Effect of Dilution Water on Pulp Consistency in Cross-directional Basis Weight Control
1Department of Electrical and Information Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China
Abstract
Cross-directional basis weight of paper based on hydraulic headbox is adjusted by the opening of dilution water valve. In order to achieve uniform concentration distribution of paper sheet, a research on the effect of dilution water added to the mainstream on concentration and flow was studied by experimental set-up and mathematical modeling in this paper. Based on the numerical simulation results, a scale model of headbox was derived to predict the water content distribution and pulp consistency distribution. Results show that these models could accurately predict the experimental flow characteristics of a pulp suspension with dilution water added. Compared with slice regulation, the inverse response problem and its improvement method were discussed. By parameters improvement of pulp consistency model, the reverse response on both sides can be well eliminated or balanced when the dilution water was adopted to cross-directional profile control system. It ensures that the influence range of a single dilution water valve is controllable, which improves the adjustment accuracy of cross-directional profile control and thus improves the paper quality.
Keywords:
Paper basis weight, cross-directional control, pulp concentration, dilution water, reverse response, numerical simulation1. Introduction
In paper making with modern high-speed paper machines, sheet properties must be continuously monitored and controlled to guarantee that the paper product quality specifications are satisfied along both the machine direction (MD) and cross direction (CD).^{1,2)} The quality of paper such as a basis weight is basically determined by uniform dispersion of fibers through a headbox which is one of the components of paper machine.
It is generally recognized that the hydraulic headbox is the most advanced headbox. Modern hydraulic headbox is equipped with dilution water supply equipment for basis weight profile control.^{3)} The task of the dilution water supply equipment is to mix dilution water as evenly as possible into the stock that flows from the inlet header, and to create a stock flow whose consistency differs from that of the inlet header flow.^{4)} This stock flow with locally adjustable consistency is used to control the web’s cross profile. The headbox is divided into CD dilution zones spaced 60 mm apart. Dilution water is supplied individually to each dilution zone. Dilution water volume is regulated by valves connected to electric actuators. These actuators are controlled by the dilution profile control system. The control system is used to locally regulate stock consistency in order to effect basis weight changes. The dilution ratio varies between 3% and 20%, with an approximate mean of 8-12%.^{5)} The hydraulic headbox systems can fine-tune the basis weight through computer controlled slice adjustments and dilution systems that compensate for profile variations in the machine direction and cross direction.
At present, more research is focused on the structure and regulation principle of hydraulic headbox with dilution water.^{6,7)} Jiang Fengwei, General Manager of Jianghe Paper Industry, and Professor Lin Meichan, headbox designer of Hangzhou Meichen Company, a famous supplier of headbox in China, reciprocally expounded the structure, design and development technology of hydraulic headbox.^{8,9)} The papers^{10,11)} discusses some details of the hydraulic headbox, such as the influence of structural parameters or external factors on the paper-making performance of the headbox, the control principle and key points of the headbox. In a more recent study, the shape optimization of headbox is carried out by integration CFD and optimization technology.
The key technology of the hydraulic headbox systems is how to add dilution water, when a dilution water valve operates, what effect it does have on the pulp consistency on both sides, how that really affects, and what expressions are used to depict? The principle of these problems are not clear. These problems involve complex measurement and control technical problems. In addition, controlling the consistency allows better basis weight profiling (profile variations in the machine direction and cross direction) than that achieved by slice bending.^{12)} It is necessary to construct a pulp consistency model to predict the amplitude relationship between the concentration and both sides of pulp flow. And analyze the effect of dilution water on slurry consistency of headbox. Furthermore, an accurate modeling strategy can reduce the costs of experiments and prototype equipment when designing headbox.
This work aims to analyze the effect of injecting dilution water on the stock concentration and fiber flow pattern in main pulp stream. The results are expected to provide a better consistency distribution, and solve the inverse response problem in the process of CD profile control. Based on the numerical simulation results, the mathematical model of pulp consistency is established by analyzing the water content of pulp, and the pulp distribution is expounded. Based on the concentration dilution mathematical model of hydraulic headbox, the inverse response problem and its improvement method are discussed. The slurry flow could be uniformly distributed as far as possible, and the influence range of diluted water added at a single point could be controlled as far as possible, so as to improve the accuracy of quantitative control of CD in hydraulic headbox with dilution water.
2. Materials and Methods
2.1 Materials
The scale headbox model is composed of inlet header, tube bank, dilution plate, equalizing chamber, turbulence generator, edge flow channels and box shell etc. Fig. 1 shows the schematics and coordinate system of headbox.
Stock flows into the headbox through an inlet header. The inlet header equalizes stock pressure across the web. The passage of the stock turns in the machine direction in the inlet header.
A perforated dilution plate and a grooved dilution plate are located between the inlet header and tube bank. They distribute the dilution water to each tube row.
The tube bank diverts the stock flow in the machine direction and smooths the pressure and jet velocity profile in the headbox cross direction by means of pressure loss.
From the tube bank the stock flows into the equalizing chamber, where the individual tube flows are blended into one homogeneous flow.
The turbulence generator combines two perforated plates and a tube bank.^{13)} The number of tube rows and the diameter of the holes in the perforated plates depend on the dimensional flow.
The whole experiment process is carried out in a hydraulic headbox with diluted water. The operation of the hydraulic headbox is shown in Fig. 2. The box body is equipped with glass holes, dilution water was used with red ink. Therefore, the flow of fluid from the point of addition to the lip of the headbox can be observed. The dilution water regulating device is shown in Fig. 3, which includes a dilution water cone pipe and a plurality of dilution water valves with automatic control units. One interface of the dilution water valve is connected with the dilution water cone pipe, and the other interface of the dilution water valve is provided with a water conveying hose, which connected with the pulping branch pipe. The amount of white water injected into the pulp by adjusting the opening of the dilution valve, the consistency of the pulp can be adjusted, thus realizing the local fine-tuning of the cross-directional basis weight of paper. The high precision valve positioner is shown in Fig. 4.
2.2 Experimental principle
The purpose of headbox is to take the stock delivered by the feed pump and transform the stock flow into an even, rectangular discharge equal in width to the paper machine and at a uniform velocity, consistency and pressure in the machine direction.^{14)} Headbox adjustments are the speed of the jet (headbox pressure), slice movements in horizontal and vertical direction and tilt. Edge flows and circulation valves are also part of the headbox controls. Dilution valves control basis weight profile. The technological process for basis weight control is shown in Fig. 5.
An alternative approach to control CD basis weight is to manipulate local flow consistency to adjust local basis weight. The idea for this profiling technique is not new. Previous designs have not incorporated a method to produce consistency variations on a finely controlled scale. An ideal system must also avoid variations in flow rate across the nozzle to avoid cross flows. Fig. 6 illustrates a consistency profiling concept that meets these requirements. The tube bank has individual injection tubes. Each tube meters low consistency white water into the header just upstream from an adjacent headbox tube. The low consistency flow turns 180 and goes directly into the adjacent flow tube. The turbulence created at the step expansion mixes the flow and channels it into the nozzle.^{15,16)} A proper tube bank and nozzle system design can keep turbulence scales down and minimize mixing. Such design can simultaneously maintain the area of adjusted consistency in a narrow band. Adjustment of the injection flow can then affect the local consistency and corresponding basis weight.
2.3 Control method for the experimental device
The peripheral control diagram of dilution headbox is shown in Fig. 7. LIC-01 is air cushion pressure control loop of steady flow chamber. PIC-01 is total pressure control loop of pulp and wire speed ratio at slice. PdIC-01 is control loop of pipe pressure difference between dilution water and dilution white slurry. On this experimental device, 10-15 dilution valves are equipped, and the corresponding control cabinet for the dilution valves is designed for modeling water concentration dilution and eliminating reverse response experiment. On the basis of these experiments, FLUENT is used to simulate the dynamic characteristics of the fluid in the headbox, so as to deepen the understanding of the turbulence mechanism in the headbox.
3. Results and Discussion
3.1 Results
The pulp consistency is measured in the experiment. In order to compare the CFD predictions with the experimental value conveniently, the water content is considered as a standard. The sum of the water content and the pulp consistency was 1. For different speeds tested (400, 650, 800, and 1,000 m·min^{-1}), the water content of the outlet is shown in Fig. 8(a-d). In Fig. 3, “EXP” represents the “experiment”, and “CFD” represents simulation. The curve of trends for the CFD predictions and the experimental values were similar.
From the comparison results, there is a small difference between the prediction of CFD and the experimental values, and the largest Error magnitude is 10^{-5}.
The main stream is injected with the dilution water at the central step diffusion tube of the headbox. The mainstream is mixed with dilution water, and the water content of the central step diffusion tube will be higher than others.^{17)} As the pulp flows forward, the difference in water content becomes smaller due to the mixing of fluids. and the water content of the outlet varies with the speed. The relationship between the speed and water content of the outlet is shown in Fig. 9. The difference in water content between 1,000 m·min^{-1} and 1,200 m·min^{-1} is small. At 800 m·min^{-1}, the water content at the center of the outlet is the highest, at 400 m·min^{-1}, the distribution curve is flatter than the other curves. Therefore, the water content does not increase with the speed. Therefore, in order to obtain the desired export concentration, the appropriate speed should be chosen.
3.2 Discussion
The basis weight regulation of CD is realized by adjusting the local pulp consistency. In order to improve the mechanism of concentration regulation and diluted water, it is necessary to establish a reasonable concentration dilution model, and the inverse response of concentration dilution in dilution hydraulic headbox should be discussed by experimental result.
For the establishment of the concentration dilution model of dilution headbox, three assumptions are followed in the research process:
a. Dilution rate is not more than 3-20% (the default average value range is 8-12%).
b. Q_{S}+Q_{D}=Q_{TOT}, and Q_{TOT} is an invariant constant for the fixed paper, namely, the fluid is uncompressed.
c. The mixture of dilution water and white pulp is evenly, and bone dry pulp remains unchanged after dilution. Namely, C_{S}Q_{S}+C_{D}Q_{D}=C_{TOT}Q_{TOT}.
Where Q_{D}, Q_{S}, Q_{TOT} are the flow of dilution water, white slurry and diluted white slurry respectively. C_{D}, C_{S}, C_{TOT} are the consistency of dilution water, white slurry and diluted white slurry respectively.
In order to clarify the relationship between the outlet water content and the outlet lateral coordinates at different speeds, processing the experimental data and establishing the model of water content of outlet as follow.
a. Take the outlet horizontal axis as the abscissa x and y_{1} as the outlet water content. The array (x, y_{1}) is filtered and denoised by MATLAB.
b. Due to the direct fitting effect is not good, reset the variables as follow
$$$\left\{\begin{array}{c}Y=\mathrm{ln}{y}_{1}\hfill \\ X={x}^{2}\hfill \end{array}\right.$$$ | [1] |
c. Fitting the new array (X,Y)，and the fitting curve is a linear curve, it is shown in Fig. 10. A linear relation is established between the X and Y, assuming
$$$Y=bX+c\mathrm{}\left(c=\mathrm{ln}\alpha \right)$$$ | [2] |
d. Then ln y=bx^{2}+ln α is obtained, analyzing the parameters, and the parameters is given by
$$$\left\{\begin{array}{c}a=\frac{1}{\sqrt{2\pi \sigma}}\hfill \\ b=-\frac{1}{2{\sigma}^{2}}\hfill \end{array}\right.$$$ | [3] |
e. Deduce the mathematical expression between x and y_{1}, finally the relationship can be expressed as
$$${y}_{1\left(x\right)=}\frac{1}{\sqrt{2\pi \sigma}}{\mathrm{e}}^{-\frac{{x}^{2}}{2{\sigma}^{2}}}$$$ | [4] |
where, σ is width of outlet flow field influenced by concentration.
The model of outlet water content is the same as the function of normal distribution. It shows that the outlet water content has a normal distribution relationship with the outlet horizontal axis.
Therefore, the distribution of the water content at the outlet of the slice lip is ideally normal at different speeds. In order to verify the above result and establish the mathematical model of pulp consistency in the CD process, a further comparison was study on the actuator response model of lip adjustment, which was proposed by Jakan Ghofraniha, M. S. Davies, Guy A. Dumont.^{18,19)}
A schematic of a two dimensional wave is shown in Fig. 11, the amplitude is a function of position x and time. It shows the slurry with the surface wave on the wire. The x-axis shows CD and y-axis shows the thickness of the slurry. The origin of the coordinate system is placed at the interface of the fluid and the bed (wire). The wire represents the porous bed. The amplitude of the wave is shown as η and the mean value as h.
The process of establishing the response model of lip adjustment process is as follows:
a. For abscissa value x (measuring point position) and ordinate value (time) y, assumes η(x, y) is the function of the “S” type wave of CD deviation profile.
b. Establishing fluid mechanics equations
As the assumptions in the part 2, analyzing pulp according to the incompressible fluid. The conservation of mass on a surface “S” that surrounds the incompressible moving fluid implies the continuity equation.
$$$\nabla u=0$$$ | [5] |
It simply states the sum of the velocity gradients in any direction should be zero. Assuming an irrotational flow then the velocity potential Ø exits, the gradient of which is the velocity of the fluid and the continuity equation (1) reduces to Laplace equation:
$$$\frac{{\partial}^{2}\varnothing}{\partial {x}^{2}}+\frac{{\partial}^{2}\varnothing}{\partial {y}^{2}}=0$$$ | [6] |
Newton’s second law of motion is then applied to fluid inside “S”:
$$$\frac{{D}_{q}}{{D}_{t}}=-\frac{1}{\rho}\nabla P+g$$$ | [7] |
The boundary condition at the surface is:
$$$\left\{\begin{array}{c}y=\eta \left(x,t\right)\hfill \\ \frac{\partial \eta}{\partial t}-{u}_{y}=0,aty=\eta \hfill \end{array}\right.$$$ | [8] |
The dynamic boundary condition is
$$$\frac{\partial \varnothing}{\partial t}+\frac{1}{2}{q}^{2}+g\eta +\frac{{P}_{\eta}}{\rho}=Const,aty=\eta $$$ | [9] |
After linearized the dynamic boundary condition, the above equation reduces to:
$$$\left\{\begin{array}{c}-\frac{\partial \varnothing}{\partial t}+g\eta =0,aty=\eta \hfill \\ -\frac{\partial \varnothing}{\partial t}={u}_{y}\hfill \\ {P}_{f}={P}_{b},aty=0\hfill \end{array}\right.$$$ | [10] |
c. Finally, solve the function of the “S” type wave of CD deviation profile by the following multi-objective model.
Objective function:
$$$\left\{\begin{array}{c}\frac{{\partial}^{2}\varnothing}{\partial {x}^{2}}+\frac{{\partial}^{2}\varnothing}{\partial {y}^{2}}=0\hfill \\ {\nabla}^{2}{P}_{b}=0\hfill \end{array}\right.$$$ | [11] |
boundary condition:
$$$\left\{\begin{array}{c}y=\eta \left(x,t\right)\hfill \\ \frac{\partial \eta}{\partial t}-{u}_{y}=0\hfill \\ \frac{\partial \varnothing}{\partial t}+\frac{1}{2}{q}^{2}+g\eta +\frac{{P}_{\eta}}{\rho}=Const\hfill \\ -\frac{\partial \varnothing}{\partial t}+g\eta =0,aty=\eta \hfill \\ -\frac{\partial \varnothing}{\partial t}={u}_{y}\hfill \\ {P}_{f}={P}_{b}\hfill \end{array}\right.$$$ | [12] |
It can be solved by a separation of variable technique with the boundary conditions. A surface disturbance of the following form is considered
$$$\eta \left(x,t\right)={A}_{0}{\text{e}}^{j\left({N}_{x}-wt\right)}$$$ | [13] |
The real part of the equation is only considered
$$$\eta \left(x,t\right)={A}_{0}{\text{e}}^{-{k}_{d}x}\mathrm{cos}\left(kx-wt\right)$$$ | [14] |
When the CD profile does not vary with time,
$$$\eta \left(x,0\right)={A}_{0}{\text{e}}^{-{k}_{d}x}\mathrm{cos}kx$$$ | [15] |
The curve of η(x,0) is shown in Fig. 12.
It can be seen that the two-dimensional “S” for CD profile is similar to the response curve of dilution valve.
After studying a number of typical response shapes obtained in the experiments with the different actuator types, it was found that the following empirical expression can well describes the shape “S” of the majority practically response.
$$$g\left(x\right)=\frac{\gamma}{2}\left\{{\text{e}}^{\frac{-\alpha {\left(x+\beta \zeta \right)}^{2}}{{\zeta}^{2}}}\mathrm{cos}\left[\frac{\pi \left(x+\beta \zeta \right)}{\zeta}\right]+{\text{e}}^{\frac{-\alpha {\left(x-\beta \zeta \right)}^{2}}{{\zeta}^{2}}}\mathrm{cos}\left[\frac{\pi \left(x-\beta \zeta \right)}{\zeta}\right]\right\}$$$ | [16] |
It is shown that the spatial response of a single controller can be represented by an extension of a set of orthogonal functions, where the spatial coordinate x is a scalar real number, γ is the gain parameter, and ζ is the width parameter. The parameters γ and ζ define the linear transformation of the response by stretching it vertically and horizontally. The attenuation parameter α changes the size of the negative lobes of the response.
The divergence parameter β defines the presence of two maxima in the response and the distance between these two maxima.
Comparing the model of outlet water content (y(x)) and the response of dilution valve (g(x)). It can be found that they are different expressions, but similar parts can be found. With mathematical simplification, g(x) can be converted to y(x). The simplification process is as follows:
a. When the error of dilution valve is approximately 0, then β=0, g(x) can be simplified to
$$$g\left(x\right)=\gamma \left\{{\text{e}}^{\frac{-\alpha {x}^{2}}{{\zeta}^{2}}}\mathrm{cos}\left(\frac{x}{\zeta}\right)\pi \right\}$$$ | [17] |
b. In the experimental simulation, taking a dilution water hydraulic headbox with 64 dilution water valves as an example, the measurements and the actuator constructed a large-scale system with dimensions of 320×64. One actuator maps five measurement points,then $$ \begin{array}{c}\mathrm{cos}\left(\frac{x}{\zeta}\right)\pi =\mathrm{cos}\left(k\pi \right)=\pm 1\\ k=\mathrm{1,2},\cdots 64\end{array}$$, and g(x) can be expressed as
$$$g\left(x\right)={\gamma}^{*}{\text{e}}^{\frac{-\alpha {x}^{2}}{{\zeta}^{2}}}$$$ | [18] |
c. Setting the parameter as
$$ \left\{\begin{array}{c}{\gamma}^{*}=\frac{1}{\sqrt{2\pi \sigma}}\hfill \\ \alpha =\frac{1}{2}\hfill \end{array}\right.$$, then
$$$g\left(x\right)=\frac{1}{\sqrt{2\pi \sigma}}{\text{e}}^{\frac{-{x}^{2}}{2{\zeta}^{2}}}={y}_{1}\left(x\right)=\frac{1}{\sqrt{2\pi \sigma}}{\text{e}}^{\frac{-{x}^{2}}{2{\sigma}^{2}}}$$$ | [19] |
The response of dilution valve is same as the function of normal distribution. It shows that the response has a normal distribution relationship with the outlet horizontal axis, as shown in Fig. 13.
In essence, the outlet water content model and the response of dilution valve represent the same effect, the two are described the concentration dilution model. This is also in line with the principle, because the water content of pulp flow is controlled by the opening of dilution valve. Finally, the results show that the concentration dilution model is a normal distribution, similar to the curve of “Λ shape”.
The sum of the water content and the pulp consistency was 1. The model of water content has been established in the above, then the model of pulp consistency can be described as
$$$C\left(x\right)=1-\frac{1}{\sqrt{2\pi \sigma}}{\text{e}}^{-\frac{{x}^{2}}{2{\sigma}^{2}}}$$$ | [20] |
The curves of pulp consistency and water content are shown in Fig. 14.
The results show that the model of pulp consistency is a reverse normal distribution. Along the machine direction of pulp flow, the deviation of water content will become smaller as the fluid advances, on the contrary, the pulp concentration will become larger. The smooth of concentration curve depends on the structure and speed of headbox.
In the slice lip adjustment system, the friction brings about boundary layer effect in the sides of slice section of headbox and makes the velocity of ejection pulp from the headbox uneven. The concentration of the corresponding longitudinal region will be reduced, the two sides of the region will not decrease, the concentration will increase instead. A obvious reverse response is shown in Fig. 15.
Previous studies have found that when a dilution valve operates, as corresponding basis weight of CD varies, so can the two sides of the valve. The action of the dilution valve is shown in the solid line in Fig. 13, which has a relatively obvious reverse response. It is hoped that the reverse response can be eliminated by optimizing the design, and the modified concentration dilution curve (dotted line in Fig. 16) can be obtained.
Suppose that the paper width is divided into n parts along the transverse direction. Then,
$$${Q}_{z}={C}_{n}\frac{W}{n}{H}_{n}{V}_{n}$$$ | [21] |
Where Q_{n}, C_{n}, H_{n}, V_{n} are the flow, pulp consistency, slice opening, velocity at the n^{th} part of slice. W is the width of the paper, which determined by production demand. V_{n} is constant in a given headbox, for CD control system with dilution water, H_{n} remains unchanged. So the basis weight of CD is determined by C_{n}. In order to eliminate the reverse response, the parameters of pulp consistency model C(x) are adjusted according to the headbox structure in this paper.
For the lineboard shape, γ=1, ζ=20, α=1.5, and β=0; for the normal shape, γ=2.5, ζ=20, α=1.5, and β=0.4; and for the Gaussian shape, γ=1, ζ=40, α=1.5, and β=0.4. It can be seen that the reverse response on both sides can be well eliminated or balanced when the dilution water is adopted to CD profile control system. The reverse response curve of the slice lip and the dilution valve response curve without reverse response as shown in Fig. 17. Fig. 18 shows the single and superimposed responses with slice and dilution control. As can be seen from the Fig. 18, whether it is regulated by a single slice lip or a combination of multiple slice lip, the pulp concentration decreases significantly when the lip opening increases (decreases). However, the pulp concentration on both sides of the corresponding downstream banner response will not decrease (high), but increase (low), it shows obvious reverse response. While the dilution response effect is better.
4. Conclusions
In this paper, the effect of dilution water added to the mainstream on concentration in headbox was studied based on the experimental and mathematical modelling. A scale model of headbox was derived to predict the water content distribution, pulp consistency dilution and pulp consistency distribution. The water content distribution was similar to the curve of “Λ”, the outlet pulp consistency were distribution of step diffusor and turbulence generator similar to the curve of “Λ” and “W” shape respectively. These models accurately predicted the experimentally flow characteristics of a pulp suspension with dilution water added.
By compare the regulation of the slice lip actuator and dilution water valve, the principle of reverse response was analyzed. By improving parameters of pulp consistency model, the reverse response on both sides could be well eliminated or balanced when the dilution water was adopted to CD profile control system. It could be ensured that the range was controlled as far as possible, the affected scope of single point addition of dilution water was discussed. In a word, by mathematical modelling the influence of dilution water added to the mainstream on concentration of both sides, a theoretical reference for the structural optimization of dilution hydraulic headbox and CD profile control strategy was provide.
Nomenclature
Q_{D} : | Dilution water flow, m^{3} |
Q_{S} : | White slurry flow, m^{3} |
Q_{TOT} : | Diluted white slurry flow, m^{3} |
C_{D} : | Dilution water consistency, % |
C_{S} : | White slurry consistency, % |
C_{TOT} : | Diluted white slurry consistency, % |
x : | CD horizontal axis |
y_{1} : | Outlet water content |
η : | Wave amplitude, m |
△h : | Change in free surface of slurry, m |
y : | Thickness of the slurry |
u : | Velocity filed, m/s |
Ø : | Velocity potential |
ρ : | Density of slurry, kg/m^{3} |
△P : | Pressure drop across mat, Kpa |
q : | Velocity term in Bernoulli’s equation |
P_{b} : | Pressure function at the mat, Kpa |
P_{f} : | Pressure in the fluid, Kpa |
μ : | Viscosity of the slurry, kg/m·sec |
A_{0}, A_{ω} : | Amplitude in wave equation |
N : | Complex wave number |
ω : | Angular frequency, Hz |
k : | Spatial frequency, cycle/m |
kd : | Damping factor in wave equation |
g(x) : | Spatial response of a single controller |
C(x) : | Slurry consistency function |
Acknowledgments
This work was partially supported by Shaanxi Key Innovation Team Project of Science and Technology (2014KCT-15) and Shaanxi Science & Technology Co-ordination & Innovation Project (2016KTCQ01-35). We sincerely thank for the funding of the project.
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