Hydraulic Diameter of an Ellipse

The hydraulic diameter, abbreviated d_h, is used to calculate the Reynolds Number, friction factor or for calculating pressure drop across two points.  For most geometries, the hydraulic diameter, or characteristic length, is not equal to the diameter of the shape.  The calculation below is an example of the hydraulic diameter when the shape is an ellipse as a flowing area. This equation will also find the hydraulic diameter of a circular flowing area if the height and width are the same values.

 

Hydraulic Diameter of an Ellipse formula
\(\ d_h \;=\;  4 \cdot w \cdot h \; ( 64 - 16\; e^2 ) \;/\; ( w + h ) \; ( 64 - 3 \; e^4 )  \)
Symbol	English	Metric
\( d_h \) = Hydraulic Diameter	\( in \)	\( mm \)
\( w \) = Width of Ellipse	\( in \)	\( mm \)
\( h \) = Height of Ellipse	\( in \)	\( mm \)
\( e \) = Euler's Constant	\( 2.7182818284 ... \)	\( 2.7182818284 ... \)