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# Introduction

Between July 19 and July 26 2015, the Department of Mathematics of the IST-UL and the Department of Mathematics of the University of Coimbra organised the ECMI-Modelling Week 2015. The aim of this week was to model a real world problem as accurate as possible and simulate it on the computer. The final goal would be achieved by presenting some of the output graphs and discuss the main conclusions of each project with the rest of the participants.

There were 10 projects overall. Each project was given by professors and researchers from the European Consortium of Mathematics in Industry (ECMI) learning institutions, that worked as mentors of each group of students that they were given. Each student was given an introduction to each project prior the Modelling Week. Given each one’s preference the groups were made based on a rule that no group shall have students from the same ECMI university.

The present work is the final report of the project group entitled Modelling drying in paper production. This project was proposed by Professor Gonçalo Pena, CMUC, Universidade de Coimbra. The students’ team was composed by Björn Nilsson, Endtmayer Bernhard, João Reis, Merina Latbi, Sonia Vivace. In this project, we aimed at developing and simulating a partial differential equations model for the drying of paper soaked in water or resin inside a drying tunnel.

In paper industry, a drying tunnel is composed of several paper dryers (or paper ovens) align in a chain and connected by small junctions. In this project we consider 7 dryers, See FIGURE. The paper rolls over several dryer cans that spin, passing through the dryers at a very high speed. The temperature in each dryer is crucial in order to obtain a certain type of paper sheets. This temperature is set independently in each dryer by an heat source, but preferable, it is higher in the middle ovens, decreasing as approaching the extremities of the chain. Each dryer is also built with isolated walls, so the inside air is independent of the outside’s. We also consider a fan placed on the top of each dryer in the same place as the heat source.

Due to the complexity of the main goal, and the short amount of time the team had to work on it, it was not possible to reach a solution for the main problem. This was mainly because the model that describes the moisture of the paper along the dryers is a formidable problem to reach in three days. Given that we hadn’t had any such model by the third day, we decided it would be worth to simplify the problem in order to proceed with a numerical simulation. That so, in this work we determine the temperature profile of the air inside the dryer’s chain, as well as its velocity along this same chain. In particular, we are interested in the profile of the temperature on the bottom of each dryer, as it is the area in contact with the paper. Due to the high velocity of the paper, we assume that the impact of the paper on the air is negligible.

The mathematical formulation of our problem is described by the following heat equation,

$\frac{\partial T}{\partial t} +\underbrace{\nabla \cdot (\mathbf{u} T)}_\text{convective term} - \underbrace{k \Delta T}_\text{diffusive term}=0$

• $$k$$ is the thermal diffusivity of air ($$1.9 \cdot 10^{-5} \, m^2/s$$)

• $$\mathbf{u}$$ m/s is the air velocity field ($$0.0001 m/s$$)

• $$T$$ is the temperature (measured in Kelvin)

Our main goal is of course to obtain $$\textbf{u}$$. The explanation of the model will be done in section SEE SECTION, however, in order to determine $$\textbf{u}$$, one shall first compute the velocity $$\textbf{v}$$. To do so, we introduce the Navier-Stokes Equations in next section. The numerical method used will be the Fnite Element Method (FEM), as it is a standard method solving such problems. A description of this method will be done in section SEE SECTION. Finally, we have used the software FreeFem++ to simulate the model using FEM, and SOMETHING I DONT REMEBER THE NAME for nicer graphs and figures.

All the students involved in this project would like to congratulate the outstanding week the ECMI, the Department of Mathematics of the IST-UL and the Department of Mathematics of the University of Coimbra has provided to us. Also, we would like to acknowledge Professor Gonçalo Pena for his guidance and support throughout all the project.

# the heat equation

The law of the temperature in the ovens is given by the heat equation: \begin{aligned} \begin{array}{ll} \frac{\partial T}{\partial t} - \alpha \Delta T= - \nabla (T v) \end{array} \label{eq_T}\end{aligned} Where T is the temperature, v the velocity and $$\alpha$$ is the diffusivity coefficient.
This equation is deduced by a balance made over the heat.
Temperature dynamics is controlled by two phenomena :

• Heat has to be harmonized and heat is directed where the temperature is the lowest. This transport term is given by the vector $$j_{diffusion} = -k \nabla T$$, it corresponds to the Fourrier Law.

• Temperature is moving through the movement of the air particules. It is given by the vector : $$j_{convective} = C_v T v$$.

Thus, the heat flow is given by: $$j_Q = j_{diffusion} + j_{convective} = -k \nabla T + C_v T v$$.
Where $$C_v$$ is the calorific capacity of the air. It’s assumed to be constant.
The heat flow is a continuous variable, and it’s continuous at the interface between the ovens.
By using the conservation law inside the field for the heat $$Q$$, we get : \begin{aligned} \frac{\partial Q}{\partial t} + div(j_Q) = 0.\end{aligned} Where $$Q =C_v T + Q_0$$
By rewritting this expression and expliciting the temperature, we get the expression (\ref{eq_T}): \begin{aligned} \begin{array}{ll} 0 = \frac{\partial (C_a T + Q_0)}{\partial t} + div(-k \nabla T + C_v T v) \\ 0 = C_a \frac{\partial T}{\partial t} - k \Delta T + C_v div(T v) \\ \text{by dividing by C_a and taking}\ \alpha = \frac{k}{C_v}: \\ 0 = \frac{\partial T}{\partial t} - \alpha \Delta T + div(T v) \end{array}\end{aligned}