---
title: "Draining Tank & Torricelli Law Interactive Lab | Sandbox Physics"
description: "Can a draining tank's falling water level show how far the outflow falls short of the ideal? Yes, through the discharge coefficient."
canonical_url: "https://sandboxphysics.com/labs/draining-tank/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/draining-tank/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## A level record contains transport losses

**Background** The tank holds a uniform liquid in constant gravity. All pressures are gauge pressures, measured relative to the surrounding air.

**Why it matters** Force balance and conservation of liquid volume predict how the tank drains. The question is how much of that you can recover from simulated sensor readings.

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 misconception** The estimate just copies the value on a reference slider.

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

#### Know the approximation

**Typical misconception** If the apparatus looks smooth, every state it shows must be physically reliable.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Acquired levels and the model reference curve answer different questions. Use either one only where the approximation holds.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)