重力流曼寧公式

概述

Manning's formula is the standard empirical equation for calculating cross-sectional average velocity and flow rate in open channels and partially filled conduits. It relates discharge to channel geometry (hydraulic radius), bed slope, and surface roughness through the Manning coefficient. The formula is widely applied in civil and environmental engineering for stormwater, sewer, and irrigation channel design.

關键公式

Mean velocity:

$v = \frac{k_n}{n} \, R_h^{2/3} \, S^{1/2}$

Volume flow rate:

$Q = A \cdot v = \frac{k_n}{n} \, A \, R_h^{2/3} \, S^{1/2}$

Hydraulic radius:

$R_h = \frac{A}{P_w}$

where $A$ is the cross-sectional flow area and $P_w$ is the wetted perimeter.

The unit conversion factor is:

變量

Manning Roughness Coefficients

計算器

Half-Filled Circular Pipe Calculator

The original source discusses partially filled circular pipes. For a half-filled circular conduit, the flow area is one half of the full circular area and the wetted perimeter is one half of the circumference, so the hydraulic radius is D/4.

單位換算器

Worked Example: Half-Circle Channel

A concrete half-circle channel (100 % filled) has the following specifications:

• Diameter: 500 mm (0.5 m)
• Material: Concrete, $n = 0.012$
• Slope: 1/100 m/m

Step 1 — Cross-sectional area:

$A = \frac{1}{2}\,\pi\!\left(\frac{D}{2}\right)^2 = \frac{1}{2}\,\pi\!\left(0.25\right)^2 = 0.0982 \text{ m}^2$

Step 2 — Wetted perimeter:

$P_w = \pi\!\left(\frac{D}{2}\right) = \pi \times 0.25 = 0.785 \text{ m}$

Step 3 — Hydraulic radius:

$R_h = \frac{A}{P_w} = \frac{0.0982}{0.785} = 0.125 \text{ m}$

Step 4 — Mean velocity:

$v = \frac{1.0}{0.012}\,(0.125)^{2/3}\,(0.01)^{1/2} = 2.09 \text{ m/s}$

Step 5 — Volume flow rate:

$Q = 0.0982 \times 2.09 = 0.205 \text{ m}^3\text{/s}$

原始源圖

The following original source images are preserved to avoid losing visual reference material. When an image contains chart or tabular data, its extracted values are represented in the page tables, calculators, or interactive charts; remaining images are retained as visual source references.

Interactive Source Diagram Data

The source velocity diagram is represented below with Manning velocity curves for common slopes. The curves use $n = 0.012$ and the same relationship shown in the source diagram:

$v = \frac{1}{n} R_h^{2/3} S^{1/2}$

The open-channel diagram basis is also represented as geometry data for a half-circle conduit. At 100% half-circle fill, $A = \pi D^2 / 8$, $P_w = \pi D / 2$, and $R_h = D / 4$.

工程要點

• Manning's equation is empirical and best suited for uniform, steady, turbulent flow in prismatic channels. Avoid applying it to rapidly varied flow or pressurised systems.
• The Manning coefficient $n$ is not dimensionless, though it is typically reported without units. Its implied dimensions are s/m^(1/3) in SI.
• $n$ values are the same numerically in SI and Imperial systems; only the leading conversion factor $k_n$ changes (1.0 vs. 1.486).
• Typical $n$ values range from about 0.010 (smooth, manufactured surfaces) to 0.060 (channels with heavy vegetation and debris).
• For partially filled circular pipes, calculate the wetted area and wetted perimeter as functions of fill depth before applying Manning's equation.
• Accuracy degrades for very shallow flows or slopes steeper than roughly 10 % — consider alternative methods in those regimes.
• Always verify results against local design codes and standards for critical infrastructure.

參考资料

• Engineering ToolBox — Manning's Formula Gravity Flow