The discharge coefficient of an orifice plate is the number that reconciles the ideal flow through a perfect constriction with the real flow that actually occurs, accounting for the way the streamlines contract past the bore and for friction. It is not a constant; it changes with the geometry of the plate and with how fast the fluid is moving. The Reader-Harris/Gallagher equation, often shortened to the RG equation, is the modern empirical correlation that computes that discharge coefficient from measurable quantities, and it is the correlation adopted by both ISO 5167 and AGA 3 for standard orifice plates. This guide explains what the equation depends on, why it has to be solved iteratively inside the flow computer because the coefficient and the Reynolds number depend on each other, and how surfacing the computed coefficient in SCADA lets an engineer confirm a meter is running inside its valid range.
Reader-Harris/Gallagher Equation in one line: The Reader-Harris/Gallagher equation, or RG equation, is the empirical correlation used in ISO 5167 and AGA 3 to compute the discharge coefficient of a standard orifice plate. It expresses the coefficient as a function of the beta ratio, the pipe Reynolds number, and the tap arrangement, capturing how the effective flow through the bore departs from the ideal. Because the Reynolds number depends on the flow rate and the flow rate depends on the discharge coefficient, the equation must be solved iteratively in the flow computer, converging on a coefficient consistent with the flow it produces.
The discharge coefficient captures the gap between the flow an ideal orifice would pass and the flow a real one passes at a given differential pressure. Several physical effects live inside that gap. As the fluid approaches the bore its streamlines converge and continue to contract just past the plate to a narrowest point called the vena contracta, so the effective flow area is smaller than the bore area, and friction along the way removes a little energy. The Reader-Harris/Gallagher equation is the correlation that quantifies the net result of all these effects as a single coefficient, built from an extensive experimental database of orifice plates rather than from first principles alone.
The first thing that drives the coefficient is the beta ratio, the bore diameter divided by the pipe diameter. Beta sets the basic geometry of the contraction: a small-beta plate presents a narrow bore in a wide pipe and contracts the flow one way, while a large-beta plate presents a wide bore and behaves differently, so the coefficient varies systematically with beta across the equation's several terms. The second major driver is the Reynolds number, a dimensionless measure of how inertia-dominated the flow is relative to viscosity. At high Reynolds numbers the coefficient becomes nearly flat and only weakly dependent on flow rate, but at lower Reynolds numbers it changes more steeply, which is one reason the standard defines a minimum Reynolds number below which the correlation is not considered valid.
The tap arrangement is the third factor, because where the pressure is sensed relative to the plate affects the differential the taps report and therefore the coefficient that reconciles it to flow. Flange taps and corner taps sit at different distances from the plate faces and see the pressure field slightly differently, so the Reader-Harris/Gallagher equation includes terms specific to the tap type. The equation is written so that supplying the beta ratio, the Reynolds number, and the correct tap terms yields the coefficient appropriate to that combination, which is why a flow computer must be configured with the actual tap type of the meter and not merely a generic setting.
There is a circular dependency at the heart of the calculation. The Reader-Harris/Gallagher equation needs the Reynolds number to compute the discharge coefficient, but the Reynolds number depends on the flow velocity, and the flow velocity is exactly what the whole calculation is trying to find using that same discharge coefficient. You cannot compute the coefficient without knowing the flow, and you cannot compute the flow without knowing the coefficient. This is not a flaw in the equation; it is an inherent feature of a coefficient that genuinely varies with flow rate, and it means the calculation cannot be done in a single pass.
The resolution is iteration. The flow computer starts with an initial guess of the discharge coefficient, uses it to compute a first estimate of the flow and hence the Reynolds number, then feeds that Reynolds number back into the Reader-Harris/Gallagher equation to get a refined coefficient. That refined coefficient produces a refined flow and Reynolds number, and the cycle repeats. With each pass the coefficient and the flow move closer to a self-consistent pair, and the process converges quickly because the coefficient's dependence on Reynolds number is gentle in the normal operating range. When the change between successive passes falls below a small tolerance, the flow computer accepts the converged coefficient and flow as the answer for that interval.
This iterative solve happens continuously inside the flow computer, once per calculation cycle, and it is one of the reasons an electronic flow computer is required for accurate orifice measurement rather than a fixed factor. The number of iterations is small and the computation is cheap, but it must actually be performed; an implementation that used a single fixed coefficient instead of converging on the flow-dependent one would be right only at the flow rate the fixed value was chosen for and increasingly wrong as the rate moved away, especially at lower Reynolds numbers where the coefficient changes most. The iteration is what keeps the coefficient honest across the meter's whole operating range.
The discharge coefficient the flow computer converges on each cycle is not just an internal intermediate; it is a diagnostic about whether the meter is operating where the correlation is valid. The Reader-Harris/Gallagher equation is only defined and only accurate above a minimum Reynolds number and within a range of beta ratios and pipe sizes, and outside that envelope the coefficient it returns is an extrapolation rather than a validated value. If a meter is running so slowly that the Reynolds number has dropped below the standard's floor, the flow it reports is being computed with a coefficient the standard does not warrant, and the measurement carries an uncertainty larger than its nameplate.
Surfacing the computed coefficient and the Reynolds number in a cloud SCADA platform such as Merobix lets a measurement engineer see this directly rather than infer it. When the coefficient and the Reynolds number are trended alongside the flow, it becomes visible when a meter spends part of its day below the valid Reynolds threshold, for example during low overnight demand on a line sized for peak flow. It also becomes visible if the coefficient is behaving oddly, sitting at a suspicious fixed value that would suggest the iteration is not running, or jumping in a way that points to a bad tap-type configuration or an unstable Reynolds input from a noisy density or viscosity value.
This visibility feeds directly into how an operator manages a meter fleet. A station that regularly dips below its valid Reynolds range is a candidate for a smaller bore, a different plate, or a turndown limit that keeps it in range, and knowing that requires seeing the Reynolds number, not just the flow. A coefficient that is stable and sitting where beta and Reynolds imply it should is quiet confirmation that the meter and its configuration are healthy. By exposing the coefficient that the Reader-Harris/Gallagher equation produces, the platform turns a buried internal calculation into an operational check that the orifice is being measured within the bounds the standard actually supports.
It calculates the discharge coefficient of a standard orifice plate, the number that relates the real flow through the plate to the ideal flow a perfect constriction would pass at the same differential pressure. It expresses that coefficient as a function of the beta ratio, the pipe Reynolds number, and the tap arrangement, based on an extensive experimental database. Both ISO 5167 and AGA 3 adopt it as the correlation for the orifice discharge coefficient.
Because the discharge coefficient depends on the Reynolds number, and the Reynolds number depends on the flow, which is itself computed using the discharge coefficient. That circular dependency cannot be resolved in one pass, so the flow computer starts with a guessed coefficient, computes a flow and Reynolds number, feeds that back to refine the coefficient, and repeats until successive passes agree within a small tolerance. The iteration converges quickly and is performed every calculation cycle.
The Reader-Harris/Gallagher equation is only validated above a minimum Reynolds number and within defined ranges of beta ratio and pipe size, so a meter running below that Reynolds floor is being computed with a coefficient the standard does not warrant. At very low flow the Reynolds number can drop under that threshold, especially on a line sized for peak flow running light overnight, and the reported flow then carries a larger uncertainty. Watching the Reynolds number tells an engineer whether the meter is operating inside its valid range.
This page references the standards, specifications, and official documentation published by the organizations below. Editions, product capabilities, and documentation change over time - confirm current requirements and specifications directly with the source.
Last reviewed: July 27, 2026. Merobix is not affiliated with, endorsed by, or sponsored by these organizations; their names are used only to identify the standards and products discussed.
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