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This paper presents a direct closed form analytical solution for the implicit type equation of Colebrook-White for friction factor calculation. Based on the developed expression, the friction factor can be directly calculated given the fluid flow parameters namely the relative roughness and the Reynolds number. Obtained expression is explicit and it does not require any iterative calculation as it is built reposing on the analytical solution of a cubic equation of real coefficients. The used cubic equation is obtained through a 3rd order Taylor series expansion of the original Colebrook-White equation around an expansion point reasonably defined. The reliability of the proposed expression is proved through comparison with original Moody chart values as well as others results issued from previous approximations found in literature and the exact solution of Colebrook-White equation. The accuracy and the robustness of the established expression over other explicit correlations were emphasized. The presented closed form expression was proved to be simple, robust and accurate. Additionally it has the advantages of covering all the range of applicability of the original Colebrook-White equation.

In this research, the surface roughness affecting the pressure drop in a pipe used as the steam generator of a PWR was studied. Based on the CFD (Computational Fluid Dynamics) technique using a commercial code named ANSYS-FLUENT, a straight pipe was modeled to obtain the Darcy frictional coefficient, changed with a range of various surface roughness ratios as well as Reynolds numbers. The result is validated by the comparison with a Moody chart to set the appropriate size of grids at the wall for the correct consideration of surface roughness. The pressure drop in a full-scale U-shaped pipe is measured with the same code, correlated with the surface roughness ratio. In the next stage, we studied a reduced scale model of a U-shaped heat pipe with experiment and analysis of the investigation into fluid-structure interaction (FSI). The material of the pipe was cut from the real heat pipe of a material named Inconel 690 alloy, now used in steam generators. The accelerations at the fixed stations on the outer surface of the pipe model are measured in the series of time history, and Fourier transformed to the frequency domain. The natural frequency of three leading modes were traced from the FFT data, and compared with the result of a numerical analysis for unsteady, incompressible flow. The corresponding mode shapes and maximum displacement are obtained numerically from the FSI simulation with the coupling of the commercial codes, ANSYS-FLUENT and TRANSIENT_STRUCTURAL. The primary frequencies for the model system consist of three parts: structural vibration, BPF(blade pass frequency) of pump, and fluid-structure interaction.

A porous metal was suggested to be used for the anode of a DCFC. Thermo-fluidic characteristics of fuel-electrolyte mixture in the porous metal should be known in order for proper design of the anode. Previous researchers investigated pressure drop and heat transfer performance of fluids flowing through metal foams that have different pore densities and porosities. Various characteristic length scales were used for the Reynolds number and friction factor of metal foams in the previous works. In the present study, we propose a realistic definition of the characteristic length scale that is applicable to pressure drop evaluation in metal foams regardless of pore density and porosity. A series of experiments was conducted to obtain friction factor of metal foam. An equivalent diameter based on permeability and porosity appeared to best fit the experimental data produced using various metal foams. The relationship between the friction factor and Reynolds number through metal foams can be classified into three regimes. In Re 2000, the value of friction factor approaches 0.17. The relationship between friction factor and Reynolds number of foam materials appeared to be similar trend to that of Moody chart.

Flow of CO 2 in the vicinity of its critical point was studied experimentally in two different flow configurations. First, a 60 cm long stainless steel pipe with 2.1 mm inner diameter was used to study near-critical CO 2 pipe flow. In terms of raw flow data, the results indicated high sensitivity of pressure drop to mass flow rate as well as to inlet conditions; i.e. pressure and temperature. Remarkably though, when friction factor and Reynolds number were defined in terms of the inlet conditions, it was established that the classical Moody chart described the flow with satisfactory accuracy. This was rationalized using shadowgraphs that visualized the process of transition from a supercritical state to a two-phase subcritical state. During the transition the two phases were separated due to density mismatch and an interface was established that traveled in the direction of the flow. This interface separated two regions of essentially single-phase flow, which explained the effective validity of the classical Moody chart. Second, Joule–Thomson throttling was studied using a 0.36-mm-diameter orifice. For conditions relevant to carbon capture and sequestration, the fluid underwent Joule–Thompson cooling of approximately 0.5 °C/bar. The temperature difference during the cooling increased with increasing inlet enthalpy. Discrepancies with previous computed and experimentally measured values of Joule–Thompson throttling were discussed in detail.

An alternate inner wall variable, for flow over a transitional rough pipe surface, is defined as the ratio of normal coordinate measured above the mean roughness level to the wall roughness scale. The Reynolds equations for mean turbulent flow in a transitional rough pipe, in two layers (inner and outer) are considered. The predictions of the mean velocity and friction factor in fully developed turbulent flow in a rough pipe flow, presented here, covers all types of roughness. The data for a particular case of the machine honed Princeton superpipe roughness, analogous to inflectional type roughness of Nikuradse, is presented, as two expressions using our roughness scale. The velocity profile and friction factor, on a transitional rough wall, are shown to be governed by the new log laws, which are explicitly independent of the transitional wall roughness. Further, the inflectional roughness has also been connected with geometric roughness parameters; like, arithmetic mean roughness, mean peak to valley heights roughness, root mean square (rms), roughness based on texture measure; and the friction factor implicit and approximate explicit formulas have also been proposed. In entire transition region between fully smooth and fully rough wall, monotonic roughness of Colebrook (Moody Chart) over estimaton the friction factor when compared with present inlectional roughness.

Despite the presence of a variety of studies dealing with the magnitude of particle Reynolds number, Re p defining transition from laminar to turbulent regime for flow through packed beds, the manner is still one of the unknowns. An approach based on the experimental data concerning upward airflow through fixed cylindrical packed beds is presented in this paper. The utilized packed beds had the following ranges of; sphericity, Φ , 0.55 ⩽ Φ ⩽ 1.00, packing material diameter to bed length ratio, D p / L , 0.04 ⩽ D p / L ⩽ 0.72, and bed porosity, e , 0.36 ⩽ e ⩽ 0.56. The test cases covered the ranges of particle Reynolds number, Re p 708 ⩽ Re p ⩽ 7772 and particle Froude number; Fr p 2.86 ⩽ Fr p ⩽ 10.39. The measurements of pressure drop through packed bed; Δ P Bed and superficial mean exit velocity; U are used to determine bed frictional effects in reference to the available literature on particle friction factors, f p . The magnitude of Re p defining transition is assumed to be 2000 with particular emphasis to the flow dynamics. The definitions of Bird et al. [R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena , John Wiley and Sons, NY, 1960] are used to calculate f p . The calculated f p for the covered test cases are given as a function of pressure coefficient, Δ P * and Re p, Fr p , Φ , e , D p / L in the approximate ranges of laminar and turbulent flow for Re p p > 2000, respectively. The proposed separate equations of f p = f p (Δ P *, Re p , Fr p , Φ , e , D p / L ) are satisfied for laminar and turbulent flows with corresponding average error margins of ±7.6% and ±18%. Furthermore ranges of transitional and fully rough flow through packed beds are estimated as 2000 ⩽ Re p ⩽ 4000 and Re p > 5000 with an analogy to the well-known Moody Chart in pipe flows.

The purpose of this paper is to present petroleum and chemical engineers with a simple means of measuring or estimating the absolute surface roughness, ϵ, and relative roughness, ϵ/D, for internally coated pipes. Our research thrust is to develop new relative roughness charts along with corresponding mathematical equations for internally coated pipes. An integral part of the frictional pressure drop due to fluid flow in pipes involves the determination of absolute surface roughness and relative roughness. An accurate determination of the pressure drop due to fluid flow in oil and gas wells is required for optimizing oil and gas production design calculations. Some of these calculations include developing tubing programs to maximize well deliverability, wellbore flow performance, sizing surface flow lines, and designing artificial lift installations. Previous multiphase fluid flow laboratory and field tests, during the past 50 years, provided correlations for calculating pressure gradients in tubulars [Chem. Eng. Prog. 45(1) (1949) 39; Hetsroni, G., 1982. Handbook of Multiphase Systems, Hemisphere/McGraw Hill, 1174 pp.; Brown, K.E., 1984. The Technology of Artificial Methods, vol. 4, Penn Well Publishing, Tulsa, OK, 447 pp.; Tengesdal, J.O., 1998. Predictions of Flow Patterns, Pressure Drop, and Liquid Holdup in Vertical Upward Two-Phase Flow. MSc Thesis, The University of Tulsa, Tulsa, OK; Farshad, F., Garber, J.D., Polaki, V., 2000. Comprehensive model for predicting corrosion rates in gas wells containing CO2. Soc. Pet. Eng. Prod. Facil. 15(3), 183–190]. All the multiphase fluid flow correlations, used in production operations to calculate these pressure gradients and flow regimes, stem from the general energy balance equation and involves the determination of friction pressure losses. Still today, these friction losses only are being evaluated by practicing engineers using the Darcy–Weisbach equation, which involves the Moody friction factor, f [Szilas, A.P., 1975. Production and Transport of Oil and Gas. Elsevier Sci. Publ., Amsterdam, p. 630; Economides, M.J., Hill, D.A., Ehlig-Economides, C., 1994. Petroleum Production Systems, Prentice Hall, Englewood Cliffs, NJ, 611 pp.]. Moody [Trans. AME 66 (1944) 671] prepared a relative roughness chart for a number of common piping materials. This initial relative roughness correlation was based on experiments on pipes artificially roughened with sand grains. At that time, internally coated pipes were not invented. Hence, Moody did not provide the relative roughness for internally coated pipes. In addition, Moody did not perform a regression analysis of the data to generate functional forms of equations by relating roughness, ϵ/D, as a function of internal pipe diameter, D. Currently, internally coated pipes are being utilized worldwide in oil field production systems. Consequently, new absolute surface roughness and relative roughness values of internally coated pipes are needed to properly model the hydrodynamics [Brown, K.E., 1984. The Technology of Artificial Methods, vol. 4, Penn Well Publishing, Tulsa, OK, 447 pp.]. The new relative roughness plots developed for internally coated pipes are presented and show that the correlation fits between the commercial steel and drawn tubing in the Moody chart.