Model of Consecutive , Steady-State Underflow for Vertical Tailing Silos

School of Mining Engineering, North China University of Science and Technology, Tangshan 063009, Hebei, China Key Laboratory of Mining and Safety Technology of Hebei Province, North China University of Science and Technology, Tangshan 063009, Hebei, China College of Resources and Safety Engineering, Central South University, Changsha 410083, Hunan, China Hunan Key Laboratory of Mineral Resources Exploitation and Hazard Control for Deep Metal Mines, Changsha 410083, Hunan, China


Introduction
To date, numerous achievements have been made in research on tailings sedimentation, particularly in the following aspects: (1) the study of the settling velocity of tailings with various influencing factors [1][2][3], (2) the study of the settling velocity of tailings in multifactor conditions [4][5][6], (3) the study of accelerating the settling velocity of finegrained tailings by adding different types of flocculant and coagulant aids [7,8], and (4) the study of the dynamic settling of tailings in thickeners [9][10][11][12].However, few studies have been conducted on discharge model vertical tailing silos.
Vertical tailing silos are critical structures used for hydraulic filling.ese silos are designed using cylinders with diameters of 8-10 m, heights of 18-20 m, and half-spherical or conical bases with certain angles.In mines, the mortar volume concentration obtained through the secondary recovery of mortar overflow ranges from 7% to 12% of the overflow.e drastic fluctuations in underflow concentration limit the increase in filling slurry concentration and increase filling cement consumption.us, high-cemented filling quality and cost cannot be guaranteed.
Continuous concentration can be achieved through the use of high-density tailings and vertical tailing silos with low-diameter settlement containers.A continuous highdensity discharge model is proposed to resolve the problems encountered in the use of the current gravity sedimentation model of vertical tailing silos.
ese problems include the need for the repeated pulping of alternative tailing discharges through the simultaneous action of multiple silos and tailing discharge from bottom flow, as well as unmanageable and drastic uctuations in density during tailing discharge.

Consecutive Underflow Model
Tail slurries present three zones when a vertical tailing silo is in the process of consecutive under ow and steady state, as shown in Figure 1, namely, clear liquid region (ϕ 0), hindered settling region (0 < ϕ < ϕ c ), and compression region (ϕ > ϕ c ) [13,14].Note that q L is the volumetric velocity in the over ow zone, q R is the volumetric velocity in the discharge zone, and q F is the feed ux.
A consecutive and stable under ow model was established under two basic assumptions: the unit is continuously fed by a singular feed source located at the interface between the clear liquid region and the hindered settling region, and the over ow has no tailings; it only has water.
Vertical tailing silos maintain their steady state when they experience consecutive under ow.Dynamic equilibrium would then exist on the condition of the temporal a ecting factor and would meet the following condition: feeding (tailings + water) over ow (water) + under ow (tailings + water).
According to the mathematical model for batch, the following formula can be deduced [14].e volumetric solid concentration is constant across each horizontal cross section, i.e., ϕ ϕ(x, t). e conservation of mass equation for solids is then given by equation ( 1), and the analogue conservation equation for the uid is as equation (2): where t is the time, v s is the solid-phase velocity, and v f is the uid-phase velocity.e continuity equation of the mixture is (z/zx)Q(x, t) 0, and it implies that e solid-uid relative velocity or slip velocity v r v s − v f for a constitutive equation will be formulated.As such, (3) Kynch's kinematic sedimentation theory is based on the assumption that the solid-uid relative velocity or slip velocity v r is v r v r (ϕ) [15].Slip velocity is commonly expressed in terms of the Kynch batch ux density function f bk .Slip velocity is expressed as v r (f bk (ϕ)/ϕ(1 − ϕ)).
us, equation (3) can take the form as follows: Based on the phenomenological theory of sedimentation [16], we derived the following equation for relative velocity v r , as follows: where Δρ > 0 denotes the solid-uid density di erence, g is the acceleration of gravity, and σ e ′ (ϕ) is the derivative of the e ective solid stress function σ e (ϕ) [17][18][19], and through which the eld equation can be obtained, as follows: ese solutions satisfy the ordinary di erential equation, which equation (7) indicates as stationary, i.e., independent of time: where C is a constant of integration, and For the corresponding concentration pro le ϕ ϕ(x), concentration with respect to the independent variable x is represented as the following equation: 3. Materials and Methods

Materials.
Tailings from the Dahongshan copper mine in Yunnan, China, were used as the experimental material.e density of the tailings was 2.897 g/cm 3 , as measured with a pycnometer.As listed in Table 1, the average particle size of  Advances in Materials Science and Engineering the tailings was 0.1165 mm, as measured through sieving or elution.e permeability coefficient of the tailings was 0.9 cm/h.

Effective Solid Stress Measurement.
Effective solid stress was measured in the compression zone.e effective solid stress can be obtained using the following equation [5,21,22]: where ϕ c is the gel point and σ 0 and k are constant parameters.Gel point (ϕ c ) is the solid volume fraction at the beginning of the compression zone, and it was estimated in accordance with the following equation [23]: We estimated the gel point (ϕ c ) as 0.296 by using equation (11), where ϕ 1 and h 1 are the initial solid volume fraction and initial height of the suspension and h c is the equilibrium height of the sediment bed.
Effective solid stress was estimated through batch centrifuge experiments that were performed by using a centrifuge (HENGNUO).e centrifuge tubes had a volume of 15 ml and a diameter of 12 mm.e centrifugal acceleration ranges from 500 to 2,000 rpm [24,25].
Batch centrifuge experiments were performed for the calculation of effective solid stress [26].e details of the experiments and calculations are listed in Table 2.
e experimental data were fitted to equation (10), and the fitted curve is shown in Figure 2.

Measurement of Solid Flux Density Function.
Michaels and Bolger presented a typical example, as follows [27,28]: e mass concentrations of six groups of mortars ranged from 15% to 40%.Tailings were subjected to batch experiments.e results are presented in Figure 3 and Table 3.
We plotted the solid flux density function curves in accordance with the batch experiment data.In the curves, the abscissa represents volume concentration, and the ordinate represents solid flux density.e experimental data were fitted to equation ( 13).e fitted curve is shown in Figure 4.

Results and Analyses
4.1.Simulation Results.Four interface levels were selected randomly between tailings and water (8.8, 9.3, 10.6, and 11.3 m), and Fluent software was utilized to simulate the underflow volume concentration as a dynamic balance.e mixture model and standard k-epsilon model were chosen for the calculation and solution, respectively.e mesh used was of double-symmetry plane to reduce the amount of calculation and reduce computing resources.
e volume distribution contours of the water phase could be drawn from the results of numerical simulation and are shown in Figure 5.
e simulations achieved accurate underflow volume concentration on the four interface levels between tailings and water via the monitoring curve (i.e., 0.5271, 0.5421, 0.5582, and 0.5703, respectively).

Results of Consecutive Underflow Model.
Four interface levels between tailings and water were selected at the gel point (ϕ c ) of 0.296 when the underflow volume concentration (0.5271, 0.5421, 0.5582, and 0.5703) was calculated using the Runge-Kutta method to verify the accuracy of the underflow consecutive models.ese can be obtained from the curve of change between volume concentration of the slurry and height of the tailing silo at dynamic sedimentation, as shown in Figure 6.
Four interface levels between tailings and water from different underflow concentrations, i.e., 8.53, 9.34, 10.31, and     Advances in Materials Science and Engineering 11.12 m, respectively, were calculated using the partial differential equation (9).

Industrial
Test.An industrial test was carried out by the rst lling station from the Dahongshan copper mine in Yunnan, China.e photographs are presented in Figure 7.
e volume concentration of the tailing slurry is 0.13, and the ow is 300 m 3 /h at feeding.Under ow volume is 0.5189, 0.5371, 0.5512, and 0.5693, respectively, when interface levels between tailings and water are 8.5, 9.3, 10.3, and 11.1 m of the industrial test result.

Analyses.
e simulation results of the interface level between tailings and water according to Fluent software were compared with the calculation results using partial di erential equation ( 9) when the volume concentrations of under ow were 0.5274, 0.5421, 0.5582, and 0.5703 for the tailing silo, as presented in Table 4 and Figure 8. e industry test results of under ow volume compared with the results of the di erential equation are presented in Table 5 and Figure 8.
Solving the di erential equation shows that the concentrations of under ow volume (0.5271, 0.5421, 0.5582, and 0.5703) corresponding to di erent heights of tailing surfaces are closely related to the results simulated using the data, and

Conclusions
is study focused on the simulation and solution of dynamic settlement of tailings in a silo and the solution of a continuous, high-concentration, and stable model.In this study, the tailings from the rst lling station in the Dahongshan copper mine in Yuxi, located in Yunnan Province, China, were taken as the experimental material.Solid ux function, e ective bulk stress, and critical compression ratios were obtained experimentally through the use of batch settling and separation trials run in a centrifuge.Taking a vertical silo with a diameter of 9 m as the geometrical model, the dynamic and static settlements of tailings in the vertical silo were simulated.In the dynamic settlement of the tailings, volume concentrations of the under ow for four di erent cumulative tailing heights were evaluated, and the variation of slurry concentration with the height of the silo was simulated.Next, the cumulative heights of tailings required for various under ow concentrations and the variation of slurry concentration with soil height were acquired by running the proposed continuous, high-concentration, and stable model.e results were all close to the simulated results, which veri ed the accuracy of the model, and then, the theoretical results were applied to the tailing lling and discharge system.e industrial testing of the model was carried out in a vertical silo (with a diameter of 9 m).
e accuracy was also evinced by comparison of the results of the industrial test and the model.
e key conclusions were as follows: (1) In existing tailing discharge systems, feeding, settlement, and discharge are alternately conducted in multiple vertical silos.Di ering from this mode of operation, a continuous, high-concentration, and stable tailing discharge model is established on the basis of the mass balance between the tailings and water.e dynamic settlement of the tailings in the silo was simulated using the same software.In the simulation, four accumulated heights of tailings were selected to record the corresponding under ow volume concentration and the variation of slurry concentration with changing height of the silo based on the preset monitoring curves.(2) e equations governing the solid ux and the e ective solid stress are derived by carrying out settling experiments and separation experiments using the centrifuge.en, these factors are substituted into the proposed continuous, highconcentration, and stable tailing discharge model to solve the model using mathematical software.In this way, the under ow volume concentrations corresponding to di erent accumulation heights and the variation of the volume concentration of the slurry with silo height are obtained.e comparison of the results and the simulated results using Fluent software showed that they were su ciently similar    e application of the proposed continuous, highconcentration, and stable tailing discharge model for vertical silos effectively overcomes the technical problems facing existing tailing discharge systems in mines.ese problems include low discharge concentration, large concentration fluctuations, and the necessity of frequently applying high-pressure flow and water.e proposed model significantly improved the work efficiency of vertical silos, as it reduced the number of working vertical silos, omitted the process of completely discharging and charging the silos, and simplified the preparation of slurry materials.With these advantages, the model guaranteed the filling efficiency and quality.Industrial tests showed that the model worked and that the underflow volume concentration met the requirements for mine production.erefore, it avoided the waste of resources and equipment and saved water and electricity.
e research provided a theoretical basis and technical guidance for the design of continuous tailing discharge and filling systems.

Figure 1 :
Figure 1: Vertical tailing silo operating at steady state in a consecutive under ow model.

3. 2 .
Methods. e solid flux density function and effective solid stress are important components of the model of consecutive steady-state underflow for vertical tailing silos.Parameters σ 0 and k in the effective solid stress function (σ e (ϕ)) are obtained through centrifuge tests.Parameters v ∞ and n in the solid flux density function (f bk ) are obtained through sedimentation experiments.e different properties of materials result in different experimental parameters [20].

Figure 2 :
Figure 2: Experimental data obtained from batch centrifuge experiments and curve of equation.

Figure 6 :
Figure 6: Curve of changing between volume concentration of the slurry and height of the tailing silo, with under ow volume concentrations of (a) 0.5274, (b) 0.5421, (c) 0.5582, and (d) 0.5703.

Figure 8 :
Figure 8: Relationship of under ow volume concentration and interface levels between tailings and water via numerical simulation by Fluent software, calculation of di erential equation, and industrial test.

Table 1 :
Physical property parameters of tailings.

Table 3 :
Results of tailing settling experiment.

Table 2 :
Results of centrifuge tests.

Table 4 :
Interface level results of numerical simulation and calculation of di erential equation.

Table 5 :
Under ow results of industrial test and calculation of di erential equation.Advances in Materials Science and Engineeringto verify the correctness of the model.e application of the model to an industrial test displays favorable effects, which validates the theoretical efficacy of the model.