# A Review of Dynamic Models of Hot-Melt Extrusion

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## Abstract

**:**

## 1. Introduction

## 2. A Simple Case Study

## 3. The Earlier Models: Static Maps

- Plug flow reactors characterized by a uniform residence time distribution for all particles.
- Continuous stirred-tank reactors (CSTRs) assuming the non-uniformity of the residence time, but a uniform composition of the flow.

- The work in [49] models the RTD of wheat flour particles using five volume elements of different sizes, each sub-divided in a series of CSTRs with identical volumes. In order to represent the tail in the RTD, three reactors with backward flows are included so as to explain material accumulation at the die (see Figure 6).
- The work in [50] uses a combination of a plug flow reactor with a cascade of CSTRs with the same volume and a backward flow ${Q}_{B}$ (see Figure 7). The plug flow reactor represents the feeding zone and is characterized by a delay ${t}_{d}$. The cascade of CSTRs models the completely filled part inducing a reflux and characterized by two parameters: the number of reactors n and the ratio between the forward and backward flows $\sigma =\frac{Q}{{Q}_{B}}$.

## 4. Reactors-In-Series Approach

**Models with Predetermined Screw Discretization**In [56,57], the filling ratio, monomer concentration, temperatures and RTD are determined for a counter-rotating twin screw extruder using a subdivision of the screws in C-chambers (see Figure 10) where mass and energy balances are expressed using continuum mechanics. If the system includes a high inter-meshing area between the screws, other volume elements may be added to model the specific flows in this zone (see [52]).An extrusion device presents a varying filling level, a function of the local screw geometry. It is generally considered that an extruder comprises two different zones, which can be partially or completely filled, inducing specific mass balance expressions. Indeed, mass flow networks are different in each zone. In partially-filled zones, only a pumping flow ${Q}_{p}$, due to the screw rotation, transports the material along the section. In completely filled zones, the presence of a pressure gradient creates leakage flows, and the resulting mass flow network is modeled by parallel C-chambers, in which the leakage locations are divided into five groups [59] (see Figure 11):- −
- the flight gap (with flow ${Q}_{f}$) between the barrel and the screw flight;
- −
- the tetrahedron gap (${Q}_{t}$) between the flight walls;
- −
- the calender gap (${Q}_{n}$) between the flight of one screw and the bottom of the other screw channel;
- −
- the side gap (${Q}_{s}$) between the flanks of the two screws flight;
- −
- the channel gap (${Q}_{c}$) down-channel flow in the screw channel.

**Models with Adjustable Discretization**In [35,53], the different screw sections can be discretized using an adjustable number of CSTRs (see Figure 12). Material can flow in the forward direction, with flow ${Q}_{f}$ due to the screw rotation, and in the backward direction, with flow ${Q}_{b}$ due to the pressure difference between two reactors, as shown in Figure 12 and Table 3 [60].Based on the filling ratio, the pressure in each reactor can be computed using a simple algebraic method. Two different situations may happen: in partially-filled reactors ($f<1$), the pressure P matches the atmospheric pressure ${P}_{0}$, and in the completely filled reactors ($f=1$), the pressure is above the atmospheric pressure; and flow continuity is verified. The problem can therefore be expressed as a system of linear algebraic equations $Ax=b$, where $x=P$ is the pressure vector and A a tridiagonal matrix.Some additional improvements are proposed in recent models, such as refined expressions of the kneading block flow in [54] or a new geometric definition of parameters ${K}^{f}$ and ${K}^{d}$ in [55], as well as the assumption that flows inside the kneading block are only described by pressure gradients.

## 5. Distributed Parameter Models

- Solid conveying zone:This zone is characterized by a varying filling ratio $f<1$. Material accumulates or flows along this zone during the transient period. Mass and energy balances are therefore expressed by the following equations:
- −
- Mass balance:$$\frac{\partial f}{\partial t}=-\xi N\frac{\partial f}{\partial x}$$
- −
- Energy balance:$$f\frac{\partial T}{\partial t}=\frac{\varphi {c}_{s}{N}^{2}{\eta}_{s}}{2V\rho {c}_{p}}+\frac{f{U}_{s}{c}_{e}\xi}{2V\rho {c}_{p}}({T}_{b}-T)-\xi Nf\frac{\partial T}{\partial x}$$

- Melt zone:Unlike the solid conveying zone, the forward flow is considered as constant since the density itself is constant and the zone completely filled. In the following equations, the backward flow is associated with the pressure flow between two parallel plates assuming that the screw gaps are much smaller than the screw diameter. The net flow rate at any location in the melt zone equals the output flow at the die, ${Q}_{d}$. The mass and energy balance equations are:
- −
- Mass balance:$${Q}_{d}=2VN\rho -B\rho \frac{1}{\eta}\frac{\partial P}{\partial x}$$Determination of this pressure is possible using the following equation:$$\frac{\partial P}{\partial x}=\frac{\eta (2VN\rho -{Q}_{d})}{\left(B\rho \right)}$$
- −
- Energy balance:$$\frac{\partial T}{\partial t}=\frac{{c}_{m}{N}^{2}\eta}{2V\rho {c}_{p}}+\frac{U{c}_{e}\xi}{2V\rho {c}_{p}}({T}_{b}-T)-{Q}_{d}\frac{\xi}{2V}\frac{\partial T}{\partial x}$$

## 6. Population Balance Modeling and Discrete Element Modeling

## 7. Data-Driven Models

**Transfer function models:**The input-output behavior of the process around a specific operating point can be described by a transfer function of the form:$$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{{b}_{m}{s}^{m}+{b}_{m-1}{s}^{m-1}+...+{b}_{0}}{{a}_{n}{s}^{n}+{a}_{n-1}{s}^{n-1}+...+{a}_{0}}$$There exist many techniques to build transfer function models based on output responses to input solicitations, and some of them are summarized in [140]. Examples of single-screw extruder modeling can also be found in [141,142,143] and serve the development of more specific techniques related to twin-screw configurations, as in [19,20,144,145].Common features of these works include:- −
- parameter identification based on input step changes;
- −
- first- or second-order transfer functions;
- −
- inclusion of time delays in some input-output relations (feed flow/die pressure, moisture content/die material temperature, etc).

In [146], a comparative study of different parameter identification methods is achieved. Different algorithms are analyzed in the context of cooking extrusion. The couple “screw rotation speed/motor load” supports relay feedback as a well-performing online method, while batch output error parameters provide an accurate modeling across a large range of operating points.The main advantages of transfer function models are their simplicity and the availability of a broad array of well-established control techniques, including PID control. However, transfer function models assume that the process behavior is linear in a limited range of operation. If several distinct operating points have to be considered, different transfer functions will usually have to be identified for each of them. In this latter case, gain-scheduling or adaptive controllers will be required.**Neural networks (NN):**Standard feedforward NNs basically define a nonlinear static map between a selected number of inputs (u) and outputs (y):$${y}_{i}=f\left({u}_{i}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\in I$$One of the most common NN architectures in system modeling is the perceptron [147] consisting of an on/off static function (called the activation function or decision function) delivering a binary output (e.g., zero or one). The sum of a weighted input linear combination is compared to a threshold separating the activation and inactivation zones as in:$${y}_{i}=\left\{\begin{array}{c}0\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}{u}_{i}\le 0\hfill \\ 1\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}{u}_{i}>0\hfill \end{array}\right.$$Perceptron networks may be built using multiple layer structures where all of the neuron outputs depend only on the inputs from the previous layer and do not interact with the same-layer neurons. These structures are called multilayer perceptrons (MLP; [148,149]; Figure 25). The nonlinearity used in the corresponding activation function is continuous (for instance, sigmoid or Gaussian functions). A significant advantage of multilayer NN models is their ability to exploit information contained in every available measurement from nonlinear processes, while, however, the main detrimental consequences are the huge required amount of data to reach acceptable training performances (i.e., the ability to reproduce correctly the outputs) and the fast increasing number of parameters following the addition of layers.The first and the last layers, respectively called the input and output (visible) layers, are distinguished from the intermediate ones, also called the hidden layers. Many other neural network structures, static or dynamic (i.e., using temporally-shifted inputs and outputs), exist, such as the radial basis function network, which, for instance, has proved quite useful in modeling bioprocesses [150] or perceptron-based structures applied to pattern recognition [151,152]. These structures generally differ from each other by the activation principle and the training rules.Steady progress in computational technology allows neural networks to be implemented on-line on many experimental plants [153,154]. SISO (single input single output) or MIMO (multi inputs multi outputs) NNs can be used to describe input-output relationships [155,156,157]. In [158], a relation between classical input variables (screw speed, feed flow rate, feed moisture, moisture content, die diameter and temperature) and outlet physical and chemical product properties (radial expansion, density, bulk density, water adsorption and solubility indexes) is established. The selected black-box model is a static three-layered feed-forward network with logistic activation functions trained by a backpropagation algorithm [148]. Results show that the method is performing well in reproducing specific material properties and could aim at obtaining generalized models predicting required input parameters to get the desired product properties.

## 8. Conclusion

- explicit solutions of the mass balance equations, i.e., functions of time describing the residence time distribution;
- systems of differential equations, resulting from mass and energy balances expressed around a representation of the systems in the form of continuous-stirred reactors in series;
- systems of partial differential equations resulting from mass and energy (possibly momentum) balances in a distributed parameter representation of the system;
- PBM-DEM methods, including material properties in differential equation systems describing each particle characteristic evolutions using Newton’s second law of motion and the angular momentum
- black-box representations of the system, e.g., transfer functions or neural networks linking input and output operational variables, inferred from experimental data and using little or no a priori physical knowledge of the system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Measurement of the residence time distribution (RTD) at $N=100$ RPM and ${Q}_{in}=0.358$ kg/h.

**Figure 4.**Plug flow reactor, continuous stirred tank reactor and actual reactor residence time distributions.

**Figure 5.**Modeling of the residence time distribution through cascades of continuous stirred-tank reactors (CSTRs) or cascades of plug-flow reactor and CSTRs.

**Figure 6.**Discretization of the extrusion device in volume elements, subdivided in cascades of ideal CSTRs.

**Figure 11.**(

**a**) Five leakage locations in a twin screw extruder; (

**b**) Flow network in a completely filled zone along one pitch length for a single-lobed screw (rectangular boxes represent C-shaped elements; square boxes represent inter-meshing elements).

**Figure 13.**CTR-in-series model validation. Solid line: model prediction; dashed line: experimental data.

**Figure 14.**Die pressure ${P}_{end}$ and outlet flow ${Q}_{out}$ evolution following a step change in the screw rotation speed.

**Figure 15.**Die pressure ${P}_{end}$ and outlet flow ${Q}_{out}$ evolution following a step change in the feed flow.

**Figure 19.**Partial differential equation (PDE) model validation. Solid line: PDE model; dashed line: experimental data.

**Figure 20.**Die pressure ${P}_{end}$ and outlet flow ${Q}_{out}$ evolutions following a step change in the screw rotation speed.

**Figure 21.**Die pressure ${P}_{end}$ and outlet flow ${Q}_{out}$ evolutions following a step change in the feed flow step.

**Figure 23.**Multi-scale approach using discrete element modeling (DEM) and population balance modeling (PBM).

**Figure 26.**Die pressure ${P}_{end}$ and outlet flow rate ${Q}_{out}$ evolution following a step change in the screw rotation speed. Solid line: transfer function model; dashed line: experimental data.

**Figure 27.**Die pressure ${P}_{end}$ and outlet flow rate ${Q}_{out}$ evolution following a step change in the input flow rate. Solid line: transfer function model; dashed line: experimental data.

**Figure 28.**Division of the extrusion system in different sections and moving boundaries of the filled zones.

Screw Geometry | Parameters | Values |
---|---|---|

Screw exterior diameter: ${D}_{ext}$ | 18 mm | |

Spacing screw: ${C}_{l}$ | 13 mm | |

Screw pitch: ξ | 11 mm | |

Screw threads number | 1 | |

Total length: L | 150 mm | |

Die geometry | Parameters | Values |

Die diameter | 2.5 mm | |

Die length | 6 mm | |

Thermal properties | Parameters | Values |

Specific heat capacity: ${c}_{p}$ | 5000 J/(kg K) | |

Thermal conductivity: ${\lambda}_{T}$ | 0.016 W/(m K) | |

Material/barrel heat exchange coefficient: U | 1200 W/(m${}^{2}$ K) | |

Barrel temperature: ${T}_{b}$ | 140 °C | |

Material property | Parameter | Value |

Density: ρ | 550 kg/m${}^{3}$ | |

Yasuda-Carreau equations | Parameters | Values |

${\eta}_{0}$ | $1.11{e}^{7}$ $Pa\xb7s$ | |

b | 0.022 | |

λ | 0.12 | |

a | 10 | |

${n}_{p}$ | 0.251 |

Parameters | Values |
---|---|

n | 4 |

${t}_{min}$ | 5.62 |

d | 0.063 |

**Table 3.**Expressions of the forward and backward flow rates where ${K}^{f}$ and ${K}^{b}$ are screw geometric parameters, P is the pressure, f the filling ratio and V the reactor volume.

Reactor i | ${\mathit{Q}}_{\mathit{i}}^{\mathit{f}}$ | ${\mathit{Q}}_{\mathit{i}}^{\mathit{b}}$ |
---|---|---|

Direct pitch | ${K}_{i}^{f}{f}_{i}{V}_{i}$ | if ${f}_{i}$ or ${f}_{i-1}=1$ : ${K}_{i}^{b}({P}_{i}-{P}_{i-1})$ |

Reverse pitch | ${K}_{i}^{f}{f}_{i}{V}_{i}$ | if ${f}_{i}$ or ${f}_{i-1}=1$ : ${K}_{i}^{b1}({P}_{i-1}-{P}_{i})$ |

if ${f}_{i}$ or ${f}_{i+1}=1$ : ${K}_{i}^{b2}({P}_{i}-{P}_{i+1})$ | ||

Die | ${K}_{n}^{f}({P}_{n}-{P}_{0})$ | ${K}_{n}^{f}({P}_{n}-{P}_{n-1})$ |

**Table 4.**Estimated parameter values for the model of [35].

Parameters | Values |
---|---|

n | 13 |

${A}_{f}$ | 0.6 |

${A}_{b}$ | 0.16 |

Parameters | Values | Absolute Confidence Intervals at 95 % |
---|---|---|

V | $2.34{e}^{-8}$ m${}^{3}$ | ± $0.03{e}^{-8}$ m${}^{3}$ |

B | $9.72{e}^{-11}$ m${}^{4}$ | ± $0.14{e}^{-11}$ m${}^{4}$ |

D | $6.64{e}^{-6}$ m${}^{2}$/s | ± $0.34{e}^{-6}$ m${}^{2}$/s |

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Grimard, J.; Dewasme, L.; Vande Wouwer, A.
A Review of Dynamic Models of Hot-Melt Extrusion. *Processes* **2016**, *4*, 19.
https://doi.org/10.3390/pr4020019

**AMA Style**

Grimard J, Dewasme L, Vande Wouwer A.
A Review of Dynamic Models of Hot-Melt Extrusion. *Processes*. 2016; 4(2):19.
https://doi.org/10.3390/pr4020019

**Chicago/Turabian Style**

Grimard, Jonathan, Laurent Dewasme, and Alain Vande Wouwer.
2016. "A Review of Dynamic Models of Hot-Melt Extrusion" *Processes* 4, no. 2: 19.
https://doi.org/10.3390/pr4020019