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8.18: Nomenclature DHLLDV Framework

  • Page ID
    32343
  • ACv

    Coefficient homogeneous regime (1.3 by default)

    -

    Ap

    Cross section of the pipe

    m2

    Ab

    Bed cross section

    m2

    A1

    Cross section restricted area above the bed

    m2

    A2

    Cross section of the bed

    m2

    Cvb

    Bed volumetric concentration

    -

    Cvb,max

    Maximum bed volumetric concentration

    -

    CvB

    Concentration at the bottom of the pipe

    -

    CD

    Particle drag coefficient

    -

    Cvs

    Spatial volumetric concentration

    -

    Cvs,x

    Spatial volumetric concentration fines

    -

    Cvs,r

    Spatial volumetric concentration without fines

    -

    Cvr

    Relative concentration Cvs/Cvb

    -

    Cvr,ldv

    Relative concentration in bed at LDV

    -

    Cvt

    Transport or delivered volumetric concentration

    -

    Cx

    Durand & Condolios coefficient

    -

    CL

    Lift coefficient

    -

    d

    Particle diameter

    m

    d0

    Particle diameter LDV transition region

    m

    dlim

    Limiting particle diameter pseudo liquid

    m

    DH

    Hydraulic diameter

    m

    DH,1

    Hydraulic diameter restricted area above the bed

    m

    Dp

    Pipe diameter

    m

    Erhg

    Relative excess hydraulic gradient

    -

    Erhg,SB

    Relative excess hydraulic gradient in the sliding bed regime

    -

    Erhg,SF

    Relative excess hydraulic gradient in the sliding flow regime

    -

    Erhg,He

    Relative excess hydraulic gradient in the heterogeneous regime

    -

    Erhg,Ho

    Relative excess hydraulic gradient in the homogeneous regime

    -

    Erhg,HeHo

    Relative excess hydraulic gradient in the heterogeneous/ homogeneous flow regimes

    -

    Erhg,Cvt

    Relative excess hydraulic gradient with constant transport concentration

    -

    Erhg,Cvs=Cvt

    Relative excess hydraulic gradient with constant spatial concentration

    -

    f

    Factor determining sliding flow

    -

    fi

    Fraction of the ith fraction in PSD

    -

    F1,l

    Force due to shear stress between liquid and wall

    kN

    F12,l

    Force due to shear stress between liquid and bed

    kN

    F2,pr

    Force due to pressure gradient on the bed

    kN

    F2,sf

    Force due to sliding friction between bed and pipe wall

    kN

    F2,l

    Force due to shear stress of liquid in the pores with the pipe wall

    kN

    FL

    Durand limit deposit velocity Froude number

    -

    FL,s

    Durand limit deposit velocity Froude number, smooth bed

    -

    FL,ss

    Durand limit deposit velocity Froude number, small particles smooth bed

    -

    FL,vs

    Durand limit deposit velocity Froude number, smooth bed, very small particles

    -

    FL,r

    Durand limit deposit velocity Froude number, rough bed, large particles

    -

    FL,ul

    Durand limit deposit velocity Froude number, upper limit

    -

    FL,ll

    Durand limit deposit velocity Froude number, lower limit

    -

    FrDC

    Durand & Condolios Froude number

    -

    Frp

    Particle Froude number

    -

    g

    Gravitational constant (9.81)

    m/s2

    h

    Thickness of bed at LDV

    m

    il, iw

    Hydraulic gradient of liquid (water)

    -

    im

    Hydraulic gradient of mixture

    -

    im,i

    Hydraulic gradient of ith fraction of PSD

    -

    im,ldv

    Hydraulic gradient mixture at LDV

    -

    im,SF

    Hydraulic gradient sliding flow

    -

    K

    Durand & Condolios constant (85)

    -

    ΔL

    Length of pipe segment considered

    m

    LR

    Lift ratio

    -

    m

    Mobilization factor homogeneous equation

    -

    mp

    Mass particle

    kg

    N

    Zandi & Govatos deposit criterion

    -

    Op

    Circumference pipe

    m

    O1

    Circumference pipe in contact with liquid

    m

    O2

    Circumference pipe in contact with bed

    m

    O12

    Width of the bed

    m

    Δpl

    Pressure loss over pipeline length ΔL

    kPa

    Δpm

    Pressure loss mixture over pipeline length ΔL

    kPa

    r

    Position in pipe starting at the bottom

    -

    Re

    Reynolds number based on velocity difference liquid flow - bed

    -

    Rep

    Particle Reynolds number

    -

    Rsd

    Relative submerged density solids in carrier liquid

    -

    Rsd,x

    Relative submerged density solids in pseudo liquid

    -

    Shr

    Settling Velocity Hindered Relative

    -

    Srs

    Slip Velocity Relative Squared

    -

    Stk

    Stokes number

    -

    u*

    Friction velocity

    m/s

    u*,ldv

    Friction velocity at the LDV

    m/s

    v1

    Average velocity above the bed

    m/s

    v2

    Velocity of the bed

    m/s

    v12

    Velocity difference bed interface (v1-v2)

    m/s

    vls

    Cross-section averaged line speed

    m/s

    vls,ldv

    Limit Deposit Velocity (LDV)

    m/s

    vls,t

    Line speed at tangent point slip ratio

    m/s

    vr

    Relative line speed vls/vls,ldv,max or vls/vsm

    m/s

    vsl

    Slip velocity (velocity difference between particle and liquid)

    m/s

    vsm

    Maximum LSDV according to Wilson

    m/s

    vt

    Particle terminal settling velocity

    m/s

    vth

    Hindered settling velocity

    m/s

    vthv

    Hindered settling velocity vehicle (Wasp model)

    vtv

    (Hindered) settling velocity in the vehicle (Wasp model)

    m/s

    vtv,ldv

    (Hindered) settling velocity in the vehicle (Wasp model) at LDV

    m/s

    X

    Fraction of fines

    -

    α

    Factor in left ratio

    -

    α

    Factor in concentration distribution -

    αE

    Coefficient homogeneous equation

    -

    αp

    LDV factor

    -

    αsm

    Factor concentration distribution

    -

    β

    Angle of bed with vertical

    rad

    β

    Power of Richardson & Zaki hindered settling factor

    -

    βsm

    Relation sediment diffusivity eddy momentum diffusivity

    -

    ε

    Critical particle diameter to pipe diameter ratio

    -

    ε

    Pipe wall roughness

    m

    φ

    Internal friction angle

    rad

    δ

    External friction angle

    rad

    δv

    Thickness viscous sub layer

    m

    λ1

    Darcy Weisbach friction factor liquid to pipe wall

    -

    λ12

    Darcy Weisbach friction factor bed interface

    -

    κ

    Von Karman constant (about 0.4)

    -

    κldv

    Slip ratio factor at the LDV

    -

    κC

    Concentration distribution constant

    -

    ρl, ρw

    Density of liquid (water)

    ton/m3

    ρx

    Density of liquid including fines (pseudo liquid)

    ton/m3

    ρm

    Mixture density

    ton/m3

    ρs

    Density of solids

    ton/m3

    \(\ v_{\mathrm{l}}\)

    Kinematic viscosity carrier liquid

    m2/s

    \(\ v_{\mathrm{x}}\)

    Kinematic viscosity pseudo liquid

    m2/s

    μl

    Dynamic viscosity liquid

    Pa·s

    μx

    Dynamic viscously liquid including fines (pseudo liquid)

    Pa·s

    μsf

    Sliding friction coefficient

    -

    \(\ \tau_{1,\mathrm{l}}\)

    Shear stress liquid-pipe wall

    kPa

    \(\ \tau_{2,\mathrm{sf}}\)

    Shear stress bed – pipe wall due to sliding friction

    kPa

    \(\ \tau_{12,\mathrm{l}}\)

    Bed shear stress

    kPa

    ξ

    Slip ratio

    -

    ξ0

    Slip ratio asymptotically for line speed zero

    -

    ξfb

    Slip ratio with fixed bed

    -

    ξldv

    Slip ratio at the LDV

    -

    ξHeHo

    Slip ratio in heterogeneous and homogeneous flow regimes

    -

    ξth

    Resulting slip ratio

    -

    ξt

    Tangent line slip ratio

    -

    ζ

    Bed fraction

    -

    ζ

    Smoothing factor lift ratio

    -

    FB

    Fixed bed regime

    -

    He

    Heterogeneous flow regime

    -

    Ho

    Homogeneous flow regime

    -

    LDV

    Limit Deposit Velocity

    -

    SB

    Sliding bed regime

    -

    SF

    Sliding flow regime

    -

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