United Formula for the Friction Factor in the Turbulent Region of Pipe Flow.

Friction factor is an important element in both flow simulations and river engineering. In hydraulics, studies on the friction factor in turbulent regions have been based on the concept of three flow regimes, namely, the fully smooth regime, the fully rough regime, and the transitional regime, since the establishment of the Nikuradze's chart. However, this study further demonstrates that combining the friction factor with Reynolds number yields a united formula that can scale the entire turbulent region. This formula is derived by investigating the correlation between friction in turbulent pipe flow and its influencing factors, i.e., Reynolds number and relative roughness. In the present study, the formulae of Blasius and Stricklerare modified to rearrange the implicit model of Tao. In addition, we derive a united explicit formula that can compute the friction factor in the entire turbulent regimes based on the asymptotic behavior of the improved Tao's model. Compared with the reported formulae of Nikuradze, the present formula exhibits higher computational accuracy for the original pipe experiment data of Nikuradze

RELATIONSHIP BETWEEN POLLUTION AND ECONOMIC GROWTH IN CHINA: EMPIRICAL EVIDENCE FROM 111 CITIES

Steady economic growth and environmental protection are two contradictory goals of top priorities in China 2016–2020 Planning Project. Cities play important role in economic and environmental development. The research on cities’ economic-pollution relationship is vital to the choices in city developing patterns. This paper investigates the relationship between economic growth and environmental pollution of 111 Chinese prefectural-level cities in the period 2004–2012 and how it might influence the choice of a city’s developing pattern. These 111 cities are classified into five different clusters, one of which has particular pollution-economic relationship and some of which coordinate the EKC theory. The paper suggests that city features, scale effect and composition effect are important in the distribution of cities’ developing patterns

Comparison between the results of the present study and the experimental data for the entire turbulence regime.

<p>Comparison between the results of the present study and the experimental data for the entire turbulence regime.</p

Friction factor of pipe flow in a rough pipe extracted from Nikuradze’s tabular and graphical presentation [2].

<p>Friction factor of pipe flow in a rough pipe extracted from Nikuradze’s tabular and graphical presentation [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154408#pone.0154408.ref002" target="_blank">2</a>].</p

Comparison between the present study and the experimental data for the smooth and rough turbulence zones.

<p>Comparison between the present study and the experimental data for the smooth and rough turbulence zones.</p

Diagram of the velocity distribution in a full-flow pipe [6].

<p>Diagram of the velocity distribution in a full-flow pipe [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154408#pone.0154408.ref006" target="_blank">6</a>].</p

Prediction Errors for Different Formulae.

<p>Prediction Errors for Different Formulae.</p

Comparison between Nikuradze’s formulae and the experimental data for the smooth and rough turbulence zones.

<p>Comparison between Nikuradze’s formulae and the experimental data for the smooth and rough turbulence zones.</p

Comparison of the curves obtained from Tao’s model for various ε values with Eq (6).

<p>Comparison of the curves obtained from Tao’s model for various ε values with <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154408#pone.0154408.e008" target="_blank">Eq (6)</a>.</p