6.15: Physical Models

In the previous chapters empirical and semi empirical models are discussed. These models are either based on a direct curve fit of experimental data or on a curve fit on dimensionless numbers based on assumed physical relations. Direct curve fit of experimental data based on the physical or geometrical parameters varied during the experiments leads to good empirical relations within the ranges the physical and geometrical parameters are varied. Outside these ranges the empirical relations are questionable. Relations based on dimensionless numbers assume physical relations between the parameters of which a dimensionless number consists, however they also pretend validity outside the range experiments were carried out. Another downside is, that the range a dimensionless parameter is varied in, does not imply the parameters the dimensionless parameter consists of can be varied in that range. An example is the Reynolds number consisting of a velocity scale, a length scale and the kinematic viscosity. Often only the velocity scale or the length scale is varied, however varying the Reynolds number also means that the kinematic viscosity can be varied, which is questionable if only 1 liquid is used during the experiments. Another example is the Froude number as used in a number of models. It consists of the velocity scale squared, divided by the length scale. Models based on the Froude number force a relation between the velocity scale and the length scale. This can however be corrected by adding additional dimensionless numbers, but this is often not the case. The advantage of physical models is, that they describe the physics involved. Maybe in the beginning the physics are limited and certain effects are neglected, but the models can be extended in the future. A number of physical models are described here, based on different points of view.

6.18.1 The Newitt et al. (1955) Model

The Newitt et al. (1955) model distinguishes different flow regimes, enabling the user to determine which flow regime is valid for which situation. Newitt et al. (1955) constructed regime diagrams based on their empirical relations. They distinguished flow with a stationary bed, flow with a moving bed, heterogeneous flow and homogeneous flow. The equations for each flow regime are still empirical, although sometimes based on physical considerations. The moving (sliding) bed regime is based on sliding friction, the heterogeneous regime on potential energy and the homogeneous regime on the Equivalent Liquid Model (ELM). The regime diagrams show that not all regimes occur for each combination of particle and pipe diameter. The Newitt et al. (1955) model however assumes that only one flow regime is present under the circumstances considered. The Newitt et al. (1955) model forms the basis of a number of physical models as discussed in the next chapters.

6.18.2 The Wasp et al. (1963) Model

The Wasp et al. (1963) model considers a combination of heterogeneous and homogeneous flow at the same time. The PSD is divided into a number of fractions. Based on the concentration profile of each fraction, following from a solution of the advection diffusion equation, the portion of solids in the vehicle is determined. After summation of the portions in the vehicle, viscosity and density of the so called vehicle liquid are adjusted for the concentration of solids in the vehicle liquid. The remainder of the solids is assumed to be in the heterogeneous regime, using the Durand & Condolios (1952) model to determine the hydraulic gradient. Often this remainder is assumed to be in the bed, however the Durand & Condolios (1952) model is developed for heterogeneous flow, which by definition means a combination of suspended and bed flow. So not all suspended flow is in the vehicle.

6.18.3 The Wilson-GIW (1979) Model

The original Wilson-GIW (1979) model is based on the force balance on a bed with pure liquid above it. After determining all forces (or shear stresses) involved, it follows whether the bed is sliding or not. The line speed where the bed starts sliding depends on the volumetric concentration and is often called the critical velocity. Since there are many definitions of the critical velocity, here this is named the Limit of Stationary Deposit Velocity (LSDV), distinguishing this from the LDV (Limit Deposit Velocity) where there is no stationary or sliding bed. The latter will always occur at a higher line speed. Based on the shear stresses determined, the hydraulic gradient (based on the hydrostatic normal stress approach) and the delivered concentration can be determined.

For the heterogeneous regime the Wilson-GIW (1979) model assumes a diminishing bed with suspension above it. With an empirical relation the portion of solids in the bed can be determined, resulting in a hydraulic gradient. Later the 4 component model of Wilson & Sellgren (2001) was developed. The slurry is divided into 4 components: Homogeneous flow, the fraction d<0.04 mm. Pseudo homogeneous flow, the fraction 0.04 mm<d<0.15·μ_r mm, where μ_r is the relative dynamic viscosity. Heterogeneous flow, the fraction 0.15·μ_r mm<d<0.015·D_p. Stratified flow, d>0.015·D_p. Not each component has to be present in the slurry. The hydraulic gradients are determined for each component separately and summed to find the total hydraulic gradient.

6.18.4 The Doron et al. (1987) and Doron & Barnea (1993) Model

Doron et al. (1987) and Doron & Barnea (1993) 2 layer model, start with solving the advection diffusion equation in order to determine the concentration profile. Based on this concentration profile it is determined whether there is a bed or not. In case of no bed, the bottom concentration is smaller than the bed concentration, the ELM is used. In case of a bed, the model of Wilson-GIW (1979) is used for the determination of the hydraulic gradient of the bed. If there is a bed however, the concentration profile has to be recalculated by solving the advection diffusion equation starting at the bed surface with the bed concentration as a boundary condition. This has to be repeated until the sum of suspended solids and bed solids matches the spatial volumetric concentration that was started with. With a no-slip condition between the solids and the liquid, apart from the hydraulic gradient, also the delivered concentration and the bed height are determined.

6.18.5 The SRC Model

At the Saskatchewan Research Council (SRC) another 2 layer model was developed. The model starts with the determination of the contact load fraction based on an empirical equation. The contact load fraction is not equal to the bed fraction, but is defined as the fraction contributing to sliding friction. The remainder is the suspended fraction. However before calculating the sliding friction, two adjustments have to be made. First the contact load fraction is submerged in a pseudo liquid consisting of the carrier liquid and the suspended particles, reducing the submerged weight of the contact load fraction and thus the sliding friction, secondly the maximum bed concentration is decreasing with increasing line speed. The latter is of influence, because the SRC model is using the hydrostatic normal stress approach as developed in the Wilson-GIW (1979) model. The hydraulic gradient of the suspended phase is determined based on the ELM.

6.18.6 The Kaushal & Tomita (2002B) Model

The Kaushal & Tomita (2002B) model is based on the Wasp et al. (1963) model, but with a different approach to determine the suspended load fraction. Where Wasp et al. (1963) determined the suspended load fraction based on a solution of the advection diffusion equation, without interaction between fraction of different particle sizes. Kaushal & Tomita (2002B) based their model on the modified Karabelas (1977) model, including interaction between different particle sizes. The original Karabelas (1977) model is based on the Hunt (1954) equation and not on the Rouse (1937) equation. Kaushal & Tomita (2002B) added the effect of hindered settling and modified the diffusivity, including the effects of particle size and grading and the effect of volumetric concentration. For the heterogeneous portion of the solids (they talk about the bed fraction), the use the Durand & Condolios (1952) model, as is used in the original Wasp et al. (1963) model.

6.18.7 The Matousek (2009) Model

The Matousek (2009) model is completely different from the other models. The previous models are all based on force equilibria and concentration distributions with the delivered concentration as an output, where the Matousek (2009) model uses the delivered concentration as an input. Based on the Meyer-Peter Muller (1948) equation, the Shields parameter is computed. The Shields parameter results in the bed shear stress. By dividing the suspended layer in a bed associated and a wall associated area, the hydraulic gradient is determined, iterating until both areas give the same hydraulic gradient. Basically, the delivered concentration is the result of the transport in a sheet flow layer and the transport in the sliding bed. The model method is a sort of reversed engineering and gives a completely different concept, although the Wilson-GIW (1979) method is applied for the sliding bed friction.

6.18.8 The Talmon (2011) & (2013) Homogeneous Regime Model

The Talmon (2011) & (2013) homogeneous regime model has been derived to prove fundamentally that the hydraulic gradient in homogeneous flow can be less than given by the ELM method. By assuming that the viscous sub-layer is particle free and thus having a different viscosity and density, compared to the turbulent layer, an equation is derived, showing the reduction of the hydraulic gradient. The reduction is increasing with an increasing Darcy Weisbach friction factor, an increasing relative submerged density of the solids and an increasing volumetric concentration. The model is developed using equations for open channel flow and requires some adjustment for pipe flow. The concept of the model however is new and proves that the ELM will, most probably, overestimate the hydraulic gradient at high to very high line speeds.