# How Rischio calculates an open channel

The units, conventions and assumptions of Rischio's open channel calculation, and its method in order, from the section to the bank-full capacity.

Rischio calculates an open channel from one cross-section, its longitudinal slope $S$ and the discharge $Q$; the [overview](https://rischio.io/docs/open-channel/) says what it reports and what it checks.

## Units and conventions

-   SI units: m, m², m³/s and m/s. The slope is in m/m: 0.002 is 0.2 %. $g = 9.81$ m/s².
-   The section is a polyline of vertices $(x, z)$, drawn from bank to bank, left to right. $x$ never decreases; two vertices with the same $x$ make a vertical wall.
-   Each segment, from one vertex to the next, has its own Manning $n$: $N$ vertices take $N - 1$ values of $n$.
-   Water levels are elevations on the datum of $z$. Depths are measured from the bed, the lowest vertex.

## Assumptions

-   Steady, uniform flow in a prismatic channel: nothing changes along the reach, and the friction slope equals $S$.
-   Where the roughness changes, the section is divided into parts, and each carries its own share: the divided channel method, as HEC-RAS does. A side steeper than 1 vertical to 5 horizontal joins the part it runs down to, so a lined channel with grassed sides stays one part.
-   Every part of the section below the water level carries flow, even a low area cut off by higher ground.

## Method

At a given water level, the wet segments make the section, [divided into parts](https://rischio.io/docs/open-channel/theory/section/) where its roughness changes. Manning’s equation gives each part’s discharge, and the [normal depth](https://rischio.io/docs/open-channel/theory/flow/) is the depth at which their sum equals $Q$; then come the Froude number, the critical depth and the specific energy. Last, Manning’s discharge with the water at the lower bank is the [bank-full capacity](https://rischio.io/docs/open-channel/theory/capacity/) $Q_{bank}$, the limit of the one check, and the rating curve gives the discharge from the bed up to that bank. The [equations](https://rischio.io/docs/open-channel/equations/) go through every step, in the order of the calculation report.

## References

-   Chow, V. T. (1959). *Open-Channel Hydraulics*. McGraw-Hill.
-   Horton, R. E. (1933). Separate roughness coefficients for channel bottom and sides. *Engineering News-Record*, 111(22), 652–653.
-   Einstein, H. A. (1934). Der hydraulische oder Profil-Radius. *Schweizerische Bauzeitung*, 103(8), 89–91.
-   Brent, R. P. (1973). *Algorithms for Minimization without Derivatives*. Prentice-Hall.
-   U.S. Army Corps of Engineers. *HEC-RAS Hydraulic Reference Manual*: cross section subdivision for conveyance calculations, and composite Manning’s n for the main channel.

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Source: https://rischio.io/docs/open-channel/theory/