Discharge Height Vertical Pipe

Overview

Engineering reference data for Discharge Height Vertical Pipe in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Main Formulas

The maximum discharge height (h) for water flowing from a vertical pipe can be calculated with the following formulas:

Imperial Units: $h = \left( \frac{Q}{5.68 \cdot k \cdot d^2} \right)^2$ where:

• h = discharge height (inches)
• Q = flow (gpm)
• d = pipe diameter (inches)
• k = coefficient (dimensionless, ranges 0.87 - 0.97)

Rearranged for Flow (Q): $Q = 5.68 \cdot k \cdot d^2 \cdot \sqrt{h}$

Example Calculation

To estimate flow from a vertical 4-inch pipe with a measured discharge height of 3 ft (36 inches) and a coefficient k = 0.97: $Q = 5.68 \times 0.97 \times (4\ \text{in})^2 \times \sqrt{36\ \text{in}}$ $Q \approx 529\ \text{gpm}$

Coefficient k

The coefficient $k$ is an empirical factor that accounts for flow characteristics. For this application, it typically ranges from *0.87 to 0.97. A value of *0.96 is often used as a default for calculations.

Unit Conversions

Key conversion factors for the Imperial system:

• $1\ \text{in} = 25.4\ \text{mm}$
• $1\ \text{gal (US)/min} = 6.30888 \times 10^{-5}\ \text{m}^3/\text{s}$
• $1\ \text{gal (US)/min} = 0.06309\ \text{dm}^3/\text{s}\ (\text{liters}/\text{s})$
• $1\ \text{gal (US)/min} = 0.227\ \text{m}^3/\text{h}$

References

• Original Source