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Ind. Eng. Chem. Res. 2005, 44, 1284-1290
Model of the Solids Deposition in Hydrotransport: An Energy Approach Dmitri Eskin* and Brian Scarlett† Particle Engineering Research Center, University of Florida, P.O. Box 116135, Gainesville, Florida 32611
A theoretical approach to calculating the deposition velocity is proposed. It is based on the hypothesis that the turbulent energy is partially spent on suspending the solids against gravity and on solids-liquid viscous friction. Only two constants of the theoretical equation are identified from the experimental data. The calculated results are compared to both the experimental data and the results obtained by other correlations. The accuracy of the results obtained by the derived correlation is sufficient for engineering applications. Introduction 1.1. Current State of the Researches on Solids Deposition. The slurry deposition velocity is a critical velocity indicating the starting point of the moving-bed regime. The velocity corresponding to the minimum pressure gradient is usually somewhat higher than the deposition velocity. Because the deposition velocity determines the lower limit to the operating velocity for most hydrotransport pipelines, its prediction is important for pipeline design. There are many correlations for estimating the deposition velocity. Durand1 proposed the following equation:
VD ) FLx2gD(s - 1)
(1)
where D is the pipe diameter, g ) 9.81 m/s2 is the gravity acceleration, s ) Fs/FL, where FL is the liquid density, Fs is the particle density, and FL is the pipe Froude number in the deposition regime. Durand’s equation can be derived from the balance of the forces acting on the granular layer:2 coulombic friction on the pipe bottom and hydrodynamic friction on the granular bed free surface. To take into account the influence of the slurry concentration, Wasp et al.3 modified Durand’s equation as follows:
VD ) 4.0(ds/D)1/6c1/5x2gD(s - 1)
(2)
where c is the mean slurry volumetric concentration and ds is the particle size. For particles with diameters close to 0.5 mm, Wilson et al.4 suggested that the (Fanning) fluid friction factor fL should be used in the correlation for FL (see eq 1):
FL ) (0.045/fL)0.13
(3)
Note that it is difficult to compare the different correlations for the deposition velocity because in most cases they assume an average particle size and do not account for solids polydispersity. We found in the literature only one approach to the calculation of the velocity Vd that to some extent takes polydispersity into account. The * To whom correspondence should be addressed. Tel.: (352) 392-6750. Fax: (352) 846-1196. E-mail:
[email protected]. † Deceased.
method based on the Durand concept1 was proposed by Gillies and Shook for hydrotransport of sands containing a significant fraction of fines (usually, submicron- and micron-sized particles). The authors5 considered such a slurry as a two-component mixture of an equivalent liquid consisting of water and fines and sand particles. In this case, the viscosity of the equivalent liquid should be measured and used in calculations instead of the water viscosity, while fines should be excluded from consideration in calculating the mean particle size. According to Gillies and Shook, the “fines” fraction is defined arbitrarily as that below 74 µm. It is clear that this approach has a restricted area of applicability. There are also a number of other empirical correlations such as equations of Zandi and Covatos6 or Kokpinar and Gogus7 containing several experimental coefficients. It is noticeable that most of correlations for the deposition velocity are based on the equation of Durand (eq 1). Therefore, improvements are usually reduced to developing complicated correlations for the Froude number FL. This simple approach does not explain the real turbulent mechanism of the solids suspension. The major deficiency of Durand’s model is that it is based on the axial momentum balance, while the radial balance determines the solids deposition. The radial momentum balance requires introduction of a model of turbulence that causes a driving force suspending solids against gravity. Thus, the methods based on Durand’s approach do not reflect the physics of the solids deposition that restricts their applicability. Most of those correlations contain a number of empirical coefficients and therefore are reliable within a certain range of experimental data only. In the present paper, we derive the physically justified correlation for Vd. The constructive energy approach to calculating the deposition velocity was proposed by Oroskar and Turian.8 According to this approach, the work that the turbulence performs by suspending particles against gravity equals some fraction of the turbulence kinetic energy. However, the authors accepted several unjustified assumptions that did not allow them to obtain a reliable theoretical solution for the deposition velocity. At first, they assumed that the average velocity of turbulent fluctuations equals the friction velocity. This approximation is valid for a pure liquid; however, there is no confirmation that it is correct for a dense slurry. To calculate the average length of a turbulent eddy, the authors used the expression derived by Davies9 that is
10.1021/ie049453t CCC: $30.25 © 2005 American Chemical Society Published on Web 12/17/2004
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1285
Figure 1. Diagram of the turbulence structure in a pipeline slurry flow.
valid for a turbulent flow of a pure liquid in a pipe. There is no proof that it is correct for a turbulent slurry flow. Moreover, Oroskar and Turian8 calculated the turbulent energy per unit of a liquid volume by assuming that particles are only a passive medium. This is not true for dense suspensions because solids also transport momentum during turbulent fluctuations. To calculate the particle settling velocity in a dense medium, the authors8 employed the correlation where the velocity is inversely proportional to the slurry porosity of unknown power m that was supposed to be determined from experiments. This correlation does not provide reliable results because the power m varies in a wide range and depends on the solids concentration, particle size, and density.10 Nonetheless, analysis performed by Oroskar and Turian8 was important because they identified the variables, which determine the deposition velocity. On the basis of 357 experimental data points, the authors derived the correlation that provides rather high accuracy (the maximum deviation was 21.82% within the range of the experimental data8):
Vd ) 1.85xgds(s - 1)c
0.1536
0.3564
(1 - c)
(
() ds D
-0.378
)
DFLxgds(s - 1) µL
×
0.09
x0.3 (4)
where x is the fraction of turbulence eddies that have the vertical component of velocity that is higher than the particle settling velocity8 and µL is the dynamic viscosity of the liquid. Our estimates show that the fraction x is very close to unity for relatively dense slurries (c > 0.05-0.1) and may be excluded from consideration. Equation 4 contains six experimental coefficients; therefore, it is almost purely empirical correlations. Note that besides the paper in ref 8 the literature review revealed several interesting theoretical works on solids deposition that analyze the turbulence mechanism of the solids suspension. Davies11 considered the force balance acting on a particle in a turbulent flow. According to his model, the particle drag force due to sedimentation equals the force associated with the dynamic pressure caused by a turbulence fluctuation. The latter force is calculated based on the fluid fluctuation velocity. However, the relative fluid-particle fluctuation velocity has to be employed for accurate calculations. Another important assumption of this model is the hypothesis that only the eddies of length equal to the particle size ds contribute to the particles suspension. In our opinion, an assumption that eddies that are
equal or larger than ds contribute to the suspension would be more plausible. The structure of the correlation for the deposition velocity obtained by Davis11 is similar to the structure of eq 4 and provides approximately the same accuracy.11 Note that Davis’s analysis11 was important for an understanding of the solids deposition mechanism. It is also important to mention the significance of the theoretical analysis of Walton,12 who considered the particle eddy diffusivity in a turbulent flow by applying dimensional analysis to the major equations describing the turbulent dispersion. His correlation for the deposition velocity provides the results, which are close to those by the Davis correlation.11 1.2. On the Turbulence Structure in a Pipeline. The model developed in this paper is based on the Kolmogoroff representation of a turbulence structure. Many researchers describing turbulence in different apparatuses have successfully employed the Kolmogoroff model of isotropic turbulence. According to this model, the turbulence can be considered as a spectrum of turbulent eddies (fluctuations) gradually decreasing in size. The directions of these fluctuations are random. The schematic of the turbulence structure in a pipeline slurry flow is shown in Figure 1. The Kolmogoroff equation determining the spectral energy density of energetic eddies is written as13
E(k) ) χturb2/3k-5/3
(5)
where k ) 1/ι is the turbulence eddy wavenumber, ι is the turbulence eddy size, χ is the coefficient, and turb is the turbulence energy dissipation rate per unit mass. By integration of eq 5 over the whole range of eddy wavenumbers k from the smallest to the largest one, the total kinetic energy of turbulence per unit mass can be obtained. The smallest wavenumber corresponds to the largest turbulence eddy that for a pipeline flow can be approximated equal to 10% of the pipe diameter.14 The largest wavenumber corresponds to the smallest (inner) turbulence scale that is determined as15
ι0 ≈ (νL3/turb)1/4
(6)
where νL is the kinematic viscosity of a liquid. In a turbulent flow, there are no eddies that are smaller than this inner turbulent scale. The kinetic energy of the smallest turbulent eddies is transformed into heat because of their laminar friction with ambient liquid. Particles transported in a pipe tend to settle on the bottom of the pipe under action of gravity. The settling is prevented by turbulence dispersion. Solid particles
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are involved in chaotic motion by being entrained by turbulence eddies. Particle fluctuation velocities differ from velocities of turbulent eddies because of solids inertia. Note that only those turbulent eddies, the sizes of which are larger than the particle size, contribute to the solids dispersion. Thus, both turbulent eddies and particles are involved in a complex three-dimensional motion, leading to the stratification of solids across the pipe. In the present paper, we restricted our analysis of slurry dynamics to the saltation regime only. This allowed us to develop the correlation for the slurry deposition velocity based on the energy balance.
The turbulent energy dissipation rate for the case of the developed turbulence is approximately equal to the total energy dissipation rate per unit mass that is calculated as
tot )
Pw ) ms
s-1 s-1 gnvy ) Fsc gvy s s
U2 dp ) 2Fslfsl dx D
where n ) 6c/πds is the numerical particle concentration and vy is the absolute vertical velocity of a particle:
msg ) FLcD(πd2/8)vt2
(8)
The velocity vy does not equal the particle settling velocity because the settling solids displace the liquid, causing its flow in the opposite direction to the velocity vL. Then the vertical particle velocity is calculated as
vy ) vt - vL
(9)
where vt is the particle settling velocity calculated on the basis of the equilibrium of gravity and the particle drag force. From the condition of continuity for a mixture, one can get the equation for the liquid vertical velocity as (for example, work by Karabelas16)
vL ) vtc
(10)
(11)
After substitution of eq 11 into eq 7, it takes the form
s-1 gvt s
Pw ) Fsc(1 - c)
(12)
To suspend solids, the total power developed by turbulence eddies for chaotic displacement (dispersion) of the flow elements must be significantly higher than the power Pw. The total power developed by eddies equals the turbulence energy dissipation rate turb. The fraction of turb that is spent on particles suspension consists of two parts: (1) the power transferred from turbulent eddies to the solids and (2) the energy dissipation rate due to the liquid-solids friction Φ. Thus, the equation of the energy dissipation rate balance is written as
Pw ) R(Fslturb - Φ)
tot ) 2fsl(U3/D)
(16)
One of the many known correlations or models (for example, ref 6 or 17) may be used to calculate the friction factor. Note that the differences in the friction factors obtained by different correlations are relatively small compared to the differences in the deposition velocities calculated by different methods. In our calculations, we used the correlation derived by Turian and Yuan17 for the saltation flow regime:
fsl ) fL + 0.9857c1.018fL1.046cD∞-0.4213
[
v∞2
(13)
where R is the empirical constant, turb is the turbulence energy dissipation rate per unit mass, and Fsl ) Fsc + FL(1 - c) is the slurry density.
]
Dg(s - 1)
-1.354
(17)
where cD∞ is the drag coefficient of a particle settling in a pure liquid, fL is the flow friction factor for a pure liquid, and v∞ is the particle settling velocity in a pure liquid. The friction factor fL can be calculated by the Blausius equation (for example, ref 18):
fL ) 0.079/Re0.25
Then the particle vertical velocity is
vy ) vt(1 - c)
(15)
where fsl is the slurry flow friction factor. Then the total energy dissipation rate per unit mass tot is
(7)
3
(14)
where dp/dx is the pressure gradient and U is the superficial slurry velocity. The pressure gradient can be calculated as
2. Model Let us calculate the specific power (per unit volume) required for suspending particles against gravity:
U dp Fsl dx
(18)
where Re ) UD/νL is the Reynolds number for a pure liquid flow in a pipe. Note that eq 17 was not used by Oroskar and Turian for deriving eq 4, which is what we used for comparative analysis in this paper (see the Calculation Examples and Discussions section). The energy dissipation due to the solids-liquid friction has been estimated based on the following model. Motion of a particle in a turbulent eddy is characterized by some particle-liquid velocity difference. Thus, the particle moves under action of the local average drag force:
F ) FLcD(πds2/8)urel2
(19)
where cD is the particle drag coefficient and urel ) uL us is the local relative liquid-particle velocity, in which uL and us are the local liquid and particle velocities, respectively. Then the energy dissipation rate due to solids-liquid friction per unit volume can be estimated as
Φ ∼ Fλξn
(20)
where λ is the average length of a particle free of motion
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1287
in a turbulent eddy and ξ is the average frequency of turbulent fluctuations. Because we consider dense slurries, the average distances between particles are small. The average clearance between particles in a particulate system is calculated as19
δ ) [(clim/c)1/3 - 1]ds
(21)
where clim is the theoretical solids packing concentration (usually, clim ≈ 0.6). Estimates show that even at relatively low solids concentration the average particle-particle clearance is so small that particles constantly collide with each other. For example, if the solids volumetric concentration is c ) 0.075, the average clearance between particles equals the particle diameter. As particles moving in a turbulent eddy collide with one another, the trajectories of both particles change significantly; therefore, we assume that the average length of a particle fluctuation caused by a turbulent eddy is proportional to the average clearance between particles, λ ∼ δ. Note that this mechanism of energy dissipation plays a significant role if particle fluctuations are rather large, i.e., when the solids concentration is relatively low and/ or particles are not very small (larger than 50-100 µm). Otherwise, the dense slurry consisting of small particles can be considered as an equivalent liquid where particles follow the turbulent fluctuations with very small (almost zero) lags because clearances between particles are smaller than those of the Kolmogoroff turbulence scale.20 To apply eq 19, one needs to determine the local relative liquid-particle velocity. Despite this velocity not being constant, it is reasonable to assume that the energy dissipation rate can be calculated on the basis of the average value. Let us employ Levich’s approach21 for estimating the maximum local relative liquid-particle velocity caused by isotropic turbulence. Following this model, we assume that the average local relative velocity is proportional to the maximum local relative velocity. The equation of motion of the particle interacting with the turbulent eddy in a coordinate system of moving liquid is written as21
durel duL 1 ) (Fs - FL)V -F Fs + F L V 2 dt dt
(
)
(22)
where V is the particle volume. The acceleration of the particle durel/dt is estimated as21
durel urel2 ∼ dt l
(23)
where l is the scale of the turbulent eddy entraining the particle. The liquid acceleration duL/dt is estimated as21
duL turb2/3 ∼ 1/3 dt l Substituting eqs 23 and 24 into eq 22 results in
(24)
urel2 turb2/3 πds2 2 1 Fs + F L V ∼ (Fs - FL)V 1/3 - FLcD u 2 l 8 rel l (25)
(
)
Note that this equation is correct only for flows with low solids concentration because it is based on the consideration of pure liquid suspending separate particles. However, because the term Φ in eq 13 rapidly decreases with an increase in the solids concentration, the error due to the inaccuracy of calculating Φ for high solids concentrations is negligibly small. The maximum value urel is estimated from the condition
durel/dl ) 0
(26)
Performing the routine mathematical treatment, we obtain
( )( )
ds1/3 Fs - FL ∼ umax rel cD1/3 F + 1F s 2 L
1/2
1 Fs + F L 2 FL
1/3
turb1/3 (27)
In the first approximation, the drag coefficient cD can be estimated by the Stokes equation with the correction on the high solids concentration:22
cD )
24 (1 - c)-2.65 Res
(28)
where the Reynolds number is based on the local relative liquid-particle velocity. Substituting this cD into eq 27 and carrying out simple mathematics, we get the estimate for the maximum relative velocity that in our case evaluates the average local relative liquid-particle velocity:
urel ∼ umax rel ∼ ds
( ) turb νL
1/2
[ ] (s - 1)3 1 s+ 2
1/4
(1 - c)1.32
(29)
To apply eq 20, we need to evaluate the average frequency of turbulent fluctuations, which according to Levich21 is
ξ ∼ turb1/3/D2/3
(30)
Substituting eqs 28 and 29 into eq 19, we obtain the estimate of the local particle-liquid drag force. Then, after substituting the estimates for the drag force, the length of the average particle fluctuation (eq 21), and the average frequency of turbulent fluctuations (eq 30) into eq 20 and performing the routine mathematical treatment, we obtain the equation of the energy dissipation rate due to particle-liquid friction:
[( )
Φ ∼ FLc
clim c
1/3
]
[ ]
- 1 (1 - c)1.32
(s - 1)3 1 s+ 2
1/4
Re5/2
νL3 D4 (31)
As one can see, the dissipation rate Φ does not depend on the particle size because we employed the Stokes equation for the particle drag coefficient. Equation 13 of the energy dissipation rate balance for the critical (deposition) flow regime takes the form
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{
Vd3 c(1 - c)(s - 1)gvt ) R 2(c(s - 1) + 1)fsl D βc
[( ) clim c
1/3
]
- 1 (1 - c)1.32
[ ] (s - 1)3 1 s+ 2
1/4
Re5/2
}
νL3 D4
(32)
where β is the empirical constant. Identification of the parameters R and β based on the set of the experimental data of Gillies and Shook23 leads to the following nonlinear equation:
{
Vd3 c(1 - c)(s - 1)gvt ) 0.27 2(c(s - 1) + 1)fsl D 16c
[( ) clim c
1/3
]
- 1 (1 - c)1.32
[ ] (s - 1)3 1 s+ 2
1/4
Re5/2
}
νL3 D4
(33)
To apply this equation, we need to calculate the particle settling velocity. We tested several correlations for vt as well as for the particle drag force coefficient.10 The Andrews and O’Rourke correlation24 for the particle drag force coefficient was selected:
c′D )
[
2/3
Rep 24 (1 - c)-2.65 + (1 - c)-1.78 Rep 6
]
(34)
where Rep is the Reynolds number calculated on the basis of the settling velocity. This correlation provided the best agreement of the calculated deposition velocity with the experimental data. Note that the settling velocities calculated by using this equation are in good agreement with those obtained by the method of Mitzmager et al. (see Kokpinar and Gogus7). Our estimations show that the Reynolds number Rep is small enough; therefore, the contribution of the second term of eq 34, which takes into account high Reynolds number effects, is relatively low. The simple analysis shows that, if the relative liquid-particle velocity is small (the Stokes equation is applicable for the particle drag force estimation), the settling velocity can be accurately calculated as if a liquid is quiescent. Thus, the application of eq 34 to particles in a slurry flow does not lead to significant errors in our case. Note that within the approach developed we do not take into account an influence of the particle oscillations on the settling velocity. In dense slurries, a particle oscillates because of the turbulence energy dissipation on a microscale,25 and the effect of this phenomenon on particle settling is an important subject for future investigations. The model developed is in line with the most recognized approach to calculating the solids concentration distribution across a pipe.16 In addition to our method, this approach considers solids equilibrium in a turbulent flow, which is represented as a mass balance of the downward particle mass flux caused by gravity and the opposite flux due to turbulent dispersion. The equation expressing this mass balance is written as16
γs
dc + vtc(1 - c) ) 0 dy
(35)
Here γs is the solids diffusivity. The concentration c in this equation is a local volumetric concentration, which is a function of the vertical coordinate y. The solids diffusivity characterizes the particle transport due to the turbulence fluctuations. In a recent paper, Kaushal et al.26 determine the solids diffusivity as the liquid diffusivity multiplied by a coefficient, which is a function of the mean solids concentration. The liquid diffusivity increases with an increase in the turbulence intensity, and it is a linear function of the friction velocity, which, in turn, is a function of both the superficial flow velocity and the liquid friction factor. It is noticeable that in our model of the solids deposition for characterizing the turbulence intensity we use the total energy dissipation rate tot playing the role, which is similar to the role of the liquid diffusivity in models of the solids stratification (for example, ref 16 or 26). Moreover, in our deposition model, the effect of the solids concentration on the energy transferred from turbulent eddies to particles has also been taken into account. The fluid-solids dissipation term Φ (see eq 20) was used for this purpose. Thus, our model of solids deposition is in conceptual agreement with the recognized models of the solids stratification in turbulent flows, which are based on the clear physical principles. 3. Calculation Examples and Discussions To illustrate the model performance and prove its validity, we compared our results with some experimental data as well as the results obtained by other correlations for the deposition velocity. Note that the experiments and calculations were conducted by assuming monodisperse solids. As mentioned above, real slurries are polydisperse and are characterized by particle size distributions, which significantly vary for different solids. One of the most important parameters for determining the deposition velocity is the settling velocity, which nonlinearly depends on the particle size; therefore, the settling velocity calculated on the basis of the mean particle size is not an accurate parameter. Moreover, the slurry friction factor also depends on the settling velocity (see eq 17). Note that the polydispersity can be taken into account if the particle size distribution is known. For example, Karabelas16 considered settling of polydispersed particles in his paper on modeling of the solids concentration distribution across a pipeline. However, some complex correlations for the deposition velocity derived based on large arrays of experimental data provide satisfactory estimations even without taking polydispersity into account. Note that such correlations are valid within the range of the experimental results only. Several coefficients, which are usually employed in such correlations, allow fitting of the experimental data despite the uncertainty of the particle size distribution. Our model based on the physically justified assumptions should work satisfactorily for turbulent slurry flows if particles are not very large (e700-800 µm). This is because in the case of larger particles the uncertain size distribution becomes so wide that a monodisperse approximation does not work satisfactorily. Some of the empirical correlations cover larger particle size ranges because they are based on fitting of coefficients determined for those size ranges (for example, ref 7). In Figure 2, we showed the deposition velocities calculated by the different correlations found in the
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1289 Table 1. Comparison of the Deposition Velocities Measured by Different Researches with the Data Obtained by Our Model and Different Correlations Vd, m/s materials (origin of data)
D, m
ds, µm
c
s
exptl data
sand (Durand, 1952)
0.150 0.150 0.150 0.102 0.152 0.152 0.052 0.052 0.052 0.052 0.052 0.052 0.108 0.108 0.108 0.108 0.108 0.108 0.108
440 440 440 450 450 880 421 421 298 596 596 596 585 585 585 585 230 230 230
0.05 0.10 0.15 0.07 0.054 0.05 0.05 0.10 0.05 0.10 0.15 0.30 0.05 0.10 0.20 0.25 0.05 0.15 0.20
2.60 2.60 2.60 2.65 2.65 2.65 2.68 2.68 2.68 1.18 1.18 1.18 2.60 2.60 2.60 2.60 2.60 2.60 2.60
2.47 2.65 2.71 1.98 2.42 2.25 1.45 1.58 1.45 0.52 0.58 0.68 1.99 2.12 2.96 2.44 1.83 2.08 2.35
sand (Graf et al., 1970) sand (Avci, 1981) anthracite sand (Yotsukura, 1961)
literature as well as some experimental results23 versus the mean solids concentration for pipes of two different diameters (D ) 0.105 and 0.495 m). The mean particle size was ds ) 620 µm. Actually, we used these six experimental data points to identify the coefficients R and β in eq 32. Therefore, for these data, we can just state that our model satisfactorily reflects the changes of the deposition velocity with variations of the pipe diameter and the slurry concentration. One can see that the correlation derived by Oroskar and Turian (eq 4) provided a good agreement with the experiments. The results obtained by Wasp’s (eq 2) and Wilson’s (eq 3) correlations did not show a high accuracy. It is important to mention that, because our model has only two empirical coefficients, the small number of experimental data was sufficient for the parameter identification. In Table 1, we showed the deposition velocities measured by different investigators as well as the results calculated by our model and by the other methods. The experimental data were taken from the table presented by Kokpinar and Gogus,7 who collected the data from various literature sources. Nineteen experimental data points in the wide range of flow parameters are the sufficient number for the model
our model
Oroskar and Turian8
Wilson et al.4
Wasp et al.3
2.02 2.39 2.49 2.02 2.14 2.59 1.51 1.79 1.34 0.45 0.48 0.48 2.00 2.38 2.46 2.37 1.41 1.71 1.66
2.02 2.20 2.30 1.80 2.09 2.32 1.25 1.37 1.18 0.43 0.45 0.46 1.81 1.98 2.11 2.14 1.55 1.77 1.81
2.17 2.17 2.17 1.80 2.22 2.22 1.27 1.27 1.27 0.40 0.40 0.40 1.82 1.82 1.82 1.82 1.82 1.82 1.82
1.80 2.07 2.25 1.73 1.88 2.07 1.29 1.48 1.22 0.51 0.56 0.64 1.70 1.95 2.24 2.34 1.45 1.81 1.91
validation. We selected only data that mainly satisfy the limitations imposed on our model: the particles are not very large, and the solids concentration is not very low. One can see that in most cases our model provided better results than other correlations. The largest deviation of our prediction from experimental results was 29.6%. These results are promising because our model has a clear physical background and significant potential for improvement because it contains the empirical correlations for the particle-liquid drag coefficient and the slurry friction factor, which may be substituted with ones that are more accurate. It is also important that our model can consider polydispersity if the particle size distribution is known. Thus, we developed the theoretical model for the determination of the deposition velocity that describes the deposition process with an accuracy sufficient for engineering purposes. 4. Conclusions The energy approach for the prediction of the slurry deposition velocity has been developed. It was assumed that the power required to suspend particles in the turbulent flow in the saltation regime is a constant fraction of the total energy dissipation rate. The model developed accounts also for the energy dissipation due to particle-liquid friction in a turbulent flow. The average relative particle-liquid velocity was estimated by Levich’s model of the relative liquid-particle fluctuation motion in a turbulent flow. It was shown that conceptually the model developed is in agreement with the recognized approach of Karabelas to the solids sedimentation in hydrotransport pipelines. The two model parameters were identified based on the experimental data. The calculations showed that the model provides a relatively high accuracy that is sufficient for engineering applications. Acknowledgment
Figure 2. Deposition velocity calculated by the different correlations, and some experimental results versus mean slurry concentration for pipes of two different diameters.
The authors acknowledge the financial support of the Particle Engineering Research Center (PERC) at the University of Florida, the National Science Foundation
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(NSF) Grant EEC-94-0289, and the Industrial Partners of the PERC. Symbols c ) mean volumetric particle concentration cD ) particle drag force based on the average liquidparticle difference in turbulent flow c′D ) particle drag force based on the particle settling velocity in a slurry cD∞ ) particle drag force coefficient based on the particle settling velocity in a pure liquid clim ) solids packing concentration D ) pipe diameter ds ) particle diameter fL ) fluid friction factor fsl ) slurry friction factor k ) turbulence eddy wavenumber, 1/m l ) turbulence scale, m ms ) particle mass, kg n ) particle numerical concentration, 1/m3 Pw ) power required for suspending particles against gravity per unit volume, W/m3 Re ) pipe Reynolds number Rep ) particle Reynolds number based on the particle settling velocity in a slurry Res ) particle Reynolds number based on the particleliquid relative velocity s ) solids-liquid density ratio U ) superficial slurry velocity, m/s uL ) local liquid velocity, m/s urel ) local relative liquid-particle velocity, m/s us ) local particle velocity, m/s V ) particle volume, m3 Vd ) deposition velocity, m/s vL ) liquid vertical velocity, m/s vt ) particle settling velocity in a slurry, m/s vy ) particle vertical velocity, m/s v∞ ) particle settling velocity in a pure liquid, m/s Greek Symbols R, β ) empirical coefficients δ ) average clearance between particles in a slurry, m tot ) total energy dissipation rate per unit mass, W/kg turb ) total turbulent energy dissipation rate, W/kg γs ) solids diffusivity, m2/s ι ) turbulent eddy size, m λ ) average length of a particle fluctuation caused by a turbulent eddy, m µL ) liquid dynamic viscosity, Pa‚s νL ) liquid kinematic viscosity, m2/s F ) liquid density, kg/m3 ξ ) average frequency of turbulent fluctuations, 1/s Φ ) energy dissipation rate due to solids-liquid friction per volume unit, W/m3 Subscripts L ) liquid s ) solids sl ) slurry
Literature Cited (1) Durand, R. Basic Relationships of the Transportation of Solids in Pipelines, Proceedings. Minnesota International Hydraulics Convention, Minneapolis, MN, 1953.
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Received for review June 22, 2004 Revised manuscript received September 20, 2004 Accepted September 29, 2004 IE049453T
