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An investigation of the effect of particle size on discharge behavior of pulverized coal
Powder Technology 284 (2015) 47–56
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
An investigation of the effect of particle size on discharge behavior of pulverized coal Yi Liu, Haifeng Lu, Xiaolei Guo, Xin Gong ⁎, Xiaolin Sun, Wei Zhao Key Laboratory of Coal Gasification and Energy Chemical Engineering of Ministry of Education, Shanghai Engineering Research Center of Coal Gasification, Institute of Clean Coal Technology, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 25 October 2014 Received in revised form 12 June 2015 Accepted 16 June 2015 Available online 23 June 2015 Keywords: Pulverized coal Particle size Flow property Stress state analysis Hopper discharge
a b s t r a c t In this paper, pulverized coal from industry was used as the experimental material. In order to study the effect of particle size, the pulverized coal was sieved into seven samples with different particle sizes. Firstly, the material properties as packing, incipient flow and wall friction properties were evaluated. Meanwhile, a transparent Perspex hopper was used to monitor the change of discharge behavior of pulverized coal caused by the increase of particle size. From the visual observation of discharge test, a progressive transition from blocking to unstable flow and to mass flow was found as the particle size increased. Hence, a complete set of physically based equations for compressibility, flowability and powder consolidation has been derived. Then, a new method, taking into account the stress state in the hopper, is introduced to assess the discharge behaviors of pulverized coal. The validity of this new method has been confirmed by comparing theoretical and experimental discharge behavior for pulverized coals. The theoretical approach based on stress state analysis, is able to provide results which correlate well with the experiments. © 2015 Published by Elsevier B.V.
1. Introduction Entrained-flow pressurized gasification process of pulverized coal, including the storage, discharge and conveying of pulverized coals, is one of the best contemporary coal gasification technologies [1,2]. However, there are several well-known problems in the handling and storage of pulverize coal, such as bridging, channeling, fluctuating flow rate or even blocking in processing equipment and storage, as the tendency of agglomeration caused by strong inter-particle forces [3–5]. The most common and serious problem is no-flow, due to arching or ratholing [6,7]. Thus, reliable information of flow properties of the pulverized coal concerned is required for reliable flow from the hopper, which is essentially crucial in handling and processing operations [8,9]. Flowing or yielding means that a powder is brought to irreversible deformation, which can be caused by an external mechanical stressing event. This stress can be produced by either force or energy. According to this method, only the trouble-free discharge of a powder out off a hopper by its dead weight is considered as good flowability. In contrast, powders that show flow problems are classified as poorly flowing or non-flowing. It is well known that there are two flow patterns in hopper or bins: mass flow and funnel flow, and most flow problems are associated with the funnel flow pattern [10,11]. To avoid funnel flow, it is important to
⁎ Corresponding author. Tel.: +86 21 6425 2521; fax: +86 21 6425 1312. E-mail address: [email protected] (X. Gong).
http://dx.doi.org/10.1016/j.powtec.2015.06.041 0032-5910/© 2015 Published by Elsevier B.V.
determine properties of pulverized coal. One of the major factors affecting flow patterns is particle size. In the case of powder with particle size less than 100 μm, van der Waals based interparticle forces, exceed the gravitational force by several orders of magnitude [12–14]. Thus, flow problems are very common in the case of cohesive powders with particle sizes less than 100 μm. To avoid this technical problem and hazard, it is really necessary to understand the effect of particle size on flow properties of pulverized coal. In order to improve a reliable and controllable discharge of powder from hoppers, effects have been taken to develop a proper method for discharge behavior of pulverized coal prediction. Powder flow characteristics are often investigated adopting a variety of methods. One of the most common of those is the shear test [15–17]. Using Mohr's circle analysis, various important parameters may be extracted from these results. These properties include the major Principal stress, the minor Principal stress, the unconfined yield stress, the cohesive, the angle of internal friction and the effective angle of internal friction. In previous studies, Jenike applied two-dimensional stress analysis to develop a numerical methodology to determine the minimum hopper slope required for mass flow and the minimum outlet dimension for unobstructed gravity flow [18]. Although this method has been widely used over the past few decades, it occasionally gives irrational results, due to the assumption of constant average material properties [19]. In fact, some of the properties, such as effective angle of internal friction and bulk density, are functions of stress state. To overcome this matter, a complete set of physically based equations for compressibility, flowability and powder consolidation
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are used in this study. And, a new approach is introduced to assess the discharge behavior for the pulverized coal. As opposed to the original Jenike criterion which only considers constant average material properties, the new method considers the effect of stress state on material properties by using these physically based equations. The validity of the new theoretical approach will be examined by comparing the theoretical findings and experimental discharge behavior. 2. Materials and methods 2.1. Materials The initial material for this study is industrial bituminous pulverized coal having a particle density of 1490 kg/m3. The initial material is sieved by an electric vibrating screen. Seven samples of pulverized coal were obtained. The physical properties of these samples are reported in Table 1 and Fig. 1. The particle size distribution of the pulverized coal in the experiments can be described using the Rosin–Rammler distribution function [20,21]. The general expression of the Rosin–Rammler function is: n D RðDÞ ¼ 100−ϕ ¼ 100 exp − De
ð1Þ
where R(D) is the distribution function, D is the particle size, De is the mean particle size and n is a measure of the spread of particle size the large value of which designates a narrower size range. This expression can be rewritten as: ð2Þ
lgf lg½100=RðDÞg ¼ n lgD þ M1
where M1 is a constant. A plot of lgD versus lg(lg(100 ∕ R(D))) will result in a straight line of slope n if the behavior of the material fits the RR model. From the observation of Fig. 2 and the corresponding linear correlation coefficient, one deduces that the RR model provides a perfect fit to the experimental PSD curve. As shown in Table 1, the samples have approximate n within the range of 0.9–1.2, indicating the same level of size range. Therefore, the particle size represents the major factor for this study. 2.2. Experimental 2.2.1. Compressibility test Compressibility is a measurement of how density changes as a function of applied normal stresses. The sample vessel specification is 50 mm × 85 ml. The measurement utilizes a vented piston to compress the sample under the increasing normal stress. Prior to each measurement, the sample was first prepared by conditioning and splitting using the standard FT4 blade and split vessel assembly to remove any residual compaction and build a uniform and loose condition in the bed of powders. After the preconditioning and slitting procedure, a sample of 85 ml is left behind and the condition bulk density ρCBD is obtained. Before the compression occurs, the blade is then exchanged for the vented piston. Fig. 3 shows the schematic of the compressibility
Fig. 1. Cumulative particle size distribution of the samples obtained with laser diffraction (Malvern 2000).
test; each normal stress will be applied for a defined time to allow the powder to reach equilibrium. The distance traveled by the piston is measured for each applied normal stress and the bulk density is automatically calculated.
2.2.2. Shear cell and wall friction test A rotational shear cell is used to quantify the incipient flow behaviors of the pulverized coals. This shear cell consists of a vessel containing the samples and a shear head to introduce both vertical and rotational stresses, as shown in Fig. 4(a). After completion of one conditional cycle, a vented piston is used to induce a precise consolidation stress in the sample. The pre-consolidation procedure is carried out prior to the splitting procedure to ensure that the sample is consolidated. Then, the vessel was split. To ensure that the surface of the sample is suitably consolidated, the sample is recompressed by the shear cell head to remove any disturbances caused by the split. After the previous operations are done, the shearing sequence could begin. Based on the normal and shear stress collected, the yield locus at a given preconsolidation condition can be acquired. Fig. 5(a) is a typical σ–τ diagram, showing two Mohr circles, yield locus, and effective yield locus. With these terms, the angle of internal friction φi, effective angle of internal friction φe, cohesive C, major Principal stress σ1, minor Principal stress σ2 and unconfined yield stress
Table 1 Physical property and packing properties of all pulverized coals used in this study. Materials
dv (μm)
n
ρCBD (kg/m3)
ρb,0 (kg/m3)
σz,0 (Pa)
N
a b c d e f g
223.8 141.3 94.2 74.9 55.9 43.2 17.7
0.91 1.02 1.11 1.12 1.21 1.27 1.21
739.0 723.0 690.6 642.3 621.9 584.3 520.5
742.4 722.1 691.2 629.9 605.6 576.2 439.7
3863.0 1475.2 1208.7 347.3 289.0 458.1 43.4
0.022 0.023 0.025 0.028 0.031 0.040 0.058
Fig. 2. Relationship between lg{lg[100/R(D)]} and lgD of pulverized coal (dv = 43.2 μm).
Y. Liu et al. / Powder Technology 284 (2015) 47–56
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Fig. 4. Image of the shear cell supplied with the FT4 powder Rheometer: (a) measuring powder flow properties and (b) measuring angle of wall friction.
3. Result and discussion 3.1. Packing properties Fig. 3. Schematic of compressibility test.
σc can be calculated. It should be noted here that the stressing prehistory of a cohesive powder flow is stationary (steady state) and results significantly in a cohesive stationary yield locus in the τ (σ) of Fig. 5(b). Where, σ0 is the isostatic tensile strength of unconsolidated powder, equals a characteristic cohesion force in an unconsolidated powder; φst is the stationary angle of internal friction, the larger is the difference between φst and φi, the more cohesive is the powder. The wall friction test is similar to the shear test. In the wall friction test, the shear head was replaced by a Perspex plate, which has the same material quality with the hopper, as shown in Fig. 4(b). Typically, a series of shear stress values for a range of reducing normal stress were measured to generate a wall yield locus, from which the angle of wall friction is determined. In this study, the pre-consolidation stress in the process of the wall friction test is 3 kPa.
2.2.3. Hopper discharge A transparent Perspex hopper was fabricated to observe the flow phenomenon of pulverized coal during the discharge process dynamically and directly, and, to be able to observe mass flow or funnel flow discharge by looking at the movement of the particle close to the hopper wall, as shown in Fig. 6. In each case, a sample weighed 0.7 kg was loaded in the hopper. Discharge was performed immediately after loading to avoid consolidation caused by long residence time. The change of powder mass was determined and recorded by a hanging weighing unit during the discharging process.
Compressibility is not directly a measurement of flowability, but nevertheless related to many process environments, for example, storage in hoppers or bins. It is useful to understand the relationship between powder bulk density ρb and applied consolidation stress σz. Tomas [22–25] has proposed such a relationship for powders by extending analogies to the adiabatic gas law for isentropic compression, as given by Eq. (3). σ z þ σ z;0 N ρb ¼ ρb;0 σ z;0
ð3Þ
where ρb,0 is the powder bulk density at negligible consolidation and σZ,0 is the pull-off stress when the unconfined yield strength is zero. The parameter N is the compressibility index, in the range of 0–1. N = 0 indicates incompressibility stiff bulk material, while N = 1 represents deal gas. The ρb(σz) functions which described the relationship between the applied normal stress and bulk density is one of the fundamental equations for stress state analysis. Fig. 7 reports the measured bulk density as a function of the applied normal stress. By adopting Eq. (3), the corresponding regression curve is demonstrated. This shows the capability of Eq. (3) to describe the packing properties of pulverized coal. Fig. 8 shows that the compressibility index N increases as the particle size decreases. Also, the smaller the particle size, the smaller are the conditioned bulk density ρCBD and the estimated loose bulk density ρb,0. This means that the pulverized coal becomes more compressible, indicating a lower pack efficiency. Furthermore, once the particle size is smaller than a certain value (100 μm), the conditioned bulk density ρCBD is significantly larger than the loose bulk density ρb,0, due to the non-ignorable gravity consolidation, as shown in Table 1.
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Fig. 5. (a) Mohr circle, yield locus and effective yield locus. (b) Stationary yield locus.
3.2. Flow properties With the normal and shear stress data, the yield locus at a given pre-consolidation level was determined by linear regression using the Mohr–Coulomb equation. τ ¼ σ tan φi þ C
ð4Þ
Then the cohesion strength C and the angle of the internal friction φi are calculated. The effective angle of internal friction is a measure for the inner friction at steady-state flow. It is calculated by measuring the
angle that a line drawn through the origin and tangential to the larger Mohr circle creates with the horizontal axis. Fig. 5(a) shows two Mohr stress circles. The smaller Mohr circle is drawn where the circumference passes through the origin and is tangential to the yield locus and represents the stress state under unconfined yield. And, the larger Mohr circle is drawn where the circumference passes through the Pre-shear data point and is tangential to the yield locus and represents the stress state during consolidation. Based on these two Mohr circles, the unconfined yield strength, minor Principal stress and major Principal stress were determined. As an instance, Fig. 9 reports graphically the yield loci and Mohr circles for pulverized coal g under different pre-consolidation conditions.
Fig. 6. Schematic of the hopper discharge system (D1 = 150 mm, D2 = 32 mm, α = 15°).
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Fig. 7. Comparison between experimental and predicted bulk density of samples in this study.
Fig. 9. Four yield loci under different consolidation stress for pulverized coal g (dv = 43.2 μm).
Pre-consolidation at different levels resulted in different yield loci. Each yield locus gave a pair of values of the unconfined yield strength σc and the major Principal stress σ1. Then, the flow function FF of the pulverized coal was determined, as shown in Fig. 10. Flow index ffc is the inverse of the flow function at any point, ffc = σ1/σc [26]. Based on the flow index, the powder materials are classified as hardened, very cohesive, cohesive, easy flowing, or free flowing [18], as shown in Fig. 10. According to Tomas [22–25], the consolidation function σc = f (σ1) can be conveniently analyzed by linear regression, Eq. (5):
slope and the intercept of the linear consolidation function (the angle of internal friction φi fluctuates by only ca. ±2° around the average, thus, an average of the values determined experimentally for all consolidation levels is used).
2 ð sin φst − sin φi Þ 2 sin φst ð1 þ sin φi Þ σ1 þ σ0 ð1 þ sin φst Þ ð1− sin φi Þ ð1 þ sin φst Þ ð1− sin φi Þ ¼ a1 σ 1 þ σ c;0 ð5Þ
Table 2 contains data on the measured flow properties and angle of wall frictions from the wall friction test. The flow index values decrease counter-clockwise from the region of free flowing to hardened powders in Fig. 10. Also, the degree of cohesiveness of the powders can be indicated by the difference between φst and φi [27]. The larger the difference between these friction angles is, the more cohesive is the powder response. As particle size decreases the flow index decreases while the difference between φst and φi increases, indicating that the cohesiveness of pulverized coal is increased. Meanwhile, wall friction measurement was also conducted for the samples. The angle of wall friction reflected the ability of pulverized coals to flow over the inner surface of the hopper. The angle of wall friction increased with the decrease of particle size. This means that the interaction between the wall and pulverized coal is gradually increased with the decrease of particle size.
σc ¼
where φst is the stationary angle of internal friction, φi is the angle of internal friction and σ0 is the isostatic tensile strength. As shown in Fig. 10, a linear relationship between the Principal stress σ1 and unconfined yield strength σc apparently exists for each of the samples defined by a certain particle size. The stationary angle of internal friction φst and the isostatic tensile strength σ0 are both basic physical material parameters for describing the flow properties of powders. The stationary angle of internal friction φst and the isostatic tensile strength σ0 can be easily obtained from the
Fig. 8. Compressibility index as a function of particle size.
φst ¼ arcsin
σ0 ¼
a1 þ ð2−a1 Þ sin φi 2−a1 ð1− sin φi Þ
ð1 þ sin φst Þ ð1− sin φi Þ σ c;0 2 ð1 þ sin φi Þ sin φst
Fig. 10. Consolidation function σc = f (σ1) of the samples used in the study.
ð6Þ
ð7Þ
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Table 2 Pulverized coal flow properties and angle of wall friction. Material
dv (μm)
σpre (kPa)
σ1 (kPa)
σ2 (kPa)
C (kPa)
σc (kPa)
φi (°)
φe (°)
φst (°)
σ0 (kPa)
φst–φi (°)
φw (°)
a
223.8
1.6
25.78
34.66
0.025
2.35
25.15
33.97
0.044
3.33
28.11
e
55.9
34.24
0.078
3.6
29.77
f
43.24
36.85
0.018
5.22
30.44
g
17.68
38.6 40.4 41.7 42.7 35.0 33.9 35.0 35.4 33.3 35.6 35.2 39.9 33.5 33.9 35.0 38.7 33.9 36.6 35.1 40.1 37.5 37.5 36.0 39.9 39.8 40.4 42.3 49.9
0.08
74.9
36.6 38.6 41.1 40.5 33.2 31.5 32.5 29.5 30.6 33.6 32.8 33 29.8 30.6 31.5 29.5 30 32.4 30.5 32.4 32.2 32.3 30.4 33.3 30.4 32.8 31.9 30.3
34.03
d
1.59 1.08 0.50 0.10 1.14 0.94 0.52 0.23 1.60 0.82 0.49 0.25 2.11 1.25 0.69 0.30 2.17 1.69 0.90 0.27 3.23 2.12 1.09 0.22 5.07 2.83 1.89 0.58
23.83
94.2
0.40 0.26 0.12 0.02 0.31 0.26 0.14 0.07 0.46 0.22 0.13 0.07 0.61 0.36 0.19 0.09 0.63 0.47 0.26 0.07 0.89 0.59 0.31 0.06 1.45 0.77 0.52 0.17
1.71
c
4.4 2.98 1.32 0.20 4.22 2.88 1.44 0.29 4.33 2.71 1.38 0.21 4.29 2.84 1.36 0.22 4.23 2.64 1.38 0.21 3.89 2.60 1.35 0.19 3.37 2.14 1.01 0.13
0.011
141.3
19.02 13.96 6.54 1.04 15.58 10.12 5.30 1.09 14.89 10.23 5.15 0.98 14.84 9.99 5.04 0.94 14.88 10.42 5.14 0.95 16.00 10.68 5.22 0.87 15.35 10.02 5.17 0.98
40.94
b
9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5
40.02
0.087
8.31
29.84
The function φe(σ1) describes the relationship between the major Principal stress σ1 and effective angle of internal friction φe. It is very important in stress state analysis. From the stationary yield locus, the following relationship is obtained: σ þ σ σ −σ 1 2 1 2 ¼ sin φst þ σ0 : 2 2
ð8Þ
In the case of cohesionless steady-state flow, which is described by the effective yield locus, the following relationship is obtained: σ −σ σ þ σ 1 2 1 2 ¼ sin φe : 2 2
ð9Þ
Combined with Eqs. (8) and (9), we obtained a simple correlation between the stress-dependent effective angle of internal friction φe, the stationary angle of internal friction φst and the center stress (σ1 + σ2) ∕ 2: sin φe ¼ sin φst 1 þ
σ0 : ðσ 1 þ σ 2 Þ=2
If the major Principal stress σ1 reaches the stationary uniaxial compressive strength σc,st the effective angle of internal friction is φe = 90°, and for σ1 → ∞ follows φe → φst, as shown in Fig. 11.
3.3. Stress state analysis and discharge behavior prediction It is clear that different particle sizes result in the different settled bed structures, and consequently affect the capability of powder to flow [28]. Thus, the effect of particle size on discharge can only be assessed when the stress state in the hopper is known. The model reported in this section describes the tensional state of the powder in the hopper using a direct method of slices inspired by Janssen's analysis. It is modified here to take into account the basic flow parameters and the fundamental equations estimated above. Then, the discharge behavior of pulverized coal was predicted by applying the Jenike criterion.
ð10Þ
The term (σ1 + σ2) ∕ 2 can be replaced by the major Principal stress σ1 flow: ðσ 1 þ σ 2 Þ σ 1 −σ 0 sin φst : ¼ 1 þ sin φst 2
ð11Þ
Then, we obtained: sin φe ¼ sin φst
σ1 þ σ0 : σ 1 − sin φst σ 0
ð12Þ
As the physical limit that φe ≤ 90°, we obtained:
σ 1 ≥σ c;st ¼
2 sin φst σ 0 : 1− sin φst
ð13Þ Fig. 11. Effective angle of internal friction as a function of major Principal stress.
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Fig. 13. Mohr's circle for the stress adjacent to the wall in a conical hopper [29].
Fig. 12. Force balance on an elemental slice of powder at depth z.
Using the usual Janssen assumptions that the stresses are constant across any cross-section, a force balance is performed on an elemental slice at depth z, as shown in Fig. 12. The force balance is: dσ z þ dz
2ðτrh Þw ¼ ρb ðσ r Þ g: d Hþ −z 2 tan α
ð14Þ
Fig. 14 shows that the calculated axial stress changes with depth (pulverized coal f). Additionally, Fig. 15 reports the relationship between the particle size and axial stress at the outlet of the hopper. Jenike [26] concludes, with the flow-no flow criterion, that a stable cohesive arch will form in a convergent channel when the unconfined yield strength σc exceeds the effective wall stress, σ′. When σ′ N σc, a forming arch will collapse, that is, the critical point is the condition σ′ = σc. This criterion is based on a force balance on the arch assuming that the only forces acting are the weight of the arch and the abutment force, see Fig. 16. Denoting the angle between the horizontal axis and tangent to the arch at the wall by β, we see that the width of the arch at the wall is δz cos β and hence the vertical force due to the effective 2
The term (τrh)w is the shear stress acting on the side of the element. It is not equal to the shear stress on the wall. Neddman considered the Mohr–Coulomb diagram for the material adjacent to the right-hand wall, as shown in Fig. 13. For the passive case, the wall plane is denoted by the point W′ on the figure. The circumferential surface of the element is an r-plane which is inclined to the wall at angle α. Therefore, it is represented on Fig. 13 by the point R. The H represents the Z-plane which lies at the opposite end of the diameter to R. From Fig. 13, the ratios σr/σz and (τrh)w/σz can be evaluated. σ r 1 þ sin φe cosðω þ φw þ 2α Þ ¼ σz 1− sin φe cosðω þ φw þ 2α Þ
ð15Þ
ðτrh Þw sin φe cosðω þ φw þ 2α Þ ¼ σz 1− sin φe cosðω þ φw þ 2α Þ
ð16Þ
where sin ω ¼
sin φw : sin φe
ð17Þ
Furthermore, the radial stresses are assumed as consolidation stress. The ρb(σr) and φe(σr) functions have already been indicated in Sections 3.1 and 3.2, as given by Eqs. (3) and (12). The boundary condition for Eq. (14) is that the top surface of the material is open to the atmosphere so that the normal stress on it may be taken to be zero. Besides, the material level H adopted in the calculation was visually obtained, as shown in Table 3. The stress state in the hopper was evaluated by solving simultaneously Eq. (14) with the boundary condition σz(z = 0) = 0. Since the effective angle of internal friction and bulk density depend on the consolidation stress, these parameters would be functions of depth z. Consequently, a numerical solution was needed. As an example,
wall stress is, πDoδz sin 2βσ′/2. The weight of the arch is πðD20 Þ ρgδzDo . The resulting equation for the effective wall stress is: σ0 ¼
Do ρb g : 2 sin 2β
ð18Þ
Since the value of β is unknown, we can only acquire the range of the value of effective wall stress by assuming the value of β. The effective wall stress reaches a minimum value, when the angle β is 45°. And, the effective wall stress reaches a maximum value, when it is perpendicular to the wall (β = 15°). Since the radial stress at the outlet was calculated, the consolidation level at the outlet was acquired. Bulk density and the unconfined yield stress were obtained by using Eqs. (3) and (5), respectively. For comparison purposes, Fig. 17 reports the calculated effective wall stress and unconfined yield stress. With the particle size larger than 100 μm, the pulverized coal may perform free discharge under gravity. When the particle size is in the range of 40 μm to 100 μm, the unconfined yield stress is higher than the minimum value of effect wall stress but lower than the maximum value. It means that unstable arches may form and erratic flow may exhibit at this particle size range. For the pulverized coal with particle size smaller than 30 μm, Table 3 The material level in the hopper for the different samples. Materials
dv (μm)
H (m)
a b c d e f g
223.8 141.3 94.2 74.9 55.9 43.2 17.7
0.174 0.176 0.179 0.186 0.188 0.193 0.216
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Fig. 16. Jenike's model for the formation of an arch. Fig. 14. Radial stress as a function of depth (coal f dv = 43.2 μm).
the unconfined yield stress is even higher than the maximum value of effect wall stress. The discharge for pulverized coal g may be prevented by the formation of a stable arch. Jenike [26] also pointed out that the maximum hopper half angle (α) for mass flow could be calculated from the angle of wall friction (φw) and the effective angle of internal friction (φe) with the following equations:
α¼
π 1 ð1− sin φe Þ 1 sin φw : − − cos−1 φw þ sin−1 sin φe 2 2 2 sin φe 2
ð19Þ
As the effective angle of internal friction is the function of major Principal stress, therefore, the slope needed for mass flow is a function of depth. For instance, Fig. 18 shows the relationship between maximum hopper half angles for mass flow and depth. As clearly visible from Fig. 18, the material at the outlet generally requires the steepest slope for mass flow. That means that the most critical condition for mass flow is found close to the outlet. According to this analysis, the hopper angle versus vertical for mass flow was calculated by Eq. (19) with the effective angle of internal friction at the outlet. The calculated results were reported in Fig. 19. For the discharge instrument used in this study, the hopper angle versus vertical is 15°. From the result of the calculation of maximum hopper half angles, it is found that pulverized coals a, b and c would
Fig. 15. The radial stress in outlet of hopper as a function of particle size.
exhibit mass flow, while pulverized coals d, e, f, and g may perform funnel flow. The above findings described that decreasing the particle size of pulverized coal would lead to the increase in cohesiveness and wall adhesion effect, causing flow problem over itself and the inner surface of the hopper. The stress state analysis indicates that the pulverized coal with particle size larger than 100 μm will perform mass flow and those with particle size smaller than 100 μm would exhibit funnel flow (unstable flow). For the pulverized coal with particle size less than 30 μm, discharge may prevented by the formation of a stable arch. 3.4. Discharge result Fig. 20 exhibits the results of pulverized coal discharge from the Perspex hopper. The flow rate is defined as M/Ma, where Ma (543.2 kg/h) is the average flow of pulverized coal with the largest particle size and M is the average flow rate of each sample. The relative flow rate profiles show a steep exponential decrease with the decrease of particle size. Based on the visual observation of discharge behaviors, three regions including arching, unstable-flow and mass flow were found. I. Mass-flow region (pulverized coal a, b). In this region, pulverized coal discharges smoothly just as a liquid. All the powder in the hopper is in motion, which shows a typical mass flow pattern. II. Unstable-flow region (pulverized coal c–f). In the unstable flow region, a dynamic arch forms and collapses alternately across
Fig. 17. Unconfined yield strength and effective wall stress at outlet as a function of particle size.
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Fig. 20. Relative rate of discharge as a function of particle size.
Fig. 18. The slope for mass flow as a function of depth (coal f dv = 43.2 μm).
III.
55
the hopper outlet. Meanwhile, the powder flows towards the outlet of the hopper in a channel formed within the powder itself. Arching region (pulverized coal g). In the arching region, a static arch forms across the hopper outlet which completely stops the powder flow. It is further found that there is an amount of pulverized coal adhering to the hopper wall even if the arch was broken artificially.
This flow region classification is also supported by the flow rate analysis. Fig. 21 reports the weighting curve and time series of relative standard deviations of flow rate for pulverized coals a, c and e. The instantaneous flow rate, which is the derivative for the weighting curve, was calculated automatically by the software. We used relative standard deviation (RSD) to quantify the complexity of changes in the time series of flow rate. RSD is defined as: M −mean RSDðiÞ ¼ i mean
ð20Þ
where Mi is the instantaneous flow rate and mean is the average value of flow rate. RSD as a dimensionless number is used to compare the complexity of changes of the flow rates for different pulverized coals. Pulverized coal with the mean particle size 224 μm shows slight fluctuation in flow rate. In contrast, a strong fluctuation in flow rate is
Fig. 19. Hopper half angle for mass flow plotted against angle of wall friction.
Fig. 21. (a) Diagram of variation of sample mass with time. (b) Relative standard deviation of flow rate.
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noticeable for pulverized coal e (43.2 μm), reflecting a typical unstable flow. Also, pulverized coal c shows moderate fluctuations in flow rate. Both the visual observation and the flow rate analysis show that the capability of pulverized coal to flow decreased with decreasing particle size. This is in agreement with the theoretical findings reported in Section 3.3 that pulverized coal with particle size smaller than 100 μm will perform unstable flow or even no flow. 4. Conclusion In this study, pulverized coal from industry was selected as the experimental material and sieved into seven samples with different particle sizes to study the effect of particle size on flowability. Our research result confirmed the effect of particle size unequivocally. As particle size decreases the bulk density of pulverized coal decreases. Moreover, the decrease of particle size causes an increase of cohesiveness and wall adhesion. Furthermore, using a complete set of physically based equations, the stress state in the hopper was analyzed. Then, the discharge behavior of pulverized coal was assessed by the Jenike criterion. Finally, theoretical findings were compared with experiments. The findings of this study were summarized in the following conclusions: (1) The relation between the particle size and bulk density of pulverized coal could be effectively described by the ρb(σz) function. It is shown that the packing efficiency of pulverized coal decreases with the decrease of particle size. (2) For pulverized coals, the relationship between major Principal stress for steady-state flow σ1 and unconfined yield stress could conveniently be analyzed by liner consolidation function. (3) Decreasing the particle size of pulverized coal leads to the increase of cohesiveness and wall adhesion effect, which causes flow problem over the powder itself and inner surface of the hopper. The mass flow will occur when the particle size is larger than 100 μm. And, a funnel flow will be observed when the particle size is in the range of 100 μm–30 μm. No discharge under gravity can be observed in the case of a particle size smaller than 30 μm.
Notation C D De Do dv ff ffc H M Mi N n p R
cohesion (Pa) particle size (m) mean particle size (m) diameter of outlet (m) volume diameter (m) flow factor flow index material level (m) flow rate (kg/h) instantaneous flow rate (kg/h) compressibility index particle distribution width mean of the major and minor Principal stresses (Pa) radius of Mohr circle (Pa)
Greek letters α hopper half angle (°) β angle defined by Eq. (18) major Principal stress for steady-state flow (Pa) σ1 unconfined yield strength (Pa) σc stationary uniaxial compressive strength (Pa) σc,st pre-consolidation stress (Pa) σpre radial stress (Pa) σr axial stress (Pa) σZ pull-off stress for zero unconfined yield strength (Pa) σZ,0 effective wall stress (Pa) σ1
ρb ρb,0 ρCBD φe φi φst φw (τrh)w ω
bulk density of powder (kg/m3) powder bulk density at negligible consolidation (kg/m3) conditioned bulk density (kg/m3) effective angle of internal friction (°) angle of internal friction (°) stationary angle of internal friction (°) angle of wall friction (°) shear stress on the side of the element (Pa) angle defined by Eq. (17) (°)
Acknowledgment This work was supported by the National Natural Science Foundation of China (21006027, 21206041), the Fundamental Research Funds for the Central Universities (WB1214012) and China Postdoctoral Science Foundation funded project (2012M520847). References [1] Z. Dai, X. Gong, X. Guo, et al., Pilot-trial and modeling of a new type of pressurized entrained-flow pulverized coal gasification technology, Fuel 87 (2008) 2304–2313. [2] X. Guo, Z. Dai, X. Gong, et al., Performance of an entrained-flow gasification technology of pulverized coal in pilot-scale plant, Fuel Process. Technol. 88 (2007) 451–459. [3] M. Mohan, B. Pitchumani, J. Tomas, Flow characterization of fine powders using material characteristic parameters, Adv. Powder Technol. 16 (2005) 123–135. [4] H. Lu, X. Guo, X. Gong, et al., Experimental study on aerated discharge of pulverized coal, Chem. Eng. Sci. 71 (2012) 438–448. [5] H. Lu, X. Guo, X. Gong, et al., Study of the flowability of pulverized coals, Energy Fuel 23 (2009) 5529–5535. [6] W. Huang, X. Gong, X. Guo, et al., Discharge characteristics of cohesive fine coal from aerated hopper, Powder Technol. 194 (2009) 126–131. [7] H. Lu, X. Guo, X. Gong, et al., Study on the fluidization and discharge characteristics of cohesive coals from an aerated hopper, Powder Technol. 207 (2011) 199–207. [8] J.J. Fitzpatrick, Particle properties and the design of solid food particle processing operations, Food Bioprod. Process. 85 (2007) 308–314. [9] J.J. Fitzpatrick, S.A. Barringer, T. Iqbal, Flow property measurement of food powders and sensitivity of Jenike's hopper design methodology to the measured values, J. Food Eng. 61 (2004) 399–405. [10] H. Purutyan, B.H. Pittenger, J.W. Carson, Solve solids handling problems by retrofitting, Chem. Eng. Prog. 94 (2013). [11] D. Schulze, Powders and Bulk Solids: Behavior, Characterization, Storage and Flow, Springer, Berlin, 2007. [12] J. Tomas, S. Kleinschmidt, Improvement of flowability of fine cohesive powders by flow additives, Chem. Eng. Technol. 32 (2009) 1470–1483. [13] R.P. Zou, A.B. Yu, Evaluation of the packing characteristics of mono-sized nonspherical particles, Powder Technol. 88 (1996) 71–79. [14] A.B. Yu, J. Bridgwater, A. Burbidge, On the modelling of the packing of fine particles, Powder Technol. 92 (1997) 185–194. [15] J.R. Johanson, The Johanson indicizer™ system vs. the Jenike shear tester, Bulk Solids Handl. 12 (1992) 237. [16] L.P. Maltby, G.G. Enstad, Uniaxial tester for quality control and flow property characterization of powders, Bulk Solids Handl. 13 (1993) 135. [17] D. Schulze, Development and application of a novel ring shear tester, Aufbereitungstechnik 35 (1994) 524–535. [18] A.W. Jenike, Storage and flow of solids, bulletin no. 123, Bull. Univ. Utah 53 (1964). [19] Chen Pan, et al., Flow properties of three fuel powders, Particuology 10 (2012) 438–443. [20] W.K. Brown, K.H. Wohletz, Derivation of the Weibull distribution based on physical principals and its connection to the Rosin–Rammler and lognormal distributions, J. Appl. Phys. 78 (1995) 2758–2763. [21] A. Macıas-Garcıa, E.M. Cuerda-Correa, M.A. Dıaz-Dıez, Application of the Rosin–Rammler and Gates–Gaudin–Schuhmann models to the particle size distribution analysis of agglomerated cork, Mater. Charact. 52 (2004) 159–164. [22] J. Tomas, Assessment of mechanical properties of cohesive particulate solids. Part 2: powder flow criteria, Part. Sci. Technol. 19 (2001) 111–129. [23] J. Tomas, Fundamentals of cohesive powder consolidation and flow, Granul. Matter 6 (2004) 75–86. [24] M. Medhe, B. Pitchumani, J. Tomas, Flow characterization of fine powders using material characteristic parameters, Adv. Powder Technol. 16 (2005) 123–135. [25] J. Tomas, Product design of cohesive powders — mechanical properties, compression and flow behavior, Chem. Eng. Technol. 27 (2004) 605–618. [26] A.W. Jenike, Gravity flow of solids, Trans. Inst. Chem. Eng. 40 (1962) 264–271. [27] O. Molerus, Theory of yield of cohesive powders, Powder Technol. 12 (1975) 259–275. [28] G. Bruni, P. Lettieri, The influence of fines size distribution on the behavior of gas fluidized beds at high temperature, Powder Technol. 163 (2006) 88–97. [29] R.M. Nedderman, Statics and Kinematics of Granular Materials, Cambridge University Press, UK, 2005.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
An investigation of the effect of particle size on discharge behavior of pulverized coal Yi Liu, Haifeng Lu, Xiaolei Guo, Xin Gong ⁎, Xiaolin Sun, Wei Zhao Key Laboratory of Coal Gasification and Energy Chemical Engineering of Ministry of Education, Shanghai Engineering Research Center of Coal Gasification, Institute of Clean Coal Technology, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 25 October 2014 Received in revised form 12 June 2015 Accepted 16 June 2015 Available online 23 June 2015 Keywords: Pulverized coal Particle size Flow property Stress state analysis Hopper discharge
a b s t r a c t In this paper, pulverized coal from industry was used as the experimental material. In order to study the effect of particle size, the pulverized coal was sieved into seven samples with different particle sizes. Firstly, the material properties as packing, incipient flow and wall friction properties were evaluated. Meanwhile, a transparent Perspex hopper was used to monitor the change of discharge behavior of pulverized coal caused by the increase of particle size. From the visual observation of discharge test, a progressive transition from blocking to unstable flow and to mass flow was found as the particle size increased. Hence, a complete set of physically based equations for compressibility, flowability and powder consolidation has been derived. Then, a new method, taking into account the stress state in the hopper, is introduced to assess the discharge behaviors of pulverized coal. The validity of this new method has been confirmed by comparing theoretical and experimental discharge behavior for pulverized coals. The theoretical approach based on stress state analysis, is able to provide results which correlate well with the experiments. © 2015 Published by Elsevier B.V.
1. Introduction Entrained-flow pressurized gasification process of pulverized coal, including the storage, discharge and conveying of pulverized coals, is one of the best contemporary coal gasification technologies [1,2]. However, there are several well-known problems in the handling and storage of pulverize coal, such as bridging, channeling, fluctuating flow rate or even blocking in processing equipment and storage, as the tendency of agglomeration caused by strong inter-particle forces [3–5]. The most common and serious problem is no-flow, due to arching or ratholing [6,7]. Thus, reliable information of flow properties of the pulverized coal concerned is required for reliable flow from the hopper, which is essentially crucial in handling and processing operations [8,9]. Flowing or yielding means that a powder is brought to irreversible deformation, which can be caused by an external mechanical stressing event. This stress can be produced by either force or energy. According to this method, only the trouble-free discharge of a powder out off a hopper by its dead weight is considered as good flowability. In contrast, powders that show flow problems are classified as poorly flowing or non-flowing. It is well known that there are two flow patterns in hopper or bins: mass flow and funnel flow, and most flow problems are associated with the funnel flow pattern [10,11]. To avoid funnel flow, it is important to
⁎ Corresponding author. Tel.: +86 21 6425 2521; fax: +86 21 6425 1312. E-mail address: [email protected] (X. Gong).
http://dx.doi.org/10.1016/j.powtec.2015.06.041 0032-5910/© 2015 Published by Elsevier B.V.
determine properties of pulverized coal. One of the major factors affecting flow patterns is particle size. In the case of powder with particle size less than 100 μm, van der Waals based interparticle forces, exceed the gravitational force by several orders of magnitude [12–14]. Thus, flow problems are very common in the case of cohesive powders with particle sizes less than 100 μm. To avoid this technical problem and hazard, it is really necessary to understand the effect of particle size on flow properties of pulverized coal. In order to improve a reliable and controllable discharge of powder from hoppers, effects have been taken to develop a proper method for discharge behavior of pulverized coal prediction. Powder flow characteristics are often investigated adopting a variety of methods. One of the most common of those is the shear test [15–17]. Using Mohr's circle analysis, various important parameters may be extracted from these results. These properties include the major Principal stress, the minor Principal stress, the unconfined yield stress, the cohesive, the angle of internal friction and the effective angle of internal friction. In previous studies, Jenike applied two-dimensional stress analysis to develop a numerical methodology to determine the minimum hopper slope required for mass flow and the minimum outlet dimension for unobstructed gravity flow [18]. Although this method has been widely used over the past few decades, it occasionally gives irrational results, due to the assumption of constant average material properties [19]. In fact, some of the properties, such as effective angle of internal friction and bulk density, are functions of stress state. To overcome this matter, a complete set of physically based equations for compressibility, flowability and powder consolidation
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are used in this study. And, a new approach is introduced to assess the discharge behavior for the pulverized coal. As opposed to the original Jenike criterion which only considers constant average material properties, the new method considers the effect of stress state on material properties by using these physically based equations. The validity of the new theoretical approach will be examined by comparing the theoretical findings and experimental discharge behavior. 2. Materials and methods 2.1. Materials The initial material for this study is industrial bituminous pulverized coal having a particle density of 1490 kg/m3. The initial material is sieved by an electric vibrating screen. Seven samples of pulverized coal were obtained. The physical properties of these samples are reported in Table 1 and Fig. 1. The particle size distribution of the pulverized coal in the experiments can be described using the Rosin–Rammler distribution function [20,21]. The general expression of the Rosin–Rammler function is: n D RðDÞ ¼ 100−ϕ ¼ 100 exp − De
ð1Þ
where R(D) is the distribution function, D is the particle size, De is the mean particle size and n is a measure of the spread of particle size the large value of which designates a narrower size range. This expression can be rewritten as: ð2Þ
lgf lg½100=RðDÞg ¼ n lgD þ M1
where M1 is a constant. A plot of lgD versus lg(lg(100 ∕ R(D))) will result in a straight line of slope n if the behavior of the material fits the RR model. From the observation of Fig. 2 and the corresponding linear correlation coefficient, one deduces that the RR model provides a perfect fit to the experimental PSD curve. As shown in Table 1, the samples have approximate n within the range of 0.9–1.2, indicating the same level of size range. Therefore, the particle size represents the major factor for this study. 2.2. Experimental 2.2.1. Compressibility test Compressibility is a measurement of how density changes as a function of applied normal stresses. The sample vessel specification is 50 mm × 85 ml. The measurement utilizes a vented piston to compress the sample under the increasing normal stress. Prior to each measurement, the sample was first prepared by conditioning and splitting using the standard FT4 blade and split vessel assembly to remove any residual compaction and build a uniform and loose condition in the bed of powders. After the preconditioning and slitting procedure, a sample of 85 ml is left behind and the condition bulk density ρCBD is obtained. Before the compression occurs, the blade is then exchanged for the vented piston. Fig. 3 shows the schematic of the compressibility
Fig. 1. Cumulative particle size distribution of the samples obtained with laser diffraction (Malvern 2000).
test; each normal stress will be applied for a defined time to allow the powder to reach equilibrium. The distance traveled by the piston is measured for each applied normal stress and the bulk density is automatically calculated.
2.2.2. Shear cell and wall friction test A rotational shear cell is used to quantify the incipient flow behaviors of the pulverized coals. This shear cell consists of a vessel containing the samples and a shear head to introduce both vertical and rotational stresses, as shown in Fig. 4(a). After completion of one conditional cycle, a vented piston is used to induce a precise consolidation stress in the sample. The pre-consolidation procedure is carried out prior to the splitting procedure to ensure that the sample is consolidated. Then, the vessel was split. To ensure that the surface of the sample is suitably consolidated, the sample is recompressed by the shear cell head to remove any disturbances caused by the split. After the previous operations are done, the shearing sequence could begin. Based on the normal and shear stress collected, the yield locus at a given preconsolidation condition can be acquired. Fig. 5(a) is a typical σ–τ diagram, showing two Mohr circles, yield locus, and effective yield locus. With these terms, the angle of internal friction φi, effective angle of internal friction φe, cohesive C, major Principal stress σ1, minor Principal stress σ2 and unconfined yield stress
Table 1 Physical property and packing properties of all pulverized coals used in this study. Materials
dv (μm)
n
ρCBD (kg/m3)
ρb,0 (kg/m3)
σz,0 (Pa)
N
a b c d e f g
223.8 141.3 94.2 74.9 55.9 43.2 17.7
0.91 1.02 1.11 1.12 1.21 1.27 1.21
739.0 723.0 690.6 642.3 621.9 584.3 520.5
742.4 722.1 691.2 629.9 605.6 576.2 439.7
3863.0 1475.2 1208.7 347.3 289.0 458.1 43.4
0.022 0.023 0.025 0.028 0.031 0.040 0.058
Fig. 2. Relationship between lg{lg[100/R(D)]} and lgD of pulverized coal (dv = 43.2 μm).
Y. Liu et al. / Powder Technology 284 (2015) 47–56
49
Fig. 4. Image of the shear cell supplied with the FT4 powder Rheometer: (a) measuring powder flow properties and (b) measuring angle of wall friction.
3. Result and discussion 3.1. Packing properties Fig. 3. Schematic of compressibility test.
σc can be calculated. It should be noted here that the stressing prehistory of a cohesive powder flow is stationary (steady state) and results significantly in a cohesive stationary yield locus in the τ (σ) of Fig. 5(b). Where, σ0 is the isostatic tensile strength of unconsolidated powder, equals a characteristic cohesion force in an unconsolidated powder; φst is the stationary angle of internal friction, the larger is the difference between φst and φi, the more cohesive is the powder. The wall friction test is similar to the shear test. In the wall friction test, the shear head was replaced by a Perspex plate, which has the same material quality with the hopper, as shown in Fig. 4(b). Typically, a series of shear stress values for a range of reducing normal stress were measured to generate a wall yield locus, from which the angle of wall friction is determined. In this study, the pre-consolidation stress in the process of the wall friction test is 3 kPa.
2.2.3. Hopper discharge A transparent Perspex hopper was fabricated to observe the flow phenomenon of pulverized coal during the discharge process dynamically and directly, and, to be able to observe mass flow or funnel flow discharge by looking at the movement of the particle close to the hopper wall, as shown in Fig. 6. In each case, a sample weighed 0.7 kg was loaded in the hopper. Discharge was performed immediately after loading to avoid consolidation caused by long residence time. The change of powder mass was determined and recorded by a hanging weighing unit during the discharging process.
Compressibility is not directly a measurement of flowability, but nevertheless related to many process environments, for example, storage in hoppers or bins. It is useful to understand the relationship between powder bulk density ρb and applied consolidation stress σz. Tomas [22–25] has proposed such a relationship for powders by extending analogies to the adiabatic gas law for isentropic compression, as given by Eq. (3). σ z þ σ z;0 N ρb ¼ ρb;0 σ z;0
ð3Þ
where ρb,0 is the powder bulk density at negligible consolidation and σZ,0 is the pull-off stress when the unconfined yield strength is zero. The parameter N is the compressibility index, in the range of 0–1. N = 0 indicates incompressibility stiff bulk material, while N = 1 represents deal gas. The ρb(σz) functions which described the relationship between the applied normal stress and bulk density is one of the fundamental equations for stress state analysis. Fig. 7 reports the measured bulk density as a function of the applied normal stress. By adopting Eq. (3), the corresponding regression curve is demonstrated. This shows the capability of Eq. (3) to describe the packing properties of pulverized coal. Fig. 8 shows that the compressibility index N increases as the particle size decreases. Also, the smaller the particle size, the smaller are the conditioned bulk density ρCBD and the estimated loose bulk density ρb,0. This means that the pulverized coal becomes more compressible, indicating a lower pack efficiency. Furthermore, once the particle size is smaller than a certain value (100 μm), the conditioned bulk density ρCBD is significantly larger than the loose bulk density ρb,0, due to the non-ignorable gravity consolidation, as shown in Table 1.
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Fig. 5. (a) Mohr circle, yield locus and effective yield locus. (b) Stationary yield locus.
3.2. Flow properties With the normal and shear stress data, the yield locus at a given pre-consolidation level was determined by linear regression using the Mohr–Coulomb equation. τ ¼ σ tan φi þ C
ð4Þ
Then the cohesion strength C and the angle of the internal friction φi are calculated. The effective angle of internal friction is a measure for the inner friction at steady-state flow. It is calculated by measuring the
angle that a line drawn through the origin and tangential to the larger Mohr circle creates with the horizontal axis. Fig. 5(a) shows two Mohr stress circles. The smaller Mohr circle is drawn where the circumference passes through the origin and is tangential to the yield locus and represents the stress state under unconfined yield. And, the larger Mohr circle is drawn where the circumference passes through the Pre-shear data point and is tangential to the yield locus and represents the stress state during consolidation. Based on these two Mohr circles, the unconfined yield strength, minor Principal stress and major Principal stress were determined. As an instance, Fig. 9 reports graphically the yield loci and Mohr circles for pulverized coal g under different pre-consolidation conditions.
Fig. 6. Schematic of the hopper discharge system (D1 = 150 mm, D2 = 32 mm, α = 15°).
Y. Liu et al. / Powder Technology 284 (2015) 47–56
51
Fig. 7. Comparison between experimental and predicted bulk density of samples in this study.
Fig. 9. Four yield loci under different consolidation stress for pulverized coal g (dv = 43.2 μm).
Pre-consolidation at different levels resulted in different yield loci. Each yield locus gave a pair of values of the unconfined yield strength σc and the major Principal stress σ1. Then, the flow function FF of the pulverized coal was determined, as shown in Fig. 10. Flow index ffc is the inverse of the flow function at any point, ffc = σ1/σc [26]. Based on the flow index, the powder materials are classified as hardened, very cohesive, cohesive, easy flowing, or free flowing [18], as shown in Fig. 10. According to Tomas [22–25], the consolidation function σc = f (σ1) can be conveniently analyzed by linear regression, Eq. (5):
slope and the intercept of the linear consolidation function (the angle of internal friction φi fluctuates by only ca. ±2° around the average, thus, an average of the values determined experimentally for all consolidation levels is used).
2 ð sin φst − sin φi Þ 2 sin φst ð1 þ sin φi Þ σ1 þ σ0 ð1 þ sin φst Þ ð1− sin φi Þ ð1 þ sin φst Þ ð1− sin φi Þ ¼ a1 σ 1 þ σ c;0 ð5Þ
Table 2 contains data on the measured flow properties and angle of wall frictions from the wall friction test. The flow index values decrease counter-clockwise from the region of free flowing to hardened powders in Fig. 10. Also, the degree of cohesiveness of the powders can be indicated by the difference between φst and φi [27]. The larger the difference between these friction angles is, the more cohesive is the powder response. As particle size decreases the flow index decreases while the difference between φst and φi increases, indicating that the cohesiveness of pulverized coal is increased. Meanwhile, wall friction measurement was also conducted for the samples. The angle of wall friction reflected the ability of pulverized coals to flow over the inner surface of the hopper. The angle of wall friction increased with the decrease of particle size. This means that the interaction between the wall and pulverized coal is gradually increased with the decrease of particle size.
σc ¼
where φst is the stationary angle of internal friction, φi is the angle of internal friction and σ0 is the isostatic tensile strength. As shown in Fig. 10, a linear relationship between the Principal stress σ1 and unconfined yield strength σc apparently exists for each of the samples defined by a certain particle size. The stationary angle of internal friction φst and the isostatic tensile strength σ0 are both basic physical material parameters for describing the flow properties of powders. The stationary angle of internal friction φst and the isostatic tensile strength σ0 can be easily obtained from the
Fig. 8. Compressibility index as a function of particle size.
φst ¼ arcsin
σ0 ¼
a1 þ ð2−a1 Þ sin φi 2−a1 ð1− sin φi Þ
ð1 þ sin φst Þ ð1− sin φi Þ σ c;0 2 ð1 þ sin φi Þ sin φst
Fig. 10. Consolidation function σc = f (σ1) of the samples used in the study.
ð6Þ
ð7Þ
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Table 2 Pulverized coal flow properties and angle of wall friction. Material
dv (μm)
σpre (kPa)
σ1 (kPa)
σ2 (kPa)
C (kPa)
σc (kPa)
φi (°)
φe (°)
φst (°)
σ0 (kPa)
φst–φi (°)
φw (°)
a
223.8
1.6
25.78
34.66
0.025
2.35
25.15
33.97
0.044
3.33
28.11
e
55.9
34.24
0.078
3.6
29.77
f
43.24
36.85
0.018
5.22
30.44
g
17.68
38.6 40.4 41.7 42.7 35.0 33.9 35.0 35.4 33.3 35.6 35.2 39.9 33.5 33.9 35.0 38.7 33.9 36.6 35.1 40.1 37.5 37.5 36.0 39.9 39.8 40.4 42.3 49.9
0.08
74.9
36.6 38.6 41.1 40.5 33.2 31.5 32.5 29.5 30.6 33.6 32.8 33 29.8 30.6 31.5 29.5 30 32.4 30.5 32.4 32.2 32.3 30.4 33.3 30.4 32.8 31.9 30.3
34.03
d
1.59 1.08 0.50 0.10 1.14 0.94 0.52 0.23 1.60 0.82 0.49 0.25 2.11 1.25 0.69 0.30 2.17 1.69 0.90 0.27 3.23 2.12 1.09 0.22 5.07 2.83 1.89 0.58
23.83
94.2
0.40 0.26 0.12 0.02 0.31 0.26 0.14 0.07 0.46 0.22 0.13 0.07 0.61 0.36 0.19 0.09 0.63 0.47 0.26 0.07 0.89 0.59 0.31 0.06 1.45 0.77 0.52 0.17
1.71
c
4.4 2.98 1.32 0.20 4.22 2.88 1.44 0.29 4.33 2.71 1.38 0.21 4.29 2.84 1.36 0.22 4.23 2.64 1.38 0.21 3.89 2.60 1.35 0.19 3.37 2.14 1.01 0.13
0.011
141.3
19.02 13.96 6.54 1.04 15.58 10.12 5.30 1.09 14.89 10.23 5.15 0.98 14.84 9.99 5.04 0.94 14.88 10.42 5.14 0.95 16.00 10.68 5.22 0.87 15.35 10.02 5.17 0.98
40.94
b
9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5 9 6 3 0.5
40.02
0.087
8.31
29.84
The function φe(σ1) describes the relationship between the major Principal stress σ1 and effective angle of internal friction φe. It is very important in stress state analysis. From the stationary yield locus, the following relationship is obtained: σ þ σ σ −σ 1 2 1 2 ¼ sin φst þ σ0 : 2 2
ð8Þ
In the case of cohesionless steady-state flow, which is described by the effective yield locus, the following relationship is obtained: σ −σ σ þ σ 1 2 1 2 ¼ sin φe : 2 2
ð9Þ
Combined with Eqs. (8) and (9), we obtained a simple correlation between the stress-dependent effective angle of internal friction φe, the stationary angle of internal friction φst and the center stress (σ1 + σ2) ∕ 2: sin φe ¼ sin φst 1 þ
σ0 : ðσ 1 þ σ 2 Þ=2
If the major Principal stress σ1 reaches the stationary uniaxial compressive strength σc,st the effective angle of internal friction is φe = 90°, and for σ1 → ∞ follows φe → φst, as shown in Fig. 11.
3.3. Stress state analysis and discharge behavior prediction It is clear that different particle sizes result in the different settled bed structures, and consequently affect the capability of powder to flow [28]. Thus, the effect of particle size on discharge can only be assessed when the stress state in the hopper is known. The model reported in this section describes the tensional state of the powder in the hopper using a direct method of slices inspired by Janssen's analysis. It is modified here to take into account the basic flow parameters and the fundamental equations estimated above. Then, the discharge behavior of pulverized coal was predicted by applying the Jenike criterion.
ð10Þ
The term (σ1 + σ2) ∕ 2 can be replaced by the major Principal stress σ1 flow: ðσ 1 þ σ 2 Þ σ 1 −σ 0 sin φst : ¼ 1 þ sin φst 2
ð11Þ
Then, we obtained: sin φe ¼ sin φst
σ1 þ σ0 : σ 1 − sin φst σ 0
ð12Þ
As the physical limit that φe ≤ 90°, we obtained:
σ 1 ≥σ c;st ¼
2 sin φst σ 0 : 1− sin φst
ð13Þ Fig. 11. Effective angle of internal friction as a function of major Principal stress.
Y. Liu et al. / Powder Technology 284 (2015) 47–56
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Fig. 13. Mohr's circle for the stress adjacent to the wall in a conical hopper [29].
Fig. 12. Force balance on an elemental slice of powder at depth z.
Using the usual Janssen assumptions that the stresses are constant across any cross-section, a force balance is performed on an elemental slice at depth z, as shown in Fig. 12. The force balance is: dσ z þ dz
2ðτrh Þw ¼ ρb ðσ r Þ g: d Hþ −z 2 tan α
ð14Þ
Fig. 14 shows that the calculated axial stress changes with depth (pulverized coal f). Additionally, Fig. 15 reports the relationship between the particle size and axial stress at the outlet of the hopper. Jenike [26] concludes, with the flow-no flow criterion, that a stable cohesive arch will form in a convergent channel when the unconfined yield strength σc exceeds the effective wall stress, σ′. When σ′ N σc, a forming arch will collapse, that is, the critical point is the condition σ′ = σc. This criterion is based on a force balance on the arch assuming that the only forces acting are the weight of the arch and the abutment force, see Fig. 16. Denoting the angle between the horizontal axis and tangent to the arch at the wall by β, we see that the width of the arch at the wall is δz cos β and hence the vertical force due to the effective 2
The term (τrh)w is the shear stress acting on the side of the element. It is not equal to the shear stress on the wall. Neddman considered the Mohr–Coulomb diagram for the material adjacent to the right-hand wall, as shown in Fig. 13. For the passive case, the wall plane is denoted by the point W′ on the figure. The circumferential surface of the element is an r-plane which is inclined to the wall at angle α. Therefore, it is represented on Fig. 13 by the point R. The H represents the Z-plane which lies at the opposite end of the diameter to R. From Fig. 13, the ratios σr/σz and (τrh)w/σz can be evaluated. σ r 1 þ sin φe cosðω þ φw þ 2α Þ ¼ σz 1− sin φe cosðω þ φw þ 2α Þ
ð15Þ
ðτrh Þw sin φe cosðω þ φw þ 2α Þ ¼ σz 1− sin φe cosðω þ φw þ 2α Þ
ð16Þ
where sin ω ¼
sin φw : sin φe
ð17Þ
Furthermore, the radial stresses are assumed as consolidation stress. The ρb(σr) and φe(σr) functions have already been indicated in Sections 3.1 and 3.2, as given by Eqs. (3) and (12). The boundary condition for Eq. (14) is that the top surface of the material is open to the atmosphere so that the normal stress on it may be taken to be zero. Besides, the material level H adopted in the calculation was visually obtained, as shown in Table 3. The stress state in the hopper was evaluated by solving simultaneously Eq. (14) with the boundary condition σz(z = 0) = 0. Since the effective angle of internal friction and bulk density depend on the consolidation stress, these parameters would be functions of depth z. Consequently, a numerical solution was needed. As an example,
wall stress is, πDoδz sin 2βσ′/2. The weight of the arch is πðD20 Þ ρgδzDo . The resulting equation for the effective wall stress is: σ0 ¼
Do ρb g : 2 sin 2β
ð18Þ
Since the value of β is unknown, we can only acquire the range of the value of effective wall stress by assuming the value of β. The effective wall stress reaches a minimum value, when the angle β is 45°. And, the effective wall stress reaches a maximum value, when it is perpendicular to the wall (β = 15°). Since the radial stress at the outlet was calculated, the consolidation level at the outlet was acquired. Bulk density and the unconfined yield stress were obtained by using Eqs. (3) and (5), respectively. For comparison purposes, Fig. 17 reports the calculated effective wall stress and unconfined yield stress. With the particle size larger than 100 μm, the pulverized coal may perform free discharge under gravity. When the particle size is in the range of 40 μm to 100 μm, the unconfined yield stress is higher than the minimum value of effect wall stress but lower than the maximum value. It means that unstable arches may form and erratic flow may exhibit at this particle size range. For the pulverized coal with particle size smaller than 30 μm, Table 3 The material level in the hopper for the different samples. Materials
dv (μm)
H (m)
a b c d e f g
223.8 141.3 94.2 74.9 55.9 43.2 17.7
0.174 0.176 0.179 0.186 0.188 0.193 0.216
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Fig. 16. Jenike's model for the formation of an arch. Fig. 14. Radial stress as a function of depth (coal f dv = 43.2 μm).
the unconfined yield stress is even higher than the maximum value of effect wall stress. The discharge for pulverized coal g may be prevented by the formation of a stable arch. Jenike [26] also pointed out that the maximum hopper half angle (α) for mass flow could be calculated from the angle of wall friction (φw) and the effective angle of internal friction (φe) with the following equations:
α¼
π 1 ð1− sin φe Þ 1 sin φw : − − cos−1 φw þ sin−1 sin φe 2 2 2 sin φe 2
ð19Þ
As the effective angle of internal friction is the function of major Principal stress, therefore, the slope needed for mass flow is a function of depth. For instance, Fig. 18 shows the relationship between maximum hopper half angles for mass flow and depth. As clearly visible from Fig. 18, the material at the outlet generally requires the steepest slope for mass flow. That means that the most critical condition for mass flow is found close to the outlet. According to this analysis, the hopper angle versus vertical for mass flow was calculated by Eq. (19) with the effective angle of internal friction at the outlet. The calculated results were reported in Fig. 19. For the discharge instrument used in this study, the hopper angle versus vertical is 15°. From the result of the calculation of maximum hopper half angles, it is found that pulverized coals a, b and c would
Fig. 15. The radial stress in outlet of hopper as a function of particle size.
exhibit mass flow, while pulverized coals d, e, f, and g may perform funnel flow. The above findings described that decreasing the particle size of pulverized coal would lead to the increase in cohesiveness and wall adhesion effect, causing flow problem over itself and the inner surface of the hopper. The stress state analysis indicates that the pulverized coal with particle size larger than 100 μm will perform mass flow and those with particle size smaller than 100 μm would exhibit funnel flow (unstable flow). For the pulverized coal with particle size less than 30 μm, discharge may prevented by the formation of a stable arch. 3.4. Discharge result Fig. 20 exhibits the results of pulverized coal discharge from the Perspex hopper. The flow rate is defined as M/Ma, where Ma (543.2 kg/h) is the average flow of pulverized coal with the largest particle size and M is the average flow rate of each sample. The relative flow rate profiles show a steep exponential decrease with the decrease of particle size. Based on the visual observation of discharge behaviors, three regions including arching, unstable-flow and mass flow were found. I. Mass-flow region (pulverized coal a, b). In this region, pulverized coal discharges smoothly just as a liquid. All the powder in the hopper is in motion, which shows a typical mass flow pattern. II. Unstable-flow region (pulverized coal c–f). In the unstable flow region, a dynamic arch forms and collapses alternately across
Fig. 17. Unconfined yield strength and effective wall stress at outlet as a function of particle size.
Y. Liu et al. / Powder Technology 284 (2015) 47–56
Fig. 20. Relative rate of discharge as a function of particle size.
Fig. 18. The slope for mass flow as a function of depth (coal f dv = 43.2 μm).
III.
55
the hopper outlet. Meanwhile, the powder flows towards the outlet of the hopper in a channel formed within the powder itself. Arching region (pulverized coal g). In the arching region, a static arch forms across the hopper outlet which completely stops the powder flow. It is further found that there is an amount of pulverized coal adhering to the hopper wall even if the arch was broken artificially.
This flow region classification is also supported by the flow rate analysis. Fig. 21 reports the weighting curve and time series of relative standard deviations of flow rate for pulverized coals a, c and e. The instantaneous flow rate, which is the derivative for the weighting curve, was calculated automatically by the software. We used relative standard deviation (RSD) to quantify the complexity of changes in the time series of flow rate. RSD is defined as: M −mean RSDðiÞ ¼ i mean
ð20Þ
where Mi is the instantaneous flow rate and mean is the average value of flow rate. RSD as a dimensionless number is used to compare the complexity of changes of the flow rates for different pulverized coals. Pulverized coal with the mean particle size 224 μm shows slight fluctuation in flow rate. In contrast, a strong fluctuation in flow rate is
Fig. 19. Hopper half angle for mass flow plotted against angle of wall friction.
Fig. 21. (a) Diagram of variation of sample mass with time. (b) Relative standard deviation of flow rate.
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Y. Liu et al. / Powder Technology 284 (2015) 47–56
noticeable for pulverized coal e (43.2 μm), reflecting a typical unstable flow. Also, pulverized coal c shows moderate fluctuations in flow rate. Both the visual observation and the flow rate analysis show that the capability of pulverized coal to flow decreased with decreasing particle size. This is in agreement with the theoretical findings reported in Section 3.3 that pulverized coal with particle size smaller than 100 μm will perform unstable flow or even no flow. 4. Conclusion In this study, pulverized coal from industry was selected as the experimental material and sieved into seven samples with different particle sizes to study the effect of particle size on flowability. Our research result confirmed the effect of particle size unequivocally. As particle size decreases the bulk density of pulverized coal decreases. Moreover, the decrease of particle size causes an increase of cohesiveness and wall adhesion. Furthermore, using a complete set of physically based equations, the stress state in the hopper was analyzed. Then, the discharge behavior of pulverized coal was assessed by the Jenike criterion. Finally, theoretical findings were compared with experiments. The findings of this study were summarized in the following conclusions: (1) The relation between the particle size and bulk density of pulverized coal could be effectively described by the ρb(σz) function. It is shown that the packing efficiency of pulverized coal decreases with the decrease of particle size. (2) For pulverized coals, the relationship between major Principal stress for steady-state flow σ1 and unconfined yield stress could conveniently be analyzed by liner consolidation function. (3) Decreasing the particle size of pulverized coal leads to the increase of cohesiveness and wall adhesion effect, which causes flow problem over the powder itself and inner surface of the hopper. The mass flow will occur when the particle size is larger than 100 μm. And, a funnel flow will be observed when the particle size is in the range of 100 μm–30 μm. No discharge under gravity can be observed in the case of a particle size smaller than 30 μm.
Notation C D De Do dv ff ffc H M Mi N n p R
cohesion (Pa) particle size (m) mean particle size (m) diameter of outlet (m) volume diameter (m) flow factor flow index material level (m) flow rate (kg/h) instantaneous flow rate (kg/h) compressibility index particle distribution width mean of the major and minor Principal stresses (Pa) radius of Mohr circle (Pa)
Greek letters α hopper half angle (°) β angle defined by Eq. (18) major Principal stress for steady-state flow (Pa) σ1 unconfined yield strength (Pa) σc stationary uniaxial compressive strength (Pa) σc,st pre-consolidation stress (Pa) σpre radial stress (Pa) σr axial stress (Pa) σZ pull-off stress for zero unconfined yield strength (Pa) σZ,0 effective wall stress (Pa) σ1
ρb ρb,0 ρCBD φe φi φst φw (τrh)w ω
bulk density of powder (kg/m3) powder bulk density at negligible consolidation (kg/m3) conditioned bulk density (kg/m3) effective angle of internal friction (°) angle of internal friction (°) stationary angle of internal friction (°) angle of wall friction (°) shear stress on the side of the element (Pa) angle defined by Eq. (17) (°)
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