Abstract
When the mass concentration exceeds 7%, pulp suspensions stop flowing and act like a solid. To investigate the fluidization characteristics of medium-high-consistency pulp suspensions and achieve pulp fluidization and pumping, experiments were carried out with waste tissue pulp and unbleached kraft pulp. The objectives of this paper were to study the rheology of medium-high-consistency pulp and to determine accurate parameters for the physical Herschel-Bulkley model. To validate this model, computational fluid dynamics (CFD) results were compared to experimental data. The simulation values were very similar and were in agreement with experimental results.
1. Introduction
Compared with low-consistency technology, medium-high-consistency technology, which uses pulp suspension mass concentrations from 5% to 40%, is an efficient, low-energy, reduced-pollution process. The fluidization of pulp suspensions is an essential consideration in pulp and paper manufacturing. One of the most important pieces of equipment in medium-high-consistency pumping is the fluidization generator, which is indispensable for generating fluidization in pulp suspensions in the pump process.
Rheologically, pulp fiber suspensions behave like yield-pseudoplastic fluids [1]. To attain fluidization, the apparent yield stress must be exceeded. When the imposed shear stress is beyond the threshold value, pulp suspensions start flowing; like a solid body, networks develop at low shear rates and break down at high shear rates.
Bennington [2, 3] and Hietaniemi and Gullichsen [4] found that fluidization could occur at a floc and fiber level because of large differences in the apparent yield stress. The latter studies measured the onset of fluidization, finding that this onset is common for turbulent flow in mixing vessels. Hemstrom [5, 6] established the relationship between the flow velocity and the consistency of pulp suspensions in a completely turbulent state. In order to produce complete fluidization in a 10% concentration pulp fiber suspension, the pulp flow velocity must reach 45–50 m/s; this velocity was clearly very difficult to achieve, and the flow loss was very large. Duffy and Titchener [7] found the relationship between the critical shear stress and the consistency of pulp suspensions in a completely turbulent state, and several corresponding coefficients were given. Based on a previous study [7], additional experimental measurements were performed by Bennington et al. [2], and similar results were found. Experiments were carried out with seven kinds of medium-consistency pulp fiber suspensions by K.-F. Chen and S.-M. Chen [8], and formulas describing the critical shear stress were obtained. The commencement regime, transition regime, and turbulent regime were analyzed. Research showed that the shear stress of the fluidization generator for medium-consistency pulp depended on the rotational speed at a given shear stress.
Bhole et al. [9] measured the power number and the axial force number of low-consistency pulp fiber suspensions. CFD was used to model impeller flow using the Bingham approximation to describe the suspension rheology; model agreement with the experimental measurements was generally good, but there was still some error. Prajapati and Ein-Mozaffari [10] used Herschel-Bulkley model to simulate yield-pseudoplastic fluids and found good agreement. Wu [11] simulated non-Newtonian fluids using the Herschel-Bulkley model and found that realizable - and standard - turbulence models reflected the fluids’ rheological properties. Derakhshandeh [12] studied low-consistency pulp fiber suspensions using ultrasonic Doppler velocimetry, and the Herschel-Bulkley parameters and critical shear stress under turbulent conditions were obtained, and some other non-Newtonian fluids, such as sewage sludge, soft glassy materials, Kaolin suspensions, and bagasse pulp, were studied by Tabuteau et al. [13], Baudez et al. [14], Cotas et al. [15], Masalova et al. [16], and Li et al. [17].
The above studies mainly investigated the fluidization of low and partial medium-consistency pulp fiber suspensions, and CFD simulations were also focused on low-medium-consistency pulp. A lack of medium-high-consistency pulp and corresponding simulations is apparent. Thus, the objective of this study was to measure the rheology of medium-high-consistency pulp and to determine accurate parameters for the physical model of medium-high-consistency pulp. To validate the model, CFD results were compared to experimental data.
2. Experimental Setup and Theory
In order to test the rheology characteristics, the experimental setup was designed as shown in the schematic diagram in Figure 1. The agitation impeller was driven by a 15 kW ABB inverter motor. The motor speed was controlled by an inverter to realize different speeds. The rotational speed, torque, and power signals were collected by a computer.
The details of the vessel and impeller dimensions are listed in Table 1.
Because the impeller-derived moment is zero in the pulp suspension, we obtain equation . The shear stress is calculated by equation . Metzner and Otto [18] found that when a non-Newtonian fluid is subjected to mechanical agitation, the average shear rate can be expressed aswhere is a constant value determined experimentally and is the impeller’s rotational speed. Ayazi Shamlou and Edwards [19] proposed a correlation to estimate that was verified by Shekhar and Jayanti [20]:where is the impeller clearance from the vessel wall and is the inside diameter of the vessel. It can be seen that is a function of the geometric parameters of the impeller and experimental vessel, not the fluid rheology.
3. CFD Simulation
In order to describe the rheology characteristics of a medium-high-consistency pulp fiber suspension and obtain mutual verification of experiments and simulations, the following assumptions were proposed:(1)The model is single-phase, and agitation in the vessel is performed under laminar flow.(2)The fluid temperature is constant at 25°C, and the fluid is non-Newtonian and, therefore, isothermal and incompressible.
3.1. Governing Equations
The governing equations for mechanical agitation under laminar flow are generally expressed aswhere is liquid density, is time, is pulp flow velocity, is static pressure, is viscous stress, and is body force.
3.2. Mesh Generation
In the present work, the flow domain in the vessel was discretized with a high-quality structured hexahedral mesh using ANSYS-ICEM, and the impeller mesh was refined as shown in Figure 2. The mesh quality has a significant effect on the accuracy of numerical results. The minimum ratio of the hexahedral mesh quality was above 0.2, and the mix-angle was larger than 40 degrees. In general, the grid had to be fine enough to calculate the flow details, especially in some complex regimes, but not so fine that excessive time was required to solve the flow field.
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Grid independency was checked as shown in Table 2 by demonstrating that additional cells did not change the torque by more than 2% compared to the fine-cell model. Five different numbers of cells were tested for 6.62% waste tissue pulp at 500 rpm. Compared to the fine-cell model, the torque deviated by 1.4% for Grid D, with cells in the mesh.
3.3. Rheology Model
The pulp fiber suspensions demonstrated yield-pseudoplastic behavior and acted as shear thinning fluids. Thus, the Herschel-Bulkley model [9, 21–23] was used to describe the pulp behavior under shear stress:where is yield viscosity, is yield stress, is thecoefficient index of consistency, is the average shear rate, and is the power law index.
3.4. CFD Methodology
All walls of the flow domain were modeled using a no-slip wall condition. The wall roughness was set to 0.03 mm. Turbulence was simulated with a laminar model, and pulp flow was simulated under laminar flow conditions. In a turbulent state, the turbulence model was the RNG - model. A frozen-rotor interface was used for the coupling connection between the stationary and rotational frames, using quality and velocity transformations. A general grid interface (GGI) between the stationary and rotational components was used for internal component connections. A scalable wall-function was used to solve the boundary layer field. The average residual convergence criterion was set to . As a simulation convergence judgment, impeller torque fluctuations within no more than 5% every 100 steps were required. The commercial software CFX 14.0 was applied to solve the 3D Reynolds-averaged Navier-Stokes equations. The simulations in this work were carried out on a cluster of twenty Xeon E5-2670V2 nodes with 128 GB of RAM. The CFD calculations are steady-state.
4. Results and Discussions
4.1. Experimental Results
In this study, the shear rate and shear stress of different consistencies of waste tissue pulp and unbleached kraft pulp were obtained using the experimental setup shown in Figure 1. Figures 3 and 4 show the relationship between the shear stress and shear rate. The shear stress increased with shear rate and pulp consistency. When the shear stress was in the turbulent regime, the shear stress increased linearly. The calculated apparent yield stress, consistency index, and power law index values predicted by the Herschel-Bulkley model by using regression analysis are shown in Tables 3 and 4. The apparent yield stress and consistency index increased with increasing consistency, but the power law index decreased. The Herschel-Bulkley model agrees well with the experimental flow data until the onset of the turbulent regime.
4.2. CFD Validation
Based on the Herschel-Bulkley model constants obtained from experimental data and shown in Tables 3 and 4, pulp suspension flows were simulated for several mass concentrations of pulp. The power number can be used as a basis of comparison; this value is given by where is power (W), is pulp density (kg/m3), is rotational speed (r/min), and is impeller diameter (mm).
Figures 5 and 6 show a comparison of the power numbers of the experimental and CFD simulation results. In order to improve the ease of analysis, logarithmic coordinates were used for both the horizontal axis (rotational speed) and vertical axis (power number). The CFD results are slightly higher than experimental values in Figures 5 and 6. The power numbers increase with increasing pulp mass concentration and decrease with impeller rotational speed. Generally, the simulated values are very similar for both waste tissue pulp and unbleached kraft pulp and are in agreement with experimental results.
4.2.1. Velocity Profiles
Figure 7 shows the velocity streamlines created by the impeller in the 11% waste tissue pulp at rotational speeds of rpm and rpm. As the rotational speed increases, the velocity magnitude increases. Because the direction of impeller rotation is positive in Figure 7(a), the pulp fiber suspension inflow is from the impeller inlet. The outflow portion is in the axial direction, with another portion in the radial direction. This is due to the rotational energy and the shape of the blades, which thus take on velocity energy. With centrifugal force increasing, the outflow direction changes to radial direction, as shown in Figure 7(b). In Figure 7(a), one vortex is at the impeller inlet, and another vortex is located on the right side of the impeller, indicated by black solid-border boxes. In Figure 7(b), one vortex is on the right side of impeller and another vortex is at the impeller outlet.
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Figure 8 shows the velocity streamlines created by the impeller in 11.46% unbleached kraft pulp at rotational speeds of rpm and rpm. In Figure 8(a), there is only one vortex at the impeller inlet. In Figure 8(b), there are two obvious vortexes, one on the right side of the impeller and another at the impeller outlet.
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Figure 9 shows the velocity streamlines created by the impeller at a speed of rpm. Figures 9(a), 9(b), and 9(c) show the streamlines for 6.62%, 11%, and 14% mass concentrations of waste tissue pulp, and Figures 9(d), 9(e), and 9(f) show the streamlines for 11.46%, 19.04%, and 31.73% mass concentrations of unbleached kraft pulp. There is only one vortex on the right side of the impeller. As the mass concentration increases, the streamlines clearly change, as shown in the solid-border boxes, and the apparent yield stress increases for both waste tissue pulp and unbleached kraft pulp, as shown in Tables 3 and 4. In the regions indicated by the dashed-border boxes, the streamline plots gradually become smoother as the mass concentration increases.
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5. Conclusions
The objective of this paper was to study the rheology of medium-high-consistency pulp and determine accurate parameters for the physical Herschel-Bulkley model. To validate this model, the CFD results were compared to the experimental data. The simulation values were very similar to the experimental results. The following conclusions were obtained based on the results in this study:(1)The shear rate and shear stress for different consistencies of both waste tissue pulp and unbleached kraft pulp were obtained experimentally. The shear stress increased with increasing shear rate and pulp consistency. The Herschel-Bulkley model was used to calculate the apparent yield stress, consistency index, and power law index values up to the onset of the turbulent regime by means of regression analysis of the experimental data. When the shear stress was in the turbulent regime, the shear stress increased linearly.(2)Based on the Herschel-Bulkley model constants from the experimental data, pulp suspension flows were simulated for several mass concentrations of pulp. The power numbers were in good agreement with experimental results.(3)The number and location of vortexes and streamlines were analyzed for different mass concentrations and speeds.
Nomenclature
| : | Impeller diameter, mm |
| : | Impeller height, mm |
| : | Inside diameter of vessel, mm |
| 1: | Outside diameter of vessel, mm |
| : | Vessel height, mm |
| : | Clearance from wall to bottom, mm |
| : | Power, W |
| : | Constant value determined experimentally |
| : | Impeller rotational speed, rpm |
| : | Impeller clearance from the vessel wall, mm |
| : | Pulp density, kg/m3. |
| : | Shear stress, Pa |
| ω: | Rotation angular velocity, rad/s |
| : | Time, s |
| : | Pulp flow velocity, m/s |
| p: | Static pressure, Pa |
| : | Viscous stress, Pa |
| : | Body force, N |
| : | Yield viscosity, Pa·s |
| : | Yield stress, Pa |
| : | Coefficient index of consistency |
| : | Average shear rate, s−1 |
| n: | Power law index. |
| : | Constant |
| : | Vessel wall |
| : | Yield. |
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The support of the National Natural Science Foundation of China (Grant no. 51379179) is gratefully acknowledged. The help from our research group is greatly appreciated.