Correlation of Settling Slurries

The literature contains many correlations for the determination of the pressure loss of horizontal pipe flow of heterogeneous slurries. Most of the correlations make use of the dimensionless excess pressure gradient Φ, which is defined as

\large \Phi = \frac{i_{m-}i_{f}}{C_{vd} i_{f}}

Where, im is the slurry frictional pressure loss in terms of meters of the fluid per meter of pipe, if is the frictional pressure loss for an equivalent volume of carrier fluid, and Cvd is the volumetric solids concentration in a homogeneous slurry. For a truly homogeneous slurry flow, Φ approaches (Ss - 1) if the friction factor for slurry is the same as that for a liquid flowing at the same velocity in the same pipe.

Durand-Condolios Correlation for a fully suspended heterogeneous flow

The first comprehensive empirical correlation was that of Durand and Condolios (1952). The Durand correlation is generally recognized as the “least effective.” However, when minimum solids data is available, it is useful in providing a generalized result. The model assumes a consistent particle size throughout the flow, and as we know, particle sizes are rarely consistent and uniform in most industrial applications. It is based on the d50 value. In terms of dimensionless excess pressure loss φ and drag coefficient CD, the correlation can be written in the form

\large \Phi =\Omega \Psi ^{n}= \Omega \left [ \frac{V_{m}^{2}\sqrt{C_{D}}}{gD \left ( S_{s}-1 \right )} \right ]^{n}

Here,

\large \Psi = \frac{V_{m}^{2}\sqrt{C_{D}}}{gD \left ( S_{s}-1 \right )}

Based on 310 data points, the constant Ω was found to have a value of 84.9 and the exponent n = -1.5, although it has since been found that the values of Ω and n need to be adjusted for different slurry system. The Durand Constant (Ω) of 82 is widely quoted in literatures.

CD is the drag coefficient for settling of the particle at its terminal velocity in the stagnant unbounded liquid. For a sphere with a diameter of dp, it is given by

\large C_{D}=\frac{4}{3}\ast \frac{gd_{p}(S_{s}-1)}{V_{s}^{2}}

The Durand correlation finds its use when modeling the heterogeneous flow regime. However, its accuracy deviates more and more from actual conditions in the other flow regimes. Experimental tests have shown that different correlations should be used in each of the identifiable flow regimes.

Over the years, research has shown that the Durand-Condolios correlation overestimates the head losses for coarse-particle slurries flowing in large pipes. It must be kept in mind that indiscriminate use of this model could potentially make viable slurry pipeline projects appear uneconomic. This approach is thus best used as a rough guide, possibly at a notional design state when limited data is available.

This correlation also tends to under-predict the pressure gradient in the low velocity region near the critical velocity for small pipe diameters and increasingly over-predicts as the pipe size is increased.

This correlation assumes horizontal pipes only.

Zandi-Govatos Correlation for a fully suspended heterogeneous flow

Doubts were raised when Zandi and Govatos (1967) showed a large discrepancy between Durand's prediction and experimental data for a coarse slurry flow exhibiting considerable stratification. Durand correlation seemed to be invalid for the solids transport by saltation; therefore, by using 990 data points, Zandi and Govatos (1967) modified the Durand-Condolios correlation and suggested a criterion for separation of the heterogeneous and saltation regimes.

\large \Phi =\begin{cases} 280\Psi ^{-1.93}& \text{ for } \Psi < 10 \\ 6.30_{\Psi }^{0.354}& \text{ for } \Psi > 10 \end{cases}

Zandi and Govatos defined an index number Ne as

\large N_{e}=\frac{V\sqrt{C_{D}}}{C_{vd}gD\left ( S_{s}-1 \right )}

When Ne < 40 saltation regime occurs, but when Ne ≥ 40 heterogeneous flow regime develops.

Newitt Correlation

Newitt et al. (1955) developed correlations for a wide range of velocities, solid loading, and particle size in a 1 inch pipe, which covered homogeneous flow, fully suspended heterogeneous flow, and flows with a moving bed. The correlations consist of a set of criteria for defining the flow regime and a regime-specific set of equations.

The transition velocities, VH = (1800gDVs)1/3 and VB = 17Vs, can be used for delineating the three flow regimes.

For the homogeneous flow regime, i.e., Vm > VH, they assumed that the friction factor for the slurry is the same as that for the carrier fluid, such that the difference in density is the only factor causing the difference in pressure loss, leading to

\large \Phi =0.6\left ( S_{s} -1\right )

Where, 0.6 is an empirical constant.

For the fully suspended heterogeneous flow regime, i.e. VB < Vm < VH, it was assumed that there was no relative velocity between the particles and the fluid, and that the extra energy required to transport the particles is proportional to the energy expended when the particles fall at their terminal velocities, Vs. The resulting equations can be written as

\large \Phi =1100 \frac{gDV_{s}}{V_{m}^{3}}\left ( S_{s} -1\right )

For sliding bed regime, i.e., Vm < VB, they assumed that the work done to overcome sliding friction between the bed and the pipe wall leads to the extra pressure loss due to the solids. The resulting expression is

\large \Phi =66\frac{gD}{V_{m}^{2}}\left ( S_{s} -1\right )

The boundary between homogeneous and fully suspended heterogeneous flow is obtained by equating the above equations.

The Newitt correlation applies only to mono-dispread systems. For solids with broad size distributions, Newitt suggested a weighted mean particle diameter as

\large d_{pm}=\Sigma _{i=1}^{n} C_{si} d_{pi} & \large d_{pm}=\Sigma _{i=1}^{n} C_{si} =1

Where, Csi is the mass fraction of solids with a partial diameter of dpi.

Turian and Yuan Correlation

The flow takes the form of different regimes. Four distinct regimes were found (flow with a stationary bed, flow with a moving bed, heterogeneous suspension, and homogeneous suspension) in slurry flow depending upon several flow parameters.

As regime identification is an important criterion for slurry pipeline design to select a suitable correlation of pressure drop for that regime, a method has been proposed to identify the regime using Turian and Yuan’s approach. The pressure drop correlations are applicable to the particular regime for which they were developed. The correlations show an ill-predicted pressure drop when they apply to other regimes. Thus, regime identification becomes important for slurry pipeline design as it is the prerequisite for selecting a suitable pressure drop correlation in that regime. The main problem arises because of the difficulty of defining the boundaries between the flow regimes. These boundaries are poorly defined because they are based on visual observations of particle motions. The approach of Turian and Yuan claims to provide a completely self-consistent definition of the flow regime boundaries. Turian and Yuan established that the excess pressure gradient in each flow regime can be correlated using an equation of the form:

\large f_{sl} - f_{f} = {K}'C^{{\alpha}'}f_{f}^{{\alpha }''}C_{D}^{\ast ^{\gamma }}F_{r}^{\delta }

The coefficients K­', α', α′′, γ and δ have values that are specific to each flow regime. The best available values of these coefficients in each flow regime are given by α.

\large C_{D}^{\ast }=\frac{432}{\Phi _{1}^{\ast }}\left ( 1+0.047\left ( \phi _{1}^{\ast } \right )^{\frac{2}{3}} \right ) + \frac{0.517}{1+154\left ( \phi_{1}^{\ast } \right )^{-\frac{1}{3}}}

\large \Phi _{1}^{\ast } = \frac{4}{3}\frac{\left ( p_{s}-p_{f} \right )p_{f}d^{3}g}{f_{f}^{2}}

\large F_{r} = \frac{V_{m}^{2}}{gD\left ( S_{s}-1 \right )}

 

The friction coefficient fsl depends on the regime according to:

Sliding bed (regime 0)

\large f_{sl}-f_{f} = 12.13C^{0.7389}f_{f}^{0.7717}C_{D}^{\ast^{-0.4054} }F_{r}^{-1.096}

Moving bed (regime 1)

\large f_{sl}-f_{f} = 107.1C^{1.018}f_{f}^{1.046}C_{D}^{\ast^{-0.4213} }F_{r}^{-1.354}

Heterogeneous suspension (regime 2)

\large f_{sl}-f_{f} = 30.11C^{0.868}f_{f}^{1.200}C_{D}^{\ast^{-0.1677} }F_{r}^{-0.6938}

Homogeneous suspension (regime 3)

\large f_{sl}-f_{f} = 8.538C^{0.5024}f_{f}^{1.428}C_{D}^{\ast^{0.1516} }F_{r}^{-0.3531}

The pressure losses of a mixture can be calculated with:

\large \Delta P_{f,sl} = 2f_{sl\rho _{w}}V_{m}^{2}\frac{L}{D}

The hydraulic gradient of the mixture flow is

\large i_{m}=\frac{\Delta P_{f,sl}}{L_{g\rho _{w}}}=2f_{sl}\left ( S_{s}-1 \right ) F_{r}

Some authors prefer to use ρsl rather than ρw.

Wilson-Addie-Sellgren-Clift (WASC)

This is generally the default calculation method used. The WASC method is more complex than the Durand method, as it takes into account the variation of particle size within the slurry by including the d50 and d85 values in the calculation. This method is generally accepted as reasonably accurate, as it is used frequently in the industry.

The following equation represents the fractional increase in pressure gradient over and above that produced by the carrier fluid if it were flowing without particles at the same velocity as the slurry.

The form of the WASC equation is:

\large \Phi = C_{vd}\left ( S_{s}-1 \right )\frac{C_{s}gD}{fV_{m}^{2}} \ast \left ( \frac{V_{50}}{V_{m}} \right )^{M}

 

Here, M is the exponent of the stratification ratio, which accounts for the distribution of particle sizes. Much like the Durand approach, the WASC method postulates that Φ (fractional increase in pressure drop due to the presence of the particles) is directly proportional to the volume fraction of solids in the slurry. The M parameter to some extent, accounts for the effect of distribution of particle sizes in the slurry and is calculated from the d50 and d85 passing size of the particle population.

\large M = \left ( 0.25 -13\sigma _{s}^{2} \right )^{-0.5}

\large \sigma = log_{10}\left ( \frac{w_{85}cosh\left ( \frac{60d_{85}}{D} \right )}{w_{50}cosh \left ( \frac{60d_{50}}{D} \right )} \right )

 

Note: It has been observed that the exponent M is often close to 1.7 for narrow particle grading and decreases as the grading becomes broader. It must be remembered that the proviso that a calculated M larger than 1.7 must be reduced to that value.

Typically, the values of M are:

0.25   fully stratified flows.

1.0     flows of complex slurries.

1.7     flows of narrow graded solids.

M has limiting upper and lower values of 1.7 and 0.25 respectively. The above relationship is applicable to horizontal pipe flow.

WASP

The WASP correlation takes the Durand and WASC methods one step further by taking into account the particle size distribution data and using this to define the individual stratified homogeneous concentration layers. The friction loss of each layer is then calculated using slurry rheology and the Durand correlation.

Calculated pressure losses for each layer are then summed up to provide a total pressure loss.

When designing slurry piping systems, engineers are often forced to model the slurry as either a heterogeneous settling slurry or a non-Newtonian, non-settling slurry, when in reality, the slurry can often exhibit a combination of both these flow types.

Four-Component and Liu Dezhong Methods

The Four-Component and Liu Dezhong methods simplify the design process by removing this decision at the project outset. Both of these methods take into account the solids data available and apply the appropriate correlation for each component part of the fluids, i.e. non-settling or heterogeneous settling.

Sellgren, Wilson, Four-Component Pressure Loss Model (v3.30)

More recent experimental tests and studies have provided a better understanding of friction losses for the size distribution of the particles forming slurry. This modeling approach classifies the slurry into four components based on particle size, with each component possessing different friction loss characteristics. Empirical comparison indicates a high level of accuracy, especially for sands, tailing, and paste systems.

The four components can be categorized as follows: 1) carrier fluid (typically water) 2) pseudo-homogeneous mixture 3) heterogeneous solids and 4) fully stratified solids.

The four components are described as follows:

Component 1: Carrier Fluid (Xf)

The carrier fluid typically approximates Newtonian fluid behavior and is considered to include particles finer than 40μm (0.04mm).

Component 2: Pseudo-Homogeneous Mixture (Xp)

The pseudo-homogeneous mixture is considered to combine the liquid and the particle size range of between 40μm and 200μm (0.04mm-0.2mm).

Component 3: Heterogeneous Solids (Xh)

The heterogeneous fraction includes the “grits” and spans particle diameters between 200μm (0.2mm) and a multiple of the internal pipe diameter “D”. Research has shown that the slurry flow regime tends to shift from heterogeneous to fully stratified load at values of 0.015D. This is the value used in fluid-flow as the upper particle diameter for heterogeneous flow, with larger particles forming the fully stratified solid fractions.

Component 4: Fully Stratified Solids (Xs)

The fully stratified solids component represents the coarser particles or “clunkers” which are for particle sizes larger than 0.015D.

The four-component model correlation defines the excess pressure gradient for the mixture as a whole, im (m water/m pipe), by the following equation:

The four-component method should be used for broadly graded slurries with a modest amount of fine particles (<200 um)

\large i_{m}=\left [ 1+{A}'\left ( S_{m}-1 \right )i_{w} \right ] + \left [ 0.22(S_{m}-1) \left ( \frac{V_{m}}{V_{50}} \right ) - M\right ] + \left [ \left ( S_{m} -1\right ) {B}' \left ( \frac{V_{m}}{0.55V_{sm}} \right ) - 0.25\right ]

Liu Dezhong Loss Model (v3.30)

This method has been developed by Professor Liu Dezhong, Beijing, who has over 50 years' experience in slurry pipeline design. This is a two-component method which has been successfully used in China for the design of short distance (less than 4 km) slurry pipelines which broadly graded size distribution. The smaller particles are considered to form a pseudo-homogeneous carrier fluid. Depending on the fluid's overall particle size distribution and concentration, it has either Newtonian or non-Newtonian Bingham Plastic Properties. The larger particles are assumed as a heterogeneous suspension.

Liu Dezhong Pressure Loss Model (v3.30)

Rather than using particle size as the defining property for two components, Liu proposes a complex method based on the change in solid concentration across the cross section of pipes—i.e. a measure of the concentration gradient from the pipe center line to just below the top of the pipe.

If the gradient is high, then the slurry will possess a higher pseudo-homogeneous fraction, while a low gradient indicates a higher heterogeneous fraction. The volumetric concentrations of both fractions are calculated from a complex set of equations, and the friction loss is calculated using the traditional carrier fluid friction loss plus the “solids effect” from the heterogeneous fraction.

The properties of the pseudo-homogeneous component are estimated using a method proposed by Tsinghua University, Beijing. A limiting volumetric concentration is determined, below which, the fluid can be considered Newtonian.

If the pseudo-homogeneous fraction volumetric concentration is below this limiting value, then the friction loss is calculated using the Darcy-Weisbach formula with the fanning friction factor, i.e. Newtonian behavior is assumed.

If the fraction is above this limiting value, the hanks formula is used to determine non-Newtonian properties for the Bingham Plastic fluid (subject to the conditions in the pipe) and the friction factor, and subsequently applied to the Darcy-Weisbach equation.

The friction loss associated with the heterogeneous (settling) fraction of slurry is calculated according to the E.J. Wasp/Durand method.

LIST OF SYMBOLS

A'      Relative density coefficient associated with the pseudo-homogeneous component of the fluid. The range for the term A' is 0 to 1.0. In case where A' = 1.0, the pseudo-homogeneous component approaches the equivalent fluid case.

B'       Particle size coefficient

C       Delivered concentration of the solid

CD     Drag coefficient

CD­­­*    Particle drag coefficient at terminal velocity

Cs      Coefficient of friction between sliding bed and pipe wall (also known as solid bed sliding coefficient)

Cvd     Volumetric solids concentration in homogeneous slurry

d        Particle diameter [m]

D       Internal pipe diameter [m]

d50     Mass median particle diameter [m]

d85     Diameter of which 85% of particles are finer [m]

f        Darcy friction factor of the fluid

Fr       Froude number

fsl       Fanning friction factor of the slurry

ff        Fanning friction factor of the carrier fluid

im       Hydraulic gradient for the flow of the slurry mixture (m water/m pipe)

iw       Hydraulic gradient for equal volumetric flow of water (m water/m pipe)

K′      Coefficient of Turian and Yuan correlation

M      Exponent to the stratification ratio which accounts for the distribution of particle sizes

Ss      Specific gravity of solids

Sm      Relative density of the slurry mixture

V       Velocity [m/s]

Vm     Mean velocity [m/s]

Vs      Terminal settling velocity of the solid particle [m/s]

V50    Value of mean velocity at which 50% of the solids are suspended by the fluid

w50    Terminal settling velocity of particle size d50 [m/s]

w85    Terminal settling velocity of particle size d85 [m/s]

 

Greek Letters

Φ       Excess pressure gradient

ϕ1*         Dimensionless number

σs       Normal stress of solid (intergranular) [kg/m.s2 ]

α', α′′, γ, δ    Coefficients of Turian correlation

ρf           Fluid density [kg/m3]

ρs       Particle density [kg/m3]

ρw      Density of carrier fluid(usually water)

 

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