M094 · Fluid mechanics

Draining Tank & Torricelli Law

Compare cylindrical and flared tanks, raise the outlet and scrub the drain time. Connect measured levels, conserved volume and a contracting ballistic jet; recover the discharge coefficient from raw observations.

Elapsed model time $\text{Ready}$

Nominal drain time $\text{Ready}$

Liquid height $\text{Ready}$

Head above outlet $\text{Ready}$

Outlet driving pressure $\text{Ready}$

Horizontal jet velocity $\text{Ready}$

Volume flow rate $\text{Ready}$

Drained volume $\text{Ready}$

Nominal retained volume $\text{Ready}$

Jet landing distance $\text{Ready}$

Reference discharge coefficient $\text{Ready}$

Orifice / surface area $\text{Ready}$

Small-head screening boundary $\text{Ready}$

Acquired stations $\text{Ready}$

Discharge coefficient from raw data $\text{Ready}$

Fit coordinate residual $\text{Ready}$

Approximation status $\text{Ready}$

GEOMETRY / PROBE / ACQUIRE

Can liquid levels reveal outlet losses?

Tank bottom radius

R_0

0.30\,m

Wall flare

f

0.00\,

Initial liquid height

z_0

1.20\,m

Outlet height above bottom

z_o

0.15\,m

Orifice diameter

d

20.00\,mm

Jet area contraction coefficient

C_c

0.62\,

Jet velocity coefficient

C_v

0.98\,

Time / nominal drain time

t/T

0.00\,

Reference fluid density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Level noise bound

\epsilon_z

0.00\,mm

The open tank and horizontal outlet share ambient pressure. Fluid height is measured from the bottom; head is above the outlet centre. Contraction changes jet area, the velocity coefficient changes jet speed, and their product changes transport. The 3D jet narrows as falling speed grows to conserve volume flow, then reaches the collection plane.

A separable volume-balance solution handles both cylindrical and flared tanks; nominal draining stops at outlet-centre height, retaining liquid below it. Constant coefficients omit Reynolds dependence, viscosity, pipe length, surface tension, vortices and breakup. Acquisition excludes head below five hole diameters and orifice area above one percent of the liquid surface. These teaching screens are not rigorous universal error bounds.

Acquire raw time and liquid-level readings. A geometry-dependent integral transforms those readings for an affine fit; its slope estimates discharge coefficient without consulting the reference coefficients. Only their product is identifiable from levels. Time scrubbing preserves data; geometry, fluid, gravity, coefficient or noise changes clear them. CSV contains raw levels, not an independently synthesized ideal reconstruction.

Quasi-steady small-orifice model with constant contraction and velocity coefficients. No pipe friction, surface tension or jet breakup. Low-head and large-orifice states are extrapolations, excluded from acquisition.

Start with the essentials

Focus question: Can liquid levels reveal outlet losses?
One-sentence intuition: Separate jet area, speed and their product before inferring transport.

Core mathematical model

Area and velocity are separate

Q=C_c A_o C_v\sqrt{2gh},\qquad C_d=C_cC_v

Equal discharge coefficients can have different jet areas and velocities.

Volume continuity

A(z)\frac{dz}{dt}=-Q,\quad A(z)=\pi R_0^2\!\left(1+\frac{fz}{H}\right)^2

The cylindrical and flared vessels use the same local volume balance.

Integrate head with the actual vessel

J(h)=\int_0^h\frac{A(z_o+u)}{\sqrt{u}}\,du

This integral is finite at zero head; constant coefficients make it affine in time.

Fit the acquired levels

-J(z_i-z_o)=\alpha t_i+\beta,\quad\widehat C_d=\frac{\widehat\alpha}{A_o\sqrt{2g}}

Only the product is inferred; levels alone cannot separate contraction and velocity loss.

A quasi-steady ballistic jet

x=v_j\tau,\quad y=y_o-\frac{g\tau^2}{2},\quad A_j(\tau)=\frac{Q}{\sqrt{v_j^2+(g\tau)^2}}

The displayed jet is an instantaneous quasi-steady streamline, with negligible flight time compared with draining.

Common difficulties

Acquire before inferring

Typical misconceptionThe estimate repeats a reference slider.

Better mental modelClear the observations: the estimate disappears. Add noise or change the reference parameter, then acquire again.

Know the approximation

Typical misconceptionA smooth apparatus proves every displayed state is physically reliable.

Better mental modelRead the domain status; extrapolated states remain visible but cannot be acquired.

Run the experiment

01
Scrub the experiment
Drag the gold time handle from full tank toward the nominal end.
What to observe: Level falls, outflow slows and the jet landing point moves inward.
02
Change storage geometry
Compare cylindrical and flared tanks with the same initial level.
What to observe: Initial head agrees; stored volume and draining time differ.
03
Infer losses
Acquire the drain sequence, add noise and export raw levels.
What to observe: The acquired coordinate slope estimates the discharge coefficient.
04
Watch the boundary
Raise the outlet and scrub into low head.
What to observe: Liquid remains below the outlet; near-empty acquisition is excluded.