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Friction factor

The friction factor, specifically the Darcy friction factor (often denoted as f or f_D), is a dimensionless quantity in fluid mechanics that characterizes the frictional resistance to fluid flow in pipes and ducts, enabling the calculation of pressure drop or head loss due to viscous shear along the conduit walls. It is a key component of the Darcy–Weisbach equation, h_f = f \frac{L}{D} \frac{V^2}{2g}, where h_f is the frictional head loss, L is the pipe length, D is the pipe diameter, V is the average flow velocity, and g is the acceleration due to gravity.^[1] This parameter accounts for both the fluid properties (via the Reynolds number) and the pipe surface characteristics (via relative roughness), making it essential for designing and analyzing piping systems in engineering applications such as water distribution, oil and gas transport, and HVAC systems.^[2] The Darcy–Weisbach equation originated in the mid-19th century, with Julius Weisbach proposing the foundational form in 1845 based on dimensional analysis and energy principles, and Henry Darcy providing experimental validation in the 1850s through measurements of head losses in pipes.^[3] The friction factor itself is defined as f_D = \frac{8 \tau_w}{\rho V^2}, where \tau_w is the wall shear stress and \rho is the fluid density, representing four times the Fanning friction factor (f_F = f_D / 4), a related but distinct parameter historically used in chemical engineering contexts.^[2] In practice, the value of f varies with flow regime: for laminar flow (Reynolds number \text{Re} < 2300), it is analytically given by f = \frac{64}{\text{Re}}, derived from the Hagen–Poiseuille solution for fully developed flow in circular pipes.^[2] For turbulent flow (\text{Re} > 4000), which predominates in most engineering applications, f depends nonlinearly on both \text{Re} (a measure of inertial versus viscous forces) and the relative roughness \epsilon / D (where \epsilon is the average roughness height). This relationship is typically determined using the implicit Colebrook equation: \frac{1}{\sqrt{f}} = -2 \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{2.51}{\text{Re} \sqrt{f}} \right), requiring iterative solution or explicit approximations like the Haaland or Swamee–Jain equations for practical computation.^[1] The seminal Moody diagram, introduced by Lewis F. Moody in 1944, graphically correlates f with \text{Re} and \epsilon / D across flow regimes, facilitating rapid estimation and remaining a standard tool despite modern computational alternatives.^[2] Beyond pipes, the friction factor concept extends to non-circular ducts (using hydraulic diameter) and multiphase flows, though adjustments are needed for accuracy. Its accurate determination is critical for optimizing energy efficiency in fluid transport systems, as even small errors in f can lead to significant over- or under-prediction of pumping requirements and operational costs. Recent research continues to refine correlations for microscale flows, biofouled surfaces, and advanced materials to enhance predictive models in emerging applications like microfluidics and sustainable energy systems.^[4]

Fundamentals of Pipe Flow Friction

Laminar and Turbulent Flow Regimes

In fluid flow through pipes, two primary regimes exist: laminar and turbulent, each characterized by distinct patterns of motion that significantly affect energy dissipation and transport phenomena. Laminar flow, also known as streamline flow, occurs when fluid particles move in smooth, parallel layers with minimal mixing between adjacent layers, resulting in orderly and predictable motion.^[5] This regime is prevalent at low flow velocities, where viscous forces dominate over inertial forces, typically corresponding to Reynolds numbers below approximately 2300.^[6] In laminar pipe flow, the velocity profile across the pipe's cross-section is parabolic, with the maximum velocity at the centerline and decreasing symmetrically to zero at the pipe walls, reflecting the no-slip boundary condition.^[5] In contrast, turbulent flow arises at higher velocities, where inertial forces overpower viscous effects, leading to chaotic, irregular motion with the formation of eddies and vortices that promote intense mixing across the flow.^[5] This regime generally occurs for Reynolds numbers exceeding 4000, though a transitional zone exists between 2300 and 4000 where flow may intermittently switch between laminar and turbulent states.^[6] The velocity profile in turbulent pipe flow is notably flatter in the core region compared to laminar flow, with a steeper gradient near the walls due to a thin viscous sublayer, resulting in higher overall energy dissipation through enhanced momentum transfer.^[5] The Reynolds number, a dimensionless parameter representing the ratio of inertial to viscous forces, serves as the key indicator for distinguishing these regimes.^[7] The distinction between these flow regimes was first experimentally demonstrated by Osborne Reynolds in 1883 through a series of pipe flow tests, where he injected a thin stream of dye into water flowing through a glass tube at varying speeds, observing the dye's straight path in laminar conditions and its rapid diffusion in turbulent ones. These foundational experiments not only visualized the transition but also underscored the practical implications for pipe design, as turbulent flow's chaotic nature amplifies shear stresses and mixing, influencing applications from chemical processing to hydraulics.^[8]

Reynolds Number and Its Role

The Reynolds number (Re) is a dimensionless quantity fundamental to fluid mechanics, representing the ratio of inertial forces to viscous forces within a flowing fluid. It is expressed as

Re = \frac{\rho v D}{\mu}

where \rho is the fluid density, v is the average velocity, D is the characteristic length (typically the pipe diameter for circular conduits), and \mu is the dynamic viscosity.^[9] This parameter enables the prediction of flow behavior without reliance on specific dimensional scales, making it applicable across varying system sizes and fluid properties.^[10] The derivation of the Reynolds number stems from dimensional analysis of the Navier-Stokes equations governing fluid motion, balancing inertial terms (on the order of \rho v^2) against viscous diffusion terms (on the order of \mu v / D).^[11] In pipe flow, this ratio highlights the competition between momentum transport by convection (inertial effects) and dissipation by internal friction (viscous effects), with higher values indicating dominance of inertia.^[12] Osborne Reynolds introduced this concept in his seminal 1883 experimental study on water flow in pipes, where he observed regime transitions using dye injection to visualize streamlines.^[13] For flow in circular pipes, the Reynolds number delineates flow regimes: laminar flow predominates at Re < 2300, where viscous forces maintain orderly, parallel streamlines; a transitional regime occurs between 2300 and approximately 4000, characterized by intermittent instabilities; and turbulent flow emerges above Re > 4000, with chaotic mixing due to inertial dominance.^[10] These thresholds, established through extensive experimentation since Reynolds' work, provide a practical basis for classifying internal flows.^[7] In non-circular ducts, such as rectangular or annular channels, the pipe diameter D is substituted with the hydraulic diameter D_h = 4A / P, where A is the cross-sectional area and P is the wetted perimeter, to preserve the validity of the Reynolds number for regime prediction.^[14] This adjustment accounts for the geometry's influence on viscous boundary layers and flow resistance.^[15] The Reynolds number is essential for selecting friction factor correlations in pipe flow analysis, as it dictates whether the flow is laminar (yielding a constant friction factor inversely proportional to Re) or turbulent (requiring empirical adjustments for surface roughness and other factors).^[16] By identifying the regime, it ensures accurate modeling of energy losses, bridging fundamental flow physics to engineering design.^[17]

Definition and Types of Friction Factor

Darcy-Weisbach Friction Factor

The Darcy-Weisbach friction factor, denoted as f_D (often simplified to f), is a dimensionless coefficient that quantifies the frictional resistance in full-flowing pipes by relating the wall shear stress to the flow's dynamic pressure. It is defined mathematically as

f_D = \frac{8 \tau_w}{\rho v^2},

where \tau_w represents the shear stress at the pipe wall, \rho is the fluid density, and v is the average velocity of the fluid.^[2] This definition captures the physical essence of friction as the retarding force exerted by the pipe surface on the flowing fluid, scaled against the inertial forces in the bulk flow.^[17] The concept originated in the mid-19th century, with Julius Weisbach introducing the foundational form of the friction equation in 1845 based on hydraulic experiments, expressing head loss in terms of a friction coefficient that accounted for velocity and pipe geometry.^[18] Henry Darcy advanced this work in the 1850s through systematic experiments on water flow in pipes, demonstrating that the friction factor varies with pipe material roughness and diameter, thereby unifying disparate empirical formulas from earlier researchers like Gaspard de Prony and Antoine Chézy into a more consistent framework.^[18] Darcy's contributions, detailed in his 1857 publication Recherches expérimentales relatives au mouvement de l'eau dans les tuyaux, emphasized practical measurements that bridged theoretical hydraulics and engineering applications.^[18] In practice, the Darcy-Weisbach friction factor is integral to the head loss equation named after its developers:

h_f = f_D \frac{L}{D} \frac{v^2}{2g},

where h_f is the frictional head loss, L is the pipe length, D is the hydraulic diameter, and g is the acceleration due to gravity.^[18] This equation enables the prediction of energy dissipation in steady, incompressible pipe flows, essential for designing water distribution systems and hydraulic networks. The factor f_D is four times the magnitude of the Fanning friction factor and is the standard choice in civil engineering and hydraulics due to its direct integration with head loss computations.^[19] Being dimensionless, f_D typically spans 0.008 to 0.1 in turbulent regimes, reflecting variations in flow conditions and surface characteristics. Its value is evaluated using the Reynolds number to distinguish flow regimes and incorporate effects like viscosity and roughness.^[2]

Fanning Friction Factor

The Fanning friction factor, denoted as f_F, is a dimensionless parameter defined as the ratio of the wall shear stress \tau_w to the dynamic pressure of the fluid, expressed as f_F = \frac{\tau_w}{\rho v^2 / 2}, where \rho is the fluid density and v is the average flow velocity.^[20] This formulation directly corresponds to the skin friction coefficient used in boundary layer analyses, emphasizing the local shear forces at the pipe wall.^[21] Introduced by American engineer John Thomas Fanning (1837–1911) in the 1870s, particularly in his 1877 publication A Practical Treatise on Water-Supply Engineering, the factor originated from compilations of empirical data on steam and water flow resistance in pipes, drawing from earlier work by Darcy and Weisbach.^[18] It became prevalent in chemical and mechanical engineering literature due to its alignment with momentum balance considerations in process equipment.^[21] The Fanning friction factor relates to the Darcy-Weisbach friction factor f_D by f_F = f_D / 4, which adjusts the pressure drop equation to \Delta P = 2 f_F \rho v^2 L / D, where L is the pipe length and D is the diameter.^[20] This version is particularly favored in heat exchanger design for non-circular ducts and when focusing on shear-driven momentum transfer in chemical processes.^[22] Typical values of f_F for turbulent pipe flow range from 0.002 to 0.025, depending on the Reynolds number and relative surface roughness, with lower values indicating smoother surfaces or higher flow rates.^[23]

Determination of Friction Factor

Analytical Solutions for Laminar Flow

In laminar flow through a circular pipe, the friction factor can be derived analytically from the principles of fluid mechanics, providing an exact closed-form expression without reliance on empirical data. This solution, known as the Hagen-Poiseuille law, was first developed theoretically by Gotthilf Hagen in 1839 and experimentally verified by Jean Poiseuille in the 1840s through measurements of water flow in capillary tubes.^[24] The derivation assumes steady, fully developed flow of an incompressible Newtonian fluid in a straight, horizontal pipe of constant circular cross-section, with no entrance or exit effects and negligible body forces other than the driving pressure gradient.^[25] The velocity profile arises from balancing the pressure force on a fluid element with the viscous shear stress. Consider a cylindrical shell of radius r and thickness dr in a pipe of radius R and length L. The net pressure force acting on the shell is \pi r^2 \Delta P, where \Delta P is the pressure drop over L. This balances the shear force $2\pi r L \tau, with wall shear stress \tau = \mu \frac{du}{dr} for a Newtonian fluid of viscosity \mu. Equating forces yields:

\pi r^2 \Delta P = 2\pi r L \mu \left| \frac{du}{dr} \right|

Solving for the velocity gradient:

\frac{du}{dr} = -\frac{\Delta P \, r}{2 \mu L}

Integrating from r to R (with u(R) = 0 at the no-slip wall) gives the parabolic velocity profile:

u(r) = \frac{\Delta P}{4 \mu L} (R^2 - r^2)

The average velocity v is obtained by integrating u(r) over the cross-section and dividing by area \pi R^2:

v = \frac{1}{\pi R^2} \int_0^R u(r) \, 2\pi r \, dr = \frac{\Delta P R^2}{8 \mu L}

This relates flow rate Q = \pi R^2 v to the pressure gradient as Q = \frac{\pi R^4 \Delta P}{8 \mu L}.^[25]^[26] To find the friction factor, substitute into the Darcy-Weisbach equation, \Delta P = f_D \frac{L}{D} \frac{\rho v^2}{2}, where D = 2R is the diameter, \rho is density, and f_D is the Darcy friction factor. From the average velocity expression, \Delta P = \frac{8 \mu L v}{R^2}. Equating and solving for f_D:

f_D = \frac{64 \mu L v}{\rho v^2 D L} = \frac{64 \mu}{\rho v D} = \frac{64 \nu}{v D} = \frac{64}{Re}

where Re = \frac{\rho v D}{\mu} is the Reynolds number and \nu = \mu / \rho is kinematic viscosity. The Fanning friction factor f_F, defined with wall shear stress \tau_w = f_F \frac{\rho v^2}{2}, relates to the Darcy factor by f_D = 4 f_F, yielding f_F = \frac{16}{Re}.^[26]^[19] This analytical solution accurately predicts pressure drop and friction losses for laminar flow when Re < 2300, matching experimental observations in smooth pipes across a wide range of Newtonian fluids./12%3A_Fluid_Dynamics_and_Its_Biological_and_Medical_Applications/12.04%3A_Viscosity_and_Laminar_Flow_Poiseuilles_Law) However, it does not apply to turbulent regimes (Re > 4000) or non-Newtonian fluids, where the velocity profile deviates from parabolic due to nonlinear effects.^[27]

Empirical Methods for Turbulent Flow

Empirical methods for determining the friction factor in turbulent pipe flow provide explicit approximations that facilitate rapid calculations, particularly as alternatives to implicit equations like the Colebrook-White equation.^[28] These correlations are derived from experimental data and theoretical insights into boundary layer behavior, offering practical solutions for engineering applications in both smooth and rough pipes.^[29] One of the earliest and most influential empirical correlations for smooth pipes is the Blasius formula, which approximates the Darcy-Weisbach friction factor f_D as

f_D \approx \frac{0.316}{\mathrm{Re}^{0.25}}

for Reynolds numbers in the range $4000 < \mathrm{Re} < 10^5.^[30] This relation stems from boundary layer theory and was originally developed by Heinrich Blasius in 1913 based on experimental measurements of turbulent flow resistance in smooth conduits.^[31] The Blasius formula provides a simple power-law dependence, emphasizing the inverse quarter-power relationship between friction factor and Reynolds number in the smooth-wall turbulent regime.^[32] For pipes with roughness, later empirical refinements extend these ideas to account for relative roughness \epsilon / D. The Swamee-Jain equation offers an explicit approximation for the Darcy-Weisbach friction factor across a wide range of turbulent conditions:

f_D = \frac{0.25}{\left[ \log_{10} \left( \frac{\epsilon}{3.7D} + \frac{5.74}{\mathrm{Re}^{0.9}} \right) \right]^2},

valid for \mathrm{Re} > 4000 and relative roughness up to \epsilon / D = 0.01.^[28] Proposed by Prabhata K. Swamee and Akshaya K. Jain in 1976, this formula was fitted to extensive experimental data to approximate implicit solutions with high accuracy, typically within 1% error for most practical cases.^[28] Similarly, the Haaland equation provides another explicit form, particularly suited for rough pipes:

\frac{1}{\sqrt{f_D}} \approx -1.8 \log_{10} \left[ \frac{(\epsilon / D)^{1.11}}{3.7} + \frac{6.9}{\mathrm{Re}} \right],

also applicable for \mathrm{Re} > 4000. Developed by S.E. Haaland in 1983, it simplifies computation while maintaining accuracy comparable to implicit methods, with deviations generally under 2% across turbulent regimes. These empirical methods, building on Blasius's foundational work for smooth pipes and extending to roughness effects in subsequent decades, enable efficient estimation of friction factors in turbulent flows without iterative solving.^[33] Their validity is confined to turbulent Reynolds numbers exceeding 4000, where they align closely with experimental observations and provide essential tools for pressure drop predictions in engineering design.^[34]

Visualization and Correlations

Moody Diagram

The Moody diagram is a graphical representation used to determine the Darcy-Weisbach friction factor (f_D) for fluid flow in pipes, plotting it against the Reynolds number (Re) and relative roughness (\epsilon / D, where \epsilon is the absolute roughness and D is the pipe diameter). This tool consolidates experimental data to illustrate the influence of flow regime and surface roughness on frictional losses, serving as a practical aid for engineers in pressure drop estimations.^[35] Developed by Lewis F. Moody and published in 1944, the diagram draws primarily from Johann Nikuradse's experimental investigations in the 1930s, which involved artificially roughening pipes with uniform sand grains to study turbulent flow resistance. Moody compiled these results alongside other empirical data to create a unified chart, aiming to provide a straightforward method for estimating friction factors in engineering applications.^[36] The diagram is constructed as a log-log plot, with the vertical axis representing f_D (typically ranging from 0.008 to 0.1) and the horizontal axis showing Re (spanning $10^3 to $10^8). Multiple curves are overlaid, each corresponding to a fixed value of relative roughness \epsilon / D (from $10^{-5} for smooth pipes to 0.1 for highly rough ones), demonstrating how roughness affects friction across flow regimes.^[35] Key regions on the diagram include the laminar flow area, depicted as a straight line following f_D = 64 / \mathrm{[Re](/page/Re)}, valid for Re below approximately 2000; a critical transition zone around Re = 2000–4000 where flow instability occurs; the smooth turbulent regime for higher Re in low-roughness pipes, where friction decreases with increasing Re; the transitional turbulent region influenced by both Re and roughness; and the fully rough regime at very high Re, where curves become horizontal, indicating friction independent of Re and solely dependent on \epsilon / [D](/page/D*).^[37] To use the diagram, one first calculates the Reynolds number based on pipe diameter, fluid velocity, and properties, then identifies the appropriate \epsilon / D value from pipe material data. The Re is located on the horizontal axis, a vertical line is drawn upward to intersect the curve for the given \epsilon / D, and f_D is read from the vertical axis at that intersection point; in ambiguous transition areas, interpolation between curves may be necessary.^[35]^[38] The primary advantages of the Moody diagram lie in its ability to visually delineate flow regimes and the interplay between Re and roughness, enabling rapid qualitative and quantitative assessments without iterative computations, which was particularly valuable in pre-digital engineering practice. However, it relies on manual reading and interpolation, potentially introducing minor inaccuracies, especially in the critical and transitional zones or for low Re values where the scale compression limits precision.^[39]^[35]

Colebrook-White Equation

The Colebrook-White equation, proposed by C. F. Colebrook in 1939 based on extensive experimental tests of fluid friction in roughened pipes conducted with C. M. White, provides the standard implicit correlation for the Darcy-Weisbach friction factor f_D in turbulent pipe flow. This equation bridges the smooth and rough pipe regimes, offering a unified expression validated against data from a wide range of pipe conditions and Reynolds numbers. The equation is expressed as

\frac{1}{\sqrt{f_D}} = -2 \log_{10} \left( \frac{\epsilon}{3.7 D} + \frac{2.51}{\mathrm{Re} \sqrt{f_D}} \right),

where \mathrm{Re} is the Reynolds number, D is the pipe diameter, and \epsilon is the absolute roughness height. It combines Prandtl's smooth pipe law, which relates friction to viscous sublayer effects, with Nikuradse's rough pipe data, where roughness protrusions dominate flow resistance; the integration relies on the law of the wall for the near-wall region and the velocity defect law in the outer flow, aligned via von Kármán's universal velocity profile to ensure continuity across regimes. The formulation accurately captures the transition from viscosity-dominated to roughness-dominated turbulence, with reported precision of ±5% over the Reynolds number range $3000 < \mathrm{Re} < 10^8.^[33] Due to its implicit nature in f_D, solution requires numerical iteration, such as the Newton-Raphson method, which converges rapidly when initialized with an explicit approximation like Haaland's formula for turbulent friction.^[40] A key input is the relative roughness \epsilon / D, which quantifies surface irregularity; typical values include \epsilon / D \approx 10^{-5}–$10^{-4} for smooth drawn tubing (corresponding to \epsilon \approx 0.0015 mm) and \epsilon / D \approx 0.001–$0.01 for rough concrete pipes (corresponding to \epsilon \approx 1–$3 mm, depending on diameter).^[41] This equation underpins graphical tools like the Moody diagram for practical visualization of friction factor trends.^[42]

Applications in Fluid Systems

Pressure Drop Calculations

The pressure drop due to friction in straight pipes is calculated using the Darcy-Weisbach equation, which relates the frictional losses to the pipe geometry, fluid properties, and flow characteristics. This equation is derived from the head loss form by multiplying by fluid density and gravity, yielding the pressure loss directly. The full Darcy-Weisbach equation for pressure drop is:

\Delta P = f_D \frac{L}{D} \frac{\rho v^2}{2}

where \Delta P is the pressure drop (Pa), f_D is the Darcy-Weisbach friction factor (dimensionless), L is the pipe length (m), D is the pipe diameter (m), \rho is the fluid density (kg/m³), and v is the average flow velocity (m/s).^[1]^[43] To perform the calculation step-by-step, first determine the Reynolds number Re = \frac{\rho v D}{\mu} using the fluid dynamic viscosity \mu, then select f_D based on Re and the relative roughness \epsilon / D (where \epsilon is the absolute roughness) via methods such as the Moody diagram or Colebrook-White equation. Next, input the pipe length L, diameter D, fluid density \rho, and velocity v (often derived from flow rate Q = v \cdot \frac{\pi D^2}{4}) into the equation. This yields the major frictional pressure loss for the straight pipe segment.^[1] For example, consider water (\rho = 1000 kg/m³, \mu = 0.001 Pa·s) flowing at 2 m/s in a 0.1 m diameter pipe over 100 m, with Re = 2 \times 10^5 and \epsilon / D = 0.001, giving f_D \approx 0.02. Substituting into the equation produces \Delta P \approx 40 kPa, illustrating typical losses in a smooth commercial pipe. Such calculations establish the scale of frictional effects, where higher velocities or roughness amplify the drop.^[1]^[44] Units must be consistent for accuracy; in SI, pressure is in pascals (Pa), lengths in meters (m), and density in kg/m³, while English units use pounds per square inch (psi) for pressure, feet (ft) for lengths, and slugs/ft³ for density, often requiring conversion factors like g_c = 32.2 lb·m/lbm·s² in the dynamic pressure term.^[45]^[43] For minor losses from fittings like elbows or valves in straight pipe runs, add equivalent lengths L_e to the total L, where each fitting contributes L_e = K \frac{D}{f_D} (with K as the loss coefficient), then recompute using the Darcy-Weisbach equation for the extended length; this approximates local turbulence as additional frictional pipe length.^[46]

Energy Losses in Piping Networks

In piping networks, the energy balance is governed by the engineering Bernoulli equation, which accounts for frictional and minor losses alongside pressure, velocity, and elevation changes. The equation in head form is expressed as \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_f + h_m, where h_f represents the major head loss due to friction in straight pipe sections, calculated using the Darcy-Weisbach equation h_f = f \frac{L}{D} \frac{v^2}{2g} with the Darcy friction factor f, and h_m denotes minor head losses from fittings and components.^[47] This form derives from conservation of energy, incorporating irreversible losses that dissipate mechanical energy as heat.^[47] Minor losses in piping networks arise from turbulence in fittings, valves, bends, and transitions, quantified as h_m = K \frac{v^2}{2g}, where K is the loss coefficient specific to the component and v is the flow velocity. For instance, a fully open globe valve has K \approx 10, significantly higher than a gate valve's K \approx 0.2.^[48] These losses can be equivalently incorporated into the frictional term by using an equivalent length L_{eq} = \frac{K D}{f}, where D is the pipe diameter and f is the Darcy friction factor, allowing the total head loss to be treated as an extended pipe length in the Darcy-Weisbach equation.^[49] In complex networks, the sum of all h_m across components often rivals or exceeds h_f, necessitating their inclusion for accurate energy budgeting.^[49] For system design, the required pump head H_p equals the total dynamic head, comprising the sum of all frictional losses h_f, minor losses h_m, and elevation differences \Delta z: H_p = \sum h_f + \sum h_m + \Delta z.^[50] Since the friction factor f depends on flow rate via the Reynolds number, iterative methods like the Hardy Cross procedure are employed to solve for flows and heads in looped networks, adjusting f until energy balances converge.^[51] In a simple loop network example, such as a municipal water distribution with pipes connecting a reservoir to demand nodes, total losses are summed along each path (e.g., 10 ft of head from friction in 500 ft of 6-inch pipe plus 5 ft from fittings), enabling pipe sizing to maintain velocities between 2-5 ft/s while minimizing pump requirements and ensuring minimum pressures like 40 psi at endpoints.^[51] For non-circular ducts in piping networks, such as rectangular or annular sections, the hydraulic diameter D_h = \frac{4A}{P} (where A is the cross-sectional area and P is the wetted perimeter) replaces the pipe diameter in friction factor calculations, including the Reynolds number \mathrm{Re} = \frac{\rho v D_h}{\mu} and relative roughness \frac{\epsilon}{D_h}.^[52] This substitution allows the standard Darcy-Weisbach equation to estimate h_f accurately, though secondary flows in non-circular geometries may slightly alter f from circular pipe correlations.^[52]

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Hagen-Poiseuille Law - an overview | ScienceDirect Topics
The Hagen–Poiseuille law is defined as the relationship that describes the flow rate (Q) of a fluid in laminar flow through smooth-walled, straight, circular ...
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Explicit Equations for Pipe-Flow Problems | Vol 102, No 5
Feb 3, 2021 · Reported herein are explicit and accurate equations for pipe diameter and head loss and a closed form solution for the discharge through the pipe.
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[PDF] On the Blasius correlation for friction factors - arXiv
The Blasius correlation is an empirical power law for turbulent pipe friction factors, derived from first principles, and extended to non-Newtonian fluids.Missing: original | Show results with:original
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A Novel Explicit Equation for Friction Factor in Smooth and Rough ...
In this paper, we propose a novel explicit equation for friction factor, which ... Blasius for pipes in 1913. The Blasius formula is. f = 0.316 Re − 0.25.Missing: original | Show results with:original
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United Formula for the Friction Factor in the Turbulent Region ... - NIH
May 2, 2016 · Blasius, H. (1913). The law of similarity of frictional processes in fluids. originally in German), ForschArbeitIngenieur-Wesen, Berlin, 131.
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Blasius Correlation - an overview | ScienceDirect Topics
The Blasius correlation is a mathematical relationship used to predict frictional resistance in fluid flow in circular tubes, established in 1913.Missing: paper | Show results with:paper
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Accurate explicit analytical solution for Colebrook-White equation
While Blasius equation is valid only for smooth pipes, results of Swamee-Jain and Haaland equations differ from original Colebrook-White equation mainly in the ...
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(PDF) Friction Factor Equations Accuracy for Single and Two-Phase ...
Nov 30, 2020 · This paper aims to investigate accuracy of 46 of those explicit equations and Colebrook implicit equation against 2397 experimental points from single-phase ...
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Friction Factor Calculations - Pipe Flow Software
The friction factor for laminar flow is calculated by dividing 64 by the Reynold's number. Friction factor (for laminar flow) = 64 / Re. Critical Flow Condition.
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[PDF] Friction Factors for Pipe Flow
This equation clearly brings out the dependence of the pipe friction phenomena upon the thickness of the laminar boundary ... originally laminar flow will remain ...
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Moody Diagram - an overview | ScienceDirect Topics
Relative roughness = 0.05/388 = 0.000129. From the Moody diagram, for R = 364,617 and (e/D) = 0.000129, we get the friction factor as f = 0.0153.Missing: plot | Show results with:plot
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Friction Factor Calculator Moody's Diagram for Smooth and Rough ...
Oct 7, 2024 · The Moody diagram (or Moody chart) is a log – log plot between three variables Reynolds Number, Relative Roughness and Friction Factor. This ...Missing: construction | Show results with:construction
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[PDF] Solving the Colebrook Equation for Friction Factors Introduction:
These equations can be solved for “f” given the Relative Roughness (? /D) and the Reynolds Number, (R), by iteration. Such iterations can be performed using an ...
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Efficient Resolution of the Colebrook Equation - ACS Publications
Industrial & Engineering Chemistry Fundamentals (1979), 18 (3), 296-7CODEN: IECFA7; ISSN:0196-4313. A single explicit equation correlating friction factor, pipe ...
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Surface Roughness Coefficients - The Engineering ToolBox
Surface coefficients that can be used to calculate friction and major pressure loss for fluid flow with surfaces like concrete, galvanized steel, corroded steel ...
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Relationship between Hazen–William and Colebrook–White ...
The current effort derives implicit equations relating C to k s that do not require additional information and compare well with published data. The exact ...
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[PDF] A Tutorial on Pipe Flow Equations - eng . lbl . gov
Aug 16, 2001 · The maximum difference in friction factor is about 17% which translates to a 8.5% difference in flow. 3. This maximum difference occurs at ...
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[PDF] Chapter 13: Head Loss in Pipes - eCommons - University of Dayton
Dec 22, 2022 · We use the Darcy-Weisbach equation to show how head loss formulas have the general form shown in Eq. (2- 264). The Darcy-. Weisbach head loss ...
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6.3.2.1: The Darcy-Weisbach Equation for Single-Segment Oil ...
The Darcy-Weisbach Equation is one of the most common equations for modeling single-phase liquid flow through pipes and tubing.
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ME 354 - Lab 3 - Minor Losses Experiment
This equivalent length may be determined by equating the Darcy-Weisbach equation (eqtn. 1) to the minor losses equation (eqtn. 6) and solving for L. This yields ...
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None
### Summary of Engineering Bernoulli Equation with Head Losses for Piping Systems
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[PDF] TABLE 10.5 Loss Coefficients for Various Transitions and Fittings
Threaded pipe fittings. Globe valve-wide open. K₁ = 10.0. (15). Angle valve-wide open. K, = 5.0. Gate valve-wide open. K₁ = 0.2. Gate valve-half open. K₁ = 5.6.
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[PDF] Pipe flow minor losses
To do: (a) Calculate the total irreversible head loss in meters and the pressure drop in kPa through this section of piping due to both major and minor losses.
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[PDF] Calculating Total Dynamic Head - Grundfos
Total Dynamic Head (TDH) is calculated by adding Elevation Head, Pressure Head, and Friction loss. Pressure is converted to feet of head.
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Pipe Network Calculator - LMNO Engineering
The pipe network analysis calculation uses the steady state energy equation, Darcy Weisbach or Hazen Williams friction losses, and the Hardy Cross method to ...
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[PDF] Non-Circular Pipe Friction
To calculate the frictional head loss non-circular pipes the method must be adapted to use the Hydraulic Diameter instead of the internal dimensions of the pipe ...