# Effect of the Concentration of Sand in a Mixture of Water and Sand Flowing through PP and PVC Elbows on the Minor Head Loss Coefficient

^{*}

## Abstract

**:**

^{−1}, 10.84 g·L

^{−1}, and 15.73 g·L

^{−1}. The tests were carried out at variable flow velocities for three elbow diameters of 63 mm, 75 mm, and 90 mm. The flow rate, pressure difference in the tested cross-sections, and temperature of the fluids were measured and automatically recorded. The results of the measurements were used to develop mathematical models for determining the minor head loss coefficient as a function of elbow diameter, sand concentration in the liquid, and Reynolds number. The mathematical model was developed by cross validation. It was shown that when the concentration of sand in the liquid was increased by 1.0 g∙L

^{−1}, the coefficient of minor head loss through the elbows increased, in the Reynolds number range of 4.6 × 10

^{4}–2.1 × 10

^{5}, by 0.3–0.01% for PP63, 0.6–0.03% for PP75, 1.1–0.06% for PP90, 0.8−0.01% for PVC63, 0.8–0.02% for PVC75, and 0.9–0.04% for PVC90. An increase in Re from 5 × 10

^{4}to 2 × 10

^{6}for elbows with diameters of 63, 75 and 90 mm caused a 7.3%, 6.8%, and 6.0% decrease in the minor head loss coefficient, respectively.

## 1. Introduction

^{−1}, ζ—dimensionless coefficient of minor head loss, ρ—fluid density, kg·m

^{−3}.

^{−2}.

^{−1}and air velocity of 15–27 m·s

^{−1}, the erosion rate resulting from the injection of sand into the gas stream was 3.3–9.26 × 10

^{−4}mm·kg

^{−1}sand. For flows of mixtures of water and air, pressure losses are affected not only by the flow velocities of both of these factors, but also by whether the flow is vertical, horizontal, or diagonal [8,9]. In the case of vertical pipelines, coalescence leads to the formation of gas voids in the elbow which reduce the cross-sectional area of liquid flow [10,11]. Air-water flow leads to a strong dissipation of the gas; the strongest dissipation was obtained between 3D and 9D upstream of an elbow (D—pipe diameter) [11]. Dissipation was found to be a linear function of Reynolds number [12].

^{5}the values of coefficient ζ are unstable, while in the transitional range 1 × 10

^{5}< Re < 4 × 10

^{5}, the value of ζ drops from around 2.3 to 1.38 along with increasing Re. In the post-critical range Re > 4 × 10

^{5}, coefficient ζ has a constant value of 1.38, regardless of changes in Re. Ma and Zhang [2] obtained ζ = 1.1 for an elbow during water flow; this value was constant for the flow velocity range of 0.5–2.4 m·s

^{−1}and Reynolds number > 6900.

## 2. Materials and Methods

_{1}= 5D downstream of the fitting and at L

_{2}= 3D upstream of the fitting, as recommended by Endress+Hauser in their installation manual for DELTABAR S differential pressure gauge, in which the manufacturer refers users to DIN 19210 recommendations for routing pressure piping. Analogously, for the flow meter (PROMAG), inlet and outlet runs were maintained to attain the specified level of accuracy of the measuring device. The tests were carried out using 63 × 3.0, 75 × 3.6 and 90 × 4.3 PN 10 PVC pipes.

^{−3}, C2 = 10.84 g·dm

^{−3}, and C3 = 15.73 g·dm

^{−3}. The concentration of sand in the mixtures was determined in accordance with reference [33]. Liquid flow rates were in the range of 5–40 m

^{3}·h

^{−1}and were increased in increments of 5 m

^{3}·h

^{−1}.

_{1}and l

_{2}stand for the inlet/outlet run lengths (distances of test cross-sections from the axis of the fitting), respectively.

_{1}− p

_{2}and flow rate Q were read from the respective gauges. Major head loss coefficient λ was calculated using the Phama formula [34], where the viscosity of water was determined, using liquid temperature measurements, from a relationship obtained by Polynomial Approximation (5) of points tabulated in a study by reference [35]. The viscosity of water with suspended solids was calculated using the Einstein Formula (6) [36].

_{z}—concentration of suspended solids in kg·m

^{−3}.

## 3. Results

#### 3.1. Hydraulic Conditions of Flow the Mixture Water and Sand

_{s}showed that the maximum grain size in the Stokes range (Re < 0.4) was 0.091 mm, and the minimum grain size in the Newton range (Re > 1000) was 3.25 mm. It follows that the particles of sand used in the experiments sedimented at rates described by the Allan model (Equation (8)). This model was used to calculate the settling rate for 0.5 mm-diameter grains, which was 0.062 m·s

^{−1}(Re = 23.8) [36].

_{s}and ρ

_{f}—sand and water densities, d—diameter grains of sand, g—acceleration of the earth and μ—dynamic coefficient of water viscosity.

_{s}= 17 × 0.062 = 1.05 m·s

^{−1}[36,37]. Under the assumed testing conditions, flow velocities for the vast majority of sand particles were higher at 0.78–4.3 m·s

^{−1}for 63 mm elbows, 0.8–3.1 m·s

^{−1}for 75 mm elbows, and 0.8–2.1 m·s

^{−1}for 90 mm elbows. In all series of tests, where the water flow rate was 10 m

^{3}·h

^{−1}and the higher limiting grain velocity determined by the Newitt formula was smaller than the actual flow velocities. Therefore, all grains were lifted with water. Only at the intensity of 5 m

^{3}·h

^{−1}, grains with a diameter of 0.385–0.5 mm could sediment and occupy the lower part of the water layer in PP63 and PVC63 and 0.395–0.5 mm in PP75, PVC75, PP90 and PVC90. It can be said that sand was practically fully dispersed in water, as observed in transparent sections of the installation. Figure 2 shows dispersion of sand in water for the particular sand concentrations C1, C2 and C3 at a flow velocity of about 1 m·s

^{−1}. Image W shows clear water.

#### 3.2. Results of Measurements

_{1}– p

_{2}allowed us to determine the coefficient of minor head loss through elbow ζ. A graph showing minor head losses through elbow PVC90 as a function of flow rate (the right part of Equation (4)) is presented in Figure 3. The points on the graph form a ‘saw-like’ curve with two types of fluctuations. A first type is related to unstable operating conditions that always occur in closed-loop pump systems and are associated with pressure pulsation. A second type is connected with the pump’s adjusting to the new operating conditions altered by changing the degree of opening of the control valve, i.e., changing the flow rate of the liquid. This second type of fluctuations was not analyzed in this study; we calculated the coefficient of minor head losses through the elbows for a constant flow rate, as regulated by the control valve, after a sufficiently long time for the system to have reached a steady state. In the test conditions (Figure 3), lower flow rates required throttling of the pump by means of a control valve. This throttling is always the source of increased flux pulsations, which in this case intensified because the system worked in a loop, i.e., a pulsating stream flowing through the discharge line after a small suppression returned to the suction tube. At higher flow rates, this phenomenon was much smaller. The moment steady-state operating conditions had been reached, the memograph was switched on, which recorded the instantaneous values of the parameters. An analysis of the results recorded by the memograph showed that some of the observation points deviated from the remaining ones located along the function curve and showed an unambiguous tendency that followed from the given parameters: flow rate, sand concentration, pipeline diameter, type of elbow, and liquid temperature. These results were rejected on the basis of criterion ζav ± 2·σ (criterion range = 2 standard deviations from the mean) [38], i.e., a region that, according to the normal distribution, comprises 95.4% of the results.

^{−1}, but at lower velocities, as follows from Newitt’s formula [36], sedimentation and dragging of sand along the bottom could have increased head loss, making the working conditions less stable. In the present study, analyses were performed at flow velocities V > 0.7 m·s

^{−1}to comply with the recommendations regarding design flow velocities in pressure sewerage systems [39,40]. The remaining results were excluded from modeling.

^{4}–3 × 10

^{4}, obtained ζ = 1.1 for copper elbows [2].

#### 3.3. Statistical Analysis of Results

^{5}to 4.7 × 10

^{4}.

#### 3.4. Appointment of Models for Calculation on the Minor Head Loss Coefficient for Elbows PP and PVC

^{2}were above 0.9. Incidentally, especially for elbow PP75, they reached a value of 0.8. Assuming that the partially generalized relationship would be based on the logarithmic function, Relationship (9) was found, which allows us to determine the minor head loss coefficient for the studied elbows as a function of Reynolds number Re and the concentration of suspended solids C

_{zaw}. Relationship (9), although it has a general form and applies to all tested fittings, differs in the values of m and k factors, which should be considered separately for each elbow. The numerical values of these coefficients obtained in the calculations are given in Table 2. The last two columns show the range of applicability of the model, i.e., the range of Re for which the tests were performed. With regard to the concentration of suspended solids, the model is valid in the range of 0–15.73 g·L

^{−1}, and sand concentrations should be substituted into the formula in these units (g·L

^{−1}).

^{2}of the determination coefficient R

^{2}show that the fit is acceptable.

^{−1}, 10.84 g∙L

^{−1}, and 15.73 g∙L

^{−1}; it was verified by calculating values ζ for C

_{zaw}= 0, i.e., that of clear water. Then, the sums of squared errors were calculated using formula Σ(ζmodel − ζmeasurement)

^{2}. These calculations were performed for each of the three concentrations of sand and for water. The total sum of squared errors across all observations, i.e., for sand concentrations of 0 g∙L

^{−1}, 5.6 g·L

^{−1}, 10.84 g·L

^{−1}, and 15.73 g·L

^{−1}, was used as the criterion of model fit. Next, a model was constructed for sand concentrations of 0 g∙L

^{−1}, 10.84 g·L

^{−1}, and 15.73 g·L

^{−1}, and the model for the concentration of 5.6 g·L

^{−1}was verified. Further models were obtained using the 10.84 g·L

^{−1}and 15.73 g·L

^{−1}sand concentrations for verification, and the remaining series of results were used to construct models. In this way, four models were obtained for each pipeline diameter, and the one that had the smallest total sum of squared errors was selected as the final model. The maximum differences between the sums of squared errors between the individual models were 14.6% for PP fittings and 13.5% for PVC fittings. The remaining ones had lower values ranging from 2.6% to 9.5%. The models with the best fit for each type of material are summarized in Equations (10) and (11):

_{in}is expressed in ‘meters’ and sand concentration C

_{zaw}in g·L

^{−1}.

^{−1}was statistically significant.

## 4. Conclusions

^{−3}higher in relation to water for PP elbows and 2.0% for PVC elbows. These differences, however, are so small that coefficients determined for water can be used to make approximate calculations.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic of the test stand for measuring minor head losses through elbows: 1—elbow, 2—thermometer, 3—pipe supplying sewage to the tank, 4—sewage tank, 5—pipe channeling sewage from the tank, 6—sewage pump, 7—needle valve, 8—sewage flow meter, 9, 10 and 12—control cable, 11—data recorder, 13—differential pressure meter, 1–1; 2–2—test cross-sections, L

_{1}, L

_{2}—distance of measuring cross-sections to the elbow.

**Figure 3.**Minor head loss coefficient ζ for a PVC90 elbow as a function of water flow velocity and three concentrations of water-and-sand mixture.

**Figure 4.**Values of coefficient ζ as a function of Reynolds number Re and concentration of sand for the test elbows. Elbows: (

**a**,

**b**)—diameter 63 mm, (

**c**,

**d**)—diameter 75 mm, (

**e**,

**f**)—diameter 90 mm.

**Figure 5.**Comparison of experimental data for ζ with the correlation found by Shiraishi et al. (2009) [1].

**Figure 6.**Longitudinal section of the test elbows: 1—inner edge of the elbow, 2—edge of the joint, 3—inner notch.

**Figure 8.**Standard deviation, skewness, and kurtosis as a function of Re for selected PP and PVC elbows. Elbows: (

**a**,

**b**)—diameter 63 mm, (

**c**,

**d**)—diameter 75 mm, (

**e**,

**f**)—diameter 90 mm.

**Figure 9.**Increase in head loss through PP elbows for water-sand mixtures compared to clear water, as a function of sand concentration and Reynolds number.

**Figure 10.**Increase in head loss through PVC elbows for water-sand mixtures compared to clear water, as a function of sand concentration and Reynolds number.

**Figure 12.**Verification of the correctness of the adopted model for PP (

**a**) and PVC (

**b**) elbows as a function of diameter d, concentration of suspended solids C, and Reynolds number Re.

**Table 1.**Means, medians and standard deviations of coefficients ζ for PP63 and PVC63 elbows at different R.

Medium | No. | W | C1 | C2 | C3 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Elbow | ζav | ζmed | ζσ | ζav | ζmed | ζσ | ζav | ζmed | ζσ | ζav | ζmed | ζσ | |

PP63 | 1 | 0.905 | 0.909 | 0.034 | 0.905 | 0.915 | 0.042 | 0.902 | 0.915 | 0.064 | 0.899 | 0.920 | 0.075 |

2 | 0.906 | 0.910 | 0.050 | 0.915 | 0.912 | 0.035 | 0.901 | 0.904 | 0.051 | 0.915 | 0.916 | 0.058 | |

3 | 0.910 | 0.909 | 0.031 | 0.913 | 0.914 | 0.033 | 0.905 | 0.905 | 0.053 | 0.923 | 0.928 | 0.058 | |

4 | 0.908 | 0.907 | 0.015 | 0.912 | 0.913 | 0.018 | 0.918 | 0.925 | 0.037 | 0.930 | 0.919 | 0.072 | |

5 | 0.930 | 0.924 | 0.019 | 0.933 | 0.938 | 0.029 | 0.933 | 0.932 | 0.013 | 0.951 | 0.952 | 0.025 | |

6 | 0.938 | 0.926 | 0.029 | 0.938 | 0.933 | 0.028 | 0.953 | 0.941 | 0.047 | 0.972 | 0.974 | 0.041 | |

7 | 0.943 | 0.944 | 0.008 | 0.967 | 0.972 | 0.012 | 0.973 | 0.961 | 0.043 | 0.969 | 0.968 | 0.008 | |

8 | 0.960 | 0.945 | 0.030 | 0.975 | 0.975 | 0.005 | 0.992 | 1.011 | 0.035 | 1.020 | 0.982 | 0.051 | |

PVC63 | 1 | 0.729 | 0.729 | 0.030 | 0.728 | 0.725 | 0.019 | 0.729 | 0.738 | 0.049 | 0.741 | 0.747 | 0.045 |

2 | 0.740 | 0.738 | 0.034 | 0.731 | 0.738 | 0.045 | 0.732 | 0.731 | 0.033 | 0.743 | 0.749 | 0.054 | |

3 | 0.736 | 0.728 | 0.029 | 0.745 | 0.742 | 0.039 | 0.726 | 0.710 | 0.043 | 0.748 | 0.754 | 0.038 | |

4 | 0.745 | 0.739 | 0.027 | 0.752 | 0.754 | 0.027 | 0.739 | 0.740 | 0.031 | 0.782 | 0.777 | 0.039 | |

5 | 0.761 | 0.759 | 0.011 | 0.774 | 0.773 | 0.011 | 0.778 | 0.777 | 0.013 | 0.791 | 0.794 | 0.013 | |

6 | 0.755 | 0.760 | 0.018 | 0.782 | 0.771 | 0.063 | 0.804 | 0.801 | 0.014 | 0.824 | 0.826 | 0.048 | |

7 | 0.767 | 0.765 | 0.010 | 0.779 | 0.783 | 0.014 | 0.834 | 0.830 | 0.034 | 0.854 | 0.836 | 0.050 | |

8 | 0.771 | 0.770 | 0.026 | 0.812 | 0.831 | 0.052 | 0.827 | 0.816 | 0.041 | 0.863 | 0.820 | 0.091 |

**Table 2.**Values of coefficients m and k for the model (Equation (7)) and ranges of the Reynolds number (applicability of the model).

Fitting | m | Value | k | Value | Re_{min} | Re_{max} |
---|---|---|---|---|---|---|

PP63 | m_{PP63}^{zaw} | −0.031306 | k_{PP63}^{zaw} | 0.661078 | 4.2 × 10^{4} | 2.6 × 10^{5} |

PP75 | m_{PP75}^{zaw} | −0.066786 | k_{PP75}^{zaw} | 0.546732 | 5.6 × 10^{4} | 2.15 × 10^{5} |

PP90 | m_{PP90}^{zaw} | −0.0085 | k_{PP90}^{zaw} | 0.428871 | 6.2 × 10^{4} | 1.7 × 10^{5} |

PVC63 | m_{PVC63}^{zaw} | −0.004077 | k_{PVC63}^{zaw} | 0.539926 | 4.6 × 10^{4} | 2.5 × 10^{5} |

PVC75 | m_{PVC75}^{zaw} | −0.00892 | k_{PVC75}^{zaw} | 0.424447 | 4.9 × 10^{4} | 2.1 × 10^{5} |

PVC90 | m_{PVC90}^{zaw} | −0.00973 | k_{PVC90}^{zaw} | 0.337607 | 5.9 × 10^{4} | 1.8 × 10^{5} |

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**MDPI and ACS Style**

Wichowski, P.; Siwiec, T.; Kalenik, M.
Effect of the Concentration of Sand in a Mixture of Water and Sand Flowing through PP and PVC Elbows on the Minor Head Loss Coefficient. *Water* **2019**, *11*, 828.
https://doi.org/10.3390/w11040828

**AMA Style**

Wichowski P, Siwiec T, Kalenik M.
Effect of the Concentration of Sand in a Mixture of Water and Sand Flowing through PP and PVC Elbows on the Minor Head Loss Coefficient. *Water*. 2019; 11(4):828.
https://doi.org/10.3390/w11040828

**Chicago/Turabian Style**

Wichowski, Piotr, Tadeusz Siwiec, and Marek Kalenik.
2019. "Effect of the Concentration of Sand in a Mixture of Water and Sand Flowing through PP and PVC Elbows on the Minor Head Loss Coefficient" *Water* 11, no. 4: 828.
https://doi.org/10.3390/w11040828