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7.12: The Bed Height

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    32133
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    7.12.1 Concentration Transformation Equations

    For a certain control volume the volumetric transport concentration can be determined if the volumetric spatial concentration and the slip velocity are known, given a certain line speed.

    \[\ \mathrm{C_{v t}=\left(1-\frac{v_{s l}}{v_{l s}}\right) \cdot C_{v s}=(1-\xi) \cdot C_{v s}}\]

    With the slip ratio:

    \[\ \xi=\frac{\mathrm{v}_{\mathrm{sl}} \mid}{\mathrm{v}_{\mathrm{ls}}}\]

    Likewise, for a certain control volume, the volumetric spatial concentration can be determined if the volumetric transport concentration and the slip velocity are known, given a certain line speed.

    \[\ \mathrm{C}_{\mathrm{v s}}=\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}}\right) \cdot \mathrm{C}_{\mathrm{v t}}=\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{C}_{\mathrm{v t}}\]

    These two equations will be used a lot in the following derivations and are considered to be well known.

    7.12.2 Fixed Bed

    Continuum Equations

    Considering a fixed bed, two phases are present, the fixed bed phase and the suspension phase. All the particles that transfer the head losses by inter-particle interactions are considered to be part of the fixed bed phase. All the particles that transfer their head loss contribution through turbulent dispersion are considered to be in the suspension phase. The flow of sediment can be expressed in terms of the pipe cross section times the volumetric spatial concentration times the line speed minus the slip velocity, where the slip velocity is considered to be the average over the whole pipe cross section. Since a fixed bed is considered, this has to be equal to the amount of solids transported in the suspension phase, which is the cross section of the suspension phase times the volumetric transport concentration of this suspension phase times the line speed.

    The volume flow of solids is (with a fixed bed the bed velocity is zero):

    \[\ \mathrm{Q}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{l s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \mathrm{v}_{\mathrm{b}}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{C}_{\mathrm{v t}, \mathrm{s}} \cdot \mathrm{v}_{\mathrm{s}}\]

    The total volume flow, liquid plus solids, is:

    \[\ \mathrm{Q}_{\mathrm{v}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{v}_{\mathrm{b}}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{s}}\]

    The volume of solids in a pipe section with length ΔL is:

    \[\ \mathrm{V}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \Delta \mathrm{L}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{s}} \cdot \Delta \mathrm{L}\]

    The total pipe cross section is the sum of the suspension cross section and the bed cross section:

    \[\ \mathrm{A}_{\mathrm{s}}+\mathrm{A}_{\mathrm{b}}=\mathrm{A}_{\mathrm{p}}\]

    Derivation

    The fraction of the pipe cross section occupied by solids is the sum of the fraction in the bed and the fraction in suspension:

    \[\ \mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}}+\left(\mathrm{A}_{\mathrm{p}}-\mathrm{A}_{\mathrm{b}}\right) \cdot \mathrm{C}_{\mathrm{v s , s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}}\right)=\mathrm{A}_{\mathrm{p}} \cdot\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{C}_{\mathrm{v t}}\]

    Now suppose that: \(\ \mathrm{C}_{\mathrm{v s}, \mathrm{s}}=\kappa_{\mathrm{ldv}} \cdot \mathrm{C}_{\mathrm{v t}} \quad\text{ with: }\kappa_{\mathrm{ldv}}=\left(\frac{\mathrm{v}_{\mathrm{l s}, \mathrm{ldv}}}{\mathrm{v}_{\mathrm{l s}, \mathrm{ld v}}-\mathrm{v}_{\mathrm{s} \mathrm{l}, \mathrm{ld v}}}\right)=\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}_{\mathrm{ldv}}}\right)\), based on the slip ratio at the Limit Deposit Velocity, this gives:

    \[\ \mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}}+\left(\mathrm{A}_{\mathrm{p}}-\mathrm{A}_{\mathrm{b}}\right) \cdot \mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot\left(\frac{\mathrm{v}_{\mathrm{l s}}}{\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}}\right)=\mathrm{A}_{\mathrm{p}} \cdot\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{C}_{\mathrm{v t}}\]

    This can be written as:

    \[\ \mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot\left(\mathrm{v}_{\mathrm{l} \mathrm{s}}-\mathrm{v}_{\mathrm{s} \mathrm{l}}\right)+\left(\mathrm{A}_{\mathrm{p}}-\mathrm{A}_{\mathrm{b}}\right) \cdot \mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \mathrm{v}_{\mathrm{ls}}\]

    Combining the terms with the bed cross section gives:

    \[\ \mathrm{A}_{\mathrm{b}} \cdot\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)+\mathrm{A}_{\mathrm{p}} \cdot \mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{l s}}\]

    Moving the terms with the full pipe cross section to the right hand side gives:

    \[\ \mathrm{A}_{\mathrm{b}} \cdot\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)=\mathrm{A}_{\mathrm{p}} \cdot\left(1-\mathrm{\kappa}_{\mathrm{l d v}}\right) \cdot \mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{l s}}+\mathrm{A}_{\mathrm{p}} \cdot \mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}} \cdot \mathrm{v}_{\mathrm{s l}}\]

    This results in an equation for the bed fraction:

    \[\ \zeta=\hat{\mathrm{A}}_{\mathrm{b}}=\frac{\mathrm{A}_{\mathrm{b}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\left(1-\kappa_{\mathrm{ldv}}\right) \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \mathrm{v}_{\mathrm{l s}}+\mathrm{\kappa}_{\mathrm{ldv}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \mathrm{v}_{\mathrm{sl}}}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{ldv}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(\mathrm{v}_{\mathrm{l s}}-\mathrm{v}_{\mathrm{s l}}\right)}\]

    This can be written as:

    \[\ \zeta=\hat{\mathrm{A}}_{\mathrm{b}}=\frac{\mathrm{A}_{\mathrm{b}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\left(1-\mathrm{\kappa}_{\mathrm{l} \mathrm{d v}}\right) \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}+\mathrm{\kappa}_{\mathrm{l} \mathrm{d} \mathrm{v}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \xi}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l} \mathrm{d} \mathrm{v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot(\mathrm{1}-\xi)}=\frac{\left(1-\mathrm{\kappa}_{\mathrm{l d v}} \cdot(1-\xi)\right) \cdot \mathrm{C}_{\mathrm{v t}}}{\left(\mathrm{C}_{\mathrm{v} \mathrm{b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot(1-\xi)}\]

    Because of the choice of κldv, the relative bed fraction is zero at a line speed equal to the Limit Deposit Velocity.

    At the LDV this gives a bed fraction:

    \[\ \begin{array}{left} \zeta_{\mathrm{ldv}}=\hat{\mathrm{A}}_{\mathrm{b}}=\frac{\mathrm{A}_{\mathrm{b}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\left(1-\mathrm{\kappa}_{\mathrm{l d v}} \cdot\left(1-\xi_{\mathrm{l} \mathrm{d} \mathrm{v}}\right)\right) \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}{\left(\mathrm{C}_{\mathrm{v} \mathrm{b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(1-\xi_{\mathrm{l d v}}\right)}=\frac{\mathrm{0}}{\left(\mathrm{C}_{\mathrm{v b}}-\mathrm{\kappa}_{\mathrm{l d v}} \cdot \mathrm{C}_{\mathrm{v t}}\right) \cdot\left(1-\xi_{\mathrm{l d v}}\right)}=\mathrm{0}\\
    \mathrm{W i t h}: \quad \mathrm{\kappa}_{\mathrm{ldv}}=\frac{1}{\left(1-\xi_{\mathrm{ldv}}\right)}\end{array}\]

    This makes sense, since the definition of the LDV is that there is no stationary or sliding bed.

    7.12.3 Sliding Bed

    Continuum Equations

    The volume flow of solids is (with a sliding bed the bed velocity is non-zero):

    \[\ \mathrm{Q}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \mathrm{v}_{\mathrm{l s}}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \mathrm{v}_{\mathrm{b}}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}, \mathrm{s}} \cdot \mathrm{v}_{\mathrm{s}}\]

    The total volume flow, liquid plus solids, is:

    \[\ \mathrm{Q}_{\mathrm{v}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{v}_{\mathrm{b}}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{s}}\]

    The volume of solids in a pipe section per unit of length is:

    \[\ \mathrm{V}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \Delta \mathrm{L}+\mathrm{A}_{\mathrm{s}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{s}} \cdot \Delta \mathrm{L}\]

    The total pipe cross section is the sum of the suspension cross section and the bed cross section:

    \[\ \mathrm{A}_{\mathrm{s}}+\mathrm{A}_{\mathrm{b}}=\mathrm{A}_{\mathrm{p}}\]

    Derivation

    The volume of solids in a pipe section per unit of length can be expressed in terms of the delivered volumetric concentrations, assuming the delivered volumetric concentration in the suspension phase is equal to this concentration at the LDV times a factor α, by:

    \[\ \mathrm{V}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{p}} \cdot\left(\frac{\mathrm{1}}{\mathrm{1 - \xi}}\right) \cdot \mathrm{C}_{\mathrm{v t}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \Delta \mathrm{L}+\mathrm{A}_{\mathrm{s}}\left(\frac{\mathrm{\alpha}}{\mathrm{1 - \xi _ { \mathrm { ldv } }}}\right) \cdot \mathrm{C}_{\mathrm{v t}} \cdot \Delta \mathrm{L}\]

    Replacing the suspension cross section by the pipe cross section minus the bed cross section gives:

    \[\ \mathrm{V}_{\mathrm{s}}=\mathrm{A}_{\mathrm{p}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{p}} \cdot\left(\frac{\mathrm{1}}{\mathrm{1}-\xi}\right) \cdot \mathrm{C}_{\mathrm{v t}} \cdot \Delta \mathrm{L}=\mathrm{A}_{\mathrm{b}} \cdot \mathrm{C}_{\mathrm{v b}} \cdot \Delta \mathrm{L}+\left(\mathrm{A}_{\mathrm{p}}-\mathrm{A}_{\mathrm{b}}\right) \cdot\left(\frac{\alpha}{\mathrm{1 - \xi}_{\mathrm{l d v}}}\right) \cdot \mathrm{C}_{\mathrm{v t}} \cdot \Delta \mathrm{L}\]

    Moving all terms with the pipe cross section to the left hand side and all terms with the bed cross section to the right hand side, gives:

    \[\ \mathrm{A_{p} \cdot\left(\frac{1}{1-\xi}-\frac{\alpha}{1-\xi_{l d v}}\right) \cdot C_{v t}=A_{b} \cdot\left(C_{v b}-\frac{\alpha \cdot C_{v t}}{1-\xi_{l d v}}\right)}\]

    This gives for the bed fraction:

    \[\ \zeta=\hat{\mathrm{A}}_{\mathrm{b}}=\frac{\mathrm{A}_{\mathrm{b}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\left(1-\alpha \cdot \kappa_{\mathrm{l} \mathrm{d} \mathrm{v}} \cdot(1-\xi)\right) \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}{\left(\mathrm{C}_{\mathrm{v} \mathrm{b}}-\frac{\alpha \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}{1-\xi_{\mathrm{ldv}}}\right)}=\frac{\left(1-\alpha \cdot \mathrm{\kappa}_{\mathrm{l} \mathrm{d} \mathrm{v}} \cdot(1-\xi)\right) \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}{\left(\mathrm{C}_{\mathrm{v} \mathrm{b}}-\alpha \cdot \mathrm{\kappa}_{\mathrm{l} \mathrm{d} \mathrm{v}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}\right) \cdot(1-\xi)}\]

    When α=1 this gives the same result as the fixed bed approach. At the LDV this only gives a bed height of zero and a bed height of 100% if α=1. A value of α>1 will give a higher bed fraction, A value of α<1 will give a lower bed fraction. So α could be a function of the line speed with a value of 1 at both zero line speed and at the LDV.

    The examples in this book are made with α=1, so with a fixed bed approach.

    Figure 7.12-1: The bed fraction and height of the bed.

    Screen Shot 2020-07-31 at 5.58.22 PM.png

    Figure 7.12-2: The bed fraction and height of the bed for different particle diameters.

    Screen Shot 2020-07-31 at 5.59.08 PM.png

    7.12.4 Some Results

    Based on the real slip velocity, the spatial concentration and the transport concentration can be converted into each other, giving the possibility to determine the spatial concentration eat each line speed, given a certain constant transport concentration. This also makes it possible to determine the fraction of the bed occupying the pipe cross section assuming a fixed bed, which is very valuable for dredging companies. Below the Limit Deposit Velocity, the bed fraction increases slowly with decreasing line speed. At a certain line speed however, the bed fraction will increase more rapidly with further decreasing line speed. Maybe the definition of the Limit Deposit Velocity should be reconsidered. Now the definition is the line speed at which a bed disappears completely with increasing line speed or appears with decreasing line speed. A definition based on an acceptable fraction of the bed occupying the pipe cross section would be more appropriate. Accepting a bed fraction of 10% would reduce the Limit Deposit Velocity from, in the example in Figure 7.12-1 and Figure 7.12-2, 6.25 m/sec to about 5 m/sec for the d=1 mm particles.

    The graphs shown are based on the fixed bed equation, so the bed height curves may differ slightly for a sliding bed.

    7.12.5 Nomenclature Bed Height

    Ab

    Bed cross section

    m2

    Ap

    Pipe cross section

    m2

    As

    Suspension cross section (cross section above the bed)

    m2

    \(\ \tilde{\mathrm{A}}_{\mathrm{b}}\)

    Bed fraction

    -

    Cvt

    Delivered (transport) volumetric concentration

    -

    Cvt,s

    Delivered (transport) volumetric concentration in suspended layer

    -

    Cvs

    Spatial volumetric concentration

    -

    Cvs,s

    Spatial volumetric concentration in suspended layer

    -

    Cvb

    Spatial volumetric concentration bed (1-n)

    -

    d

    Particle diameter

    m

    Dp

    Pipe diameter

    m

    ΔL

    Pipe length

    m

    n

    Porosity

    -

    Qs

    Total solids flow

    m3/s

    Qv

    Total flow, liquid + solids

    m3/s

    Rsd

    Relative submerged density

    -

    vls

    Line speed

    m/s

    vb

    Bed velocity

    m/s

    vs

    Suspension velocity

    m/s

    vls,ldv

    Limit Deposit Velocity

    m/s

    vsl

    Slip velocity

    m/s

    vsl,ldv

    Slip velocity at the LDV

    m/s

    Vs

    Volume of solids in a pipe section

    m3

    α

    Slip ratio factor

    -

    κldv

    Reversed slip ratio at LDV

    -

    μsf

    Sliding friction factor

    -

    ξ

    Slip ratio

    -

    ξldv

    Slip ratio at LDV

    -

    ζ

    Bed fraction

    -

    ζldv

    Bed fraction at the LDV

    -


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