M094 · Fluid mechanics

Draining Tank & Torricelli Law

Can a draining tank's falling water level show how far the outflow falls short of the ideal? Yes, through the discharge coefficient. How fast the level drops, times the tank's cross-section, gives the outflow rate. Divide that by the ideal Torricelli rate, the hole area times the speed of a fall from the surface to the hole. The result combines two losses: the jet narrows after leaving the hole, and it comes out slower than ideal.

Drain a cylindrical or a flared tank, raise the outlet and step through the drain in time. See how the falling level, the volume that leaves and the narrowing jet, which falls like a projectile, fit together. Then work out the discharge coefficient from the raw level readings.

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 $\text{Ready}$

Set the geometry, then measure

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 the horizontal outlet are both at the surrounding air pressure. Liquid height is measured from the tank bottom, and head is measured from the outlet centre. The contraction coefficient scales the jet area, the velocity coefficient scales the jet speed, and their product scales the outflow. The 3D jet narrows as it falls faster, keeping the volume flow the same, until it reaches the collection plane.

One volume balance, solved by separating variables, covers both cylindrical and flared tanks. Draining nominally stops when the level reaches the outlet centre, and the liquid below stays in the tank. The coefficients are constant and ignore Reynolds-number dependence, viscosity, pipe length, surface tension, vortices and jet breakup. Acquisition is blocked when the head is below five hole diameters or the hole area exceeds one percent of the liquid surface. These cutoffs are teaching checks, not rigorous error bounds that hold in general.

An acquisition records raw times and liquid levels. An integral that depends on the tank shape turns those readings into a straight-line fit, and the slope gives the discharge coefficient without looking at the reference coefficients. The levels reveal only the product of the two coefficients, not each one separately. Scrubbing through time keeps the data; changing the geometry, fluid, gravity, coefficients or noise clears it. The CSV holds the raw levels; it contains no separately computed ideal curve.

The outlet is treated as a small orifice in quasi-steady flow, with constant contraction and velocity coefficients. Pipe friction, surface tension and jet breakup are left out. States with low head or a large orifice are extrapolations: they are shown but cannot be acquired.

Start with the essentials

Focus question: Can liquid levels reveal outlet losses?
One-sentence intuition: The falling level reveals only the product of jet contraction and velocity loss, not each one separately.

Core mathematical model

Flow rate from jet area and jet speed

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

Two outlets with the same discharge coefficient can still have different jet areas and speeds.

Volume continuity

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

Both tanks obey the same rule: the outflow equals the liquid cross-section times the rate at which the level falls.

Head integral for the actual tank shape

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

The integral stays finite at zero head. With constant coefficients it changes linearly with 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}}

The fit gives only the product of the two coefficients. Levels alone cannot tell contraction apart from velocity loss.

The jet as a projectile path

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 jet shown is the quasi-steady streamline at that instant. Its flight time is negligible compared with the time the tank takes to drain.

Common difficulties

Acquire before inferring

Typical misconceptionThe estimate just copies the value on a reference slider.

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

Know the approximation

Typical misconceptionIf the apparatus looks smooth, every state it shows must be physically reliable.

Better mental modelCheck the domain status. Extrapolated states stay visible, but you cannot acquire data from them.

Run the experiment

01
Move through the drain
Drag the gold time handle from full tank toward the nominal end.
What to observe: The level falls, the outflow slows and the jet lands closer to the tank.
02
Change the tank shape
Compare cylindrical and flared tanks with the same initial level.
What to observe: The starting head is the same, but the stored volume and the draining time differ.
03
Infer losses
Acquire the drain sequence, add noise and export raw levels.
What to observe: In the head-integral coordinate the acquired levels fall on a straight line in time, and its slope gives the discharge coefficient.
04
Look near empty
Raise the outlet and drag the time handle into low head.
What to observe: Liquid below the outlet stays in the tank, and no data are acquired near empty.