Coefficient Of Discharge Calculator - Actual Flow vs Theoretical Flow

A coefficient of discharge calculator backs out Cd, the dimensionless number that turns an ideal Torricelli discharge into the flow you actually measure, using the definition Cd = Q_actual / Q_theoretical where Q_actual is the volume you collect over a measured time and Q_theoretical is the ideal discharge A * sqrt(2 g H) from the orifice area and hydraulic head.

• • Undergraduate fluid-mechanics labs: Measure Cd of a tank orifice directly by timing how long a bucket takes to fill, then compare the value with the textbook range of 0.60 to 0.65 for a sharp-edged opening.
• • Orifice plate calibration: Confirm that a manufactured plate matches its published discharge coefficient instead of trusting the catalog number.
• • Reverse-checking a flow estimate: When an orifice flow result looks too high or too low, this calculator shows whether the chosen Cd or the measured head is the problem.
• • Teaching the vena contracta: Let students see that a real jet contracts and loses energy at the edge, so measured flow is always below the ideal prediction.

The coefficient of discharge is the single correction factor that makes the simple orifice equation match reality. On its own, Q = A * sqrt(2 g H) assumes no viscosity, no jet contraction, and no friction at the edge, so it always over-predicts the real flow.

Use this coefficient of discharge calculator when you have measured an actual flow (a bucket and a stopwatch are enough) and want the Cd of the orifice rather than assuming a textbook value. It is the inverse of a standard orifice flow calculation, where Cd is supplied and the discharge is found.

If you already know Cd and want the discharge instead of the discharge coefficient, orifice flow calculator runs the forward form Q = Cd * A * sqrt(2 g H) with the same head, area, and gravity inputs.

The calculator combines the circular area formula A = pi d^2 / 4 with the ideal Torricelli velocity sqrt(2 g H) to build the theoretical discharge, divides that into the bucket-derived actual discharge V / t, and reports Cd together with the supporting numbers so you can see where the gap comes from.

• Cd: Dimensionless coefficient of discharge, the ratio of actual to theoretical discharge.
• Q_act: Actual (measured) volumetric discharge, in cubic meters per second, from volume V collected over time t.
• Q_th: Theoretical discharge from ideal flow, in cubic meters per second.
• A: Cross-sectional area of the orifice, computed from the diameter as A = pi d^2 / 4 in square meters.
• d: Internal diameter of the circular orifice, in meters.
• g: Local gravitational acceleration, in m/s^2 (ISO 80000-3 standard is 9.80665 m/s^2).
• H: Hydraulic head, the vertical distance from the upstream free water surface to the orifice centerline, in meters.
• V: Volume collected in the catch bucket during the timed run, in cubic meters.
• t: Time in seconds the bucket took to fill.

The calculation is a ratio of two discharges. Theoretical discharge comes from conservation of energy between the free surface and the jet: ideal velocity v_th = sqrt(2 g H) times the geometric area. Actual discharge comes straight from the bucket: volume collected divided by time.

A Cd near 0.62 for a sharp-edged orifice is the expected result. If your measured Cd comes out above 1.0, the head or area was under-measured, or the orifice is not sharp-edged; if it reads near zero, check that the bucket volume and time were entered correctly and that the head was non-zero.

According to Omni Calculator, Coefficient of Discharge, the coefficient of discharge is the ratio of actual discharge to theoretical discharge, with Q_th = A * sqrt(2 g h) and Q_act = V / t.

The exact value of Cd drifts with Reynolds number because viscous losses at the orifice edge change with the flow regime, so when you want to quantify that regime explicitly Reynolds number calculator is the next step after this one.

Three ideas explain why a real orifice never reaches Cd = 1.0, and they tell you what to check when a measured value looks wrong.

The same gravity-driven head-to-discharge idea appears in free-surface channels, where a control section replaces the orifice, so open channel flow calculator is a useful companion when the geometry stops being a simple tank and hole.

Five steps take you from a tank experiment to a measured discharge coefficient. Defaults match a standard sharp-edged circular orifice at sea-level gravity, so most lab users only fill in the measured fields.

1. 1 Measure the orifice diameter: Use the nominal diameter from the spec, or measure across the opening with calipers, and convert to meters before entering d.
2. 2 Read the hydraulic head: From the upstream free water surface, drop a vertical line to the orifice centerline. That distance in meters is H; keep the head roughly constant while you measure.
3. 3 Collect a bucket of flow: Place a bucket under the jet, start a stopwatch, and record the volume V (in cubic meters) and the fill time t. A longer run reduces timing error.
4. 4 Confirm gravity: Leave g at 9.80665 m/s^2 for sea-level work, or edit it for your location.
5. 5 Read Cd and the two discharges: The calculator returns Cd, the theoretical and actual discharge, the orifice area, and the ideal velocity. Compare Cd with the 0.60 to 0.65 range for a sharp-edged orifice.

For a 50 mm sharp-edged orifice at 200 mm head with 2.411 L collected in exactly 1 s, this coefficient of discharge calculator returns Cd = 0.620, with Q_th = 3.89 x 10^-3 m^3/s, Q_act = 2.41 x 10^-3 m^3/s, and v_th = 1.981 m/s.

When the upstream velocity is not negligible compared with the jet velocity, the simple Q_th = A sqrt(2 g H) form over-predicts and a full Bernoulli energy balance between the two sections is required, which is exactly what Bernoulli equation calculator is set up to compute.

The calculator is built for the bucket-and-stopwatch measurement that comes up in fluid-mechanics labs and field checks, not just for plugging in a known Cd.

• • Cd from raw measurements: Enter the diameter, head, collected volume, and fill time and the discharge coefficient comes straight out, no spreadsheet or hand algebra required.
• • Theoretical and actual side by side: Q_th and Q_act are both shown, so the calculation is transparent and you can see immediately which term drove the final Cd.
• • Built-in Torricelli sanity check: The ideal velocity v_th = sqrt(2 g H) is reported, so a head or gravity typo shows up as an obviously wrong velocity before you trust the Cd.
• • Standard gravity pre-filled: g is pre-loaded with the ISO 80000-3 standard value of 9.80665 m/s^2 but can be edited for planetary, high-altitude, or historical unit work.
• • Unit-clean inputs: Volume is entered in cubic meters with a 1 L = 0.001 m^3 conversion in the help text, so lab data drops in without unit scrambling.

Both Cd and the Darcy friction factor are empirical loss coefficients that correct an ideal result, and the same boundary-layer reasoning behind one shows up in the other, which is what friction factor calculator is set up to compute.

Cd = Q_act / Q_th looks simple, but several physical effects move your measured value away from the textbook range. Check these before reporting a number.

• • The calculator assumes steady, incompressible flow with a fully developed upstream velocity profile. Pulsating flows, two-phase mixtures, and choked compressible flows need a different formulation.
• • Submerged outlet conditions (orifice below the downstream water level) are not auto-corrected; subtract the downstream submergence from H before entering the head.
• • Non-circular orifices (rectangular, triangular, segmental) need a different area formula and Cd. Use this calculator only for circular orifices.

According to Wikipedia, Orifice plate, the coefficient of discharge for a sharp-edged orifice plate is typically between 0.6 and 0.85, with a first approximation of about 0.62 for fully developed flow.

Once you have a measured Cd, convert Q_act to a mean jet speed with the pipe velocity calculator, since that speed is exactly what the discharge coefficient is implicitly correcting.