \sim \dot{V}_z \sim b h D N \tag{9}$$
For constant material values and a constant helix angle as well as by introducing the at first purely formal correlations of
$$v \sim D N \quad b \sim D \quad h \sim D^{\psi} \quad N \sim D^{-\chi} \tag{10}$$
for the mass throughput results:
$$\dot{m} = \dot{V} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi} \tag{11}$$
By considering the specified requirements and based on a variable screw length to diameter ratio (L/D), the model laws illustrated in the table above are determined with the characteristic values specified in 3 to 5. In doing so, two cases are separated: Case 1: $T_1$ respectively $T_1-T_0$ = constant Case 2: $T_1$ respectively $T_1-T_0$ = variable
Whereas the first case is based on the assumption of a constant melt temperature at the screw tip and a constant melt temperature difference for the homogenizing section respectively, this is allowed to be variable in the second case. It is thereby assumed that the difference of the melt temperatures at the screw tip is a power function of the shear rate.
$$T_1 - T_{1,0} \sim \dot{\gamma}^{-\xi} \tag{12}$$
Table: Model laws
| Parameter | Case 1 $T_1-T_0$ = constant | Case 2 $T_1-T_0$ = variable |
|---|---|---|
| $\frac{L}{L_0}$ | $\left(\frac{D}{D_0}\right)^{1+\omega}$ | $\left(\frac{D}{D_0}\right)^{1+\omega}$ |
| $\frac{h}{h_0}$ | $\left(\frac{D}{D_0}\right)^{\psi}$ | $\left(\frac{D}{D_0}\right)^{\psi}$ |
| $\frac{N}{N_0}$ | $\left(\frac{D}{D_0}\right)^{-\chi}$ | $\left(\frac{D}{D_0}\right)^{-\chi}$ |
| $\frac{\dot{m}}{\dot{m}_0}$ | $\left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ | $\left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ |
| $\frac{P}{P_0}$ | $\left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ | $\left(\frac{D}{D_0}\right)^{\psi(1+\kappa_p)-\chi(1-\kappa_p)+\kappa_p(2-\xi)}$ |
| $\frac{\dot{Q}}{\dot{Q}_0}$ | $\left(\frac{D}{D_0}\right)^{2+\psi-\chi}$ | $\left(\frac{D}{D_0}\right)^{\psi(1+\kappa_p)-\chi(1-\kappa_p)+\kappa_p(2-\xi)}$ |
| $\frac{Md}{Md_0}$ | $\left(\frac{D}{D_0}\right)^{2+\psi}$ | $\left(\frac{D}{D_0}\right)^{\psi(1+\kappa_p)-\chi_d(1-\kappa_p)+2}$ |
| $\frac{p}{p_0}$ | $\left(\frac{D}{D_0}\right)^{0}$ | $\left(\frac{D}{D_0}\right)^{-\xi+\kappa_p+\kappa_q}$ |
| $\frac{T_1 - T_0}{T_{1,0} - T_{0,0}}$ | $\left(\frac{D}{D_0}\right)^{0}$ | $\left(\frac{D}{D_0}\right)^{-\xi+\kappa_p+\kappa_q}$ |
To be able to make use of the available model laws the assigned exponents have to be linked with each other. For this purpose, exclusively those screw zones are examined, in which melt can be found. Deriving from the energy balance of the melt conveying zone the dimensionless parameters result as illustrated in the table. Energetic similarity can be observed in each case, in which the characteristic values of a set of characteristic values are constant.
Table: Characteristic values and assumptions for the melt conveying zone.
| Set | $\Pi_{\dot{Q}}$ | $\Pi_\kappa$ | $\Pi_7$ | $\Pi_{12}$ | $\Pi_{11}$ | Assumptions |
|---|---|---|---|---|---|---|
| Set 1 | $\Pi_{\dot{Q}} = \frac{\eta v L_{sz}}{h^2 \rho c_v \Delta T_{sz}}$ | $\Pi_\kappa = \frac{\dot{q} L_{sz} \left(1 - \frac{\beta}{\pi}\right)}{h v \rho c_v \Delta T_{sz}}$ | $\Pi_7 = \frac{\Delta p}{\rho c_v \Delta T}$ | - | - | $\dot{\gamma} \sim \frac{v}{h} \quad h \sim D^{\psi} \quad N \sim D^{-\chi} \quad v \sim D N \quad L_{sz} \sim D^{1+\omega}$ |
| Set 2 | $\Pi_{\dot{Q}} = \frac{\eta v L_{sz}}{h^2 \rho c_v \Delta T_{sz}}$ | $\Pi_7 = \frac{\Delta p_{sz}}{\rho c_v \Delta T_{sz}}$ | - | $\Pi_{12} = \frac{\lambda \Delta T_w L_{sz} \left(1 - \frac{\beta}{\pi}\right)}{h^2 v \rho c_v \Delta T_{sz}}$ | $\Pi_{11} = \frac{\alpha h}{\lambda}$ | $\dot{\gamma} \sim \frac{v}{h} \quad h \sim D^{\psi} \quad N \sim D^{-\chi} \quad v \sim D N \quad L_{sz} \sim D^{1+\omega}$ |
In the case of polymer melts, the viscosity emerging in the characteristic values is dependent on the temperature and the shear rate. For the relevant processing range it is sufficient to represent the viscosity by means of a power function. In the general case, at a machine magnification the viscosity function is running in direction of the arrow as depicted in the figure.
Figure: Viscosity as a function of the shear rate and the temperature.
Viscosity and temperature changes can be determined by means of the Equations 13 and 14.
$$\frac{\eta}{\eta_0} = \left(\frac{\dot{\gamma}}{\dot{\gamma}_0}\right)^{-\kappa} \text{ with } \kappa = 1-n \text{ for } T = \text{const.} \tag{13}$$
$$\frac{T}{T_0} = \left(\frac{\dot{\gamma}}{\dot{\gamma}_0}\right)^{-\xi} \tag{14}$$
In reference to the melt conveying zone, with the prerequisites specified in the table the underlying principles for the screw speed exponent and for the flight depth exponent thus derive as described in the table. In doing so, the material values ρ, λ and $c_v$ were assumed invariant.
Table: Screw speed and channel depth exponents
$$\psi_{T_w} = \frac{(1+\omega)(2-\kappa)}{(4-3\kappa)+2\xi} \leq \psi = 1 + \frac{\omega-\chi(1-\kappa+\xi)}{2-\kappa+\xi} \leq \psi_q = \frac{(1+\omega)(2-\kappa)}{3-2\kappa+\xi} \tag{15}$$
$$\chi_{T_w} = \frac{\psi_{T_w}(2+\xi)-(ω+\xi)}{1-\xi} \geq \chi \geq \chi_q = \frac{\psi_q(1+\xi)-(ω+\xi)}{1-\xi} \tag{16}$$
The equations in the table can serve for the design of twin screw extruders. Are both the model design and the main design already determined for a trans-fer, it has to be taken into account that the flight depth exponent is unequivocally defined because of the closely intermeshing geometry.
In this case, the flight depth exponent results from its definition equation:
$$\psi = \frac{\lg\left(\frac{h}{h_0}\right)}{\lg\left(\frac{D}{D_0}\right)} \tag{17}$$
For closely intermeshing machines the flight depth deriving therein is a function of the wheel base, as by neglecting the clearance the following equation applies:
$$h_{max} = D - a \tag{18}$$
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 [Chr94] it appears expedient to differentiate three cases in doing so.
Model transfer
Constant Specific Energy
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:
$$\frac{\eta}{\eta_0} = \left(\frac{D}{D_0}\right)^{2\psi+\chi-2-\omega} \tag{19}$$
Furthermore, the shear rate can be described by the following relation:
$$\frac{\dot{\gamma}}{\dot{\gamma}_0} = \left(\frac{D}{D_0}\right)^{1-\psi-\chi} \tag{20}$$
If both terms are logarithmized and the diameter ratio is eliminated, one receives:
$$\lg\frac{\eta}{\eta_0} = \frac{2\psi + \chi - 2 - \omega}{1 - \psi - \chi} \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{21}$$
According to Equation 1, at a constant temperature for the viscosity function it also applies:
$$\lg\frac{\eta}{\eta_0} = (n - 1) \lg\frac{\dot{\gamma}}{\dot{\gamma}_0} \tag{22}$$
By that method the screw speed exponent for a constant specific energy input derives:
$$\chi = \frac{n - n\psi + 1 + \omega - \psi}{n} \tag{23}$$
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.
Constant Heat Flow
Based on the energy balance of the melt zone for the heating capacity results
$$\Pi_6 = \frac{\dot{q} L_{sz} \left(1 - \frac{\beta}{\pi}\right)}{h v \rho c_v \Delta T_{sz}} \tag{24}$$
The expression in brackets in Equation 6 can be considered as an approximately constant value [Pot81]. Are, furthermore, constant material values assumed and is the velocity replaced, the ratio of heat flow densities of the model design and the main design derive:
$$\frac{\dot{q}}{\dot{q}_0} = \frac{L_{sz,0} h D N}{L_{sz} h_0 D_0 N_0} = \frac{\Delta T_{sz}}{\Delta T_{sz,0}} = \left(\frac{D}{D_0}\right)^{1-\psi-\chi} \frac{\Delta T_{sz}}{\Delta T_{sz,0}} \tag{25}$$
By assuming a constant melt temperature at the screw tip for both the model design and the main design, it consequently has to apply
$$\chi = \psi - \omega \tag{26}$$
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 $\frac{\dot{m}}{\dot{m}_0} = \frac{\dot{V}}{\dot{V}_0} = \left(\frac{D}{D_0}\right)^{2+\psi-\chi}$.
Constant Residence Time
The medium residence time can be described as a quotient of the free channel volume and the volume throughput.
$$\bar{t} = \frac{V_{free}}{\dot{V}} \tag{27}$$
The substitution of both values by
$$\dot{V} \sim b h D N \text{ and } V_{free} \sim b h L \tag{28}$$
and the formation of the quotient for the model design and for the main design leads to Equation 29.
$$\frac{\bar{t}}{\bar{t}_0} = \frac{L D_0 N_0}{L_0 D N} = \left(\frac{D}{D_0}\right)^{\psi+\chi} \tag{29}$$
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
$$\chi = -\omega \tag{30}$$
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.
Nomenclature Scale-up
Latin Symbols
| Symbol | Meaning |
|---|---|
| $a$ | Center distance line |
| $b$ | Channel width |
| $c_v$ | Specific heat capacity at a constant volume |
| $D$ | Screw diameter |
| $h$ | Channel depth |
| $h_{max}$ | Maximum channel depth |
| $L$ | Length |
| $M_d$ | Torque |
| $\dot{m}$ | Mass flow rate |
| $N$ | Screw speed |
| $n$ | Exponent of the power law |
| $p$ | Pressure at the screw tip |
| $\Delta p$ | Pressure difference |
| $P$ | Power |
| $\dot{Q}$ | Heat flow |
| $\dot{q}$ | Heat flow density |
| $t$ | Time |
| $\bar{t}$ | Medium residence time |
| $T$ | Temperature |
| $T_1$ | Temperature at the screw tip |
| $T_0$ | Starting temperature |
| $\Delta T_w$ | Difference between barrel wall temperature and medium melt temperature |
| $\Delta T_{SZ}$ | Temperature difference of the melt zone |
| $v$ | Peripheral velocity |
| $V_{free}$ | Free channel volume |
| $\dot{V}$ | Volume flow rate |
| $\dot{V}_s$ | Volume flow rate resulting from drag flow |
| $\dot{V}_p$ | Volume flow rate resulting from pressure flow |
Greek Symbols
| Symbol | Meaning |
|---|---|
| $\alpha$ | Heat transfer coefficient |
| $\beta$ | Pressure angle |
| $\dot{\gamma}$ | Shear rate |
| $\eta$ | Viscosity |
| $\varphi_s$ | Helix angle |
| $\chi$ | Screw speed exponent |
| $\kappa$ | Material exponent |
| $\Pi$ | Dimensionless parameter |
| $\lambda$ | Heat conductivity |
| $\rho$ | Density |
| $\omega$ | Exponent of the screw length to diameter ratio (L/D) |
| $\xi$ | Temperature exponent |
| $\psi$ | Flight depth exponent |
Indices
| Index | Meaning |
|---|---|
| $0$ | Model (design) |
| $SZ$ | Melt zone |
| $T_w$ | For constant wall temperature difference |
| $\dot{q}$ | For constant heat flow density |
References
[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
[Erd69] Erdmenger, R.: Mehrwellenschnecken der Verfahrenstechnik, Chem.-Ing.-Techn. 36, S.175-185, 1969
[May84] Mayer, A.: Extruderbaureihen - Ein Beitrag zur Auslegung und Optimierung von Einschneckenextrudern, Dissertation RWTH Aachen 1984
[Paw71] Pawlowski, I.: Die Ähnlichkeitstheorie in der physikalisch-technischen Forschung, Springer-Verlag 1971
[Pot81] Potente, H.: Auslegung von Schneckenmaschinen-Baureihen, Carl Hanser Verlag 1981