Four explicit formulae for friction factor calculations in pipe flow

Abstract

This paper presents four explicit formulae to calculate the friction factor for all flow regimes present at Moody diagram without iterations, including the critical zone, for four different problems. The first formula is used to calculate the head losses of pipes given its discharges, lengths, diameters, absolute roughness, the kinematic viscosity of the fluid that flows in the pipes and the gravity acceleration. The second formula is used to calculate the discharges of pipes given its head losses, lengths, diameters, absolute roughness, the kinematic viscosity of the fluid that flows in the pipes and the gravity acceleration. The third formula is used to calculate the diameters of pipes given its discharges, lengths, head losses, absolute roughness, the kinematic viscosity of the fluid that flows in the pipes and the gravity acceleration. The fourth formula is used to calculate the diameters of pipes given its lengths, head losses, absolute roughness, the kinematic viscosity of the fluid that flows in the pipes, the fluid velocities in the pipes and the gravity acceleration. The calculated friction factors are used to calculate the discharges, head losses and diameters of pipes for the steady state and for the extended period state. A case study is presented where it was possible to apply the first two formulae to calculate the friction factors for the steady state in a water main with nine pipes. The results show that the two applied formulae are working well because the calculated pressure heads of some nodes of the water main were compared to the pressure heads gauged in situ on the same nodes of the water main and the results were close.

Using maximum entropy to develop explicit formulae for friction factor calculation in pipe flow

“…Second, because the three other formulae were developed based on the first formula. Another reason to conclude that the four formulae are correct it is because the problems these new formulae solve are the same problems presented in Diniz and Souza [2] solved with other formulae to calculate the friction factors as already mentioned. These authors have already applied the four problems to a real hydraulic network and the results were very reliable.…”

“…Diniz and Souza [2] presented four explicit formulae to calculate the friction factor for four different problems without iterations for all flow regimes present at Moody diagram, including the critical zone. Now, it will be presented four explicit formulae to calculate the friction factor also without iterations for the same four problems, but only for three flow regimes present at Moody diagram and using the maximum entropy model.…”

“…It has given rise to many researchers, almost all over the world, to strive themselves in finding explicit equations, which could be used as alternatives to the Colebrook-White equation, to calculate the friction factor. Some more compact and simple, easier to be memorized, but with large deviations, others less compact and complex, more difficult to be memorized, but with minor deviations and some others matching simplicity and accuracy, with errors well reduced, in relation to the friction factor calculated with the Colebrook-White equation (Diniz and Souza [2]). …”

Accurate Explicit Equations for the Determination of Pipe Diameters

“…There are numerous papers dealing with the explicit formulat ion of f in type II problems (computation of head loss, h f ) when Q, D, ε,ν and L are given e.g. Jain [14]; Swa mee and Jain [21]; Barr [1]; Haaland [12]; Ro meo et al [19]; Imb rahim [13]; Sonnad and Goudar [20]; Valiantzas [24]; Clamond [5]; Din iz and Souza [8]; Yildirim [27]; Brkić [3]; Brkić [4]; Danish et al [7]; Giustolisi et al [10]; Li et al [15].…”

“…In diameter estimation problems, the comb ination of continuity equation (1) and Darcy -Weisbach equation (2) gives: (7) where: (8) and S=h f /L. Substitution of Eq.…”