# Experimental Determination of the Friction Factor in a Tube with Internal Helical Ribs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Friction Factor in Smooth and Rifled Tubes

^{5}), the McAdams Equation [30] can be used:

_{c}in contact with the fluid. For triangular and rectangular ribs, the parameter takes the value of 2; for rounded ribs, it approaches infinity.

_{n}—tube cross-section surface area (ribs not taken into account);

_{xs}—tube cross-section surface area minus the surface area occupied by ribs:

_{i}= 15.54 mm were tested to develop Equation (14). Equation (14) holds when $0.0212\text{}\text{}\frac{e}{{d}_{i}}\text{}\text{}0.0354$, $18\text{}\text{}N\text{}\text{}45$, ${25}^{\xb0}\text{}\text{}\mathsf{\beta}{\text{}\text{}45}^{\xb0}$, and $15,000\text{}\text{}\mathit{Re}\text{}\text{}50,000$.

## 3. Test Stand for Experimental Determination of Pressure Losses in Rifled Tubes

^{2};

^{2}, and the tube perimeter, including the ribs, is 114.44 mm. The hydraulic diameter calculated using Equation (15) is 32.305 mm. This value is used in further calculations to compare data obtained from measurements with theoretical computations related to smooth tubes.

## 4. Experimental Determination of The Friction Factor

_{F}:

_{i}=1/34.9.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | rib width at the base, mm |

A | area, mm^{2} |

A_{n} | tube cross-section surface area (ribs not taken into account), mm^{2} |

A_{xs} | tube cross-section surface area minus the surface area occupied by ribs, mm^{2} |

b | rib average width, mm |

d_{h} | hudraulic diameter, mm |

d_{o} | outer diameter, mm |

d_{i} | inner diameter (without ribs), mm |

d_{min} | minimum diameter, mm |

e | rib height, mm |

f | friction factor of smooth tubes |

f_{F} | friction factor of rifled tubes,- |

g | wall thickness, mm |

N | number of ribs in the cross-section, - |

O | wetted perimeter, m |

p | pitch, mm |

Re | Reynolds number |

u | uncertainty |

V | volumetric flow, m^{3}/h |

Greek symbols | |

α | rib apex angle, |

β | rib helical angle, |

Δp | differential pressure, Pa |

ρ | density, kg/m^{3} |

ν | kinematic viscosity coefficient, m^{2}/s |

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**Figure 1.**Geometrical dimensions of rifled tubes: (

**a**) Transverse section, (

**b**) longitudinal section, (

**c**) rib detail. a—Rib width at the base; b—rib average width; d

_{o}—outer diameter; d

_{i}—inner diameter (with no ribs); d

_{min}—minimum diameter with ribs; e—rib height; g—wall thickness; p—rib pitch; α—rib extension apex angle; β—rib helical angle.

**Figure 2.**Comparison of the friction factor curves plotted for tubes with internal helical ribs and smooth tubes. Regarding internally ribbed tubes: 1—Equation (11); 2—Equation (14); 3—Equation (9). Regarding smooth tubes: 4—Equation (2).

**Figure 3.**Diagram of the test stand. A—rifled tube (to determine the heat transfer coefficient); B, C —other tested objects; 1—cooler; 2—data acquisition system; 3—circulating water containers; 4—contamination filter; 5—circulating pump; 6—bypass; 7—flowmeter; 8—horizontal segment for frictional loss testing; 9—differential pressure meter; 10—heating elements power control system.

**Figure 6.**The PXWD differential pressure transducer [35].

**Figure 7.**Friction factor values obtained experimentally and their comparison with smooth tubes; 1—experimental data; 2—the proposed equation (Equation (20)); 3—the Blasius equation (Equation (2)).

**Figure 9.**Comparison of the friction factor values obtained by means of a PXWD transducer and a U-tube in relation to the equation values; 1—PXWD transducer; 2—U-tube; 3—Equation (20).

**Figure 10.**Comparison of the experimental data with the correlations defining the friction factor in rifled tubes with internal helical ribs and smooth tubes; 1—Equation (13); 2—Equation (11); 3—Equation (14);.4—Equation (9); 5—Equation (2); 6—the proposed equation (Equation (20)); 7—experimental data.

Researcher | Working Fluid | Inner Diameter d _{i} [mm] | Pitch p [mm] | Rib Height e [mm] |
---|---|---|---|---|

Ackerman [11] | Supercritical pressure water | 18 | 21.8 | 0.9 |

Gee and Webb [12] | Air | 25.4 | 3.81 | 0.25 |

Zimparov et al. [13] | Water | 25.0 | 6.5–16.9 | 0.44–1.18 |

Ravigururajan and Bergles [14] | Water and air | 14.0–22.86 | 4.16–10.16 | 0.89–1.78 |

Cheng et al. [15] | Oil | 11.0 | 5.5 | 0.5 |

Dong et al. [16] | Water and oil | 19.0–25.0 | 10.0–12.0 | 0.39–0.8 |

Barba et al. [17] | Water | 14.5 | 11.5 | 1.5 |

Vicente et al. [18] | Water and glycol | 18.0 | 10.9–22.1 | 0.42–1.03 |

Wang et al. [19,20] | Supercritical pressure water | 18.6 | 11.6 | 1.2 |

Yang et al. [21] | Two-phase mixture | 21.0 | 22.7 | 0.85 |

Khoeini et al. [22] | R-134a | 9.52 | 8.0 | 1.5 |

Ji et al. [23] | Water | 18.9–19.1 | - | 0.39–0.45 |

Lu et al. [24] | Liquid salt | 10.2 | 3.2 | 0.38–0.76 |

Li et al. [25] | Supercritical CO_{2} | 16.5 | - | 0.85 |

Yang et al. [26] | Supercritical CO_{2} | 9.0 | 10.0–20.0 | 0.25–0.90 |

Weiguo et al. [27] | Therminol 55 | 14.2 | 21.0 | 0.85 |

Zhang et al. [28] | Supercritical pressure water | 20.62 | 12.87 | 1.25 |

Shen et al. [29] | Subcritical pressure water | 18.96 | 18.1 | 1.24 |

**Table 2.**Characteristic dimensions of the experimentally-tested tube with internal helical ribs [34].

Characteristic Dimension | Value |
---|---|

Outer diameter, d_{o} | 50.8 mm |

Inner diameter (without ribs), d_{i} | 34.9 mm |

Minimum diameter, d_{min} | 32.9 mm |

Wall thickness, g | 7.95 mm |

Rib height, e | 1 mm |

Pitch, p | 30 mm |

Rib width at the base, a | 5 mm |

Rib average width, b | 4.5 mm |

Rib apex angle, α | 45° |

Rib helical angle, β | 30° |

Number of ribs in the cross-section, N | 6 |

**Table 3.**Friction factor values obtained experimentally and calculated values of the measurement uncertainty.

No. | Measured Flow V m ^{3}/h | Fluid Temperature t °C | Density ρ kg/m ^{3} | Kinematic Viscosity Coefficient ν m ^{2}/s | Reynolds Number Re | Measured Differential Pressure Δp Pa | Friction Factor f _{F} | Uncertainty u(f) |
---|---|---|---|---|---|---|---|---|

1 | 8.028 | 22.8 | 997.59 | 9.43 × 10^{−7} | 93195 | 2326.58 | 0.0204 | 0.0003 |

2 | 7.800 | 23.0 | 997.55 | 9.40 × 10^{−7} | 90888 | 2184.68 | 0.0203 | 0.0003 |

3 | 7.580 | 23.2 | 997.50 | 9.35 × 10^{−7} | 88751 | 2038.65 | 0.0200 | 0.0003 |

4 | 7.331 | 23.4 | 997.45 | 9.31 × 10^{−7} | 86231 | 1962.47 | 0.0206 | 0.0003 |

5 | 7.079 | 23.6 | 997.41 | 9.27 × 10^{−7} | 83623 | 1831.70 | 0.0206 | 0.0003 |

6 | 6.811 | 23.7 | 997.39 | 9.25 × 10^{−7} | 80613 | 1715.03 | 0.0209 | 0.0003 |

7 | 6.564 | 23.7 | 997.37 | 9.24 × 10^{−7} | 77790 | 1592.57 | 0.0208 | 0.0003 |

8 | 6.294 | 23.9 | 997.34 | 9.21 × 10^{−7} | 74849 | 1477.20 | 0.0210 | 0.0003 |

9 | 6.054 | 24.0 | 997.29 | 9.17 × 10^{−7} | 72282 | 1369.14 | 0.0211 | 0.0003 |

10 | 5.763 | 23.9 | 997.32 | 9.19 × 10^{−7} | 68648 | 1261.42 | 0.0214 | 0.0003 |

11 | 5.512 | 24.1 | 997.27 | 9.15 × 10^{−7} | 65934 | 1150.97 | 0.0214 | 0.0003 |

12 | 5.147 | 21.7 | 997.85 | 9.67 × 10^{−7} | 58279 | 1088.23 | 0.0232 | 0.0003 |

13 | 4.942 | 21.5 | 997.89 | 9.70 × 10^{−7} | 55768 | 986.23 | 0.0228 | 0.0003 |

14 | 4.679 | 21.8 | 997.83 | 9.65 × 10^{−7} | 53072 | 881.18 | 0.0227 | 0.0003 |

15 | 4.454 | 21.8 | 997.83 | 9.65 × 10^{−7} | 50523 | 815.94 | 0.0232 | 0.0003 |

16 | 4.176 | 21.9 | 997.80 | 9.62 × 10^{−7} | 47529 | 723.04 | 0.0234 | 0.0003 |

17 | 3.946 | 21.9 | 997.80 | 9.62 × 10^{−7} | 44894 | 656.44 | 0.0238 | 0.0003 |

18 | 3.694 | 21.9 | 997.80 | 9.62 × 10^{−7} | 42040 | 601.64 | 0.0249 | 0.0003 |

19 | 3.441 | 22.0 | 997.79 | 9.61 × 10^{−7} | 39214 | 536.43 | 0.0255 | 0.0003 |

20 | 3.156 | 22.2 | 997.74 | 9.56 × 10^{−7} | 36142 | 466.61 | 0.0264 | 0.0004 |

21 | 2.912 | 22.0 | 997.78 | 9.60 × 10^{−7} | 33211 | 397.07 | 0.0264 | 0.0004 |

22 | 2.654 | 22.0 | 997.78 | 9.60 × 10^{−7} | 30266 | 361.31 | 0.0289 | 0.0004 |

23 | 2.400 | 22.1 | 997.75 | 9.58 × 10^{−7} | 27426 | 307.55 | 0.0301 | 0.0004 |

24 | 2.079 | 22.5 | 997.67 | 9.50 × 10^{−7} | 23968 | 243.62 | 0.0318 | 0.0005 |

25 | 2.074 | 22.4 | 997.68 | 9.51 × 10^{−7} | 23883 | 246.31 | 0.0323 | 0.0005 |

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**MDPI and ACS Style**

Grądziel, S.; Majewski, K. Experimental Determination of the Friction Factor in a Tube with Internal Helical Ribs. *Energies* **2019**, *12*, 257.
https://doi.org/10.3390/en12020257

**AMA Style**

Grądziel S, Majewski K. Experimental Determination of the Friction Factor in a Tube with Internal Helical Ribs. *Energies*. 2019; 12(2):257.
https://doi.org/10.3390/en12020257

**Chicago/Turabian Style**

Grądziel, Sławomir, and Karol Majewski. 2019. "Experimental Determination of the Friction Factor in a Tube with Internal Helical Ribs" *Energies* 12, no. 2: 257.
https://doi.org/10.3390/en12020257