# Mathematics in paper

- Jari Hämäläinen
^{1}Email author

**2**:2

https://doi.org/10.1186/2190-5983-2-2

© Hämäläinen; licensee Springer 2012

**Received: **6 July 2012

**Accepted: **27 July 2012

**Published: **6 August 2012

## Abstract

This paper aims to give a general overview of industrial mathematics applied to papermaking. Modelling challenges vary from computational fluid dynamics (CFD) to finite-element analyses (FEA) when the paper web transforms from a multiphase flow to a solid fibre network - to a ready paper. Also, different length scales are present from fibre level to machine level problems, *i.e.* from millimetres up to one hundred metres. Mathematical modelling of papermaking is aiming at optimizing the process and the end-product. Thus, computational tools for optimal shape design and optimal control purposes have been developed.

## 1 Introduction

*in*paper, that is, what mathematics is behind papermaking. Paper is made from wood fibres mixed in water with some chemical and filler ingredients called as fibre suspension. A modern paper machine is producing a two kilometres long and ten metres wide paper web in a minute with extremely evenly distributed solid materials (see Figure 1). A papermaking production line begins from a so-called headbox which distributes a dilute (1% solids content) fibre suspension coming from a pump evenly across the width of a paper machine. A ten metre wide and one centimetre thick free jet flies from a headbox to a forming section between porous wires and most of water are removed in fractions of a second. After mechanical dewatering the solid contents of a wet paper web increases up to 50%. The rest of water is removed by evaporation in a drying section. Paper machines vary depending on paper grades they produce, that is, a paper machine for tissue paper is different from a machine for copy paper or newspaper, but the basic concept is the same from a headbox to a dryer.

## 2 Mathematical modelling of fibre suspension flows in multiple scales

## 3 Optimized fluid flows

*i.e.*, one flow field simulation, takes a few seconds in HOCS Fibre when full three-dimensional Navier-Stokes solution takes hours. HOCS Fibre has been used to tens of paper mills and the results are illustrated in Figure 4. The red markers illustrate the fibre orientation misalignment before optimization and the green markers after optimization. These are measured from a ready paper before and after HOCS Fibre optimizations.

There are also aims to model the whole paper mill and to optimize it. Now, it is not anymore so clear what optimal means, that is, there are plenty of objectives to be considered simultaneously. This leads to multi-objective optimization which is a basis for decision-support systems for papermaking [4]. Mathematical models of the whole mill are further simplified from the model of the headbox fluid flow optimizers. In addition, paper quality parameters are also prediction based on statistical correlations derived from measured data. So, there are tens of algebraic correlations in addition to first-principle process models.

These optimization examples illustrate that there are not ‘good’ or ‘bad’ models, but different levels of complexity can be used in different applications.

## 4 From rheology of fluids to rheology of solids

High-concentration fibre suspension can be considered as a generalized-Newtonian fluid obeying a shear-thinning viscosity model. It may also include memory effects. Rheological behaviour can also be detected for a ready paper [5]. Several studies have shown that a paper sheet is an elasto-visco-plastic material. And, mechanical properties and corresponding material parameters of the paper depend strongly on the fibrous structure of the paper sheet, which in turn, has been determined by fluid dynamics in the beginning of the papermaking process, in the headbox.

## 5 Concluding remarks

Papermaking is full of mathematics. Some examples of fluid dynamical and solid mechanical modelling were given in this article. Model-based optimization tools are also natural applications for mathematical models. But, plenty of mathematical modelling of papermaking was left out, *e.g.* the whole papermaking chemistry as well as free-surface flows, fluid-structure interactions (FSI) and phase-change problems (*e.g.* evaporation). And, totally different area of mathematics is the economy of papermaking. Pulp and paper industry is facing outstanding challenges in reducing investment and operating costs while aiming at top-quality products and increased production. Today this is further challenged by competition of raw materials with bio-energy and bio-fuel productions and other new wood-based products.

When you are writing mathematical formulae on a paper sheet next time, you may remember that there is already a lot of mathematics in the paper sheet.

## Declarations

## Authors’ Affiliations

## References

- Hämäläinen J, Lindström S, Hämäläinen T, Niskanen H:
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**CFD-based optimization for complete industrial process: papermaking.**In*Optimization and Computational Fluid Dynamics*. Edited by: Thevenin D, Janiga G. Springer, Berlin; 2008.Google Scholar - Avikainen M, Hämäläinen J, Tarvainen P:
**HOCS Fibre: CFD-based software for fibre orientation profile optimization for conventional and dilution headboxes.***Nord Pulp Pap Res J*2010,**25**(4):456–462. 10.3183/NPPRJ-2010-25-04-p456-462View ArticleGoogle Scholar - Hämäläinen J, Madetoja E, Ruotsalainen H:
**Simulation-based optimization and decision support for conflicting objectives in papermaking.***Nord Pulp Pap Res J*2010,**25**(3):405–410. 10.3183/NPPRJ-2010-25-03-p405-410View ArticleGoogle Scholar - Hämäläinen J, Eskola R, Erkkilä A-L, Leppänen T:
**Rheology in papermaking - from fibre suspension flows to mechanics of solid paper.***Korea-Australia Rheol J*2011,**23**(4):211–217. 10.1007/s13367-011-0026-2View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.