Failure probability estimation of the gas supply using a data-driven model in an integrated energy system

Risk analysis of an underground gas storage facility using a physics-based system performance model and Monte Carlo simulation

Empirical Colebrook equation from 1939 is still accepted as an informal standard to calculate friction factor during the turbulent flow through pipes from smooth with almost negligible relative roughness to the very rough inner surface. The Colebrook equation contains flow friction factor in implicit logarithmic form where it is, aside of itself, a function of the Reynolds number Re and the relative roughness of inner pipe surface. To evaluate the error introduced by many available explicit approximations to the Colebrook equation, it is necessary to determinate value of the friction factor from the Colebrook equation as accurate as possible. The most accurate way to achieve that is using some kind of iterative methods. Usually classical approach also known as simple fixed point method requires up to 8 iterations to achieve the high level of accuracy, but does not require derivatives of the Colebrook function as here presented accelerated Householder approach (3rd order, 2nd order: Halley and Schroder method and 1st order: Newton-Raphson) which needs only 3 to 7 iteration and three-point iterative methods which needs only 1 to 4 iteration to achieve the same high level of accuracy. Strategies how to find derivatives of the Colebrook function in symbolic form, how to avoid use of the derivatives (Secant method) and how to choose optimal starting point for the iterative procedure are shown. Householder approach to the Colebrook equations expressed through the Lambert W-function is also analyzed. One approximation to the Colebrook equation based on the analysis from the paper with the error of no more than 0.0617% is shown.

/pdf/advanced-iterative-procedures-for-solving-the-implicit-5duty818iv.pdf

Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (e/D ⟶ 0) to very rough (up to e/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface e/D: λ = f(λ, Re, e/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, e/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schroder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

/pdf/advanced-iterative-procedures-for-solving-the-implicit-4qf9sg7vgo.pdf

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

An integrated gas supply reliability evaluation method of the large-scale and complex natural gas pipeline network based on demand-side analysis

Impact of compressor failures on gas transmission network capability

Linepack planning models for gas transmission network under uncertainty

https://publications.anl.gov/anlpubs/2023/03/181468.pdf

Edward Kent

Papers

Impact of compressor failures on gas transmission network capability

Linepack planning models for gas transmission network under uncertainty

Deploying Extended Reality (XR) for Digital Operations and Maintenance (O&M) at the Mechanisms Engineering Test Loop (METL)