Fact-checked by Grok 2 weeks ago

Discharge coefficient

The discharge coefficient, denoted as C_d, is a dimensionless parameter in fluid mechanics defined as the ratio of the actual mass or volume flow rate through a restriction—such as an orifice, nozzle, or valve—to the theoretical flow rate assuming ideal, frictionless conditions.^[1]^[2] This coefficient quantifies deviations from ideal flow due to effects like flow contraction at the vena contracta, boundary layer friction, and turbulence, which reduce the effective discharge area and overall efficiency.^[3] Typical values range from approximately 0.6 for sharp-edged orifices to 0.98 or higher for optimized venturi meters, reflecting the degree of flow recovery and minimal losses in well-designed geometries.^[4]^[3] In engineering practice, the discharge coefficient serves as a critical correction factor in flow rate equations, enabling precise calculations for devices like orifice plates and flow nozzles used in pipeline metering.^[5] It is influenced by factors such as the Reynolds number, geometric ratios (e.g., orifice-to-pipe diameter), and fluid properties, with higher Reynolds numbers generally yielding C_d values closer to 1.0 due to reduced viscous effects.^[3] Applications span hydraulic structures like weirs and spillways for open-channel flow measurement, as well as industrial systems in chemical processing and HVAC for controlling fluid throughput.^[6]^[7] The parameter's determination often involves empirical calibration or theoretical models, such as those incorporating approach flow conditions and pressure ratios, to ensure accuracy in high-stakes environments like propulsion nozzles or leak detection in pressure vessels.^[8]^[9] By bridging theoretical predictions and experimental realities, the discharge coefficient enhances the reliability of fluid system designs across civil, mechanical, and aerospace engineering disciplines.^[1]

Definition and Fundamentals

Definition

The discharge coefficient, denoted as C_d, is a dimensionless parameter in fluid mechanics defined as the ratio of the actual volumetric flow rate Q_{\text{actual}} to the theoretical volumetric flow rate Q_{\text{theoretical}} based on Bernoulli's principle: C_d = \frac{Q_{\text{actual}}}{Q_{\text{theoretical}}}.^[10]^[11] This coefficient quantifies the efficiency of fluid discharge through restrictions such as orifices or nozzles by accounting for real-world losses, including viscous friction along flow boundaries, stream contraction, and the vena contracta effect where the jet cross-section minimizes downstream of the opening.^[1] Values of C_d typically range from 0.6 for sharp-edged orifices to 0.98 for optimized nozzles, varying with device geometry and flow regime.^[10] The concept originated in the 19th century, introduced by hydraulic engineers such as James B. Francis through his experiments on weirs, to enable precise volumetric measurements in engineering applications.^[12] As a dimensionless quantity, C_d applies universally across scales, fluid types, and measurement systems without dependence on specific units.

Theoretical Basis

The theoretical foundation of the discharge coefficient arises from the application of Bernoulli's equation to ideal fluid flow through an orifice or constriction, assuming incompressible, steady, and inviscid conditions where viscous losses are neglected. Under these assumptions, the velocity of the fluid exiting the orifice, derived from Torricelli's theorem, is given by v = \sqrt{\frac{2 \Delta P}{\rho}}, where \Delta P is the pressure difference across the orifice, and \rho is the fluid density. The theoretical discharge rate Q_{\text{theoretical}} is then the product of this velocity and the orifice area A:

Q_{\text{theoretical}} = A \sqrt{\frac{2 \Delta P}{\rho}}.

This expression represents the ideal volumetric flow rate without accounting for real-world deviations such as flow contraction or energy dissipation.^[13]^[14] In actual flows, the discharge coefficient C_d corrects the theoretical rate to match observed conditions, defined as C_d = \frac{Q_{\text{actual}}}{Q_{\text{theoretical}}}. It decomposes into the product of the contraction coefficient C_c and the velocity coefficient C_v: C_d = C_c \cdot C_v. The contraction coefficient C_c (typically 0.6 to 0.8) accounts for the vena contracta effect, where streamlines converge and the effective cross-sectional area of the jet narrows to A_c = C_c A due to lateral velocity components near the orifice edges, reducing the flow area below the geometric orifice size. This phenomenon occurs because the flow separates from the sharp edges, forming a minimum area jet shortly downstream.^[14]^[13] The velocity coefficient C_v (typically 0.95 to 0.99) addresses deviations from the ideal velocity due to frictional and other energy losses. From the energy equation, applying Bernoulli's principle between upstream and vena contracta points yields the actual velocity v_{\text{actual}} = C_v \sqrt{\frac{2 \Delta P}{\rho}}, where C_v = \sqrt{\frac{h - h_{\text{loss}}}{h}} and h_{\text{loss}} is the head loss across the orifice (with h = \Delta P / (\rho g) for hydrostatic cases). In real flows, h_{\text{loss}} arises from viscous effects and turbulence, violating the inviscid assumption. When the head loss is expressed as h_{\text{loss}} = K \frac{v_{\text{actual}}^2}{2g} using the loss coefficient K, solving for C_v gives C_v = \frac{1}{\sqrt{1 + K}}; for configurations where contraction is negligible (C_c \approx 1), this approximates C_d \approx \sqrt{\frac{1}{1 + K}}, highlighting how losses diminish the effective discharge.^[14]

Applications in Fluid Flow

Orifice and Nozzle Flow

In closed conduit flows, the discharge coefficient is essential for metering and controlling fluid through orifices and nozzles, accounting for deviations from ideal flow due to geometric and viscous effects. Sharp-edged orifices, often implemented as thin-plate inserts in pipes, exhibit a typical discharge coefficient of approximately 0.61 for high Reynolds number flows, attributed to the vena contracta phenomenon where the fluid stream contracts to about 0.61 times the orifice area immediately downstream, reducing the effective flow area.^[15] This value corrects the theoretical flow prediction based on Bernoulli's principle, yielding the actual volumetric flow rate as

Q_{\text{actual}} = C_d A \sqrt{\frac{2 \Delta P}{\rho}}

where C_d is the discharge coefficient, A is the orifice area, \Delta P is the pressure differential across the orifice, and \rho is the fluid density.^[11] The discharge coefficient for such orifices remains relatively stable above Reynolds numbers of about 10,000, though it increases slightly at lower values due to enhanced viscous effects.^[15] Nozzle designs, such as conical or rounded entrances, achieve higher discharge coefficients ranging from 0.95 to 0.99 by minimizing flow contraction through smooth convergence, which suppresses vena contracta formation and reduces energy losses.^[10] These elevated values are particularly beneficial in applications requiring efficient atomization and minimal pressure drop, such as fuel injectors in internal combustion engines and spray nozzles in industrial processes, where precise control of liquid discharge enhances combustion efficiency and spray uniformity.^[16] The International Organization for Standardization (ISO) 5167 series, particularly Part 2, establishes guidelines for orifice plates in pressurized conduits, providing empirical equations and calibration curves for the discharge coefficient based on the beta ratio \beta = d/D (orifice diameter d to pipe diameter D). The Reader-Harris/Gallagher equation in ISO 5167-2 computes C_d as a function of \beta, pipe Reynolds number, and tap positions, typically yielding values from 0.59 to 0.62 for \beta between 0.2 and 0.75, ensuring traceability and accuracy within ±1% for flow measurements.^[17] A practical application is seen in pipeline flow metering for industries like oil and gas, where orifice plates with known discharge coefficients enable direct computation of mass flow rates from differential pressure readings, facilitating non-intrusive monitoring and billing without velocity probes.^[18] For instance, in natural gas transmission lines, this method supports volumetric accuracies of 1-2% over wide operating ranges, integrating seamlessly with SCADA systems for real-time custody transfer.^[18]

Open Channel and Weir Flow

In open channel flows, the discharge coefficient plays a crucial role in quantifying flow rates over hydraulic structures like weirs and spillways, where gravity drives the free-surface overflow under atmospheric pressure. Unlike pressurized conduit flows, these applications involve subcritical approach conditions where the Froude number (Fr) of the upstream flow typically remains low (Fr < 1), influencing the coefficient through velocity distribution and contraction effects. The discharge coefficient, C_d, adjusts the idealized theoretical discharge to account for real-world losses, contractions, and nappe formation, enabling accurate measurement in irrigation, flood control, and dam operations. For rectangular sharp-crested weirs, the seminal Francis formula provides the discharge as Q = C_d L H^{3/2}, where Q is the discharge, L is the weir length, H is the head over the crest, and C_d \approx 0.62 (in consistent units, derived from empirical calibration in customary systems as approximately 3.33 for the coefficient in Q = 3.33 L H^{3/2} fps). This value stems from James B. Francis's 19th-century experiments on thin-plate weirs, balancing contraction losses and vena contracta formation. End contractions in fully contracted setups reduce the effective length to L_e = L - 0.2 H (for two sides), further modulated by the subcritical approach Froude number, which can decrease C_d by up to 5% if Fr exceeds 0.1 due to altered streamlines and increased energy dissipation.^[12]^[19] Variations in weir geometry lead to adjusted C_d values. For Cippoletti (trapezoidal) weirs with 1:4 side slopes, C_d ranges from 0.58 to 0.62, reflecting the integrated rectangular and triangular contributions, with the formula Q \approx 3.37 L H^{3/2} in fps units showing ±5% accuracy under standard conditions. Broad-crested weirs, where the crest width sustains critical flow (Fr = 1 at the crest), exhibit higher C_d ≈ 0.85 for head-to-width ratios (h/L) between 0.1 and 0.3, as the parallel streamlines minimize contraction losses compared to sharp crests.^[20]^[21] Historically, the Francis formula has been applied to dam spillways for flood routing and capacity estimation, with empirical adjustments for velocity of approach to correct for non-negligible upstream velocities. The modified discharge incorporates the approach head h_a (v^2 / 2g, where v is upstream velocity) as Q = C_d L (H + h_a)^{3/2} H^{1/2}, accounting for the increased total energy while scaling by the square root of the measured head to refine accuracy in high-flow scenarios. This correction, validated in U.S. Bureau of Reclamation practices, ensures reliable predictions for spillway designs like those on the Hoover Dam.^[12]^[22]

Venturi and Other Devices

The Venturi meter is a differential pressure flow measurement device featuring a gradual contraction followed by a diverging expansion section, which minimizes flow disturbances and enables high discharge coefficients typically ranging from 0.984 to 0.995 depending on construction (e.g., as-cast or machined convergent sections).^[23] This high value arises from the streamlined geometry that reduces vena contracta effects and frictional losses compared to sharp-edged devices. The volumetric flow rate Q through a Venturi meter for incompressible fluids is calculated as

Q = C_d \cdot \frac{\pi}{4} d^2 \sqrt{\frac{2 \Delta P}{\rho (1 - \beta^4)}}

where C_d is the discharge coefficient, d is the throat diameter, \Delta P is the pressure differential, \rho is the fluid density, and \beta = d / D is the diameter ratio with D as the pipe diameter; uncertainties in C_d are approximately ±0.7% to ±1% under standard turbulent conditions (Reynolds number Re_D \geq 2 \times 10^5).^[23] Flow nozzles represent another low-loss device suited for high-velocity flows in pipes, with discharge coefficients generally between 0.95 and 0.99, increasing toward 0.995 at high Reynolds numbers (Re > 10^6) and varying with the beta ratio (0.2 to 0.8).^[11] These nozzles feature a rounded inlet and abrupt outlet, providing better pressure recovery than orifices while maintaining accuracy within ±2% as per ASME standards.^[24] Pitot tubes, used for direct velocity measurement, incorporate a discharge coefficient of approximately 0.98 to 1.00 to account for velocity profile non-uniformities and tube interference, with standard designs achieving a coefficient near unity (0.99) without significant corrections.^[25]^[26] Annular orifices, formed between concentric cylinders, find applications in HVAC systems for flow control in ducts and valves, where the discharge coefficient varies with the area ratio (typically 0.6 to 0.8 for gap-to-pipe ratios under turbulent conditions) and Reynolds number effects.^[27] In aerospace, variable area meters such as adjustable nozzles or rotameter-like devices adjust the effective area for thrust vectoring or fuel metering, with discharge coefficients depending on the instantaneous area ratio (often 0.85 to 0.95) and calibrated empirically to handle compressible flows.^[28] These devices prioritize adaptability in dynamic environments like aircraft propulsion systems.^[8] A key advantage of Venturi meters, flow nozzles, and similar devices over sharp-edged orifices is their minimal permanent pressure loss—often recovering 80-90% of the differential pressure—allowing repeated use in process industries without excessive energy penalties.^[11] This efficiency supports applications in continuous monitoring of liquids and gases in pipelines.

Determination and Measurement

Experimental Methods

The volumetric method is a fundamental experimental technique for determining the discharge coefficient C_d by directly measuring the actual discharge Q_\text{actual} and comparing it to the theoretical discharge predicted from pressure differences and device geometry. In this approach, fluid is collected in a calibrated tank over a precise time interval, yielding Q_\text{actual} = V / t, where V is the collected volume and t is the time. Pressure transducers or manometers measure the upstream head or differential pressure to compute the theoretical flow, allowing C_d = Q_\text{actual} / Q_\text{theoretical}. This method is particularly suitable for laboratory settings with orifices or nozzles, achieving uncertainties as low as ±1-2% when using high-precision timing and volume measurement equipment.^[29]^[2] Tracer dilution and velocity profiling methods provide alternative ways to quantify Q_\text{actual} in both laboratory scale models and field applications, such as weirs or orifices in open channels. Tracer dilution involves injecting a known concentration of dye or salt upstream, allowing complete mixing, and sampling downstream to measure dilution; the discharge is then calculated from the injection rate and concentration ratio, enabling C_d evaluation against head measurements. Velocity profiling employs Pitot tubes or current meters to map cross-sectional velocities, integrating the area-velocity product to obtain Q_\text{actual}, which is compared to theoretical values for C_d. These techniques are effective for turbulent flows where direct volumetric collection is impractical, with tracer methods offering accuracies of ±5% in streams.^[30]^[31] Calibration rigs standardized by organizations like ASME and ISO facilitate precise C_d determination under controlled conditions of known heads and pressures. These setups typically feature a test section with the flow device (e.g., orifice plate), upstream settling chambers, and reference flow meters or volumetric collectors to validate measurements; differential pressure is monitored via taps, and flow rates are varied to assess C_d across regimes. ASME MFC-3M outlines procedures for orifice, nozzle, and Venturi calibrations, emphasizing pipe Reynolds numbers above $10^5 for stable results, while ISO 5167 specifies geometric tolerances and uncertainty analyses targeting ±0.5-1% for C_d. Such rigs are essential for industrial certification, incorporating error propagation from instrumentation like transducers (±0.1% full scale).^[32]^[33] Advancements since 2000 have introduced non-intrusive optical techniques like laser Doppler velocimetry (LDV) and particle image velocimetry (PIV) for C_d measurement in complex geometries where traditional probes disrupt flow. LDV uses laser beams to detect Doppler shifts from seeded particles, providing point-wise velocity data that integrates to Q_\text{actual} for comparison with pressure-based theory; it excels in high-speed, turbulent orifice jets with resolutions down to 1 mm/s. PIV captures instantaneous planar velocity fields by imaging particle displacements between laser pulses, enabling whole-field integration for discharge in irregular shapes like submerged weirs, with typical uncertainties of ±2-5% in validation studies. These methods, often combined with high-speed cameras, have improved C_d accuracy in non-steady flows influenced by Reynolds number variations.^[34]^[35]

Theoretical and Empirical Models

Theoretical and empirical models provide predictive frameworks for estimating the discharge coefficient C_d in various flow configurations, enabling computation without direct experimental measurement. For orifice plates in pipe flow, empirical correlations derived from extensive calibration data are widely used. The Reader-Harris/Gallagher equation, incorporated in the ISO 5167-2 standard (as of 2022), calculates C_d as a function of the diameter ratio \beta = d/D (where d is the orifice diameter and D is the pipe diameter), the Reynolds number \mathrm{Re}, and pressure tap positions L_1 and L_2':

C = 0.5961 + 0.026 \beta^2 - 0.216 \beta^8 + 10^6 \beta^{3.5} \left( \frac{1}{\mathrm{Re}_D} \right)^{0.7} \left( \frac{10^6}{\mathrm{Re}_D} \right)^{0.3} + (0.0188 + 0.0063 A) \beta^{1.1} \frac{\beta}{1 - \beta^4} - (0.043 + 0.080 e^{-10 L_1} - 0.123 e^{-7 L_1}) (1 - 0.11 A) \frac{\beta^4}{1 - \beta^4} - 0.031 (M_2' - 0.8) \beta^{1.3}

where A = \frac{0.00016 \times 10^6 \beta}{\mathrm{Re}_D}, M_2' = \frac{2 L_2'}{\beta^2} (1 - 0.15 (\frac{L_2'}{0.5})^ {3.5} ), L_1 = l_1 / D, L_2' = l_2' / (1 - \beta), and \mathrm{Re}_D is the pipe Reynolds number. For D < 71.12 mm, an additional term + 0.011 (0.75 - \beta) (2.8 - D)/0.34 is included (dimensions in mm). This formulation unifies data across corner, D and D/2, and flange pressure tappings, with uncertainties typically below 1% for standard geometries.^[3]^[36]^[37] Theoretical models leverage computational fluid dynamics (CFD) to solve the Navier-Stokes equations for flow through orifices and nozzles, directly computing C_d from simulated velocity and pressure fields. These simulations incorporate turbulence closures such as the standard k-\epsilon model to capture Reynolds stresses in high-Re flows, with grid refinement near the orifice edge essential for accuracy. Validation against experimental data shows CFD predictions within 2-5% of measured C_d for \beta ranging from 0.2 to 0.75, particularly useful for non-standard designs where empirical data is scarce.^[38]^[39] For open-channel weirs, the Kindsvater-Carter method provides an empirical adjustment to the discharge coefficient for partially and fully contracted rectangular thin-plate weirs, accounting for end contractions, approach velocity, and small heads relative to weir length L and channel wall height P. The effective discharge coefficient C_e is determined from graphical or tabular data as a function of the head-to-height ratio h_1/P and contraction ratio L/B, with small corrections for effective head H_e = H + 0.003 ft and effective length L_e = L + 0.003 (L/B) ft to account for surface tension and velocity of approach effects. This model extends Francis' formula, with applicability to H/L < 0.5 and contraction ratios up to 0.2.^[40]^[41] These models are generally valid for Reynolds numbers \mathrm{Re} > 10^4, where viscous effects are negligible and flow is fully turbulent; below this threshold, C_d increases due to boundary layer growth. For non-standard geometries, such as irregular contractions or low \beta, empirical and CFD approaches require experimental validation to ensure accuracy within 1-3%.^[3]^[38]

Influencing Factors and Variations

Geometric Effects

The geometry of flow control devices significantly influences the discharge coefficient by affecting flow contraction, separation, and energy losses. In orifice plates, the sharpness of the upstream edge plays a critical role in determining the contraction coefficient, which directly impacts the overall discharge coefficient. For sharp-edged orifices, the contraction coefficient is approximately 0.61 due to the vena contracta formation, resulting in a discharge coefficient around 0.61.^[42] In contrast, rounded edges minimize flow contraction and separation, elevating the discharge coefficient to nearly 0.98 by promoting more uniform velocity profiles.^[42] The beta ratio, defined as the ratio of orifice diameter to pipe diameter (β = d/D), further modulates the discharge coefficient in orifice meters. Optimal beta ratios between 0.2 and 0.6 provide minimum uncertainty and stable discharge coefficients by balancing pressure drop and flow symmetry.^[43] Lower beta ratios (e.g., below 0.2) amplify wall effects, increasing viscous losses and reducing coefficient stability, while the discharge coefficient generally increases linearly with beta in typical ranges. These geometric sensitivities can interact with flow conditions to alter performance, though shape effects dominate in isolation. For weirs, crest geometry and contractions alter the effective discharge coefficient by influencing the nappe shape and wetted perimeter. Rounding the weir crest typically enhances discharge capacity compared to sharp crests, but excessive rounding or surface irregularities can introduce scale effects that slightly reduce the coefficient by 1-2% relative to ideal sharp-crested designs.^[45] Side contractions, where the weir length is less than the channel width, reduce the effective discharge by 5-10% (depending on head and geometry) through adjustment of the effective crest length for end contractions and flow asymmetry, without altering the discharge coefficient itself.^[12] Triangular V-notch weirs, with their converging sides, exhibit discharge coefficients ranging from 0.58 to 0.62 for a 90° notch, offering precise measurement for low flows due to the sensitive head-discharge relationship.^[46] Nozzle profiles optimize discharge coefficients by streamlining acceleration and reducing boundary layer separation. Converging angles less than 20° (e.g., 10°-15°) yield higher discharge coefficients, often exceeding 0.95, by minimizing flow divergence and losses during contraction.^[47] Bell-mouth inlets, with their smoothly curved profiles, approach a discharge coefficient of 1.0 by eliminating sharp edges and achieving near-ideal isentropic flow.^[48] These geometric considerations guide device design to maximize accuracy and efficiency in flow measurement, with sharp-edged orifices suited for standard applications and rounded or bell-mouth variants preferred for high-precision scenarios.

Flow Regime and Conditions

The discharge coefficient C_d in fluid flow through orifices and nozzles is strongly influenced by the Reynolds number (Re), which characterizes the ratio of inertial to viscous forces. At high Reynolds numbers, typically Re > $10^5, C_d approaches an asymptotic value, often around 0.6 for sharp-edged orifices, where inertial effects dominate and viscous losses become negligible.^[18] In contrast, at low Re (e.g., below 250), viscous dominance leads to significant deviations, with C_d dropping by 10-20% compared to high-Re values due to increased boundary layer effects and flow contraction alterations.^[49] For compressible gas flows in nozzles, C_d is adjusted by the isentropic flow factor, accounting for thermodynamic expansion. When the pressure ratio across the nozzle falls below the critical value (approximately 0.528 for diatomic gases like air), the flow becomes choked at sonic conditions, and C_d approaches 1 for ideal isentropic nozzles with minimal losses.^[50] This ideal behavior assumes one-dimensional flow without boundary layer or heat transfer effects, though real nozzles exhibit C_d values of 0.99 or higher at high Re under choked conditions.^[50] Upstream flow disturbances, including turbulence, can impact C_d in orifice flows by altering the velocity profile and vena contracta.^[51] In high-speed liquid flows, cavitation further exacerbates this reduction by introducing vapor pockets that impede effective flow area, leading to additional C_d decreases of several percent in the cavitating regime.^[52] Fluid properties modulated by temperature, such as viscosity, indirectly affect C_d through changes in Re. For water, higher temperatures reduce viscosity, thereby increasing Re and shifting C_d toward more stable, asymptotic values; for instance, a temperature rise from 80°F to 120°F can alter viscosity by 54%, resulting in up to 0.5% variation in C_d.^[18] This effect is more pronounced in liquids than gases, where viscosity changes are smaller.

Corrections and Standards

The ISO 5167 series of international standards provides detailed specifications for the discharge coefficient in pressure differential flow measurement devices, including polynomials for calculating the coefficient applicable to orifice plates, nozzles, and Venturi tubes. These polynomials, such as the Reader-Harris/Gallagher equation for orifice plates in ISO 5167-2, account for geometric parameters like the diameter ratio β and Reynolds number, ensuring uncertainties typically below 1% under ideal conditions. The standards also address pressure tap positions, distinguishing between corner taps, D and D/2 taps, and flange taps, with separate polynomial forms for each to minimize measurement errors due to differing pressure recovery characteristics.^[53] Corrections for non-ideal conditions are integral to the ISO 5167 framework, particularly for compressible gases where temperature and pressure effects require an expansion factor ε to adjust the discharge coefficient for density variations along the device. For multiphase flows, density corrections involve effective mixture density models or adjusted coefficients to account for phase interactions, reducing errors in gas-liquid mixtures by up to 10-15% when calibrated properly. Upstream disturbances, such as pipe bends within 10-20 diameters, can introduce errors of approximately +1% in the discharge coefficient; standards recommend minimum straight-run lengths or flow conditioners to mitigate these, with empirical factors applied based on disturbance type and location.^[33]^[54] ASME MFC-3M and API MPMS Chapter 14.3 standards adapt similar principles for petroleum metering applications, mandating certified discharge coefficients with uncertainties less than 1% for orifice-based natural gas measurements, often aligning with ISO polynomials but emphasizing field calibration for hydrocarbons. These evolved from 1930s adaptations of Hagen-Poiseuille principles for laminar flows, transitioning to turbulent empirical models in early AGA Report No. 3, which formed the basis for modern API guidelines focused on custody transfer accuracy.^[55] Post-2020 updates to ISO 5167, including the 2022 editions, incorporate minor refinements for consistency across parts, with enhanced uncertainty analyses and validation data supporting CFD simulations for device performance. These revisions facilitate corrections for emerging technologies like additive-manufactured flow devices, where CFD-validated adjustments address surface roughness and geometric deviations, improving coefficient accuracy in non-traditional fabrication.^[56]^[57]^[58]

References

[1]
Discharge Coefficient - an overview | ScienceDirect Topics
The discharge coefficient, C_D, is defined as the ratio of the cross-sectional area at the choke point (vena contracta) to the area at the discharge point.<|control11|><|separator|>
[2]
[PDF] Discharge Coefficient
An important element in this process is the discharge coefficient, which is defined as: “In a nozzle or other constriction, the ratio of the mass flow rate at ...
[3]
USBR Water Measurement Manual - Chapter 14
The effective discharge coefficient for venturi meters ranges from 0.9 to about unity (Streeter, 1951; American Society of Mechanical Engineers, 1983) with ...
[4]
4_7aupdate
For a wide range of sharp or square edged orifices and weirs the dimensionless discharge coefficient as defined above has a value close to 0.6. This is true for ...
[5]
[PDF] Discharge coefficients of square-edged orifices for measuring the ...
Discharge coefficients are quantities used in orifice meter computations, relating discharge rate to temperature, static pressure, and pressure drop. Tables of ...
[6]
[PDF] DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS
An explicit relation between the discharge coefficient and these variables is, of course, not known. In the following sections of this report, the effect of ...
[7]
Discharge coefficient – Knowledge and References - Taylor & Francis
The discharge coefficient, is defined as the ratio of the actual or experimental flow rate to the theoretical flow rate. The discharge coefficient does not ...
[8]
[PDF] discharge coefficients for thick plate orifices with approach flow ...
(3) Determine the discharge coefficient from the curve of discharge coefficient againstthe velocity headratio (shownin fig. 8)if the approachflow Math ...
[9]
[PDF] Appendix D Leak Rate vs. Crack Length Calculations
Determination of a pseudo discharge coefficient relating the flow rate through the actual crack to the flow rate through an ideal isentropic nozzle. D.2.
[10]
Discharge Coefficient for Nozzles and Orifices - Neutrium
Feb 11, 2015 · The discharge coefficient is a dimensionless number used to characterise the flow and pressure loss behaviour of nozzles and orifices in fluid systems.
[11]
Orifice, Nozzle, and Venturi Flow Meters: Principles, Calculations ...
A discharge coefficient cd = 0.975 can be indicated as standard, but the value varies noticeably at low values of the Reynolds number. The pressure recovery is ...
[12]
[PDF] WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS
... James B. Francis ................... ..... 23. Experiments and formula of ... discharge over weirs of irregular cross section ...... 62. Bazin's ...Missing: history Julius
[13]
[PDF] The History of the Darcy-Weisbach Equation for Pipe Flow Resistance
Abstract. The historical development of the Darcy-Weisbach equation for pipe flow resistance is examined. A concise examination of the evolution of the ...
[14]
Flow Through an Orifice - Richard Fitzpatrick
The jet narrows over a short distance beyond the orifice that is comparable with the jet diameter to form what is generally known as a vena contracta.
[15]
[PDF] INVESTIGATION OF THE DISCHARGE COEFFICIENT FOR ...
The purpose of this study is to examine the discharge coefficient as it pertains to flow through a circular orifice cut into a thin-walled vertical riser pipe.
[16]
Discharge Coefficient - an overview | ScienceDirect Topics
The discharge coefficient is defined as a dimensionless number that quantifies the flow efficiency of a fluid through an orifice, with a typical value of 0.61 ...
[17]
Spray A Nozzle Geometry - Engine Combustion Network
The table below provides measured nozzle geometry information obtained via various techniques as discussed in Kastengren, 2012.Missing: conical 0.95-0.99
[18]
The calculation of the uncertainty of the orifice-plate discharge ...
Using ISO 5167–2: 2003 the discharge coefficient is given by the Reader-Harris/Gallagher (1998) Equation [5]:(2a) C = 0.5961 + 0.0261 β 2 − 0.216 β 8 C ∞ t e r ...
[19]
[PDF] Improving Flow Measurement by Real-Time Flow Calculation in ...
The discharge coefficient is an empirical term describing the amount of flow that actually passes through the orifice meter. It corrects the theoretical flow ...Missing: pipeline study
[20]
[PDF] Demystifying the Discharge Coefficient for Flow over Thin Weirs and ...
Aug 8, 2025 · Using dimensional analysis, we show that the discharge coefficient, Cd, in the classical weir- discharge equation is best understood as a weir ...
[21]
USBR Water Measurement Manual - WEIRS, Section 12. Cipoletti Weir
The accuracy of the discharge coefficient is +5 percent. (b) Discharge of Cipoletti Weirs ... rectangular weir apply to the Cipoletti weir. The height of ...
[22]
[PDF] Discharge coefficient of a rectangular sharp-edged broad-crested weir
... broad-crested weir without lateral contraction. ... It determines the constant value of the discharge coefficient at C = 0.85 in ranges of 0.1 ≤ h/t ≤ 0.3.
[23]
Chapter 7 - WEIRS, Section 14. Velocity of Approach Corrections
Then, the Kindsvater-Carter relationship is used to calculate discharge for both the fully contracted discharge case and the partially contracted case, causing ...Missing: formula | Show results with:formula
[24]
[PDF] ISO 5167-4
Mar 1, 2003 · The relative uncertainty of the discharge coefficient as given in 5.5.3 is equal to 1 %. 5.7.3 Classical Venturi tube with a rough-welded sheet- ...
[25]
Most Probable Discharge Coefficients for ASME Flow Nozzles
In this paper, various rational, semirational, and, empirical approximations for the ASME long-radius, flow-nozzle discharge coefficients are first reviewed ...
[26]
MEASUREMENTS IN PRESSURE CONDUITS, Section 10. Pitot ...
Properly constructed and undamaged standard pitot-static tubes have a coefficient very close to unity and can be used without corrections for tube interference ...Missing: discharge | Show results with:discharge
[27]
[PDF] Method 2 Determination of Stack Gas Velocity and Volumetric Flow ...
Jul 1, 1999 · Type S pitot tube coefficient. Cp(std). = Standard pitot tube coefficient; use 0.99 if the coefficient is unknown and the tube is designed ...Missing: discharge | Show results with:discharge
[28]
Flow Through Annular Orifices | J. Fluids Eng.
Feb 14, 2022 · Coefficients for the annular orifice formed between a circular disk and a cylindrical tube are reported for twenty-one orifices.Missing: HVAC | Show results with:HVAC
[29]
Discharge Coefficients of Ports with Stepped Inlets - MDPI
Sep 19, 2018 · The area ratio of the port to the annulus, β, then determines the annulus approach velocity. (2) Similarly, the core inlet mass flow is ...
[30]
[PDF] Discharge Coefficient Experiment - ASEE PEER
The experiment uses a vertical tube with nozzles, measuring water volume and time to determine discharge coefficients for straight and counter-bored holes.
[31]
[PDF] Measurement of discharge using tracers
Measurement of stream discharge by dilu- tion methods depends on the determination of the degree of dilution of an added tracer solu- tion by the flowing water.
[32]
Methods of Measuring Flows in Open Channels - Openchannelflow
The tracer-dilution method consists of adding a known amount of concentrated tracer at a constant rate to the flow stream. Chemical analysis is used to ...
[33]
None
Error: Could not load webpage.<|separator|>
[34]
[PDF] The Engineer's Guide to DP Flow Measurement | Emerson
This chapter focuses on two main types of primary element technologies: area meters and averaging pitot tubes or sampling meters. Chapter 8 – Transmitter ...
[35]
The Discharge Coefficient of a Planar Submerged Slit-Jet
Specifically, a single channel laser Doppler velocimeter (LDV) was used to measure the time average velocity on the centerline, at. The spatial-average ...
[36]
A Particle Image Velocimetry-Based Investigation of the Flow Field ...
Sep 27, 2013 · The investigated geometry produces a highly complex flow field which was investigated employing particle image velocimetry (PIV).Particle Image Velocimetry · Results · Time-Mean Flow Fields<|control11|><|separator|>
[37]
[PDF] THE ORIFICE PLATE DISCHARGE COEFFICIENT EQUATION
This report describes the final work undertaken to achieve the equation which is being balloted in ISOrrC 30/SC 2 for inclusion in ISO 5167-1.
[38]
[PDF] Estimation of Discharge Coefficient in Orifice Meter by ...
Aug 1, 2018 · In this work, analysis of flow through an orifice is simulated using ANSYS Fluent computational fluid dynamics (CFD) software. Three cases with ...
[39]
[PDF] Paper 6.3 Overview of CFD Modelling of Orifice Plate Flometers
This paper provides an overview of CFD simulation of orifice-plates for flow-measurement. Engineers. It introduces basic concepts of CFD modelling, and compares ...
[40]
WEIRS, Section 6. Partially and Fully Contracted Rectangular Weirs
The Kindsvater-Carter method is used for rating rectangular weirs, including partially contracted ones, and is useful when full contractions are difficult. It ...
[41]
[PDF] Calculating flow over rectangular sharp-edged weirs
Sep 12, 2015 · 3/2 . (3). As U = q/d this is actually a transcendental equation for q which may be solved by direct iteration. Commonly, the approach velocity ...
[42]
[PDF] CRITICAL FLOW RESTRICTING ORIFICES - OSTI.GOV
The discharge coefficient for a sharp edged orifice is 0.61 and that of a rounded edge orifice is 0.98. Upon close examination of the orifice plate, the edge ...Missing: sharpness | Show results with:sharpness
[43]
[PDF] AGA Report No. 3
Minimum uncertainty of the orifice plate coefficient of discharge (Ca) is achieved with ẞ, between 0.2 and 0.6 and orifice bore diameters greater than or equal ...
[44]
(PDF) Effects of Beta Ratio and Reynold's Number on Coefficient of ...
Aug 6, 2025 · It was found that C d had a positive linear relationship with beta ratio where its dependency was stronger on β than R e in case of low flow rate.
[45]
Effects of Crest Rounding on Sharp-Crested Weirs - Openchannelflow
When combined with surface roughness, the increase in discharge error due to weir crest rounding can be 1-2% - or doubling the standard 2% error of thin-plate ...
[46]
[PDF] Discharge Characteristics of Triangular-notch Thin-plate Weirs
The dependent ratio in equation 3 is proportional to the coefficient of discharge ... 2 Equation 6 is described as the Kindsvater-Shen equation in the ...
[47]
Effect of Nozzle Geometry on the Flow Dynamics and Resistance ...
Mar 8, 2022 · The experimental results show that the nozzle with a converging angle of 10° or 15° has a higher discharge coefficient and a better cluster capacity.<|separator|>
[48]
[PDF] Computational Investigation of the Discharge Coefficient of ... - CORE
The discharge coefficient is a critical parameter for accurately calculating flow rate in any flow meter which operates by means of creating a pressure ...
[49]
A novel comprehensive correlation for discharge coefficient of ...
A major problem associated with the use of orifice flow meters at low Reynolds number flows is the significant variation of discharge coefficient (Cd) as a ...
[50]
[PDF] experimental and analytical sonic nozzle discharge coefficients for ...
There are two main factors that cause the sonic discharge coefficient to be less than unity. First, fluid acceleration near the wall and the resulting inertia!
[51]
Turbulence effects on the discharge coefficient and mean flow rate ...
This experimental study uses a wind tunnel and scale model to investigate turbulence effects on the discharge coefficient and mean flow rate of wind-driven ...
[52]
An Experimental and Numerical Study on the Cavitation and Spray ...
Feb 22, 2024 · Their results indicated that an increase in the number of cavitation bubbles led to a decrease in the discharge coefficient. Mithun et al. [27] ...<|separator|>
[53]
ISO 5167-3:2020 - Measurement of fluid flow by means of pressure ...
This standard specifies the geometry and use of nozzles and Venturi nozzles for measuring fluid flow in full conduits, including three standard nozzle types.
[54]
(PDF) Influence of flow disturbances on the performance of industry ...
Dec 9, 2020 · The results indicate that the correction models used to transfer the water calibration to cryogenic conditions (using LIN) can potentially ...Missing: discharge | Show results with:discharge
[55]
[PDF] a review of api mpms chapter 14.3 / aga report no. 3 − part 2 | asgmt
The first U.S. metering standard was produced by the American Gas Association in. 1930…AGA Report No. 1 (which later evolved into API MPMS, Chapter 14.3 or AGA ...Missing: Hagen- Poiseuille
[56]
https://www.solartronisa.com/-/media/ameteksolartronisa/files/white-papers/additional-data-supporting-the-2022-updates-to-the-iso-5167-standards.pdf
[57]
ISO 5167: Major developments in fluid flow measurement | Fuji Electric
The2022 edition of ISO 5167 brings some important updates. It incorporates new data and improved methods for calculating flow rate and associated uncertainty.Missing: CFD additive
[58]
[PDF] Correction for Orifice Plates installed in Reverse Orientation, CFD or ...
Oct 24, 2024 · This paper examines the impact on the flow measurement in caused by the sequential installation of two orifice plates, one after the other ...Missing: additive | Show results with:additive