8.12: Constructing the Transport Concentration Curves

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There are 4 theoretical equations derived for the slip ratio, the region of line speeds below the LDV, the region of line speeds around the LDV and the region of line speeds above the LDV.

The slip ratio ξ_HeHo in the heterogeneous and homogeneous regime above the LDV is:

\[\ \xi_{\mathrm{HeHo}}=\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{ls}}}=\mathrm{8 .5} \cdot\left(\frac{\mathrm{1}}{\sqrt{\lambda_{\mathrm{l}}}}\right) \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{5 / 3} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right) \cdot \frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{v}_{\mathrm{l s}}}\]

The slip ratio ξ_ldv around the LDV can be approximated by, based on the 3LM model for a sliding bed with sheet flow:

\[\ \begin{array}{left} \xi_{\mathrm{aldv}}=\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{l s}}}=\left(1-\mathrm{C}_{\mathrm{v r}}\right) \cdot \mathrm{e}^{\left(-\left(\mathrm{0 .8 3 +} \frac{\mu_{\mathrm{sf}}}{4}+\left(\mathrm{C}_{\mathrm{vr}}-\mathrm{0 .5 - 0 . 0 7 5 \cdot D _ { \mathrm { p } }}\right)^{2}+\mathrm{0 . 0 2 5 \cdot D _ { \mathrm { p } }}\right) \cdot \mathrm{D}_{\mathrm{p}}^{0.025} \cdot\left(\frac{\mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}}{\mathrm{v}_{\mathrm{ls}, \mathrm{lsdv}}}\right)^{\alpha} \cdot \mathrm{C}_{\mathrm{vr}}^{0.65} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{\mathrm{1 . 5 8 5}}\right)^{0.1}\right)} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}}{\mathrm{v}_{\mathrm{l s}}}\right)^{4}\\
\text{With : } \alpha=0.58 \cdot \mathrm{C_{\mathrm{vr}}^{-0.42}}\\
\text{At the LDV:}\\
\xi_{\mathrm{ldv}}=\left(1-\mathrm{C}_{\mathrm{vr}}\right) \cdot \mathrm{e}^{\left(-\left(0.83+\frac{\mu_{\mathrm{sf}}}{4}+\left(\mathrm{C}_{\mathrm{vr}}-0.5-0.075 \cdot \mathrm{D}_{\mathrm{p}}\right)^{2}+0.025 \cdot \mathrm{D}_{\mathrm{p}}\right) \cdot \mathrm{D}_{\mathrm{p}}^{0.025} \cdot\left(\frac{\mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}}{\mathrm{v}_{\mathrm{ls}, \mathrm{lsdv}}}\right)^{\alpha} \cdot \mathrm{C}_{\mathrm{vr}}^{0.65} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{1.585}\right)^{0.1}\right)}\end{array}\]

The slip ratio ξ_fb based on a fixed bed with suspension above the bed is:

\[\ \begin{array}{left} \xi_{\mathrm{fb}}=\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{l s}}}=1-\frac{\mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}-\mathrm{v}_{\mathrm{l s}}\right)+\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}}\\
\mathrm{w i t h}: \mathrm{\kappa}_{\mathrm{l d v}}=\left(\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}}{\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}-\mathrm{v}_{\mathrm{s l}, \mathrm{l d v}}}\right)=\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}_{\mathrm{l d v}}}\right)\end{array}\]

The slip ratio according to the 3LM model is:

\[\ \begin{array}{\left} \xi_{3 \mathrm{LM}}=\frac{\mathrm{v}_{\mathrm{sl}}}{\mathrm{v}_{\mathrm{ls}}}=\left(1-\mathrm{C}_{\mathrm{vr}}\right) \cdot \mathrm{e}^{\left(-\left(0.83+\frac{\mu_{\mathrm{sf}}}{4}+\left(\mathrm{c}_{\mathrm{vr}}-0.5-0.075 \cdot \mathrm{D}_{\mathrm{p}}\right)^{2}+0.025 \cdot \mathrm{D}_{\mathrm{p}}\right) \cdot \mathrm{D}_{\mathrm{p}}^{0.025} \cdot\left(\frac{\mathrm{v}_{\mathrm{ls}}}{\mathrm{v}_{\mathrm{ls}, \mathrm{lsdv}}}\right)^{\alpha} \cdot \mathrm{C}_{\mathrm{vr}}^{0.65} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{1.585}\right)^{0.1}\right)}\\
\text{With : }\quad \alpha=0.58 \cdot \mathrm{C}_{\mathrm{vr}}^{-0.42}\end{array}\]

The resulting theoretical slip ratio curve is determined by:

\[\ \begin{array}{left}\mathrm{\xi_{f b}<\xi_{a l d v}} &\Rightarrow \quad\mathrm{\xi_{t h}=\xi_{f b}}\\
\mathrm{\xi_{f b} \geq \xi_{a l d v}} &\Rightarrow \quad\mathrm{\xi_{t h}=\xi_{a l d v}}\\
\mathrm{\xi_{H e H o}>\xi_{a l d v}} &\Rightarrow \quad\mathrm{\xi_{t h}=\xi_{H e H o}}\end{array}\]

The tangent line equation ξ_tis, tangent to the slip ratio around the LDV:

\[\ \mathrm{\xi_t=\frac{v_{sl}}{v_{ls}}=\left(1-\frac{C_{vt}}{C_{vb}} \right)-4 \cdot\left(1-\frac{C_{vt}}{C_{vb}} \right)\cdot e^{\left(-\left(0.83+\frac{\mu_{sf}}{4}+\left(C_{vr-0.5-0.075\cdot D_p} \right)^2 +0.025 \cdot D_p \right)\cdot D_p^{0.025}\cdot \left(\frac{v_{ls,ldv}}{v_{ls,lsdv}} \right)^{\alpha}\cdot C_{vr}^{0.65}\cdot \left(\frac{R_{sd}}{1.585} \right)^{0.1} \right)\cdot \left(\frac{v_{ls,ldv}}{v_{ls,t}} \right)^4 \cdot \left(\frac{v_{ls}}{v_{ls,t}} \right)}} \]

The tangent point v_ls,t is now:

\[\ \left.\mathrm{v}_{\mathrm{l s}, \mathrm{t}}=\left(\mathrm{5} \cdot \mathrm{e}^{\left(-\left(\mathrm{0 . 8 3}+\frac{\mu_{\mathrm{sf}}}{4}+\left(\mathrm{C}_{\mathrm{vr}}-\mathrm{0 . 5 - 0 . 0 7 5} \cdot \mathrm{D}_{\mathrm{p}}\right)^{2}+\mathrm{0 . 0 2 5} \cdot \mathrm{D}_{\mathrm{p}}\right)\right.} \cdot \mathrm{D}_{\mathrm{p}}^{0.025} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{ldv}}}{\mathrm{v}_{\mathrm{l s}, \mathrm{lsdv}}}\right)^{\alpha} \cdot \mathrm{C}_{\mathrm{vr}}^{0.65} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{\mathrm{1 . 5 8 5}}\right)^{0.1}\right)\right)^{1 / 4} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}\]

Giving for the tangent line:

\[\ \xi_{\mathrm{t}}=\frac{\mathrm{v}_{\mathrm{sl}} }{\mathrm{v}_{\mathrm{ls}}}=\left(1-\frac{\mathrm{C}_{\mathrm{v t}}}{\mathrm{C}_{\mathrm{vb}}}\right) \cdot\left(1-\frac{4}{\mathrm{5}} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{t}}}\right)\right)\]

The approximation of the slip ratio is now a weighed slip ratio according to:

\[\ \begin{array}{left}\xi_{\mathrm{SBHeHo}}=\xi_{\mathrm{th}} \cdot\left(1-\left(\frac{\mathrm{v}_{\mathrm{ls}}}{\mathrm{v}_{\mathrm{ls}, \mathrm{t}}}\right)^{\alpha}\right)+\xi_{\mathrm{t}} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l s}, \mathrm{t}}}\right)^{\alpha}& \text{if}\quad \mathrm{v}_{\mathrm{l s}}<\mathrm{v}_{\mathrm{l s}, \mathrm{t}}\\\xi_{\mathrm{SBHeHo}}=\xi_{\mathrm{th}}&\text{if}\quad \mathrm{v}_{\mathrm{ls}} \geq \mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{t}}\\
\text{With: }\quad \alpha=0-1\\
\xi_{\mathrm{SBHeHo}}>\xi_{3 \mathrm{LM}} \Rightarrow \xi_{\mathrm{SBHeHo}}=\xi_{3 \mathrm{LM}}\end{array}\]

In the case of sliding flow, the slip ratio has to be corrected. A pragmatic approach to determine the relative excess hydraulic gradient in the sliding flow regime is to use a weighted average between the heterogeneous regime and the sliding bed regime. First the factor between particle size and pipe diameter is determined:

\[\ \mathrm{f}=\frac{4}{3}-\frac{1}{3} \cdot \frac{\mathrm{d}}{\mathrm{r}_{\mathrm{d} / \mathrm{D} \mathrm{p}} \cdot \mathrm{D}_{\mathrm{p}}} \quad\text{ with: }\mathrm{0} \leq \mathrm{f} \leq \mathrm{1}\]

Secondly the weighted average slip ratio is determined:

\[\ \xi_{\mathrm{SF}}=\xi_{\mathrm{SBHeHo}} \cdot \mathrm{f}+\xi_{3 \mathrm{LM}} \cdot(1-\mathrm{f})\]

The resulting curve can be adjusted by changing the power α, however a value of 0.5 gives good results. The resulting slip ratio can never be larger than the slip ratio from the 3LM model. The resulting relative excess hydraulic gradient can be determined by multiplying the E_rhg curve for constant C_vs with the factor κ=1/(1-ξ). The volumetric spatial concentration on a constant delivered volumetric concentration curve equals:

\[\ \mathrm{C}_{\mathrm{v s}}(\xi)=\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{C}_{\mathrm{v t}} \quad \Rightarrow \quad \mathrm{E}_{\mathrm{r h g}, \mathrm{C v t}}=\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{E}_{\mathrm{r h g}, \mathrm{C v s}=\mathrm{C v t}}\]

Doing so, one should use the sliding bed equation also in the stationary bed range of the constant spatial volumetric concentration curve. If the stationary bed regime curve intersects with the heterogeneous regime curve below the sliding bed curve, one should continue the heterogeneous curve up to the intersection with the sliding bed curve and from there with decreasing line speed follow the sliding bed curve. The hydraulic gradient for the constant delivered volumetric concentration curve is now:

\[\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{E}_{\mathrm{r h g}, \mathrm{C v t}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}=\mathrm{i}_{\mathrm{l}}+\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{E}_{\mathrm{r h g}, \mathrm{C v s}=\mathrm{C v t}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}\]

This gives for the pressure difference:

\[\ \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L} \cdot \mathrm{E}_{\mathrm{r h g}, \mathrm{C v t}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}=\Delta \mathrm{p}_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L} \cdot\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{E}_{\mathrm{r h g}, \mathrm{C v s}=\mathrm{C v t}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}\]