Borda–Carnot equation (Sudden contraction of a pipe)

Description

Borda–Carnot equation is an empirical description of the mechanical energy losses of the fluid due to a (sudden) flow expansion. It describes how the total head reduces due to the losses. In case of a sudden reduction of pipe diameter, without streamlining, the flow is not able to follow the sharp bend into the narrower pipe. As a result, there is flow separation, creating recirculating separation zones at the entrance of the narrower pipe. The main flow is contracted between the separated flow areas, and later on expands again to cover the full pipe area.

There is not much head loss between cross section 1, before the contraction, and cross section 3, the vena contracta at which the main flow is contracted most. But there are substantial losses in the flow expansion from cross section 3 to 2. These head losses can be expressed by using the Borda–Carnot equation, through the use of the coefficient of contraction μ=A3/A2. The energy loss ΔE per unit of fluid volume and due to the pipe contraction is depended on the three cross section areas and the The mean flow velocity at section 1.

Related formulas

Variables

ΔEThe fluid's mechanical energy loss per unit of fluid volume (J/m3)
ρDensity of the fluid (kg/m3)
μ coefficient of contraction = A3/A2 (dimensionless)
A1The cross-sectional area at section 1 (m2)
A2The cross-sectional area at section 2 (m2)
v1The mean flow velocity at section 1 (m/s)