Mathematical modeling of mechanical dewatering in the press section - Paper Technology International 2023 - Paper Technology International 2023 - Flipbook - Page 84

Mathematical modeling of mechanical dewatering in the press section - Paper Technology International 2023

PAPERTECHNOLOGYINTERNATIONAL
Figure 3: Multi scale
approach to the
simulation of the
papermaking process.
Simulation approach
Our modeling covers the full paper machine from the
initial dewatering of the paper suspension, over the mechanical
dewatering of the preformed paper web up to the thermal treatment
of the paper web in the dryer section. To achieve this, we use a
multiscale approach, as shown schematically in reference (see
Figure 3) with models covering the whole range of length scales
from the micrometer scale of fibers and fillers to the several
hundred meters of a full-scale paper machine. To obtain the needed
geometry and parameters for our models, lab measurements
especially tomographic imaging as well as virtually generated
structures are used. On the micrometer scale, the effect of fines
and fillers is included by modifying, parameters such as dewatering
behavior or the pore volume distribution.
Different kinds of mathematical modeling and simulation
tools for different length scales enable us to determine the
necessary parameters and derive the required material models as
described in the following sections. On the paper machine scale we
use an in-house process simulation library that allows us to simulate
complete sections of the paper machine. Here, we follow a modular
concept starting from individual flow and material transport modules
that are then combined to larger units, e.g., suction boxes or a press
nip. It ensures fast and efficient calculations by reducing the level
of detail and defining effective parameters describing the important
aspects of the materials. Finally, we combine the different section
models to describe the full length of the paper machine.
Image-based structural analysis and flow analysis
Realistic material parameters for both press fabrics and
the paper web are required as input parameters for the process
simulation model of the press section. The availability of highaccuracy tomographic imaging together with fast and efficient
numerical methods to study the mechanical and fluid dynamic
properties of the fiber web and paper machine fabrics enable a
detailed view of the structures and properties of the materials. In
CFD simulations, mass, momentum and energy balance equations
are solved with numerical methods based on a calculation grid which
describes the porous structures of the press fabric or the paper
web. As a result, flow velocities v, concentrations c and pressure
p distributions inside the pore volume of the press fabrics can be
visualized in detail (see Figure 4). Studies of the fabric in different
compression stages allow the determination of the compressed
pore structure and its physical properties. The absolute and relative
permeabilities κ and κ_r, which are related to pore size distribution
and water saturation respectively, are derived from Darcy’s law [3]
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with μ being the dynamic viscosity of the fluid and ∂p/∂x being
the local pressure gradient. Darcy’s law describes the two-phase
flow of water and air, where water is described as incompressible
medium, while air is compressible following the ideal gas law. These
parameters are then used to describe the dewatering capability of
the porous material during the process of mechanical dewatering
within the process simulation.
Image-based analyses provide additional information on the pore
size distribution, porosity and capillary pressure curves as a function
of the water saturation. Porosity and permeability are related, e.g. by
power-laws, depending on the results of the image analysis. For the
process simulation of the press section this relationship is of great
importance.
Figure 4: Flow Field of press fabric subject to a laminar flow.
Effective material models
With the small-scale simulations described in the above
section, we obtain the effective parameters for the process
simulation, describing the important aspects of the materials
involved in the dewatering process. This is necessary because the
complete process of paper dewatering cannot be modeled on this
high level of detail. One of these models is described by Terzaghi’s
principle [5], where the total pressure is divided into hydrostatic
pressure and structural stress and thus coupling flow and structural
properties of the porous materials.
Other examples of our material models, as shown in
Figure 5, are capillary pressure relations and relative permeability
curves. These flow properties are archived by fitting, for example, a
Brooks-Corey model [6] on the results obtained by CFD simulations
described in the section above.

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