Continuity Equation 3D Simulator — Mass Balance & Branching Manifold Interactive

Interactive 3D continuity-equation simulator with an Equipment laboratory workbench (metered inlet reducer, three-outlet branching manifold, two delivery meters with a biasable indicator and a controlled leak path), a control-volume audit console, a Curves & measurements analysis tab with live charts and model equations, an Experiments tab with four guided fixtures and a model-verification bench, and a Learn & assess tab with lessons, a knowledge-check quiz and referenced scope notes.

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About the Continuity Equation 3D Simulator

This simulator applies the continuity equation (Q = AV, conservation of mass) to an inlet reducer feeding a three-outlet branching manifold with a controlled leak path. Adjust inlet flow, bore sizes, flow-split allocation, leak fraction and a deliberate branch-meter measurement bias to separate true physical mass balance from what instruments actually report.

What the simulator shows

• A real-time 3D cutaway workbench of the metered inlet and reducing spool, a transparent throat section, a three-outlet distribution manifold, delivery branch A with a biased flow indicator, delivery branch B with a reference meter, a controlled leak valve and drain cup, and a control-volume audit console, with home view, focus-selected-part, full-enclosure toggle, exploded view, auto-rotate, expand and show/hide labels controls. • Eight laboratory controls: imposed inlet flow (0–5 L/s), inlet bore (30–100 mm), reducer throat bore (15–60 mm), branch A bore (20–80 mm), branch B bore (20–80 mm), branch A share of non-leak flow (0–100%), inlet flow diverted to the leak outlet (0–30%) and branch A indicated-flow bias (−30% to +30% of true A flow). • Playback controls: pause/resume, 0.1 s and 1 s step advances, four speeds, plus 'Close leak outlet' and 'Clear branch A meter bias' quick actions. • A Curves & measurements tab with an inlet/A-flow/B-flow chart, a raw/corrected/true-residual chart, the full equation set (Q=AV, leak/split allocation, indicated-flow bias, raw vs. corrected vs. true residual) and snapshot readouts. • An Experiments tab with four guided fixtures (balanced delivery manifold, narrow throat, uncounted leak path, biased branch meter) and a Model verification bench with an independent-check run, timestamped event log and copyable trial report. • A Learn & assess tab with guided lessons, a knowledge-check quiz with reset, and a written model-scope statement with a reference link.

How the continuity equation governs the branching manifold

For steady incompressible flow, Q = AV holds at every unbranched section — narrowing the throat at fixed inlet flow raises velocity there without changing volume flow, as the 'Narrow throat' experiment demonstrates (throat velocity becomes four times its default value at a 15 mm bore). At the manifold, the leak fraction is removed from inlet flow first, then the remainder is split between branches A and B by the selected allocation — a prescribed allocation rather than a solved pressure-resistance network.

The true mass-balance residual, Qin − QA − QB − Qleak, is always exactly zero in this model, because the simulator enforces conservation directly rather than deriving it from a resistance network solution.

Separating true flow from instrument readings

The simulator distinguishes a raw residual (Qin minus indicated A and B flows), a corrected residual (raw residual minus the known leak), and the true physical residual (always zero). In the 'Uncounted leak path' experiment, a 20% leak fraction produces a nonzero raw residual that closes once the leak is included in the audit. In the 'Biased branch meter' experiment, a +20% indicator bias on branch A makes the measured balance nonzero even though true mass conservation is exact — the key lesson is that a zero measured residual does not by itself prove there is no leak, since instrument bias and an uncounted outlet can offset each other.

Frequently asked questions

What is the continuity equation?

The continuity equation, Q = AV, expresses conservation of mass for steady incompressible flow: volume flow equals cross-sectional area times mean velocity at every section. The simulator applies it through an inlet reducer and a branching three-outlet manifold.

Does narrowing a pipe section increase the total flow?

No. Continuity conserves volume flow Q through a reducer; only mean velocity increases in the narrower section. The "Narrow throat" experiment shows throat velocity rising fourfold at an unchanged 30 mm-bore default flow.

Does a zero measured flow balance prove there is no leak?

No. The "Biased branch meter" experiment shows a branch meter reading 20% high can offset a leak or otherwise make a measured balance appear closed, even though the simulator's true physical residual is exactly zero. Several measurement errors can mask real uncounted outlets.

How does the leak path affect the mass balance calculation?

The leak is modeled as an explicit third outlet. If excluded from the audit, inlet flow minus branches A and B shows an apparent residual equal to the leak flow; including the leak in the calculation closes the balance, as shown in the "Uncounted leak path" experiment.

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