7.3: A Head Loss Model for Fixed Bed Slurry Transport
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7.3.1 The Basic Equations for Flow and Geometry
In order to understand the fixed bed model, first all the geometrical parameters are defined. The cross section of the pipe with a particle bed as defined in the Wilson et al. (1992) two layer model has been illustrated by Figure 6.20-1, here Figure 7.3-1.
The geometry is defined by the following equations.
The length of the liquid in contact with the whole pipe wall if there is no bed is:
\[\ \mathrm{O}_{\mathrm{p}}=\pi \cdot \mathrm{D}_{\mathrm{p}}\]
The length of the liquid or the suspension in contact with the pipe wall:
\[\ \mathrm{O}_{\mathrm{1}}=\mathrm{D}_{\mathrm{p}} \cdot(\pi-\boldsymbol{\beta})\]
The length of the fixed or sliding bed in contact with the wall:
\[\ \mathrm{O}_{2}=\mathrm{D}_{\mathrm{p}} \cdot \boldsymbol{\beta}\]
The top surface length of the fixed or sliding bed:
\[\ \mathrm{O}_{\mathrm{1 2}}=\mathrm{D}_{\mathrm{p}} \cdot \sin (\boldsymbol{\beta})\]
The cross sectional area A_{p} of the pipe is:
\[\ \mathrm{A}_{\mathrm{p}}=\frac{\pi}{4} \cdot \mathrm{D}_{\mathrm{p}}^{\mathrm{2}}\]
The cross sectional area A_{2} of the fixed or sliding bed is:
\[\ \mathrm{A}_{2}=\frac{\pi}{4} \cdot \mathrm{D}_{\mathrm{p}}^{2} \cdot\left(\frac{\beta-\sin (\beta) \cdot \cos (\beta)}{\pi}\right)\]
The cross sectional area A_{1} above the bed, where the liquid or the suspension is flowing, also named the restricted area:
\[\ \mathrm{A _ { 1 }}=\mathrm{A _ { p }}-\mathrm{A _ { 2 }}\]
The hydraulic diameter D_{H,1} of the cross-sectional area above the bed as function of the bed height, is equal to four times the cross sectional area divided by the wetted perimeter:
\[\ \mathrm{D}_{\mathrm{H}, \mathrm{1}}=\frac{\mathrm{4} \cdot \mathrm{A}_{\mathrm{1}}}{\mathrm{O}_{\mathrm{1}}+\mathrm{O}_{\mathrm{1} 2}} \quad\text{ or simplified: }\quad \mathrm{D}_{\mathrm{H}, \mathrm{1}}=\sqrt{\frac{\mathrm{4} \cdot \mathrm{A}_{\mathrm{1}}}{\pi}}\]
The volume balance gives a relation between the line speed v_{ls}, the velocity in the restricted area above the bed v_{r} or v_{1} and the velocity of the bed v_{b} or v_{2}.
\[\ \mathrm{v}_{\mathrm{l} \mathrm{s}} \cdot \mathrm{A}_{\mathrm{p}}=\mathrm{v}_{\mathrm{1}} \cdot \mathrm{A}_{\mathrm{1}}+\mathrm{v}_{\mathrm{2}} \cdot \mathrm{A}_{\mathrm{2}}\]
Thus the velocity in the restricted area above the bed is:
\[\ \mathrm{v}_{1}=\frac{\mathrm{v}_{\mathrm{l s}} \cdot \mathrm{A}_{\mathrm{p}}-\mathrm{v}_{\mathrm{2}} \cdot \mathrm{A}_{\mathrm{2}}}{\mathrm{A}_{\mathrm{1}}}\]
Or the velocity of the bed is (for a fixed bed v_{2}=0):
\[\ \mathrm{v}_{2}=\frac{\mathrm{v}_{\mathrm{l s}} \cdot \mathrm{A}_{\mathrm{p}}-\mathrm{v}_{\mathrm{1}} \cdot \mathrm{A}_{\mathrm{1}}}{\mathrm{A}_{2}}\]
7.3.2 The Shear Stresses Involved
In order to determine the forces involved, first the shear stresses involved have to be determined. The general equation for the shear stresses is:
\[\ \tau=\rho_{\mathrm{l}} \cdot \mathrm{u}_{*}^{2}=\frac{\lambda_{\mathrm{l}}}{4} \cdot \frac{1}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}^{2}\]
The force F on the pipe wall over a length ΔL is now:
\[\ \mathrm{F}=\tau \cdot \pi \cdot \mathrm{D}_{\mathrm{p}} \cdot \Delta \mathrm{L}=\frac{\lambda_{\mathrm{l}}}{4} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}^{2} \cdot \pi \cdot \mathrm{D}_{\mathrm{p}} \cdot \Delta \mathrm{L}\]
The pressure Δp required to push the solid-liquid mixture through the pipe is:
\[\ \Delta \mathrm{p}=\frac{\mathrm{F}}{\mathrm{A}_{\mathrm{p}}}=\frac{\frac{\lambda_{\mathrm{l}}}{\mathrm{4}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}^{2} \cdot \pi \cdot \mathrm{D}_{\mathrm{p}} \cdot \Delta \mathrm{L}}{\frac{\pi}{4} \cdot \mathrm{D}_{\mathrm{p}}^{2}}=\lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}^{2}\]
This is the well-known Darcy Weisbach equation. Over the whole range of Reynolds numbers above 2320 the Swamee Jain equation gives a good approximation for the friction coefficient:
\[\ \lambda_{\mathrm{l}}=\frac{1.325}{\left(\ln \left(\frac{0.27 \cdot \varepsilon}{\mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}} \quad\text{ with: }\quad \mathrm{Re}=\frac{\mathrm{v} \cdot \mathrm{D}_{\mathrm{p}}}{v_{\mathrm{l}}}\]
This gives for the shear stress on the pipe wall for clean water:
\[\ \tau_{\mathrm{l}}=\frac{\lambda_{\mathrm{l}}}{4} \cdot \frac{1}{2} \cdot \rho_{1} \cdot v_{\mathrm{ls}}^{2} \quad\text{ with: }\quad \lambda_{\mathrm{l}}=\frac{1.325}{\left(\ln \left(\frac{0.27 \cdot \varepsilon}{\mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}} \quad\text{ and }\quad \mathrm{Re}=\frac{\mathrm{v}_{\mathrm{ls}} \cdot \mathrm{D}_{\mathrm{p}}}{v_{\mathrm{l}}}\]
For the flow in the restricted area, the shear stress between the liquid and the pipe wall is:
\[\ \tau_{1,\mathrm{l}}=\frac{\lambda_{1}}{4} \cdot \frac{1}{2} \cdot \rho_{\mathrm{l}} \cdot v_{1}^{2} \quad\text{ with: }\quad \lambda_{1}=\frac{1.325}{\left(\ln \left(\frac{0.27 \cdot \varepsilon}{\mathrm{D}_{\mathrm{H}}}+\frac{5.75}{\mathrm{R} \mathrm{e}^{0.9}}\right)\right)^{2}} \quad\text{ and }\quad \operatorname{Re}=\frac{\mathrm{v}_{1} \cdot \mathrm{D}_{\mathrm{H}}}{v_{\mathrm{l}}}\]
For the flow in the restricted area, the shear stress between the liquid and the bed is:
\[\ \tau_{12,\mathrm{l}}=\frac{\lambda_{12}}{4} \cdot \frac{1}{2} \cdot \rho_{\mathrm{l}} \cdot v_{1}^{2} \quad\text{ with: }\quad \lambda_{12}=\frac{\alpha \cdot 1.325}{\left(\ln \left(\frac{0.27 \cdot \mathrm{d}}{\mathrm{D}_{\mathrm{H}}}+\frac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}} \quad\text{ and }\quad \mathrm{Re}=\frac{\mathrm{v}_{1} \cdot \mathrm{D}_{\mathrm{H}}}{v_{\mathrm{l}}}\]
The factor α as used by Wilson et al. (1992) is 2 or 2.75, depending on the publication and version of his book. Televantos et al. (1979) used a factor of 2. For the flow between the liquid in the bed and the pipe wall, the shear stress between the liquid and the pipe wall is:
\[\ \tau_{2,\mathrm{l}}=\frac{\lambda_{2}}{4} \cdot \frac{1}{2} \cdot \rho_{\mathrm{l}} \cdot v_{2}^{2} \quad\text{ with: }\quad \lambda_{2}=\frac{1.325}{\left(\ln \left(\frac{0.27 \cdot \varepsilon}{\mathrm{d}}+\frac{5.75}{\operatorname{Re}^{0.9}}\right)\right)^{2}} \quad\text{ and }\quad \operatorname{Re}=\frac{\mathrm{v_{2}} \cdot \mathrm{d}}{v_{\mathrm{l}}}\]
Wilson et al. (1992) assume that the sliding friction is the result of a hydrostatic normal force between the bed and the pipe wall multiplied by the sliding friction factor. The average shear stress as a result of the sliding friction between the bed and the pipe wall, according to the Wilson et al. (1992) normal stress approach is:
\[\ \tau_{2, \mathrm{sf}}=\frac{\mu_{\mathrm{sf}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \mathrm{A}_{\mathrm{p}}}{\beta \cdot \mathrm{D}_{\mathrm{p}}} \cdot \frac{2 \cdot(\sin (\beta)-\beta \cdot \cos (\beta))}{\pi}\]
It is however also possible that the sliding friction force results from the weight of the bed multiplied by the sliding friction factor. For low volumetric concentrations, there is not much difference between the two methods, but at higher volumetric concentrations there is. The average shear stress as a result of the sliding friction between the bed and the pipe wall, according to the weight normal stress approach is:
\[\ \tau_{2, \mathrm{sf}}=\frac{\mu_{\mathrm{sf}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vb}} \cdot \mathrm{A}_{\mathrm{p}}}{\beta \cdot \mathrm{D}_{\mathrm{p}}} \cdot \frac{(\beta-\sin (\beta) \cdot \cos (\beta))}{\pi}\]
7.3.3 The Forces Involved
First the equilibrium of the forces on the liquid above the bed is determined. This is necessary to find the correct hydraulic gradient.
The resisting shear force on the pipe wall O_{1} above the bed is:
\[\ \mathrm{F}_{1,\mathrm{l}}=\tau_{1,\mathrm{l}} \cdot \mathrm{O}_{1} \cdot \Delta \mathrm{L}\]
The resisting shear force on the bed surface O_{12} is:
\[\ \mathrm{F}_{\mathrm{1 2}, \mathrm{l}}=\tau_{\mathrm{1 2}, \mathrm{l}} \cdot \mathrm{O}_{\mathrm{1 2}} \cdot \mathrm{\Delta} \mathrm{L}\]
The pressure Δp on the liquid above the bed is:
\[\ \Delta \mathrm{p}=\Delta \mathrm{p}_{2}=\Delta \mathrm{p}_{1}=\frac{\tau_{1,\mathrm{l}} \cdot \mathrm{O}_{1} \cdot \Delta \mathrm{L}+\tau_{12,\mathrm{l}} \cdot \mathrm{O}_{12} \cdot \Delta \mathrm{L}}{\mathrm{A}_{1}}=\frac{\mathrm{F}_{1,\mathrm{l}}+\mathrm{F}_{12, \mathrm{l}}}{\mathrm{A}_{1}}\]
The force equilibrium on the liquid above the bed is shown in Figure 6.20-2.
Secondly the equilibrium of forces on the bed is determined as is shown in Figure 6.20-3.
The driving shear force on the bed surface is:
\[\ \mathrm{F}_{12,\mathrm{l}}=\tau_{12,\mathrm{l}} \cdot \mathrm{O}_{12} \cdot \mathrm{\Delta L}\]
The driving force resulting from the pressure Δp on the bed is:
\[\ \mathrm{F}_{2, \mathrm{p r}}=\Delta \mathrm{p} \cdot \mathrm{A}_{2}\]
The resisting force between the bed and the pipe wall due to sliding friction is:
\[\ \mathrm{F}_{2, \mathrm{s f}}=\tau_{2, \mathrm{s f}} \cdot \mathrm{O}_{2} \cdot \Delta \mathrm{L}\]
The resisting shear force between the liquid in the bed and the pipe wall is:
\[\ \mathrm{F}_{2,\mathrm{l}}=\tau_{2,\mathrm{l}} \cdot \mathrm{O}_{2} \cdot \mathrm{n} \cdot \Delta \mathrm{L}\]
This shear force is multiplied by the porosity n, in order to correct for the fact that the bed consists of a combination of particles and water. There is an equilibrium of forces when:
\[\ \mathrm{F}_{\mathrm{1 2}, \mathrm{l}}+\mathrm{F}_{\mathrm{2}, \mathrm{p r}}=\mathrm{F}_{\mathrm{2}, \mathrm{s f}}+\mathrm{F}_{\mathrm{2}, \mathrm{l}}\]
Below the Limit of Stationary Deposit Velocity, the bed is not sliding and the force F_{2,l} equals zero. Since the problem is implicit with respect to the velocities v_{1} and v_{2}, it has to be solved with an iteration process.
The mixture pressure is now:
\[\ \Delta \mathrm{p}_{\mathrm{m}}=\frac{\lambda_{\mathrm{1}} \cdot \mathrm{O}_{\mathrm{1}}+\lambda_{\mathrm{1} 2} \cdot \mathrm{O}_{\mathrm{1 2}}}{\mathrm{4} \cdot\left(\mathrm{1}-\mathrm{C}_{\mathrm{v r}}\right) \cdot \mathrm{A}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{1}}^{\mathrm{2}} \cdot \Delta \mathrm{L} \quad \text{ with}: \quad \mathrm{C}_{\mathrm{v r}}=\frac{\mathrm{C}_{\mathrm{v s}}}{\mathrm{C}_{\mathrm{v} \mathrm{b}}}\]
The excess pressure or excess hydraulic gradient can be written as:
\[\ \begin{array}{left} \Delta \mathrm{p}_{\mathrm{m}}-\Delta \mathrm{p}_{\mathrm{l}}=\left(\left(\lambda_{1} \cdot \mathrm{O}_{1}+\lambda_{12} \cdot \mathrm{O}_{12}\right) \cdot\left(\frac{\mathrm{1}}{\mathrm{1}-\mathrm{C}_{\mathrm{v r}}}\right)^{\mathrm{3}}-\lambda_{\mathrm{l}} \cdot \mathrm{O}_{\mathrm{p}}\right) \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2} \cdot \frac{\Delta \mathrm{L}}{\mathrm{4} \cdot \mathrm{A}_{\mathrm{p}}}\\
\text{or}\\
\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}=\left(\left(\lambda_{1} \cdot \mathrm{O}_{1}+\lambda_{12} \cdot \mathrm{O}_{12}\right) \cdot\left(\frac{\mathrm{1}}{\mathrm{1}-\mathrm{C}_{\mathrm{v r}}}\right)^{\mathrm{3}}-\lambda_{\mathrm{l}} \cdot \mathrm{O}_{\mathrm{p}}\right) \cdot \frac{\mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{8} \cdot \mathrm{g} \cdot \mathrm{A}_{\mathrm{p}}}\end{array}\]
In terms of the relative excess hydraulic gradient this can be written as:
\[\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{vs}}}=\left(\left(\lambda_{1} \cdot \mathrm{O}_{1}+\lambda_{12} \cdot \mathrm{O}_{12}\right) \cdot\left(\frac{\mathrm{1}}{\mathrm{1 - C}_{\mathrm{v r}}}\right)^{3}-\lambda_{1} \cdot \mathrm{O}_{\mathrm{p}}\right) \cdot \frac{\mathrm{v}_{\mathrm{l s}}^{2}}{\mathrm{8 \cdot g \cdot A _ { \mathrm { p } } \cdot \mathrm { R } _ { \mathrm { s d } } \cdot \mathrm { C } _ { \mathrm { v s } }}}\]
7.3.4 The Relative Roughness
In the Wilson (1992) approach the Darcy-Weisbach friction factor between the liquid and the top of the bed is crucial, together with the multiplication factor as applied by Televantos et al. (1979) of 2-2.75. In this approach the particle diameter d is used as a bed roughness k_{s} and the resulting Darcy-Weisbach friction factor multiplied by 2 or 2.75. Another approach found in literature is the approach of making the effective bed roughness a function of the Shields parameter. Many researchers developed equations for this purpose, but the fact that many equations exist usually means that the physics are not understood properly. Following is a list of existing equations in order of time.
Nielsen (1981)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{1 9 0} \cdot \sqrt{\theta-\mathrm{\theta}_{\mathrm{c}}}\]
Grant & Madsen (1982)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=4 \mathrm{3 0} \cdot(\sqrt{\theta}-\mathrm{0 . 7} \cdot \sqrt{\theta_{\mathrm{c}}})^{2}\]
Wilson (1988) based his equation on experiments in closed conduits.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{5} \cdot \theta\]
Wikramanayake & Madsen (1991)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{6 0} \cdot \mathrm{\theta}\]
Wikramanayake & Madsen (1991)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{3 4 0} \cdot(\sqrt{\theta}-\mathrm{0 . 7} \cdot \sqrt{\theta_{\mathrm{c}}})^{2}\]
Madsen et al. (1993)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{1 5}\]
Van Rijn (1993)
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{3} \cdot \theta\]
Camenen et al. (2006) collected many data from literature and found a best fit equation. The data however was a combination of experiments in closed and not closed conduits.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{0 .6 + 1.8} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}^{*}} ^{1.2}}{\mathrm{F r}^{\mathrm{2 .4}}}\right) \cdot \mathrm{\theta}^{\mathrm{1 .7}}\]
Or:
\[\ \begin{array}{left}\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{0 .6}+\mathrm{2 . 4} \cdot\left(\frac{\theta}{\theta_{\text {cr }, \mathrm{u r}}}\right)^{1.7}\\
\text{With: }\theta_{\mathrm{cr}, \mathrm{ur}}=1.18 \cdot \frac{\mathrm{Fr}^{1.4}}{\mathrm{v}_{\mathrm{t}^{*}}^{0.7}}\end{array}\]
\[\ \text{With : }\mathrm{v}_{\mathrm{t}^{*}}=\left(\frac{\mathrm{R}_{\mathrm{s d}}^{2}}{\mathrm{g} \cdot v_{\mathrm{l}}}\right)^{1 / 3} \cdot \mathrm{v}_{\mathrm{t}} \quad \text{&} \quad \mathrm{F r}=\frac{\mathrm{v}_{\mathrm{l s}}}{\sqrt{\mathrm{g} \cdot \mathrm{R}_{\mathrm{H}}}}\]
Matousek (2007) based his first equation on a limited amount of experiments in a closed conduit.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{1.3 \cdot} \theta^{\mathrm{1 .6 5}}\]
Matousek & Krupicka (2009) improved his relation based on more experiments.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{2 6 0} \cdot\left(\frac{\mathrm{R}}{\mathrm{d}}\right)^{\mathrm{1}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{v}_{\mathrm{r}}}\right)^{2.5} \cdot \theta^{\mathrm{1 .7}}\]
Krupicka & Matousek (2010) improved their relation again and gave it a form similar to the Camenen et al. (2006) equation.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\mathrm{1.7} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}^{*}}^{\mathrm{1 . 1}}}{\mathrm{F r}^{\mathrm{2 . 3}}}\right) \cdot\left(\frac{\mathrm{R}}{\mathrm{d}}\right)^{\mathrm{0 . 3 2}} \cdot \mathrm{\theta}^{\mathrm{1.4}}\]
Krupicka & Matousek (2010) also gave a more explicit equation to determine the Darcy-Weisbach friction factor λ_{b} without having to use the bed roughness k_{s}/d.
\[\ \lambda_{12}=\mathrm{0.25} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}^{*}}^{0.58}}{\mathrm{F r}^{1.3}}\right) \cdot\left(\frac{\mathrm{R}}{\mathrm{d}}\right)^{-0.36} \cdot \theta^{0.74}\]
By using the standard equation for the Shields parameter:
\[\ \theta=\frac{\frac{\lambda_{12}}{8} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{g} \cdot \mathrm{d}}\]
This can be written explicitly as:
\[\ \lambda_{12}^{0.26}=\mathrm{0.25 \cdot\left(\frac{v_{t^{*}}^{0.58}}{F r^{1.3}}\right) \cdot\left(\frac{R}{d}\right)^{-0.36} \cdot\left(\frac{\frac{1}{8} \cdot\left(v_{1}-v_{2}\right)^{2}}{R_{s d} \cdot g \cdot d}\right)^{0.74}}\]
Whether this is the purpose of this equation is not clear, but mathematically it’s correct.
Camenen & Larson (2013) wrote a technical note on the accuracy of equivalent roughness height formulas in practical applications. They already concluded that most equations are based on a relation between the relative roughness k_{s}/d and the Shields parameter.
The relative roughness is a parameter that often has nothing to do with the real roughness of the bed, but it is a parameter to use in calculations to estimate an equivalent roughness value in the case of sheet flow. Sheet flow is a layer of particles flowing with a higher speed than the bed and with a velocity gradient, from a maximum velocity at the top to the bed velocity at the solid bed.
Camenen & Larson (2013) also concluded that the equations are implicit and have to be solved by iteration, since the Shields parameter depends on the relative roughness through the Darcy-Weisbach friction factor.
\[\ \lambda_{12}=\mathrm{8} \cdot\left(\frac{\kappa}{\ln \left(\frac{3.7 \cdot \mathrm{D}_{\mathrm{H}}}{\mathrm{k}_{\mathrm{s}}}\right)}\right)^{2}=8 \cdot\left(\frac{\kappa}{\ln \left(\frac{14.8 \cdot \mathrm{R}_{\mathrm{H}}}{\mathrm{k}_{\mathrm{s}}}\right)}\right)^{2}\]
The Darcy-Weisbach friction factor as applied here is for very large Reynolds numbers. Camenen & Larson (2013) stated that this implicit equation is difficult to solve and that it has either two solution or no solution at all. Mathematically this is not correct. There are 3 solutions or there is 1 solution as is shown in Figure 7.3-6. Figure 7.3-6 (v_{ls}=2 m/sec & R=0.0525 m) shows the calculated k_{s}/d versus the input k_{s}/d for the Wilson (1988) equation, the Matousek (2009) equation, the improved Matousek (2010) equation and the Camenen et al. (2006) equation. The Wilson (1988) , Matousek (2009) and Camenen et al. (2006) equations show 3 intersection points with the k_{s,calculated}=k_{s,input} line (y=x). Matousek (2010) only shows 1 intersection point. It is clear that the intersection point right from the peaks is a point for k_{s,input}>14.8·R_{h} which is physical nonsense, so this solution should be eliminated. Still in a numerical solver this could be output.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\frac{\mathrm{3 .7} \cdot \mathrm{D}_{\mathrm{H}}}{\mathrm{d}} \cdot \mathrm{e}^{-\mathrm{\kappa} \cdot \sqrt{\frac{\mathrm{8}}{\lambda_{\mathrm{b}}}}}=\frac{\mathrm{1 4 . 8 \cdot R}_{\mathrm{H}}}{\mathrm{d}} \cdot \mathrm{e}^{-\mathrm{\kappa} \cdot \sqrt{\frac{\mathrm{8}}{\lambda_{\mathrm{b}}}}}\]
Now there are either 2 or 0 solutions left, depending on the different parameters and the model chosen. Figure 7.3-7 shows the relative roughness versus the Shields parameter for the 4 models as used above and mathematical solutions for a number of velocities above the bed (assuming the bed has no velocity) using the following equation.
\[\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{d}}=\frac{\mathrm{14 .8 \cdot R}_{\mathrm{H}}}{\mathrm{d}} \cdot \mathrm{e}^{-\mathrm{\kappa} \cdot \sqrt{\frac{\mathrm{v}_{\mathrm{a b}}^{2}}{\theta \cdot \mathrm{R}_{\mathrm{sd}} \mathrm{g} \cdot \mathrm{d}}}}\]
Also this figure shows either two intersection points with a specific velocity, or no intersection point. It also shows that if a model intersects with a constant velocity curve close to the tangent point, the solution is very sensitive to small variations in the parameters or there is no solution at all. From analyzing a number of the models it appeared that each model has solutions up to a maximum velocity above the bed depending on the particle diameter d and the bed associated hydraulic radius R_{H}. Of course other parameters like the relative submerged density of the particles R_{sd} and the kinematic viscosity ν_{l} of the carrier liquid also play a role. Now suppose one of the models is correct, then above this maximum velocity no solution exists. But since this is true for all models, there exists a velocity above the bed above which no solutions exist at all. Figure 7.3-8 and Figure 7.3-9 clearly show the lower and upper solution for the Wilson (1988) equation in two different coordinate systems. This together with the fact that below this maximum velocity always two solutions exist, leaving us with the question which of the two solutions should be chosen, gives us no other choice than to reject the hypothesis that an equivalent roughness should be used as a function of the Shields parameter. Apparently this does not work. The question is, why all the researchers didn’t relate the Darcy-Weisbach friction factor directly to the parameters involved, skipping the relative roughness and the Shields parameter. Most probably because in erosion and sediment transport it’s a custom to use these parameters.
7.3.5 The Darcy-Weisbach friction factor first attempt
Analyzing Figure 7.3-4, Figure 7.3-5 and the latest developments of the relative roughness equations shows that the relative roughness depends on the bed associated hydraulic radius, on the terminal settling velocity of the particles, on the Froude number of the flow and on the ratio between the particle diameter and the bed associated hydraulic radius. In Figure 7.3-4 different hydraulic radii are shown with different colors and this shows that a different hydraulic radius forms a group of data points within a certain band width. The Darcy-Weisbach friction factor increases exponentially with increasing line speed and also increases with decreasing bed associated hydraulic radius.
Figure 7.3-8 shows the Wilson (1988) experiments with the relative Darcy-Weisbach friction factor versus the velocity above the bed. Also in this graph the different bed associated hydraulic radii can be distinguished. Wilson (1988) used a multiplication factor of 2.75 and later 2.0, which in this graph equals 1.75 and 1.0 on the vertical axis. From the graph it is clear that this factor can be somewhere between 0.1 and 5.0, giving a multiplication factor from 1.1 to 6.0. It is however important what the value of this factor is at the Limit of Stationary Deposit Velocity, the moment the bed starts sliding. Based on the graph a relative factor of 1-2 or a multiplication factor from 2-3 seems reasonable. The graph however gives more information. Figure 7.3-9 shows the same data points but now with the Darcy-Weisbach friction factor on the vertical axis. Both graphs also show the lower and upper solution of the Wilson (1988) equation. Other equations would give a similar shape of the lower and upper solution.
Based on the Wilson (1988) and the Krupicka & Matousek (2010) experiments, complemented with (still confidential) experiments in the Laboratory of Dredging Engineering an empirical explicit equation has been developed for the relation between the Darcy-Weisbach friction factor and the different parameters involved. This equation is:
\[\ \lambda_{12}=\mathrm{0 .0 6} \cdot \sinh \left(48 \cdot \mathrm{Fr}^{2.83} \cdot \mathrm{Re}^{-0.33} \cdot \mathrm{v}_{\mathrm{t}^{*}}^{-0.5}\right)\]
\[\ \text{With : }\mathrm{v}_{\mathrm{t}^{*}}=\left(\frac{\mathrm{R}_{\mathrm{s d}}^{\mathrm{2}}}{\mathrm{g} \cdot v_{\mathrm{l}}}\right)^{\mathrm{1 / 3}} \cdot \mathrm{v}_{\mathrm{t}}, \quad \mathrm{F r}=\frac{\mathrm{v}_{\mathrm{a b}}}{\sqrt{\mathrm{g} \cdot \mathrm{2} \cdot \mathrm{D}_{\mathrm{H}}}}, \quad \mathrm{R} \mathrm{e}=\frac{\mathrm{v}_{\mathrm{a b}} \cdot \mathrm{2} \cdot \mathrm{D}_{\mathrm{H}}}{v_{\mathrm{l}}}\]
Figure 7.3-4, Figure 7.3-5, Figure 7.3-8 and Figure 7.3-9 show the resulting curves for RH/d=35, 55 and 75 for the Wilson (1988) experiments. If the resulting Darcy-Weisbach friction factor is smaller than the result of equation (7.3-18), this equation is used. The curves can be extended for higher velocities above the bed, but are limited here to the maximum velocity of the solutions based on the Wilson (1988) equation. One can see that the resulting curves match the data points well and also match the curvature through the data points much better than the Wilson (1988) equation. The factor 2 in both the Froude number and the Reynolds number is to compensate for the fact that in the old equations the bed associated radius is used. The bed associated radius depends not only on the real hydraulic radius, but also on the contribution of the bed friction to the total friction. This bed associated radius can only be determined based on experiments. At high velocities where the bed friction dominates the total friction, the bed associated radius may get a value of 2 times the real hydraulic radius. Since here we are looking for an explicit expression, the real hydraulic radius or hydraulic diameter is used, compensated with this factor 2.
At small line speeds this factor may be near 1, but at line speeds that matter, the factor of 2 gives a good estimation.
7.3.6 Conclusion & Discussion
For the modeling of the Darcy-Weisbach friction factor on a bed with high velocity above the bed, usually relations between the equivalent relative roughness k_{s}/d and the Shields parameter θ are used. This approach has some complications. There are either 3 solutions or just 1 solution, where the solution with the highest relative roughness is physically impossible and unreasonable. Leaving either 2 or 0 solutions. Now Camenen & Larson (2013) suggested to use the lower solution, but in literature (Krupicka & Matousek (2010)) also data points are found on the upper branch of the solution.
Probably for relatively small Shields parameters, this method gives satisfactory results, but surely not for larger Shields parameters. Another point of discussion is, that most equations are based on experiments, where the Shields parameter was measured and the relative roughness was determined with an equation similar to equation (7.3-51). In engineering practice however this is not possible since the Shields parameter is not an input but is supposed to be an output. So besides a number of mathematical issues, the method is also not suitable for engineering practice. This is the reason for a first attempt to find an explicit practical equation for the Darcy-Weisbach friction factor directly. As long as the flow over a bed does not cause particles to start moving, the standard Darcy-Weisbach friction factor equation (7.3-18) is used, where the roughness is replaced by the particle diameter. Some use the particle diameter times a factor, but based on the experiments used here, a factor of 1 seems suitable. As soon as the top layer of the bed starts sliding, while the bed itself is still stationary, the Darcy-Weisbach friction factor increases according to equation (7.3-52). The higher the velocity difference between the flow above the bed and the bed, the thicker the layer of sheet flow and the higher the Darcy-Weisbach friction factor. To investigate the influence of this new approach, two simulations were carried out. The first simulation with a fixed factor of 2 for the Darcy-Weisbach friction factor, as is shown in Figure 7.3-10. A second simulation with the new approach as described here as is shown in Figure 7.3-11.
With the new approach there were some issues with the convergence of the numerical method, resulting in a maximum relative spatial volumetric concentration of 0.95. The difference between the two simulations is significant. The maximum Limit of Stationary Deposit Velocity (the velocity where the bed starts sliding) is about 6.8 m/sec with the fixed Darcy-Weisbach friction factor, while this is about 4.6 m/sec with the new approach. But the shape of the Limit of Stationary Deposit Velocity curves are different, especially at higher concentrations. Also the maximum occurs at a lower concentration. It is thus very important to have a good formulation for the friction on the top of the bed due to sheet flow, in order to have a good prediction of the Limit of Stationary Deposit Velocity. Of course, this is a first attempt to find an explicit formulation for the Darcy-Weisbach friction factor, so improvements are expected in the near future.
7.3.7 The Darcy Weisbach friction factor second attempt
The most promising equations in terms of the relative roughness as a function of the Shields parameter are the equations of Camenen et al. (2006) and Krupicka & Matousek (2010) because they are based on the extensive experimental databases. Only one relation is available in an explicit form, the Miedema (2014) relation, equation (7.3-52). The results of this equation are shown in Figure 7.3-9. The Krupicka & Matousek (2010) equation (7.3-45) is almost explicit, although it still uses the bed associated hydraulic radius. The two relations differ in the fact that one is concave and the other convex. Still in the region of the data points both may give a good correlation coefficient. The Krupicka & Matousek (2010) equation will give a rapid increase of the Darcy-Weisbach friction factor at low velocities, where the increase decreases with increasing velocity. The Miedema (2014) equation gives a continuous increasing Darcy-Weisbach friction factor with an increasing increase with the velocity. The downside of the Miedema (2014) equation is that it is derived based on the bed associated hydraulic radius and applied for the hydraulic radius by applying a factor 2, which might sometimes be the case, but not always. Looking at the lower solution of the Wilson (1988) equation (7.3-35) in Figure 7.3-9, its shape shows more similarity with the Miedema (2014) equation. The purpose of this research is to find an explicit formulation of the bed friction factor as a function of known variables like the pipe diameter or hydraulic radius (based on the bed height without sheet flow), the relative submerged density of the particles, the velocity difference between the flow above the bed and the bed itself, the Darcy-Weisbach friction factors of clean water with the pipe wall λ_{1} and clean water with the bed without sheet flow, the particle diameter and the terminal settling velocity of the particle.
solids and size [mm] |
solids density [kg/m^{3}] |
pipe B x H [mm] |
data source |
sand - 0.7 |
2670 |
93.8 x 93.8 |
Nnadi & Wilson 1992 |
nylon - 3.94 |
1140 |
93.8 x 93.8 |
Nnadi & Wilson 1992 |
bakelite - 0.67 |
1560 |
93.8 x 93.8 |
Nnadi & Wilson 1992 |
bakelite - 1.05 |
1560 |
93.8 x 93.8 |
Nnadi & Wilson 1992 |
ballotini - 0.18 |
2450 |
50.8 x 51.2 |
Matousek et al. 2013 |
sand - 0.125 |
2650 |
88 x 288 |
Bisschop et al. 2014 |
Based on the experimental data in Table 7.3-1, regressions are carried out on different types of equations. In all cases the input quantities were the measured hydraulic gradient, the velocity above the bed and the dimensions of the restricted area above the bed.
The equations tested are exponential and power equations with the velocity above the bed, the hydraulic radius of the discharge area above the bed, the relative submerged density, the Darcy-Weisbach friction factor for pipe wall in case of flow of water in a pipe, the Darcy-Weisbach friction factor of clean water flow above the bed, the particle diameter and the terminal settling velocity as input parameters. The exponential equation was chosen because at larger values of the argument it has the same behavior as equation (7.3-52). Both types of equations give the same correlation coefficient of about 0.87. An important difference between the exponential and the power approach is that the exponential equation will have an offset bigger than zero for very small velocities, while the power approach has an offset equal to zero. As a first thought the offset should be the Darcy-Weisbach friction factor based on the particle diameter according to equation (7.3-18). However applying this, reduced the correlation coefficient considerably (see Table 7.3-2). Applying the clean water wall Darcy-Weisbach friction factor, did increase the correlation coefficient to a value of 0.91 when it was multiplied by 0.8. The resulting power equation containing all parameters is:
\[\ \lambda_{12}=\mathrm{0.8 \cdot \lambda _ { 1 } + \mathrm { 0 . 0 3 6 } \cdot \frac { ( \mathrm { v } _ { 1 } - \mathrm { v } _ { 2 } ) ^ { 2 . 4 6 2 } \cdot \mathrm { d } ^ { 0 . 8 0 9 } } { \mathrm { R } _ { \mathrm { H } } ^ { 1 .1 2 } \cdot \mathrm { R } _ { \mathrm { sd } } ^ { 0 . 8 4 7 } \cdot \mathrm { v } _ { \mathrm { t } } ^ { 0 . 4 6 1 } }} \quad\text{ with: }\quad \lambda_{1}=\frac{\mathrm{1 . 3 2 5}}{\left(\ln \left(\frac{\mathrm{0 . 2 7} \cdot \varepsilon}{\mathrm{D}_{\mathrm{H}}}+\frac{\mathrm{5 . 7 5}}{\mathrm{R e}^{0.9}}\right)\right)^{2}}\]
Adding the clean water Darcy-Weisbach friction factor of the pipe wall to the power term in this equation did not increase the correlation coefficient. Wilson (1988) stated that the bed friction factor of sheet flow does not depend on the particle diameter and thus also not on the terminal settling velocity. To investigate this, the regression was also carried out omitting the particle diameter and the terminal settling velocity. The result is the following equation, which gives a correlation coefficient of 0.90, almost the same as the above equation.
\[\ \lambda_{12}=0.8 \cdot \lambda_{1}+0.000527 \cdot \frac{\left(\mathrm{v_{1}-v_{2}}\right)^{2.422}}{\mathrm{R_{H}}^{1.017} \cdot \mathrm{R_{s d}}^{1.474}} \text{ or }\lambda_{12}=0.8 \cdot \lambda_{1}+0.004936 \cdot \frac{\mathrm{F r}^{2.292} \cdot \mathrm{R e}^{0.129}}{\mathrm{R_{s d}}^{1.474}}\]
The Darcy-Weisbach friction factors found from the experiments, combined with the predicted Darcy-Weisbach friction factors based on the power equation and the exponential equation are shown in Figure 7.3-12. The coverage of the experimental data point by the predicted data points is reasonable, but sufficient for the purpose of this study. The scatter of the experimental data points is much larger than the predicted data points, which makes sense, because normally a curve fit approach narrows the scatter.
One can question whether the Reynolds numbers should contain the average velocity in the cross section above the bed v_{1} or the velocity difference between this average velocity and the bed velocity (v_{1}-v_{2}). However since the experiments used were carried out with a stationary bed, v_{2} was zero, so it does not make a difference here.
Equation |
Value of correlation coefficient |
Remarks |
Exponential |
0.914 |
With particle parameters |
Exponential |
0.906 |
Without particle parameters |
Power (equation (7.3-54)) |
0.911 |
With particle parameters |
Power (equation (7.3-55)) |
0.893 |
Without particle parameters |
Dimensionless (eqn. (7.3-56)) |
0.864 |
Without particle parameters |
Dimensionless (eqn. (7.3-57)) |
0.909 |
With particle parameters |
Figure 7.3-13 shows the predicted versus the measured Darcy-Weisbach bed friction factors. In order to compare the resulting equations with the original equation of Wilson (1988), the relative roughness based on the bed associated hydraulic radius has been determined for the original data and for the power and the exponential equations. The result is shown in Figure 7.3-14. The power fit and exponential fit give the same image as the original data. The data points with very high Shields numbers are from Bisschop et al. (2014) with velocities above the stationary bed up to 6 m/s. Equation (7.3-55) is also shown in dimensionless notation as a function of the Froude number, the Reynolds number and the relative submerged density. The dependency on the Reynolds number is weak and on the relative submerged density a bit more than half the power of the Froude number. To simplify the equation, the Froude number as used by Durand & Condolios (1952) can be used. After re-evaluating (optimizing) the first term in the equation for the highest correlation coefficient leading to a factor 0.7, this gives:
\[\ \lambda_{12}=0.7 \cdot \lambda_{1}+0.0476 \cdot\left(\frac{\left(\mathrm{v}_{1}-\mathrm{v}_{2}\right)}{\sqrt{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{H}} \cdot \mathrm{R}_{\mathrm{sd}}}}\right)^{2.58}=0.7 \cdot \lambda_{1}+0.0476 \cdot \mathrm{Fr}_{\mathrm{DC}}^{2.58}\]
The correlation coefficient for this equation is 0.865, a bit less than the 0.9 of the previous equation, but still acceptable (Miedema & Matousek (2014)).
Finally the idea came up that in a sheet flow layer the energy losses do not only depend on the submerged weight of the particles, but kinetic energy losses will also depend on the mass of the particles. To add a dimensionless term including the particle mass, the mass of the particle m_{p} is divided by the weight of 1 m3 of the carrier liquid. Miedema & Ramsdell (2014) found the following equation:
\[\ \lambda_{12}=\mathrm{0 . 8 3} \cdot \lambda_{1}+\mathrm{0 .37} \cdot\left(\frac{\left(\mathrm{v}_{1}-\mathrm{v}_{2}\right)}{\sqrt{\mathrm{2 \cdot g \cdot D}_{\mathrm{H}} \cdot \mathrm{R}_{\mathrm{s d}}}}\right)^{2.73} \cdot\left(\frac{\rho_{\mathrm{s}} \cdot \frac{\pi}{6} \cdot \mathrm{d}^{3}}{\rho_{\mathrm{l}} \cdot \mathrm{1}^{\mathrm{3}}}\right)^{\mathrm{0 . 0 9 4}}=\mathrm{0 . 8 3 \cdot \lambda _ { 1 } + \mathrm { 0 . 3 7 } \cdot \mathrm { F r } _ { \mathrm { D C } } ^ { 2 . 7 3 }} \cdot\left(\frac{\mathrm{m _ { \mathrm { p } }}}{\rho_{\mathrm{l}}}\right)^{\mathrm{0 . 0 9 4}}\] |
Figure 7.3-15 shows the resulting Durand Froude number maximum LSDV curves, without the effect of suspended particles. Also the curves showing the theoretical transition between the sliding bed regime and the heterogeneous regime are shown. If this curve is below the LSDV curve, the stationary bed will transit directly to the heterogeneous regime. If the curve is above the LSDV curve, there will be a sliding bed. The LSDV curves can be approximated with the following equation:
\[\ \mathrm{F}_{\mathrm{L}}=\mathrm{0 .947} \cdot \mathrm{d}^{-0.056 \cdot \mathrm{D}_{\mathrm{p}}^{0.05}} \cdot \ln \left(\mathrm{e} \cdot \mathrm{D}_{\mathrm{p}}^{0.014}\right)=\mathrm{0 .9 4 7} \cdot \mathrm{d}^{-0.056 \cdot \mathrm{D}_{\mathrm{p}}^{0.05}} \cdot\left(\mathrm{1}+\mathrm{0 . 0 1 4} \cdot \ln \left(\mathrm{D}_{\mathrm{p}}\right)\right)\]
The graph and the equation assume that all particles are in the bed, resulting in decreasing curves. This makes sense, since smaller particles resulting in a smaller bed friction, so the velocity where the bed starts sliding, the LSDV, decreases with increasing particle diameter.
Since both the wall shear stress and the bed interface shear stress depend linear on the liquid density and the velocity above the bed squared, a higher liquid density resulting from a smaller stratification ratio, will result in a smaller LSDV, ignoring viscosity effects. An additional effect is the decrease of the relative submerged density with increasing liquid density. This decrease in relative submerged density results in a decrease of the weight of the bed and thus of the sliding friction force. Resulting in a lower LSDV. Including the viscosity effect reduces the Reynolds number and increases the Darcy Weisbach friction factor, reducing the LSDV even more. All these effects however are not enough to reproduce the strong decrease of the LSDV with decreasing particle diameter of the famous Wilson et al. (2006) demi McDonald.
7.3.8 Conclusions & Discussion
The goal of this study, finding an explicit relation between the bed Darcy-Weisbach friction factor and the known variables has been reached. Four equations have been found by regression, the first one based on 6 variables, the second and third based on 4 variables and the fourth again based on 6 variables. The 7 variables involved are the velocity difference between the bed and the flow above the bed u, the hydraulic radius of the flow above the bed R_{H}, the relative submerged density R_{sd}, the particle diameter d, the terminal settling velocity of the particles v_{t}, the mass of the particle m_{p} and the wall Darcy-Weisbach friction factor λ_{1}. The second equation is independent on the particle related variables, the particle diameter d and the terminal settling velocity v_{t}. Both equations have about the same correlation coefficient, 0.91 and 0.90. This supports the hypothesis of Wilson (1988) that the excess head losses due to sheet flow hardly depend on the particle related variables.
It should be considered however that the bed Darcy-Weisbach friction factor λ_{12} is determined from the hydraulic gradient, keeping the wall Darcy-Weisbach friction factor λ_{1} constant based on clean water flow and the pipe wall roughness. So all the additional head losses are considered to be caused by the increasing bed Darcy-Weisbach friction factor. In reality, the sheet flow will also influence the wall Darcy-Weisbach friction factor, especially at high velocities where the sheet flow may reach or almost reach the wall. For the purpose of determining hydraulic gradients this is not to relevant if the sheet flow is considered a black box with inputs (v_{1}-v_{2}), R_{H}, R_{sd}, d, v_{t} and λ_{1} and an output λ_{12}. However if the internal structure of the sheet flow has to be known in order to determine for example the delivered concentration, the equations may not be sufficient.
Matousek (2009) uses the internal structure in his head loss model to determine the delivered concentration. Miedema & Ramsdell (2013) based their model on the spatial concentration, using a holdup function to determine the delivered concentration (not yet published).
At first 4 equations were derived, exponential and power, with and without particle related variables. The correlation coefficients did not differ much. The author has chosen to use the two power equations in this chapter because of the applicability in their models.
Based on the two resulting equations the conclusion can be drawn that the bed Darcy-Weisbach friction factor depends on the wall Darcy-Weisbach friction factor at low velocities, depends weakly on the particle related variables, depends strongly on the velocity difference between bed and flow above the bed with a power of about 2.4-2.5, depends reversely proportional on the hydraulic radius of the discharge area above the bed to a power of about 1 and depends reversely proportional on the relative submerged density of the solids to a power of about 1.5. A further simplification, using the Durand & Condolios (1952) Froude number to the power 2.58 still gives an acceptable correlation coefficient. Adding the mass of the particle m_{p} to the equation increases the correlation coefficient to about 0.91, which seems to be the maximum achievable with the given dataset. This last equation (7.3-57) is the equation used in the DHLLDV Framework.
Showing the measured and predicted data in the k_{s}/d versus Shields parameter coordinate system, Figure 7.3-14, leads to the conclusion that the predicted k_{s}/d values match equation (7.3-35) well. Although at small values (up to about 2) of the Shields parameter equation (7.3-35) overestimates the k_{s}/d values, while at larger values of the Shields parameter equation (7.3-35) underestimates the k_{s}/d values. But this was already the case with the original experimental data of Nnadi & Wilson (1992).
The resulting equations are satisfying, but the coefficients and powers will probably change slightly if more experimental data is available. The resulting equations are derived for rectangular cross sections. Circular cross section may also result in different coefficients and powers, however some first tests on limited data show similar tendencies.
7.3.9 Nomenclature Fixed Bed Regime
A_{p} |
Cross section pipe |
m^{2} |
A_{1} |
Cross section above bed |
m^{2} |
A_{2} |
Cross section bed |
m^{2} |
C_{vb} |
Volumetric bed concentration |
- |
C_{vs} |
Spatial volumetric concentration |
- |
C_{vr} |
Relative spatial volumetric concentration |
- |
d |
Particle diameter |
m |
D_{H} |
Hydraulic diameter |
m |
D_{p} |
Pipe diameter |
m |
E_{rhg} |
Relative excess hydraulic gradient |
- |
F |
Force |
kN |
F_{1,l} |
Force between liquid and pipe wall |
kN |
F_{12}_{,l} |
Force between liquid and bed |
kN |
F_{2,pr} |
Force on bed due to pressure |
kN |
F_{2,sf} |
Force on bed due to friction |
kN |
F_{2,l} |
Force on bed due to pore liquid |
kN |
Fr |
Froude number |
- |
F_{r}_{DC} |
Durand Froude number |
m/s^{2} |
i_{l} |
Hydraulic gradient liquid |
- |
i_{m} |
Hydraulic gradient mixture |
- |
i_{plug} |
Hydraulic gradient plug flow |
- |
k_{s} |
Bed roughness (input=measured versus calculated=predicted) |
m |
ΔL |
Length of pipe section |
m |
n |
Porosity |
- |
O_{p} |
Circumference pipe |
m |
O_{1} |
Circumference pipe above bed |
m |
O_{2} |
Circumference pipe in bed |
m |
O_{12} |
Width of bed |
m |
p |
Pressure |
kPa |
Δp |
Pressure difference |
kPa |
Δp_{1} |
Pressure difference cross-section 1 |
kPa |
Δp_{2} |
Pressure difference cross-section 2 |
kPa |
Δp_{m} |
Pressure difference mixture |
kPa |
Re |
Reynolds number |
- |
R_{sd} |
Relative submerged density |
- |
R |
Bed associated radius |
m |
R_{H} |
Hydraulic radius |
m |
u_{*} |
Friction velocity |
m/s |
v |
Velocity |
m/s |
v_{r} |
Relative velocity |
- |
v_{1,m} |
Velocity in cross-section 1 |
m/s |
v_{2,m} |
Velocity in cross-section 2 |
m/s |
v_{t} |
Terminal settling velocity |
m/s |
v_{t*} |
Dimensionless terminal settling velocity |
- |
v_{ls} |
Line speed |
m/s |
v_{1} |
Velocity above bed |
m/s |
v_{2} |
Velocity bed |
m/s |
α |
Multiplication factor |
- |
β |
Bed angle |
rad |
ε |
Pipe wall roughness |
m |
κ |
Von Karman constant (0.4) |
- |
ρ_{l} |
Density carrier liquid |
ton/m^{3} |
ρ_{s} |
Density solids |
ton/m^{3} |
θ |
Shields parameter |
- |
θ_{cr} |
Critical Shields parameter |
- |
λ |
Darcy-Weisbach friction factor |
- |
λ_{l } |
Darcy-Weisbach friction factor liquid-pipe wall |
- |
λ_{1 } |
Darcy-Weisbach friction factor with pipe wall |
- |
λ_{2 } |
Darcy-Weisbach friction factor with pipe wall |
- |
λ_{12 } |
Darcy-Weisbach friction factor on the bed |
- |
\(\ v_{\mathrm{l}}\) |
Kinematic viscosity |
m^{2}/s |
\(\ \tau\) |
Shear stress |
kPa |
\(\ \tau_{\mathrm{l}}\) |
Shear stress liquid-pipe wall |
kPa |
\(\ \tau_{1,\mathrm{l}}\) |
Shear stress liquid-pipe in bed |
kPa |
\(\ \tau_{12,\mathrm{l}}\) |
Shear stress bed-liquid |
kPa |
\(\ \tau_{ 2,\mathrm{l}}\) |
Shear stress liquid-pipe in bed |
kPa |
\(\ \tau_{\mathrm{2, sf}}\) |
Shear stress from sliding friction |
kPa |
μ_{sf} |
Sliding friction coefficient |
- |