# Friction Loss Models

The Slurry module allows for eight different friction loss modules: ▪Durand ▪Wasp ▪Wilson, Addie, Sellgren and Clift ▪Sellgren, Wilson Four-Component Model ▪Liu Dezhong ▪Vsm ▪V50 ▪4CM Durand The method used in the Slurry module is more properly called the Durand-Condolios-Worster correlation. This correlation defines the excess pressure gradient Φ as: Φ = (r P s - r P f ) / r P f = Ω Cv ( Cd 0.5 Fr)-1.5 where: r P s is the pressure drop due to friction for the slurry mixture. r P f is the pressure drop due to friction for the carrier fluid alone. Ω is the Durand-Condolios-Worster coefficient. Cv is the Volumetric fraction of solids in suspension. Cd is the drag coefficient at terminal settling velocity. Fr is the Froude number. The value of Ω is uncertain with values between 65 and 150. FluidFlow uses a value of 82 for this coefficient. The Durand method uses the d50 particle size only. Size distribution data entry is not used in the calculation. A Newtonian carrier fluid must be defined as the vehicle when using this modelling approach. Wasp This method is an expansion/refinement of the Durand method using the entered size distribution data to define stratified homogeneous concentration layers. The friction loss of each layer is then calculated using slurry rheology and the Durand correlation. Calculated pressure losses of each layer are summed to provide a total pressure loss. A Newtonian carrier fluid must be defined as the vehicle when using this modelling approach. Wilson, Addie, Sellgren and Clift By default this is the calculation method used by FluidFlow. The total pressure loss for the flow of heterogeneous settling slurry is given by the following equation: r P s / L = r P f / L + (1 + Φ) where: r P s is the pressure drop due to friction for the slurry mixture. r P f is the pressure drop due to friction of the carrier fluid. L is the pipe length. Φ is the frictional increase in pressure drop due to the presence of solid particles. r P s represents the frictional pressure loss and does not include pipe end elevation differences. FluidFlow displays Total Pressure Loss which includes pressure recovery. The Φ term is calculated from the equation Φ = Cvd (Ss - Sf ) Cs g D / (f Vm 2) (V50 / Vm)M where: Cvd is the volumetric concentration of solids. Ss is the relative density of the solids. Sf is the relative density of the carrier fluid. Cs is the coefficient of friction between the bed flow and the pipe wall (defined in the Solids database as the Solid Bed Siding Coefficient). g is acceleration due to gravity. D is the inside diameter of the pipe. f is the Darcy friction factor for the carrier fluid. Vm is the average velocity of the mixture. V50 is the value of Vm at which 50% of the solids are in suspension. M is the exponent to the stratification ratio which accounts for the distribution of particle sizes. M has limiting upper and lower values of 1.7 and 0.25 respectively. Between those limits its value is a function of the terminal settling velocity of the particles. The above relationship holds for horizontal flow. In vertical and sloping pipe sections the relationship changes, FluidFlow uses the following relationship for flow in vertical sections and pipes where the slope is greater than 30 degrees. r P s = 2 ρ f Vf 2 f L / D where: Vf is the actual velocity of the carrier fluid in the vertical section and is a function of in-situ concentration and solids hold up. Settling Velocity: The Slurry module uses the methods described in Wilson, Addie, Sellgren and Clift to determine terminal settling velocities. The Solid Shape Factor is included. Solid Shape Factor takes into account the difference in settling velocity between a solid of the same functional diameter in the shape of a sphere and the same solid of an irregular shape. The Calculation Dialog tab 'Non-Newtonian & Slurry' has a tick box allowing the shape factor to be ignored in a calculation, thereby assuming the particle is spherical. A Newtonian carrier fluid must be defined as the vehicle when using this modelling approach. Sellgren, Wilson Four-Component Pipeline Friction Loss Model More recent experimental tests and studies have provided a better understanding of friction losses for the size distribution of the particles forming a slurry. This modelling approach considers the slurry to be made up of four component parts: 1) a carrier fluid (typically water), 2) pseudo-homogeneous mixture, 3) heterogeneous solids and 4) fully stratified solids. The four components can be described as follows: Component 1: Carrier Fluid (Xf Fraction). The carrier fluid typically approximates to Newtonian fluid behavior and is considered to include particles finer than 40µm. These solids are assumed to "combine" with the liquid and are considered only to the extent that they affect the average density and viscosity of the carrier fluid "mixture". Component 2: Pseudo-homogeneous mixture (Xp Fraction). The pseudo-homogeneous mixture is considered to combine the liquid and the particle size range of between 40µm and 200µm. Component 3: Heterogeneous solids (Xh Fraction). The heterogeneous solids include the "Grits" and include particle sizes between 200 µm and 0.015D. Component 4: Fully stratified solids (Xs Fraction). 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: im = [1 + A'(Sm - 1)iw] + [0.22(Sm - 1)(V/V50)-M] + [(Sm - 1)B'(V/0.55Vsm)-0.25] where: im is the hydraulic gradient for the flow of the slurry mixture (m water/m pipe). A' is the relative density coefficient associated with the pseudo-homogeneous component of the fluid. The range for the term A' is 0 to 1.0. In the case where A' = 1.0, the pseudo-homogeneous component approaches the equivalent fluid case. Sm is the relative density of the slurry mixture. iw is the hydraulic gradient for equal volumetric flow of water (m water/m pipe). V is the mean velocity. V50 is the value of mean velocity at which 50% of the solids are suspended by the fluid. M is the exponent to the stratification ratio which accounts for the distribution of particle sizes. B' is a particle size coefficient. Design Notes: The Four-Component method should be used for broadly graded slurries with a modest amount of fine particles (sub 200 micron). A Newtonian carrier fluid must be defined as the vehicle when using this modelling approach. Liu Dezhong Pipe Friction Loss Model The method proposed by Liu Dezhong is a two-component method, and has been successfully used in China for the design of short distance (less than 4 km) slurry pipelines with 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. Rather than use particle size as the defining property for the two components, Liu proposes a complex method based on the change in concentration across the cross section of the pipe - i.e., a measure of the concentration gradient from the pipe centerline 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 calculated using the traditional carrier fluid friction loss plus "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 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 friction factor, and subsequently applied to the Darcy-Weisbach equation. The friction loss associated with the heterogeneous (settling) fraction of the slurry is calculated according to the E.J. Wasp/Durand method. Vsm Friction Loss Model The Vsm friction loss method is developed to model fully stratified flow in settling slurry systems.Moreover, this correlation can be used for performing hydraulic calculations where particle size is unavailable or for determining the upper pressure loss limit of a slurry transport system. Among the different methods, only the Vsm model does not utilize particle size as input and only relies on the solids density and concentration for its calculation. When this correlation is applied, other settling slurry correlations are automatically selected. These are: ▪Vertical Pipe Friction Loss: 4CM ▪Inclined Pipe Correlation Selection: Wilson - Tse 1984 Chart ▪Deposition Velocity Correlation: WASC Generalized Relationship As particle size data is not applied in the Vsm method, these input fields for inlet boundary nodes will be hidden: ▪Use Size Distribution Data ▪D50 Mean Diameter ▪D85 Diameter ▪% Retention on various mesh sizes Moreover, centrifugal pump solids derating method would also become limited to the following options because of the lack of solids particle size: •Fixed •ANSI2021 Monosize V50 Friction Loss Model The V50 friction loss correlation is an updated form of the Wilson-Addie-Sellgren-Clift method. In general they are essentially the same but with the following distinctions: ▪V50 and Stratification Ratio exponent can either be calculated or derived from scale-up tests. By selecting V50 as friction loss correlation in the calculation options, additional dropdown fields for V50 and stratification ratio exponent (M) would appear. The user has the option to have the software calculate or manually define V50 and the "M" value obtained from scale up tests. ▪The new calculation method for V50 does not account for the pipe size nor friction factor but rather based only on solid characteristics and carrier fluid viscosity. ▪Stratification Ratio Exponent (M) for the V50 model has a limit of 1.0 compared to 1.7 of the previous. 4CM Friction Loss Model The 4CM friction loss model is an update of the Sellgren, Wilson Four-Component Model. This has been developed for use with settling slurries of any grading profile and is thus applied as the default friction loss correlation. This correlation assumes that fine particles sized 40 microns and smaller combine with the liquid phase to form a Newtonian carrier fluid that flows turbulently and above the limit of stationary deposition. Under 4CM, the total friction loss for the slurry is the sum of the friction losses contributed by each component category as characterized below: ▪Carrier fluid which also includes solid volumetric fraction for particles 40 x 2.65 / Solids Density microns ▪Pseudo – homogeneous, where solid particles sized 40 to 200 microns are present ▪Heterogenous, component which contain particles 200 microns up to a size equivalent to 1.5% of pipe diameter and ▪Fully stratified, fraction which accounts for the solids larger that 1.5% of the pipe diameter. IMPORTANT NOTE When comparing FluidFlow's Slurry module calculations with examples from other programs or references it is important to ensure that those calculations include the shape factor correction, if this correction is included in the FluidFlow calculation. FluidFlow Slurry by default includes shape factor correction; this is not always the case with other calculation methods and examples.