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The effective regurgitant orifice area is a flow-equivalent area for a leaking heart valve: the instantaneous backward volume flow rate $Q_R(t)$ divided by the corresponding jet speed $u_j(t)$,It characterizes the effective stream passing through the leak. It need not equal the anatomical opening traced on an image, because flow contracts and its velocity is not uniform across every geometric surface. The vena contracta and the upstream convergence pattern provide complementary flow information.^[ref]

The proximal isovelocity surface area method illustrates how an area can be inferred without drawing the hole itself. Color Doppler identifies an upstream surface at a selected aliasing speed $v_a$. In an ideal hemispherical convergence model of radius $r$, conservation of mass givesContinuous-wave Doppler echocardiography supplies the jet speed. The convergence radius and jet speed must refer to matching times. The hemisphere is an assumption about flow geometry, not a property guaranteed by the instrument.^[ref]

For an original model calculation, let $r=0.8$ cm, $v_a=30$ cm/s and $u_j=500$ cm/s. Then $Q_R\simeq 120.64$ mL/s and $A_{\mathrm{eff}}\simeq 0.2413\mathrm{c}{\mathrm{m}}^2$. At otherwise fixed inputs a ten-percent radius overestimate multiplies the inferred area by $1.1^2=1.21$. The radius-squared dependence explains why a small geometric measurement error can matter substantially.

An instantaneous effective area and an integrated regurgitant volume answer different questions:If the effective area is constant, this becomes $V_R=A_{\mathrm{eff}}\mathrm{VTI}$ using the velocity-time integral. A stipulated VTI of 150 cm with the calculated constant area gives 36.19 mL. A single peak area generally cannot be pulled outside the integral when the leak changes during systole. Alternatively, $V_R/\mathrm{VTI}$ defines a velocity-weighted mean effective area over that interval. The technical report explicitly distinguishes instantaneous convergence measurements from temporally varying regurgitation.^[ref]

An orifice is a localized opening through which a fluid passes, often from a vessel into a region of lower pressure. A simple inviscid incompressible estimate for an opening of area $A$ iswhen the reservoir velocity and elevation change are negligible and $\Delta p$ is the driving pressure difference. The relation follows from a Bernoulli equation estimate of exit speed.

Real jets contract and dissipate energy, so a discharge coefficient $C_d$ is often introduced through $Q=C_d{Q}_{\mathrm{ideal}}$. The vena contracta can have a smaller area than the opening itself. Measuring collected fluid per unit time and the driving head tests the coefficient for a specified geometry and regime. For substantial gas compression or choked flow, the incompressible formula must be replaced by an appropriate compressible-flow model.

An ideal fluid issuing from a small hole a vertical depth $h$ below a free surface has speed

Here $h$ is the vertical distance from the free surface to the centre of the orifice, and $g$ is the positive magnitude of gravitational acceleration. The ideal prediction assumes incompressible flow, the same ambient pressure at the free surface and emerging jet, negligible velocity of the reservoir's free surface, and negligible viscous dissipation and surface tension effects. For a falling free surface, a quasi-steady approximation additionally assumes that the local outflow adjusts much faster than $h$ changes. It applies the Bernoulli equation at each instant, rather than claiming that the whole drainage is steady flow.^[ref]

A short proof uses a streamline connecting the reservoir's free surface at elevation $z_s$ to the emerging jet at elevation $z_o$. With constant mass density $\rho$, negligible surface speed and no mechanical energy losses, the Bernoulli equation givesEqual pressures cancel, leaving $v^2/2=gh$: a unit mass gains the same kinetic energy as a body in free fall through height $h$. This makes Torricelli's law a useful first estimate before the detailed velocity field is known. At one quarter of the original head, ideal jet speed is one half as large; it is not proportional to head.

Far from the accelerating outlet region, the reservoir has approximately hydrostatic pressure $p_{\mathrm{ambient}}+\rho gh$ at the outlet's elevation. The emerging jet instead has approximately ambient pressure when surface tension is negligible. Do not assign that bulk hydrostatic pressure to the emerging jet as well as adding the height difference: that would count the available mechanical energy twice. A sealed reservoir, a submerged outlet or significant surface tension changes the pressure conditions and requires a different balance.

Actual volumetric flow rate is commonly represented by a discharge coefficient:where $a$ is the geometric area of the orifice. This definition compares volumetric flow rates, not just speeds. If a nearly uniform jet at its vena contracta has area $a_j$ and speed $v_j$, define the two separate ratios $C_c=a_j/a$ and $C_v=v_j/\sqrt{2gh}$. Then $Q=a_j{v}_j$ gives $C_d=C_c{C}_v$. A contracted jet can consequently have a speed close to $\sqrt{2gh}$ while $C_d$ is appreciably below one. No particular $C_d$ is universal: calibrate it for the opening and operating range.

The vena contracta is the narrowest jet section after an opening. Its area may be smaller than the opening area, so the jet speed and volumetric flow rate require separate comparisons with ideal flow.