Hydraulic Diameter

The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channel length, it is defined as :

D_\text = \frac,

where : is the

cross-sectional Cross-sectional data, or a cross section of a study population, in statistics and econometrics, is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. The anal ...

area of the flow, : is the

wetted perimeter The wetted perimeter is the perimeter of the cross sectional area that is "wet". The length of line of the intersection of channel wetted surface with a cross sectional plane normal to the flow direction. The term wetted perimeter is common in ci ...

of the cross-section. More intuitively, the hydraulic diameter can be understood as a function of the hydraulic radius , which is defined as the cross-sectional area of the channel divided by the wetted perimeter. Here, the wetted perimeter includes all surfaces acted upon by shear stress from the fluid. :

R_\text = \frac,

D_\text = 4R_\text,

Note that for the case of a circular pipe, :

D_\text =\frac=2R

The need for the hydraulic diameter arises due to the use of a single dimension in case of

dimensionless quantity A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

such as

Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...

, which prefer a single variable for flow analysis rather than the set of variables as listed in the table below.^{his sentence needs work/i>} The
Manning formula The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
contains a quantity called the
hydraulic radius The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
. Despite what the name may suggest, the hydraulic diameter is ''not'' twice the hydraulic radius, but four times larger. Hydraulic diameter is mainly used for calculations involving
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
.
Secondary flow In fluid dynamics, flow can be decomposed into primary plus secondary flow, a relatively weaker flow pattern superimposed on the stronger primary flow pattern. The primary flow is often chosen to be an exact solution to simplified or approximated ...
s can be observed in non-circular ducts as a result of turbulent
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
in the turbulent flow. Hydraulic diameter is also used in calculation of heat transfer in internal-flow problems.

Non-uniform and non-circular cross-section channels
In the more general case, channels with non-uniform non-circular cross-sectional area, such as the
Tesla valve A Tesla valve, called a valvular conduit by its inventor, is a fixed-geometry passive check valve. It allows a fluid to flow preferentially in one direction, without moving parts. The device is named after Nikola Tesla, who was awarded in 192 ...
, the hydraulic diameter is defined as: : $D_\text = \frac,$ where : is the total wetted volume of the channel, : is the total wetted surface area. This definition is reduced to $\frac$ for uniform non-circular cross-section channels, and $2R$ for circular pipes.

List of hydraulic diameters
For a fully filled duct or pipe whose cross-section is a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
, the hydraulic diameter is equivalent to the diameter $D$ of a circle
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
within the
wetted perimeter The wetted perimeter is the perimeter of the cross sectional area that is "wet". The length of line of the intersection of channel wetted surface with a cross sectional plane normal to the flow direction. The term wetted perimeter is common in ci ...
. This can be seen as follows: The $N$ -sided regular polygon is a union of $N$ triangles, each of height $D/2$ and base $B = D \tan(\pi/N)$ . Each such triangle contributes $BD/4$ to the total area and $B$ to the total perimeter, giving : $D_\text = 4\frac = D$ for the hydraulic diameter.

References
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See also
*
Equivalent spherical diameter The equivalent spherical diameter of an irregularly shaped object is the diameter of a sphere of equivalent geometric, optical, electrical, aerodynamic or hydrodynamic behavior to that of the particle under investigation. The particle size of a pe ...
*
Hydraulic radius The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
*
Darcy friction factor Darcy, Darci or Darcey may refer to: Science * Darcy's law, which describes the flow of a fluid through porous material * Darcy (unit), a unit of permeability of fluids in porous material * Darcy friction factor in the field of fluid mechanics * ...
Fluid dynamics Heat transfer Hydrology Hydraulics