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Studies on pinched sluice concentration. Part II: Characterization of flow over a pinched sluice
Minerals Engineering 15 (2002) 437–446 This article is also available online at: www.elsevier.com/locate/mineng
Studies on pinched sluice concentration. Part II: Characterization of flow over a pinched sluice L. Erg€ un
a,*
, S. Ersayın
b
a
b
Mining Engineering Department, Hacettepe University, Beytepe 06532, Ankara, Turkey Coleraine Minerals Research Laboratory, University of Minnesota Duluth, Duluth, MN 55722, USA Received 21 December 2001; accepted 25 March 2002
Abstract A series of tests were carried out to determine characteristics of the flows over pinched sluices. Effects of volumetric flow rate, feed % solids and slope angle on flow height were measured. An empirical model was developed to define the relationship between operating variables and flow height. Within the experimental range, Reynolds numbers varied between 1200 and 5000, implying that the flow could be classified as upper transitional to turbulent depending on operating conditions. Data also show that turbulence can be suppressed by increasing feed % solids. On the basis of Froude numbers the whole range is classified as supercritical since they varied between 2.6 and 6.7. The abilities of laminar flow, Manning and smooth turbulent flow equations to predict mean flow velocities were tested. None of the equations provided a satisfactory fit to experimental data. The Manning equation was successfully modified into a model to define the relationship between operating variables and mean flow velocities. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Gravity concentration
1. Introduction Although gravity concentration methods have been used for centuries, their separation mechanism has not been completely understood, and satisfactory quantitative models are yet to be developed. Due to similarity of the processes and governing forces, the equations developed in open channel hydraulics and sediment transport are being adapted to flowing film concentration. These equations were successfully modified and applied to spiral concentration (Holland-Batt, 1989) and were then used in several studies (Holland-Batt and Holtham, 1991; Holtham, 1992). Later, Loveday and Cilliers (1994) proposed a model for fluid flow on spirals including the effect of the Bagnold force. Recently, Kapur and Meloy (1998) described an extended approach to include the geometry of spiral. On the other hand, computational fluid dynamics (CFD) offers potential for modelling of fluid behaviour in flowing film concentration. Using a CFD based model coupled with particle image velocimetry (PIV) data, Golab et al. (1998) * Corresponding author. Tel.: +90-312-2977646; fax: +90-3122992155. E-mail address: [email protected] (L. Erg€ un).
found good correlation with measured and predicted behaviour of a flow on a spiral concentrator. In contrast to spirals, there are only a few fundamental studies on pinched sluices. One reason for this is the decreasing use of pinched sluices after the development of Reichert cones, which have much higher capacity, facilitate multi-stage concentration in one unit and provide easier handling of material. However, pinched sluices are the simplest gravity equipment. As known, their main deficiency is that they have low upgrading ratio in a single pass. It is believed that this could be improved by better understanding of flow and separation characteristics. Such knowledge would also be useful for the operation of Reichert cones providing separations based on the same principles. Furthermore, a better understanding of the separation on this simple device would greatly contribute to the theoretical analysis of the separation on more complicated flowing film concentrators such as spirals. Naguib (1971) was the first to propose a theoretical approach based on open channel hydraulics and sediment transport for separation in a pinched sluice. However, he oversimplified the process and did not provide any experimental evidence for his approach. Although the idea of converging sidewalls was interesting, the
0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 2 - 6 8 7 5 ( 0 2 ) 0 0 0 5 9 - 6
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use of a Lamflo sluice, which was the product of his theoretical analysis, has been diminished. Abdinegoro and Partridge obtained vertical concentration profiles of both ilmenite and quartz variation along the flow height by carrying out test work at a range of flow rates and two different pulp solids contents. However, they performed their tests under extremely high flow rates in order to provide a flow height thick enough to be divided into nine streams at the discharge end. This made the applicability of their findings into industrial practice questionable. Subasinghe and Kelly (1984) carried out the most detailed study found in the literature. They proposed a semi-empirical model tentatively based on the fundamental equations of fluid dynamics and hydraulics for predicting the quality of separation. Later, the model was modified by Jeyadevan and Subasinghe (1990) so as to include solid parameters. The test conditions covered a large range, and a considerable portion of their feed % solids (FPS) and slope angle ranges were outside of industrial interest. It is believed that the relationship developed for such a large range decreased sensitivity for the range of practical use. In both studies, the main aim was to predict performance; consequently the flow characteristics were not thoroughly examined. The aim of this study was to characterize the flow over a pinched sluice using Reynolds and Froude numbers as criteria, to investigate the effects of operating parameters on flow characteristics, to analyse the capability of existing equations for mean velocity to predict the measured values, and consequently, if necessary, to develop a model for accurately predicting the mean flow velocity for a given set of operating conditions. For this purpose, the flow heights at the discharge end were measured for a range of flow rates, FPS and sluice angles using a pinched sluice with geometry similar to a Reichert tray. Then data were used for the study of flow characteristics over pinched sluices. 1.1. Characterization of flow over a pinched sluice Since a pinched sluice is basically an open channel with a decreasing width in the direction of flow, Reynolds and Froude Numbers can be used for flow characterization. The Reynolds number is described as the ratio of inertial forces to viscous forces (Chow, 1959), and for an open channel is calculated as Re ¼
qVR ; l
where q V
Q
specific gravity of fluid mean velocity of the stream at the discharge end of sluice, for a channel of rectangular cross section V ¼ Q=ðb hÞ volumetric flow rate
b h R
width of sluice at the discharge end flow height at the discharge end of sluice hydraulic radius of channel, for a channel of rectangular cross section R ¼ ðb hÞ=ð2h þ bÞ l viscosity of fluid The following equation is obtained by substitution: qQ : ð1Þ Re ¼ ð2h þ bÞl
It is well known that open channel flows are classified using the Reynolds number as follows (Chow, 1959): Re < 500 ) laminar flow;
If
500 < Re < 2000 ) transitional flow; Re > 2000 ) turbulent flow: On the other hand, the Froude number (Fr) describes the ratio of inertial forces to gravity forces (Chow, 1959) and, for horizontal open channels, is expressed as: V Fr ¼ pffiffiffiffiffi : gh For a channel of large slope angle, it may be defined as Fr ¼
bh1:5
Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g cos a
ð2Þ
where g gravitational acceleration a slope angle Based on Froude number, open channel flows are classified as follows: If
Fr < 1 ) subcritical flow; Fr > 1 ) supercritical flow:
In the subcritical region, the flow is under the effect of gravity, and, hence, steady flow is observed. In the supercritical region, the flow shows a tendency to form eddies. 1.2. Calculation of flow velocity in open channels There are a number of equations in the literature for the calculation of flow velocity in channels and rivers and numerous books describing their theoretical basis (Chow, 1959; Raudkivi, 1976; Graf, 1971). An adaptation of these equations to spiral concentrators was presented by Holland-Batt (1989). In this study, the three equations were selected for examining their ability to predict the measured mean velocities. These were: 1. Laminar flow equation which was originally developed by Gaudin for the theoretical treatment of separation in flowing film concentration (Holland-Batt, 1989) V hV : ¼ V 3m
ð3Þ
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2. Conventional Manning equation commonly used for the turbulent flows in open channels (Chow, 1959) pffiffiffiffiffiffiffiffiffi 1 V ¼ R2=3 sin a: ð4Þ n 3. Velocity equation for smooth turbulent flows developed for turbulent flows in which viscous resistance has considerable effect (Holland-Batt, 1989) V 1 9hV ln ¼ 1 ; ð5Þ V j m where V n R
mean velocity of flow (cm/s) roughness coefficient known as Manning’s n hydraulic radius of channel (cm), as defined above pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V friction or shear velocity, V ¼ gh sin a j roughness coefficient known as Karman coefficient m kinematic viscosity (gr/cm s) g gravitational acceleration (cm=s2 ) Initially, the fit of these equations to the experimental data was examined. Then some modifications were introduced to obtain a better fit.
2. Experimental studies 2.1. Material and experimental setup Tests were carried out using a single sluice with a similar geometry to the Reichert tray unit (Subasinghe and Kelly, 1984). Its dimensions were 80 cm in length, 25 cm in width at the feed end and 5.5 cm at the discharge end. An artificial mixture of magnetite ()0.3 mm) with a specific gravity of 5.17 and quartz ()0.5 mm) with a specific gravity of 2.64 was used as the test material. Both materials were deslimed ð 38 lmÞ by repeated decantation. The makeup feed contained only 1.4% of 38 lm size material. Feed grade was kept constant during all tests at 5% magnetite by weight. The ranges of experimental conditions for the tests are presented in Table 1. During initial tests, the stability of the system was tested by varying FPS from 0% to 70% by weight under a range of flow rates. It was found that flow height, product flow rates and % solids measurements were reproducible.
Table 1 The ranges of variables studied during the tests
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Pulp prepared at predetermined conditions was pumped to the T-shaped feed distributor at the top of the sluice so as to have zero forward velocity and was then fed uniformly over the sluice. Flow height measurements were made by inserting a thin ruler (0.15 mm thick) into the flowing pulp without causing significant disturbance. All measurements were taken at the centre point along the width of the sluice, and triplicated. The height of flow at both sidewalls of the sluice was slightly higher than the one measured at the centre. Repeated measurements indicated that the sensitivity of flow height measurements was within 0.1 cm. 2.2. Calculation of mean velocity and hydrodynamic criteria Following the flow height measurements, the mean velocity (V) for each volumetric flow rate (Q) was calculated using the continuity equation given below. Q ¼ A:V and hence V ¼
Q ; bh
where A is the cross sectional area of flow, and b is the width of the sluice at the discharge end. Since the width is constant, velocity could be calculated using the volumetric flow rate and flow height (h) measured in a test. Then, for each operating condition, Re and Fr were calculated using Eqs. (1) and (2), respectively. Calculation of Re requires knowledge of pulp viscosity. In this study, viscosity of the pulp was not directly measured. Instead, it was calculated by using the following empirical equation developed by Laapas (1985): l ¼ l0 þ
CCv ð1 Cv Þ
4:8
;
ð6Þ
where l viscosity of slurry (Pa s) l0 viscosity of water (Pa s) 1:91 C 1:8 þ 1:67 10 2 ½ð1 WÞ=W W sphericity of solid particles Cv fractional volumetric solids content of slurry This equation requires a definition of particle sphericity. Microscopic examination of quartz and magnetite revealed that both materials could be classified as angular, and therefore, C of the mixture was taken as 1.83, which was the value used by Laapas for crushed quartz. 2.3. Calculation of linear particle concentration (k)
Feed % solids (by weight)
Volumetric feed flow rate (cc/s)
Slope angle (°)
0–60
200–636
13–19
For the evaluation of the effect of FPS, a criteria defined by Bagnold (1954) was used. This was linear particle concentration defined as the ratio of particle
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diameter to mean free dispersion distance, and calculated by the following equation: k¼ where C0
C0 Cv
1 1=3
;
ð7Þ
1
maximum possible fractional volumetric solids content Cv fractional volumetric solids content Bagnold used uniform spheres and calculated C0 as 0.74. However, the value of k is expected to change with particle size and shape, and Allen (1985) reported that C0 could be as low as 0.558. In this study, C0 was measured experimentally. For this purpose, the material was put in a transparent graduated cylinder and shaken vigorously. Then, the cylinder was placed on a vibrating plate and vibrated till no more compaction occurred (approx. 40 min). The total volume of fully compacted material with voids between particles (TV) was measured. Then, from a known volume, water was carefully added until all the voids between the particles (VV) were filled. The cylinder was visually observed for penetration of water into all voids avoiding any channelling etc. It was assumed that the void volume (VV) was equal to the volume of water used for filling. Then, C0 is estimated as ½ðTV VVÞ=TV. The procedure was repeated several times and good reproducibility was observed. The mean value of C0 was 0.696. Finally, k values were calculated using Eq. (7).
3. Results The following empirical equation was developed in an earlier study by the authors (Erg€ un and Ersayın, 1998) to define the relationship between flow height (h) and main operating variables: h ¼ a þ b Q þ c ðFPSÞ3 d a; where Q volumetric feed flow rate FPS feed % solids a slope angle of the sluice a; b; c; d empirical constants This equation is only valid for a range of flow rates, since it implies that there would be a certain flow height even when the flow rate is nil. To overcome this deficiency and obtain better fit to the experimental data, the subject was reconsidered by examining the flow rate–flow height relationship for each slope angle and FPS separately. It was found that the relationship between flow height and volumetric flow rate could be described as h / Qa with the parameter ‘a’ being smaller than unity. The finding was implemented into the above equation with some success, but then it was realized that other modifications were needed for further improvement, and eventually the following equation was developed using residual regression analysis technique:
Fig. 1. Flow rate–flow height relationship at different FPS and slope angles.
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441
Fig. 2. Measured vs calculated (Eq. (8)) flow heights including all the experimental data. The dotted lines show 0.1 cm boundary from ideal line.
h¼
a Q 1000
sin a þ c
n
n
ða þ bðfpsÞ 1 þ dðfpsÞ 2 Þ;
ð8Þ
where h flow height (cm) Q volumetric flow rate (cc/s) a slope angle of the sluice (°) fps fractional feed solids content (by weight) a; b; c; d; n1 ; n2 model constants a¼ 0:761 0:023 b¼ 4:125 2:769 c¼ 0:409 0:017 d¼ 0:159 0:071 n1 ¼ 5:195 1:503 n2 ¼ 0:176 0:227 This equation provided better fit with R2 of 0.95 as compared to 0.91 of the previous model. The residual sum of squares for 114 flow height measurements was 0.24. The flow heights measured at different slope angles are presented in Fig. 1. Measured vs predicted values are shown in Fig. 2. The dotted lines indicate a boundary of measured values of 0.1 cm. Considering the accuracy of measured values, it may be suggested that Eq. (8) describes the relationship very well. In the rest of the paper, predicted rather than measured flow heights are used. This facilitated comparisons at the selected volumetric flow rates and FPS.
a sluice angle of 17° are presented in the rest of the paper. 3.2. The effect of operating variables on the mean velocity As shown in Fig. 4, increasing the volumetric flow rate results in increased mean velocity. It decreases with the increase in FPS up to 10% solids by weight, stays almost constant when FPS is further increased from 10% to 30%, and sharply decreases above 50% solids (Fig. 5). Increasing slope angle increases the mean flow velocity (Fig. 6).
3.1. The effect of operating parameters on flow height The relationships between flow height and main operating parameters are illustrated in Fig. 3. As expected, flow height increases with flow rate. A similar effect was created by increasing FPS, but its effect was more pronounced above 50% solids by weight. It is also shown that slope angle has relatively low effect on flow height. Therefore, only the data corresponding to
Fig. 3. The simulated (Eq. (8)) effects of operating variables on flow height.
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Fig. 4. Variation of the mean flow velocity with the volumetric flow rate.
Fig. 6. Variation of the mean flow velocity with the slope angle.
3.3. The effect of operating variables on hydrodynamic criteria Fig. 7 shows the Reynolds number as a function of the volumetric flow rate and FPS. Within the FPS range of 0–30%, the flow is classified as turbulent, i.e. Re > 2000. It becomes transitional or turbulent depending on the volumetric flow rate when FPS is higher than 50%. On the basis of Froude numbers, the flow within the experimental range is classified as supercritical. Only a slight decrease in Froude numbers with an increase in volumetric flow rate is observed (Fig. 8). Increasing FPS decreases the mean velocity and, consequently results in a sharp decrease in Froude numbers. Fig. 7. Effect of flow rate on the Reynolds number at different FPS.
3.4. Mean velocity models The abilities of three available equations, namely laminar flow, Manning and smooth turbulent flow, to predict the measured mean velocities were examined. Manning and smooth turbulent flow equations were similar in nature since both are suitable for turbulent flows and include a coefficient defining surface roughness. Having established that the flow over pinched
Fig. 8. Effect of flow rate on the Froude number at different FPS.
Fig. 5. Variation of the mean flow velocity with the FPS.
sluices is not laminar within the range of test conditions, the laminar flow equation was not expected to provide a good fit. Nevertheless, none of the equations provided a satisfactory fit to the experimental data. Manning and laminar flow equations grossly overestimated the actual velocities, while the smooth turbulent flow equation provided underestimated values.
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4. Discussion Performance of a pinched sluice might be viewed as an interaction of two separate processes. These are stratification of heavy minerals while the slurry flows over a pinched sluice and separation of the slurry stream into two products after it leaves the sluice. The quality of stratification as well as position of the splitter defines performance. This paper deals with the factors influencing the former process. Stratification is expected to be strongly dependent on the flow conditions. Before this link is quantified, an accurate knowledge is required of flow characteristics over a pinched sluice and their variation with operating conditions. Mean flow velocity is the dominant variable defining the flow characteristics. This variable is not directly measured, but calculated from measured flow heights at the discharge end and volumetric flow rates. Since flow height was a variable dependent on the operating conditions, an accurate model defining the relationship was needed to calculate mean velocity for any given set of operating conditions. The relationship between flow height and operating variables of a pinched sluice was first examined by Subasinghe and Kelly (1984), who proposed an equation by assuming that the mean velocity would be equal to the Manning equation. This equation implied that FPS had no effect on flow height. Although their experimental data indicated a vague relationship between the two, they attributed such differences to experimental errors. Later, Jeyadevan and Subasinghe (1990) realised that this assumption may not be valid and modified the equation to include parameters defining solids concentration and particle size. It is highly arguable that this modified equation has some physical basis as suggested by the authors. It is rather empirical in nature, since it includes six model parameters to be fitted. Erg€ un and Ersayın (1998) first tried the model developed by Jeyadevan and Subasinghe (1990) and found that it did not provide a satisfactory fit to their data despite that the model had six model fitting parameters. They developed a purely empirical model containing four parameters. Their model provided a better fit, but it was valid only within the experimental range. In this study, this empirical model was successfully modified to extend its range and improve the fit. The present model provided a reliable base for further study.
443
they were over 10,000 within a very large range of conditions. The former defines the conditions as upper transitional to turbulent in contrast to the latter’s claim of highly turbulent. Reynolds numbers in this study are close to, but slightly lower than, that given by Abdinegoro and Partridge (1979). It appears that the difference is due to higher flow rate/unit area experienced by them. This led to questioning of Subasinghe and Kelly’s findings. A close examination of their data indicated that larger Reynolds numbers could be caused by misinterpretation of hydraulic radius as defined by Eq. (1). Instead, they used flow height, which amplified the actual values. Another contributing factor could be viscosity, for which no reference was made in their paper. Results show that flow velocity decreases with increased FPS and this effect is less pronounced at relatively high volumetric flow rates. Within FPS range of industrial interest, i.e. 50–65% solids by weight, Reynolds numbers indicate that the flow is either transitional or turbulent. Beyond FPS of 60%, the flow is classified as transitional for the complete range of volumetric flow rates studied. Up to 30% solids, increasing the volumetric flow rate increases the degree of turbulence. When FPS is 50% or higher, the positive effect of volumetric flow rate on turbulence is relatively small. Fig. 9 illustrates the variation of the boundary flow rate, i.e., Re ¼ 2000, with FPS for a slope angle of 17°. While the flow is classified as turbulent at 940 L/h and higher flow rates with 50% solids by weight, the boundary flow rate became 1150 L/h at 54% solids and 1480 and 2030 L/h for 58% and 62% solids, respectively. These results show that turbulence can be depressed by increasing FPS and are in conformity with the findings of Abdinegoro and Partridge (1979). Although a statistical analysis showed that Reynolds numbers and performance could not be directly correlated, it was also found that performance at low Reynolds numbers, i.e., transitional flow conditions, was considerably better (Erg€ un and Ersayın, 1998). The lack of direct
4.1. Flow conditions over a pinched sluice With regard to characterization of flow over pinched sluices on the basis of Reynolds numbers, there are two contradictory data available in the literature. While Abdinegoro and Partridge (1979) found that Reynolds numbers varied between 1200 and 5000 depending on operating conditions, Subasinghe and Kelly claimed that
Fig. 9. Variation of transitional to turbulent flow boundary with flow rate and FPS.
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correlation is thought to be due to the interaction between stratification and splitter position. Froude numbers varied between 2.6 and 6.7. This range of Froude numbers is classified as supercritical and confirms the earlier finding by Subasinghe and Kelly (1984). Although Froude numbers slightly decrease with the increasing volumetric flow rates, this does not change the flow region defined by this criterion. An increase in Froude numbers could be interpreted as an increase in the tendency to form turbulent eddies. As expected, decreased velocity originating from increased solids concentration results in a decrease in Froude numbers. No significant changes in Froude numbers were observed when solids concentration was increased beyond 50% solids. It may be suggested that the Froude number, which was originally developed for flow characterization of water in open channels, might not be a suitable criteria for slurries containing a high proportion of solids. The Froude number is defined as the ratio of inertial (velocity dependent) forces to gravity forces. When it is applied to the slurry flow, an increase in flow height with solids concentration is interpreted as an increase in gravitational force and, consequently, despite an increase in velocity, creates negligible change in the Froude number. 4.2. Mean flow velocity modelling None of the three mean velocity equations tested on experimental data provided a satisfactory fit. This failure initiated a study to develop a modified equation to predict mean flow velocities accurately. The laminar flow equation was eliminated from the search for a better model due to the lack of both theoretical and practical grounds to pursue further. Since the Manning equation was originally developed for water or very dilute suspension over an open channel, its failure could be due two factors: (1) narrowing of the sluice in the direction of flow, which could increase the resistance and slow down the flow, and (2) presence of a high proportion of solids, which could create a similar effect by increasing the friction between the slurry and sluice surface. To eliminate the effect of solids in the slurry, its fit to water only test data was separately examined. Initially, the roughness coefficient for the galvanized surface was used; this resulted in a very large overestimation of the actual velocities. The next step was to define whether the fit could be improved by manipulating the roughness coefficient in the equation. Using a non-linear regression algorithm, the best-fit value of the roughness coefficient was calculated as n ¼ 0:00229, which was approximately six times lower than the original value of n ¼ 0:014. Although this modification improved the fit, there remained large deviations between the measured and calculated velocities. Fig. 10 shows that such modification has a tendency to
Fig. 10. The fit of the Manning (Vm ) and smooth turbulent (Vs ) flow equations with a best-fit roughness coefficients to the mean flow velocities of water only test data.
underestimate at low velocity and overestimate in high velocity range. The same type of approach was also tested on the data from the tests with slurry flows. It was assumed that the roughness coefficient would be dependent on FPS and, a separate best-fit roughness coefficient was calculated for each FPS condition. The quality of fit is illustrated in Fig. 11. Although the fit was improved, the deviations were still large. However, it should be noted that the quality of fit increased with FPC. While the regression coefficient, R2 , for water only test data was 0.5, it became 0.71 for 60% solids. A similar methodology was also applied to improve the quality of flow velocity predictions obtained using the smooth turbulent flow equation (Eq. (5)). Although this equation provided the closest estimates to the measured velocities, it overestimated the actual values when the roughness coefficient for galvanized steel ðj ¼ 0:4Þ was used. Its best-fit values for water only tests and for each FPS condition were separately calculated. Improved fit provided by this procedure is illustrated in Figs. 10 and 12 for water only and slurry tests, respectively. In general, the fit was inferior to that with the
Fig. 11. The fit of the Manning equation to mean flow velocities when a separate best-fit roughness coefficient was calculated for each FPS.
L. Erg€un, S. Ersayın / Minerals Engineering 15 (2002) 437–446
Fig. 12. The fit of the smooth turbulent flow equation to mean flow velocities when a separate best-fit roughness coefficient was calculated for each FPS.
Manning equation. However, similar trends were observed, i.e., the fit was improved as FPS increased. The regression coefficient (R2 ) increased from 0.41 for water only tests to 0.50 for the tests with 60% solids in feed. Since the Manning equation showed better potential for improvement, the rest of the study focused on further improving the predictions of this equation. Failure of the Manning equation with a modified roughness coefficient to predict mean flow velocities from water only tests implied that a narrowing effect could not be simulated by increasing this parameter. However, results also showed that it had a very good potential for reflecting the changes in FPS. Therefore, it was considered as a compact parameter defining the effects of solid concentration, particle shape and specific gravity. As a second step toward developing an improved equation, it was suggested that the narrowing effect could be embedded into the equation by manipulating the exponent of hydraulic radius, m, which is normally equal to 2/3. The best-fit values of both parameters were calculated using a non-linear regression algorithm. As expected, m varied within a short range of 0.364–0.395. Since initially it was assumed that m was a compact parameter defining the geometry of pinched sluices, its mean value, 0.379, was considered a suitable value for the sluice used in the tests, and the best-fit values of the roughness coefficient were re-calculated. Its fit to experimental data was beyond any expectation (Fig. 13). The regression coefficients, R2 , calculated separately for each solid concentration varied between 0.996 and 0.998. The roughness coefficient, km , also exhibited a meaningful relationship decreasing with the increase in solid concentration. However, it was thought that Bagnold’s k would be a more suitable indicator of such an effect since it is a compact parameter including % solid, shape and size distribution. Its relationship with km was examined and it is found that a linear relationship between the two exists, if the data from the water only tests
445
Fig. 13. The fit of the modified Manning equation (Eq. (9)) to experimental mean flow velocities.
were ignored (Fig. 14). The following velocity equation was obtained for the material used in the test by substituting km with the linear equation defining its relationship with k: pffiffiffiffiffiffiffiffiffi V ¼ ð310:67 29:929kÞR0:379 sin a: ð9Þ Eq. (9) did not deteriorate the fit obtained when km was used instead. It is believed that considerable progress could be made by determining the model parameters of Eq. (9) by using feed materials having different k, and pinched sluices of different geometry. It is expected that m will be coefficient for a given geometry and will not change even when the material characteristics are changed. On the other hand, pinched sluices of different geometry are expected to produce different m values probably decreasing, for example, with the length. 4.3. Velocity profile Flow characteristics affect the quality of stratification. Performance, however, also depends on the position of the splitter and the trajectory that the slurry
Fig. 14. The relationship between the roughness coefficient of the modified Manning equation ðkm Þ and linear particle concentration ðkÞ.
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follows right after the edge of a sluice. This in turn is a function of vertical velocity profile that exists within the slurry. Prediction of performance requires coupling of all the knowledge. Unfortunately, information regarding vertical velocity profile is scarce. Available data indicate that, unlike other flowing film devices, the zone of maximum velocity is near the middle of the slurry (Abdinegoro and Partridge, 1979). It is also known that the logarithmic velocity profile developed for water flows in open channels could not be used for slurry flows containing a large proportion of solids (Subasinghe and Kelly, 1984). Therefore, further study is needed to define what happens after the slurry leaves the edge of a sluice. Particle image velocimetry appears to have a potential as a measuring device to fill this knowledge gap.
5. Conclusion It was found that volumetric flow rate, solids concentration and sluice angle have a significant effect on flow height. It increases with flow rate and FPS, but decreases with an increase in sluice angle. Within the range of volumetric flow rates studied, the flow is classified as turbulent when pulp solids concentration is below 50% by weight. Above this limit, the flow is either transitional or turbulent. Froude numbers indicated that, for the complete range of test conditions, the flow is supercritical, implying that it has a tendency to form eddies. Such behaviour is in conformity with visual observations made during test work. Since specific value of Froude number varies within a short range, its usefulness as a criterion is questionable. The mean velocity of flow decreased with increase in solids concentration and this effect was more pronounced above 50% solids by weight. As expected, the mean velocity increased with the volumetric flow rate, but slope angle did not have a strong effect on it. Neither Manning nor smooth turbulent flow equations provided a satisfactory fit to the experimental mean velocities, even when the parameters defining roughness were varied to improve their fit. A modified form of the Manning equation is proposed in this study to define the mean flow velocity over a pinched sluice. It provided excellent fit to the experimental data. It appears that it could also be used for the determination of mean flow velocities on a pinched
sluice of different geometry, operating at a wide range of conditions.
Acknowledgements The authors would like to thank The Scientific and Technical Research Council of Turkey for the financial support through the Project No. YBAG 0069.
References Abdinegoro, S., Partridge, A.C., 1979. Flow characteristics of a pinched sluice. In: Proc. Australian IMM Conference, Western Australia, pp. 79–83. Allen, J.R.L., 1985. Principles of Physical Sedimentology. Allen & Unwin, London. Bagnold, R.A., 1954. Experiments on gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proceedings of Royal Society of London, Series A 225, 49–63. Chow, V.T., 1959. Open-Channel Hydraulics. McGraw-Hill, New York. Erg€ un, L., Ersayın, S., 1998. Characterization of flow on a pinched sluice and its effect on separation. In: Innovation in Physical Separation Technologies, Richard Mozley Symposium Volume, IMM, London, pp. 85–98. Golab, K.J., Holtham, P.N., Wu, J., 1998. Validation of a computer model of fluid flow on the spiral separator. In: Innovation in Physical Separation Technologies, Richard Mozley Memorial Volume, IMM, London, pp. 176–186. Graf, W.H., 1971. Hydraulics of Sediment Transport. McGraw-Hill, New York. Holland-Batt, A.B., 1989. Spiral separation: theory and simulation. Transactions of IMM, Section C January–April, C8–C16. Holland-Batt, A.B., Holtham, P.N., 1991. Particle and fluid motion on spiral separators. Minerals Engineering 7 (3/4), 457–482. Holtham, P.N., 1992. Particle transport in gravity concentrators and the Bagnold effect. Minerals Engineering 5 (2), 205–221. Jeyadevan, B., Subasinghe, G.K.N.S., 1990. The effect of solid parameters in a pinched sluice concentrator. International Journal of Mineral Processing 29, 31–50. Kapur, P.C., Meloy, T.P., 1998. Spirals observed. International Journal of Mineral Processing 53, 15–28. Laapas, H.R., 1985. Rheology of fast settling mineral slurries. In: Proc. of 15th International Mineral Processing Congress, Cannes, Part 3, pp. 28–40. Loveday, G.K., Cilliers, J.J., 1994. Fluid flow modelling in spiral concentrators. Minerals Engineering 7 (2/3), 223–237. Naguib, A.G., 1971. New considerations in design and application of sluice separators. Mining Congress Journal September, 123–128. Raudkivi, A.J., 1976. Loose Boundary Hydraulics. Pergamon Press, Oxford. Subasinghe, G.K.N.S., Kelly, E.G., 1984. Modelling of pinched sluice type concentrators. In: Control’84 Mineral/Metallurgical Processing. AIME/SME/TMS, pp. 87–95.
Studies on pinched sluice concentration. Part II: Characterization of flow over a pinched sluice L. Erg€ un
a,*
, S. Ersayın
b
a
b
Mining Engineering Department, Hacettepe University, Beytepe 06532, Ankara, Turkey Coleraine Minerals Research Laboratory, University of Minnesota Duluth, Duluth, MN 55722, USA Received 21 December 2001; accepted 25 March 2002
Abstract A series of tests were carried out to determine characteristics of the flows over pinched sluices. Effects of volumetric flow rate, feed % solids and slope angle on flow height were measured. An empirical model was developed to define the relationship between operating variables and flow height. Within the experimental range, Reynolds numbers varied between 1200 and 5000, implying that the flow could be classified as upper transitional to turbulent depending on operating conditions. Data also show that turbulence can be suppressed by increasing feed % solids. On the basis of Froude numbers the whole range is classified as supercritical since they varied between 2.6 and 6.7. The abilities of laminar flow, Manning and smooth turbulent flow equations to predict mean flow velocities were tested. None of the equations provided a satisfactory fit to experimental data. The Manning equation was successfully modified into a model to define the relationship between operating variables and mean flow velocities. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Gravity concentration
1. Introduction Although gravity concentration methods have been used for centuries, their separation mechanism has not been completely understood, and satisfactory quantitative models are yet to be developed. Due to similarity of the processes and governing forces, the equations developed in open channel hydraulics and sediment transport are being adapted to flowing film concentration. These equations were successfully modified and applied to spiral concentration (Holland-Batt, 1989) and were then used in several studies (Holland-Batt and Holtham, 1991; Holtham, 1992). Later, Loveday and Cilliers (1994) proposed a model for fluid flow on spirals including the effect of the Bagnold force. Recently, Kapur and Meloy (1998) described an extended approach to include the geometry of spiral. On the other hand, computational fluid dynamics (CFD) offers potential for modelling of fluid behaviour in flowing film concentration. Using a CFD based model coupled with particle image velocimetry (PIV) data, Golab et al. (1998) * Corresponding author. Tel.: +90-312-2977646; fax: +90-3122992155. E-mail address: [email protected] (L. Erg€ un).
found good correlation with measured and predicted behaviour of a flow on a spiral concentrator. In contrast to spirals, there are only a few fundamental studies on pinched sluices. One reason for this is the decreasing use of pinched sluices after the development of Reichert cones, which have much higher capacity, facilitate multi-stage concentration in one unit and provide easier handling of material. However, pinched sluices are the simplest gravity equipment. As known, their main deficiency is that they have low upgrading ratio in a single pass. It is believed that this could be improved by better understanding of flow and separation characteristics. Such knowledge would also be useful for the operation of Reichert cones providing separations based on the same principles. Furthermore, a better understanding of the separation on this simple device would greatly contribute to the theoretical analysis of the separation on more complicated flowing film concentrators such as spirals. Naguib (1971) was the first to propose a theoretical approach based on open channel hydraulics and sediment transport for separation in a pinched sluice. However, he oversimplified the process and did not provide any experimental evidence for his approach. Although the idea of converging sidewalls was interesting, the
0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 2 - 6 8 7 5 ( 0 2 ) 0 0 0 5 9 - 6
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use of a Lamflo sluice, which was the product of his theoretical analysis, has been diminished. Abdinegoro and Partridge obtained vertical concentration profiles of both ilmenite and quartz variation along the flow height by carrying out test work at a range of flow rates and two different pulp solids contents. However, they performed their tests under extremely high flow rates in order to provide a flow height thick enough to be divided into nine streams at the discharge end. This made the applicability of their findings into industrial practice questionable. Subasinghe and Kelly (1984) carried out the most detailed study found in the literature. They proposed a semi-empirical model tentatively based on the fundamental equations of fluid dynamics and hydraulics for predicting the quality of separation. Later, the model was modified by Jeyadevan and Subasinghe (1990) so as to include solid parameters. The test conditions covered a large range, and a considerable portion of their feed % solids (FPS) and slope angle ranges were outside of industrial interest. It is believed that the relationship developed for such a large range decreased sensitivity for the range of practical use. In both studies, the main aim was to predict performance; consequently the flow characteristics were not thoroughly examined. The aim of this study was to characterize the flow over a pinched sluice using Reynolds and Froude numbers as criteria, to investigate the effects of operating parameters on flow characteristics, to analyse the capability of existing equations for mean velocity to predict the measured values, and consequently, if necessary, to develop a model for accurately predicting the mean flow velocity for a given set of operating conditions. For this purpose, the flow heights at the discharge end were measured for a range of flow rates, FPS and sluice angles using a pinched sluice with geometry similar to a Reichert tray. Then data were used for the study of flow characteristics over pinched sluices. 1.1. Characterization of flow over a pinched sluice Since a pinched sluice is basically an open channel with a decreasing width in the direction of flow, Reynolds and Froude Numbers can be used for flow characterization. The Reynolds number is described as the ratio of inertial forces to viscous forces (Chow, 1959), and for an open channel is calculated as Re ¼
qVR ; l
where q V
Q
specific gravity of fluid mean velocity of the stream at the discharge end of sluice, for a channel of rectangular cross section V ¼ Q=ðb hÞ volumetric flow rate
b h R
width of sluice at the discharge end flow height at the discharge end of sluice hydraulic radius of channel, for a channel of rectangular cross section R ¼ ðb hÞ=ð2h þ bÞ l viscosity of fluid The following equation is obtained by substitution: qQ : ð1Þ Re ¼ ð2h þ bÞl
It is well known that open channel flows are classified using the Reynolds number as follows (Chow, 1959): Re < 500 ) laminar flow;
If
500 < Re < 2000 ) transitional flow; Re > 2000 ) turbulent flow: On the other hand, the Froude number (Fr) describes the ratio of inertial forces to gravity forces (Chow, 1959) and, for horizontal open channels, is expressed as: V Fr ¼ pffiffiffiffiffi : gh For a channel of large slope angle, it may be defined as Fr ¼
bh1:5
Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g cos a
ð2Þ
where g gravitational acceleration a slope angle Based on Froude number, open channel flows are classified as follows: If
Fr < 1 ) subcritical flow; Fr > 1 ) supercritical flow:
In the subcritical region, the flow is under the effect of gravity, and, hence, steady flow is observed. In the supercritical region, the flow shows a tendency to form eddies. 1.2. Calculation of flow velocity in open channels There are a number of equations in the literature for the calculation of flow velocity in channels and rivers and numerous books describing their theoretical basis (Chow, 1959; Raudkivi, 1976; Graf, 1971). An adaptation of these equations to spiral concentrators was presented by Holland-Batt (1989). In this study, the three equations were selected for examining their ability to predict the measured mean velocities. These were: 1. Laminar flow equation which was originally developed by Gaudin for the theoretical treatment of separation in flowing film concentration (Holland-Batt, 1989) V hV : ¼ V 3m
ð3Þ
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2. Conventional Manning equation commonly used for the turbulent flows in open channels (Chow, 1959) pffiffiffiffiffiffiffiffiffi 1 V ¼ R2=3 sin a: ð4Þ n 3. Velocity equation for smooth turbulent flows developed for turbulent flows in which viscous resistance has considerable effect (Holland-Batt, 1989) V 1 9hV ln ¼ 1 ; ð5Þ V j m where V n R
mean velocity of flow (cm/s) roughness coefficient known as Manning’s n hydraulic radius of channel (cm), as defined above pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V friction or shear velocity, V ¼ gh sin a j roughness coefficient known as Karman coefficient m kinematic viscosity (gr/cm s) g gravitational acceleration (cm=s2 ) Initially, the fit of these equations to the experimental data was examined. Then some modifications were introduced to obtain a better fit.
2. Experimental studies 2.1. Material and experimental setup Tests were carried out using a single sluice with a similar geometry to the Reichert tray unit (Subasinghe and Kelly, 1984). Its dimensions were 80 cm in length, 25 cm in width at the feed end and 5.5 cm at the discharge end. An artificial mixture of magnetite ()0.3 mm) with a specific gravity of 5.17 and quartz ()0.5 mm) with a specific gravity of 2.64 was used as the test material. Both materials were deslimed ð 38 lmÞ by repeated decantation. The makeup feed contained only 1.4% of 38 lm size material. Feed grade was kept constant during all tests at 5% magnetite by weight. The ranges of experimental conditions for the tests are presented in Table 1. During initial tests, the stability of the system was tested by varying FPS from 0% to 70% by weight under a range of flow rates. It was found that flow height, product flow rates and % solids measurements were reproducible.
Table 1 The ranges of variables studied during the tests
439
Pulp prepared at predetermined conditions was pumped to the T-shaped feed distributor at the top of the sluice so as to have zero forward velocity and was then fed uniformly over the sluice. Flow height measurements were made by inserting a thin ruler (0.15 mm thick) into the flowing pulp without causing significant disturbance. All measurements were taken at the centre point along the width of the sluice, and triplicated. The height of flow at both sidewalls of the sluice was slightly higher than the one measured at the centre. Repeated measurements indicated that the sensitivity of flow height measurements was within 0.1 cm. 2.2. Calculation of mean velocity and hydrodynamic criteria Following the flow height measurements, the mean velocity (V) for each volumetric flow rate (Q) was calculated using the continuity equation given below. Q ¼ A:V and hence V ¼
Q ; bh
where A is the cross sectional area of flow, and b is the width of the sluice at the discharge end. Since the width is constant, velocity could be calculated using the volumetric flow rate and flow height (h) measured in a test. Then, for each operating condition, Re and Fr were calculated using Eqs. (1) and (2), respectively. Calculation of Re requires knowledge of pulp viscosity. In this study, viscosity of the pulp was not directly measured. Instead, it was calculated by using the following empirical equation developed by Laapas (1985): l ¼ l0 þ
CCv ð1 Cv Þ
4:8
;
ð6Þ
where l viscosity of slurry (Pa s) l0 viscosity of water (Pa s) 1:91 C 1:8 þ 1:67 10 2 ½ð1 WÞ=W W sphericity of solid particles Cv fractional volumetric solids content of slurry This equation requires a definition of particle sphericity. Microscopic examination of quartz and magnetite revealed that both materials could be classified as angular, and therefore, C of the mixture was taken as 1.83, which was the value used by Laapas for crushed quartz. 2.3. Calculation of linear particle concentration (k)
Feed % solids (by weight)
Volumetric feed flow rate (cc/s)
Slope angle (°)
0–60
200–636
13–19
For the evaluation of the effect of FPS, a criteria defined by Bagnold (1954) was used. This was linear particle concentration defined as the ratio of particle
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diameter to mean free dispersion distance, and calculated by the following equation: k¼ where C0
C0 Cv
1 1=3
;
ð7Þ
1
maximum possible fractional volumetric solids content Cv fractional volumetric solids content Bagnold used uniform spheres and calculated C0 as 0.74. However, the value of k is expected to change with particle size and shape, and Allen (1985) reported that C0 could be as low as 0.558. In this study, C0 was measured experimentally. For this purpose, the material was put in a transparent graduated cylinder and shaken vigorously. Then, the cylinder was placed on a vibrating plate and vibrated till no more compaction occurred (approx. 40 min). The total volume of fully compacted material with voids between particles (TV) was measured. Then, from a known volume, water was carefully added until all the voids between the particles (VV) were filled. The cylinder was visually observed for penetration of water into all voids avoiding any channelling etc. It was assumed that the void volume (VV) was equal to the volume of water used for filling. Then, C0 is estimated as ½ðTV VVÞ=TV. The procedure was repeated several times and good reproducibility was observed. The mean value of C0 was 0.696. Finally, k values were calculated using Eq. (7).
3. Results The following empirical equation was developed in an earlier study by the authors (Erg€ un and Ersayın, 1998) to define the relationship between flow height (h) and main operating variables: h ¼ a þ b Q þ c ðFPSÞ3 d a; where Q volumetric feed flow rate FPS feed % solids a slope angle of the sluice a; b; c; d empirical constants This equation is only valid for a range of flow rates, since it implies that there would be a certain flow height even when the flow rate is nil. To overcome this deficiency and obtain better fit to the experimental data, the subject was reconsidered by examining the flow rate–flow height relationship for each slope angle and FPS separately. It was found that the relationship between flow height and volumetric flow rate could be described as h / Qa with the parameter ‘a’ being smaller than unity. The finding was implemented into the above equation with some success, but then it was realized that other modifications were needed for further improvement, and eventually the following equation was developed using residual regression analysis technique:
Fig. 1. Flow rate–flow height relationship at different FPS and slope angles.
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441
Fig. 2. Measured vs calculated (Eq. (8)) flow heights including all the experimental data. The dotted lines show 0.1 cm boundary from ideal line.
h¼
a Q 1000
sin a þ c
n
n
ða þ bðfpsÞ 1 þ dðfpsÞ 2 Þ;
ð8Þ
where h flow height (cm) Q volumetric flow rate (cc/s) a slope angle of the sluice (°) fps fractional feed solids content (by weight) a; b; c; d; n1 ; n2 model constants a¼ 0:761 0:023 b¼ 4:125 2:769 c¼ 0:409 0:017 d¼ 0:159 0:071 n1 ¼ 5:195 1:503 n2 ¼ 0:176 0:227 This equation provided better fit with R2 of 0.95 as compared to 0.91 of the previous model. The residual sum of squares for 114 flow height measurements was 0.24. The flow heights measured at different slope angles are presented in Fig. 1. Measured vs predicted values are shown in Fig. 2. The dotted lines indicate a boundary of measured values of 0.1 cm. Considering the accuracy of measured values, it may be suggested that Eq. (8) describes the relationship very well. In the rest of the paper, predicted rather than measured flow heights are used. This facilitated comparisons at the selected volumetric flow rates and FPS.
a sluice angle of 17° are presented in the rest of the paper. 3.2. The effect of operating variables on the mean velocity As shown in Fig. 4, increasing the volumetric flow rate results in increased mean velocity. It decreases with the increase in FPS up to 10% solids by weight, stays almost constant when FPS is further increased from 10% to 30%, and sharply decreases above 50% solids (Fig. 5). Increasing slope angle increases the mean flow velocity (Fig. 6).
3.1. The effect of operating parameters on flow height The relationships between flow height and main operating parameters are illustrated in Fig. 3. As expected, flow height increases with flow rate. A similar effect was created by increasing FPS, but its effect was more pronounced above 50% solids by weight. It is also shown that slope angle has relatively low effect on flow height. Therefore, only the data corresponding to
Fig. 3. The simulated (Eq. (8)) effects of operating variables on flow height.
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Fig. 4. Variation of the mean flow velocity with the volumetric flow rate.
Fig. 6. Variation of the mean flow velocity with the slope angle.
3.3. The effect of operating variables on hydrodynamic criteria Fig. 7 shows the Reynolds number as a function of the volumetric flow rate and FPS. Within the FPS range of 0–30%, the flow is classified as turbulent, i.e. Re > 2000. It becomes transitional or turbulent depending on the volumetric flow rate when FPS is higher than 50%. On the basis of Froude numbers, the flow within the experimental range is classified as supercritical. Only a slight decrease in Froude numbers with an increase in volumetric flow rate is observed (Fig. 8). Increasing FPS decreases the mean velocity and, consequently results in a sharp decrease in Froude numbers. Fig. 7. Effect of flow rate on the Reynolds number at different FPS.
3.4. Mean velocity models The abilities of three available equations, namely laminar flow, Manning and smooth turbulent flow, to predict the measured mean velocities were examined. Manning and smooth turbulent flow equations were similar in nature since both are suitable for turbulent flows and include a coefficient defining surface roughness. Having established that the flow over pinched
Fig. 8. Effect of flow rate on the Froude number at different FPS.
Fig. 5. Variation of the mean flow velocity with the FPS.
sluices is not laminar within the range of test conditions, the laminar flow equation was not expected to provide a good fit. Nevertheless, none of the equations provided a satisfactory fit to the experimental data. Manning and laminar flow equations grossly overestimated the actual velocities, while the smooth turbulent flow equation provided underestimated values.
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4. Discussion Performance of a pinched sluice might be viewed as an interaction of two separate processes. These are stratification of heavy minerals while the slurry flows over a pinched sluice and separation of the slurry stream into two products after it leaves the sluice. The quality of stratification as well as position of the splitter defines performance. This paper deals with the factors influencing the former process. Stratification is expected to be strongly dependent on the flow conditions. Before this link is quantified, an accurate knowledge is required of flow characteristics over a pinched sluice and their variation with operating conditions. Mean flow velocity is the dominant variable defining the flow characteristics. This variable is not directly measured, but calculated from measured flow heights at the discharge end and volumetric flow rates. Since flow height was a variable dependent on the operating conditions, an accurate model defining the relationship was needed to calculate mean velocity for any given set of operating conditions. The relationship between flow height and operating variables of a pinched sluice was first examined by Subasinghe and Kelly (1984), who proposed an equation by assuming that the mean velocity would be equal to the Manning equation. This equation implied that FPS had no effect on flow height. Although their experimental data indicated a vague relationship between the two, they attributed such differences to experimental errors. Later, Jeyadevan and Subasinghe (1990) realised that this assumption may not be valid and modified the equation to include parameters defining solids concentration and particle size. It is highly arguable that this modified equation has some physical basis as suggested by the authors. It is rather empirical in nature, since it includes six model parameters to be fitted. Erg€ un and Ersayın (1998) first tried the model developed by Jeyadevan and Subasinghe (1990) and found that it did not provide a satisfactory fit to their data despite that the model had six model fitting parameters. They developed a purely empirical model containing four parameters. Their model provided a better fit, but it was valid only within the experimental range. In this study, this empirical model was successfully modified to extend its range and improve the fit. The present model provided a reliable base for further study.
443
they were over 10,000 within a very large range of conditions. The former defines the conditions as upper transitional to turbulent in contrast to the latter’s claim of highly turbulent. Reynolds numbers in this study are close to, but slightly lower than, that given by Abdinegoro and Partridge (1979). It appears that the difference is due to higher flow rate/unit area experienced by them. This led to questioning of Subasinghe and Kelly’s findings. A close examination of their data indicated that larger Reynolds numbers could be caused by misinterpretation of hydraulic radius as defined by Eq. (1). Instead, they used flow height, which amplified the actual values. Another contributing factor could be viscosity, for which no reference was made in their paper. Results show that flow velocity decreases with increased FPS and this effect is less pronounced at relatively high volumetric flow rates. Within FPS range of industrial interest, i.e. 50–65% solids by weight, Reynolds numbers indicate that the flow is either transitional or turbulent. Beyond FPS of 60%, the flow is classified as transitional for the complete range of volumetric flow rates studied. Up to 30% solids, increasing the volumetric flow rate increases the degree of turbulence. When FPS is 50% or higher, the positive effect of volumetric flow rate on turbulence is relatively small. Fig. 9 illustrates the variation of the boundary flow rate, i.e., Re ¼ 2000, with FPS for a slope angle of 17°. While the flow is classified as turbulent at 940 L/h and higher flow rates with 50% solids by weight, the boundary flow rate became 1150 L/h at 54% solids and 1480 and 2030 L/h for 58% and 62% solids, respectively. These results show that turbulence can be depressed by increasing FPS and are in conformity with the findings of Abdinegoro and Partridge (1979). Although a statistical analysis showed that Reynolds numbers and performance could not be directly correlated, it was also found that performance at low Reynolds numbers, i.e., transitional flow conditions, was considerably better (Erg€ un and Ersayın, 1998). The lack of direct
4.1. Flow conditions over a pinched sluice With regard to characterization of flow over pinched sluices on the basis of Reynolds numbers, there are two contradictory data available in the literature. While Abdinegoro and Partridge (1979) found that Reynolds numbers varied between 1200 and 5000 depending on operating conditions, Subasinghe and Kelly claimed that
Fig. 9. Variation of transitional to turbulent flow boundary with flow rate and FPS.
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correlation is thought to be due to the interaction between stratification and splitter position. Froude numbers varied between 2.6 and 6.7. This range of Froude numbers is classified as supercritical and confirms the earlier finding by Subasinghe and Kelly (1984). Although Froude numbers slightly decrease with the increasing volumetric flow rates, this does not change the flow region defined by this criterion. An increase in Froude numbers could be interpreted as an increase in the tendency to form turbulent eddies. As expected, decreased velocity originating from increased solids concentration results in a decrease in Froude numbers. No significant changes in Froude numbers were observed when solids concentration was increased beyond 50% solids. It may be suggested that the Froude number, which was originally developed for flow characterization of water in open channels, might not be a suitable criteria for slurries containing a high proportion of solids. The Froude number is defined as the ratio of inertial (velocity dependent) forces to gravity forces. When it is applied to the slurry flow, an increase in flow height with solids concentration is interpreted as an increase in gravitational force and, consequently, despite an increase in velocity, creates negligible change in the Froude number. 4.2. Mean flow velocity modelling None of the three mean velocity equations tested on experimental data provided a satisfactory fit. This failure initiated a study to develop a modified equation to predict mean flow velocities accurately. The laminar flow equation was eliminated from the search for a better model due to the lack of both theoretical and practical grounds to pursue further. Since the Manning equation was originally developed for water or very dilute suspension over an open channel, its failure could be due two factors: (1) narrowing of the sluice in the direction of flow, which could increase the resistance and slow down the flow, and (2) presence of a high proportion of solids, which could create a similar effect by increasing the friction between the slurry and sluice surface. To eliminate the effect of solids in the slurry, its fit to water only test data was separately examined. Initially, the roughness coefficient for the galvanized surface was used; this resulted in a very large overestimation of the actual velocities. The next step was to define whether the fit could be improved by manipulating the roughness coefficient in the equation. Using a non-linear regression algorithm, the best-fit value of the roughness coefficient was calculated as n ¼ 0:00229, which was approximately six times lower than the original value of n ¼ 0:014. Although this modification improved the fit, there remained large deviations between the measured and calculated velocities. Fig. 10 shows that such modification has a tendency to
Fig. 10. The fit of the Manning (Vm ) and smooth turbulent (Vs ) flow equations with a best-fit roughness coefficients to the mean flow velocities of water only test data.
underestimate at low velocity and overestimate in high velocity range. The same type of approach was also tested on the data from the tests with slurry flows. It was assumed that the roughness coefficient would be dependent on FPS and, a separate best-fit roughness coefficient was calculated for each FPS condition. The quality of fit is illustrated in Fig. 11. Although the fit was improved, the deviations were still large. However, it should be noted that the quality of fit increased with FPC. While the regression coefficient, R2 , for water only test data was 0.5, it became 0.71 for 60% solids. A similar methodology was also applied to improve the quality of flow velocity predictions obtained using the smooth turbulent flow equation (Eq. (5)). Although this equation provided the closest estimates to the measured velocities, it overestimated the actual values when the roughness coefficient for galvanized steel ðj ¼ 0:4Þ was used. Its best-fit values for water only tests and for each FPS condition were separately calculated. Improved fit provided by this procedure is illustrated in Figs. 10 and 12 for water only and slurry tests, respectively. In general, the fit was inferior to that with the
Fig. 11. The fit of the Manning equation to mean flow velocities when a separate best-fit roughness coefficient was calculated for each FPS.
L. Erg€un, S. Ersayın / Minerals Engineering 15 (2002) 437–446
Fig. 12. The fit of the smooth turbulent flow equation to mean flow velocities when a separate best-fit roughness coefficient was calculated for each FPS.
Manning equation. However, similar trends were observed, i.e., the fit was improved as FPS increased. The regression coefficient (R2 ) increased from 0.41 for water only tests to 0.50 for the tests with 60% solids in feed. Since the Manning equation showed better potential for improvement, the rest of the study focused on further improving the predictions of this equation. Failure of the Manning equation with a modified roughness coefficient to predict mean flow velocities from water only tests implied that a narrowing effect could not be simulated by increasing this parameter. However, results also showed that it had a very good potential for reflecting the changes in FPS. Therefore, it was considered as a compact parameter defining the effects of solid concentration, particle shape and specific gravity. As a second step toward developing an improved equation, it was suggested that the narrowing effect could be embedded into the equation by manipulating the exponent of hydraulic radius, m, which is normally equal to 2/3. The best-fit values of both parameters were calculated using a non-linear regression algorithm. As expected, m varied within a short range of 0.364–0.395. Since initially it was assumed that m was a compact parameter defining the geometry of pinched sluices, its mean value, 0.379, was considered a suitable value for the sluice used in the tests, and the best-fit values of the roughness coefficient were re-calculated. Its fit to experimental data was beyond any expectation (Fig. 13). The regression coefficients, R2 , calculated separately for each solid concentration varied between 0.996 and 0.998. The roughness coefficient, km , also exhibited a meaningful relationship decreasing with the increase in solid concentration. However, it was thought that Bagnold’s k would be a more suitable indicator of such an effect since it is a compact parameter including % solid, shape and size distribution. Its relationship with km was examined and it is found that a linear relationship between the two exists, if the data from the water only tests
445
Fig. 13. The fit of the modified Manning equation (Eq. (9)) to experimental mean flow velocities.
were ignored (Fig. 14). The following velocity equation was obtained for the material used in the test by substituting km with the linear equation defining its relationship with k: pffiffiffiffiffiffiffiffiffi V ¼ ð310:67 29:929kÞR0:379 sin a: ð9Þ Eq. (9) did not deteriorate the fit obtained when km was used instead. It is believed that considerable progress could be made by determining the model parameters of Eq. (9) by using feed materials having different k, and pinched sluices of different geometry. It is expected that m will be coefficient for a given geometry and will not change even when the material characteristics are changed. On the other hand, pinched sluices of different geometry are expected to produce different m values probably decreasing, for example, with the length. 4.3. Velocity profile Flow characteristics affect the quality of stratification. Performance, however, also depends on the position of the splitter and the trajectory that the slurry
Fig. 14. The relationship between the roughness coefficient of the modified Manning equation ðkm Þ and linear particle concentration ðkÞ.
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L. Erg€un, S. Ersayın / Minerals Engineering 15 (2002) 437–446
follows right after the edge of a sluice. This in turn is a function of vertical velocity profile that exists within the slurry. Prediction of performance requires coupling of all the knowledge. Unfortunately, information regarding vertical velocity profile is scarce. Available data indicate that, unlike other flowing film devices, the zone of maximum velocity is near the middle of the slurry (Abdinegoro and Partridge, 1979). It is also known that the logarithmic velocity profile developed for water flows in open channels could not be used for slurry flows containing a large proportion of solids (Subasinghe and Kelly, 1984). Therefore, further study is needed to define what happens after the slurry leaves the edge of a sluice. Particle image velocimetry appears to have a potential as a measuring device to fill this knowledge gap.
5. Conclusion It was found that volumetric flow rate, solids concentration and sluice angle have a significant effect on flow height. It increases with flow rate and FPS, but decreases with an increase in sluice angle. Within the range of volumetric flow rates studied, the flow is classified as turbulent when pulp solids concentration is below 50% by weight. Above this limit, the flow is either transitional or turbulent. Froude numbers indicated that, for the complete range of test conditions, the flow is supercritical, implying that it has a tendency to form eddies. Such behaviour is in conformity with visual observations made during test work. Since specific value of Froude number varies within a short range, its usefulness as a criterion is questionable. The mean velocity of flow decreased with increase in solids concentration and this effect was more pronounced above 50% solids by weight. As expected, the mean velocity increased with the volumetric flow rate, but slope angle did not have a strong effect on it. Neither Manning nor smooth turbulent flow equations provided a satisfactory fit to the experimental mean velocities, even when the parameters defining roughness were varied to improve their fit. A modified form of the Manning equation is proposed in this study to define the mean flow velocity over a pinched sluice. It provided excellent fit to the experimental data. It appears that it could also be used for the determination of mean flow velocities on a pinched
sluice of different geometry, operating at a wide range of conditions.
Acknowledgements The authors would like to thank The Scientific and Technical Research Council of Turkey for the financial support through the Project No. YBAG 0069.
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