It should be noted that most models are based on constant delivered volumetric concentration experiments. For the high line speed heterogeneous and homogeneous regimes the spatial and delivered volumetric concentrations are close, almost equal. However for the sliding bed regime and the low line speed heterogeneous regimes they differ and the lower the line speed the larger the difference. This often results in a more negative power of the excess pressure gradient related to the line speed.
6.26.1 The Newitt et al. (1955) Model
The Newitt et al. (1955) models already distinguish 3 main flow regimes. The sliding bed regime, the heterogeneous regime and the homogeneous regime. In the sliding bed regime, the excess hydraulic gradient is independent from the line speed, it mainly depends on the weight of the bed and a sliding friction coefficient. In the heterogeneous regime the excess hydraulic gradient depends reversely on the line speed. In the homogeneous regime the excess hydraulic gradient is proportional to the line speed squared and is similar to the ELM. Based on the formulations of the 3 flow regimes a regime diagram is constructed, but still with sharp transitions between the flow regimes.
6.26.2 The Wasp et al. (1963) Model
One of the shortcomings of the Newitt et al. (1955) models is the sharp transition between the flow regimes and the inability to deal with graded solids. The Wasp et al. (1963) model deals with the transition of the heterogeneous regime and the homogeneous regime. The graded solids are divided into fractions. For each fraction the portion in suspension is determined, based on the advection diffusion equation. By summation of these portions the total amount of solids in suspension is determined. The amount of solids in suspension behave according to the ELM, while the remaining amount of solids behave heterogeneously according to Durand & Condolios (1952). This way a smooth transition is achieved between the heterogeneous and homogeneous flow regime and the grading of the solids is taken into account. The method works well for solids with a lot of fines, while coarse uniform solids behave according to Durand & Condolios (1952).
6.26.3 The Wilson-GIW (1979) Model
The Wilson-GIW (1979) model started as a 2 layer model with a sliding or stationary bed and pure liquid above it. Based on an equilibrium of forces acting on the bed, the bed velocity and the hydraulic gradient can be determined. First however it is determined whether the bed is sliding or is stationary, resulting in a LSDV curve. The method is based on the spatial volumetric concentration and outputs the delivered volumetric concentration. By iteration constant delivered concentration curves can be constructed. The main shortcoming of the original model is the inability to deal with suspended particles. This model deals with the sliding bed regime solely. The model uses a hydrostatic approach to determine the normal stress between the bed and the pipe wall. This approach is questionable.
Later a model for the heterogeneous regime is added, based on the so called stratification ratio. A stratification ratio of 1 means that all the solids are in the bed, a stratification ratio of 0 means all particles are suspended. Based on a velocity where the stratification ratio is 50% and a power function for the stratification ratio, the excess hydraulic gradient can be determined. The excess hydraulic gradient is reversely proportion to the line speed to a power between 0.25 for very graded solids up to 1.7 for uniform solids.
Recently a 4 component model was introduced, based on 4 components, particle size regions. Very fine particles behave homogeneously, with or without correction of the liquid properties. Fine to medium particles behave pseudo homogeneously, according to the ELM. Medium to coarse particles behave heterogeneously and very coarse particles behave stratified, according to the sliding bed regime. By splitting a PSD into 4 fractions, determining the excess hydraulic gradient for each fraction and adding up the excess hydraulic gradients, the total excess hydraulic gradient is determined. The downside is, that the division between the 4 fractions depends on the particles size and the pipe diameter and not on the line speed or relative submerged density. So this model is only applicable for sands in a certain line speed region.
6.26.4 The Doron et al. (1987) and Doron & Barnea (1993) Model
The Doron & Barnea (1993) model started as a 2 layer model, with suspension above a sliding bed. The portion of the solids in suspension is determined with the concentration distribution above the bed, based on the advection diffusion equation for open channel flow. This is an addition compared to the Wilson-GIW (1979) model, which only has pure liquid above the sliding bed. The 2 layer model gives good predictions, however always predicts a sliding bed for constant delivered volumetric concentrations. To deal with this a 3 layer model was developed containing a stationary bed at the bottom of the pipe, a moving bed on top of it and a heterogeneous layer above the moving bed. Based on a set of continuity and force equilibrium equations, the amount of solids in the heterogeneous layer and the thickness and the velocity of the moving bed layer are determined. If the friction on the stationary layer exceeds the available sliding friction, the whole bed is sliding, if not, the stationary layer is assumed to be real stationary. The 3 layer model predicts a stationary bed at very low line speeds.
6.26.5 The SRC Model
The SRC model is also developed for graded solids with a lot of fines, comparable to the Wasp et al. (1963) model. The main difference is, that the SRC model uses the sliding bed model, comparable to the Wilson-GIW (1979) model for the bed fraction. The model does not use an advection diffusion equation to determine the suspended fraction, but instead it uses an empirical equation to determine the contact load fraction. The remainder is the solids in suspension. The solids in suspension are also assumed to be present in the bed and increase the liquid density, resulting in a lower relative submerged density of the solids. The maximum bed concentration depends on the line speed and decreases with increasing line speed. At a certain line speed, the bed concentration is so low that one cannot call it a bed anymore. Still it moves over the bottom of the pipe and has a sort of sliding bed behavior. This will be named the sliding flow regime.
6.26.6 The Kaushal & Tomita (2002B) Model
Kaushal & Tomita (2002B) have modified the Wasp et al. (1963) model. They noticed that the way of determining the suspended fraction, does not always give proper results. Especially for coarser particles, the Wasp model is just the Durand & Condolios (1952) model. By using the Karabelas (1977) approach for graded solids in a circular pipe and modifying the diffusivity, they managed to overcome the shortcomings of the Wasp method. The diffusivity now depends on the size of a particle fraction and on the volumetric concentration. The concentration distribution model now also gives good results for coarse particles. In the heterogeneous regime the model is as good as the Durand & Condolios (1952) model, just as the Wasp model.
6.26.7 The Matousek (2009) Model
The Matousek (2009) model is a sort of reversed engineering model. The starting point is the delivered volumetric concentration. Based on the Meyer-Peter Muller equation, the Shields parameter is determined. From the Shields parameter the bed shear stress is determined. Based on the continuity and force equilibrium equations, the bed velocity is determined. By iteration an equilibrium situation has to be determined where the total delivered concentration matches the input. The methodology also outputs the bed height. The model is suitable for the sliding bed regime and the stationay bed regime, but not for the heterogeneous or homogeneous regimes.
6.26.8 The Talmon (2011) & (2013) Homogeneous Regime Model
Many researchers have found that at high line speeds the flow regime is the homogeneous regime, following the ELM. However they also noticed that the hydraulic gradient is often between the pure liquid hydraulic gradient and the ELM. In terms of the excess hydraulic gradient this means that the excess hydraulic gradient is between zero and the excess hydraulic gradient of the ELM. One should keep in mind that measurements in the homogeneous regime are mostly at line speeds just above the heterogeneous regime. For very fine and fine particles almost real homogeneous behavior is observed, but for medium and coarse particles this is more pseudo homogeneous behavior, which is still a transition region from heterogeneous to true homogeneous behavior. Talmon (2011) & (2013) developed a method to prove that the excess hydraulic gradient is slightly lower than the ELM excess hydraulic gradient. The method is based on the assumption that the viscous sub-layer is particle free, while outside the viscous sub-layer there is a uniform distribution of particles. The method is based on the velocity distribution of open channel flow, but gives promising results.