Abstract

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ=ζ(Re, ε∗, λ)which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as threeor two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džuni´c–Petkovi´c–Petkovi´c; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

Abstract

The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, with the known Reynolds number R e and the known relative roughness of a pipe inner surface ε*; λ= ξ (R e, ε*, λ). It is based on logarithmic law in the form that captures the unknown flow friction factor λ in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Padé polynomials with only one l o g-call in total for the whole procedure (expensive l o g-calls are substituted with Padé polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution

Abstract

This paper provides a new unified formula for Newtonian fluids valid for all pipe flow regimes from laminar to fully rough turbulent flow.This includes laminar flow; the unstable sharp jump from laminar to turbulent flow; and all types of turbulent regimes, including the smooth turbulent regime, the partial non-fully developed turbulent regime, and the fully developed rough turbulent regime. The new unified formula follows the inflectional form of curves suggested in Nikuradse’s experiment rather than the monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula uses switching functions and interchangeable formulas for the laminar, smooth turbulent, and fully rough turbulent flow regimes. Thus, the formulation presented below represents a coherent hydraulic model suitable for engineering use. This new flow friction model is more flexible than existing literature models and provides smooth and computationally cheap transitions between hydraulic regimes.

Abstract

This paper presents evolutionary optimization of explicit approximations of the empirical Colebrook’s equation that is used for the calculation of the turbulent friction factor (λ), i.e., for the calculation of turbulent hydraulic resistance in hydraulically smooth and rough pipes including the transient zone between them. The empirical Colebrook’s equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of the inner pipe surface (ε/D). It is implicit in the unknown friction factor (λ). The implicit Colebrook’s equation cannot be rearranged to derive the friction factor (λ) directly, and therefore, it can be solved only iteratively [λ = f(λ, R, ε/D)] or using its explicit approximations [λ≈f(R, ε/D)], which introduce certain error compared with the iterative solution. The optimization of explicit approximations of Colebrook’s equation is performed with the aim to improve their accuracy, and the proposed optimization strategy is demonstrated on a large number of explicit approximations published up to date where numerical values of the parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After that improvement, the computational burden stays unchanged while the accuracy of approximations increases in some of the cases very significantly.

Abstract

Abstract

When designing systems for the transportation and distribution of gas, safety is one of the main issues to be considered.  In order to minimize potential hazards, this subject is treated in international regulations. Safety concerns related to the presence of real and potential corrosion type defects in gas pipes are presented. An actual gas pipeline, which has been feeding natural gas to the city of Córdoba (Argentina) for more than forty years, is considered as a study case. In order to determine the properties of this gas pipeline, mechanical tests from material samples were carried out. The stress state associated with “volumetric” type defects is determined using a model that is based on the flux stress of the material of the pipe, which is appropriate for studying the behavior of corroded pipelines. This model allows computing, depending on the dimensions of the pipe and the length of the defect, the transition pressure which separates possible defects into two categories: those which cause failure by breakage of the pipe and those which only cause gas leaks. The pipe failure pressure for passing and non-passing defects can also be determined using this model. The system safety conditions are defined considering the size of the defects and the work pressure. The range of lengths of possible defects, the size of the defects that are critical, and the size of the defects which are tolerable for a fixed safety coefficient, are determined. Finally, the problem of determining in a quick way, the reduced pressure acceptable for the safe operation until repairs, of a system under the presence of flaws of “non-tolerable size”, is addressed.

Abstract

When designing systems for the transportation and distribution of gas, safety is
one of the main issues to be considered. In order to minimize potential hazards, this subject
is treated in international regulations. Manufacturing characteristics of gas pipes and their
typical defects are described in the first part of this work. Safety concerns related to the
presence of real and potential crack type defects in gas pipes are presented. An actual gas
pipeline, which has been feeding natural gas to the city of Córdoba (Argentina) for more than
forty years, is considered as a study case. In order to determine the properties of this gas
pipeline, mechanical tests from material samples were carried out. The stress state associated
with “plane” type defects is determined using a model that is based on the material toughness of
the pipe, which is appropriate for studying the behavior of cracks. This model allows computing,
depending on the dimensions of the pipe, the transition pressure which separates possible
defects into two categories: defects which cause failure by breakage of the pipe and defects
which only cause gas leaks. The pipe failure pressure for passing and non-passing defects can
also be determined using this model. The system safety conditions are defined considering the
size of the defects and the work pressure. The range of lengths of possible defects, the size of
the defects that are critical, and the size of the defects which are tolerable for a fixed safety
coefficient, are determined. Finally, the problem of determining in a quick way, the reduced
pressure acceptable for the safe operation until repairs, of a system under the presence of a
crack of “non-tolerable size”, is addressed.
 

Abstract

Empirical Colebrook equation implicit in unknown flow friction factor (λ) is an accepted standard forcalculation of hydraulic resistance in hydraulically smooth and rough pipes. The Colebrook equation givesfriction factor (λ) implicitly as a function of the Reynolds number (Re) and relative roughness (ε/D) of innerpipe surface; i.e. λ0=f(λ0, Re, ε/D). The paper presents a problem that requires iterative methods for thesolution. In particular, the implicit method used for calculating the friction factor λ0is an application of fixed-point iterations. The type of problem discussed in this "in the classroom paper" is commonly encountered influid dynamics, and this paper provides readers with the tools necessary to solve similar problems. Students’task is to solve the equation using Excel where the procedure for that is explained in this “in the classroom”paper. Also, up to date numerous explicit approximations of the Colebrook equation are available where as an additional task for students can be evaluation of the error introduced by these explicit approximations λ≈f(Re,ε/D) compared with the iterative solution of implicit equation which can be treated as accurate.