# 7.11: The Transition Heterogeneous vs. Homogeneous in Detail


## 7.11.1 The Transition Heterogeneous-Homogeneous

In the previous paragraphs the heterogeneous regime and the homogeneous regime are discussed. The question is now, is there a sharp transition between the two flow regimes, or is this more complicated. Figure 7.11-2, Figure 7.11-3, Figure 7.11-4, Figure 7.11-5, Figure 7.11-6 and Figure 7.11-7 show experiments of Clift et al. (1982) in pipes with 0.2032 m and 0.44 m diameters. The particle diameters in these experiments were 0.29 mm, 0.42 mm and 0.68 mm. The graphs show that the behavior near the ELM curve depends on the pipe diameter. The experiments in the 0.2 m diameter pipe show a rather smooth transition, while the experiments in the 0.44 diameter pipe show a collapse of the relative excess hydraulic gradient near the intersection point of the two flow regimes. Wilson et al. (1992) explain this phenomena with near wall lift, based on the ratio of the submerged weight of a particle to the lift force. There may, however be other phenomena influencing this behavior.

## 7.11.2 The Lift Ratio

When the particle diameter approaches the thickness of the viscous sub-layer at the transition velocity heterogeneous-homogeneous a peculiar phenomenon occurs. On one hand the kinetic losses collapse due to the lift forces available just above the viscous sub-layer, assuming there are no lift forces in the viscous sub-layer, on the other hand the eddies in the turbulent layer are not yet strong enough to integrate the particles with the main flow. When the line speed increases, the lift forces are strong to counteract the submerged weight and kinetic energy of the particles, but not enough to integrate the particles in the eddy behavior. This results in a sort of delay between the collapse of the kinetic losses and the mobilization of the RELM (Reduced ELM).

Wilson et al. (2010) introduced the lift force on a particle as:

$\ \mathrm{F}_{\mathrm{L}}=\mathrm{C}_{\mathrm{L}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{u}_{*}^{\mathrm{2}} \cdot \frac{\pi}{4} \cdot \mathrm{d}^{2}$

And for the submerged weight of the particle (including the shape factor):

$\ \mathrm{F}_{\mathrm{G}}=\left(\rho_{\mathrm{s}}-\rho_{\mathrm{l}}\right) \cdot \mathrm{g} \cdot \frac{\pi}{\mathrm{6}} \cdot \mathrm{d}^{\mathrm{3}} \cdot \Psi$

Giving, assuming a shape factor of about 0.75:

$\ \mathrm{L}_{\mathrm{R}}=\frac{\mathrm{F}_{\mathrm{L}}}{\mathrm{F}_{\mathrm{G}}}=\frac{\mathrm{C}_{\mathrm{L}} \cdot \frac{1}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{u}_{*}^{2} \cdot \frac{\pi}{4} \cdot \mathrm{d}^{2}}{\left(\rho_{\mathrm{s}}-\rho_{\mathrm{l}}\right) \cdot \mathrm{g} \cdot \frac{\pi}{6} \cdot \mathrm{d}^{3} \cdot \mathrm{\Psi}}=\mathrm{C}_{\mathrm{L}} \cdot \frac{\mathrm{3}}{4} \cdot \frac{\mathrm{u}_{*}^{2}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g} \cdot \mathrm{d} \cdot \Psi}=\mathrm{C}_{\mathrm{L}} \cdot \frac{\mathrm{3}}{32} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{g} \cdot \mathrm{d} \cdot \mathrm{\Psi}}=\mathrm{C}_{\mathrm{L}} \cdot \mathrm{\theta}$

The difference in kinetic energy of a particle with a horizontal and a vertical velocity component (the hindered settling velocity) compared with the kinetic energy of a particle with only the horizontal velocity component is:

$\ \mathrm{E}_{\mathrm{K}}=\frac{\mathrm{1}}{2} \cdot \mathrm{m}_{\mathrm{p}} \cdot \mathrm{v}_{\mathrm{t h}}^{\mathrm{2}}=\frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{s}} \cdot \frac{\pi}{6} \cdot \mathrm{d}^{\mathrm{3}} \cdot \Psi \cdot \mathrm{v}_{\mathrm{t h}}^{2}$

The force to reduce the vertical component of a particle to zero over a certain distance x equals:

$\ \mathrm{F}_{\mathrm{K}}=\frac{\mathrm{1}}{\mathrm{2}} \cdot \mathrm{m}_{\mathrm{p}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{\mathrm{2}}}{\mathrm{x}}=\frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{s}} \cdot \frac{\pi}{\mathrm{6}} \cdot \mathrm{d}^{\mathrm{3}} \cdot \Psi \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{\mathrm{2}}}{\mathrm{x}}=\frac{\mathrm{E}_{\mathrm{K}}}{\mathrm{x}}$

Which is the kinetic energy divided by the distance x, so it also matches the principle that the work carried out equals the force times the distance is equal to the destroyed kinetic energy.

To prevent a particle from having a collision with the pipe wall, the lift force has to counteract both the submerged weight of the particle and the kinetic energy force, giving for the lift ratio:

$\ \mathrm{L}_{\mathrm{R}}=\frac{\mathrm{F}_{\mathrm{L}}}{\mathrm{F}_{\mathrm{G}}+\mathrm{F}_{\mathrm{K}}}=\frac{\mathrm{C}_{\mathrm{L}} \cdot \frac{1}{2} \cdot \mathrm{\rho}_{\mathrm{l}} \cdot \mathrm{u}_{*}^{2} \cdot \frac{\pi}{4} \cdot \mathrm{d}^{2}}{\left(\rho_{\mathrm{s}}-\rho_{\mathrm{l}}\right) \cdot \mathrm{g} \cdot \frac{\pi}{6} \cdot \mathrm{d}^{3} \cdot \Psi+\frac{1}{2} \cdot \mathrm{\rho}_{\mathrm{s}} \cdot \frac{\pi}{6} \cdot \mathrm{d}^{3} \cdot \Psi \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{\mathrm{2}}}{\mathrm{x}}} \approx \frac{\mathrm{C}_{\mathrm{L}} \cdot \mathrm{u}_{*}^{2}}{\mathrm{d} \cdot\left(\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g}+\frac{1}{2} \cdot \frac{\rho_{\mathrm{s}}}{\rho_{\mathrm{l}}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{2}}{\mathrm{x}}\right)}$

Now suppose the distance to stop the particle equals a number of times the thickness of the viscous sub layer, this gives:

$\ \mathrm{L}_{\mathrm{R}}=\frac{\mathrm{C}_{\mathrm{L}} \cdot \mathrm{u}_{*}^{2}}{\mathrm{d} \cdot\left(\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g}+\frac{1}{2} \cdot \frac{\rho_{\mathrm{s}}}{\rho_{\mathrm{l}}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}^{2} \cdot \mathrm{u}_{*}}{\boldsymbol{\alpha} \cdot \mathrm{1 1 . 6} \cdot \mathrm{v}_{\mathrm{l}}}\right)} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{C}_{\mathrm{v} \mathrm{b}}}\right)$

The concentration term is added, because the higher the concentration, the smaller the capacity of the turbulent flow to generate lift forces. When the spatial concentration Cvs equals the bed concentration Cvb, there will not be any lift forces anymore. For now a linear relation is chosen.

Figure 7.11-1 shows this lift ratio at the intersection line speed between the heterogeneous and homogeneous flow regimes for α=10, Cvs=0.175 and Cvb=0.55 (which is also used in the graphs), where the intersection line speed can be computed with:

$\ \frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{0 . 1 7 5} \cdot(1+\beta)}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+8.5^{2} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{ls}}}\right)^{2}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}$

In reality the intersection line speed will be a bit higher, due to the reduced ELM because of the particle poor viscous sub layer (reduction ELM 50%-60%). From the figure it is shown that for very small particles the submerged weight dominates and the lift ratio hardly reaches a value of 1. For medium sized particles the near wall lift dominates giving a lift ratio above 1. It must be mentioned that for small diameter pipes the lift ratio never reaches 1, while the lift ratio increases with increasing pipe diameter. For large particles both submerged weight and kinetic energy dominate, resulting in a lift ratio never reaching 1. So the conclusion can be drawn that very small and very large particles do not show a collapse of the heterogeneous Erhg, but medium sized particles do, where the collapse is stronger in larger diameter pipes.

So how to model this collapse? The kinetic energy contribution in the heterogeneous flow regime equation is subject to the influence of the near wall lift force, so only this term is reduced. Based on the experiments considered, the reduction is not linear with the lift ratio, but quadratic, resulting in:

$\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{0 . 1 7 5} \cdot(1+\beta)}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+\mathrm{8 .5}^{2} \cdot\left(\frac{\mathrm{l}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{ls}}}\right)^{2} \cdot\left(1-\mathrm{L}_{\mathrm{R}}^{2}\right)$

When the lift ratio has a value close to 1, theoretically there are no more collisions with the wall. However not all particles will have exactly the same kinetic energy, so even when the lift ratio is larger than 1, still some particles will have collisions. Therefore a smoothing function is proposed for lift ratio’s larger than 70% (ζ=0.5), giving:

$\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\mathrm{0 . 1 7 5} \cdot(1+\beta)}\right)^{\beta}}{\mathrm{v}_{\mathrm{ls}}}+8.5^{2} \cdot\left(\frac{\mathrm{1}}{\lambda_{\mathrm{l}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right)^{2} \cdot(1-\zeta) \cdot \frac{\zeta}{\mathrm{L}_{\mathrm{R}}^{2}}$

Figure 7.11-2, Figure 7.11-3, Figure 7.11-4, Figure 7.11-5, Figure 7.11-6 and Figure 7.11-7 show the decrease of the heterogeneous relative excess hydraulic gradient (heterogeneous flow with near wall lift), based on the lift ratio reduction. It is clear that a larger pipe diameter results in a larger decrease. The reason the Erhg does not go to zero, is that the potential energy losses are not affected by the lift ratio.

## 7.11.3 Limit Deposit Velocity & Concentration Distribution

Now that the collapse of the heterogeneous Erhg has been modelled, the question is, how does the homogeneous Erhg behave? The background of the ELM is that small particles rotate with turbulent eddies. This implies that the kinetic rotational energy of an eddy increases if the mass of the eddy increases. Since the mass of an eddy increases with the mixture density, the rotational energy also increases with the mixture density. So at high line speeds, where turbulence has fully developed, head losses will be proportional with the mixture density. Because of the particle poor viscous sub layer there is a lubrication effect, resulting in a smaller increase, as has been described with equation (7.6-40). This equation however assumes that turbulence has fully developed and particles follow the eddies. This may be true at high line speeds, but certainly not at low line speeds. It is also the question whether larger particles will follow this principle. At lower line speeds, the RELM effect will not be fully mobilized. The question is, how much of the RELM effect is mobilized as a function of the line speed?

To answer this question, first the Limit Deposit Velocity (LDV) is considered. The Limit Deposit Velocity is defined as the line speed above which there is no stationary or sliding bed. The definition of Wilson et al. (1992) is the line speed where a bed starts sliding, the Limit of Stationary Deposit Velocity (LSDV). The latter will only occur if the particle diameter to pipe diameter ratio exceeds a certain value. For small particles, the bed will never start sliding, because it already dissolves into heterogeneous transport, before the shear stress on the bed is large enough to initiate a sliding bed. The two different definitions also give different results. The LDV found by Durand & Condolios (1952) gives higher values than the LSDV found by Wilson et al. (1992). Goedde (1978) investigated the LDV by measuring the bed height at different line speeds below the LDV and found the LDV by drawing a straight line through the data points in order to obtain the line speed where the bed height is zero. He carried out experiments on plastic, coal, sand and iron ore. He concluded that the LDV found matched the findings of Durand & Condolios (1952) very well. Durand & Condolios (1952) in their publications only mention that their LDV is the line speed above which there is no deposit, but with the findings of Goedde (1978) this definition can be made more explicit, no deposit means, nor a stationary deposit, nor a sliding bed.

Miedema & Ramsdell (2015A) developed a model for the LDV based on the definition of Goedde (1978) and Durand & Condolios (1952). This definition implies that at the LDV, the spatial concentration at the pipe bottom equals the bed concentration.

The advection diffusion equation when in equilibrium shows the balance between the upwards flow of particles due to diffusion and the downwards flow of particles due to gravity (the terminal settling velocity). Wasp et al. (1977) and Doron et al. (1987) use the solution of the advection diffusion equation for low concentrations, while Karabelas (1977) and Kaushal & Tomita (2002) use the Hunt (1954) approach with upwards liquid flow. Hindered settling is not yet included in the basic solutions, but added by replacing the terminal settling velocity vt by the hindered terminal settling velocity vth.

For the diffusivity and the relation between the sediment diffusivity and the turbulent eddy momentum diffusivity different approaches are possible. Using the Lane & Kalinske (1941) approach, the following equation can be derived for pipe flow:

$\ \mathrm{C}_{\mathrm{v s}}(\mathrm{r})=\mathrm{C}_{\mathrm{v B}} \cdot \mathrm{e}^{-\mathrm{1 2} \cdot \frac{\mathrm{v}_{\mathrm{t h}}}{\beta_{\mathrm{sm}} \cdot \kappa \cdot \mathrm{u}_{*}} \cdot \frac{\mathrm{r}}{\mathrm{D}_{\mathrm{p}}}}$

Now based on the assumption that the diffusivity has to have a value such that at the LDV the concentration at the bottom of the pipe equals the bed concentration (the definition of the LDV), the following equation is derived by Miedema & Ramsdell (2015B):

$\ \mathrm{C}_{\mathrm{v s}}(\mathrm{r})=\mathrm{C}_{\mathrm{v B}} \cdot \mathrm{e}^{-\frac{\alpha_{\mathrm{s m}}}{\mathrm{C}_{\mathrm{v r}}} \cdot \frac{\mathrm{u}_{*, \mathrm{ldv}}}{\mathrm{u}_{*}} \cdot \frac{\mathrm{v}_{\mathrm{t h}}}{\mathrm{v}_{\mathrm{t h}, \mathrm{ldv}}} \cdot \frac{\mathrm{r}}{\mathrm{D}_{\mathrm{p}}}} \quad\text{ with: }\quad \alpha_{\mathrm{sm}}=\mathrm{1 .0 0 4 6 + 0 .1 7 2 7 \cdot \mathrm { C } _ { \mathrm { v r } } - \mathrm { 1 .1 9 0 5 } \cdot \mathrm { C } _ { \mathrm { v r } } ^ { 2 }}$

The settling velocity vth is the hindered settling velocity of the particle, based on the properties of the liquid, adjusted for the homogeneous fraction, resulting in the vehicle liquid according to Wasp et al. (1977). The correction factor αsm appears to depend only on the relative concentration Cvr. The bottom concentration CvB is now for line speeds above the LDV:

$\ \mathrm{C_{v B}=C_{v b} \cdot \frac{u_{*, ld v}}{u_{*}} \cdot \frac{v_{t h}}{v_{t h, l d v}}}$

Figure 7.11-9 shows the concentration profiles for different relative concentrations, adjusted for the circular shape of the pipe at the LDV, compared with data from Kaushal et al. (2005) for a 0.44 mm particle, giving a reasonable match. Since hindered settling is applied for the average vehicle concentration, applying the local concentration may alter the profiles slightly. Especially at higher concentrations, the hindered settling in the upper half of the pipe will be less than in the bottom half of the pipe, resulting in lower concentrations in the upper half of the pipe and higher concentrations in the lower half of the pipe, as shown in Figure 7.11-9.

Now how does this contribute to the mobilization of the RELM? Assume this mobilization m depends on the ratio of the concentration at two levels r1 and r2 in the pipe, this gives:

$\ \mathrm{m=\frac{C_{vs}(r_2)}{C_{vs}(r_1)}=\frac{e^{-\frac{\alpha_{sm}}{C_{vr}}\cdot \frac{u_{*,ldv}}{u_*}\cdot \frac{v_{th}}{v_{th,ldv}}\cdot\frac{r_2}{D_p}}}{e^{-\frac{\alpha_{sm}}{C_{vr}}\cdot \frac{u_{*,ldv}}{u_*}\cdot \frac{v_{th}}{v_{th,ldv}}\cdot \frac{r_1}{D_p}}}=e^{-\left( \frac{r_2}{D_p}-\frac{r_1}{D_p} \right)\cdot \frac{\alpha_{sm}}{C_{vr}}\cdot \frac{u_{*,ldv}}{u_*}\cdot \frac{v_{th}}{v_{th,ldv}}}}$

Wasp et al. (1977) use r1=0.5·Dp and r2=0.92·Dp to determine the vehicle fraction, which is not the same as the mobilization factor of the RELM. Different values are tested and the best outcome was found choosing r1=0.45·Dp and r2=0.55·Dp, giving:

$\ \mathrm{m=\frac{C_{vs}(r_2)}{C_{vs}(r_1)}=\frac{e^{-\frac{\alpha_{sm}}{C_{vr}}\cdot \frac{u_{*,ldv}}{u_*}\cdot \frac{v_{th}}{v_{th,ldv}}\cdot0.55}}{e^{-\frac{\alpha_{sm}}{C_{vr}}\cdot\frac{u_{*,ldv}}{u_*}\cdot \frac{v_{th}}{v_{th,ldv}}\cdot 0.45}}=e^{-0.1 \cdot \frac{\alpha_{sm}}{C_{vr}}\cdot\frac{u_{*,ldv}}{u_*}\cdot\frac{v_{th}}{v_{th,ldv}}}}$

Basically this shows the concentration gradient at the center of the pipe. Figure 7.11-4 and Figure 7.11-5 show the mobilized homogeneous flow Erhg according to:

$\ \mathrm{E_{rhg}=\frac{i_m-i_l}{R_{sd}\cdot C_{vs}}=m \cdot i_l \cdot \left(1-\left(1-\frac{1+R_{sd}\cdot C_{vs}-\left(\frac{A_{C_v}}{\kappa}\cdot ln\left(\frac{\rho_m}{\rho_l} \right)\cdot \sqrt{\frac{\lambda_l}{8}}+1 \right)^2}{R_{sd}\cdot C_{vs}\cdot \left(\frac{A_{C_v}}{\kappa}\cdot ln\left(\frac{\rho_m}{\rho_l} \right)\cdot\sqrt{\frac{\lambda_l}{8}}+1 \right)^2} \right)\left(1-\left(\frac{\delta_v}{d} \right) \right) \right)}$

## 7.11.4 Resulting Relative Excess Hydraulic Gradient Curves

Figure 7.11-10 gives an example of the resulting relative excess hydraulic gradient curves for a 0.2032 m diameter pipe and particles ranging from 0.1 mm to 10 mm, showing the different flow regimes for the transport (delivered) concentration case. Figure 7.11-11 shows the resulting relative excess hydraulic gradient curves for a 0.44 m diameter pipe for the transport (delivered) concentration case. From these two figures it is clear that the larger pipe diameter shows a steeper decrease of the Erhg in the heterogeneous regime for particles of 0.2 mm, 0.3 mm and 0.5 mm, so medium sands. Very small particles are dominated by the submerged weight, while larger particles are dominated by their kinetic energy.

## 7.11.5 Conclusions & Discussion

The transition of different flow regimes is important, but the transition of the heterogeneous flow regime with the homogeneous flow regime is very important, due to the fact that this coincides often with operational line speeds. To model this transition it is necessary to have appropriate models for the heterogeneous and the homogeneous flow regimes. The homogeneous flow regime is modelled as a reduced equivalent liquid model (RELM) with mobilization of the reduction based on the ratio between the thicknesses of the viscous sub layer to the particle diameter. For very small particles there is no reduction at low line speeds. The reduction is in effect at higher line speeds. Medium and large particles encounter this reduction however also at lower line speeds. The heterogeneous flow regime is modelled based on potential and kinetic energy losses, where the potential energy losses are reversely proportional to the line speed and the kinetic energy losses (based on collisions) are reversely proportional to the line speed squared.

At the transition (intersection line speed) of the two flow regimes, the heterogeneous head losses collapse in larger pipe diameters for medium sized particles. It appears that these particles encounter the highest lift forces, compared to their submerged weight and kinetic energy. For very small particles the submerged weight dominates, due to the fact that transition line speed is very small and there is hardly lift. For very large particles the kinetic energy dominates and the lift is not capable to decelerate the particles. But for medium sized particles, the lift is stronger than the combined effect of submerged weight and kinetic energy. In a 0.2032 m diameter pipe the collapse of the collisions is still weak, but in a 0.44 m diameter pipe this is already strong. Larger pipe diameter will most probably show a stronger effect, explaining also why dredging companies state that medium sized particles in large diameter pipes have head losses close to the head losses of pure water.

The RELM is not yet fully mobilized at low line speeds, due to the fact that turbulent eddies are not yet capable of integrating particles in the rotation of the eddies. The higher the line speed, the more the particles become an integrated part of the turbulence (if they are not too large). Based on the definition of the LDV and the concentration distribution equation from the advection diffusion equation, a mobilization factor has been defined for the mobilization of the RELM. The concentration distribution is such that at the LDV the concentration at the bottom of the pipe equals the bed concentration.

Resuming it can be stated that the new model explains for the homogeneous behavior of very small particles, regarding the mobilization of the lubrication effect of the particle poor viscous sub layer. It can also be stated that an explanation is found for the collapse of the heterogeneous head losses of medium sized particles in larger pipes, based on near wall lift and the mobilization of the RELM.

## 7.11.6 Nomenclature

 Acv Coefficient RELM (default 3) - CD Particle drag coefficient - CL Lift coefficient - Cvt Delivered (transport) volumetric concentration - Cvs Spatial volumetric concentration - Cvb Spatial volumetric concentration bed (1-n) - Cvr Relative spatial concentration - d Particle diameter m Dp Pipe diameter m ELM Equivalent liquid model - Erhg Relative excess hydraulic gradient - Ek Kinetic energy in vertical direction N·m FL Lift force on particle N FG Submerged weight of particle N FK Kinetic energy deceleration force N g Gravitational constant 9.81 m/s2 m/s2 il Pure liquid hydraulic gradient m/m im ​​​​​​​ Mixture hydraulic gradient m/m LDV Limit Deposit Velocity m/s LSDV Limit of Stationary Deposit Velocity m/s LR Lift ratio - m Mobilized RELM factor - mp Mass particle kg n Porosity - RELM Reduced equivalent liquid model - r,r1,r2 Vertical distance in pipe m R Stratification ratio Wilson - Rsd Relative submerged density - Shr Settling velocity Hindered Relative - Srs Slip velocity Relative Squared - u* Friction velocity m/s u*,ldv Friction velocity at LDV m/s vls Line speed m/s vsl Slip velocity m/s vt Terminal settling velocity particle m/s vth Hindered terminal settling velocity particle m/s vth,ldv Hindered terminal settling velocity particle at LDV m/s vδv Velocity at viscous sub layer thickness m/s x Distance to decelerate particle m α Number of times thickness viscous sub layer - αsm Coefficient concentration distribution - β Richardson & Zaki hindered settling power - βsm Diffusivity factor - δv Viscous sub layer thickness m ε Pipe wall roughness - κ Von Karman constant (about 0.4) - κC Concentration eccentricity factor - λl Darcy Weisbach friction factor liquid - μsf Sliding friction factor - ρl Density liquid ton/m3 ρm Density mixture ton/m3 ρs Density solid ton/m3 $$\ v_{\mathrm{l}}$$ Kinematic viscosity liquid m2/s ψ Shape factor - ζ Smoothing parameter lift ratio -

7.11: The Transition Heterogeneous vs. Homogeneous in Detail is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Sape A. Miedema via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.