Hydraulic Design of Standard Culvert (3) Pipe culverts

A culvert is a covered channel of relatively short length designed to pass water through an embankment (e.g. highway, railroad and dam). It is a hydraulic structure. From a hydraulic aspect, a dominant feature of a culvert is whether it runs full or not.
The hydraulic design of a culvert is basically the selection of an optimum compromise between discharge capacity and head loss or afflux, and of course the total costs of the structure (Chanson 2004a, pp. 440-451). The latter encompassing the design, construction, maintenance & operation, environmental, upstream flooding, risks of failure…
The basic design process takes place in two stages. First, a system analysis must be carried out to determine the objectives of the culvert, the design data, the constraints ... Second, the barrel size is selected by test-and-trial procedure. Both inlet-control and outlet-control calculations are performed for Q = Qdes. The optimum size is smallest barrel size allowing for inlet control operation at design discharge Qdes for the maximum acceptable afflux.
In a free-surface flow in a circular pipe, the free-surface width B, cross-section area A, wetted perimeter Pw, and hydraulic diameter DH are functions of the filling ratio d/D. The optimum design of a standard pipe culvert follows the same steps as for a standard box culvert.
The hydraulic design of culverts is a seminal application of the basic principles of fluid mechanics and hydraulic engineering to hydraulic structures. These are discussed in a number of relevant Youtube video movies in the same Playlist at:
{   / @hubert_chanson  }
Finally, the design of a culvert that is intended to be constructed would require the certification of a professional civil engineer.
 
Fluid mechanics and hydraulics in Hubert Chanson Youtube channel {   / @hubert_chanson  }
Fundamentals of open channel hydraulics [Playlist]
Advanced hydraulics of open channel flows [Playlist]
Environmental hydraulics of open channel flows [Playlist]

Hydraulic Design of Standard Culvert (1) Introduction
Hydraulic Design of Standard Culvert (2) Box culverts
Hydraulic Design of Standard Culvert (3) Pipe culverts
Hydraulic Design of Standard Culvert (4) high tailwater conditions (dtw larger than D)

Acknowledgements
Concrete Pipe Association of Australasia
Professor Colin APELT

References
Concrete Pipe Association of Australasia (1991). "Hydraulics of Precast Concrete Conduits." Jenkin Buxton Printers, Australia, 3rd edition, 72 pages.
Concrete Pipe Association of Australasia (2012). "Hydraulics of Precast Concrete Conduits." CPAA Design Manual, Australia, 64 pages.
CHANSON, H. (2004a). "The Hydraulics of Open Channel Flow: An Introduction." Butterworth-Heinemann, 2nd edition, Oxford, UK, 630 pages (ISBN 978 0 7506 5978 9).
CHANSON, H. (2004b). "Environmental Hydraulics of Open Channel Flows." Elsevier-Butterworth-Heinemann, Oxford, UK, 483 pages (ISBN 978 0 7506 6165 2).
CHANSON, H., and LENG, X. (2021). "Fish Swimming in Turbulent Waters. Hydraulics Guidelines to assist Upstream Fish Passage in Box Culverts." CRC Press, Taylor and Francis Group, Leiden, The Netherlands, 202 pages and 19 video movies (DOI: 10.1201/9781003029694) (ISBN 978-0-367-46573-5 [Hardback]; 978-1-003-02969-4 [e-book]).