PSI - Issue 59

Mariia Serediuk et al. / Procedia Structural Integrity 59 (2024) 763–770 Mariia Serediuk et al. / Structural Integrity Procedia 00 (2019) 000 – 000

769

7

Table 3. Reynolds number determination results Re T2 . Source Equation

Result

1.143 27

Seredyuk et al. (2002)

314490

T2 Re 



 10

2  T Re

Altschul (1970)

240880

8.15 Re 0.0032 0.221Re  



Loitsiansky (1973)

36100

0.237

T2

T2

VNTP 2-86 (1987)

28000

–

D k e

– relative roughness of the pipe surface

 

Universal mathematical models for the coefficient of hydraulic resistance, which provide reliable results for all three friction zones of the turbulent regime, have become widely used in global pipeline transport engineering. An example of such a universal model is the Colebrook-White formula

1

2,51

e k

  

  

(7)

2lg





3.71

D

Re





The Colebrook-White formula is a transcendental equation and does not explicitly solve for the coefficient of hydraulic resistance. The solution of equation (7) involves the use of the method of successive approximations, which complicates the hydrodynamic calculations of pipelines. This drawback is eliminated in Xofer's formula, which is an approximation of the Colebrook-White formula

(8)

1 4.518 Re

 

2

  

  

7 3.71 e k 

  

  

     

2lg

lg

Re

D

Considering the above, we have proposed an improved method of hydrodynamic calculation of pipelines. It provides for the use of formula (3) for the coefficient of hydraulic resistance for the laminar regime at Reynolds numbers up to Re ʹ кр =2041; application of formula (4) for the transitional regime, i.e. from Re ʹ кр =2041 to the Reynolds number Re T ,, which separates the transitional and turbulent regimes. For Reynolds numbers greater than Re T , the coefficient of hydraulic resistance is calculated by Hofer's formula regardless of the friction zone of the turbulent regime. In the proposed method of hydrodynamic calculation of the pipeline, the transition Reynolds number Re T depends on the diameter and the absolute equivalent roughness of the pipe or on its relative roughness. The results of determining the value of ReT for standard pipeline diameters with an absolute roughness of k e = 0.1 mm (for design calculations) and k e =0,2 mm (for operational calculations) are given in Table 4. Table 4. The value of the transition Reynolds number Re T , which divides the scope of application of formulas (5) and (8) for the coefficient of hydraulic resistance. Outer diameter, mm 1220 1020 820 720 530 426 377 325 273 219 159 Value Re T with k e = 0.1 mm 2840 2841 2842 2843 2844 2848 2850 2854 2857 2863 2870 Value Re T with k e = 0.2 mm 2845 2847 2849 2851 2857 2861 2863 2870 2877 2887 2905

Processing of the data in Table 4 made it possible to obtain the following analytical dependence of the transition Reynolds number Re T on the relative roughness of the pipe: