Terminate a draining solution at its physical endpoint
For a large tank with surface area Atank and a small outlet, a quasisteady Torricelli model gives outflow Cd a sqrt(2gH). Mass continuity then gives Atank dH/dt = −Cd a sqrt(2gH). The discharge coefficient is stipulated, and surface kinetic energy is neglected under the disclosed small outlet-to-tank area ratio.
Learning outcome: Derive the separated solution, identify its endpoint and compare Euler event times on two grids without hiding the signed error.
Starting recap
Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
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At outlet area 0.001 m², exact emptying time is about 74.5356 s. Refining the Euler grid changes its numerical event error, not this analytic target. A changed discharge coefficient would instead change the model itself.
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Why must the squared exact-height expression stop at emptying?
Experiment
Derive the separated solution, identify its endpoint and compare Euler event times on two grids without hiding the signed error.
Integrate a draining-height law without stepping below the tank bottom
Worked teaching model · synthetic values · Large tank area.1 m²,initial height1 m,orifice discharge coefficient.6,g=10. Quasisteady Torricelli outflow with inlet-surface kinetic energy neglected. Small orifice/tank area ratio.01–.03; emptying ends the model. No physical vessel or plumbing.
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Exact draining event and bounded numerical steps
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: exact until empty (solid).2: Euler with within-step zero event (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Exact draining event and bounded numerical steps
exact until empty
| time (s) | liquid height (m) |
|---|---|
| 0 | 1 |
| 0.4658475 | 0.9751563 |
| 0.931695 | 0.950625 |
| 1.397542 | 0.9264063 |
| 1.86339 | 0.9025 |
| 2.329237 | 0.8789063 |
| 2.795085 | 0.855625 |
| 3.260932 | 0.8326562 |
| 3.72678 | 0.81 |
| 4.192627 | 0.7876562 |
| 4.658475 | 0.765625 |
| 5.124322 | 0.7439063 |
| 5.59017 | 0.7225 |
| 6.056017 | 0.7014063 |
| 6.521865 | 0.680625 |
| 6.987712 | 0.6601563 |
| 7.45356 | 0.64 |
| 7.919407 | 0.6201562 |
| 8.385255 | 0.600625 |
| 8.851102 | 0.5814062 |
| 9.31695 | 0.5625 |
| 9.782797 | 0.5439063 |
| 10.24864 | 0.525625 |
| 10.71449 | 0.5076563 |
| 11.18034 | 0.49 |
| 11.64619 | 0.4726563 |
| 12.11203 | 0.455625 |
| 12.57788 | 0.4389062 |
| 13.04373 | 0.4225 |
| 13.50958 | 0.4064062 |
| 13.97542 | 0.390625 |
| 14.44127 | 0.3751563 |
| 14.90712 | 0.36 |
| 15.37297 | 0.3451562 |
| 15.83881 | 0.330625 |
| 16.30466 | 0.3164063 |
| 16.77051 | 0.3025 |
| 17.23636 | 0.2889062 |
| 17.7022 | 0.275625 |
| 18.16805 | 0.2626562 |
| 18.6339 | 0.25 |
| 19.09975 | 0.2376562 |
| 19.56559 | 0.225625 |
| 20.03144 | 0.2139063 |
| 20.49729 | 0.2025 |
| 20.96314 | 0.1914062 |
| 21.42898 | 0.180625 |
| 21.89483 | 0.1701562 |
| 22.36068 | 0.16 |
| 22.82653 | 0.1501562 |
| 23.29237 | 0.140625 |
| 23.75822 | 0.1314063 |
| 24.22407 | 0.1225 |
| 24.68992 | 0.1139062 |
| 25.15576 | 0.105625 |
| 25.62161 | 0.09765625 |
| 26.08746 | 0.09 |
| 26.55331 | 0.08265625 |
| 27.01915 | 0.075625 |
| 27.485 | 0.06890625 |
| 27.95085 | 0.0625 |
| 28.4167 | 0.05640625 |
| 28.88254 | 0.050625 |
| 29.34839 | 0.04515625 |
| 29.81424 | 0.04 |
| 30.28009 | 0.03515625 |
| 30.74593 | 0.030625 |
| 31.21178 | 0.02640625 |
| 31.67763 | 0.0225 |
| 32.14348 | 0.01890625 |
| 32.60932 | 0.015625 |
| 33.07517 | 0.01265625 |
| 33.54102 | 0.01 |
| 34.00687 | 0.00765625 |
| 34.47271 | 0.005625 |
| 34.93856 | 0.00390625 |
| 35.40441 | 0.0025 |
| 35.87026 | 0.00140625 |
| 36.3361 | 0.000625 |
| 36.80195 | 0.00015625 |
| 37.2678 | 0 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
Euler with within-step zero event
| time (s) | liquid height (m) |
|---|---|
| 0 | 1 |
| 1.86339 | 0.9 |
| 3.72678 | 0.8051317 |
| 5.59017 | 0.7154025 |
| 7.45356 | 0.6308211 |
| 9.31695 | 0.5513968 |
| 11.18034 | 0.4771407 |
| 13.04373 | 0.4080654 |
| 14.90712 | 0.3441854 |
| 16.77051 | 0.285518 |
| 18.6339 | 0.2320842 |
| 20.49729 | 0.183909 |
| 22.36068 | 0.1410244 |
| 24.22407 | 0.1034712 |
| 26.08746 | 0.07130427 |
| 27.95085 | 0.04460141 |
| 29.81424 | 0.02348236 |
| 31.67763 | 0.008158406 |
| 33.36072 | 0 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Exact emptying time | 37.2678 | s |
| Euler detected emptying time | 33.36072 | s |
| Euler emptying-time error | -3.907084 | s |
| Euler time step | 1.86339 | s |
| Orifice to tank area ratio | 0.02 | ratio |
Emptying is a domain endpoint, not permission to square a negative height root and refill the tank.
Continuity gives Atank dH/dt=−Cd a sqrt(2gH). Integrating yields sqrtH=sqrtH0−kt/2 only until it reaches zero, at2sqrtH0/k. Euler predicts decreasing positive heights; if its next step would be negative, interpolate to zero within that step and terminate. That numerical event time is approximate and is compared with the exact analytic event, not hidden by replacing it.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Emptying is a domain endpoint, not permission to square a negative height root and refill the tank.
Continuity gives Atank dH/dt=−Cd a sqrt(2gH). Integrating yields sqrtH=sqrtH0−kt/2 only until it reaches zero, at2sqrtH0/k. Euler predicts decreasing positive heights; if its next step would be negative, interpolate to zero within that step and terminate. That numerical event time is approximate and is compared with the exact analytic event, not hidden by replacing it.
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Which observation would test the assumed discharge model rather than the numerical step size?
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