# Real Values of Local Resistance Coefficients during Water Flow through Welded Polypropylene T-Junctions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}= 50 mm and outlet diameter D

_{2}= 25 mm and cone convergence angles θ = 3°, 5°, 10°, 20°, and 90°; and two edge reducers with the inlet and outlet diameters D

_{1}/D

_{2}= 50 mm/25 mm and D

_{1}/D

_{2}= 68 mm/25 mm. The investigations were performed with use of a non-Newton liquid which was a coal–water mix with a mass concentration 57%–62%. The authors proved that during the flow of the coal–water mix with a given mass concentration through the tested fittings, the parameters of these fittings, i.e., R/D, θ, and β, had an insignificant impact on the values of the local resistance coefficient. As the Reynolds number increased to 1000 and the mass concentration of the coal–water mix increased, the values of the local resistance coefficient of the tested fittings decreased, whereas for higher values of the Reynolds number (Re > 1000), they presented various behaviors (trends) due to particular rheological properties of the coal–water mix, wherein shear stresses increased in line with the rise in the Reynolds number.

## 2. Materials and Methods

#### 2.1. Description of Test Rig

^{3}⋅min

^{−1}. At the inlet and outlet of the T-junction (7) were mounted impulse hoses (8) connected to ball valves (9). The hydraulic resistance of the individual T-junctions was measured using a DELTABAR 230 piezoelectric pressure difference meter (10) of Endress+Hauser with a measurement range of 0.0–500 mbar. The T-junctions were mounted at the test rig in the vertical position with use of a screwed joint (5) and a short polypropylene pipe section (6). At the highest points of the measurement system, vents (11) were mounted. Values of water flow rate were controlled by a needle valve (12), beyond which an electronic resistance thermometer (24) was mounted to measure water temperature. The rig operated in an open system and the water flowing out was carried into the sewerage system (13).

#### 2.2. Methodology of Investigations

^{3}⋅min

^{−1}within the range 5–25 dm

^{3}⋅min

^{−1}. The water temperature T during the measurements was equal to 12 °C. The tests were conducted using 30 randomly welded polypropylene T-junctions with the internal diameter 13.2 mm, divided into three groups according to the welding quality. There were 10 T-joints in each group (Figure 3). The first group contained properly warmed up and properly pressed T-junctions (Figure 3a), the second one contained poorly warmed up and poorly pressed T-junctions (Figure 3b), and the third one contained excessively warmed up and excessively pressed T-junctions (Figure 3c).

^{3}⋅s

^{−1}, g is the gravitational acceleration in m⋅s

^{2}, Δp is the pressure difference in Pa, V is the water flow velocity in m⋅s

^{−1}, ρ is the water mass density in kg⋅m

^{−3}, d is the internal diameter of the T-junction in mm.

## 3. Results and Discussion

^{2}from sample were over 0.98, indicating that the hydraulic resistance of the welded T-junctions depended at least 98% on the water flow rate and manufacturing precision of the T-junction joint, and only 2% on other factors like water temperature or gravitational acceleration.

_{1p}, ζ

_{2p}, ζ

_{3p}, ζ

_{4p}, ζ

_{5p}, ζ

_{6p}, ζ

_{7p}, ζ

_{8p}, and ζ

_{9p}were determined and are presented in Figure 6 as a function of the Reynolds number. The Reynolds number depends on the water flow velocity, internal diameter of a T-junction, and the water kinematic viscosity coefficient; hence, it described the hydraulic flow conditions of water in the tested T-junctions very well. Both the local resistance coefficient and the Reynolds number are dimensionless quantities; thus, researchers who determine local resistance coefficients ζ in fittings analyze them as a function of the Reynolds number [2,4,11,12,21,22,23,24,25,26,31,32,33,34,35,36]. The values of the local resistance coefficients decreased along with the increase of the Reynolds number (Figure 6), which was the trend expected according to the literature data [17,32]. The lowest values of the local resistance coefficients occurred for the properly warmed up and properly pressed T-junctions, and the highest for the excessively warmed up and excessively pressed ones. The trend (regression) of the measured values of ζ was of an exponential type, and the values of the determination coefficient R

^{2}from sample were over 0.64, indicating that the local resistance coefficient depended at least 64% on the Reynolds number Re, i.e., on the water flow velocity V, the T-junction diameter d, the water kinematic viscosity coefficient ν, and manufacturing precision of the T-junction joint, and depended 36% on the pressure difference Δp occurring in the T-junction during the water flow.

_{1p}= 0.44, the divergent water flow ζ

_{2p}= 1.37, and the convergent water flow ζ

_{3p}= 1.65. The mean value of the local resistance coefficient from the measurements made for the too poorly warmed up and too poorly pressed T-junctions with the through-run water flow was equal to ζ

_{4p}= 0.80, the divergent water flow ζ

_{5p}= 1.75, and the convergent water flow ζ

_{6p}= 2.29. The mean value of the local resistance coefficient from the measurements made for the excessively warmed up and excessively pressed T-junctions with the through-run water flow was equal to ζ

_{7p}= 1.47, the divergent water flow ζ

_{8p}= 6.26, and the convergent water flow ζ

_{9p}= 7.26. Table 2 presents the values determined from the nomograms proposed by Rennels and Hudson [26] used to calculate values of the local resistance coefficient in T-junctions for engineering purposes, i.e., for designing systems and installations of water supply.

_{1w}: local resistance coefficient for the properly warmed up and properly pressed T-junctions with the through-run water flow; ζ

_{2w}: local resistance coefficient for the properly warmed up and properly pressed T-junctions with the divergent water flow; ζ

_{3w}: local resistance coefficient for the properly warmed up and properly pressed T-junctions with the convergent water flow; ζ

_{4w}: local resistance coefficient for the poorly warmed up and poorly pressed T-junctions with the through-run water flow; ζ

_{5w}: local resistance coefficient for the poorly warmed up and poorly pressed T-junctions with the divergent water flow; ζ

_{6w}: local resistance coefficient for the poorly warmed up and poorly pressed T-junctions with the convergent water flow; ζ

_{7w}: local resistance coefficient for the excessively warmed up and excessively pressed T-junctions with the through-run water flow; ζ

_{8w}: local resistance coefficient for the excessively warmed up and excessively pressed T-junctions with the divergent water flow; ζ

_{9w}: local resistance coefficient for the excessively warmed up and excessively pressed T-junctions with the convergent water flow; Re: Reynolds number.

_{w}, whereas the abscissae were the values obtained from the measurements ζ

_{p}(Figure 6). The obtained points were approximated by a linear function crossing the origin of coordinates; thus, the correctness of choice of the mathematical model was verified by the slope coefficient of the linear function. The analysis of the dependence presented in Figure 7 allowed us to state that the exponential mathematical model described reality well, as the slope coefficient of the linear function was equal to 1.

^{−1}, ν is the kinematic viscosity coefficient of water in m

^{2}⋅s

^{−1}, and d is the internal diameter of the T-junction in m.

_{1p}, ζ

_{2p}, ζ

_{3p}, ζ

_{4p}, ζ

_{5p}, ζ

_{6p}, ζ

_{7p}, ζ

_{8p}, ζ

_{9p}(Figure 6) and calculated from Equations (13)–(21), ζ

_{1cal}, ζ

_{2cal}, ζ

_{3cal}, ζ

_{4cal}, ζ

_{5cal}, ζ

_{6cal}, ζ

_{7cal}, ζ

_{8cal}, and ζ

_{9cal}, were indeed significant. First, the normality of distribution was checked using the Shapiro–Wilk test, and then the homogeneity of variance was tested using the Levene test. Calculations of normality of distribution and homogeneity of variance were done with STATISTICA software and the obtained results are presented in Table 3. In both tests for individual groups, the values of calculated probability p

_{cal}were greater than the assumed significance level α = 0.05 (Table 3), meaning that the conditions of normal distribution and homogeneity of variance in the examined groups were satisfied. Student’s t-test was then used for two populations; a zero hypothesis (H

_{0}: n

_{1}= n

_{2}) stated that the mean values were statistically equal, and an alternative hypothesis (H

_{1}: n

_{1}≠ n

_{2}) stated that the mean values were statistically different. Calculations of the Student t-statistic value |t

_{cal}| were performed using the computer software STATISTICA (StatSoft Polska Sp. zo.o, Poland); the obtained results are gathered in Table 4.

_{cal}| ≥ t

_{α = 0.05}and, for v = n

_{1}+ n

_{2}– 2 = 40 degrees of freedom and α = 0.05, i.e., a selected 5% risk of error (significance level), the critical value t

_{α = 0.05}= 2.021 was read from the tables. Analysis of Table 4 allowed us to state that |t

_{cal}| ≤ t

_{α=0.05}, i.e., the zero hypothesis could not be rejected; thus, the differences between the mean values of the results of the local resistance coefficients obtained from measurements, ζ

_{1p}, ζ

_{2p}, ζ

_{3p}, ζ

_{4p}, ζ

_{5p}, ζ

_{6p}, ζ

_{7p}, ζ

_{8p}, and ζ

_{9p}(Figure 6) and calculated from Equations (13)–(21), ζ

_{1cal}, ζ

_{2cal}, ζ

_{3cal}, ζ

_{4cal}, ζ

_{5cal}, ζ

_{6cal}, ζ

_{7cal}, ζ

_{8cal}, and ζ

_{9cal}, were statistically insignificant, i.e., equal to each other. This is also confirmed by the fact that the calculated probability value, p

_{cal}, was greater than 0.05 (assumed significance level).

## 4. Conclusions

_{4p}, ζ

_{5p}, ζ

_{6p}) and excessively warmed up and excessively pressed T-junctions (ζ

_{7p}, ζ

_{8p}, ζ

_{9p}) allowed us to state that for the through-run water flow, the value of ζ

_{4p}increased by 45%, ζ

_{7p}by 70.1%, in the divergent water flow ζ

_{5p}increased by 21.7% and ζ

_{8p}by 78.1%, and in the convergent water flow, ζ

_{6p}increased by 27.9% and ζ

_{9p}by 77.3%.

_{1p}increased by 88.6%, in the divergent water flow ζ

_{2p}increased by 4.4%, and in the convergent water flow ζ

_{3p}increased by 32.1%; for poorly warmed up and poorly pressed T-junctions, in the through-run water flow, the value of ζ

_{4p}increased by 93.8%, in the divergent water flow, ζ

_{5p}increased by 25.1%, and in the convergent water flow, ζ

_{6p}increased by 51.1%; for the excessively warmed up and excessively pressed T-junctions, in the through-run water flow, the value of ζ

_{7p}increased by 96.6%, in the divergent water flow, ζ

_{8p}increased by 79.1%, and in the convergent water flow, ζ

_{9p}increased by 84.6%.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Scheme of the test rig for investigations of hydraulic resistance of polypropylene T-junctions: 1–water supplying pipe; 2–pump; 3–electromagnetic water flow meter; 4–needle valve; 5: screwed joints; 6–polypropylene pipe sections; 7–T-junction; 8: impulse hoses; 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23–ball cut-off valves; 10: piezoelectric pressure difference meter; 11–vents; 12–needle valve for water flow control; 13–floor sewerage inlet; 24–electronic resistance thermometer; A–through-run water flow; B–divergent water flow; C–convergent water flow.

**Figure 2.**Polyfusion welder: 1–temperature signal lamps; 2–heating plate; 3–heating mandrel; 4–heating sleeve.

**Figure 3.**Three types of polypropylene T-junction with visible fashes resulting from the pressing of the welded elements to each other: (

**a**) properly warmed up and properly pressed T-junction, (

**b**) poorly warmed up and poorly pressed T-junction, (

**c**) excessively warmed up and excessively pressed T-junction; 1: fash resulting from the pressing of the welded elements to each other.

**Figure 4.**Dependence between the real pressure difference Δp in polypropylene T-junctions with an internal diameter of 13.2 mm and the water flow rate q: (

**a**) properly warmed up and properly pressed T-junction, (

**b**) poorly warmed up and poorly pressed T-junction, (

**c**) excessively warmed up and excessively pressed T-junction.

**Figure 5.**Dependence between the mean pressure difference Δp in the screwed joints and polypropylene pipe sections and the water flow rate q.

**Figure 6.**Dependence between the local resistance coefficient ζ in polypropylene T-junctions with an internal diameter of 13.2 mm and Reynolds number Re: (

**a**) properly warmed up and properly pressed T-junction, (

**b**) poorly warmed up and poorly pressed T-junction, (

**c**) excessively warmed up and excessively pressed T-junction.

**Figure 7.**Verification of the correctness of the exponential mathematical model for calculations of the local resistance coefficient ζ.

**Figure 8.**Nomogram for determining values of the local resistance coefficient ζ in welded polypropylene T-junctions with an internal diameter of 13.2 mm, depending on water flow velocity V: (

**a**) properly warmed up and properly pressed T-junction, (

**b**) poorly warmed up and poorly pressed T-junction, (

**c**) excessively warmed up and excessively pressed T-junction.

**Table 1.**Basic statistics of the local resistance coefficients ζ determined from the measurements of the polypropylene T-junctions.

Local Resistance Coefficient | Statistics | ||||
---|---|---|---|---|---|

Minimum | Maximum | Mean | Median | Standard Deviation | |

ζ_{1p} | 0.38 | 0.54 | 0.44 | 0.44 | 0.06 |

ζ_{2p} | 1.22 | 1.67 | 1.37 | 1.35 | 0.08 |

ζ_{3p} | 1.47 | 1.98 | 1.65 | 1.62 | 0.12 |

ζ_{4p} | 0.71 | 1.02 | 0.80 | 0.79 | 0.05 |

ζ_{5p} | 1.53 | 2.16 | 1.75 | 1.71 | 0.12 |

ζ_{6p} | 2.03 | 2.70 | 2.29 | 2.25 | 0.17 |

ζ_{7p} | 1.35 | 1.83 | 1.47 | 1.45 | 0.09 |

ζ_{8p} | 5.74 | 7.55 | 6.26 | 6.15 | 0.36 |

ζ_{9p} | 6.62 | 8.62 | 7.26 | 7.12 | 0.49 |

**Table 2.**Values of the local resistance coefficient for the polypropylene T-junctions determined from the nomograms [26].

Flow Direction | ζ_{n} |
---|---|

Through-run, ζ_{10}_{n} | 0.05 |

Divergent, ζ_{11}_{n} | 1.31 |

Convergent, ζ_{12}_{n} | 1.12 |

**Table 3.**Results of calculations of the statistics using the Shapiro–Wilk test and Levene test. Differences of the mean values are significant with the probability p > 0.05.

Local Resistance Coefficient ζ | Probability Value Calculated with the Shapiro–Wilk Test p_{cal} | Probability Value Calculated with the Levene Test p_{cal} |
---|---|---|

ζ_{1p} ^{1} | 0.175 | 0.536 |

ζ_{1cal} ^{2} | 0.136 | |

ζ_{2p} ^{1} | 0.071 | 0.715 |

ζ_{2cal} ^{2} | 0.144 | |

ζ_{3p} ^{1} | 0.096 | 0.836 |

ζ_{3cal} ^{2} | 0.127 | |

ζ_{4p} ^{1} | 0.111 | 0.992 |

ζ_{4cal} ^{2} | 0.132 | |

ζ_{5p} ^{1} | 0.107 | 0.654 |

ζ_{5cal} ^{2} | 0.132 | |

ζ_{6p} ^{1} | 0.114 | 0.986 |

ζ_{6cal} ^{2} | 0.127 | |

ζ_{7p} ^{1} | 0.157 | 0.967 |

ζ_{7cal} ^{2} | 0.144 | |

ζ_{8p} ^{1} | 0.106 | 0.890 |

ζ_{8cal} ^{2} | 0.132 | |

ζ_{9p} ^{1} | 0.074 | 0.971 |

ζ_{9cal} ^{2} | 0.140 |

^{1}ζ

_{1p}–ζ

_{9p}: local resistance coefficients obtained in measurements (Figure 6);

^{2}ζ

_{1cal}–ζ

_{9cal}: local resistance coefficients calculated in Equations (13)–(21).

**Table 4.**Results of calculations of the statistics using Student’s t-test. Differences of the mean values were significant with the probability p < 0.05.

Local Resistance Coefficientζ | Statistics | ||||
---|---|---|---|---|---|

Mean | Standard Deviation | Calculated Value of the Student’s t-Test |t_{cal}| | Calculated Probability Value p_{cal} | Calculated Value of the Student’s t-Test Read from Tables for p = 0.05 and v = 40 t_{α = 0.05} | |

ζ_{1p} ^{1} | 0.459 | 0.025 | −1.875 | 0.075 | 2.021 |

ζ_{1cal} ^{2} | 0.461 | 0.029 | |||

ζ_{2p} ^{1} | 1.352 | 0.080 | 1.820 | 0.084 | |

ζ_{2cal} ^{2} | 1.347 | 0.072 | |||

ζ_{3p} ^{1} | 1.643 | 0.124 | 1.810 | 0.085 | |

ζ_{3cal} ^{2} | 1.639 | 0.119 | |||

ζ_{4p} ^{1} | 0.790 | 0.054 | 1.482 | 0.154 | |

ζ_{4cal} ^{2} | 0.789 | 0.054 | |||

ζ_{5p} ^{1} | 1.801 | 0.119 | –1.978 | 0.062 | |

ζ_{5cal} ^{2} | 1.812 | 0.123 | |||

ζ_{6p} ^{1} | 2.288 | 0.171 | 1.639 | 0.117 | |

ζ_{6cal} ^{2} | 2.280 | 0.166 | |||

ζ_{7p} ^{1} | 1.506 | 0.088 | –0.882 | 0.388 | |

ζ_{7cal} ^{2} | 1.511 | 0.080 | |||

ζ_{8p} ^{1} | 6.320 | 0.359 | −1.943 | 0.066 | |

ζ_{8cal} ^{2} | 6.338 | 0.368 | |||

ζ_{9p} ^{1} | 7.268 | 0.494 | −0.828 | 0.418 | |

ζ_{9cal} ^{2} | 7.273 | 0.494 |

^{1}ζ

_{1p}–ζ

_{9p}: local resistance coefficients obtained in measurements (Figure 6);

^{2}ζ

_{1cal}–ζ

_{9cal}: local resistance coefficients calculated using Equations (13)–(21).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kalenik, M.; Chalecki, M.; Wichowski, P.
Real Values of Local Resistance Coefficients during Water Flow through Welded Polypropylene T-Junctions. *Water* **2020**, *12*, 895.
https://doi.org/10.3390/w12030895

**AMA Style**

Kalenik M, Chalecki M, Wichowski P.
Real Values of Local Resistance Coefficients during Water Flow through Welded Polypropylene T-Junctions. *Water*. 2020; 12(3):895.
https://doi.org/10.3390/w12030895

**Chicago/Turabian Style**

Kalenik, Marek, Marek Chalecki, and Piotr Wichowski.
2020. "Real Values of Local Resistance Coefficients during Water Flow through Welded Polypropylene T-Junctions" *Water* 12, no. 3: 895.
https://doi.org/10.3390/w12030895