# 6.19: The Jufin and Lopatin (1966) Model


## 6.8.1 Introduction

The Jufin & Lopatin (1966) model was constructed as a proposal for the Soviet technical norm in 1966. The authors did not submit a new model but selected the best combination of correlations for the frictional head loss and the critical velocity from four models submitted by different Soviet research institutes. The four models submitted were tested by a large experimental database collected by a number of researchers. The database contained data from both laboratory and field measurements (including data from dredging installations). The data covered a wide range of pipeline sizes (24 – 900 mm) and particle sizes (sand and gravel, 0.25 - 11 mm). Some of the data on which the model is based can be found in the chapters about Silin, Kobernik & Asaulenko (1958), Kazanskij (1978) and on the website www.dhlldv.com.

Kazanskij (1972) gave a summary and sort of manual for the use of the Jufin-Lopatin model. First of all sands and gravels are divided into 4 groups, according to Table 6.8-1. The ψ* parameter characterizes the particles and is comparable with the Durand & Condolios (1952) Cx parameter.

$\ \Psi^{*}=\mathrm{F r}_{\mathrm{p}}^{\mathrm{3} / 2}=\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{3} / \mathrm{2}}$

 Group Range ψ* A d<0.06 mm - B d60<10 mm All d10<10 mm1.5, d0>2.5 mm D d10>10 mm -

The particle diameter d0 (sometimes named dmf) is the average particle diameter, not a weighted particle diameter, and can be determined by:

$\ \mathrm{d}_{0}=\frac{\sum_{\mathrm{i}=1}^{100} \mathrm{d}_{\mathrm{i}}}{100} \quad\text{ or }\quad \mathrm{d}_{0}=\frac{\sum_{\mathrm{i}=10}^{90} \mathrm{d}_{\mathrm{i}}}{9}$

So each fraction has the same weight in the determination of the d0 value. For uniform sands and gravels, the d0 is equal to the particle diameter.

## 6.8.2 Group A: Fines

Group A covers the fines, silt. For silt Jufin & Lopatin (1966) use the ELM without the Thomas (1965) viscosity, so:

$\ \Delta \mathrm{p}_{\mathrm{m}}=\lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{ls}}^{\mathrm{2}}$

The hydraulic gradient im (for mixture) is now:

$\ \mathrm{i}_{\mathrm{m}}=\frac{\Delta \mathrm{p}_{\mathrm{m}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}$

## 6.8.3 Group B: Sand

Group B covers fine and medium sands, with possibly some fine gravel. The equation found by Jufin & Lopatin (1966) was based on the empirical experience, suggesting that the minimum hydraulic gradient at the velocity vmin was independent of the mixture flow properties and it was 3 times higher than the hydraulic gradient of water flow at the same velocity in a pipeline. This was also experienced in the American dredging industry (see Turner (1996)). Now most frictional head models follow the equation:

$\ \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{2} \cdot \mathrm{\Omega} \cdot\left(\frac{\mathrm{1}}{\mathrm{v}_{\mathrm{ls}}}\right)^{3}\right) \quad\text{ with: }\quad \mathrm{v}_{\min }=\Omega^{1 / 3}$

The minimum is found at the cube root of Ω, as is the case with the Durand/Condolios/Gibert, Newitt et al. and Fuhrboter models. The frictional-head-loss correlation by Jufin & Lopatin is:

$\ \Delta \mathrm{p}_{\mathrm{m}}=\Delta \mathrm{p}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{2} \cdot\left(\frac{\mathrm{v}_{\mathrm{m i n}}}{\mathrm{v}_{\mathrm{l s}}}\right)^{\mathrm{3}}\right)$

With, for quarts particles (sometimes a factor 5.3 is used instead of 5.5) :

$\ \mathrm{v_{\min }=5.5 \cdot\left(C_{v t} \cdot \Psi^{*} \cdot D_{p}\right)^{1 / 6}=3.76 \cdot\left(C_{v t} \cdot \Psi^{*} \cdot g \cdot D_{p}\right)^{1 / 6}=3.46 \cdot\left(C_{v t} \cdot \Psi^{*} \cdot g \cdot D_{p} \cdot R_{s d}\right)^{1 / 6}}$

This can be written in a more general form for the hydraulic gradient according to:

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{2} \cdot \mathrm{4 1 . 3 5} \cdot\left(\mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \Psi^{*} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 2}\left(\frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l s}}}\right)^{3}\right)$

Figure 6.8-1 shows the ψ* parameter of Jufin & Lopatin (1966) according to Kazanskij (1972). This parameter is compared with the equivalent parameters of Gibert (1960), Fuhrboter (1961) and the DHLLDV Framework (Miedema S. A., 2014). The trends are similar, but especially for medium sands, the values differ. The table of ψ* values (the black line with yellow circles) does not match equation (6.8-1) (the green line) well. The thick dashed brown line representing the DHLLDV Framework implementation, is closer to the table values. A power of 3 seems more appropriate than a power of 3/2.

Figure 6.8-2 shows a comparison in terms of the Gibert (1960) Cx value. Of course the trends are similar compared to Figure 6.8-1.

Assuming the experiments are carried out with quarts this can be written as:

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{2} \cdot \mathrm{4 1 . 3 5} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{3} / \mathrm{4}} \cdot\left(\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 2} \cdot\left(\mathrm{C}_{\mathrm{v t}}\right)^{1 / 2}\left(\frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l s}}}\right)^{3}\right)$

The term vmin should have the dimension of velocity, but in equation (6.8-7) it has the dimension of the cube root of velocity. This has to be compensated without violating the model of Jufin Lopatin. Now the product of kinematic viscosity $$\ v$$ and the gravitational constant g has the dimension of velocity to the 3rd power. It is not clear whether Jufin & Lopatin carried out experiments in liquids with different viscosities, but for dredging purposes it is neutral using this. This gives for vmin, using a kinematic viscosity of 10-6 m2/sec and a gravitational constant of 9.81 m/sec2:

$\ \mathrm{v}_{\min }=44.88 \cdot\left(\mathrm{C_{v t} \cdot \Psi^{*} \cdot g \cdot D_{p} \cdot R_{s d}}\right)^{1 / 6} \cdot\left({v_{\mathrm{l}}} \cdot \mathrm{g}\right)^{2 / 9}$

Substituting this in equation (6.8-6) gives the equation for the hydraulic gradient.

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}} \cdot\left(\mathrm{1}+\mathrm{2} \cdot \mathrm{9 0 3 8 9} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{3} / 4} \cdot\left(\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 2} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3} \cdot\left(\mathrm{C}_{\mathrm{v t}}\right)^{1 / 2}\left(\frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l s}}}\right)^{3}\right)$

## 6.8.4 The Limit Deposit Velocity

Jufin & Lopatin (1966) defined the Limit Deposit Velocity as (sometimes a value of 8 is used instead of 8.3):

$\ \mathrm{v}_{\mathrm{l s}, \mathrm{l d v}}=\mathrm{8 . 3} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \Psi^{*}\right)^{1 / 6} \cdot \mathrm{D}_{\mathrm{p}}^{\mathrm{1 / 3}}$

It is clear that this Limit Deposit Velocity also does not have the dimension of velocity, but the cube root of length. To give this Limit Deposit Velocity the dimension of velocity, the equation is modified to (for quarts and water):

$\ \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}=9.23 \cdot\left(\mathrm{C}_{\mathrm{vt}} \cdot \Psi^{*}\right)^{1 / 6} \cdot\left(2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{sd}}\right)^{1 / 3} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 9}$

Which can be written as:

$\ \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}=9.23 \cdot\left(\mathrm{C}_{\mathrm{vt}}\right)^{1 / 6} \cdot\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{1 / 4} \cdot\left(2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{sd}}\right)^{1 / 3} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 9}$

Giving:

$\ \mathrm{F_{L}=\frac{v_{l s, l d v}}{\left(2 \cdot g \cdot D_{p} \cdot R_{s d}\right)^{1 / 2}}}{=9.23 \cdot \frac{\mathrm{\left(C_{v t}\right)^{1 / 6}} \cdot\left(\frac{\mathrm{v_{t}}}{\sqrt{\mathrm{g \cdot d}}}\right)^{1 / 4} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{1 / 9}}{\left(2 \cdot \mathrm{g} \cdot \mathrm{D_{p} \cdot R_{s d}}\right)^{1 / 6}}}$

The effect of a broad particle size distribution is taken into account by determining an average value of the modified particle Froude number from values of the modified Froude number for soil fraction pi of different size di. The values for ψ* can also be taken from Table 6.8-1 or Figure 6.8-1.

$\ \Psi^{*}=\mathrm{F r_{v t}^{1.5}}=\mathrm{\sum_{i=1}^{n} F r_{v t, i}^{15} \cdot p_{i}=\sum_{i=1}^{n} \Psi^{*}\left(d_{i}\right) \cdot p_{i}}$

 size fraction of solids, d [mm] particle settling parameter, ψ* Jufin & Lopatin (1966) particle settling parameter, ψ* Jufin (1971) 0.05 - 0.10 0.0204 0.02 0.10 - 0.25 0.0980 0.1 0.25 - 0.50 0.4040 0.4 0.50 - 1.00 0.7550 0.8 1.0 - 2.0 1.1550 1.2 2.0 - 3.0 1.5000 1.5 3.0 - 5.0 1.7700 1.8 5 - 10 1.9400 1.9 10 - 20 1.9700 2.0 >20 2.0000 2.0

## 6.8.6 Group C: Fine Gravel

Group C is a transition between medium sized sand and coarse gravel. The equation for vmin has to be corrected according to:

\ \begin{aligned} \mathrm{v}_{\mathrm{m i n}} &=\mathrm{5 .5} \cdot \mathrm{b} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \Psi^{*} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 6}=3.7 \mathrm{6} \cdot \mathrm{b} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \Psi^{*} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 6} \\ &=\mathrm{3 .4 6} \cdot \mathrm{b} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \Psi^{*} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 6}=44.8 \mathrm{8} \cdot \mathrm{b} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \Psi^{*} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 6} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 9} \end{aligned}

The correction factor b can be determined with:

$\ \mathrm{b}=1+\frac{(\Psi-1.5)}{(2.0-1.5)} \cdot(\mathrm{a}-1)$

Where the factor a can be found in Table 6.8-2.

 d0 10 mm20 mm ρm (ton/m3) 1.02 1.05 1.10 1.20 1.02 1.05 1.10 1.20 Dp<400 mm 1.01 1.18 1.34 1.48 1.11 1.30 1.48 1.68 400 mm

## 6.8.7 Group D: Coarse Gravel

For Group D the correction factor is just a, according to Table 6.8-2.

\ \begin{aligned} \mathrm{v}_{\min } &=\mathrm{5 .5} \cdot \mathrm{a} \cdot\left(\mathrm{C}_{\mathrm{v} \mathrm{t}} \cdot \mathrm{2} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 6}=\mathrm{3} .7 \mathrm{6} \cdot \mathrm{a} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}\right)^{1 / 6} \\ &=\mathrm{3} . \mathrm{46} \cdot \mathrm{a} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 6}=\mathrm{4 4 . 8 8} \cdot \mathrm{a} \cdot\left(\mathrm{C}_{\mathrm{v t}} \cdot \mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 6} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 9} \end{aligned}

## 6.8.8 Conclusions & Discussion

The model of Jufin & Lopatin (1966) for Group A is the ELM model without a viscosity correction.
The models for Groups B, C and D are similar, but Groups C and D have a correction factor. In order to make the Jufin & Lopatin (1966) model comparable with other models, the basic equation is written in terms of the liquid hydraulic gradient plus the solids effect.

\ \begin{aligned} \mathrm{i}_{\mathrm{m}}&=\mathrm{i}_{\mathrm{l}}+\mathrm{2} \cdot \mathrm{9 0 3 8 9} \cdot\left(\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \Psi^{*}\right)^{1 / 2} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{2 / 3} \cdot\left(\mathrm{C}_{\mathrm{v t}}\right)^{1 / 2}\left(\frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l s}}}\right)^{3} \cdot \frac{\lambda_{1} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}} \\ &= \mathrm{i}_{\mathrm{l}}+\mathrm{9 0 3 8 9}{ \cdot \frac { \lambda _ { 1 } \cdot ( \Psi ^ { * } \cdot \mathrm { R } _ { \mathrm { s d } } ) ^ { 1 / 2 } \cdot ( v_ { \mathrm { l } } \cdot \mathrm { g } ) ^ { 2 / 3 } } { ( \mathrm { g } \cdot \mathrm { D } _ { \mathrm { p } } ) ^ { 1 / 2 } \cdot ( \mathrm { C } _ { \mathrm { v t } } ) ^ { 1 / 2 } } \cdot \frac { \mathrm { C } _ { \mathrm { v } \mathrm { t } } } { \mathrm { v } _ { \mathrm { ls } } }} \end{aligned}

The Darcy-Weisbach friction factor for a smooth pipe can be approached by:

$\ \lambda_{1}=\alpha \cdot\left(\mathrm{v_{l s}}\right)^{\alpha_{1}} \cdot\left(\mathrm{D_{p}}\right)^{\alpha_{2}}$

With:

$\ \alpha=0.01216\text{ and }\alpha_{1}=-0.1537 \cdot\left(\mathrm{D_{p}}\right)^{-0.089}\text{ and }\alpha_{2}=-0.2013 \cdot\left(\mathrm{v_{l s}}\right)^{-0.088}$

For laboratory conditions both powers are close to -0.18, while for real life conditions with higher line speeds and much larger pipe diameters this results in a power for the line speed of about α1=-0.155 and for the pipe diameter of about α2=-0.175. This should be considered when analyzing the models for heterogeneous transport.

This gives for the Darcy-Weisbach friction factor in a dimensionless form:

$\ \mathrm{\lambda_{1}=0.01216 \cdot\left(v_{1 s}\right)^{-0.155} \cdot\left(D_{p}\right)^{-0.175} \approx 0.1233 \cdot\left(v_{1 s}\right)^{-0.155} \cdot\left(g \cdot D_{p}\right)^{-0.172}} \cdot\left(v_{1} \cdot \mathrm{g}\right)^{1 / 6}$

Substitution gives:

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{1 1 1 4 5} \cdot \frac{\left(\Psi^{*} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 2} \cdot\left(v_{\mathrm{l}} \cdot \mathrm{g}\right)^{\mathrm{5} / \mathrm{6}}}{\left(\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}\right)^{0.672} \cdot\left(\mathrm{C}_{\mathrm{v} \mathrm{s}}\right)^{1 / 2}} \cdot \frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l} \mathrm{s}} \mathrm{0 . 1 5 5}} \cdot \frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{v}_{\mathrm{ls}}}$

With the solids effect factor Sk (to compare with Fuhrboter) defined as:

$\ \mathrm{i}_{\mathrm{m}}=\mathrm{i}_{\mathrm{l}}+\mathrm{S}_{\mathrm{k}} \cdot \frac{\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{v}_{\mathrm{l}}} \quad\text{ with: }\quad \mathrm{S}_{\mathrm{k}}=\mathrm{1 1 1 4 5} \cdot \frac{\left(\Psi^{*} \cdot \mathrm{R}_{\mathrm{s d}}\right)^{1 / 2} \cdot\left(v_{1} \cdot \mathrm{g}\right)^{5 / 6}}{\left(\mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}\right)^{0.672} \cdot\left(\mathrm{C}_{\mathrm{v s}}\right)^{1 / 2}} \cdot \frac{\mathrm{1}}{\mathrm{v}_{\mathrm{l s}} \mathrm{0 . 1 5 5}}$

The Sk curve of Jufin-Lopatin matches the original Fuhrboter curve reasonably for a Dp=0.1016 m pipe and 25% spatial volumetric concentration, just as the DHLLDV Framework and the Gibert data. However, the Jufin-Lopatin equation contains the pipe diameter and the concentration and will thus give different results for other pipe diameters and concentrations.

This page titled 6.19: The Jufin and Lopatin (1966) Model is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Sape A. Miedema (TU Delft Open Textbooks) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.