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Rheological behavior and wall slip of concentrated coal water slurry in pipe flows
Chemical Engineering and Processing 48 (2009) 1241–1248
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Rheological behavior and wall slip of concentrated coal water slurry in pipe flows Liangyong Chen ∗ , Yufeng Duan, Changsui Zhao, Liguo Yang School of Energy and Environment, Southeast University, No. 2 Si Pai Lou, Nanjing 210096, China
a r t i c l e
i n f o
Article history: Received 19 September 2008 Received in revised form 9 April 2009 Accepted 12 May 2009 Available online 19 May 2009 Keywords: Coal water slurry Rheology Wall slip Pipe flow
a b s t r a c t Experimental investigations were carried out on a pilot scale slurry transportation apparatus to investigate wall-slip phenomenon and rheological properties of concentrated coal water slurries (CWSs) in pipe flows. Straight pipes with different inner diameters (25 mm, 32 mm and 50 mm) were selected to characterize the flow behaviors of CWSs with various solid contents (65.3 wt%, 67.1 wt% and 68.2 wt%). A procedure based on a new slip model was developed to determine the true rheological behavior (independent from pipe diameter) and the slip velocity at the pipe wall. The results suggested that 65.3 wt% CWSs exhibited a Newtonian fluid behavior and the slurry flow was free from wall-slip effects. But at 67.1 wt% and 68.2 wt% solid contents, the slurry flows were strongly affected by wall slip and with the increase of wall shear stress, the slurries exhibited their true rheological behaviors firstly as a shear-thinning fluid and then as a shear thickening fluid. The existence of a minimum value of slippage contribution was indicative of the transition of flow behavior from shear thinning to shear thickening. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The transportation of concentrated coal water slurries by pipelines plays an important role in numerous industrial applications, ranging from coal combustion to gasification processes. The rheological properties are very important in characterization of flow behavior of coal water slurries. The existing experimental studies on rheological behavior of CWSs emphasize the effects of particle size, volume fraction, coal type, surfactants and temperature [1–7]. Coal water slurries tend to be filled with coal particles at concentrations approaching their maximum packing fraction and exhibit a complex non-Newtonian behavior which would vary exquisitely with slight increase in solid concentration. In the meantime, wall slip usually occurs in the flow of highly concentrated slurries. Sometimes it becomes the controlling factor [8]. The presence of wall slip makes any flow of slurries easier due to the lubrication effects. Thus accounting accurately for energy losses in CWS flow with the presence of wall slip is only possible if both of the rheological data of bulk slurries and the slip boundary conditions are available. In spite of industrial importance of pipeline flow of CWSs, only few researchers [9–11] utilized large-diameter pipe to investigate both the rheological behavior and the wall-slip behavior over a wide range of wall shear stresses. They found that, if wall-slip
∗ Corresponding author. Tel.: +86 25 83795652; fax: +86 25 83795652. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Duan), [email protected] (C. Zhao), ylg [email protected] (L. Yang). 0255-2701/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2009.05.002
effects were not taken into account, the viscosity of the materials or the pressure gradients in pipes would be seriously underestimated. However, their results could not provide a definitive demonstration of slip flow behavior of concentrated coal water slurries in pipe flows. The purpose of the present work is to investigate the true rheological behaviors of highly filled CWSs, the slip boundary conditions and the effects of wall slip on pipe flow. All the results were based on pressure drop versus flow rate experiments carried out on a pilot scale slurry pipeline facility. The common explanation for wall slip is that, in the flow of concentrated slurries, a liquid-rich thin layer is created near the pipe wall as a result of static wall depletion effects or shear-induced particle migration from the solid boundary [12,13]. In the liquid-rich thin layer (also named slip layer), the local solid concentration and the shear viscosity of slurries are much lower than those in the bulk slurries. Large velocity gradients are produced in the liquidrich thin layer, resulting in apparent slippage of the bulk slurries. Thus, the creation of slip layer makes any flow of slurries easier due to the lubrication effects [12]. It is almost impossible to infer the wall-slip behavior from the mechanism of the formation of a slip layer. Currently, there is no a priori method of estimating and predicting the wall-slip behavior of concentrated slurries [14]. Although various methods are available to carry out direct observation or measurement on wall slippage [14–16], they generally require highly sophisticated equipments. An alternative method is to develop a simple phenomenological slip model to infer wall-slip velocity from the pressure drop versus flow rate data. The classical Mooney method [17], in which it is assumed that the slip
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velocity at the wall is only dependent on wall shear stress, has been widely used to determine the wall-slip velocity and the true rheological behavior of various materials such as polymer solutions, particle suspensions and emulsions. However, this method is not always successful. In this study, we attempted to analyze these data which was clearly indicative of the presence of wall slip using the Mooney method. Negative intercepts were observed on the apparent shear rate axis of the Mooney plots. Negative intercepts on the apparent shear rate axis were also reported in many literatures [18,19] in which the Mooney plots were applied. In this case, it remains possible to deduce a slip velocity. But the slip velocity would be higher than the mean velocity of the fluid, what is physically unreasonable in a pressure driven flow. Jastrzebski [20] proposed an improved Mooney method to evaluate slip velocities of Kaolinite platelets and water mixture flowing in tubes. In the improved Mooney method, it proposed that wall-slip velocity was proportional to wall shear stress and inversely proportional to the pipe radius at a given stress level. Meng et al. [9], Lu and Zhang [11] and Khan et al. [21] applied the improved slip model to investigate the true rheological behavior of coal water mixture and ceramic paste in pressure driven flow. All of them obtained excellent fits to their experimental data. However, the improved Mooney method could not give satisfactory results when it was applied to our experimental data. Recently, in the investigations on flow behavior of polymeric materials, Crawford et al. [22] put forward a new slip model in which it is assumed that the wall-slip velocity was proportional to wall shear stress and inversely proportional to Rd , where R is the pipe radius and d is an index parameter to be determined by experiments. In our study, a procedure was developed based on this new slip model to determine the true rheological behavior of highly filled CWSs and the slip velocity at the wall. The excellent linear fit of each apparent shear rate to 1/R1+d and the overlap of slip-corrected rheological diagrams obtained from different diameter pipes demonstrated the validity of the new slip model. The resulting true rheological behaviors and slip boundary conditions permitted a reliable extrapolation technique for pressure drop predictions versus flow rate for larger diameter pipes that account for the wall slippage phenomenon. These results also provide slip boundary conditions for computational fluid dynamics simulations. The slip effects were investigated and the results displayed some important slurry flow characteristics which are very important to pipeline design and slurry transportation process.
Fig. 1. Schematic diagram of experimental setup.
conditions. The flow rate and the pressure drop were measured by an electro-magnetic flow meter (measuring range: 0–3 m3 /h) and an electric differential-pressure manometer (measuring range: 0–200 kPa), respectively. The basic error of two types of measuring device is ±1% of measuring range. Both signals were converted from analog to digital one by an A/D converter and then recorded in a computer database for later retrieval and analysis. A heat exchanger incorporated in the end of the test loop was used to control the slurry temperature. 2.2. Materials Coal water slurries used in this study were supplied by BaYi Coal Water Slurry Manufacturing Ltd., in Shandong Province of China. The pulverized coal contained in the tested slurries had a specific gravity of 1.338. Its size distribution exhibited a mean particle diameter of 33.5 m and 90% of coal particles were smaller than 100 m. Flow rate versus pressure drop measurements were carried out at mass fractions of 65.3%, 67.1% and 68.2%, respectively. To prevent conglomeration of coal particles, additives (anionic surface active agents, provided by Nanjing University) were added to the test slurries. The ratio of additive to coal in weight was maintained at 6‰ for all test slurries. The concentration deviations were controlled smaller than 0.2%. During testing, the slurry temperature was maintained at 15 ± 1.5 ◦ C. The signals of the slurry temperature were also recorded on-line during testing. Each test was repeated at least two times for more accuracy.
2. Experimental 3. Theoretical 2.1. Experimental set-up The efforts to determine the wall slippage and the rheological behavior of concentrated CWSs were done on a pilot scale experimental setup with computerized data acquisition system. The schematic diagram of the experimental set-up is shown in Fig. 1. It consists of a slurry tank, a test loop, measuring devices and a data acquisition system. The tested slurries were stored in the slurry tank. Stirring was accomplished using three impellers connected to the stirring shaft. During testing, the slurries were under stirring to prevent settling of solid particles and the slurry tank was well covered to prevent water loss. The driving force for pumping test slurries was provided by a screw pump with a rated flow rate of 3 m3 /h, driven by a YCT motor. The flow rate in the test loop was regulated by controlling the speed of the YCT motor. Steel pipes with different inner diameters of 25 mm, 32 mm and 50 mm were selected to characterize the flow behavior and the wall-slip effects. All of them were interchangeable in the test loop. For each diameter, the test section was 3300 mm long and sufficient entrance before the test section was used to achieve fully developed flow
The mechanism of slip flow of concentrated CWSs in fully developed laminar flow regime in a straight pipe is depicted in an exaggerated manner in Fig. 2. The slip layer near the inner wall
Fig. 2. Schematic of slip flow of coal water slurries in pipes.
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allows the bulk slurries to slide over the channel surface. Here, the bulk slurry is assumed to be a non-Newtonian fluid with yield stress. It consists of deforming zone and plug flow region in Fig. 2. The thickness of slip layer is estimated to be significantly small than the pipe diameter [14]. Furthermore, the slip layer has substantially lower viscosity than the bulk slurries. When slurries were sheared, large velocity gradients are produced in the slip layer. These give rise to a step change in the slope of velocity distribution. Thus, the wall-slip velocity is defined as a relative velocity between the bulk slurries at the wall and the wall itself. The velocity at any points in the bulk slurries can be expressed as the sum of the wall-slip velocity us , and the shear velocity of bulk slurries, ub . According to Crawford et al. [22], the wall-slip velocity is supposed to be proportional to wall shear stress and inversely proportional to Rd . It is expressed by us =
w
(1)
Rd
where R is the radius of pipe; d is a index parameter to be determined by experiments; is slip coefficient which is only a function of the wall shear stress for the same slurries. The total volumetric flow rate Q can be given by
R
Q = Qs + Qb = 2
(us + ub )r dr =
R2 w
0
Rd
+ 2
R
ub r dr
(2)
0
where Qb and Qs are the volumetric flow rate of the bulk slurries and that due to slip respectively; r is the radius from pipe centre line. Here, 2
Qs = R us =
R2 w
R
Qb = 2
ub rdr = 0
(3)
Rd
R3
w
2 f ()d
3 w
(4)
0
In Eq. (4), based on the facts of strong particle–particle interaction and the dispersing effects of the additive, it is assumed that the bulk slurry is homogeneous and the shear stress is related to the shear rate by a constitutive equation = f(). One may express Eq. (2) as follows: 4Q 1 4 = 4w + 3 R3 R1+d w
w
2 f ()d
(5)
0
In Eq. (5), (4Q)/R3 is apparent shear rate; the second term on the right hand side is independent of the pipe radius at constant wall shear stress. Taking the derivative of apparent shear rate with respect to 1/R1+d at constant wall shear stress generates the following expression:
∂(4Q/R3 ) ∂(1/R1+d )
= 4w
(6)
w
According to Eq. (6), plots of 4Q/R3 versus 1/R1+d at constant wall shear stress will have a gradient of 4w . Then the wall-slip velocity can be obtained from: 4us =
4w Rd
(7)
As is mentioned by Crawford et al. [22], it is necessary to run an algorithm to find the parameter d; d is varied so as to minimize the overall error between the data points and the best-fit lines on 4Q/R3 versus 1/R1+d plots. However, it is difficult to find the best parameter d. In the present investigation, an average deviation of less than 5% between the data points and the fit lines was imposed for each plot and d was adjusted so that all the resulting rheological diagrams (plots of w versus tw ) obtained from different pipes overlapped. Here, a try-and-error method was used to determine
Fig. 3. w versus w obtained from different pipes at 65.3 wt% solid content.
index parameter d and it is considered that the rheological diagrams obtained from different pipes coincide with each other, provided that the fractional deviation between the resulting wall shear stress and the fitting curve of these points is less than 5%. Although this was somewhat subjective, the results would be given in sufficient accuracy for engineering purpose. The true wall shear rate of the bulk slurries at constant wall shear stress would be given by Eq. (8) [23]: tw =
Q − Q
d ln[(Q − Qs )/R3 ] 3+ d ln w
s
R3
(8)
where tw is the true wall shear rate of the bulk slurries. 4. Results and discussion 4.1. Manifestation of wall slip The Rabinowitsch–Mooney method is generally used to determine the rheological behavior of any time independent fluid in pipe flow based on the assumption that there is no slippage between the fluid and the wall. The shear stress and the wall shear rate are given by Eqs. (9) and (10): w = w =
DP 4L
(9)
32Q 1 + 3n D3
4n
=
8V 1 + 3n D
4n
(10)
Here w is wall shear rate; P is the pressure drop across a pipe of length L; V is mean velocity and n = [d ln(DP/4L)]/[d ln(8V/D)]. If there is no slippage at the wall, the apparent rheological diagrams (plots of w versus w ) obtained from different diameter pipes would coincide with each other, or the rheological models obtained from different diameter pipes are identical for the same slurries. Figs. 3–5 show w versus w from different pipes at 65.3 wt%, 67.1 wt% and 68.2 wt% solid contents respectively. For 65.3 wt% CWSs, the apparent rheological diagrams are independent of pipe diameter in the range of the flow rate observed. In this case, no-slip boundary condition is valid and the rheological diagrams obtained from different pipes are the true rheological data. However, as for 67.1 wt% and 68.3 wt% CWSs, the apparent rheological diagram exhibit a higher wall shear rate at constant wall shear stress when the pipe diameter decreases, as shown in Figs. 4 and 5. The sensitive dependences of apparent rheological diagrams on pipe diameter are indicative of the presence of significant wall slip [11,14,24].
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Fig. 6. Wall shear stress versus true wall shear rate.
Fig. 4. w versus w obtained from different pipes and the true rheological data at 67.1 wt% solid content.
4.2. True rheological behavior To determine the wall-slip velocity and the true rheological behavior of CWSs in pipe flow with the presence of wall slip, plots of 4Q/R3 against 1/R1+d at constant wall shear stresses were drawn with a selected value of d. When d = 2.26 for 67.1 wt% CWSs and d = 1.15 for 68.2 wt% CWSs, each plotting data on 4Q/R3 versus 1/R1+d plots was found so fit in a line with the correlation coefficients more than 0.98. As is implied by Eq. (6), the slope of the linear regression line was equal to 4w . Then the wall-slip velocity and the true wall shear rate at each constant wall shear stress were calculated by using Eqs. (7) and (8), respectively. The true (or slip-corrected) rheological diagrams of 67.1 wt% and 68.2 wt% CWSs are shown together with that of 65.3 wt% CWSs in Fig. 6. Obviously, for the same slurries, the slip-corrected rheological diagrams from different pipes show a high degree of overlap in the range of wall shear stress observed. This validates the proposed slip model and the selected value of d. The dependence of the wall shear stress on the true wall shear rate would be well represented by the three parameters Herschel–Bulkley model:
and n = 1, the slurries behavior as a Newtonian fluid. In our analysis, the values of K and n were determined by fitting the results of the slip-corrected rheological diagram with the least square method. The expressions for the true rheological behavior are listed together with the correlation coefficients in Table 1. 65.3 wt% CWSs behaved as a Newtonian fluid, while 67.1 wt% and 68.2 wt% CWSs exhibited their flow behaviors as remarkable non-Newtonian fluids which were manifested by large yield stresses and serious deflections of flow behavior index from 1. Fig. 7 gives the true shear viscosity as a function of true wall shear rate. Here, the true shear viscosity was calculated by =
w tw
(12)
In Eq. (11), y is the yield stress of bulk slurries; K and n are consistency coefficient and flow behavior index, respectively. When y = 0
As expected, the true shear viscosity increased with increasing solid concentration. Both 67.1 wt% CWSs and 68.2 wt% CWSs showed complex flow behaviors, to be precise, shear thinning at low shear rate and then shear thickening at high shear rate. The sharp decrease in shear viscosity at low shear rate was mainly due to destroy of colloidal structure and the rapid increase of alignment of the particles in the flow direction with the increase of shear. While at high shear stress, higher deformations occur in the bulk slurries, implying particle rearrangements and increase in average distances between layers of particles. Thus the forces opposing to the flow caused an increase in shear viscosity [25].
Fig. 5. w versus w obtained from different pipes and the true rheological data at 68.2 wt% solid content.
Fig. 7. True shear viscosity versus true wall shear rate.
n w = y + Ktw
(11)
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Table 1 Index parameter for determination of wall slip and rheological model. Solid loading (wt%)
d
True rheological model
Rheological model without slip correction 2
Model
R
D (mm)
Model
R2 >0.988
65.3
∞
w = 0.625tw
0.998
25, 32, 50
w = 0.625w
67.1
2.26
w = 33.6 + 0.092tw 1.637
0.997
25 32 50
w = 0.791w w = 19.59 + 0.345w 1.205 w = 20.97 + 0.445w 1.225
0.998 0.999 0.999
68.2
1.15
w = 42.3 + 0.243tw 1.488
0.996
25 32 50
w = 1.151w w = 20.84 + 0.365w 1.245 w = 26.42 + 0.542w 1.278
0.997 0.999 0.994
It would be noted that in the experimental range, both the true rheological behavior of the bulk slurries and the boundary conditions (described below) varied exquisitely with the slight increase of solid concentration. At a relatively low solid concentration, the slurries were Newtonian and the flow was free from wall-slip effects. However, when the solid concentration was up to a critical value which is very close to the maximum packing fraction (here, the critical value is between 65.3 wt% and 67.1 wt %), the addition of more solid particles would cause a drastic increase of interparticle force in bulk slurries, which led to an abrupt transition from a Newtonian to a remarkable non-Newtonian behavior. At the same time, the drastic increase of interparticle force in bulk slurries led to an abrupt change of interaction between the bulk slurries and the inner wall which indicated by a transition from a no-slip to a slip boundary condition. A similar observation has been reported by other workers in the investigations on wall-slip phenomenon of concentrated suspensions [8,26]. Soltani and Yilmazer [8] believed that the slip layer thickness would be very small or zero at low concentrations, but it suddenly increased when concentration was up to a certain value. For comparison, the slip-corrected rheological diagrams of 67.1 wt% and 68.2 wt% CWSs are shown together with the apparent rheological diagrams in Figs. 4 and 5, respectively. Obviously, there are great discrepancies between the slip-corrected and the apparent rheological diagrams obtained from different diameter pipes. The apparent rheological data would seriously underestimate the viscosity of the bulk slurries. At a given wall shear stress, the discrepancy between the true wall shear rate and the wall shear rate determined by Eq. (10) was observed to increase with decreasing pipe diameter. The rheological models obtained from the apparent rheological diagrams are also shown in Table 1.
as follows. At very high concentration, the structural arrangement of the coal particles in bulk slurries would be affected by the pipe size. As the structural arrangement changed, the amount of water available to make up the slip layer was changed accordingly. Obviously, the change in structural arrangement was closely related to the interparticle distance. So the dependence of wall-slip velocity on pipe size would be influenced by solid concentration. The slip coefficients for 67.1 wt% and 68.2 wt% CWSs are plotted against the wall shear stress in Figs. 8 and 9, respectively. Obviously, there are great discrepancies between the two slip coefficients in magnitude and changing trend. This observation partly suggested that the slip velocity could be seriously influenced by solid concentration. In the new slip model, the slip coefficient is used as a
4.3. Index parameter, slip coefficient and wall-slip velocity
Fig. 8. Slip coefficient versus wall shear stress at 67.1 wt% solid content.
The slip model given by Eq. (1) is purely empirical and the model parameters would be determined only by experiments. In Ref. [21], Crawford did not give any theoretical explanation for pipe size dependence of wall-slip velocity. But, for his experimental data as well as ours, this empirical model can give much more linear plots of apparent shear rate against 1/R1+d than the classical and the improved Mooney methods [17,20], and eventually give a high degree of overlap of these slip-corrected rheological diagrams. Simultaneously, the independence of the slip-corrected rheological diagrams on pipe size validated the slip model. Another advantage of the slip model is that it allows for a more general dependence of wall-slip velocity on pipe radius. One can see, when d = 0 and d = 1, the analysis methods for wall slip become equivalent to the classical Mooney method [17] and the improved Mooney method proposed by Jastrzebski [20], respectively. The index parameters d for our test slurries are shown in Table 1. When d was assumed to be infinity for 65.3 wt% CWSs, the slope of each 4Q/R3 versus 1/R1+d plot was zero and no-slip boundary condition was well satisfied. Obviously, the value of d varied with solid concentration. A possible explanation is
Fig. 9. Slip coefficient versus wall shear stress at 68.2 wt% solid content.
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Table 2 Expressions for wall-slip velocity as functions of wall shear stress. D (mm)
67.1 wt%
68.2 wt% 2
Equation 25 32 50
R −2
−4
us = 6.81 × 10 + 3.9 × 10 w 1.318 us = 5.33 × 10−2 + 1.9 × 10−4 w 1.343 us = 2.30 × 10−2 + 0.7 × 10−4 w 1.340
parameter to determine wall-slip velocity and it is assumed to be only a function of the wall shear stress for the same slurries. Actually, the product of the slip coefficient and the wall shear stress is used to describe the interaction between the bulk slurries and the inner wall. As is mentioned above, when the solid concentration was beyond the critical value, a slight increase in solid concentration caused a rapid increase of interparticle force, and then the rapid increase of interparticle force would led to a drastic change of interaction between the bulk slurries and the inner wall. Thus the slip coefficient is also closely related to solid concentration. When solid concentration increased, it would show great discrepancies in magnitude and changing trend. In Figs. 10 and 11, the wall-slip velocities in each pipe are shown as a function of wall shear stress for 67.1 wt% and 68.2 wt% CWSs, respectively. The wall-slip velocity consistently increased with increasing wall shear stress and at a constant wall shear stress,
R2
Equation −2
−3
us = 1.14 × 10 + 1.02 × 10 w 1.089 us = 1.06 × 10−2 + 7.1 × 10−4 w 1.103 us = 8.02 × 10−3 + 4.2 × 10−4 w 1.108
0.998 0.999 0.999
0.999 0.999 0.999
it increased with decreasing pipe diameter. However, as concentration increased from 67.1 wt% to 68.2 wt%, the wall-slip velocities became less sensitive to changes in pipe size. Various forms of wallslip velocity and wall shear stress relationship had been suggested, but in our studies, the dependence of wall-slip velocity on wall shear stress can be described very well by the following empirical equation: ˇ
us = us0 + ˛w
(13)
Here us0 , ˛ and ˇ are fitting parameters. The resulting expressions together with the correlation coefficients are summarized in Table 2. It seems that, for the same slurries, the value of ˇ kept constant when pipe diameter increased. All the values of us0 are positive, indicating that the bulk slurries flow like a plug with the lubrication effects of the slip layer at very low wall shear stress. 4.4. Slippage contribution Because using wall-slip velocity as a single indicator will not provide the overall explanation for the effects of wall slip, the contribution of wall slippage to the total flow rate is usually considered as an important analysis method. It can be represented by the following expression: ε=
Qs us w = = Q V VRd
(14)
Fig. 10. Wall-slip velocity versus wall shear stress at 67.1 wt% solid content.
The roles played by wall slip are shown in Figs. 12 and 13, in which the contributions of wall slippage are plotted as functions of wall shear stress for 67.1 wt% and 68.2 wt% CWSs, respectively. In the experimental range, three flow patterns were observed for each pipe. At very low wall shear stress (lower than the yield stress of bulk slurries), the slurry flow was completely dominated by the wall slip and there was no shear deformation in the bulk slurries [12]. In this case, the bulk slurries lubricated with slip layer at the wall, flowed like a plug and the experimental value of slippage contribution was around unity. When the wall shear stress was large
Fig. 11. Wall-slip velocity versus wall shear stress at 68.2 wt% solid content.
Fig. 12. Slippage contribution versus wall shear stress for various pipes at 67.1 wt% coal content.
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(2) In the experimental range, both the true rheological behavior of the bulk slurries and the boundary conditions varied exquisitely with slight increase of solid concentration. 65.3 wt% CWSs exhibited a Newtonian fluid behavior and was free from wall-slip effects. But at 67.1 wt% and 68.2 wt% solid content, coal water slurries exhibited their true rheological behaviors firstly as a shear-thinning fluid and then as a shear thickening fluid. At the same time, the slurry flow was strongly affected by slip at the wall. (3) For 67.1 wt% and 68.2 wt% CWSs, as wall shear stress increased, a complex changing trend of slippage contribution was observed due to the complex rheological behaviors of the bulk slurries. For each pipe, there was a minimum value of slippage contribution, indicating the transition of the flow behavior from shear thinning to shear thickening. Acknowledgement Fig. 13. Slippage contribution versus wall shear stress for various pipes at 68.2 wt% coal content.
This study was funded by the State Basic Research Development Program (973 Plan) of China (No. 2004CB217701).
enough to induce a significant shear deformation in the bulk slurries, the slippage contribution decreased dramatically. At this stage, the bulk slurries behaved as a shear-thinning fluid. When the wall shear stress increased further, shear thickening in the bulk slurries started and the slippage contribution increased gradually. Due to the coincidence of the yield stress of the bulk slurries with the wall shear stress at which the value of slippage contribution became less than 1, the transition from a plug flow to a deformation flow can be used to fix the yield stress of the bulk slurries in pipe flows with the presence of wall slip [14]. The yield stress can be estimated directly from the plots of slippage contribution against the wall shear stress, as shown in Figs. 12 and 13. The critical shear stresses at 67.1 wt% and 68.2 wt% coal contents were around 30 Pa and 40 Pa, respectively, which were very close to the fitting value of yield stresses of the bulk slurries shown in Table 1. From Figs. 12 and 13, it was observed that, for each pipe there was a minimum value of slippage contribution at the shear stress level marked by dash lines. Generally, the behavior of decrease in slippage contribution with increase of wall shear stress is associated with the shear-thinning nature of the bulk slurries, while for shearthickening nature, slippage contribution increases with increase of wall shear stress [12,14,27]. Thus, the existence of the minimum value of slippage contribution in Figs. 12 and 13 was indicative of transition of flow behavior from shear thinning to shear thickening. From Figs. 12 and 13, it can be seen that the slippage contribution became increasingly more important for smaller pipe. However, it must be noted that for the larger pipe (D = 50 mm), there was a 20–30% contribution of slippage to the total flow at high wall shear stress.
Appendix A. Nomenclature
5. Conclusions Flow behaviors of concentrated coal water slurries were characterized by using a set of steel pipes. Based on the new slip model, the true rheological behaviors and wall-slip velocities were obtained for slurries with various solid contents. The main conclusions are as follows: (1) The proposed slip model allows a more general dependence of wall-slip velocity on pipe radius. The excellent linear fit of each plot of apparent shear rate to 1/R1+d and the overlap of the slipcorrected rheological diagrams obtained from different pipes validated the new slip model and the selected value of index parameter.
d n r D K L P Q R u us0 V
index parameter flow behavior index radius from pipe centre (m) pipe diameter (m) consistency coefficient (Pa sn ) pipe length (m) pressure drop (Pa) total volumetric flow rate (m3 /s) pipe radius (m) velocity (m/s) parameter, Eq. (13) mean velocity (m/s)
Greek letters ˛ parameter, Eq. (13) ˇ parameter, Eq. (13) slip coefficient shear stress (Pa) y yield stress (Pa) shear rate (s−1 ) true shear viscosity (Pa s) ε slippage contribution Subscripts a apparent b bulk slurries s slip w wall tw true value on the wall References [1] S.C. Tsai, E.W. Knell, Viscometry and rheology of coal water slurry, Fuel 65 (1986) 566–571. [2] N.S. Roh, D.H. Shin, D.C. Kim, J.D. Kim, Rheological behavior of coal-water mixtures. 1. Effects of coal type, loading and particle size, Fuel 74 (1995) 1220–1225. [3] N.S. Roh, D.H. Shin, D.C. Kim, J.D. Kim, Rheological behavior of coal-water mixtures. 2. Effects of surfactants and temperature, Fuel 74 (1995) 1313–1318. [4] R. Yavuz, S. Kucukbayrak, Effect of particle size distribution on rheology of lignite–water slurry, Energy Sources 20 (1998) 787–794. [5] R.M. Turian, J.F. Attal, D.J. Sung, L.E. Wedgewood, Properties and rheology of coal–water mixtures using different coals, Fuel 81 (2002) 2019–2033. [6] S.K. Majumder, K. Chandna, D.S. De, G. Kundu, Studies on flow characteristics of coal–oil–water slurry system, Int. J. Miner. Process. 79 (2006) 217–224.
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[7] G. Ahmet, A. Metin, D. Cetin, K. Semra, B. Ramis, An investigation on effects of various parameters on viscosities of coal–water mixture prepared with Erzurum–Askale lignite coal, Fuel Process. Technol. 87 (2006) 821–827. [8] F. Soltani, U. Yilmazer, Slip velocity and slip layer thickness in flow of concentrated suspensions, J. Appl. Polym. Sci. 70 (1998) 515–522. [9] L.J. Meng, L. Kong, M.Y. Zhang, Similitude criterion and resistance characteristics of coal–water slurry flowing in pipe, J. Chem. Ind. Eng. (China) 46 (1995) 298–303. [10] L.J. Meng, M.Y. Zhang, (CWM) of high concentration and it effects on flow characteristics in pipe, J. Eng. Thermal Energy Power 11 (1996) 85–88. [11] P. Lu, M.Y. Zhang, Rheology of coal–water paste, Powder Technol. 150 (2005) 189–195. [12] H.A. Barnes, A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure, J. Non-Newtonian Fluid Mech. 56 (1995) 221–251. [13] M. Allende, D.M. Kalyon, Assessment of particle-migration effects in pressuredriven viscometric flows, J. Rheol. 44 (2000) 79–90. [14] D.M. Kalyon, Apparent slip and viscoplasticity of concentrated suspensions, J. Rheol. 49 (2005) 621–640. [15] B. Ouriev, Investigation of the wall effect in highly concentrated disperse system by means of non-invasive UVP-PD method in the pressure driven shear flow, Colloid J. 64 (2002) 740–745. [16] M. Han, C. Kim, M. Kim, S. Lee, Particle migration in tube flow of suspensions, J. Rheol. 43 (1999) 1157–1174. [17] M. Mooney, Explicit formulas for slip and fluidity, J. Rheol. 2 (1931) 210–222.
[18] Ph. Mourniac, J.F. Agassant, B. Vergnes, Determination of the wall slip velocity in the flow of a SBR compound, Rheol. Acta 31 (1992) 565–574. [19] P.J. Halliday, A.C. Smith, Estimation of the wall slip velocity in the capillary flow of Potato granule paste, J. Rheol. 39 (1995) 139–149. [20] Z.D. Jastrzebski, Entrance effects and wall effects in an extrusion rheometer during the flow of concentrated suspensions, Ind. Eng. Chem. Fundam. 6 (1967) 445–454. [21] A.U. Khan, B.J. Briscoe, P.F. Luckham, Evaluation of slip in capillary extrusion of ceramic pastes, J. Eur. Ceram. Soc. 21 (4) (2001) 483–491. [22] B. Crawford, J.K. Watterson, P.L. Spedding, S. Raghunathan, W. Herron, M. Proctor, Wall slippage with siloxane gum and silicon rubbers, J. Non-Newtonian Fluid Mech. 129 (2005) 38–45. [23] D.M. Kalyon, Y.L. Yeow, Letter to the editor: comments on “A new method of processing capillary viscometry data in the presence of wall slip” [J. Rheol. 47 (2003) 337–348] (multiple letters), J. Rheol. 47 (2003) 1087–1089. [24] Y. Cohen, A.B. Metzner, Apparent slip flow of polymer solutions, J. Rheol. 29 (1985) 67–102. [25] S.M. Olhero, J.M.F. Ferreira, Influence of particle size distribution on rheology and particle packing of silica-based suspensions, Powder Technol. 139 (2004) 69–75. [26] S.A. Gulmus, U. Yilmazer, Effect of volume fraction and particle size on wall slip in flow of polymeric suspensions, J. Appl. Polym. Sci. 98 (2005) 439– 448. [27] D.M. Kalyon, P. Yaras, B. Aral, U. Yilmazer, Rheological behavior of a concentrated suspension: a solid rocket fuel stimulant, J. Rheol. 37 (1993) 35–53.
Contents lists available at ScienceDirect
Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Rheological behavior and wall slip of concentrated coal water slurry in pipe flows Liangyong Chen ∗ , Yufeng Duan, Changsui Zhao, Liguo Yang School of Energy and Environment, Southeast University, No. 2 Si Pai Lou, Nanjing 210096, China
a r t i c l e
i n f o
Article history: Received 19 September 2008 Received in revised form 9 April 2009 Accepted 12 May 2009 Available online 19 May 2009 Keywords: Coal water slurry Rheology Wall slip Pipe flow
a b s t r a c t Experimental investigations were carried out on a pilot scale slurry transportation apparatus to investigate wall-slip phenomenon and rheological properties of concentrated coal water slurries (CWSs) in pipe flows. Straight pipes with different inner diameters (25 mm, 32 mm and 50 mm) were selected to characterize the flow behaviors of CWSs with various solid contents (65.3 wt%, 67.1 wt% and 68.2 wt%). A procedure based on a new slip model was developed to determine the true rheological behavior (independent from pipe diameter) and the slip velocity at the pipe wall. The results suggested that 65.3 wt% CWSs exhibited a Newtonian fluid behavior and the slurry flow was free from wall-slip effects. But at 67.1 wt% and 68.2 wt% solid contents, the slurry flows were strongly affected by wall slip and with the increase of wall shear stress, the slurries exhibited their true rheological behaviors firstly as a shear-thinning fluid and then as a shear thickening fluid. The existence of a minimum value of slippage contribution was indicative of the transition of flow behavior from shear thinning to shear thickening. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The transportation of concentrated coal water slurries by pipelines plays an important role in numerous industrial applications, ranging from coal combustion to gasification processes. The rheological properties are very important in characterization of flow behavior of coal water slurries. The existing experimental studies on rheological behavior of CWSs emphasize the effects of particle size, volume fraction, coal type, surfactants and temperature [1–7]. Coal water slurries tend to be filled with coal particles at concentrations approaching their maximum packing fraction and exhibit a complex non-Newtonian behavior which would vary exquisitely with slight increase in solid concentration. In the meantime, wall slip usually occurs in the flow of highly concentrated slurries. Sometimes it becomes the controlling factor [8]. The presence of wall slip makes any flow of slurries easier due to the lubrication effects. Thus accounting accurately for energy losses in CWS flow with the presence of wall slip is only possible if both of the rheological data of bulk slurries and the slip boundary conditions are available. In spite of industrial importance of pipeline flow of CWSs, only few researchers [9–11] utilized large-diameter pipe to investigate both the rheological behavior and the wall-slip behavior over a wide range of wall shear stresses. They found that, if wall-slip
∗ Corresponding author. Tel.: +86 25 83795652; fax: +86 25 83795652. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Duan), [email protected] (C. Zhao), ylg [email protected] (L. Yang). 0255-2701/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2009.05.002
effects were not taken into account, the viscosity of the materials or the pressure gradients in pipes would be seriously underestimated. However, their results could not provide a definitive demonstration of slip flow behavior of concentrated coal water slurries in pipe flows. The purpose of the present work is to investigate the true rheological behaviors of highly filled CWSs, the slip boundary conditions and the effects of wall slip on pipe flow. All the results were based on pressure drop versus flow rate experiments carried out on a pilot scale slurry pipeline facility. The common explanation for wall slip is that, in the flow of concentrated slurries, a liquid-rich thin layer is created near the pipe wall as a result of static wall depletion effects or shear-induced particle migration from the solid boundary [12,13]. In the liquid-rich thin layer (also named slip layer), the local solid concentration and the shear viscosity of slurries are much lower than those in the bulk slurries. Large velocity gradients are produced in the liquidrich thin layer, resulting in apparent slippage of the bulk slurries. Thus, the creation of slip layer makes any flow of slurries easier due to the lubrication effects [12]. It is almost impossible to infer the wall-slip behavior from the mechanism of the formation of a slip layer. Currently, there is no a priori method of estimating and predicting the wall-slip behavior of concentrated slurries [14]. Although various methods are available to carry out direct observation or measurement on wall slippage [14–16], they generally require highly sophisticated equipments. An alternative method is to develop a simple phenomenological slip model to infer wall-slip velocity from the pressure drop versus flow rate data. The classical Mooney method [17], in which it is assumed that the slip
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velocity at the wall is only dependent on wall shear stress, has been widely used to determine the wall-slip velocity and the true rheological behavior of various materials such as polymer solutions, particle suspensions and emulsions. However, this method is not always successful. In this study, we attempted to analyze these data which was clearly indicative of the presence of wall slip using the Mooney method. Negative intercepts were observed on the apparent shear rate axis of the Mooney plots. Negative intercepts on the apparent shear rate axis were also reported in many literatures [18,19] in which the Mooney plots were applied. In this case, it remains possible to deduce a slip velocity. But the slip velocity would be higher than the mean velocity of the fluid, what is physically unreasonable in a pressure driven flow. Jastrzebski [20] proposed an improved Mooney method to evaluate slip velocities of Kaolinite platelets and water mixture flowing in tubes. In the improved Mooney method, it proposed that wall-slip velocity was proportional to wall shear stress and inversely proportional to the pipe radius at a given stress level. Meng et al. [9], Lu and Zhang [11] and Khan et al. [21] applied the improved slip model to investigate the true rheological behavior of coal water mixture and ceramic paste in pressure driven flow. All of them obtained excellent fits to their experimental data. However, the improved Mooney method could not give satisfactory results when it was applied to our experimental data. Recently, in the investigations on flow behavior of polymeric materials, Crawford et al. [22] put forward a new slip model in which it is assumed that the wall-slip velocity was proportional to wall shear stress and inversely proportional to Rd , where R is the pipe radius and d is an index parameter to be determined by experiments. In our study, a procedure was developed based on this new slip model to determine the true rheological behavior of highly filled CWSs and the slip velocity at the wall. The excellent linear fit of each apparent shear rate to 1/R1+d and the overlap of slip-corrected rheological diagrams obtained from different diameter pipes demonstrated the validity of the new slip model. The resulting true rheological behaviors and slip boundary conditions permitted a reliable extrapolation technique for pressure drop predictions versus flow rate for larger diameter pipes that account for the wall slippage phenomenon. These results also provide slip boundary conditions for computational fluid dynamics simulations. The slip effects were investigated and the results displayed some important slurry flow characteristics which are very important to pipeline design and slurry transportation process.
Fig. 1. Schematic diagram of experimental setup.
conditions. The flow rate and the pressure drop were measured by an electro-magnetic flow meter (measuring range: 0–3 m3 /h) and an electric differential-pressure manometer (measuring range: 0–200 kPa), respectively. The basic error of two types of measuring device is ±1% of measuring range. Both signals were converted from analog to digital one by an A/D converter and then recorded in a computer database for later retrieval and analysis. A heat exchanger incorporated in the end of the test loop was used to control the slurry temperature. 2.2. Materials Coal water slurries used in this study were supplied by BaYi Coal Water Slurry Manufacturing Ltd., in Shandong Province of China. The pulverized coal contained in the tested slurries had a specific gravity of 1.338. Its size distribution exhibited a mean particle diameter of 33.5 m and 90% of coal particles were smaller than 100 m. Flow rate versus pressure drop measurements were carried out at mass fractions of 65.3%, 67.1% and 68.2%, respectively. To prevent conglomeration of coal particles, additives (anionic surface active agents, provided by Nanjing University) were added to the test slurries. The ratio of additive to coal in weight was maintained at 6‰ for all test slurries. The concentration deviations were controlled smaller than 0.2%. During testing, the slurry temperature was maintained at 15 ± 1.5 ◦ C. The signals of the slurry temperature were also recorded on-line during testing. Each test was repeated at least two times for more accuracy.
2. Experimental 3. Theoretical 2.1. Experimental set-up The efforts to determine the wall slippage and the rheological behavior of concentrated CWSs were done on a pilot scale experimental setup with computerized data acquisition system. The schematic diagram of the experimental set-up is shown in Fig. 1. It consists of a slurry tank, a test loop, measuring devices and a data acquisition system. The tested slurries were stored in the slurry tank. Stirring was accomplished using three impellers connected to the stirring shaft. During testing, the slurries were under stirring to prevent settling of solid particles and the slurry tank was well covered to prevent water loss. The driving force for pumping test slurries was provided by a screw pump with a rated flow rate of 3 m3 /h, driven by a YCT motor. The flow rate in the test loop was regulated by controlling the speed of the YCT motor. Steel pipes with different inner diameters of 25 mm, 32 mm and 50 mm were selected to characterize the flow behavior and the wall-slip effects. All of them were interchangeable in the test loop. For each diameter, the test section was 3300 mm long and sufficient entrance before the test section was used to achieve fully developed flow
The mechanism of slip flow of concentrated CWSs in fully developed laminar flow regime in a straight pipe is depicted in an exaggerated manner in Fig. 2. The slip layer near the inner wall
Fig. 2. Schematic of slip flow of coal water slurries in pipes.
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allows the bulk slurries to slide over the channel surface. Here, the bulk slurry is assumed to be a non-Newtonian fluid with yield stress. It consists of deforming zone and plug flow region in Fig. 2. The thickness of slip layer is estimated to be significantly small than the pipe diameter [14]. Furthermore, the slip layer has substantially lower viscosity than the bulk slurries. When slurries were sheared, large velocity gradients are produced in the slip layer. These give rise to a step change in the slope of velocity distribution. Thus, the wall-slip velocity is defined as a relative velocity between the bulk slurries at the wall and the wall itself. The velocity at any points in the bulk slurries can be expressed as the sum of the wall-slip velocity us , and the shear velocity of bulk slurries, ub . According to Crawford et al. [22], the wall-slip velocity is supposed to be proportional to wall shear stress and inversely proportional to Rd . It is expressed by us =
w
(1)
Rd
where R is the radius of pipe; d is a index parameter to be determined by experiments; is slip coefficient which is only a function of the wall shear stress for the same slurries. The total volumetric flow rate Q can be given by
R
Q = Qs + Qb = 2
(us + ub )r dr =
R2 w
0
Rd
+ 2
R
ub r dr
(2)
0
where Qb and Qs are the volumetric flow rate of the bulk slurries and that due to slip respectively; r is the radius from pipe centre line. Here, 2
Qs = R us =
R2 w
R
Qb = 2
ub rdr = 0
(3)
Rd
R3
w
2 f ()d
3 w
(4)
0
In Eq. (4), based on the facts of strong particle–particle interaction and the dispersing effects of the additive, it is assumed that the bulk slurry is homogeneous and the shear stress is related to the shear rate by a constitutive equation = f(). One may express Eq. (2) as follows: 4Q 1 4 = 4w + 3 R3 R1+d w
w
2 f ()d
(5)
0
In Eq. (5), (4Q)/R3 is apparent shear rate; the second term on the right hand side is independent of the pipe radius at constant wall shear stress. Taking the derivative of apparent shear rate with respect to 1/R1+d at constant wall shear stress generates the following expression:
∂(4Q/R3 ) ∂(1/R1+d )
= 4w
(6)
w
According to Eq. (6), plots of 4Q/R3 versus 1/R1+d at constant wall shear stress will have a gradient of 4w . Then the wall-slip velocity can be obtained from: 4us =
4w Rd
(7)
As is mentioned by Crawford et al. [22], it is necessary to run an algorithm to find the parameter d; d is varied so as to minimize the overall error between the data points and the best-fit lines on 4Q/R3 versus 1/R1+d plots. However, it is difficult to find the best parameter d. In the present investigation, an average deviation of less than 5% between the data points and the fit lines was imposed for each plot and d was adjusted so that all the resulting rheological diagrams (plots of w versus tw ) obtained from different pipes overlapped. Here, a try-and-error method was used to determine
Fig. 3. w versus w obtained from different pipes at 65.3 wt% solid content.
index parameter d and it is considered that the rheological diagrams obtained from different pipes coincide with each other, provided that the fractional deviation between the resulting wall shear stress and the fitting curve of these points is less than 5%. Although this was somewhat subjective, the results would be given in sufficient accuracy for engineering purpose. The true wall shear rate of the bulk slurries at constant wall shear stress would be given by Eq. (8) [23]: tw =
Q − Q
d ln[(Q − Qs )/R3 ] 3+ d ln w
s
R3
(8)
where tw is the true wall shear rate of the bulk slurries. 4. Results and discussion 4.1. Manifestation of wall slip The Rabinowitsch–Mooney method is generally used to determine the rheological behavior of any time independent fluid in pipe flow based on the assumption that there is no slippage between the fluid and the wall. The shear stress and the wall shear rate are given by Eqs. (9) and (10): w = w =
DP 4L
(9)
32Q 1 + 3n D3
4n
=
8V 1 + 3n D
4n
(10)
Here w is wall shear rate; P is the pressure drop across a pipe of length L; V is mean velocity and n = [d ln(DP/4L)]/[d ln(8V/D)]. If there is no slippage at the wall, the apparent rheological diagrams (plots of w versus w ) obtained from different diameter pipes would coincide with each other, or the rheological models obtained from different diameter pipes are identical for the same slurries. Figs. 3–5 show w versus w from different pipes at 65.3 wt%, 67.1 wt% and 68.2 wt% solid contents respectively. For 65.3 wt% CWSs, the apparent rheological diagrams are independent of pipe diameter in the range of the flow rate observed. In this case, no-slip boundary condition is valid and the rheological diagrams obtained from different pipes are the true rheological data. However, as for 67.1 wt% and 68.3 wt% CWSs, the apparent rheological diagram exhibit a higher wall shear rate at constant wall shear stress when the pipe diameter decreases, as shown in Figs. 4 and 5. The sensitive dependences of apparent rheological diagrams on pipe diameter are indicative of the presence of significant wall slip [11,14,24].
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Fig. 6. Wall shear stress versus true wall shear rate.
Fig. 4. w versus w obtained from different pipes and the true rheological data at 67.1 wt% solid content.
4.2. True rheological behavior To determine the wall-slip velocity and the true rheological behavior of CWSs in pipe flow with the presence of wall slip, plots of 4Q/R3 against 1/R1+d at constant wall shear stresses were drawn with a selected value of d. When d = 2.26 for 67.1 wt% CWSs and d = 1.15 for 68.2 wt% CWSs, each plotting data on 4Q/R3 versus 1/R1+d plots was found so fit in a line with the correlation coefficients more than 0.98. As is implied by Eq. (6), the slope of the linear regression line was equal to 4w . Then the wall-slip velocity and the true wall shear rate at each constant wall shear stress were calculated by using Eqs. (7) and (8), respectively. The true (or slip-corrected) rheological diagrams of 67.1 wt% and 68.2 wt% CWSs are shown together with that of 65.3 wt% CWSs in Fig. 6. Obviously, for the same slurries, the slip-corrected rheological diagrams from different pipes show a high degree of overlap in the range of wall shear stress observed. This validates the proposed slip model and the selected value of d. The dependence of the wall shear stress on the true wall shear rate would be well represented by the three parameters Herschel–Bulkley model:
and n = 1, the slurries behavior as a Newtonian fluid. In our analysis, the values of K and n were determined by fitting the results of the slip-corrected rheological diagram with the least square method. The expressions for the true rheological behavior are listed together with the correlation coefficients in Table 1. 65.3 wt% CWSs behaved as a Newtonian fluid, while 67.1 wt% and 68.2 wt% CWSs exhibited their flow behaviors as remarkable non-Newtonian fluids which were manifested by large yield stresses and serious deflections of flow behavior index from 1. Fig. 7 gives the true shear viscosity as a function of true wall shear rate. Here, the true shear viscosity was calculated by =
w tw
(12)
In Eq. (11), y is the yield stress of bulk slurries; K and n are consistency coefficient and flow behavior index, respectively. When y = 0
As expected, the true shear viscosity increased with increasing solid concentration. Both 67.1 wt% CWSs and 68.2 wt% CWSs showed complex flow behaviors, to be precise, shear thinning at low shear rate and then shear thickening at high shear rate. The sharp decrease in shear viscosity at low shear rate was mainly due to destroy of colloidal structure and the rapid increase of alignment of the particles in the flow direction with the increase of shear. While at high shear stress, higher deformations occur in the bulk slurries, implying particle rearrangements and increase in average distances between layers of particles. Thus the forces opposing to the flow caused an increase in shear viscosity [25].
Fig. 5. w versus w obtained from different pipes and the true rheological data at 68.2 wt% solid content.
Fig. 7. True shear viscosity versus true wall shear rate.
n w = y + Ktw
(11)
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Table 1 Index parameter for determination of wall slip and rheological model. Solid loading (wt%)
d
True rheological model
Rheological model without slip correction 2
Model
R
D (mm)
Model
R2 >0.988
65.3
∞
w = 0.625tw
0.998
25, 32, 50
w = 0.625w
67.1
2.26
w = 33.6 + 0.092tw 1.637
0.997
25 32 50
w = 0.791w w = 19.59 + 0.345w 1.205 w = 20.97 + 0.445w 1.225
0.998 0.999 0.999
68.2
1.15
w = 42.3 + 0.243tw 1.488
0.996
25 32 50
w = 1.151w w = 20.84 + 0.365w 1.245 w = 26.42 + 0.542w 1.278
0.997 0.999 0.994
It would be noted that in the experimental range, both the true rheological behavior of the bulk slurries and the boundary conditions (described below) varied exquisitely with the slight increase of solid concentration. At a relatively low solid concentration, the slurries were Newtonian and the flow was free from wall-slip effects. However, when the solid concentration was up to a critical value which is very close to the maximum packing fraction (here, the critical value is between 65.3 wt% and 67.1 wt %), the addition of more solid particles would cause a drastic increase of interparticle force in bulk slurries, which led to an abrupt transition from a Newtonian to a remarkable non-Newtonian behavior. At the same time, the drastic increase of interparticle force in bulk slurries led to an abrupt change of interaction between the bulk slurries and the inner wall which indicated by a transition from a no-slip to a slip boundary condition. A similar observation has been reported by other workers in the investigations on wall-slip phenomenon of concentrated suspensions [8,26]. Soltani and Yilmazer [8] believed that the slip layer thickness would be very small or zero at low concentrations, but it suddenly increased when concentration was up to a certain value. For comparison, the slip-corrected rheological diagrams of 67.1 wt% and 68.2 wt% CWSs are shown together with the apparent rheological diagrams in Figs. 4 and 5, respectively. Obviously, there are great discrepancies between the slip-corrected and the apparent rheological diagrams obtained from different diameter pipes. The apparent rheological data would seriously underestimate the viscosity of the bulk slurries. At a given wall shear stress, the discrepancy between the true wall shear rate and the wall shear rate determined by Eq. (10) was observed to increase with decreasing pipe diameter. The rheological models obtained from the apparent rheological diagrams are also shown in Table 1.
as follows. At very high concentration, the structural arrangement of the coal particles in bulk slurries would be affected by the pipe size. As the structural arrangement changed, the amount of water available to make up the slip layer was changed accordingly. Obviously, the change in structural arrangement was closely related to the interparticle distance. So the dependence of wall-slip velocity on pipe size would be influenced by solid concentration. The slip coefficients for 67.1 wt% and 68.2 wt% CWSs are plotted against the wall shear stress in Figs. 8 and 9, respectively. Obviously, there are great discrepancies between the two slip coefficients in magnitude and changing trend. This observation partly suggested that the slip velocity could be seriously influenced by solid concentration. In the new slip model, the slip coefficient is used as a
4.3. Index parameter, slip coefficient and wall-slip velocity
Fig. 8. Slip coefficient versus wall shear stress at 67.1 wt% solid content.
The slip model given by Eq. (1) is purely empirical and the model parameters would be determined only by experiments. In Ref. [21], Crawford did not give any theoretical explanation for pipe size dependence of wall-slip velocity. But, for his experimental data as well as ours, this empirical model can give much more linear plots of apparent shear rate against 1/R1+d than the classical and the improved Mooney methods [17,20], and eventually give a high degree of overlap of these slip-corrected rheological diagrams. Simultaneously, the independence of the slip-corrected rheological diagrams on pipe size validated the slip model. Another advantage of the slip model is that it allows for a more general dependence of wall-slip velocity on pipe radius. One can see, when d = 0 and d = 1, the analysis methods for wall slip become equivalent to the classical Mooney method [17] and the improved Mooney method proposed by Jastrzebski [20], respectively. The index parameters d for our test slurries are shown in Table 1. When d was assumed to be infinity for 65.3 wt% CWSs, the slope of each 4Q/R3 versus 1/R1+d plot was zero and no-slip boundary condition was well satisfied. Obviously, the value of d varied with solid concentration. A possible explanation is
Fig. 9. Slip coefficient versus wall shear stress at 68.2 wt% solid content.
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Table 2 Expressions for wall-slip velocity as functions of wall shear stress. D (mm)
67.1 wt%
68.2 wt% 2
Equation 25 32 50
R −2
−4
us = 6.81 × 10 + 3.9 × 10 w 1.318 us = 5.33 × 10−2 + 1.9 × 10−4 w 1.343 us = 2.30 × 10−2 + 0.7 × 10−4 w 1.340
parameter to determine wall-slip velocity and it is assumed to be only a function of the wall shear stress for the same slurries. Actually, the product of the slip coefficient and the wall shear stress is used to describe the interaction between the bulk slurries and the inner wall. As is mentioned above, when the solid concentration was beyond the critical value, a slight increase in solid concentration caused a rapid increase of interparticle force, and then the rapid increase of interparticle force would led to a drastic change of interaction between the bulk slurries and the inner wall. Thus the slip coefficient is also closely related to solid concentration. When solid concentration increased, it would show great discrepancies in magnitude and changing trend. In Figs. 10 and 11, the wall-slip velocities in each pipe are shown as a function of wall shear stress for 67.1 wt% and 68.2 wt% CWSs, respectively. The wall-slip velocity consistently increased with increasing wall shear stress and at a constant wall shear stress,
R2
Equation −2
−3
us = 1.14 × 10 + 1.02 × 10 w 1.089 us = 1.06 × 10−2 + 7.1 × 10−4 w 1.103 us = 8.02 × 10−3 + 4.2 × 10−4 w 1.108
0.998 0.999 0.999
0.999 0.999 0.999
it increased with decreasing pipe diameter. However, as concentration increased from 67.1 wt% to 68.2 wt%, the wall-slip velocities became less sensitive to changes in pipe size. Various forms of wallslip velocity and wall shear stress relationship had been suggested, but in our studies, the dependence of wall-slip velocity on wall shear stress can be described very well by the following empirical equation: ˇ
us = us0 + ˛w
(13)
Here us0 , ˛ and ˇ are fitting parameters. The resulting expressions together with the correlation coefficients are summarized in Table 2. It seems that, for the same slurries, the value of ˇ kept constant when pipe diameter increased. All the values of us0 are positive, indicating that the bulk slurries flow like a plug with the lubrication effects of the slip layer at very low wall shear stress. 4.4. Slippage contribution Because using wall-slip velocity as a single indicator will not provide the overall explanation for the effects of wall slip, the contribution of wall slippage to the total flow rate is usually considered as an important analysis method. It can be represented by the following expression: ε=
Qs us w = = Q V VRd
(14)
Fig. 10. Wall-slip velocity versus wall shear stress at 67.1 wt% solid content.
The roles played by wall slip are shown in Figs. 12 and 13, in which the contributions of wall slippage are plotted as functions of wall shear stress for 67.1 wt% and 68.2 wt% CWSs, respectively. In the experimental range, three flow patterns were observed for each pipe. At very low wall shear stress (lower than the yield stress of bulk slurries), the slurry flow was completely dominated by the wall slip and there was no shear deformation in the bulk slurries [12]. In this case, the bulk slurries lubricated with slip layer at the wall, flowed like a plug and the experimental value of slippage contribution was around unity. When the wall shear stress was large
Fig. 11. Wall-slip velocity versus wall shear stress at 68.2 wt% solid content.
Fig. 12. Slippage contribution versus wall shear stress for various pipes at 67.1 wt% coal content.
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(2) In the experimental range, both the true rheological behavior of the bulk slurries and the boundary conditions varied exquisitely with slight increase of solid concentration. 65.3 wt% CWSs exhibited a Newtonian fluid behavior and was free from wall-slip effects. But at 67.1 wt% and 68.2 wt% solid content, coal water slurries exhibited their true rheological behaviors firstly as a shear-thinning fluid and then as a shear thickening fluid. At the same time, the slurry flow was strongly affected by slip at the wall. (3) For 67.1 wt% and 68.2 wt% CWSs, as wall shear stress increased, a complex changing trend of slippage contribution was observed due to the complex rheological behaviors of the bulk slurries. For each pipe, there was a minimum value of slippage contribution, indicating the transition of the flow behavior from shear thinning to shear thickening. Acknowledgement Fig. 13. Slippage contribution versus wall shear stress for various pipes at 68.2 wt% coal content.
This study was funded by the State Basic Research Development Program (973 Plan) of China (No. 2004CB217701).
enough to induce a significant shear deformation in the bulk slurries, the slippage contribution decreased dramatically. At this stage, the bulk slurries behaved as a shear-thinning fluid. When the wall shear stress increased further, shear thickening in the bulk slurries started and the slippage contribution increased gradually. Due to the coincidence of the yield stress of the bulk slurries with the wall shear stress at which the value of slippage contribution became less than 1, the transition from a plug flow to a deformation flow can be used to fix the yield stress of the bulk slurries in pipe flows with the presence of wall slip [14]. The yield stress can be estimated directly from the plots of slippage contribution against the wall shear stress, as shown in Figs. 12 and 13. The critical shear stresses at 67.1 wt% and 68.2 wt% coal contents were around 30 Pa and 40 Pa, respectively, which were very close to the fitting value of yield stresses of the bulk slurries shown in Table 1. From Figs. 12 and 13, it was observed that, for each pipe there was a minimum value of slippage contribution at the shear stress level marked by dash lines. Generally, the behavior of decrease in slippage contribution with increase of wall shear stress is associated with the shear-thinning nature of the bulk slurries, while for shearthickening nature, slippage contribution increases with increase of wall shear stress [12,14,27]. Thus, the existence of the minimum value of slippage contribution in Figs. 12 and 13 was indicative of transition of flow behavior from shear thinning to shear thickening. From Figs. 12 and 13, it can be seen that the slippage contribution became increasingly more important for smaller pipe. However, it must be noted that for the larger pipe (D = 50 mm), there was a 20–30% contribution of slippage to the total flow at high wall shear stress.
Appendix A. Nomenclature
5. Conclusions Flow behaviors of concentrated coal water slurries were characterized by using a set of steel pipes. Based on the new slip model, the true rheological behaviors and wall-slip velocities were obtained for slurries with various solid contents. The main conclusions are as follows: (1) The proposed slip model allows a more general dependence of wall-slip velocity on pipe radius. The excellent linear fit of each plot of apparent shear rate to 1/R1+d and the overlap of the slipcorrected rheological diagrams obtained from different pipes validated the new slip model and the selected value of index parameter.
d n r D K L P Q R u us0 V
index parameter flow behavior index radius from pipe centre (m) pipe diameter (m) consistency coefficient (Pa sn ) pipe length (m) pressure drop (Pa) total volumetric flow rate (m3 /s) pipe radius (m) velocity (m/s) parameter, Eq. (13) mean velocity (m/s)
Greek letters ˛ parameter, Eq. (13) ˇ parameter, Eq. (13) slip coefficient shear stress (Pa) y yield stress (Pa) shear rate (s−1 ) true shear viscosity (Pa s) ε slippage contribution Subscripts a apparent b bulk slurries s slip w wall tw true value on the wall References [1] S.C. Tsai, E.W. Knell, Viscometry and rheology of coal water slurry, Fuel 65 (1986) 566–571. [2] N.S. Roh, D.H. Shin, D.C. Kim, J.D. Kim, Rheological behavior of coal-water mixtures. 1. Effects of coal type, loading and particle size, Fuel 74 (1995) 1220–1225. [3] N.S. Roh, D.H. Shin, D.C. Kim, J.D. Kim, Rheological behavior of coal-water mixtures. 2. Effects of surfactants and temperature, Fuel 74 (1995) 1313–1318. [4] R. Yavuz, S. Kucukbayrak, Effect of particle size distribution on rheology of lignite–water slurry, Energy Sources 20 (1998) 787–794. [5] R.M. Turian, J.F. Attal, D.J. Sung, L.E. Wedgewood, Properties and rheology of coal–water mixtures using different coals, Fuel 81 (2002) 2019–2033. [6] S.K. Majumder, K. Chandna, D.S. De, G. Kundu, Studies on flow characteristics of coal–oil–water slurry system, Int. J. Miner. Process. 79 (2006) 217–224.
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