Article pubs.acs.org/IECR
Two-Layer Model for Horizontal Pipe Flow of Newtonian and Non-Newtonian Settling Dense Slurries Mario R. Rojas and A. Eduardo Sáez* Department of Chemical and Environmental Engineering, The University of Arizona, Tucson, Arizona 85721, United States ABSTRACT: The steady-state flow of Newtonian and non-Newtonian dense aqueous slurries in horizontal pipes has been analyzed using a two-layer model, consisting of a top layer of flowing suspension and a bottom bed of stationary or moving particles. The coarse solids used have a wide range of particle density and particle size distributions. The fluids studied were designed to simulate U.S. Department of Energy Hanford site waste slurries. The most important changes from previous models include an independent settling analysis for different particle size fractions, effects of particle shape on settling velocity, and a new correlation for the turbulent particle dispersivity. The results indicate that the turbulent dispersivity of settling particles is sensitive to particle size and density. The model gives good estimation of the critical deposition velocity as the minimum of the pressure drop versus superficial slurry velocity relation. The existence of a stationary layer can be observed and predicted by the model under laminar and turbulent flow conditions.
1. INTRODUCTION The transport of slurries in horizontal pipes is a process with widespread application in practice. Even though it is generally desirable to operate at velocities that ensure complete suspension of solids, some processes operate under conditions at which a granular deposit occupies part of the cross section of the pipe. In fact, for slurries composed of dense particles, it might be attractive to operate in a regime in which a granular deposit exists. Typically, these processes involve slurries with wide particle size distributions (PSDs) in turbulent flows. Several flow regimes may occur in horizontal pipeline flow of slurries, depending on slurry characteristics and flow conditions: (i) fully suspended flow, with all solids being transported by the flowing fluid; (ii) flow with a moving bed, in which a granular deposit slides on the bottom section of the pipe; and (iii) flow with a stationary bed, in which a granular deposit of settled particles occupies the bottom of the pipe. It is also possible to have coexisting moving and stationary beds. Analysis and modeling of these flows usually relies on mass, momentum (or force) balances applied to the flowing suspension and the moving and stationary beds separately. This approach, originally attributed to Wilson1 has been widely employed as a semiempirical basis to develop models to predict pressure drops and solids distribution in slurry flows. Gillies et al.,2 among others, proposed a two-layer model to predict pressure losses for settling slurries in horizontal pipes. The two layers are a stationary solid deposit at the bottom of the pipe, and a suspension (Newtonian fluid) flowing at uniform velocity on top. This model was capable of representing solids concentration profiles over the cross section and their relation to the mean flow velocity and the settling particle velocity. The same research group has used the two-layer model in different applications, such as the study of frictional losses in concentrated slurry flows3 and the modeling of heterogeneous slurries at relatively high fluid velocities,4 with successful results. Doron and Barnea5 proposed a three-layer model as an extension of their own two-layer model6 and other published models. They proposed that the main limitation of the © 2012 American Chemical Society
two-layer model is its inability to predict accurately the existence of a stationary bed at low flow rates: in some cases when a stationary bed was observed, model results indicated flow with a moving bed. This also leads to reduced reliability of the pressure drop calculations for low flow rates at which a stationary bed can be expected. Using the three-layer model, the authors were able to quantify the critical deposition velocity as the limit when the stationary bed height approaches zero. The value obtained can be viewed as an upper limit for the critical velocity since, in practice, a bed layer can be considered to vanish when its height is of the order of the particle size. According to their results, model predictions are in fairly close agreement with the Turian et al.7 expression and the correlation proposed by Gillies et al.8 for critical velocity, which were derived from semiempirical analyses. The Doron and Barnea model incorporates a vertical solids mass balance to quantify the solid distribution in the pipe under steady-state conditions. The balance leads to a one-dimensional version of the sedimentation-dispersion equation, whose solution yields the vertical solids concentration profile in the cross section of the moving layer. The solids settling velocities and dispersion coefficients are vital for the accurate performance of this prediction. Gorji and Ghorbani9 applied two and three-layer model conceptualization to predict successfully the pressure losses in slurry pipeline flow. Their results show that, as long as there is a stable bed at the bottom of the pipe, the pressure losses are independent of the slurry velocity, which is not always the case. More recently, Matoušek10 presented a new two-layer model for pressure drop predictions assuming a stationary layer at the bottom of the pipe. This semiempirical approach follows the same analysis as previous two-layer models based on a force Received: Revised: Accepted: Published: 7095
July 29, 2011 March 6, 2012 May 2, 2012 May 2, 2012 dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
alumina and stainless steel particles have irregular shapes. PSDs range from 1 to 200 μm (Figure 2). The average particle size of the distribution corresponds to a hydraulic diameter, calculated from
balance with the incorporation of empirical correlations to calculate different parameters such as shear stresses and solid transport rates for engineering applications. The studies mentioned above are limited to Newtonian slurries. Models to predict solids transport and pressure drops in pipes for slurries with non-Newtonian fluids are in limited number in the literature, even though important applications such as drilling fluid transport rely on non-Newtonian matrices. Ramadan et al.11 proposed a modified three-layer model for such a case. The model considers carrier fluids with rheology represented by the power-law model and focuses on the calculation of solids transport. In this work, we modify the two-layer model to characterize the flow of dense/concentrated Newtonian and nonNewtonian slurries with broad PSDs in flow through horizontal pipes, in an attempt to represent real nuclear waste slurries of the U.S. Department of Energy Hanford site. Slurries that will be transported from underground tanks to the DOE Waste Treatment Plant have particular composition and rheological properties, including possible high salt and colloidal matter concentrations, and solids with broad PSD (below 200 μm) and densities (ranging from 2000 to 11000 kg/m3).12 For the case of Newtonian fluids, the most important changes of the model presented here with respect to previous works include independent settling analysis for different particle size fractions, effects of the shape of the particles on the settling velocity calculation, and a new correlation to represent turbulent particle dispersivities. We extend the model to non-Newtonian carrier slurries with a yield stress, using the Casson model to represent their rheology.
MM
HL
HH
11
138
86
25
150
9.8 2500 1000 1.0 1.00 1.00
7.4 2500 1000 1.0 1.00 1.00
8.7 3770 1000 1.0 0.45 1.17
9.3 7950 1000 1.0 0.90 1.15
3.0 7950 1000 1.0 0.90 1.15
60
60
50
60
40
π
(2)
and the volume-equivalent-sphere diameter or nominal diameter, calculated from the particle volume (V),
dn =
3
6V /π
(3)
Experimental Setup. The slurries were prepared in a 1500-L mixing tank connected to a flow loop system. During the experiment, the slurry is transported through the system by a single 15-hp/1800 rpm centrifugal pump (Georgia Iron Works) from the tank to the test pipe. The main section of the test pipe consists of 3-in schedule 40 stainless steel straight horizontal pipeline on which different pressure ports were installed. Pressure drops were measured with a differential pressure transducer over a pipe length of 5.7 m. The slurry is recirculated to the feed tank, and measurements start when the flow reaches steady state (approximately 30−60 min). Before it enters the main horizontal measurement section, the slurry flows through a Coriolis flow meter (Micro-Motion, F-series). A chiller connected to the mixing-tank was used to keep the temperature constant during the experiment (20 °C). All the tests were conducted starting with a relatively high superficial velocity of 3−4 m/s, and then the velocity was decreased in 0.15 m/s steps until a rise in differential pressure was detected, indicating the presence of a settled bed of particles; this point was considered as the experimental critical deposition velocity. Early tests were performed to ensure repeatability of results (not shown) yielding variations in pressure drop measurements lower than 10%. After the Coriolis flow meters, the slurry entered an electrical resistance tomography (ERT) probe (Industrial Tomography Systems) that recorded cross-sectional maps of the slurry electrical conductivity in real time. Since conductivity is a function of solids concentration, these maps yield a representation of the solids distribution over the cross section, including observation of the deposited bed layer.
simulant particle hydraulic diameter, dph (μm) solids content (vol%) particle density (kg/m3) carrier fluid density (kg/m3) carrier fluid viscosity (mPa s) particle circularity, c surface-equivalent-sphere to nominal diameter, dA/dn bed layer solid concentration, Cb (vol %)
(1)
4A p
dA =
Table 1. Properties of the Newtonian Simulantsa
LH
∑i n id pi2
where ni and dpi are the number and diameter of particles in a given size range, respectively. To include the effect of particle shape in drag coefficient calculations, we followed the approach presented by Tran-Cong et al.13 Since their correlation for particle drag coefficient is based on the definition of particle circularity (c) and surfaceequivalent sphere to nominal diameter ratio, these have been included in Table 1. The surface-equivalent sphere to nominal diameter ratio (dA/dn), is the ratio between dA, the particle diameter defined in terms of the projected area of the particle (Ap)
2. EXPERIMENTAL SECTION Materials. The fluids used were simulants designed to match specific physical properties, such as rheology and PSD, of actual waste slurries of the Hanford site.12 All fluids were aqueous-based and all experiments were performed at 20 °C. Glass (Spheriglass, Potters Industry), alumina (Washington Mills), and stainless steel 316 (Aemtek) were selected to represent the coarse particles in the experiments. These materials have densities of 2500, 3770, and 7950 kg/m3, respectively, and were qualified here as “low”, “medium”, and “high” particle density. Other properties of the simulants are listed in Tables 1 and 2. Examples of particle morphologies are shown in Figure 1. The glass particles are spherical, but the
LL
∑i n id pi3
d ph =
3. MODEL 3.1. General Equations. All multilayer models make use of friction coefficients as a way to calculate the shear stresses involved (pipe walls and interfaces) as well as drag coefficients to study the settling of the particles under turbulent conditions.
a Simulant identification (two-letter code): first letter identifies the particle density (low, medium, or high), and the second letter refers to relative average particle size.
7096
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
Table 2. Properties of the Non-Newtonian Simulantsa simulant particle hydraulic diameter, dph (μm) solids content (vol %) particle density (kg/m3) infinite shear viscosity (mPa s) yield stress (Pa) particle circularity, c surface-equiv-sphere to nom. diameter, dA/dn bed layer solid concent., Cb (vol %) a
LL1
LL2
LH1
LH2
MM1
MM2
HL1
HL2
HH1
HH2
27 9.7 2500 2.7 1.7 1.00 1.00 60
16 10.9 2500 2.2 4.8 1.00 1.00 60
116 8.4 2500 1.8 2.4 1.00 1.00 60
153 10.7 2500 2.6 4.4 1.00 1.00 60
83 9.5 3770 2.3 2.4 0.45 1.17 50
86 9.7 3770 2.4 4.7 0.45 1.17 50
23 9.5 7950 2.5 2.0 0.90 1.15 60
21 9.8 7950 3.3 5.1 0.90 1.15 60
156 3.9 7950 1.4 2.2 0.90 1.15 40
177 4.8 7950 2.0 5.2 0.90 1.15 40
Numbers refer to low yield stress (1) and high yield stress (2).
an irregular shape and broad size distributions are considered. Fluids with dense particles can also be difficult to model since they have a tendency to accumulate in the bottom of the pipe even at relatively high velocities. Initially, we formulated a three-layer model, allowing for the presence of a sliding bed between the stationary bed and the fully suspended flow region. However, our calculations showed that only two layers were formed and, in most situations, the bottom layer was stagnant. These facts were confirmed by experimental observation. For this reason, the model finally applied and presented below was a two-layer model in which the bottom layer could be either a stationary bed or a moving solids layer. Our model is based on mass balances and momentum balances along the flow direction over the cross section of the pipe. The contribution of each layer to the force balance consists of shear stresses at the pipe wall and at the interface between layers. More details about the formulation of the twolayer model for Newtonian fluids are presented elsewhere.2,5 Figure 3 shows a schematic description of the two-layer model. A bed of particles, occupying a surface area Ab on the
Figure 1. Micrographs of (a) LH, (b) HL, (c) MM, and (d) HH simulant particles.
Figure 3. Geometry and stress distribution in the two-layer model.
cross section, coexists with a fully suspended flow layer of surface area Ah. We consider the flow to be at steady state, and that the size of the two layers is independent of longitudinal position, which is in agreement with experimental observations. At steady state, the fully suspended layer carries all solids that are fed to the pipe. Owing to the presence of relatively large particles, it is expected that the solids concentration in the suspended layer will be a function of vertical position (y). Vertical particle transport is governed by the convectiondispersion equation, which balances the settling flux of particles with the turbulent dispersive flux,
Figure 2. Simulant number PSD (Newtonian simulants).
ε
Results are sensitive to the proper calculation of key parameters such as the particles settling velocity, drag coefficient, and particle dispersivity in the upper layer. The calculation of these parameters becomes more complex when coarse particles with
d2C dC +w =0 2 dy dy
(4)
where w is the terminal settling velocity of the particles, ε is the turbulent dispersivity, and C is the solids concentration 7097
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
The effective settling velocity of the particles used in the integration of eq 4 is
(expressed as solids volume fraction). Integrating eq 4 over the cross-section of the upper layer yields the average solids concentration in the upper layer, C D2 Ch = b 2Ah
∫θ
π /2 b
⎛ wD ⎞ exp⎜ − [sin γ − sin(θ b)]⎟cos2 γ dγ ⎝ 2ε ⎠
w=
where D is the pipe diameter, θb is the angle associated with the stationary bed location (Figure 3), and Cb is the concentration of solids in the stationary bed, used as boundary condition in the integration of eq 4 (Tables 1 and 2), along with a no flux condition at the pipe surface. We will consider that the PSD can be discretized into narrow ranges, each characterized by its own dispersivity and settling velocity. The calculation of settling velocity is one of the most important modifications proposed here. It includes effects of the solids concentration, turbulence intensity, and particle shape, which affect the estimation of the drag coefficient for the particles. Settling velocities were calculated considering narrow portions of the PSD separately as follows. First, the particle settling velocity in the absence of interparticle effects is calculated from the balance between gravitational, and buoyancy and drag forces from
(6)
(12)
dP = −τhS h − τiSi dx
(13)
dP = −Fmb − τbS b + τiSi dx
(14)
where Fb is the friction force exerted on the settled bed layer by the pipe wall. The shear stress τb in eq 14 is a dynamic shear stress that applies when the bottom layer slides along the pipe. The static friction force is proportional to a normal force that includes the weight of the settled particles and the transmission of normal stresses acting on the interface between the two layers. Integration of the normal forces acting on the surface of the pipe leads to6
( ) n
⎧ ⎛ D ⎞2 ⎡⎛ 2y ⎞⎛ π⎞ Fmb = η⎨2C b⎜ ⎟ (ρs − ρl )g ⎢⎜ b − 1⎟⎜θ b + ⎟ ⎝ ⎝ ⎠ ⎠ 2 2⎠ ⎣⎝ D ⎩
0.42
⎪
−1.16 ⎤
( ) dA Re i dn
⎥ ⎦
⎪
⎤ τS ⎫ + cos θ b⎥ + i i ⎬ tan ϕ ⎭ ⎦
(7)
⎪
The values of particle circularity (c) and dA/dn are provided in Tables 1 and 2 for each simulant tested in this work. Once the drag coefficients are known and values of w0i were calculated for each fraction of the PSD, we calculated a hindered settling velocity into the cluster for each of the fractions using the equation developed by Cheng16 ⎡ ⎤1.5 wi 1 − 2C h ⎢ 25 + 1.2d*′pi − 5 ⎥ = w0i 2 − 3C h ⎢⎣ 25 + 1.2d*pi − 5 ⎥⎦
⎪
(15)
where Fmb is the static friction force, η is the dry friction coefficient, yb is the bed height (Figure 3), ϕ is the internal friction angle, and g is the acceleration of gravity. The angle (θb) and perimeters (Sh, Sb, Si) in the previous equations can be expressed as functions of yb. Equations 5, 11, 12, and the equality of the pressure gradient between eqs 13 and 14 can be solved having yb, Ch, Uh, and Ub as unknowns for the case of a moving bottom layer. If the system exhibits a stationary bed (as is the case in most of our simulations), the friction force in eq 14 is lower than the static friction force (eq 15). In this case, Ub = 0 and eq 14 is no longer applicable. Under these conditions, eqs 5, 11, and 12 are solved for yb, Ch, and Uh, and then eq 13 can be used to calculate the pressure gradient. The systems of nonlinear algebraic equations for each case were solved using the iterative multivariable Newton’s method, programed in MATLAB.
(8)
where ⎡ (s − 1)g ⎤1/3 d*pi = ⎢ ⎥ d pi ⎣ ν2 ⎦
UhAh + UbAb = UA s
Ab
⎡ ⎞0.687 ⎤ ⎛ ⎥ ⎢1 + 0.15 ⎜ dA Re ⎟ i dA ⎥⎦ ⎢ c d ⎠ ⎝ n Re i ⎣ d ⎡ c ⎢1 + 4.25 × 104 ⎣
(11)
where τh and τi represent shear stresses acting on the surface of the pipe in contact with the fully suspended layer and at the interface between the two layers, respectively (Figure 3), and S represents perimeter. If there is a moving bottom layer present, the pressure gradient in eq 13 must equal the pressure gradient given by a momentum balance in the moving bed,
24
+
UhC hAh + UbC bAb = UC s sA
Ah
where CDi is the drag coefficient for particle size range i, calculated using the particle Reynolds number based on w0i (Rei), and s is the ratio between solid and fluid densities, ρs/ρl. Values of drag coefficient were obtained using the modified form of the Clift et al. correlation14 provided by Tran-Cong et al.,13 which takes into consideration particle shape. Turbulence effects on the drag coefficient were incorporated from the Brucato et al. correlation.15 The result is the relation C Di =
(10)
where Us is the superficial velocity of the feed slurry, Ub is the velocity of liquid moving with the moving bed (when present), and Cs is the feed solids concentration. 3.2. Newtonian Simulants. A momentum balance in the flow direction applied to the fully suspended layer leads to6
4(s − 1)d pig 3C Di
Ch
Steady-state mass balances in solid and liquid phases yield
(5)
w0i =
∑i (wiC hi)
(9)
and for the calculation of d*′pi, the kinematic viscosity of the liquid (ν) and the liquid density are replaced in this equation by the kinematic viscosity and density of the suspension, respectively. 7098
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research Wall shear stresses (τh and τb) are evaluated from 1 τ = ρU 2f 2
Article
scales of dissipative microscopic eddies, the theory predicts a thickening of the viscous sublayer, which tends to increase throughput velocity and thus promotes drag reduction. The Wilson−Thomas theory has been adapted to different rheological behaviors, including Casson fluids, for which it can been shown that the superficial velocity (U) achieved by the fluid for a given shear stress is related to the velocity of an equivalent Newtonian (Un) fluid by
(16)
where
ρ = ρs C + ρl (1 − C)
(17) 6
and friction coefficients are calculated from
f = αRe−β
⎛ ⎞ 1−ξ ⎟ + u* U = Un + 2.5u*ln⎜⎜ 2 1 ⎟ + ξ + ξ 1 ⎝ 3 3 ⎠ ⎡ ⎛2 1 ⎞⎤ × ⎢ξ(2.5 + 1.25ξ) + 11.6⎜ ξ + ξ ⎟⎥ ⎝3 ⎣ 3 ⎠⎦
(18)
where α and β are constants that depend of the flow regime: turbulent (α = 0.046, β = 0.2) and laminar (α = 16, β = 1). The Reynolds number in eq 18 is defined by
Re =
ρUD h μ
where ξ is the ratio of the yield stress to the wall shear stress (τc/τw), and u* is the shear velocity, calculated from
(19)
where the hydraulic diameter (Dh) is four times the crosssectional area of the layer divided by its perimeter. The interface shear stress is given by
u* =
1 τi = ρh Uh2fi (20) 2 where the friction factor is calculated from the Colebrook correlation, using the particle size as an equivalent roughness, ⎛ d 1 2.51 ph = −0.86ln⎜⎜ + 2fi Reh 2fi ⎝ 3.7D h
⎞ ⎟ ⎟ ⎠
Ah
(22)
(23)
D h2ρh τc μc2
(24)
The transition flow velocity is then calculated from Ret =
Us =
D h Utρh μc
dP = −τw(S h + Si) dx
(28)
For turbulent flow, the Newtonian superficial velocity (Un), is calculated using τi = τw in eq 20 to determine f i and then the nonlinear Colebrook’s eq 21 is solved to determine Un as the velocity in the Reynolds number. Once Un is known, U = Uh is calculated from eq 26. Once the velocity in the upper layer (Uh) is calculated, the total mass balance is employed to calculate the slurry feed velocity
where Ca is the Casson number, Ca =
(27)
Since the bottom layer is stationary, Ub = 0, and eqs 11 and 12 imply Ch = Cs. Knowledge of Ch allows us to determine the geometry of the bottom stationary layer directly from eq 5 (note that all geometric parameters in that equation can be related to yb, which is the only unknown that defines the geometry.) Application of the model proceeds via an iterative technique that starts by assuming a value for the pressure gradient. Given that the geometry of the system is defined, τw is calculated directly from eq 26. Since the yield stress is known, ξ = τc/τw is calculated. At this point, the effective viscosity of the Casson fluid is evaluated from μc μ= (1 − ξ )2 (29)
(21)
where τc is the yield stress and μc is the Casson viscosity. The model and calculation procedure depend on whether the flow is laminar or turbulent. The transition between laminar and turbulent flow is determined using the correlation19 0.4 ⎡ ⎛ Ca ⎞ ⎤ ⎟ ⎥ Ret = 1050⎢1 + ⎜1 + ⎝ 370 ⎠ ⎦ ⎣
τw ρ
We assume that the wall shear stress in this equation equals the shear stresses in the upper flow region; that is, τw = τh = τi. The momentum balance, eq 13, simplifies to
3.3. Non-Newtonian Simulants. A new approach was developed to treat non-Newtonian simulants. The model presented here assumes that the bottom layer is stationary, following experimental observation. In addition, independent measurements in a shear rheometer suggested that the shear rheology (shear stress τ vs shear rate γ̇) of the non-Newtonian simulants was well represented by the Casson model,12
τ1/2 = τc1/2 + (μc γ̇)1/2
(26)
UhAh A
(30)
If the velocity calculated from this equation is not equal to the inlet slurry velocity, a new pressure gradient is assumed and iterations proceed until convergence is achieved. Laminar Flow. The flowing (top) layer will have a plug-flow region because of the existence of a yield stress. In this case, eq 5 is not used, and the solids concentration is assumed to be uniform. From eqs 11 and 12 we conclude that (since Ub = 0) Ch = Cs. The upper layer (Ah) is approximated to be a circular region consisting of a core plug flow region and an annular frictional sublayer. The plug flow region diameter is expressed
(25)
In these equations, Dh is the hydraulic diameter of the top layer. In what follows, we present model equations and calculation procedures for each case separately. Turbulent Flow. The Wilson and Thomas17,18 theory for non-Newtonian fluids was adapted to the two-layer model. This theory considers the effect of variable fluid viscosity on the velocity profiles of the viscous sublayers in turbulent flow. Through the analysis of viscosity effects at the time and length 7099
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
the coefficient a in eq 33 depends on Reynolds number and Archimedes number of the particles as follows
as a function of the hydraulic diameter of the upper layer section from the stress distribution in a cylindrical pipe with a yield stress fluid, D τ Dp = h c τw
a = kAr nRe m
where n and m were found to be 0.41 and 0.75, respectively, k is a fitting constant, and Ar is the Archimedes number of the particles, defined by
(31)
Here we assume that the plug flow region occupies most of the effective area for flow in the top layer. The analytical solution for the velocity field in a pipe of circular cross section for a Casson fluid leads to the following relation for the average velocity ⎛ τ ⎞⎡ 8Uh 16 4 1 4⎤ = ⎜⎜ w ⎟⎟⎢1 − ξ + ξ− ξ ⎥ ⎦ ⎣ μ Dh 7 3 21 ⎝ c⎠
(34)
Ar =
gd p3(s − 1)ρh 2 μh 2
(35)
We extended the correlation to include both Newtonian and non-Newtonian simulant data (Figure 5). Note that particles
(32)
Combination of this equation with eq 30 yields a relation between the geometry of the top layer (represented by Ah and Dh) and the shear stress τw for a given slurry feed velocity Us. Use of this relation in combination with eqs 28 and 31 allows for the calculation of the top layer geometry and the pressure drop as a function of slurry feed velocity.
4. RESULTS AND DISCUSSION The application of the model presented above to experimental data required knowledge of turbulent dispersivities. In the original Doron and Barnea model, this parameter is calculated from Taylor’s equation20 ε = aD hu*
Figure 5. Particle dispersivity correlation for Newtonian and nonNewtonian simulants.
(33)
where a = 0.026. However, this approach does not appropriately take into account particle characteristics (density, shape, PSD). Analysis of our data suggested that the coefficient in Taylor’s equation is related to particle properties, and that there is a stronger dependence of the dispersivity on fluid velocity for our slurries than that represented by eq 33. It is important to recall that Taylor’s correlation applies to solutes in turbulent flow and not to settling particles. Our results indicate that turbulent transport of particles differs from transport of molecular species, which could be a consequence of increase of energy dissipation in eddies of size comparable to the solid particles. Figure 4 shows the dependence of the dispersion coefficient calculated by fitting our model to experimental data on pressure
with highly different values of densities and PSDs but with similar Archimedes numbers possess similar values of a, which supports the proposed form of the correlation. Turbulent dispersivities have been found to correlate with Ar and Re in previous works (albeit in different applications). For instance, Wen and Yu21 and Reganathan et al.22 used such a correlation to represent solids dispersion coefficients in liquid− solid fluidized beds. The two-layer model was used to predict the pressure drop over the 5.7 m of straight horizontal pipe for all the simulants using the proposed correlation for particle dispersivity. The results on pressure drop versus superficial velocity for the Newtonian stimulants suggest that the model is capable of predicting the observed trends in flow of concentrated slurries (Figures 6 and 7). At high fluid velocities, when the solid is fully suspended, the pressure drop decreases monotonically with a
Figure 4. Particle dispersivity as obtained from fitting the model to experimental data. Figure 6. Pressure drop as a function of superficial velocity for low density/small particle size Newtonian simulants. Solid lines represent the model. Dashed lines indicate the predicted critical velocity.
drops (detailed results shown below). Different relations were obtained depending on particle properties. We postulated that 7100
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
superficial velocity range in which a stationary bed was observed and predicted by the model. However, even when a solid deposit was observed, there is no evidence of a critical velocity for any of these two fluids even at relatively low fluid velocities. The two-layer model predicts a small bed layer that vanishes at approximately 0.5 m/s for both fluids. Experimental observations confirm that the pressure drop increases continuously with fluid velocity since the critical velocity approaches zero and most of the particles can be suspended with a minimum amount of shear. Under these flow conditions a steady state is difficult to reach (and to measure experimentally) and the bed layer is unstable. Therefore, any additional perturbation can help modify the flow pattern and as a consequence lift the solids remaining at the bottom of the pipe. The pressure drops are significantly different for the two fluids at low fluid superficial velocities because of their differences in yield stress and apparent viscosity. The difference becomes smaller as the velocity increases, which makes the viscous component of the flow less important and inertial effects predominate. Note that differences between the transition velocities to turbulent flow for both fluids because of their difference in rheological properties have predominant effects under laminar flow conditions. The hydraulic diameter approximation for the upper layer has proven to be a good approach for calculating the apparent Reynolds number, but in some cases, especially at low fluid velocities, our approach for the calculation of the transition velocity predicts laminar flow when the observed flow is turbulent. In those cases, the regime prediction fails because some solids start to accumulate more at the bottom of the pipe and when the hydraulic diameter reduces to a certain point, turbulence is achieved, and therefore, the bed size reduces again. As changes in bed height are small and this is a dynamic process that happens relatively quickly, there is actually a pseudo steady state in turbulent flow that can still be modeled by the two-layer model. Results for the non-Newtonian simulants with intermediate particle density and size (MM) present an interesting behavior in terms of their critical velocities (Figure 9). The fluid with the
Figure 7. Pressure drop as a function of superficial velocity for medium and high density simulants. Solid lines represent the model. Dashed lines indicate the predicted critical velocity.
reduction in velocity. Additional decreases in fluid velocity lead to the formation of a stationary or moving layer at the bottom of the pipe, whose growth as the velocity is reduced causes a decrease in suspension flow area and a consequent increase in the pressure drop. At low fluid superficial velocities, the pressure drop continues to rise as the bed thickness increases uniformly. The minimum observed in the pressure drop curves yields the critical deposition velocity of the suspension. Figure 6 shows that, for low particle densities (glass particles), the modified model proposed here predicts the pressure drops satisfactorily in the whole velocity range evaluated. For both simulants, the modified model predicts accurately the critical velocity (dashed lines). The minimum in the predicted curves is close to the point at which the model predicts that the stationary bed disappears with any additional increment in fluid velocity. For higher particle densities, the pressure drops are also well predicted by the model (Figure 7). The HL simulant has a relative small Arquimedes number (1.07) because of its small particle size; even so, it exhibits a behavior analogous to fluids loaded with bigger particles (MM). Here, the density of the particles plays an important role that can also be well described with the correlation proposed. Figure 8 shows the pressure drops of low density/small particle size non-Newtonian simulants (glass beads, LL). The
Figure 9. Pressure drop as a function of superficial velocity for medium density/medium particle size non-Newtonian simulants. Solid lines represent the model.
higher yield stress (MM2) exhibits a higher pressure drop at low fluid velocities. Although there is an important difference in the rheological properties for these two fluids, the critical velocities are fairly comparable. In this case, the flow is always turbulent, and the model once again gives an accurate representation of the experimental data.
Figure 8. Pressure drop as a function of superficial velocity for low density/small particle size simulants. Solid lines represent the model; no critical velocity was observed.
two simulants shown differ in rheological properties (Table 2). Both simulants exhibited a laminar flow regime in all the 7101
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
turbulent flow, the ones with higher yield stresses show consistently thicker bed layers at the same fluid velocity. In turbulent flow, having a higher viscosity and yield stress reduces the turbulent intensity by decreasing the Reynolds number in the upper layer. Figure 12 shows examples of ERT images along with the bed layer thickness prediction of the model for the low density
Fluids based on stainless steel particles (HH) present a high density and large particle size that pose a challenge to the modeling approach. However, as shown in Figure 10, the model
Figure 10. Pressure drop as a function of superficial velocity for high density/large particle size simulants. Solid lines represent the model; dashed lines represent the critical velocities.
Figure 12. Bed height predicted by the model as a function of superficial velocity for LH non-Newtonian simulants. ERT images are shown for the indicated experiments.
is capable of capturing the pressure drop trends, including a sharp change in pressure drops at low velocities because of presence of relatively thick stationary beds. Once again the fluid with the highest viscosity and yield stress shows the higher pressure gradient at the same superficial fluid velocity. The results presented suggest that the application of the Wilson−Thomas model (WTM) along with the two-layer concept and the new dispersion coefficient correlation provide a good representation of the data when two layers coexist. Furthermore, the WTM equation for turbulent flow works satisfactorily at high velocities when the flow becomes homogeneous and there is no longer a bed layer at the bottom. The height of the bottom bed layer can be predicted as a function of superficial velocity (Figure 11). For practical
non-Newtonian simulants. These images provide a way to study the evolution of the bed layer at the bottom of the pipe. The results show the presence of a stationary bed over practically all the range of fluid velocities studied. While LH2 exhibits a more linear behavior with a relatively thick bed layer even at high fluid velocities, the LH1 simulant exhibits a rapid decrease in bed thickness and a practically homogeneous flow at high velocities. The critical velocity that marks the transition between the two-layer regime and homogeneous flow (Uc) can be expressed in dimensionless form in terms of a Froude number, Fr =
Uc g ·D·(s − 1)
(36)
This parameter has been used to study the equivalences of critical deposition velocities among slurries with different densities and particle sizes. Previous studies8,12,23 have identified that Froude number and Archimedes number are directly correlated: at low Ar, there is an increasing trend between Fr and Ar mainly attributed to the effect of gravitational forces, but a slight decreasing trend is observed beyond Ar = 80 owing to predominance of viscous forces.12 Figure 13 shows the relation obtained in this work for all simulants employed. In the case of Figure 11. Predicted bed heights as a function of superficial velocity.
purposes, it can be considered that the bed disappears when its height approaches the particle diameter, which usually happens at velocities in the range 0.3−1.5 m/s, according to our observations. However, for the relatively dense/large particle size slurry (HH), experimental observations and ERT images suggest that the stagnant bottom layer only disappears at the maximum velocity used. As discussed before, the LL nonNewtonian simulants exhibit a slight solid accumulation under laminar flow conditions at very low fluid velocities. Notice that under laminar flow, viscous effects are more important and the fluid with the higher viscosity and yield stress has the smaller bed, whereas for all the rest of the simulants studied under
Figure 13. Critical Froude number as a function of Archimedes number for all the simulants. 7102
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
Industrial & Engineering Chemistry Research
Article
non-Newtonian fluids, the Casson infinite shear viscosity was used as an approximation to real viscosity to calculate the Archimedes number. Even though trends obtained follow expectations based on previous works, there is a clear difference between Newtonian and non-Newtonian simulants. For Ar < 80, non-Newtonian simulants exhibit higher Froude number, which reflects their lower solids transport capacity because of predominance of laminar flows in the experiments. However, when viscous effects become more important (Ar > 80 in this case), the critical Froude number is appreciably lower for non-Newtonian slurries.
(9) Gorji, M.; Ghorbani, N. The Influences of Velocity on Pressure Losses in Hydrated Slurries. Int. J. Numer. Methods Heat Fluid Flow 2008, 18, 5. (10) Matoušek, V. Predictive Model for Frictional Pressure Drop in Settling-Slurry Pipe with Stationary Deposit. Powder Technol. 2009, 192, 367. (11) Ramadan, A.; Skalle, P.; Saasen, A. Application of a Three-Layer Modeling Approach for Solids Transport in Horizontal and Inclined Channels. Chem. Eng. Sci. 2005, 60, 2557. (12) Poloski, A. P.; Etchells, A. W.; Chun, J.; Adkins, H. E.; Casella, A. M.; Minette, M. J.; Yokuda, S. T. A Pipeline Transport Correlation for Slurries with Small but Dense Particles. Can. J. Chem. Eng. 2010, 88, 182. (13) Tran-Cong, S.; Gay, M.; Michaelides, E. Drag Coefficients of Irregularly Shaped Particles. Powder Technol. 2004, 139, 21. (14) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: New York, 1978. (15) Brucato, A.; Grisafi, F.; Montante, G. Particle Drag Coefficients in Turbulent Flow. Chem. Eng. Sci. 1998, 53, 3295. (16) Cheng, N. J. Effects of Concentration on Settling Velocity of Sediment Particles. J. Hydraulic Eng. 1997, 123, 728. (17) Wilson, K. C.; Thomas, A. A New Analysis of the Turbulent Flow of Non-Newtonian Fluids. Can. J. Chem. Eng. 1985, 63, 539. (18) Wilson, K. C.; Thomas, A. New Analysis of Non-Newtonian Turbulent Flow Yield-Power-Law Fluids. Can. J. Chem. Eng. 1987, 65, 335. (19) Poloski, A. P.; Adkins, H. E.; Abreah, J.; Casella, A. M.; Hohimer, R. E.; Nigl, F.; Minette, M. J.; Toth, J. J.; Tingey, J. M.; Yokuda, S. T. Deposition Velocities of Newtonian and Non-Newtonian Slurries in Pipelines; DOE Publication No. WTP-RPT-175;Pacific Northwest National Laboratory: Richland, WA, 2008. (20) Taylor, G. I. The Dispersion of Matter in Turbulent Flow through a Pipe. Proc. R. Soc., London 1954, A223, 446. (21) Wen, C. Y.; Yu, Y. H. Mechanics of Fluidization. Chem. Eng. Progress Symp. Series 1966, 66, 101. (22) Renganathan, T.; Krishnaiah, K. Liquid Phase Mixing in 2-Phase Liquid−Solid Inverse Fluidized Bed. Chem. Eng. J. 2004, 98, 213. (23) Gillies, R. G.; Schaan, J.; Summer, R. J.; McKibben, M. J.; Schook, C. A. Deposition Velocities for Newtonian Slurries in Turbulent Flow. Can. J. Chem. Eng. 2000, 78, 704.
5. CONCLUSIONS A two-layer model has been modified to study Newtonian and non-Newtonian slurries loaded with dense particles of broad PSDs and irregular shapes. To include the non-Newtonian properties of the simulants and their effects on the flow, the twolayer model has been coupled with the Wilson−Thomas turbulence equation for Casson fluids. The model predicts accurately experimental pressure drops, critical deposition velocities, and the thickness of the stationary or moving bed when it is present, demonstrating its potential applicability to simulate the hydrodynamics of complex slurries at the U.S. Department of Energy’s Hanford site. Previous two-layer models could not represent the Newtonian simulant data obtained in this work, specifically because of the wide PSD employed, as well as the relatively high density of the particles. This necessitated development of a new correlation for the solids dispersion coefficient in turbulent flows that depends on the particle Archimedes and Reynolds numbers. This correlation allows the two-layer model to reproduce accurately the experimental data.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: (520)6215369. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by Battelle. The authors are grateful to Adam Poloski and Harold Adkins of Pacific Northwest National Laboratory in Richland, WA, for helpful discussions and for providing experimental results.
■
REFERENCES
(1) Wilson, K. C. Stationary Deposits and Sliding Beds in Pipes Transporting Solids. Proceedings of the 1st International Conference on Hydraulic Transport of Solids in Pipes; BHRA Fluid Engineering: Cranfield, U.K., 1970; Vol. C3, p 28. (2) Gillies, R. G.; Shook, C. A.; Wilson, K. C. An Improved Two Layer Model for Horizontal Slurry Pipeline Flow. Can. J. Chem. Eng. 1991, 69, 173. (3) Gillies, R. G.; Shook, C. A. Modelling High Concentration Settling Slurry Flows. Can. J. Chem. Eng. 2000, 78, 709. (4) Gillies, R. G.; Shook, C. A.; Xu, J. Modelling Heterogeneous Slurry Flows at High Velocities. Can. J. Chem. Eng. 2004, 82, 1060. (5) Doron, P.; Barnea, D. A Three-Layer Model for Solid-Liquid Flow in Horizontal Pipes. Int. J. Multiphase Flow 1993, 19, 1029. (6) Doron, P.; Granica, D.; Barnea, D. Slurry Flow in Horizontal Pipes-Experimental and Modeling. Int. J. Multiphase Flow 1987, 13, 535. (7) Turian, R. M.; Yuan, T. F. Flow of Slurries in Pipelines. AIChE J. 1997, 23, 232. (8) Gillies, R. G.; Shook, C. A. A Deposition Velocity Correlation for Water Slurries. Can. J. Chem. Eng. 1991, 69, 1225. 7103
dx.doi.org/10.1021/ie201667k | Ind. Eng. Chem. Res. 2012, 51, 7095−7103
