Torricelli's Theorem Calculator

Derivation from Bernoulli's Principle

Torricelli's theorem states that the exit velocity of a fluid from a small hole in a tank is:

v = √(2gh)

Where: - v = exit velocity (m/s) - g = acceleration due to gravity (9.8 m/s²) - h = height of fluid surface above exit (m)

**Derivation from Bernoulli's Equation** Applying Bernoulli's principle between the surface (point 1) and exit (point 2):

P_1 + (1)/(2)\rho v_1^2 + \rho gh_1 = P_2 + (1)/(2)\rho v_2^2 + \rho gh_2

**Assumptions** - **Large tank**: v_1 ≈ 0 (surface velocity negligible) - **Same pressure**: P_1 = P_2 = P_atm (both exposed to atmosphere) - **Small hole**: Exit area much smaller than tank area - **Steady flow**: Constant conditions

**Simplification** With v_1 = 0, P_1 = P_2, and h_1 - h_2 = h:

\rho gh = (1)/(2)\rho v^2

Solving for velocity:

v = √(2gh)

**Physical Interpretation** - Exit velocity equals **free-fall velocity** from height h - All potential energy converts to kinetic energy - Independent of fluid density - Depends only on height and gravity

v = √(2gh)

Energy Conservation Perspective

Torricelli's theorem can be derived from energy conservation:

**Potential Energy** At the surface:

PE = mgh

**Kinetic Energy** At the exit:

KE = (1)/(2)mv^2

**Conservation** Assuming no losses:

mgh = (1)/(2)mv^2

Solving:

v = √(2gh)

**Key Points** - **Energy conversion**: Potential → Kinetic - **No losses**: Ideal case (no friction) - **Same as free fall**: Identical to dropping from height h - **Independent of mass**: Mass cancels out

**Real-World Considerations** - **Friction losses**: Reduce actual velocity - **Vena contracta**: Effective area smaller than hole - **Viscous effects**: Energy dissipation - **Turbulence**: Additional losses

Flow Rate and Discharge

The volumetric flow rate through the hole:

**Flow Rate**

Q = Av = A√(2gh)

where A is the exit area.

**Key Observations** - Flow rate **decreases** as tank empties - Proportional to **square root of height** - Depends on **hole area** - Independent of **fluid density**

**Time to Empty** For a cylindrical tank:

t = \fracA_tankA_hole√(\frac2h_0)g

where h_0 is initial height.

**Practical Applications** - **Tank design**: Sizing drain holes - **Flow control**: Regulating discharge - **Drainage systems**: Designing outlets - **Irrigation**: Water distribution

Limitations and Corrections

Real-world deviations from Torricelli's theorem:

**1. Vena Contracta** - Effective area smaller than hole - Contraction coefficient C_c ≈ 0.61-0.64 - Actual velocity: v = C_v√(2gh) - Discharge coefficient: C_d = C_c C_v

**2. Viscous Losses** - Friction reduces velocity - More significant for small holes - Viscous fluids affected more

**3. Surface Tension** - Small holes affected - Creates additional resistance - More important for small scales

**4. Turbulence** - Energy dissipation - Reduces efficiency - Creates losses

**5. Non-Steady Flow** - Height changes with time - Velocity decreases as tank empties - Requires integration for exact solution

Applications in Engineering

Torricelli's theorem finds many applications:

**1. Tank Design** - Sizing drain holes - Determining flow rates - Designing outlets - Emergency drainage

**2. Water Systems** - Reservoir design - Irrigation systems - Drainage networks - Water distribution

**3. Chemical Processing** - Tank emptying - Batch operations - Flow control - Process design

**4. Hydraulics** - Flow calculations - System design - Pressure analysis - Energy considerations

Historical Context

Torricelli's theorem:

**Discovery** - Discovered by Evangelista Torricelli (1608-1647) - Italian physicist and mathematician - Student of Galileo - Inventor of barometer

**Significance** - Early application of energy conservation - Connected fluid mechanics to mechanics - Foundation for Bernoulli's principle - Important in hydraulics

**Modern Relevance** - Still widely used - Foundation for many calculations - Teaching tool for energy concepts - Practical engineering applications

Worked Example

Inputs

• height: 2

Result: Exit Velocity: 6.26 m/s

Explanation

For a fluid height h = 2 m:

Calculate exit velocity using Torricelli's theorem: v = √(2gh) v = √(2 × 9.8 × 2) v = √(39.2) v ≈ 6.26 m/s

This is the velocity at which fluid exits the tank, equal to the free-fall velocity from 2 m height.

Second Scenario

Inputs

• height: 1.5

Result: Exit Velocity: 6.26 m/s

Explanation

This scenario uses different inputs (height = 1.5) to show how changing one variable affects the torricelli's theorem result. Run the calculator above with these values to get the exact updated output with step-by-step work.