Performance of explicit approximations of the coefficient of head loss for pressurized conduits

ABSTRACT One of the parameters involved in the design of pressurized hydraulic systems is the pressure drop in the pipes. The verification of the pressure drop can be performed through the Darcy-Weisbach formulation, which considers a coefficient of head loss (f) that can be estimated by the implicit Colebrook-White equation. However, for this determination, it is necessary to use numerical methods or the Moody diagram. Because of this, numerous explicit approaches have been proposed to overcome such limitation. In this sense, the objective of this study was to analyze the explicit approximations of the f for pressurized conduits in comparison to the Colebrook-White formulation, determining the most precise ones so that they can be used as an alternative solution that is valid for the turbulent flow regime. Twenty nine explicit equations found in the literature were analysed, determining the f through the Reynolds number in the range of 4 × 103 ≤ Re ≤ 108 and a relative roughness (Ɛ/D) of 10-6 ≤ Ɛ/D ≤ 5 × 10-2, and obtaining 160 points for each equation. The performance index and relative error of the formulations were analyzed in relation to the Colebrook-White equation. Considering the equations analyzed, we found seven that presented excellent performance and high precision, highlighting the formulation of Offor & Alabi, which can be used as an alternative to the Colebrook-White standard equation.

Review of explicit approximations to the Colebrook relation for flow friction

Because of Moody's chart has demonstrated applicability of the Colebrook equation over a very wide range of Reynolds number and relative roughness values, this equation becomes the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can cause a problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is the main reason for attempting to develop a relationship that is a reasonable approximation for the Colebrook equation but which is explicit in friction factor. A review of existing explicit approximation of the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative solution of implicit Colebrook equation is shown for the entire range of turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that most of the available approximations of the Colebrook equation, with a few exceptions, are very accurate with deviations of no more than few percentages.

On the History, Science, and Technology Included in the Moody Diagram

This paper is a historical review of the science, both experimental and theoretical, behind the iconic Moody diagram used to avoid tedious iterations choosing pumps and pipes. The large body of historical pipe flow measurements and the choice of dimensionless groups and the Buckingham-Π theorem are also discussed. The traditional use of the Moody diagram to solve common pipe flow problem is discussed. Alternatives to the Moody diagram from the literature and novel ones presented here are shown to produce a solution without iteration for any type of pipe loss problem.

Development of the Energy Simulator for the Water Hydraulic System Under Flow Condition Changes

The new energy simulator we developed is based on a hydraulic servosystem dynamic flow model introducing flow coefficients determined by Reynolds number. One is the pipe flow coefficient flow determined by the Moody diagram and the other is the servovalve flow coefficient based on flow model experiments. The motor dynamic model is introduced to determine efficiency such as coil resistance or rotor viscosity loss. Leakage of the hydraulic servovalve was also determined by the leak model. The feasibility of the proposed simulator was verified using computer simulation and experiments, showing differences from conventional simulators that depend on manually set parameters such as flow coefficients. Simulation and experiment results agreed well, and the proposed simulator determines hydraulic servosystem energy consumption. New simulator concepts, calculation models, and experiment results are also discussed.

Efficient Computation of Three-Dimensional Flow in Helically Corrugated Hoses Including Swirl

In this article we propose an efficient method to compute the friction factor of helically corrugated hoses carrying flow at high Reynolds numbers. A comparison between computations of several turbulence models is made with experimental results for corrugation sizes that fall outside the range of validity of the Moody diagram. To do this efficiently we implement quasi-periodicity. Using the appropriate boundary conditions and matching body force, we only need to simulate a single period of the corrugation to find the friction factor for fully developed flow. A second technique is introduced by the construction of an appropriately twisted wedge, which allows us to furthermore reduce the problem by a further dimension while accounting for the Beltrami symmetry that is present in the full three-dimensional problem. We make a detailed analysis of the accuracy and time-saving that this novelty introduces. We show that the swirl inside the flow, which is introduced by the helical boundary, has a positive effect on the friction factor. Furthermore, we give a prediction for which corrugation angles the assumption of axisymmetry is no longer valid. It then has to make place for Beltrami-symmetry if accurate results are required.

Calculations of Pressure Drop in a Circular Duct Using the Traditional Method Compared to That Using the Constricted Flow Diameter

The traditional method of calculating pressure drop in a pipe that conveys a fluid involves use of the Moody Diagram. This diagram is a correlation of data with friction factor plotted as a function of Reynolds number and relative roughness. A new graphical representation of these data has been formulated, and makes use of what is known as the constricted flow diameter. The background for this new correlation is based on using a pipe diameter less twice the average roughness height. A new “modified” Moody Diagram has been produced based on the constricted flow diameter. The presence of roughness features on the inside pipe wall has an effect on the flow along the pipe surface which is not accounted for in the traditional Moody diagram. The new diagram accounts for this effect. To demonstrate the use of the new diagram, several example problems have been formulated and solved using the traditional and the modified diagrams. Calculations indicate that at the smaller pipe sizes, the use of the constricted flow diameter yields significantly different results from those obtained in the traditional way. These results have a major influence on modeling flows in mini- and in micro-channels. Laminar and turbulent flows are both affected.

Effects of Roughness on Turbulent Flow in Microchannels and Minichannels

The effect of roughness ranging from smooth to 24% relative roughness on laminar flow has been examined in previous works by the authors. It was shown that using a constricted parameter, εFP, the laminar results were predicted well in the roughened channels ([1],[2],[3]). For the turbulent regime, Kandlikar et al. [1] proposed a modified Moody diagram by using the same set of constricted parameters, and using the modification of the Colebrook equation. A new roughness parameter εFP was shown to accurately portray the roughness effects encountered in laminar flow. In addition, a thorough look at defining surface roughness was given in Young et al. [4]. In this paper, the experimental study has been extended to cover the effects of different roughness features on pressure drop in turbulent flow and to verify the validity of the new parameter set in representing the resulting roughness effects. The range of relative roughness covered is from smooth to 10.38% relative roughness, with Reynolds numbers up to 15,000. It was found that using the same constricted parameters some unique characteristics were noted for turbulent flow over sawtooth roughness elements.

An Alternative Approach for Teaching Turbulent Flow

Teaching of turbulence in undergraduate and early graduate level fluid mechanics and heat transfer courses is a difficult undertaking. The approach taken in typical texts requires the students to accept a number of basic concepts without much quantitative justifications. This paper presents an alternative approach, one in which most of the salient features of the turbulent flow are derived by using numerical solutions and experimental results, as opposed to simply having them presented. In this approach, Prandtl’s mixing length model is used to obtain the velocity distribution for fully developed pipe flow. By comparing the numerical calculations with the experimental results, students determine the value of κ that best fits the experimental data on their own. In addition, deficiency of the mixing length in the transition region is shown. It is also shown that other models like Van Driest’s do a better job. The Logarithmic Law of the wall as well as 7th power law are also proven. The different models are used to determine the friction factor for pipe flow and the results are compared with the values obtained from the Moody diagram.