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8.11: Determining the Limit Deposit Velocity

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    32217
  • 8.11.1 Introduction

    Figure 8.11-1: The algorithm to determine the Limit Deposit Velocity.

    Screen Shot 2020-08-01 at 11.03.46 PM.png

    Figure 8.11-1 shows the algorithm to determine the Limit Deposit Velocity. The different steps are discussed in the next chapters.

    8.11.2 Very Small & Small Particles

    For very small particles, smaller than about 50% of the thickness of the viscous sub layer, the LDV and the Froude number FL are:

    \[\ \begin{array}{left}\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}=\mathrm{1. 4} \cdot ( v _ { \mathrm{l} } {\cdot \mathrm { R } _ { \mathrm { s d } } \cdot \mathrm { g } ) ^ { 1 / 3 } \cdot \sqrt { \frac { \mathrm { 8 } } { \lambda _ { \mathrm{l} } } }}\\
    \mathrm{F}_{\mathrm{L}, \mathrm{v s}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{ld v}}}{\left(\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}=\frac{\mathrm{1 .4} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g}\right)^{1 / 3} \cdot \sqrt{\frac{\mathrm{8}}{\lambda_{\mathrm{l}}}}}{\left(\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}\end{array}\]

    For small particles and a smooth bed, in the case of sand particles smaller than about d=0.15 mm, this gives for the Limit Deposit Velocity:

    \[\ \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}^{3}=\alpha_{\mathrm{p}}^{3} \frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\kappa_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{C}_{\mathrm{vs}} \cdot\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}\right)}{\lambda_{\mathrm{l}}}\]

    In terms of the Durand & Condolios LDV Froude number FL factor this can be written as:

    \[\ \begin{array}{left} \mathrm{F_{\mathrm{L}, \mathrm{ss}}}=\frac{\mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}}{\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}=\alpha_{\mathrm{p}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\kappa_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{C}_{\mathrm{vs}}}{\lambda_{\mathrm{l}} \cdot\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}\right)^{1 / 3}\\
    \text{with : }\quad \alpha_{\mathrm{p}}=3.4 \cdot\left(\frac{1.65}{\mathrm{R}_{\mathrm{sd}}}\right)^{2 / 9}, \quad \kappa_{\mathrm{C}}=0.175 \cdot(1+\beta)\end{array}\]

    The coefficient αp=3.4 is an upper limit. The minimum found is about 3.0, the average 3.2. To be on the safe side, the value of 3.4 should be used. To find the highest correlation with experimental data, the value of 3.2 should be used. With the following conditions the Froude number FL for very small and small particles can be determined:

    \[\ \begin{array}{llll}\text {If } & \mathrm{F}_{\mathrm{L}, \mathrm{v} \mathrm{s}}>\mathrm{F}_{\mathrm{L}, \mathrm{s s}} & \Rightarrow & \mathrm{F}_{\mathrm{L}, \mathrm{s}}=\mathrm{F}_{\mathrm{L}, \mathrm{vs}} \\ \mathrm{I f} & \mathrm{F}_{\mathrm{L}, \mathrm{v} \mathrm{s}} \leq \mathrm{F}_{\mathrm{L}, \mathrm{ss}} & \Rightarrow &\mathrm{F}_{\mathrm{L}, \mathrm{s}}=\mathrm{F}_{\mathrm{L}, \mathrm{ss}}\end{array}\]

    8.11.3 Large & Very Large Particles

    The Limit Deposit Velocity LDV is for medium and large particles and a rough bed:

    \[\ \mathrm{v}_{\mathrm{ls}, \mathrm{l d v}}^{\mathrm{3}}=\alpha_{\mathrm{p}}^{\mathrm{3}} \cdot \frac{\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\mu_{\mathrm{s f}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \frac{\pi}{\mathrm{8}}\right)^{1 / 2} \cdot \mathrm{C}_{\mathrm{v r}, \mathrm{l d v}}^{\mathrm{1 / 2}}}{\lambda_{\mathrm{l}}} \cdot\left(\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{3 / 2}\]

    And the Durand & Condolios LDV Froude number:

    \[\ \mathrm{F}_{\mathrm{L}, \mathrm{r}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{ld v}}}{\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}=\alpha_{\mathrm{p}} \cdot\left(\frac{\left(1-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\mu_{\mathrm{s} \mathrm{f}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \frac{\pi}{\mathrm{8}}\right)^{1 / 2} \cdot \mathrm{C}_{\mathrm{v r}, \mathrm{ld v}}^{1 / 2}}{\lambda_{\mathrm{l}}}\right)^{1 / 3}\]

    The bed fraction at the Limit Deposit Velocity is, depending on the particle diameter to pipe diameter ratio:

    \[\ \begin{array}{ll}\mathrm{d} \leq 0.015 \cdot \mathrm{D}_{\mathrm{p}} & \mathrm{C}_{\mathrm{vr}, \mathrm{ldv}}=0.0002 \cdot \mathrm{D}_{\mathrm{p}}^{-1} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{1.65}\right)^{-1}=\frac{0.0065}{2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}} \\ \mathrm{d}>0.015 \cdot \mathrm{D}_{\mathrm{p}} & \mathrm{C}_{\mathrm{vr}, \mathrm{ldv}}=0.0002 \cdot \mathrm{D}_{\mathrm{p}}^{-1} \cdot\left(\frac{\mathrm{R}_{\mathrm{sd}}}{1.65}\right)^{-1} \cdot\left(\frac{\mathrm{d}}{0.015 \cdot \mathrm{D}_{\mathrm{p}}}\right)^{1 / 2}=\frac{0.053}{2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{sd}} \cdot \mathrm{D}_{\mathrm{p}}} \cdot\left(\frac{\mathrm{d}}{\mathrm{D}_{\mathrm{p}}}\right)^{1 / 2}\end{array}\]

    8.11.4 The Resulting Upper Limit Froude Number

    The resulting upper limit of the Froude number FL,ul value can now be determined according to (for sand):

    \[\ \begin{array}{lll}\mathrm{F}_{\mathrm{L}, \mathrm{s}} \leq \mathrm{F}_{\mathrm{L}, \mathrm{r}} & \Rightarrow\quad \mathrm{F}_{\mathrm{L}, \mathrm{u l}}=\mathrm{F}_{\mathrm{L}, \mathrm{s}} \\ \mathrm{F}_{\mathrm{L}, \mathrm{s}}>\mathrm{F}_{\mathrm{L}, \mathrm{r}} & \Rightarrow\quad \mathrm{F}_{\mathrm{L}, \mathrm{u l}}=\mathrm{F}_{\mathrm{L}, \mathrm{s}} \cdot \mathrm{e}^{-\mathrm{d} / \mathrm{d}_{0}}+\mathrm{F}_{\mathrm{L}, \mathrm{r}} \cdot\left(\mathrm{1}-\mathrm{e}^{-\mathrm{d} / \mathrm{d}_{0}}\right) \\ \mathrm{d}>\mathrm{d}_{\mathrm{r o u g h}} & \Rightarrow\quad \mathrm{F}_{\mathrm{L}, \mathrm{u l}}=\mathrm{F}_{\mathrm{L}, \mathrm{r}} \\ \mathrm{w i t h}: \quad \mathrm{d}_{0}&= \mathrm{0 . 0 0 0 5} \cdot\left(\frac{\mathrm{1 .6 5}}{\mathrm{R}_{\mathrm{s d}}}\right)\end{array}\]

    8.11.5 The Lower Limit

    The lower limit of the LDV is the transition velocity between the sliding bed regime and the heterogeneous regime, resulting in the transition velocity at:

    \[\ \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}^{2}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\kappa_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}+\frac{\mathrm{8 . 5}^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{1}{\sqrt{\mathrm{C}_{\mathrm{x}}}}\right)^{10 / 3} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3}}{\mu_{\mathrm{sf}}}\]

    This equation shows that the transition between the sliding bed regime and the heterogeneous regime depends on the sliding friction coefficient. The equation derived is a second degree function and can be written as:

    \[\ -\mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}^{2}+\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\kappa_{\mathrm{C}}}\right)^{\beta}}{\mu_{\mathrm{sf}}} \cdot v_{\mathrm{ls}, \mathrm{ldv}}+\frac{\frac{8.5^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{1}{\sqrt{\mathrm{C}_{\mathrm{x}}}}\right)^{10 / 3} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3}}{\mu_{\mathrm{sf}}}=0\]

    \[\ \begin{array}{l}\text{With}\\
    \text { A }=-1 \\
    \text { B }=\frac{\mathrm{v_{t}} \cdot\left(1-\frac{\mathrm{C_{v s}}}{\kappa_{\mathrm{C}}}\right)^{\beta}}{\mu_{\text {sf }}}\\
    \text { C }=\frac{\frac{8.5^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{1}{\sqrt{\mathrm{C_{x}}}}\right)^{10 / 3} \cdot\left(v_{\mathrm{l}} \cdot g\right)^{2 / 3}}{\mu_{\text {sf }}} \\
    \mathrm{v}_{\text {ls,ldv}}=\frac{\mathrm{-B-\sqrt{B^{2}-4 \cdot A \cdot C}}}{\mathrm{2 \cdot A}}
    \end{array}\]

    In terms of the Durand & Condolios LDV Froude number FL factor this can be written as:

    \[\ \mathrm{F}_{\mathrm{L}, \mathrm{ll}}=\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}}{\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 2}}\]

    8.11.6 The Resulting Froude Number

    The resulting Froude FL value can now be determined according to:

    \[\ \begin{array}{lll}\mathrm{F}_{\mathrm{L}, \mathrm{u l}} \geq \mathrm{F}_{\mathrm{L}, \mathrm{ll}} & \Rightarrow & \mathrm{F}_{\mathrm{L}}=\mathrm{F}_{\mathrm{L}, \mathrm{u l}} \\ \mathrm{F}_{\mathrm{L}, \mathrm{ll}}>\mathrm{F}_{\mathrm{L}, \mathrm{ul}} & \Rightarrow & \mathrm{F}_{\mathrm{L}}=\mathrm{F}_{\mathrm{L}, \mathrm{ll}}\end{array}\]

    For small particles and/or concentrations near 20% and/or large pipe diameters, usually the upper limit Froude number will be valid. For large particles and/or low concentrations and/or very small to small pipe diameters, usually the lower limit Froude number will be valid.

    Figure 8.11-2: The resulting FLcurves.

    Screen Shot 2020-08-02 at 12.09.59 AM.png

    Figure 8.11-2 shows the resulting LDV curves for a number of volumetric concentrations, including the Durand & Condolios (1952) data. The graph matches the graph as published by Durand (1953) very well. The use of the lower limit based on the transition sliding bed regime to heterogeneous regime is not exact, since this transition velocity will not be exact. It is possible that this lower limit should be set to 90% or 95% of this transition velocity. Here the theoretical transition velocity is used.

    8.11.7 The Transition Fixed Bed – Sliding Bed (LSDV)

    The transition fixed bed – sliding bed is not considered to be a real Limit Deposit Velocity, but it is a regime change and thus will be discussed here. This transition is named the Limit of Stationary Deposit Velocity (LSDV) resulting from 2LM or 3LM analysis like the Wilson et al. (1992) model. This transition will only occur above a certain particle diameter to pipe diameter ratio. Very small particles will never have a sliding bed. But medium sized particles that will have a sliding bed in small pipe diameters, may not have a sliding bed in large pipe diameters. Mathematically however, the transition line speed can always be determined, even though the transition will not occur in reality. In such a case the bed is already completely suspended before it could start sliding.

    The total hydraulic gradient of a sliding bed im,sb is considered to be equal to the hydraulic gradient required to move clear liquid through the pipe il and the sliding friction hydraulic gradient resulting from the friction force between the solids and the pipe isf, is:

    \[\ \mathrm{i}_{\mathrm{m}, \mathrm{s} \mathrm{b}}=\mathrm{i}_{\mathrm{l}}+\mathrm{i}_{\mathrm{s f}}=\lambda_{\mathrm{l}} \cdot \frac{\mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}+\mu_{\mathrm{s f}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \mathrm{R}_{\mathrm{s d}}\]

    The hydraulic gradient im,fb due to the flow through the restricted area AH above the bed is:

    \[\ \mathrm{i}_{\mathrm{m}, \mathrm{f b}}=\lambda_{\mathrm{r}} \cdot \frac{\mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{r}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{H}}}=\lambda_{\mathrm{r}} \cdot \frac{\mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{H}}} \cdot\left(\frac{\mathrm{A}_{\mathrm{p}}}{\mathrm{A}_{\mathrm{H}}}\right)^{\mathrm{2}}\]

    This gives for the transition line speed:

    \[\ \mathrm{v}_{\mathrm{ls}, \mathrm{FBSB}}^{2}=\frac{2 \cdot \mu_{\mathrm{sf}} \cdot \mathrm{g} \cdot \mathrm{C}_{\mathrm{vs}} \cdot \mathrm{R}_{\mathrm{sd}}}{\frac{\lambda_{\mathrm{r}}}{\mathrm{D}_{\mathrm{H}}} \cdot\left(\frac{\mathrm{A}_{\mathrm{p}}}{\mathrm{A}_{\mathrm{H}}}\right)^{2}-\frac{\lambda_{\mathrm{l}}}{\mathrm{D}_{\mathrm{p}}}}\]

    8.11.8 The Transition Heterogeneous – Homogeneous (LDV Very Fine Particles)

    For very fine particles, the Limit Deposit Velocity is close to the transition line speed between the heterogeneous regime and the homogeneous regime. Values found in literature (Thomas (1976)) are between 80% and 100% of this transition velocity. This transition velocity can be determined by making the relative excess pressure contributions of both regimes equal, according to:

    \[\ \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}=\frac{\left(2 \cdot \mathrm{g} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{D}_{\mathrm{p}}\right)}{\lambda_{\mathrm{l}}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{v}_{\mathrm{l s}}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{v s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}+\frac{\mathrm{8 . 5}^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(\frac{\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 3}}{\mathrm{v}_{\mathrm{l s}}}\right)^{2}\right)\]

    This gives for the transition velocity between the heterogeneous regime and the homogeneous regime:

    \[\ \mathrm{v}_{\mathrm{l s}, \mathrm{H} \mathrm{e} \mathrm{Ho}}^{4}=\frac{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}{\lambda_{\mathrm{l}}} \cdot\left(\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}, \mathrm{H} \mathrm{e} \mathrm{Ho}}+\frac{\mathrm{8 .5}^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3}\right)\]

    This equation implies that the transition line speed depends reversely on the Darcy Weisbach viscous friction coefficient λl. Since the viscous friction coefficient λl depends reversely on the pipe diameter Dp with a power of about 0.2, the transition line speed will depend on the pipe diameter with a power of about (1.2/4) =0.3. It is advised to use the Thomas (1965) viscosity correction for this transition line speed, otherwise to high transition velocities may be found. The equation derived is implicit and has to be solved iteratively.

    8.11.9 The Transition Sliding Bed – Heterogeneous (LDV Coarse Particles)

    When a sliding bed is present, particles will be in suspension above the sliding bed. The higher the line speed, the more particles will be in suspension. The interaction between the particles in suspension and the particles in the bed will still be by inter particle interactions, reason that the sliding bed is still carrying the weight of all the particles in suspension. Apparently the weight of all the particles is resulting in sliding friction. At a certain line speed all the particles will be in suspension and the sliding bed regime transits to heterogeneous flow. The particles now interact with the pipe wall by collisions and not by sliding friction anymore.

    At the transition line speed the excess pressure losses of both regimes should be equal, giving

    \[\ \mu_{\mathrm{sf}}=\frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{v}_{\mathrm{ls}}} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{vs}}}{\kappa_{\mathrm{C}}}\right)^{\beta}+\frac{8.5^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(\frac{(v \cdot \mathrm{g})^{1 / 3}}{\mathrm{v}_{\mathrm{ls}}}\right)^{2}\]

    Resulting in the transition velocity at:

    \[\ \mathrm{v}_{\mathrm{l s}, \mathrm{S B H e}}^{2}=\frac{\mathrm{v _ { t }} \cdot(\mathrm{1 - \frac { C _ { \mathrm { v s } } } { \kappa _ { \mathrm { C } } }})^{\beta} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{S B H e}}+\frac{\mathrm{8 . 5}^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / \mathrm{3}}}{\mu_{\mathrm{s f}}}\]

    This equation shows that the transition between the sliding bed regime and the heterogeneous regime depends on the sliding friction coefficient. Implicitly Newitt et al. (1955) already found this, but didn’t explicitly mention this, because they assumed that potential energy is responsible for all the excess head losses in heterogeneous flow. The equation derived is a second degree function and can be written as:

    \[\ -\mathrm{v}_{\mathrm{l s}, \mathrm{SBHe}}^{2}+\frac{\mathrm{v _ { t }} \cdot\left(1-\frac{\mathrm{C}_{\mathrm{v s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mu_{\mathrm{s f}}} \cdot \mathrm{v}_{\mathrm{l s}, \mathrm{S B H e}}+\frac{\frac{\mathrm{8 .5}^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{1 0 / 3}} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3}}{\mu_{\mathrm{s f}}}=\mathrm{0}\]

    \[\ \begin{array}{left} \text{With:}\\
    \mathrm{A}=-\mathrm{1} \\
    \mathrm{B}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mu_{\mathrm{s f}}} \\
    \mathrm{C}=\frac{\frac{8.5^{2}}{\lambda_{\mathrm{l}}} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{10 / 3} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3}}{\mu_{\mathrm{sf}}} \\
    \mathrm{V}_{\mathrm{l s}, \mathrm{S B H e}}=\frac{-\mathrm{B}-\sqrt{\mathrm{B}^{2}-\mathrm{4} \cdot \mathrm{A} \cdot \mathrm{C}}}{\mathrm{2 \cdot A } }
    \end{array}\]

    The lower limit of the LDV is found to be about 80% of this transition line speed. This also implies that the transition of a sliding bed to heterogeneous transport is not sharp, an intersection point, but it’s a gradual process starting at about 80% of the transition line speed.