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7.6: A Head Loss Model for Homogeneous Slurry Transport

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    29232
  • 7.6.1 Homogeneous Transport – The Equivalent Liquid Model (ELM)

    Slurry transport in horizontal and vertical pipelines is one of the major means of transport of sands and gravels in the dredging industry. There exist 5 main flow regimes, the fixed or stationary bed regime, the sliding bed regime, the heterogeneous flow regime, the homogeneous flow regime and the sliding flow regime. Of course the transitions between the regimes are not very sharp, depending on parameters like the particle size distribution. In the case of very fine particles and/or very high line speeds, the mixture is often considered to be a liquid with the liquid density ρl equal to the mixture density ρm, where the liquid density ρl can be replaced by the mixture density ρm in the hydraulic gradient equations. The velocity profile in a cross section of the pipe is considered to be symmetrical and the slip between the particles and the liquid is considered negligible. The concentration is assumed to be uniform over the cross section. Thus, the transport (or delivered) concentration and the spatial concentration are almost equal and will be named Cv. This is often referred to as the equivalent liquid model (ELM).

    Since the pressure losses are often expressed in terms of the hydraulic gradient, first some basic equations for the hydraulic gradient and the relative excess hydraulic gradient (solids effect) are given. The hydraulic gradient according to the equivalent liquid model is:

    \[\ \mathrm{i}_{\mathrm{m}}=\frac{\Delta \mathrm{p}_{\mathrm{m}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \quad\text{ with: }\quad \lambda_{\mathrm{m}}=\lambda_{\mathrm{l}}\]

    Where it is assumed that the Darcy-Weisbach friction factors for liquid λl and mixture λm are equal. This can also be written as:

    \[\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}\right)\]

    Newitt et al. (1955) found that only 60% of the solids weight should contribute to the mixture density in order to obtain the equivalent liquid model, but this depends on the line speed and possibly on other parameters as well. Many others also found hydraulic gradients below the ELM at high line speeds. Wilson et al. (1992) explain this with the effect of near wall lift resulting in an almost particle free viscous sub layer. However for very small particles values are found giving a higher value of the hydraulic gradient, which is often explained by correcting (increase) the apparent kinematic viscosity, for example with the Thomas (1965) equation. The pressure losses can be shown in an almost dimensionless form in a double logarithmic graph with the relative excess hydraulic gradient Erhg as the ordinate and the hydraulic liquid gradient il as the abscissa. In terms of the relative excess hydraulic gradient, Erhg, the above equation can be written as:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\mathrm{i}_{\mathrm{l}}\]

    So in the Erhg(il) graph the above equation results in a straight line giving Erhg=il. Figure 7.6-1 shows experimental data of Thomas (1976) of d=0.04 mm iron ore in a Dp=0.1075 m horizontal pipe versus the Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework, where the 4 term Thomas (1965) viscosity equation and the homogeneous flow correction equation (7.6-37) with ACv=3 are implemented. The match is remarkable.

    7.6.2 Approach

    In order to test the Talmon (2011) & (2013) method of incorporating a particle free viscous sub-layer and to check if there are alternative methods the following approach is followed:

    1. Method 1: First the Talmon (2011) & (2013) method is discussed briefly.
    2. Method 2: Since the Talmon (2011) & (2013) method uses a 2D approach, with von Driest damping (Schlichting, 1968), but without a real concentration profile, in this second method the equations are derived for pipe flow with the Nikuradse (1933) mixing length equation, without von Driest damping (Schlichting, 1968) and without a real concentration profile. The results are corrected for the volume flow.
    3. Method 3: Von Driest damping (Schlichting, 1968) is added to method 2, resulting in a velocity profile comparable and very close to the Talmon (2011) & (2013) method 1. So method 1 and method 3 are equivalent.
      Figure 7.6-1: The Thomas (1976) experimental data in a im(vls) graph without Thomas (1965) viscosity.

      Screen Shot 2020-07-28 at 10.22.55 AM.png

      Figure 7.6-27.6-3: The Thomas (1976) experimental data in a im(vls) graph with Thomas (1965) viscosity.

      Screen Shot 2020-07-28 at 10.24.20 AM.png

    4. Method 4: The “Law of the Wall” 2D approach without von Driest damping (Schlichting, 1968).
    5. The 4 methods are compared and an equation describing the average behavior is derived.
    6. Method 5: Finally a concentration profile is added to method 2. This method can simulate all previous methods, depending on the concentration profile chosen.
    7. Based on experiments a value for the parameters of the concentration profile is chosen.

    7.6.3 Method 1: The Talmon (2011) & (2013) Homogeneous Regime Equation

    Talmon (2011) & (2013) derived an equation to correct the homogeneous equation (the ELM model) for the slurry density, based on the hypothesis that the viscous sub-layer hardly contains solids at very high line speeds in the homogeneous regime. This theory results in a reduction of the resistance compared with the ELM, but the resistance is still higher than the resistance of clear liquid. Talmon (2011) & (2013) used the Prandtl approach for the mixing length, which is a 2D approach for open channel flow with a free surface.

    The Prandtl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968). The Talmon (2011) & (2013) approach resulted in the following equation, with αh=6.7:

    \[\ \mathrm{\frac{\lambda_m}{\lambda_l}=\frac{1}{\left(\alpha_h \cdot \sqrt{\frac{\lambda_l}{8}}\cdot R_{sd} \cdot C_v +1 \right)^2} \quad\text{and}\quad E_{rhg}=i_l \cdot \frac{R_{sd}\cdot C_v +1 - \left(\alpha_h \cdot \sqrt{\frac{\lambda_l}{8}}\cdot R_{sd}\cdot C_v +1\right)^2}{R_{sd}\cdot C_v \cdot \left(\alpha_h \cdot \sqrt{\frac{\lambda_l}{8}}\cdot R_{sd}\cdot C_v +1\right)^2}=\alpha_E \cdot i_l} \]

    This equation underestimates the hydraulic gradient (overestimates the effect of a particle free viscous sub layer) in a number of cases (small and large particles) as Talmon (2011) & (2013) proves with the examples shown in his papers. Only for d50=0.37 mm and Dp=0.15 m (medium particles) there is a good match. The philosophy behind this theory, combining a viscous sub-layer with liquid with a kernel with mixture, is however very interesting, because it explains fundamentally why the hydraulic gradient can be lower than the hydraulic gradient according to the ELM, as has been shown by many researchers. The model has been derived using the standard mixing length equation for 2D flow and without a concentration distribution. When reproducing this method it was found that the coefficient αh is not a constant but this coefficient depends on the value of Rsd·Cv according to Figure 6.25-1. The value of 6.7 is found for a value of about 0.6 of the abscissa.

    Figure 7.6-4: The coefficient αh as a function of Rsd·Cv for method 1.

    Screen Shot 2020-07-28 at 10.50.25 AM.png

    7.6.4 Method 2: The Approach using the Nikuradse (1933) Mixing Length

    The concept of Talmon (2011) & (2013) is adopted, but modified by using pipe flow with the Prandtl (1925) and Nikuradse (1933) mixing length equations, a linear shear stress distribution with a maximum at the pipe wall and zero in the center and a concentration distribution, assuming that in the homogeneous regime the mixture can be considered a Newtonian liquid with properties slightly different from those of water. The shear stress between the mixture, the slurry, and the pipe wall is the sum of the viscous shear stress and the turbulent shear stress:

    \[\ \tau=\tau_{v}+\tau_{\mathrm{t}}=\mu_{v} \cdot \frac{\partial \mathrm{u}}{\partial \mathrm{z}}+\mu_{\mathrm{t}} \cdot \frac{\partial \mathrm{u}}{\partial \mathrm{z}}=\rho_{\mathrm{m}} \cdot v_{\mathrm{m}} \cdot \frac{\partial \mathrm{u}}{\partial \mathrm{z}}+\rho_{\mathrm{m}} \cdot v_{\mathrm{t}} \cdot \frac{\partial \mathrm{u}}{\partial \mathrm{z}} \quad\text{ with: }\quad v_{\mathrm{t}}=\ell^{2} \cdot\left|\frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right|\]

    Now the shear stress can be expressed as (with z the distance from the pipe wall):

    \[\ \tau=\rho_{\mathrm{m}} \cdot\left(\mathrm{u}_{*}\right)^{2} \cdot\left(\frac{\mathrm{R}-\mathrm{z}}{\mathrm{R}}\right)=\rho_{\mathrm{m}} \cdot\left(v_{\mathrm{m}}+\ell^{2} \cdot\left|\frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right|\right) \cdot \frac{\partial \mathrm{u}}{\partial \mathrm{z}} \quad\text{ with: }\quad \mathrm{R}=\mathrm{D}_{\mathrm{p}} / 2\]

    This is a second degree function of the velocity gradient. Solving this with respect to the velocity gradient gives:

    \[\ \mathrm{\frac{\partial u}{\partial z}=\frac{-\frac{\mu_m}{\rho_m}+\sqrt{\left(\frac{\mu_m}{\rho_m} \right)^2+4 \cdot \ell^2 \cdot (u_*)^2 \cdot \left(\frac{R-z}{R} \right)}}{2 \cdot \ell^2}=\frac{2 \cdot (u_*)^2 \cdot \left(\frac{R-z}{R} \right)}{\frac{\mu_m}{\rho_m}+\sqrt{\left(\frac{\mu_m}{\rho_m} \right)^2 +4 \cdot \ell^2 \cdot (u_*)^2 \cdot \left(\frac{R-z}{R} \right)}}} \]

    A required condition for pipe flow is, that the integral of the velocity over the pipe cross-section equals the average line speed times the cross-section, so:

    \[\ \mathrm{ \int_{0}^{R} \frac{\partial u}{\partial z} \cdot 2 \cdot \pi \cdot(R-z) \cdot d z=v_{l s} \cdot \pi \cdot R^{2}}\]

    Figure 7.6-5: The resulting velocity distributions in a Dp=1 m pipe, corrected for the flow.

    Screen Shot 2020-07-28 at 11.37.16 AM.png

    The Nikuradse (1933) equation for the mixing length in pipe flow for large Reynolds numbers is:

    \[\ \frac{\ell}{\mathrm{R}}=\mathrm{0 .1 4 - 0 .0 8 \cdot \left( 1 - \frac { \mathrm { z } } { \mathrm { R } } \right) ^ { 2 } - \mathrm { 0 .0 6 } \cdot \left( \mathrm { 1 - \frac { z } { \mathrm { R } } } \right) ^ { 4 }}=\mathrm{0 .1 4 \cdot \left( 1 - \frac { 4 } { 7 } \cdot \left( \mathrm { 1 } - \frac { \mathrm { z } } { \mathrm { R } } \right) ^ { 2 } - \frac { \mathrm { 3 } } { 7 } \cdot \left( \mathrm { 1 } - \frac { \mathrm { z } } { \mathrm { R } } \right) ^ { 4 } \right)}\]

    The velocity profile can be determined by numerical integration. This velocity profile however, should result in an average velocity equal to the line speed used to determine the friction velocity. This appeared to be valid for very high line speeds (Reynolds numbers) in the range of 1300-1500 m/sec, which is far above the range dredging companies operate (3-10 m/sec). For line speeds in the range 3-10 m/sec, the velocity profile resulted in an average velocity smaller than the line speed with a factor 0.8-1.0. Introducing a factor β and an extra term in the mixing length equation solves this problem. The original Nikuradse (1933) equation is multiplied with a factor β and an extra term is added to ensure that l/R=κ·z for z=0, like is the case in the original equation. The factor β is determined for each calculation in such a way, that the flow following from the line speed times the cross section of the pipe, equals the flow from integration of the velocity profile.

    \[\ \frac{\ell}{\mathrm{R}}=\beta \cdot\left(\mathrm{0.14}-\mathrm{0.08} \cdot\left(\mathrm{1}-\frac{\mathrm{z}}{\mathrm{R}}\right)^{2}-\mathrm{0 . 0 6} \cdot\left(\mathrm{1}-\frac{\mathrm{z}}{\mathrm{R}}\right)^{4}\right)+(\mathrm{1}-\boldsymbol{\beta}) \cdot \mathrm{\kappa} \cdot \frac{\mathrm{z}}{\mathrm{R}} \cdot \mathrm{e}^{-\frac{\mathrm{z}}{\mathrm{0} . \mathrm{0 0 0 0 2} \cdot \mathrm{R}}}\]

    Figure 7.6-5 shows the resulting velocity distributions for a 1 m diameter pipe. Now the concept is, that a mixture flow with liquid as a carrier liquid in the viscous sub layer will have a lower resistance than a mixture flow with mixture in the viscous sub-layer. One can also say that in order to get the same pipeline resistance, the velocity in the center of the pipe of mixture with liquid in the viscous sub-layer um has to be higher than the case with mixture or liquid in the whole pipe ul. Assuming that the dynamic viscosity of the mixture is equal to the dynamic viscosity of the carrier liquid, μm=μl, in the viscous sub-layer and the boundary layer where no solids are present, gives:

    \[\ \frac{\partial \mathrm{u}}{\partial \mathrm{z}}=\frac{-\frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}} \cdot v_{\mathrm{l}}+\sqrt{\left(\frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}} \cdot v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(\mathrm{u}_{*}\right)^{2} \cdot\left(\frac{\mathrm{R}-\mathrm{z}}{\mathrm{R}}\right)}}{\mathrm{2} \cdot \ell^{2}}\]

    Now assuming that the term with the density ratio is relevant only near the pipe wall and not in the center of the pipe, this equation will simulate a mixture with liquid in the viscous sub-layer. In fact, the density ratio reduces the effect of the kinematic viscosity, which mainly affects the viscous sub-layer. The velocity difference in the center of the pipe between mixture and liquid, um-ul, can now be determined with:

    \[\ \mathrm{u}_{\mathrm{m}}-\mathrm{u}_{\mathrm{l}}=\int_{\mathrm{0}}^{\mathrm{R}}\left(\frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right)_{\mathrm{m}} \cdot \mathrm{d z}-\int_{\mathrm{0}}^{\mathrm{R}}\left(\frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right)_\mathrm{1} \cdot \mathrm{d z}=\int_{\mathrm{0}}^{\mathrm{R}} \frac{-\frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}} \cdot v_{\mathrm{l}}+\sqrt{\left(\frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}} \cdot v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(\mathrm{u}_{*}\right)^{2} \cdot\left(\frac{\mathrm{R}-\mathrm{z}}{\mathrm{R}}\right)}}{\mathrm{2} \cdot \ell^{2}} \cdot \mathrm{d} \mathrm{z}
    -\int_{0}^{R} \frac{-v_{\mathrm{l}}+\sqrt{\left(v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(u_{*}\right)^{2} \cdot\left(\frac{\mathrm{R-z}}{\mathrm{R}}\right)}}{2 \cdot \ell^{2}} \cdot \mathrm{dz}\]

    This velocity difference, in the center of the pipe, is about equal to the difference of the average line speeds, however both can be determined numerically. Further it appears from the numerical solution of this equation, that dividing the velocity difference by the average liquid velocity ul or vls,l results in a factor F, which only depends on the volumetric concentration Cv, the relative submerged density Rsd and slightly on the line speed vls in the range 3-10 m/sec and on the pipe diameter Dp through the Darcy-Weisbach friction factor λl, according to:

    \[\ \mathrm{F}=\frac{\mathrm{u}_{\mathrm{m}}-\mathrm{u}_{\mathrm{l}}}{\mathrm{u}_{\mathrm{l}}}=\frac{\mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{m}}-\mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{l}}}{\mathrm{v}_{\mathrm{ls}, \mathrm{l}}}=\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}=\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot\left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}-\mathrm{1}\right)\]

    The shear stress at the pipe wall of a Newtonian liquid is by definition:

    \[\ \rho_{\mathrm{l}} \cdot\left(\mathrm{u}_{*}\right)^{2}=\frac{\lambda_{\mathrm{l}}}{\mathrm{8}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls,l}}^{2} \quad\text{ and }\quad \rho_{\mathrm{m}} \cdot\left(\mathrm{u}_{*}\right)^{2}=\frac{\lambda_{\mathrm{m}}}{\mathrm{8}} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{ls}, \mathrm{m}}^{2}\]

    From this a relation for the ratio of the Darcy-Weisbach friction coefficients of a flow with mixture in the center and carrier liquid in the viscous sub-layer to a flow with 100% liquid can be derived.

    \[\ \frac{\lambda_{\mathrm{l}}}{\mathrm{8}} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{l}}^{2}=\frac{\lambda_{\mathrm{m}}}{\mathrm{8}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{m}}^{2} \quad \Rightarrow \quad \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l}}^{2}}{\mathrm{v}_{\mathrm{l s}, \mathrm{m}}^{2}} \quad\Rightarrow \quad\frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l}}^{\mathrm{2}}}{\left(\mathrm{F} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{l}}+\mathrm{v}_{\mathrm{l s}, \mathrm{l}}\right)^{2}}=\frac{\mathrm{1}}{(\mathrm{F}+\mathrm{1})^{2}}\]

    Equation (7.6-15) is independent of the method used, but the factor F, the velocity ratio, is. Substituting the factor F from equation (7.6-13) gives:

    \[\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{1}{(\mathrm{F}+1)^{2}}=\frac{1}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}}+1\right)^{2}}\]

    This ratio depends on the homogeneous factor αh, the Darcy-Weisbach friction factor λl, the volumetric concentration Cv and the relative submerged density Rsd. The ratio of the hydraulic gradients is now:

    \[\ \frac{\mathrm{i}_{\mathrm{m}}}{\mathrm{i}_{\mathrm{l}}}=\frac{\lambda_{\mathrm{m}} \cdot \rho_{\mathrm{m}}}{\lambda_{\mathrm{l}} \cdot \rho_{\mathrm{l}}}=\frac{\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}} \quad\Rightarrow \quad\mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}\]

    This gives for the excess hydraulic gradient im-il (the solids effect):

    \[\ \mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}-\mathrm{i}_{\mathrm{l}} \cdot \frac{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}} =\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}-\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}} \]

    The relative excess hydraulic gradient Erhg is now:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}-\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}} \cdot\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}\right)^{2}}=\mathrm{\alpha}_{\mathrm{E}} \cdot \mathrm{i}_{\mathrm{l}}\]

    The limiting value for the relative excess hydraulic gradient Erhg for a volumetric concentration Cv approaching zero, becomes:

    \[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}=\mathrm{i}_{\mathrm{l}} \cdot\left(1-\mathrm{2} \cdot \alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}}\right)\]

    For sand and gravel with a density of 2.65 ton/m3, the factor αh is about 9.3, almost independent of the pipe diameter Dp and the line speed vls for pipes with diameters of 0.5 m up to 1.2 m and line speeds from 2 m/sec up to 10 m/sec. For very small pipes and very low line speeds, like Dp=0.1 m and vls=1 m/sec, this factor decreases to about 8.5. The factor αh is not 100% linear with the term Rsd·Cv for sands with a density of 2.65 ton/m3 and volumetric concentrations up to 35-40%. Since the solution depends on Rsd·Cv combined, the factor αh also depends on this and not on Rsd and Cv separately. Figure 7.6-6 shows the dependency of the factor αh on the relative excess density Rsd·Cv. The factor αE decreases with increasing concentration and relative submerged density of the solids and increases with increasing line speed. At normal line speeds (3-6 m/sec) and concentrations (0.1-0.3) this factor is about 0.74-0.78 (see Figure 7.6-11). A larger pipe gives less reduction. This is caused by the smaller Darcy-Weisbach friction coefficient λl of larger pipes.

    Figure 7.6-6: The homogeneous factor αh as a function of Rsd·Cv for method 2.

    Screen Shot 2020-07-28 at 12.32.37 PM.png

    7.6.5 Method 3: Adding the von Driest (Schlichting, 1968) Damping to Method 2

    Talmon (2013) used the Prandtl approach for the mixing length, which is a 2D approach for open channel flow with a free surface. The Prandtl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968):

    \[\ \begin{array}{left}\text{Prandl: } &\ell=\kappa \cdot \mathrm{z}\\
    \text{von Driest : }&\ell=\kappa \cdot \mathrm{z} \cdot\left(1-e^{-z^{+} / \mathrm{A}}\right) \quad\text{ with: }\quad \mathrm{z}^{+}=\frac{\mathrm{z} \cdot \mathrm{u}_{*}}{v_{\mathrm{l}}} \quad \mathrm{A}=\mathrm{2 6}\end{array}\]

    Figure 7.6-9 and Figure 7.6-7 show the velocity profile and the mixing length profile of the Talmon (2013) approach with von Driest (Schlichting, 1968) damping and the Nikuradse (1933) approach without damping. In both cases, the mixing length equations have been corrected in order to get the correct volume flow. There is a clear difference of the velocity profiles. Applying the von Driest (Schlichting, 1968) damping to the Nikuradse (1933) equation (7.6-9) for the mixing length in pipe flow for large Reynolds numbers according to:

    \[\ \frac{\ell}{\mathrm{R}}=\mathrm{0 .1 4} \cdot\left(\mathrm{1 - \frac { 4 } { 7 }} \cdot \left(\mathrm{1 - \frac { z } { \mathrm { R } }}\right)^{\mathrm{2}}-\frac{\mathrm{3}}{\mathrm{7}} \cdot\left(\mathrm{1}-\frac{\mathrm{z}}{\mathrm{R}}\right)^{4}\right) \cdot\left(\mathrm{1 - e}^{-\mathrm{z}^{+} / \mathrm{A}}\right)\]

    Gives almost exactly the same results as the Talmon (2013) approach, although the mixing length is completely different as is shown in Figure 7.6-8. Only very close to the wall, where the viscous effects dominate, the same mixing lengths are found. Apparently, the von Driest damping, effective close to the pipe wall, dominates the effect of a particle free viscous sub layer, as expected. The results are almost independent of the pipe diameter Dp and the line speed vls. Figure 7.6-10 shows that the velocity profiles determined with equation (7.6-21) (Talmon) and equation (7.6-22) (Miedema) are almost the same and the behavior with respect to the hydraulic gradient reduction is equivalent.

    7.6.6 Method 4: The Law of the Wall Approach

    Often for open channel flow the so called “Law of the Wall” equations are used. Since in dredging the pipe wall is assumed to be smooth due to the continuous sanding of the pipe wall, the smooth wall approach is discussed here. Based on the following assumption for the mixing length by Prandtl (1925) and the assumption that the viscous shear stress is negligible in the turbulent region, the famous logarithmic velocity equation, “Law of the Wall” for the turbulent flow is derived:

    \[\ \tau=\tau_{\mathrm{wall}} \cdot\left(1-\frac{\mathrm{z}}{\mathrm{R}}\right)=\rho_{\mathrm{l}} \cdot \ell^{2} \cdot\left(\frac{\mathrm{du}}{\mathrm{dz}}\right)^{2}\text{ and }\ell=\kappa \cdot \mathrm{z} \cdot\left(1-\frac{\mathrm{z}}{\mathrm{R}}\right)^{0.5}\]

    This "Law of the Wall" is also a 2D approach for open channel flow and does not correct for pipe flow. The general equation for the velocity profile as a function of the distance to the smooth wall is:

    \[\ \mathrm{u}(\mathrm{z})=\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{z}}{\mathrm{z}_{0}}\right) \quad\text{ with: }\quad \mathrm{z}_{0}=\mathrm{0 .1 1} \cdot \frac{v_{\mathrm{l}}}{\mathrm{u}_{*}}\]

    For the 100% liquid (or mixture) the velocity profile is defined as:

    \[\ \mathrm{u}_{\mathrm{l}}(\mathrm{z})=\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{z}}{\mathrm{z}_{\mathrm{0}, \mathrm{l}}}\right) \quad\text{ with: }\quad \mathrm{z}_{\mathrm{0}, \mathrm{1}}=\mathrm{0 .1 1} \cdot \frac{v_{\mathrm{l}}}{\mathrm{u}_{*}}\]

    For the mixture with liquid in the viscous sub-layer the velocity profile can be defined as:

    \[\ \mathrm{u}_{\mathrm{m}}(\mathrm{z})=\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{z}}{\mathrm{z}_{0, \mathrm{m}}}\right) \quad\text{ with: }\quad \mathrm{z}_{0, \mathrm{m}}=\mathrm{0 .1 1} \cdot \frac{v_{\mathrm{l}}}{\mathrm{u}_{*}} \cdot \frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}}\]

    The velocity difference at the center of the pipe is now:

    \[\ \begin{array}{left}\mathrm{u}_{\mathrm{m}}(\mathrm{R})-\mathrm{u}_{\mathrm{l}}(\mathrm{R})&=\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{R}}{\mathrm{z}_{0, \mathrm{m}}}\right)-\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{R}}{\mathrm{z}_{0, \mathrm{l}}}\right)\\
    &=\frac{\mathrm{u}_{*}}{\kappa} \cdot \ln \left(\frac{\mathrm{R} \cdot \mathrm{u}_{*}}{\mathrm{0} \cdot \mathrm{1 1} \cdot v_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right)-\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\mathrm{R} \cdot \mathrm{u}_{*}}{\mathrm{0} . \mathrm{1 1} \cdot v_{\mathrm{l}}}\right)=\frac{\mathrm{u}_{*}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right)\end{array}\]

    This gives for the Darcy-Weisbach friction coefficient ratio:

    \[\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{I}}}=\frac{\mathrm{v}_{\mathrm{ls}, \mathrm{l}}^{2}}{\mathrm{v}_{\mathrm{ls}, \mathrm{m}}^{2}}=\frac{\mathrm{u}_{\mathrm{l}}^{2}}{\mathrm{u}_{\mathrm{m}}^{2}}=\frac{\mathrm{u}_{\mathrm{l}}}{\left(\frac{\mathrm{u}_{*}}{\kappa} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right)+\mathrm{u}_{\mathrm{l}}\right)^{2}}=\frac{1}{\left(\frac{1}{\kappa} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{8}}+1\right)^{2}} \quad\text{ with: }\quad \mathrm{u}_{*}=\sqrt{\frac{\lambda_{\mathrm{l}}}{8}}{ \cdot \mathrm{u}_{\mathrm{l}}}\]

    The relative excess hydraulic gradient Erhg is now:

    \[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}}-\left(\frac{\mathrm{1}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+1}\right)^{2}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}} \cdot\left(\frac{1}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+1}\right)^{2}}=\alpha_{\mathrm{E}} \cdot \mathrm{i}_{\mathrm{l}}\]

    Figure 7.6-7: The mixing length versus the distance to the wall for a Dp=1 m pipe at a line speed vls=5 m/sec, κPrandtl=0.4, κvonDriest=0.3915 & κNikuradse=0.4, without von Driest (Schlichting, 1968) damping.

    Screen Shot 2020-07-28 at 3.43.32 PM.png

    Figure 7.6-8: The mixing length versus the distance to the wall for a Dp=1 m pipe at a line speed vls=5 m/sec, κPrandtl=0.4, κvonDriest=0.3915 & κNikuradse=0.4, with von Driest (Schlichting, 1968) damping.

    Screen Shot 2020-07-28 at 3.45.11 PM.png

    Figure 7.6-9: The velocity versus the distance to the wall for a Dp=1 m pipe at a line speed vls=5 m/sec, κPrandtl=0.4, κvonDriest=0.3915 & κNikuradse=0.4, without von Driest (Schlichting, 1968) damping.

    Screen Shot 2020-07-28 at 3.46.23 PM.png

    Figure 7.6-10: The velocity versus the distance to the wall for a Dp=1 m pipe at a line speed vls=5 m/sec, κPrandtl=0.4, κvonDriest=0.3915 & κNikuradse=0.4, with von Driest (Schlichting, 1968) damping.

    Screen Shot 2020-07-28 at 3.47.29 PM.png

    7.6.7 Comparison of the Models

    Now 3 formulations are found for the reduction of the Darcy-Weisbach friction factor and the relative excess hydraulic gradient Erhg for slurry transport of a mixture with pure carrier liquid (water) in the viscous sub-layer, these are equations (7.6-4), (7.6-16) & (7.6-19) and (7.6-28) & (7.6-29):

    Law of the Wall (no damping)

    Nikuradse (no damping)

    Pr andl von Driest (damping)

    \(\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{1}}{\left(\frac{1}{\mathrm{\kappa}} \cdot \ln \left(1+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+\mathrm{1}}\right)^{2}}\) \(\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{1}{\left(\alpha_{\mathrm{h}} \cdot \lambda_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}}+1\right)^{2}}\) \(\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{1}}{\left(\alpha_{\mathrm{h}} \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{ \cdot \mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}+\mathrm{1}}\right)^{2}}\)

    κ = 0.4

    αh = 9.3

    αh = 6.7

    Miedema

    Miedema

    Talmon

    Since the above solutions are not (very) sensitive for changes in the pipe diameter Dp or the line speed vls, but mainly for changes of the density ratio ρml, a comparison is made for a Dp=1 m diameter pipe at a line speed of vls=5 m/sec in sand with a solids density of 2.65 ton/m3 and a virtual solid with a density of 10 ton/m3. Figure 7.6-11 and Figure 7.6-12 show the Darcy-Weisbach friction factor ratios and the relative excess hydraulic gradient coefficient αE. The methods 2 and 4 without damping do not differ too much, both give a reduction on the solids effect of about 18-26% in sand for medium concentrations. The methods 1 and 3 with damping however give a reduction on the solids effect of about 55-65% in sand, almost 3 times as much. For the virtual solid with a solids density of 10 ton/m3, the reductions are 18-30% and 65-80%. Based on the data as shown by Talmon (2013), the reduction of the solids effect of 55-65% with method 1 with damping is overestimating the reduction, while the two methods without damping seem to underestimate the reduction, assuming that the reduction measured is caused by the effect of a lubricating viscous sub-layer.

    If damping is added to the Nikuradse (1933) mixing length equation, method 3, the same results are obtained as the Prandtl mixing length equation with von Driest (Schlichting, 1968) damping, method 1. Apparently the mixing length damping dominates the difference between the 3 methods. The von Driest modification is an empirical damping function that fits experimental data, and also changes the near-wall asymptotic behavior of the eddy viscosity νt, from z2 to z4. Although neither of them are correct (DNS-data gives νt proportional to z3), the von Driest damping generally improves the predictions. It has, since its first appearance, repeatedly been used in turbulence models to introduce viscous effects in the near-wall region. The von Driest damping however has never been developed to deal with the problem of a lubricating viscous sub-layer as is elaborated in this chapter.

    Because of the overestimation of methods 1 and 3 and the underestimation of methods 2 and 4, an average between Prandtl without damping, method 3, and Prandtl with damping, method 1, could be used according to (with αh about 6.7):

    \[\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{\mathrm{1}}{\left(\left(\frac{\left.\frac{\mathrm{1}}{\mathrm{\kappa}} \cdot \ln \left(1+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}\right)+\alpha_{\mathrm{h}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}\right)}{2} \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+\mathrm{1}}\right)^{2}\right.}\]

    The results of this equation are also shown in Figure 7.6-11 and Figure 7.6-12. For sands with a solids density of 2.65 ton/m3 this gives a reduction of about 35-45%, on average 40%. The downside of this equation is, that the equation gives a fixed result for a fixed Rsd·Cv value and is not adaptable to more experimental data. Reason to investigate the possibility of applying a concentration profile, where the concentration equals zero at the pipe wall and increases, with a sort of von Driest damping function, to a maximum value at the center of the pipe. This concentration profile has to be corrected, based on numerical integration, to ensure that the average concentration matches a given value.

    Figure 7.6-11: The Darcy-Weisbach friction coefficient ratio λml and the factor αE as a function of the volumetric concentration Cv at Dp=1 m, vls=5 m/sec and solids density of 2.65 ton/m3.

    Screen Shot 2020-07-28 at 4.13.10 PM.png

    Figure 7.6-12: The Darcy-Weisbach friction coefficient ratio λml and the factor αE as a function of the volumetric concentration Cv at Dp=1 m, vls=5 m/sec and solids density of 10 ton/m3.

    Screen Shot 2020-07-28 at 4.18.59 PM.png

    7.6.8 Method 5: Applying a Concentration Profile to Method 2

    The original Talmon (2013) concept assumes a constant density ratio for the whole cross section of the pipe. This of course is not in agreement with the physical reality. The concept assumes carrier liquid in the viscous sub-layer and mixture in the remaining part of the cross-section, but uses a constant density ratio. In order to correct this a damping factor for the density ratio \(\ \alpha_{\rho}\) is proposed. This density ratio damping factor takes care that there is only carrier liquid very close to the wall. The factor \(\ \mathrm{A}_{\rho}\) determines the thickness of this carrier liquid layer. If \(\ \mathrm{A}_{\rho}\) equals zero, the solution obtained with Prandtl or Nikuradse with von Driest damping is found, methods 1 and 3. If \(\ \mathrm{A}_{\rho}\) equals 4.13 the solution of the “Law of the Wall” is found, method 4, and if \(\ \mathrm{A}_{\rho}\) equals 3.02 the solution of the Nikuradse equation without damping is found, method 2. The concentration profile and the density ratio are defined as:

    \[\ \mathrm{C}_{\mathrm{v}, \mathrm{z}}=\mathrm{C}_{\mathrm{v}, \max } \cdot\left(\mathrm{1 - e}^{-\mathrm{A}_{\rho} \cdot \frac{\mathrm{z}}{\delta_{\mathrm{v}}}}\right) \quad\text{ and }\quad \alpha_{\rho}=\frac{\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}, \mathrm{z}}}{\mathrm{1}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}\]

    The maximum concentration Cv,max in the concentration profile is found by integrating the concentration profile over the cross-section of the pipe and making it equal to the average concentration multiplied with the cross section of the pipe according to:

    \[\ \int_{\mathrm{0}}^{\mathrm{R}} \mathrm{C}_{\mathrm{v}, \mathrm{z}} \cdot \mathrm{2} \cdot \pi \cdot(\mathrm{R}-\mathrm{z}) \cdot \mathrm{d} \mathrm{z}=\mathrm{2} \cdot \boldsymbol{\pi} \cdot \mathrm{C}_{\mathrm{v}, \mathrm{m a x}} \cdot \int_{\mathrm{0}}^{\mathrm{R}}\left(\mathrm{1}-\mathrm{e}^{-\mathrm{A}_{\rho} \cdot \frac{\mathrm{z}}{\delta_{\mathrm{v}}}}\right) \cdot(\mathrm{R}-\mathrm{z}) \cdot \mathrm{d} \mathrm{z}=\mathrm{C}_{\mathrm{v}} \cdot \pi \cdot \mathrm{R}^{2}\]

    The maximum concentration Cv,max is now equal to average concentration Cv times a correction factor.

    \[\ \mathrm{C}_{\mathrm{v}, \mathrm{m a x}}=\mathrm{C}_{\mathrm{v}} \cdot \frac{\pi \cdot \mathrm{R}^{2}}{2 \cdot \pi \cdot\left(\frac{\mathrm{R}^{2}}{2}-\frac{\mathrm{R} \cdot \delta_{\mathrm{v}}}{\mathrm{A}_{\rho}}+\left(\frac{\delta_{\mathrm{v}}}{\mathrm{A}_{p}}\right) \cdot\left(1-\mathrm{e}^{-\mathrm{A}_{\rho} \cdot \frac{\mathrm{R}}{\delta_{\mathrm{v}}}}\right)\right)}\]

    The velocity gradient, including the concentration profile, is now:

    \[\ \frac{\partial \mathrm{u}}{\partial \mathrm{z}}=\frac{-\alpha_{\rho} \cdot v_{\mathrm{l}}+\sqrt{\left(\alpha_{\rho} \cdot v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(\mathrm{u}^{*}\right)^{2} \cdot\left(\frac{\mathrm{R-z}}{\mathrm{R}}\right)}}{2 \cdot \ell^{2}}\]

    The integrated velocity difference um-ul is now:

    \[\ \begin{array}{left} \mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{m}}-\mathrm{v}_{\mathrm{ls,l}} \approx \mathrm{u}_{\mathrm{m}}-\mathrm{u}_{\mathrm{l}}&=\int_{\mathrm{0}}^{\mathrm{R}} \frac{-\alpha_{\rho} \cdot v_{\mathrm{l}}+\sqrt{\left(\alpha_{\rho} \cdot v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(\mathrm{u}^{*}\right)^{2} \cdot\left(\frac{\mathrm{R}-\mathrm{z}}{\mathrm{R}}\right)}}{2 \cdot \ell^{2}} \cdot \mathrm{d} \mathrm{z} \\ &-\int_{\mathrm{0}}^{\mathrm{R}} \frac{-v_{\mathrm{l}}+\sqrt{\left(v_{\mathrm{l}}\right)^{2}+4 \cdot \ell^{2} \cdot\left(\mathrm{u}^{*}\right)^{2} \cdot\left(\frac{\mathrm{R}-\mathrm{z}}{\mathrm{R}}\right)} }{\mathrm{2} \cdot \ell^{2}}\cdot \mathrm{d} \mathrm{z} \end{array}\]

    For the resulting Darcy-Weisbach friction factor ratio this can be approximated by:

    \[\ \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{l}}}=\frac{1}{\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+1}\right)^{\mathrm{2}}}\]

    Where ACv depends on the value of Aρ. The relative excess hydraulic gradient Erhg is now:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+\mathrm{l}}\right)^{2}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+\mathrm{1}}\right)^{2}}=\alpha_{\mathrm{E}} \cdot \mathrm{i}_{\mathrm{l}}\]

    Now, from numerical solutions, it appears that equations (7.6-36) and (7.6-37) give a very good approximation of all 4 methods for the range of parameters as normally used in dredging. The factor ACv=1 for the “Law of the Wall” (method 4), ACv=1.25 for the Nikuradse solution without damping (method 2) and ACv=3.4 for the Prandtl and Nikuradse solutions with von Driest damping (methods 1 and 3). The average equation (7.6-30) has a coefficient of ACv=2.2 and Aρ=1.05. Table 7.6-1 gives an overview of these values.

    Figure 7.6-14 shows a lower limit of the data, an upper limit of the data and the curve of Talmon (2013), method 1, compared with experimental data of Talmon (2011) in a vertical pipe. The lower and upper limit are determined for the particles from d=0.345 mm to d=0.750 mm. The finest particles of d=0.125 mm show less or even a reversed influence, probably because of the Thomas (1965) viscosity effect.

    Figure 7.6-13: The concentration distribution for the cases considered from Table 7.6-1.

    Screen Shot 2020-07-28 at 5.15.22 PM.png

    Table 7.6-1: Some Aρ and ACv values.

    Aρ

    ACv
    Law of the Wall 4.13

    1.00

    Nikuradse (no damping)

    3.02

    1.25

    Prandtl (damping)

    0.01

    3.40

    Average

    1.05

    2.20

    Lower limit of data

    5.43

    0.80

    Upper limit of data

    1.67

    1.80

    Figure 7.6-13 shows the concentration profiles for Dp=1 m, vls=5 m/sec, δ=0.088 mm for the cases considered in.

    Figure 7.6-14: Talmon (2011) data compared with the theory.

    Screen Shot 2020-07-28 at 5.19.21 PM.png

    Figure 7.6-1 shows experimental data of Thomas (1976) of iron ore in a horizontal pipe, where the theoretical curve contains both the Thomas (1965) viscosity and equation (7.6-37) with ACv=1.3, the average of the lower and upper limit.

    7.6.9 Applicability of the Model

    Homogeneous transport is defined as transport where the concentration distribution is close to being uniform and the head losses behave similar to the Darcy Weisbach head losses, but with some correction.

    The basis of the homogeneous transport regime model is the equivalent liquid model (ELM). In terms of the relative excess hydraulic gradient, Erhg, this can be written as:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\mathrm{i}_{\mathrm{l}}\]

    Talmon (2013) derived an equation to correct the homogeneous equation (the ELM model) for the slurry density, based on the hypothesis that the viscous sub-layer hardly contains solids at very high line speeds in the homogeneous regime. This theory results in a reduction of the resistance compared with the ELM, but the resistance is still higher than the resistance of clear water. Talmon (2013) used the Prandl approach for the mixing length, which is a 2D approach for open channel flow with a free surface. The Prandl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968). Miedema (2015A) improved the equation for pipe flow and a concentration distribution giving for the relative excess hydraulic gradient Erhg:

    \[\ \mathrm{E}_{\mathrm{r h g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\mathrm{i}_{\mathrm{l}} \cdot \frac{\mathrm{1}+\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}-\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}+\mathrm{1}\right)^{\mathrm{2}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\frac{\mathrm{A}_{\mathrm{C}_{\mathrm{v}}}}{\mathrm{\kappa}} \cdot \ln \left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}}\right) \cdot \sqrt{\frac{\lambda_{\mathrm{l}}}{\mathrm{8}}}{+1}\right)^{2}}\]
    Figure 7.6-15: Experimental data of Whitlock et al. (2004).

    Screen Shot 2020-07-28 at 5.27.28 PM.png

    Figure 7.6-16: Experiments of Blythe & Czarnotta (1995).

    Screen Shot 2020-07-28 at 5.28.20 PM.png

    The resulting equation (7.6-39) with ACv=3 gives a good average behavior based on the data of Talmon (2011) and Thomas (1976). Since the model is based on a particle free viscous sub-layer and the viscosity of the carrier liquid, it may not give good predictions for very small particles. Very small particles may influence the viscosity and fit completely in the viscous sub layer, especially at low line speeds. It is observed that very small particles behave according to the ELM, if necessary corrected for the viscosity and density. Medium and large particles show the reduction according to equation (7.6-39). The transition of the ELM to the reduced ELM appears to depend on the ratio of the thickness of the viscous sub layer to the particle diameter with a maximum of 1. This ratio is an indication of the concentration reduction in the viscous sub layer, giving:

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

    Figure 7.6-15 shows experimental data of Whitlock et al. (2004) showing that very small particles (d=0.085 mm) have a rather sharp transition from the heterogeneous regime to the ELM, while at higher line speeds the reduction due to a lower concentration in the homogeneous regime is mobilized. Larger particles (d=0.4 mm) however seem to have some overshoot. Figure 7.6-16 shows experiments of Blythe & Czarnotta (1995). From these experiments it is clear that the relative excess hydraulic gradient crosses the ELM curve, after which it tends to go back to the direction of the homogeneous curve (reduced ELM or RELM) asymptotically.

    7.6.10 Conclusions

    The concept of Talmon (2013) is applicable for determining the pressure losses in the homogeneous regime, however this concept has to be modified with respect to the shear stress distribution, the concentration distribution and a check on conservation of volume flow and concentration. The resulting equations (7.6-36) and (7.6-37) with ACv=1.3 give a good average behavior based on the data of Talmon (2011) and Thomas (1976). The original factor ACv=3.4 of Talmon (2013) seems to overestimate the reduction of the solids effect. It should be mentioned that the experiments as reported by Talmon (2011) were carried out in a vertical pipe ensuring symmetrical flow. For horizontal pipes the results may differ, since the velocity and concentration profiles are not symmetrical at the line speeds common in dredging. Since the model is based on a particle free viscous sub-layer and the viscosity of the carrier liquid, it may not give good predictions for very small or large particles. Very small particles may influence the viscosity, while very large particles are not influenced by the viscous sub-layer.

    Using the resulting equations (7.6-36) and (7.6-37), implies using von Driest damping in combination with a concentration profile. The resulting equations (7.6-36) and (7.6-37) are flexible in use.
    The error of using ACv=1-3 is difficult to define. With respect to the relative excess hydraulic gradient Erhg the accuracy is about +/- 10%. With respect to the hydraulic gradient im, which is of interest for the dredging companies, the accuracy is better to much better, since this hydraulic gradient equals im=il+Erhg·Rsd·Cv.

    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.

    Equations (7.6-39) and (7.6-40) are implemented in the Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework with a default value of ACv=3 (see Miedema & Ramsdell (2014)).

    The latest calibrations with experiments show that ACv=2.5-3.

    7.6.11 Nomenclature Homogeneous Regime

    A

    Von Driest damping factor (26)

    -

    ACv

    Concentration factor

    -

    Aρ

    Density factor

    -

    Cv

    Concentration averaged over the cross section of the pipe

    -

    Cv,z

    Concentration at distance z of the pipe wall

    -

    Cv,max

    Maximum concentration in the center of the pipe

    -

    d

    Particle diameter

    m

    d50

    50% passing particle diameter

    M

    Dp

    Pipe diameter

    m

    Erhg

    Relative excess hydraulic gradient

    -

    F

    Homogeneous reduction factor

    -

    g

    Gravitational constant (9.81)

    m/s2

    ΔL

    Length of pipe segment considered

    m

    im

    Mixture hydraulic gradient

    m/m

    il

    Liquid hydraulic gradient

    m/m

    Δpm

    Pressure loss mixture over a length ΔL

    kPa

    R

    Pipe radius

    m

    Rsd

    Relative submerged density (sand 1.65)

    -

    u

    Velocity

    m/s

    ul

    Velocity liquid

    m/s

    um

    Velocity mixture

    m/s

    u*

    Friction velocity

    m/s

    vls

    Line speed

    m/s

    vls,l

    Line speed liquid

    m/s

    vls,m

    Line speed mixture

    m/s

    z

    Distance to the wall

    m

    z+

    Dimensionless distance to the wall

    -

    z0

    Constant velocity profile liquid

    m

    z0,m

    Constant velocity profile mixture

    m

    αh

    Homogeneous factor

    -

    αE

    Homogeneous factor Erhg value

    -

    β

    Nikuradse correction factor

    -

    δv

    Thickness viscous sub-layer

    m

    λl

    Darcy-Weisbach friction factor liquid

    -

    λm

    Darcy-Weisbach friction factor mixture

    -

    ρl

    Density liquid

    ton/m3

    ρm

    Density mixture

    ton/m3

    κ

    Von Karman constant (0.4)

    -

    \(\ \tau\)

    Shear stress

    kPa

    \(\ \tau_{\text{wall}}\)

    Shear stress at the wall

    kPa

    \(\ \tau_{v}\)

    Viscous shear stress

    kPa

    \(\ \tau_{\text{t}}\)

    Turbulent shear stress

    kPa

    \(\ \mu_{\text{v}}\)

    Viscous dynamic viscosity

    Pa·s

    \(\ \mu_{\text{t}}\)

    Turbulent dynamic viscosity

    Pa·s

    \(\ \mu_{\text{l}}\)

    Dynamic viscosity liquid

    Pa·s

    \(\ \mu_{\text{m}}\)

    Dynamic viscosity mixture

    Pa·s

    \(\ v_\mathrm{l}\)

    Kinematic viscosity liquid

    m2/s

    \(\ v_\mathrm{m}\)

    Kinematic viscosity mixture

    m2/s

    \(\ v_\mathrm{t}\)

    Turbulence viscosity

    m2/s

    \(\ \ell \)

    Mixing length

    m