Dominguez et al. Equation for Deposition Velocity Estimation in a Subcritical Flow Regime 2

[Robert_Liang]

An equation below for deposition velocity estimation is introduced in Slurry Systems Handbook. This equation was published by Dominguez et al (1996).

(6-52)
where:
g - gravity acceleration
R_H - hydraulic radius
ρ_S - solids density
ρ_m - slurry density
d₈₅ - solid particle size

The relation of V_D and d₈₅ is illustrated below, R_H is 0.1 ft.

The relation chart of V_D (ft/s) and R_H (ft) is illustrated below, d85 is 0.5 mm

This chart meets our normal observation: the bigger the pipe/launder, the higher the deposition velocity.

For the slurry carrier’s viscosity different from water, the following equation is derived by Dominguez et al. (1996):

(6-53)

Where:
_μm - slurry absolute viscosity
_μL - carrier absolute viscosity

The book Slurry Systems Handbook has not mentioned the units of hydraulic radius and absolute viscosity, and the valid range of particle size, solids concentration and carrier viscosity.
Below is a chart of deposition velocity (ft/s) related to hydraulic radius (ft) where the viscosity is considered. d85 is 0.5 mm, μL is 0.00167 Pa•s, CV is 52%

This chart shows a negative relation of VD and RH: the bigger the pipe/launder, the lower the deposition velocity, which looks ridiculous. So, I doubt the correctness of the equation (6-53) mentioned in Slurry Systems Handbook

I have not access to the original paper of Dominguez et al. (1996). Does anybody have the paper and let me know the correct equation? I have asked the author of Slurry Systems Handbook by email for this question on June 11, 2019 but no reply yet so far.

Thanks a lot,
Robert Liang

 

[IRstuff]

What's the name of the paper or where it was published?

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

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[IRstuff]

Maybe useful?

http://www.dhlldv.com/

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

faq731-376 forum1529 Entire Forum list

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[Mech85]

Hi Robert,

Please find attached a file with screenshots of the "Deposit velocity of slurry flow in open channels" paper by Dominguez et al from 1996. There is a small mistake in the formula in Slurry Systems Handbook when the slurry viscosity is to be included. The superscript "3" is missing from the last part of the equation.

Kind Regards

 

[Robert_Liang]

Thanks a lot to IRstuff for the book you recommended. It is a great reference.

Thanks Mech85 for the clarification, I'll verify my calculation with your correction and report the result.

Robert Liang

 

[Robert_Liang]

Hi Mech85,

The calculation is done with the corrected equation.

(by Dominguz et al. 1996)

(from Slurry System Handbook)

Where:
μ_m slurry absolute viscosity (Pa•s)
μ_L carrier absolute viscosity (Pa•s)
ρ_m slurry density (kg/m³)

Below are charts of deposition velocity V_D (m/s) related to hydraulic radius R_H (m) or particle size d₈₅ (mm) where the viscosity is considered. d₈₅ is 0.554 mm, μ_L is 0.00167 Pa•s, C_V is 9.64%

The relation between deposition velocity and particle size illustrated below where R_H = 0.06536 m and C_Vf = 4%

The relation between deposition velocity and hydraulic radius illustrated below where C_vf = 4% and d₈₅ = 0.554 mm

This chart shows the relation of V_D and R_H: the bigger the pipe/launder, the higher the deposition velocity, which looks regular. So, I think the equation above for deposition velocity of slurry in open channel is rational.

 

[Robert_Liang]

The equation for slurry dynamic viscosity calculation is also checked with solid concentration by volume where liquid dynamic viscosity μ_L = 0.00167 Pa•s

(from Slurry Systems Handbook)

 

[Robert_Liang]

I have done a study of Darcy friction factor formula to find the relation between open channel slope and channel wall roughness or solids concentration.

(6-56) in Slurry System Handbook by Green et al. 1978
Where Reynolds number

(page 6.31 of Slurry Systems Handbook)

slurry dynamic viscosity

(from Slurry System Handbook)

slurry density

Slope

(6-6) in Slurry System Handbook

The chart below shows the strong relation of slope and roughness where
C_V = 52.14%
R_H = 0.06536 m
V = 2.32 m/s

The slope changes 37.81° from 1.87° to 39.68° while effective linear roughness changes from 1x10^-3 mm to 265x10^-3 mm.

I found the slope for slurry gravity flow in open channel is only related to the wall roughness, from the calculation of Darcy friction factor vs solid concentration by volume.
Below is the relation between slope and solid concentration by volume C_V
where :
• the effective linear roughness k_S = 0.013 mm
• hydraulic radius R_H = 0.06536 m
• flow velocity V = 2.32 m/s
• solids density ρ_S = 2095 kg/m³
• liquid density ρ_L = 1234 kg/m³

The slope only changes 0.03815° from 4.284° to 4.332° while C_V changes from 0.44% to 52.14% which means the slurry dynamic viscosity changes from 1.7 cP to 34.6 cP. This concludes that the slope has little relation with solid concentration.