8.18: Nomenclature DHLLDV Framework
 Page ID
 32343
A_{Cv} 
Coefficient homogeneous regime (1.3 by default) 
 
A_{p} 
Cross section of the pipe 
m^{2} 
A_{b} 
Bed cross section 
m^{2} 
A_{1} 
Cross section restricted area above the bed 
m^{2} 
A_{2} 
Cross section of the bed 
m^{2} 
C_{vb} 
Bed volumetric concentration 
 
C_{vb}_{,max} 
Maximum bed volumetric concentration 
 
C_{vB} 
Concentration at the bottom of the pipe 
 
C_{D} 
Particle drag coefficient 
 
C_{vs} 
Spatial volumetric concentration 
 
C_{vs}_{,x} 
Spatial volumetric concentration fines 
 
C_{vs}_{,r} 
Spatial volumetric concentration without fines 
 
C_{vr} 
Relative concentration C_{vs}/C_{vb} 
 
C_{vr}_{,ldv} 
Relative concentration in bed at LDV 
 
C_{vt} 
Transport or delivered volumetric concentration 
 
C_{x} 
Durand & Condolios coefficient 
 
C_{L} 
Lift coefficient 
 
d 
Particle diameter 
m 
d_{0} 
Particle diameter LDV transition region 
m 
d_{lim} 
Limiting particle diameter pseudo liquid 
m 
D_{H} 
Hydraulic diameter 
m 
D_{H,1} 
Hydraulic diameter restricted area above the bed 
m 
D_{p} 
Pipe diameter 
m 
E_{rhg} 
Relative excess hydraulic gradient 
 
E_{rhg,SB} 
Relative excess hydraulic gradient in the sliding bed regime 
 
E_{rhg}_{,SF} 
Relative excess hydraulic gradient in the sliding flow regime 
 
E_{rhg,He} 
Relative excess hydraulic gradient in the heterogeneous regime 
 
E_{rhg,Ho} 
Relative excess hydraulic gradient in the homogeneous regime 
 
E_{rhg,HeHo} 
Relative excess hydraulic gradient in the heterogeneous/ homogeneous flow regimes 
 
E_{rhg,Cvt} 
Relative excess hydraulic gradient with constant transport concentration 
 
E_{rhg,Cvs=Cvt} 
Relative excess hydraulic gradient with constant spatial concentration 
 
f 
Factor determining sliding flow 
 
f_{i} 
Fraction of the ith fraction in PSD 
 
F_{1,l} 
Force due to shear stress between liquid and wall 
kN 
F_{12}_{,l} 
Force due to shear stress between liquid and bed 
kN 
F_{2,pr} 
Force due to pressure gradient on the bed 
kN 
F_{2,sf} 
Force due to sliding friction between bed and pipe wall 
kN 
F_{2,l} 
Force due to shear stress of liquid in the pores with the pipe wall 
kN 
F_{L} 
Durand limit deposit velocity Froude number 
 
F_{L,s} 
Durand limit deposit velocity Froude number, smooth bed 
 
F_{L,ss} 
Durand limit deposit velocity Froude number, small particles smooth bed 
 
F_{L,vs} 
Durand limit deposit velocity Froude number, smooth bed, very small particles 
 
F_{L,r} 
Durand limit deposit velocity Froude number, rough bed, large particles 
 
F_{L,ul} 
Durand limit deposit velocity Froude number, upper limit 
 
F_{L,ll} 
Durand limit deposit velocity Froude number, lower limit 
 
F_{r}_{DC} 
Durand & Condolios Froude number 
 
Fr_{p} 
Particle Froude number 
 
g 
Gravitational constant (9.81) 
m/s^{2} 
h 
Thickness of bed at LDV 
m 
i_{l}, i_{w} 
Hydraulic gradient of liquid (water) 
 
i_{m} 
Hydraulic gradient of mixture 
 
i_{m,i} 
Hydraulic gradient of i_{th} fraction of PSD 
 
i_{m,ldv} 
Hydraulic gradient mixture at LDV 
 
i_{m,SF} 
Hydraulic gradient sliding flow 
 
K 
Durand & Condolios constant (85) 
 
ΔL 
Length of pipe segment considered 
m 
L_{R } 
Lift ratio 
 
m 
Mobilization factor homogeneous equation 
 
m_{p } 
Mass particle 
kg 
N 
Zandi & Govatos deposit criterion 
 
O_{p } 
Circumference pipe 
m 
O_{1 } 
Circumference pipe in contact with liquid 
m 
O_{2 } 
Circumference pipe in contact with bed 
m 
O_{12}_{ } 
Width of the bed 
m 
Δp_{l}_{ } 
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 
 
Re_{p } 
Particle Reynolds number 
 
R_{sd} 
Relative submerged density solids in carrier liquid 
 
R_{sd}_{,x } 
Relative submerged density solids in pseudo liquid 
 
S_{hr} 
Settling Velocity Hindered Relative 
 
S_{rs} 
Slip Velocity Relative Squared 
 
Stk 
Stokes number 
 
u_{* } 
Friction velocity 
m/s 
u_{*,ldv } 
Friction velocity at the LDV 
m/s 
v_{1 } 
Average velocity above the bed 
m/s 
v_{2 } 
Velocity of the bed 
m/s 
v_{12}_{ } 
Velocity difference bed interface (v_{1}v_{2}) 
m/s 
v_{ls}_{ } 
Crosssection averaged line speed 
m/s 
v_{ls,ldv } 
Limit Deposit Velocity (LDV) 
m/s 
v_{ls,t } 
Line speed at tangent point slip ratio 
m/s 
v_{r } 
Relative line speed v_{ls}/v_{ls}_{,ldv,max} or v_{ls}/v_{sm} 
m/s 
v_{sl}_{ } 
Slip velocity (velocity difference between particle and liquid) 
m/s 
v_{sm } 
Maximum LSDV according to Wilson 
m/s 
v_{t } 
Particle terminal settling velocity 
m/s 
v_{th}_{ } 
Hindered settling velocity 
m/s 
v_{thv}_{ } 
Hindered settling velocity vehicle (Wasp model) 

v_{tv}_{ } 
(Hindered) settling velocity in the vehicle (Wasp model) 
m/s 
v_{tv}_{,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   
α_{E } 
Coefficient homogeneous equation 
 
α_{p} 
LDV factor 
 
α_{sm} 
Factor concentration distribution 
 
β 
Angle of bed with vertical 
rad 
β 
Power of Richardson & Zaki hindered settling factor 
 
β_{sm} 
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 
δ_{v} 
Thickness viscous sub layer 
m 
λ_{1 } 
Darcy Weisbach friction factor liquid to pipe wall 
 
λ_{12}_{ } 
Darcy Weisbach friction factor bed interface 
 
κ 
Von Karman constant (about 0.4) 
 
κ_{ldv}_{ } 
Slip ratio factor at the LDV 
 
κ_{C } 
Concentration distribution constant 
 
ρ_{l}, ρ_{w } 
Density of liquid (water) 
ton/m^{3} 
ρ_{x}_{ } 
Density of liquid including fines (pseudo liquid) 
ton/m^{3} 
ρ_{m } 
Mixture density 
ton/m^{3} 
ρ_{s } 
Density of solids 
ton/m^{3} 
\(\ v_{\mathrm{l}}\) 
Kinematic viscosity carrier liquid 
m^{2}/s 
\(\ v_{\mathrm{x}}\) 
Kinematic viscosity pseudo liquid 
m^{2}/s 
μ_{l} 
Dynamic viscosity liquid 
Pa·s 
μ_{x} 
Dynamic viscously liquid including fines (pseudo liquid) 
Pa·s 
μ_{sf} 
Sliding friction coefficient 
 
\(\ \tau_{1,\mathrm{l}}\) 
Shear stress liquidpipe 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 
 
ξ_{0 } 
Slip ratio asymptotically for line speed zero 
 
ξ_{fb}_{ } 
Slip ratio with fixed bed 
 
ξ_{ldv} 
Slip ratio at the LDV 
 
ξ_{HeHo}_{ } 
Slip ratio in heterogeneous and homogeneous flow regimes 
 
ξ_{th} 
Resulting slip ratio 
 
ξ_{t } 
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 
 