Chezy Formula

Overview

The Chezy formula is an early open channel flow equation relating velocity to hydraulic radius and channel slope. It is less commonly used than Manning's formula but historically significant.

Formula

$v = C \sqrt{R_h S}$

Calculator

Notes

• Results are approximate and should be verified for critical applications
• Input values should be within reasonable engineering ranges

Manning's Relationship for Chezy's Coefficient

The Chezy coefficient $C$ can be estimated using the empirical Manning relationship:

$C = \frac{1}{n} R_h^{1/6}$

where $n$ is the Manning coefficient of roughness, which depends on the channel material and condition. For example, for concrete conduits, a common value is $n = 0.02$ or directly $C = 50 \, \text{m}^{1/2}/\text{s}$.

Example: Flow in a Concrete Conduit

Consider a rectangular concrete conduit with dimensions $1 \, \text{m} \times 1 \, \text{m}$ and a slope $S = 1/100 \, \text{m/m}$. The conduit is half filled with water.

Step 1: Calculate the cross-sectional area $A$

The filled height is $0.5 \, \text{m}$, so:

$A = 0.5 \times 1 \, \text{m} \times 1 \, \text{m} = 0.5 \, \text{m}^2$

Step 2: Calculate the wetted perimeter $P$

For a rectangular channel half filled:

$P = 2 \times 0.5 \, \text{m} + 1 \, \text{m} = 2 \, \text{m}$

Step 3: Calculate the hydraulic radius $R_h$

$R_h = \frac{A}{P} = \frac{0.5 \, \text{m}^2}{2 \, \text{m}} = 0.25 \, \text{m}$

Step 4: Calculate the mean flow velocity $v$

Using Chezy's formula with $C = 50 \, \text{m}^{1/2}/\text{s}$:

$v = C \sqrt{R_h S} = 50 \sqrt{0.25 \times \frac{1}{100}} = 50 \times \sqrt{0.0025} = 50 \times 0.05 = 2.5 \, \text{m/s}$

Step 5: Calculate the volume flow rate $q$

$q = A v = 0.5 \, \text{m}^2 \times 2.5 \, \text{m/s} = 1.25 \, \text{m}^3/\text{s}$

This example illustrates the application of Chezy's formula in a practical scenario.

Factors Influencing Chezy's Coefficient

The Chezy coefficient $C$ is not a fixed material property. It is a function of the flow regime and channel characteristics, primarily determined by:

1. Flow Reynolds Number ($Re$): Indicates whether the flow is laminar or turbulent.
2. Relative Roughness ($\epsilon / R_h$): The ratio of the characteristic roughness element height ($\epsilon$) to the hydraulic radius ($R_h$).

For practical engineering applications, empirical relationships like Manning's formula are used to estimate $C$ based on measurable channel properties, as described in the Manning's Relationship for Chezy's Coefficient section.

References

• Original Source