Theoretical analysis of LC-refining – pressure screening systems in TMP

Abstract

LC refining of mechanical pulps has proven to save energy in the production of TMP pulps. However, the specific role of LC refining as part of a TMP system has not been thoroughly studied since it is difficult to conceive any particular system at industrial-scales and impractical at pilot-scales. In this study, pressure screening and LC refining models that describe fibre length distributions, together with correlations to predict refining power were used to model three basic refining systems. From the simulation results, the impact of important variables such as reject ratio, refiner gap and refining net-power was studied. Performance curves of length-weighed average fibre length were generated from simulation results and were used to assess each system behaviour and also to make comparisons between systems. Data from an industrial scale TMP mill sub-system was gathered and compared to simulation results showing relative errors between 0–18 % on the predicted variables.

Introduction

The main advantage of Thermomechanical Pulp (TMP) is its high yield, where between 94–98 % of wood chips are converted to pulp. However, TMP processes are very inefficient. Theoretical estimations claim that a very small part of the energy spent, about 13 %, is actually used for wood chip defibration and fibre development (Kerekes 2015).

Pilot-plant scale studies have shown that it is possible to reduce energy consumption by replacing the secondary High Consistency (HC) stage with Low Consistency (LC) for a given tensile strength target. In general, energy savings between 100–200 kWh/t are achieved by using LC either in the main line or reject line (Cannell 2002, Sandberg et al. 2009, Luukkonen et al. 2010, Andersson 2011, Gorski et al. 2012, Sandberg et al. 2017a). Despite the benefits in energy savings, fibre shortening during LC refining reduces tensile strength, limiting the amount of energy that can be saved. Moreover, it has been also observed that LC refining produces a slightly inferior pulp in terms of tear index and light scattering (at the same tensile strength index) whereas the freeness level is similar compared to the HC refined pulps.

LC refining is becoming more popular and extensively used nowadays at different positions in TMP lines. Despite its current popularity, the holistic role of LC refining as part of a TMP system has not been thoroughly understood mainly because there is no systematic way of studying refining systems. Perhaps the logistic and economic demand of conceiving a particular system at industrial scales discourages researchers and slows down the pace at which meaningful results are generated. Moreover, refining and pressure screening can be combined in a wide range of ways that leads to a significant amount of possible configurations. Additionally, mechanical pulping is carried out according to specific needs to meet pulp quality specifications that vary from case to case which translates to a large amount of systems to consider. Finally, complex interactions arise when LC refiners and fractionation stages are combined exacerbating the problem from a system-design point of view.

These are some of the most relevant studies concerning refining system-design at industrial scale: Sandberg et al. (2009) reported their experiences installing a brand-new LC refiner in a TMP line. In general, they observed the electrical energy consumption to be 40–50 % lower than the conventional HC refining at a certain tensile index. Additionally they observed lower light scattering values and increased screening availability. Later on, Sandberg et al. (2017a) studied several TMP configurations at industrial scale to assess energy efficiency and pulp quality for each configuration. They observed that LC improved the energy efficiency at a given tensile strength index when used in both the main and reject line but produced lower light scattering and fibre length. Recently, Sandberg et al. (2017b) described modifications done to a conventional TMP line to produce newsprint grade pulp at 1500 kWh/t; following process intensification concepts, they managed to remove all fractionation stages by implementing chip impregnation, improved chip refining and LC refining resulting in less unit operations.

There have also been studies done at pilot scales involving some aspects of system-design. For instance, Andersson et al. (2012) studied the effect of long fibre concentration (measured by the R14+R28 Bauer McNett fractions) on LC refining of mechanical pulps; they observed that increasing the long fibre concentration allows better refiner loadability and higher tensile index increase. Miller et al. (2017) studied the effects of fractionation via pressure screening in multistage refining; they concluded that fractionation allows a better refiner loadability and enables harsh treatments to fibres without excessive fibre shortening. Lemrini et al. (2015) fractionated and HC and LC refined pulp to then further refine the screened pulp in LC–HC stages at different energy split ratios. They found that fractionation allowed more energy to be applied via LC refining without excessive fibre shortening but lower tensile strength index was observed compared to HC refined pulp. The term loadability is recurrent in some refining studies and it is used to describe the increase in refining power due to an increase on the refiner’s feed fibre length.

The objective of this work is to broaden the understanding of refining systems when LC refiners and pressure screens are coupled. This is done by using existing models and correlations to simulate and study systems composed of refining and screening operations.

System modeling

A system composed merely of an LC refiner is a complex system to analyze since it is by nature a heterogeneous process involving a heterogeneous raw material. Additionally, developing a mathematical model capable of accurately describing certain pulp properties changes due to refining is challenging. Part of the difficulties faced during modeling is that often times the properties of interest are not conservative (e. g. tensile, bulk, freeness). On the other hand, modeling conservative properties (e. g. fibre length) is an easier task. In the case of fibre length, screening models and comminution models (for LC refining) actually describe Fibre Length Distribution (FLD) changes using mass balances as a main tool of analysis.

In this work, three basic refining systems are studied. The systems consist of one LC refiner and one pressure screen arranged in different configurations namely Reject-refining, Feed-back rejects and Feed-back reject-refining. Figure 1 illustrates these three systems and provide further details on the configurations and description. In the subsequent sections pressure screening and LC refining comminution model are reviewed, followed by other important definitions and correlations needed for this study.

Figure 1.

Block diagrams of the three refining systems studied. (a) Reject-refining: Initial pulp is screened and the reject fraction is then refined to be later on re-combined with the accept fraction. (b) Feed-back rejects: Initial pulp is refined, then fractionated and the rejects are pumped back to the refiner inlet. (c) Feed-back reject-refining: Initial pulp is screened and the reject fraction is then refined to be later on pumped back to the screen feed.

Screening models

Pressure screening is a fractionation method where a pulp feed stream is separated into two different streams, each with different FLD. In pressure screening fibres are sorted mainly by fibre length (Scott and Abubakr 1994) and to some low extent based on fibre flexibility (Young and Olson 2004). Figure 2 shows a block diagram of a pressure screening operation.

Figure 2.

Block diagram depicting a pressure screening operation. A initial pulp stream is fed to the screen and is fractionated into an accepts stream and rejects stream. Q is volumetric flow rate; C is consistency; $\{{\mathit{y}}_{\mathit{i}}\}$ is FLD vector; the superscripts f, a and r refer to feed, accepts and rejects respectively.

Olson and coworkers (1998, 1998, 2000, 2001) developed models for pressure screening by performing simple mass balances around a cross section element of a screen assuming plug flow. Thus, performance equations are derived to describe the accepts and rejects FLD in terms of volumetric reject ratio ${\mathit{R}}_{\mathit{v}}$ and a passage ratio function $\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}$. The former is just the ratio between the reject volumetric flow rate ${\mathit{Q}}_{\mathit{r}}$ and feed volumetric flow rate ${\mathit{Q}}_{\mathit{f}}$ (${\mathit{R}}_{\mathit{v}}={\mathit{Q}}_{\mathit{r}}\mathrm{/}{\mathit{Q}}_{\mathit{f}}$). The latter one characterizes the fibre passage through a single screen aperture as a function of fibre length (or bin size). It is worth mentioning that $\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}$ is dependent on the screen basket geometry, rotor design and speed among others, and must be experimentally determined. Assuming a constant coarseness for all fibres within the pulp and denoting ${{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}$, ${{\mathit{y}}_{\mathit{i}}}^{\mathit{r}}$ and ${{\mathit{y}}_{\mathit{i}}}^{\mathit{a}}$ elements in the FLD of the feed, rejects and accepts respectively, according to Olson et al. (1998) one has:

$$\begin{array}{rl}{{\mathit{y}}_{\mathit{i}}}^{\mathit{r}}& =\left({\displaystyle \frac{{{\mathit{R}}_{\mathit{v}}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}-1}}{\sum _{\mathit{i}=1}^{\mathit{p}}{{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}{{\mathit{R}}_{\mathit{v}}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}-1}}}\right){{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}\end{array}$$ (1a)

$$\begin{array}{rl}{{\mathit{y}}_{\mathit{i}}}^{\mathit{a}}& =\left({\displaystyle \frac{1-{{\mathit{R}}_{\mathit{v}}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}}}{1-\sum _{\mathit{i}=1}^{\mathit{p}}{{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}{{\mathit{R}}_{\mathit{v}}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}}}}\right){\mathit{y}}_{\mathit{i}}^{\mathit{f}}\end{array}$$ (1b)

In addition to sort fibres, pressure screens change the consistency (Olson et al. 1998); rejects are thickened and accepts are diluted. These changes in consistency can also be described using $\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}$, ${\mathit{R}}_{\mathit{v}}$ and ${{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}$ as:

$$\begin{array}{rl}{\displaystyle \frac{{\mathit{C}}_{\mathit{r}}}{{\mathit{C}}_{\mathit{f}}}}& ={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{p}}{{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}{{\mathit{R}}_{\mathit{v}}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}-1}\end{array}$$ (2a)

$$\begin{array}{rl}{\displaystyle \frac{{\mathit{C}}_{\mathit{a}}}{{\mathit{C}}_{\mathit{f}}}}& ={\displaystyle \frac{1-{\mathit{R}}_{\mathit{v}}{\mathit{C}}_{\mathit{r}}\mathrm{/}{\mathit{C}}_{\mathit{f}}}{1-{\mathit{R}}_{\mathit{v}}}}\end{array}$$ (2b)

The ratio ${\mathit{C}}_{\mathit{r}}\mathrm{/}{\mathit{C}}_{\mathit{f}}$ is usually regarded as the reject thickening ratio T. Thus, by knowing the passage ratio function $\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}$, ${\mathit{R}}_{\mathit{v}}$ and ${{\mathit{y}}_{\mathit{i}}}^{\mathit{f}}$, it is possible to fully describe changes in FLD during pressure screening.

Comminution model in LC refining

Fibres undergo a series of strains during LC refining resulting in various structural changes, including fibre length shortening. Changes in FLD can be analyzed using a comminution model. Rubiano Berna et al. (2018a) developed a parametrization of the comminution model to predict FLD changes during LC refining. The parameters describing the selection function and breakage function were found to be strongly correlated to refiner gap and plate geometry. The comminution model described by Rubiano Berna et al. (2018a) considers fibres in the range $\mathit{l}=[0.2\mathrm{,}4.0]$ mm because objects smaller than 0.2 mm are generated by other mechanisms than cutting (fines generation by removing fractions of cellulose from the fibre wall). Thus, the fines generation can not be modelled using a comminution model and therefore were not taken into consideration in this study.

By denoting ${{\mathit{y}}_{\mathit{i}}}^{\mathit{o}}$ and ${{\mathit{y}}_{\mathit{i}}}^{\mathit{r}\mathit{r}}$ elements in the FLD of the initial and refined pulp respectively, based on Figure 3, the comminution model equations are:

$$\begin{array}{rl}\{{\mathit{y}}_{\mathit{i}}^{\mathit{r}\mathit{r}}\}& ={exp}_{\text{m}}[\mathit{\pi }\mathit{\alpha }\mathit{\beta }{\displaystyle \frac{\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2}{\mathit{Q}}}\mathrm{(}1-{\mathit{\lambda }}^2\mathrm{)}\mathbf{\text{A}}]\{{\mathit{y}}_{\mathit{i}}^{\mathit{o}}\}\end{array}$$ (3a)

$$\begin{array}{rl}\mathbf{\text{A}}& =\mathit{f}\mathrm{(}\text{gap}\mathrm{,}\mathit{\alpha }\mathrm{)}\end{array}$$ (3b)

$$\begin{array}{rl}\mathit{\lambda }& ={\displaystyle \frac{{\mathit{R}}_{\mathit{i}}}{{\mathit{R}}_{\mathit{o}}}}\end{array}$$ (3c)

Where ${exp}_{\text{m}}$ denotes the matrix exponential operation, Q is the volumetric flow rate, ω is the rotational speed of the refiner, ${\mathit{R}}_{\mathit{i}}$ and ${\mathit{R}}_{\mathit{o}}$ the refiner inner and outer radius respectively, $\mathit{\alpha }={\mathit{B}}_{\mathit{w}}\mathrm{/}{\mathit{B}}_{\mathit{w}}+{\mathit{G}}_{\mathit{w}}$ is plate geometry constant describing the coarseness of the plate and $\mathit{\beta }=2{\mathit{G}}_{\mathit{w}}{\mathit{D}}_{\mathit{w}}\mathrm{/}{\mathit{B}}_{\mathit{w}}+{\mathit{G}}_{\mathit{w}}$ describes the free area of flow. ${\mathit{B}}_{\mathit{w}}$, ${\mathit{G}}_{\mathit{w}}$ and ${\mathit{D}}_{\mathit{w}}$ are the plate bar width, groove width and groove depth respectively. Lastly, the function f which defines the comminution model matrix A are a family of correlations that help to define the selection function and breakage function and, as indicated in Equation (3b), it depends on gap and plate geometry.

Figure 3.

Block diagram of a single LC refining stage. $\{{\mathit{y}}_{\mathit{i}}\}$ is FLD vector; the superscripts o, and $\mathit{r}\mathit{r}$ refer to the initial pulp, and refined pulp.

Refining size aspects

Pulp refiners are usually sized and operated following field domain and heuristic. Manufacturers usually recommend operation conditions and pre-establish upper and lower limits to allow some flexibility during normal operation. As a result, there is a close relationship between the refiner size and the normal operation conditions namely flow rate Q and rotational speed ω. Figure 4 shows the recommended flow rates and rotational speeds for various refiner sizes according to Aikawa Fibre Technologies. Based on this information, the following relationships can be made:

$$\begin{array}{rl}\mathit{\omega }& ={\displaystyle \frac{31.47}{{{\mathit{R}}_{\mathit{o}}}^{0.75}}}[\text{rad}\mathrm{/}\text{s}]\end{array}$$ (4a)

$$\begin{array}{rl}\mathit{Q}& =0.37{{\mathit{R}}_{\mathit{o}}}^{2.37}[{\text{m}}^3\mathrm{/}\text{s}]\end{array}$$ (4b)

Combining Equations 4a and 4b

$$\begin{array}{rl}{\displaystyle \frac{\mathit{\omega }}{\mathit{Q}}}& ={\displaystyle \frac{85.05}{{{\mathit{R}}_{\mathit{o}}}^{3.12}}}[\text{rad}\mathrm{/}{\text{m}}^3]\end{array}$$ (4c)

Where ${\mathit{R}}_{\mathit{o}}$ is in meters. Hence the ratio $\mathit{\omega }\mathrm{/}\mathit{Q}$ is defined as a function of the external radius ${\mathit{R}}_{\mathit{o}}$.

Figure 4.

Twin-Flow disc refiner recommended flow rates and rotational speeds for various refiner sizes. Bottom x-axis: refiner outer radius ${\mathit{R}}_{\mathit{o}}$ in m; top x-axis: refiner outer diameter ${\mathit{D}}_{\mathit{o}}$ in inches; left y-axis: flow rate Q in gpm and ${\mathrm{m}}^3{\mathrm{s}}^{-1}$; right y-axis: rotational speed ω in $\text{rad}{\mathrm{s}}^{-1}$ and RPM. Dashed lines correspond to the upper and lower limits for the flow rate, whereas the solid lines correspond to the recommended values. Data taken from Aikawa Fibre Technologies.

Refining power consumption

Rubiano Berna et al. (2018b) developed a correlation to predict refining dimensionless power as a function of plate gap, refiner’s feed length-weighed mean fibre length ${\mathit{l}}_{\mathit{w}}$ and plate geometry. The use of dimensionless power proved to be suitable at comparing performance across different refiners sizes as the correlation was built entirely on pilot-scale data and validated using industrial-scale data. The correlation is written as:

$$\begin{array}{rl}{\displaystyle \frac{{\mathit{P}}_{\mathit{d}}}{{\mathit{\alpha }}^2}}& ={\left({\displaystyle \frac{{\mathit{l}}_{\mathit{w}}}{{{\mathit{l}}_{\mathit{w}}}^{\mathit{o}}}}\right)}^{{\mathit{c}}_1}{\mathit{G}}^{\mathrm{(}2-{\mathit{c}}_2\mathrm{)}}\end{array}$$ (5a)

$$\begin{array}{rl}{\mathit{P}}_{\mathit{d}}& ={\displaystyle \frac{{\mathit{P}}_{\mathit{n}\mathit{e}\mathit{t}}}{\mathit{\rho }{\mathit{\omega }}^3\mathrm{(}{\mathit{R}}_{\mathit{o}}^5-{\mathit{R}}_{\mathit{i}}^5\mathrm{)}}}\end{array}$$ (5b)

$$\begin{array}{rl}\mathit{G}& =\left({\displaystyle \frac{{\text{gap}}_{\mathit{o}}}{\text{gap}}}-1\right)\end{array}$$ (5c)

Where ${\mathit{l}}_{\mathit{w}}$ and gap are in mm, ${\text{gap}}_{\mathit{o}}=2.5$ mm, ${{\mathit{l}}_{\mathit{w}}}^{\mathit{o}}=4.664$, ${\mathit{c}}_1=2.701$ and ${\mathit{c}}_2=0.955$.

Methodology

Figure 5 shows a representative block diagram of an arbitrary system where input and output variables are indicated. This arbitrary system can be simulated using the models described before. Moreover, ideal mixing was assumed at each mixing point to predict the mixed-flows FLD. Models were implemented in MATLAB^® Simulink^® toolbox. This environment was then used to simulate the three different refining systems shown in Figure 1.

Figure 5.

Block diagram showing an arbitrary model. Input variables: initial pulp FLD $\{{\mathit{y}}_{\mathit{i}}^{\mathit{o}}\}$ and consistency ${\mathit{C}}_{\mathit{o}}$, gap and ${\mathit{R}}_{\mathit{v}}$; Fixed parameters: screen size, refiner size (${\mathit{R}}_{\mathit{i}}$ and ${\mathit{R}}_{\mathit{o}}$), plate geometry (α and β), Q and ω; Output variables: flows FLD $\{{\mathit{y}}_{\mathit{i}}^{\mathit{k}}\}$ and consistencies ${\mathit{C}}_{\mathit{k}}$, dimensionless power ${\mathit{P}}_{\mathit{d}}$ and thickening factor T.

Although it is possible to derive explicit governing equations for all systems in consideration (see Appendix A for an example), using a software to aid in the calculations is faster and more convenient. Furthermore, there may be cases where it is mathematically impossible to derive explicit governing equations (e. g. system with various refiners and screens with intricate flow combinations and recirculations).

For all systems, the input variables were varied between $\text{gap}=[0.2\mathrm{,}2.0]\text{mm}$ and $\mathit{R}\mathit{v}=[0.1\mathrm{,}0.5]$. Additionally, it was set that the initial flow and feed to the refiner were 4 % consistency, as the comminution model was built upon refining data at that consistency. Thus, it was necessary to implement a dilution flow before each screening operation and adjust it accordingly to always achieve a reject consistency of 4 %. Finally, regarding refiner size, the inner radius was set to ${\mathit{R}}_{\mathit{i}}=0.6{\mathit{R}}_{\mathit{o}}$ (a common practice done by manufacturers). and the plate characteristics were set to ${\mathit{B}}_{\mathit{w}}=1.65$, ${\mathit{G}}_{\mathit{w}}=2.29$ and $\mathit{D}\mathit{w}=6.35\text{mm}$. The screen basket was set to 0.8 mm holes.

The FLD were determined for each single flow of the system. Later on, ${\mathit{l}}_{\mathit{w}}$ values were calculated from the FLD and used to construct performance curves for each system.

Results and discussion

Refiner size effect on fibre shortening

A system composed of a single refining stage was simulated to realize the impact of the refiner size on the fibre shortening. Five different refiner sizes were considered and simulated using the comminution model. The resulting ${\mathit{l}}_{\mathit{w}}$ values were plotted against gap as shown in Figure 6. As a side note, all figures presented in this section plot ${\mathit{l}}_{\mathit{w}}$ in the y-axis and, unless otherwise is stated, ${\mathit{l}}_{\mathit{w}}$ refers to the fibre length leaving the system in consideration.

Figure 6.

Curves of ${\mathit{l}}_{\mathit{w}}$ vs. gap for five different refiner sizes. As can be seen, if manufacturer recommended volumetric flow rates and rotational speed are followed, small refiners are prone to cut to a higher extent than big refiners.

Interestingly, simulated results shows that large refiners produce less fibre shortening than small refiners. The reason for this difference in performance is due to the ratio $\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2\mathrm{/}\mathit{Q}$ which is in the argument of the matrix exponential operator of the comminution model (see Equation 3a). From Equation 3a and 4c one can derive the following relationship:

$$\begin{array}{c}{\displaystyle \frac{\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2}{\mathit{Q}}}\propto {\displaystyle \frac{1}{{{\mathit{R}}_{\mathit{o}}}^{1.2}}}\end{array}$$ (6)

The ratio $\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2\mathrm{/}\mathit{Q}$ can be split into two components: (1) a measure of pulp residence time represented in ${{\mathit{R}}_{\mathit{o}}}^2\mathrm{/}\mathit{Q}$ and (2) frequency of bar-bar crossings represented in ω. Hence, $\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2\mathrm{/}\mathit{Q}$ accounts for the probability of fibres being captured by bar-bar crossings during their time spent inside the refiner. Therefore, if manufacturer recommended flow rates and rotational speed are followed, FLD changes due to refining decrease with refiner size as deduced from Equation 6. This is perhaps, one of the reasons why some pilot-plant refining trials results do not match industrial-scale results.

Split-flows and recirculation aspects

Data presented in this subsection and onwards correspond to simulation results considering a system with a refiner size ${\mathit{R}}_{\mathit{o}}=0.254$ m (${\mathit{D}}_{\mathit{o}}=20$ inches). The idea is to depict the general behaviour of each system rather than present specific data for each refiner size considered. The systems shown in Figure 1 were simulated following the methodology described in the previous section. These three systems were compared to the performance of a single refining stage regarded from now on as the reference case (black solid-line).

Introducing a recirculation flow allows for some share of fibres to get refined more than once. This recirculation not only has an impact on the fibre shortening (as will be seen in the following sections) but also in the overall energy that is transferred to pulp. A similar situation happens in Reject-refining, where the initial pulp is split into accepts and rejects. After refining the rejects and re-combining with the accepts, the energy content of the re-combined flow is affected by the mass reject ratio at the screen. In order to describe the effect of the split ratios and recirculations on the system behaviour, a ratio E defined as the mass flow rate flowing through the refiner over the mass flow rate leaving the system $\mathit{E}={\dot{\mathit{m}}}_{\mathit{R}}\mathrm{/}{\dot{\mathit{m}}}_{\mathit{S}}$ is introduced.

In a system like Reject-refining, this ratio is fairly straight-forward to realize, however for a system with recirculation this is not done in a single step. To further explain the concept of E, consider the Figure 7 where Reject-refining is shown. It is fairly easy to conclude that $\mathit{E}={\dot{\mathit{m}}}_3\mathrm{/}{\dot{\mathit{m}}}_5$. Furthermore, if the mass flow rates are expressed as the product of volumetric flow rate and pulp consistency, one gets $\mathit{E}={\mathit{R}}_{\mathit{v}}\mathit{T}$. The product ${\mathit{R}}_{\mathit{v}}\mathit{T}$ in pressure screening is known as the mass reject ratio.

Figure 7.

Scheme of Reject-refining used to derive equations for the ratio E. ${\dot{\mathit{m}}}_{\mathit{i}}$ and ${\mathit{e}}_{\mathit{i}}$ refer to the mass flow rate and energy content respectively; ${\mathit{e}}_{\mathit{R}}$ is the energy supplied by the refiner (note the dashed-line used to represent an energy flow and to differentiate from material flows).

One can derive an alternative definition of E as shown in Equation 7. Starting by setting the initial flow to one (${\dot{\mathit{m}}}_1={\dot{\mathit{m}}}_5=1$) one has:

$$\begin{array}{c}{\dot{\mathit{m}}}_3={\dot{\mathit{m}}}_4={\mathit{R}}_{\mathit{v}}\mathit{T}\end{array}$$ (7a)

From an energy balance around the refiner:

$$\begin{array}{c}{\mathit{e}}_4={\mathit{e}}_3+{\mathit{e}}_{\mathit{R}}\end{array}$$ (7b)

Where ${\mathit{e}}_{\mathit{R}}$ is the energy supplied by the refiner or Specific Energy Consumption (SEC) (${\mathit{e}}_{\mathit{R}}$ is used to facilitate the notation) and ${\mathit{e}}_{\mathit{i}}$ refers to the energy content of the i flow or Specific Refining Energy (SRE) (again, ${\mathit{e}}_{\mathit{i}}$ is used to facilitate the notation). Performing an energy balance at the mixing point yields:

$$\begin{array}{c}{\mathit{e}}_5={\dot{\mathit{m}}}_2{\mathit{e}}_2+{\dot{\mathit{m}}}_4{\mathit{e}}_4\end{array}$$ (7c)

Now, the initial flow energy content is zero (${\mathit{e}}_1=0$) since the initial pulp is unrefined. As a consequence, ${\mathit{e}}_1={\mathit{e}}_2={\mathit{e}}_3=0$, combining Equation 7a, 7b and 7c yields:

$$\begin{array}{c}{\displaystyle \frac{{\mathit{e}}_5}{{\mathit{e}}_{\mathit{R}}}}={\mathit{R}}_{\mathit{v}}\mathit{T}=\mathit{E}\end{array}$$ (7d)

Since ${\mathit{e}}_5$ represents the energy content of the flow leaving the system (${\mathit{e}}_5={\mathit{e}}_{\mathit{S}}$), it can be concluded that E is also the ratio between the energy content of the flow leaving the system over the energy supplied by the refiner $\mathit{E}={\mathit{e}}_{\mathit{S}}\mathrm{/}{\mathit{e}}_{\mathit{R}}$ (alternatively, it can be written as $\mathit{E}=\mathit{S}\mathit{R}\mathit{E}\mathrm{/}\mathit{S}\mathit{E}\mathit{C}$). It turns out that the ratio E has the following mathematical expressions depending on the system configuration:

• –Reject-refining: $\mathit{E}={\mathit{R}}_{\mathit{v}}\mathit{T}$
• –Feed-back rejects: $\mathit{E}=1\mathrm{/}1-{\mathit{R}}_{\mathit{v}}\mathit{T}$
• –Feed-back reject-refining: $\mathit{E}={\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{/}1-{\mathit{R}}_{\mathit{v}}\mathit{T}$

Therefore, E is dependent of ${\mathit{R}}_{\mathit{v}}$ and T. The first dependency is quite obvious given that the reject ratio has a direct impact on the flows in the system. On the other hand, the dependency of E with T is a bit more complex since T itself has other dependencies as shown in Equation 2a. These dependencies are illustrated in Figure 8 where the ratio E is plotted as a function of gap for various ${\mathit{R}}_{\mathit{v}}$.

Figure 8.

Ratio E as a function of gap for different values of ${\mathit{R}}_{\mathit{v}}$. E is mass flow rate flowing through the refiner over the mass flow rate leaving the system. Alternatively, E is the ratio of the energy content in the flow leaving the system over the energy supplied by the refiner. The reference case is a system of a single refining stage (black solid-line) where $\mathit{E}=1$ always. In principle, the E dependency with ${\mathit{R}}_{\mathit{v}}$ is quite obvious since it directly affects the flows in the system. However the E dependency with gap is not straight forward to deduce. The latter dependency is linked to the reject thickening factor T which describes the mass split ratio in pressure screening and it is function of the feed’s FLD and ${\mathit{R}}_{\mathit{v}}$ for a given screen.

From Figure 8a is seen that in Reject-refining E does not depend on gap but only on ${\mathit{R}}_{\mathit{v}}$. This is natural since, in this system, screening is totally independent from refining. In contrast, in Feed-back rejects (Figure 8b) and Feed-back reject-refining (Figure 8c), E is indeed dependent of both ${\mathit{R}}_{\mathit{v}}$ and gap. This is because screening and refining are coupled via the recirculation flow.

Values of E for the Reject-refining system are below the reference case, between the range $\mathit{E}=[0.3\mathrm{,}0.6]$, because the initial flow is split and only a portion of the pulp is refined. Conversely, Feed-back rejects and Feed-back reject-refining have E ratios greater than the reference case, in the range $\mathit{E}=[1.5\mathrm{,}4.0]$ and $\mathit{E}=[0.8\mathrm{,}3.0]$ respectively.

At this point, it is imperative to recall that the refiner size was the same for all systems and therefore the refiner’s flow rate was also the same. With this in mind, the ratio E helps to realize the system’s throughput if the first definition is used ($\mathit{E}={\dot{\mathit{m}}}_{\mathit{R}}\mathrm{/}{\dot{\mathit{m}}}_{\mathit{S}}$). In general, the system’s throughput ${\dot{\mathit{m}}}_{\mathit{S}}$ is the refiner’s flow rate ${\dot{\mathit{m}}}_{\mathit{R}}$ (i. e. recommendend flow rate or refiner’s capacity) over E. Analogously, E can also be used to characterize a system in terms of energy contents if the second definition is used ($\mathit{E}={\mathit{e}}_{\mathit{S}}\mathrm{/}{\mathit{e}}_{\mathit{R}}$). The energy content of the flow leaving the system ${\mathit{e}}_{\mathit{S}}$ is E times the energy supplied by the refiner ${\mathit{e}}_{\mathit{R}}$.

For instance, for cases where $\mathit{E}\mathrm{<}1$ (e. g. Reject-refining), the system’s throughput ${\dot{\mathit{m}}}_{\mathit{S}}$ is higher than the refiner’s flow rate but the energy content of the flow leaving the system is lower. Conversely, when $\mathit{E}\mathrm{>}1$, the system’s throughput is lower than the refiner’s capacity and the energy content of the flow leaving the system is higher. The definitions of E and the aforementioned statements will become handy in the next subsection when the systems are assessed in terms of overall energy content.

It is worth noting that the previous systems assessments using E are based on having the same refiner size across the systems. As explained before, throughputs and flow energy contents depend on their configuration. A different way of assessing systems would be setting the same throughput for all systems. If this would be the case, the refiners’ flow rates would be vary between systems and therefore refiner sizes would be different. (based on manufacturers’ recommendations). Since refiner size has an impact on fibre shortening, the systems would behave differently and comparisons between them would not be possible under the same grounds.

For the sake of the discussion, refiner’s flow rate could be set to a specific value so that refiner size do not have an impact on fibre shortening (e. g. same fibre shortening regardless of refiner size). In doing so, it would be possible to assess the systems based on a target throughput using E. In this hypothetic case, E would give an idea on the system’s refining capacity demands. For instance, if $\mathit{E}\mathrm{<}1$, the system would require a small refiner whereas for cases where $\mathit{E}\mathrm{>}1$, the refiner size would be considerable bigger.

Fibre length as function of gap

Figure 9 shows curves of ${\mathit{l}}_{\mathit{w}}$ vs gap at different ${\mathit{R}}_{\mathit{v}}$ values. In these curves, at wide gaps fibre length remains practically unchanged, but as gap start to decrease changes in fibre length are seen. This is the typical trend seen in single-stage refining trials. Although the behaviours are similar, ${\mathit{l}}_{\mathit{w}}$ changes with ${\mathit{R}}_{\mathit{v}}$ and the system configuration. For instance, Reject-refining produces a pulp with higher ${\mathit{l}}_{\mathit{w}}$ as seen in Figure 9a but the effect of ${\mathit{R}}_{\mathit{v}}$ seems to be quite limited. This is because only a share of the whole pulp, the long fibre fraction, is refined. This behaviour has been observed by Miller et al. (2017) at pilot-plant scales. Figure 9b and 9c differ drastically from their counterpart Figure 9a because (1) all values of ${\mathit{l}}_{\mathit{w}}$ are bellow the reference line and (2) the effect of ${\mathit{R}}_{\mathit{v}}$ on ${\mathit{l}}_{\mathit{w}}$ is more pronounced. This is due to the recirculation flow involved in these two systems allowing a share of fibres to get refined more than once which leads to higher fibre shortening. Since the recirculation flow is directly affected by the ${\mathit{R}}_{\mathit{v}}$, it is observed that higher values of ${\mathit{R}}_{\mathit{v}}$ produce lower fibre length in systems with recirculation (again, high ${\mathit{R}}_{\mathit{v}}$ values allow a greater share of fibres get refined more than once due to recirculation).

Figure 9.

Performance curves in terms of gap for all systems described in Figure 1. Each system is compared to a reference case of a single refiner (black solid-line).

Fibre length as function of dimensionless power

The correlation developed by Rubiano Berna et al. (2018b) (presented in previous sections) was used to transform the curves presented in Figure 9 into curves that relate ${\mathit{l}}_{\mathit{w}}$ with a dimensionless quantity ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$. It is worth mentioning that refining processes are usually assessed in terms of the energy supplied by the refiner (${\mathit{e}}_{\mathit{R}}$) which is the ratio of the net-power ${\mathit{P}}_{\mathit{n}\mathit{e}\mathit{t}}$ over the mass flow rate flowing through the refiner ${\dot{\mathit{m}}}_{\mathit{R}}$. Therefore, the dimensionless quantity ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$ is directly related to the refining net-power, thus for a given refiner size, it is safe to assume that:

$$\begin{array}{c}{\mathit{e}}_{\mathit{R}}={\displaystyle \frac{{\mathit{P}}_{\mathit{n}\mathit{e}\mathit{t}}}{{\dot{\mathit{m}}}_{\mathit{R}}}}\propto {\displaystyle \frac{{\mathit{P}}_{\mathit{d}}}{{\mathit{\alpha }}^2}}\end{array}$$ (8)

Where ${\dot{\mathit{m}}}_{\mathit{R}}={\mathit{C}}_{\mathit{R}}{\mathit{Q}}_{\mathit{R}}$. Thus, ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$ is, in a way, a measure of ${\mathit{e}}_{\mathit{R}}$

Figure 10 shows values of ${\mathit{l}}_{\mathit{w}}$ as function of ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$ for different values ${\mathit{R}}_{\mathit{v}}$ for the three systems. For illustrative purposes, a secondary x-axis of SEC in kWh/t was added. Dashed-lines of constant gap were added to the plots as reference and to ease the comparisons between Figure 9 and 10. Moreover, in Figure 10a and 10b, the dashed-lines have a circle indicating where it intersects the reference curve. In Reject-refining, this intersection point corresponds to ${\mathit{R}}_{\mathit{v}}\to 1$; at reject-ratio equals one, this system becomes a single stage refining and the performance curve corresponds to the reference case. Similarly, for the Feed-back rejects system, the intersection point corresponds to ${\mathit{R}}_{\mathit{v}}\to 0$; if this is the case, again, the system will behave as a single stage refining system. This situation does not happen in Feed-back reject-refining. If ${\mathit{R}}_{\mathit{v}}\to 0$ the system will behave as a single screening operation. On the other hand, if ${\mathit{R}}_{\mathit{v}}\to 1$, the system behaves as batch refining system. For this reason there is no indication of the performance curves intersecting the reference case line.

Figure 10.

Performance curves in terms of dimensionless quantity $\frac{{\mathit{P}}_{\mathit{d}}}{\mathit{\alpha }}$ for all systems described in Figure 1. Each system is compared to a reference case of a single refiner (black solid-line). Dashed-black lines are lines of constant gap. Values of SEC (top x-axis) were calculated from the dimensionless power ${\mathit{P}}_{\mathit{d}}$ for Twin-Flow disc refiner with ${\mathit{R}}_{\mathit{o}}=0.2540$ m (${\mathit{D}}_{\mathit{o}}=20$ in) and following recommended volumetric flow rate Q and rotational speed ω values from Figure 4.

Although ${\mathit{l}}_{\mathit{w}}$ decreases as the dimensionless quantity ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$ in the x-axis increases as expected, the systems seem to behave differently between one and another. As mentioned before, Reject-refining, which produces higher ${\mathit{l}}_{\mathit{w}}$ than the reference case, now has more distinctive differences across ${\mathit{R}}_{\mathit{v}}$ as seen in Figure 10a. As for the Feed-back rejects, the curves show a drastic decrease in ${\mathit{l}}_{\mathit{w}}$ in the first region of the x-axis whereas for Feed-back reject-refining this decrease is less drastic.

Interestingly, there is an optimum operation point in the Feed-back rejects system; for each gap curve (more evident at $\text{gap}=0.15$), there is a point where the maximum power is consumed. This point arises between the perfect trade-off between the enriching in long fibres of the inner flow (feed to the refiner) and fibre shortening due to refining. An increased concentration of long fibres in the inner flow allows a better loadability (Andersson et al. 2012, Sandberg et al. 2017a, Miller et al. 2017). This is desired because it allows to widen the gap to re-establish the desired energy consumption and thus decreasing fibre shortening due to the wider gap (Rubiano Berna et al. 2018a). Conversely, large recirculations leads to excessive cutting and thus a decreased long fibre concentration in the inner flow. Although the ${\mathit{R}}_{\mathit{v}}$ value of these optimum points seems to be fairly low compared to values in industry, these optimums are the result of the combination of screen type, initial pulp, refiner size, etc and is unique for each particular system. Thus, varying the system variables could help to find an operationally attractive optimum point and ultimately help to achieve the main objective of the LC refining implementation in TMP; to consume as much as possible power via LC refining without excessive fibre shortening.

Overall system performance curves

The performance curves in terms of ${\mathit{P}}_{\mathit{d}}\mathrm{/}{\mathit{\alpha }}^2$ are useful to compare the systems based on SEC as explained by Equation 8. However, this metric does not provide information about the energy content of the flow leaving the system, which is also important. As explained before, split-flows and recirculation affect the energy content of the flow leaving the system. To account for split-flows and recirculation, the previously defined $\mathit{E}={\dot{\mathit{m}}}_{\mathit{R}}\mathrm{/}{\dot{\mathit{m}}}_{\mathit{S}}$ ratio is used together with Equation 8 to find an expression for the energy content of the flow leaving the system ${\mathit{e}}_{\mathit{S}}$ as:

$$\begin{array}{c}{\mathit{e}}_{\mathit{S}}={\displaystyle \frac{{\mathit{P}}_{\mathit{n}\mathit{e}\mathit{t}}}{{\dot{\mathit{m}}}_{\mathit{S}}}}={\displaystyle \frac{{\mathit{P}}_{\mathit{n}\mathit{e}\mathit{t}}}{{\dot{\mathit{m}}}_{\mathit{R}}}}{\displaystyle \frac{{\dot{\mathit{m}}}_{\mathit{R}}}{{\dot{\mathit{m}}}_{\mathit{S}}}}\propto \mathit{E}{\displaystyle \frac{{\mathit{P}}_{\mathit{d}}}{{\mathit{\alpha }}^2}}\end{array}$$ (9)

Performance curves in terms of the dimensionless quantity $\mathit{E}\frac{{\mathit{P}}_{\mathit{d}}}{{\mathit{\alpha }}^2}$ are shown in Figure 11 where, again for illustrative purposes, secondary x-axis of ${\text{SRE}}_{\mathit{s}\mathit{y}\mathit{s}}$ in kWh/t was added. These curves are helpful to evaluate the performance of a system since it gives an idea about the refining degree of pulp and allows direct comparisons across systems.

Figure 11.

Performance curves in terms of dimensionless quantity $\mathit{E}\frac{{\mathit{P}}_{\mathit{d}}}{\mathit{\alpha }}$ for all systems described in Figure 1. Each system is compared to a reference case of a single refiner (black solid-line). Dashed-black lines are lines of constant gap. Values of SRE (top x-axis) were calculated from the dimensionless power ${\mathit{P}}_{\mathit{d}}$ for a Twin-Flow disc refiner with ${\mathit{R}}_{\mathit{o}}=0.2540$ m (${\mathit{D}}_{\mathit{o}}=20$ in) and following recommended volumetric flow rate Q and rotational speed ω values from Figure 4.

As can be seen in Figure 11a, Reject-refining produces the pulp with greater ${\mathit{l}}_{\mathit{w}}$ than the reference case, yet it produces pulp with the lowest energy content as it is not able to surpass values of 1.2 at gap values of 0.15 mm. However, given that ${\mathit{l}}_{\mathit{w}}$ is higher than the reference, this allows the pulp to possibly undergo a second refining stage to achieving higher energy contents without experiencing excessive fibre shortening. This system configuration should be implemented if at least two refining stages in series are installed.

From Figure 11b is seen that the Feed-back rejects system can produce large changes in ${\mathit{l}}_{\mathit{w}}$ but is also capable of achieving high energy contents at relatively wide gaps. This could be particularly useful for applications where the pulp needs to undergo gentle treatments. Lastly, in Figure 11c is seen that Feed-back reject-refining achieves fairly high energy contents at a reasonable decrease in fibre length partially because only a share of the flow is refined. In some sense, this specific system is a hybrid between Reject-refining and Feed-back rejects. As only the reject fraction is being refined, it would be suitable in cases where the main objective is to develop properties of the long fibre fraction material or effectively reduce shives.

Industrial scale data comparisons

A subsystem of the Holmen Paper TMP mill in Braviken, Sweden was compared to simulation results. The industrial refining system subject to study was composed of an Andritz 72" Twin-Flow disc refiner (TF-72) and three pressure screening units (two Ahlstom-F4 and one Ahlstom-F5). The system configuration is exactly the one presented in Figure 1b with the particularity of having the three screens operating in parallel. The pulp feeding the system was primary HC refined spruce pulp with ${\mathit{l}}_{\mathit{w}}=1.40\text{mm}$ and 4 % consistency. The TF-72 operated at $\mathit{\omega }=325$ RPM, $\mathit{Q}=252$ L/ s and $\text{gap}=0.14\text{mm}$. The refining plates used had the following characteristics: ${\mathit{R}}_{\mathit{i}}=0.405$ and ${\mathit{R}}_{\mathit{o}}=0.914\text{m}$. ${\mathit{B}}_{\mathit{w}}=1.60$, ${\mathit{G}}_{\mathit{w}}=3.20$ and ${\mathit{D}}_{\mathit{w}}=7.35\text{mm}$. The screens were provided with 0.15 mm slot AFT macroflow baskets and AFT-GHC rotors and operated at a passing speed of 20 m/s. Two of the screens (F4(2) and F5) had reject dilution to prevent plugging. For the screens F4(1), F4(2) and F5, the ${\mathit{R}}_{\mathit{v}}$ values were 0.17, 0.15 and 0.11 respectively. Once the system achieved a steady state, flow rates and consistencies of certain flows were measured together with refining power. Samples were collected and analyzed to measure their FLD. In order to be consistent with the simulations results, the FLD were bounded to fibres within the range $\mathit{l}=[0.2\mathrm{,}4.0]\text{mm}$ and values of ${\mathit{l}}_{\mathit{w}}$ were calculated. Table 1 summarizes the experimental data, simulation results and relative errors.

Table 1.

Summary of the experimental data (exp) and simulation results (sim). In general, relative errors were reasonable low, between 0–18 %. Simulated thickening factor T values for the F4(2) and F5 screens are higher than the experimental values. During operation, reject dilution were used in these two screens to prevent plugging. Dilutions were not measured nor accounted in the simulations.

Flow	${\mathit{l}}_{\mathit{w}}$ (mm)			C (%)
	Exp	Sim	Error (%)	Exp	Sim	Error (%)
Feed refiner	1.41	1.44	2.34
Refined pulp	1.34	1.38	2.99
Rejects F4(1)	1.49	1.49	0.00	4.39	4.38	0.23
Rejects F4(2)	1.52	1.51	0.07	3.80	4.38	14.37
Rejects F5	1.58	1.50	5.19	3.10	3.67	18.87
Accepts F4(1)	1.15	1.27	11.13	1.34	1.18	11.64

Screen	T			Refiner	${\mathit{P}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ ( MW)
	Exp	Sim	Error (%)		Exp	Sim	Error (%)
F4(1)	2.523	2.517	0.23	TF72	3.0	2.8	6.00
F4(2)	2.360	2.699	14.37
F5	2.711	3.222	18.87

Relative errors were reasonable low for the fibre length values ${\mathit{l}}_{\mathit{w}}$, between 0–11 % which falls within the analysis error. However, the simulation over-predicted the consistencies and reject thickening ratio values related to the F4(2) and F5 screens. This discrepancy is associated to the reject dilution (a water flow) used during normal operation to prevent plugging; these water flows were not experimentally measured nor accounted in the simulation. A dilution, of course, leads to lower consistencies which is seen in the experimental data. As a side note, reject dilutions are usually in the order of 5 % of the reject volumetric ratio so it is improbable to affect the screening performance.

Conclusions

Refining systems were theoretically analyzed from a system-design point. In general it was found that:

• –Refiner size affects fibre shortening if manufacturer recommended flow rates and rotational speed are followed. Small refiners cut fibres to a greater extent than large refiners at a given gap.
• –Recirculation greatly affects the system performance. It allows to achieve higher energy contents but also could drastically decrease fibre length.
• –Combining refining and screening allows to increase the fibre length feeding the refiner thus increasing the loadability.

And specifically for each system:

• –Reject-refining could be beneficial in cases where the pulp can go through at least two refining stages in series. Moreover, pulp can undergo harsh treatments without excessive fibre shortening as only the long fibre fraction of pulp is refined. This system’s throughput is higher compared to the refiner’s capacity.
• –Feed-back rejects could be suitable for applications requiring gentle treatments (wide gaps) or where only a single stage of refining is possible since it can achieve high energy contents. However the system throughput is significantly lower compared to the refiner’s capacity.
• –Feed-back reject-refining could be regarded as a hybrid between the previous two systems. Achieves relatively high energy contents without excessive fibre shortening. This system could be suitable to efficiently reduce shives and develop properties of the long fibre fraction.

Finally, the models implementation predicted industrial data reasonable well, proving that simulation results can aid in system design and evaluation.

Appendix A. Mathematical analysis of a system

Flows fibre length distributions

The following mathematical procedure includes matrix operations. To simplify the notation, matrices for the screening and refining operations and vectors for the fibre length distributions are defined and presented in bold fonts.

$$\begin{array}{c}\mathbf{\text{H}}=\text{diag}\left[{\displaystyle \frac{{\mathit{R}}_{\mathit{v}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}-1}}{\sum _{\mathit{i}=1}^{\mathit{p}}{\mathit{R}}_{\mathit{v}}^{\mathit{P}\mathrm{(}{\mathit{l}}_{\mathit{i}}\mathrm{)}-1}}}\right]\end{array}$$ (A.1a)

For refining:

$$\begin{array}{c}\mathbf{\text{J}}={exp}_{\text{m}}\left[\mathit{\pi }\mathit{\alpha }\mathit{\beta }{\displaystyle \frac{\mathit{\omega }{{\mathit{R}}_{\mathit{o}}}^2}{\mathit{Q}}}\mathrm{(}1-{\mathit{\lambda }}^2\mathrm{)}\mathbf{\text{A}}\right]\end{array}$$ (A.1b)

For the fibre length distribution of the flow k:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{k}}=\{{{\mathit{y}}_{\mathit{i}}}^{\mathit{k}}\}\end{array}$$ (A.1c)

Consider the Feed-back reject-refining system with the notation shown in the Figure A.1. The rejects are related to the screen feed as:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{r}}=\mathbf{\text{H}}{\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}\end{array}$$ (A.2)

The accepts are related to the screen feed as:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{a}}=\mathrm{(}\mathbf{\text{I}}-{\mathit{R}}_{\mathit{v}}\mathbf{\text{H}}\mathrm{)}{\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}\end{array}$$ (A.3)

The refined pulp is related to the screen rejects as:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}=\mathbf{\text{J}}{\mathbf{\text{Y}}}_{\mathit{r}}\end{array}$$ (A.4)

Figure A.1.

Feed-back reject-refining.

Combining Equation A.2 and A.4 yields:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}=\mathbf{\text{J}}\mathbf{\text{H}}{\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}\end{array}$$ (A.5)

Now, from a mass balance at the mixing point:

$$\begin{array}{c}{\mathit{Q}}_{\mathit{f}\mathit{s}}{\mathit{C}}_{\mathit{f}\mathit{s}}{\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}={\mathit{Q}}_{\mathit{f}}{\mathit{C}}_{\mathit{f}}{\mathbf{\text{Y}}}_{\mathit{f}}+{\mathit{Q}}_{\mathit{r}\mathit{r}}{\mathit{C}}_{\mathit{r}\mathit{r}}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}\end{array}$$ (A.6)

Dividing Equation A.6 by ${\mathit{Q}}_{\mathit{f}\mathit{s}}{\mathit{C}}_{\mathit{f}\mathit{s}}$, the FLD of the screen feed ${\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}$ is obtained as:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}=\mathrm{(}1-{\mathit{R}}_{\mathit{v}}\mathrm{)}\mathit{T}{\mathbf{\text{Y}}}_{\mathit{f}}+{\mathit{R}}_{\mathit{v}}\mathit{T}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}\end{array}$$ (A.7)

Replacing Equation A.5 in Equation A.7:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}=\mathbf{\text{J}}\mathbf{\text{H}}[\mathrm{(}1-{\mathit{R}}_{\mathit{v}}\mathrm{)}\mathit{T}{\mathbf{\text{Y}}}_{\mathit{f}}+{\mathit{R}}_{\mathit{v}}\mathit{T}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}]\end{array}$$ (A.8)

And finally, finding an explicit expression for ${\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}$ in terms of the initial pulp ${\mathbf{\text{Y}}}_{\mathit{f}}$:

$$\begin{array}{c}{\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}=\mathrm{(}1-{\mathit{R}}_{\mathit{v}}\mathrm{)}\mathit{T}\mathbf{\text{J}}\mathbf{\text{H}}{[\mathbf{\text{I}}-{\mathit{R}}_{\mathit{v}}\mathit{T}\mathbf{\text{J}}\mathbf{\text{H}}]}^{-1}{\mathbf{\text{Y}}}_{\mathit{f}}\end{array}$$ (A.9)

Where I is the identity matrix. Once ${\mathbf{\text{Y}}}_{\mathit{r}\mathit{r}}$ is known, it is used in Equation A.7 to calculate ${\mathbf{\text{Y}}}_{\mathit{f}\mathit{s}}$. Subsequently it is possible to calculate ${\mathbf{\text{Y}}}_{\mathit{r}}$ and ${\mathbf{\text{Y}}}_{\mathit{a}}$ using Equation A.2 and A.3 respectively.

Calculation of E

If the mass flow rate flowing through the screen is set to 1, one has:

$$\begin{array}{c}1={\dot{\mathit{m}}}_{\mathit{a}}+{\dot{\mathit{m}}}_{\mathit{r}}\end{array}$$ (A.10a)

Expressing the mass flow rates as the product of volumetric flow rate times consistency:

$$\begin{array}{rl}& 1={\mathit{Q}}_{\mathit{a}}{\mathit{C}}_{\mathit{a}}+{\mathit{Q}}_{\mathit{r}}{\mathit{C}}_{\mathit{r}}\end{array}$$ (A.10b)

$$\begin{array}{rl}& {\displaystyle \frac{1}{{\mathit{Q}}_{\mathit{a}}{\mathit{C}}_{\mathit{a}}}}=1+{\displaystyle \frac{{\mathit{Q}}_{\mathit{r}}{\mathit{C}}_{\mathit{r}}}{{\mathit{Q}}_{\mathit{a}}{\mathit{C}}_{\mathit{a}}}}\end{array}$$ (A.10c)

$$\begin{array}{rl}& {\displaystyle \frac{1}{1-{\mathit{R}}_{\mathit{v}}\mathit{T}}}-1={\displaystyle \frac{{\mathit{Q}}_{\mathit{r}}{\mathit{C}}_{\mathit{r}}}{{\mathit{Q}}_{\mathit{a}}{\mathit{C}}_{\mathit{a}}}}\end{array}$$ (A.10d)

Since the mass flow leaving the system ${\mathit{Q}}_{\mathit{a}}{\mathit{C}}_{\mathit{a}}$ is the same as the mass flow feeding the system ${\mathit{Q}}_{\mathit{f}}{\mathit{C}}_{\mathit{f}}$:

$$\begin{array}{c}{\displaystyle \frac{{\mathit{Q}}_{\mathit{r}}{\mathit{C}}_{\mathit{r}}}{{\mathit{Q}}_{\mathit{f}}{\mathit{C}}_{\mathit{f}}}}={\displaystyle \frac{{\mathit{R}}_{\mathit{v}}\mathit{T}}{1-{\mathit{R}}_{\mathit{v}}\mathit{T}}}\end{array}$$ (A.10e)

On the left-hand side one has the ratio mass flow rate flowing through the refiner over the mass flow rate leaving the system which is the definition of E.

$$\begin{array}{c}\mathit{E}={\displaystyle \frac{{\mathit{R}}_{\mathit{v}}\mathit{T}}{1-{\mathit{R}}_{\mathit{v}}\mathit{T}}}\end{array}$$ (A.10f)

The alternate definition of E in terms of energy content is also derived. Setting ${\dot{\mathit{m}}}_{\mathit{f}\mathit{s}}=1$ and ${\mathit{e}}_{\mathit{f}}=0$.

$$\begin{array}{c}{\mathit{e}}_{\mathit{r}\mathit{r}}={\mathit{e}}_{\mathit{r}}+{\mathit{e}}_{\mathit{R}}\end{array}$$ (A.11a)

Where ${\mathit{e}}_{\mathit{R}}$ is the energy supplied by the refiner (SEC). At the mixing point:

$$\begin{array}{c}{\mathit{e}}_{\mathit{f}\mathit{s}}={\dot{\mathit{m}}}_{\mathit{r}\mathit{r}}{\mathit{e}}_{\mathit{r}\mathit{r}}\end{array}$$ (A.11b)

Since ${\dot{\mathit{m}}}_{\mathit{r}\mathit{r}}={\dot{\mathit{m}}}_{\mathit{r}}={\mathit{R}}_{\mathit{v}}\mathit{T}$, combining Equation A.11a and A.11b leads to:

$$\begin{array}{rl}& {\mathit{e}}_{\mathit{f}\mathit{s}}={\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{(}{\mathit{e}}_{\mathit{r}}+{\mathit{e}}_{\mathit{R}}\mathrm{)}\end{array}$$ (A.11c)

$$\begin{array}{rl}& {\mathit{e}}_{\mathit{f}\mathit{s}}-\mathrm{(}{\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{)}{\mathit{e}}_{\mathit{r}}=\mathrm{(}{\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{)}{\mathit{e}}_{\mathit{R}}\end{array}$$ (A.11d)

As ${\mathit{e}}_{\mathit{f}\mathit{s}}={\mathit{e}}_{\mathit{r}}={\mathit{e}}_{\mathit{a}}$, the interest is on finding an expression for the ratio ${\mathit{e}}_{\mathit{a}}\mathrm{/}{\mathit{e}}_{\mathit{R}}$ (energy content of the flow leaving the system over the energy supplied by the refiner), which is the alternate definition of E.

$$\begin{array}{rl}& {\mathit{e}}_{\mathit{a}}\mathrm{(}1-{\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{)}=\mathrm{(}{\mathit{R}}_{\mathit{v}}\mathit{T}\mathrm{)}{\mathit{e}}_{\mathit{R}}\end{array}$$ (A.11e)

$$\begin{array}{rl}& \mathit{E}={\displaystyle \frac{{\mathit{R}}_{\mathit{v}}\mathit{T}}{1-{\mathit{R}}_{\mathit{v}}\mathit{T}}}\end{array}$$ (A.11f)

Funding Statement

This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) Collaborative Research and Development (CRD) grant CRDPJ 437223-12 and the support of the industrial partners AB Enzymes, Alberta Newsprint Company, Andritz, BC Hydro, Canfor, Catalyst Paper, FPInnovations, Holmen, Meadow Lake Pulp (Paper Excellence), Millar Western, NORPAC, West Fraser, Westcan Engineering and Winstone Pulp International who are thanked for their support.