Bernoulli's Equation — Question 6

Question 6

A siphon is limited by absolute crest pressure

A primed siphon of uniform tube area draws water from a large open reservoir to an atmospheric outlet $d=2.0\,\mathrm{m}$ below its free surface. The crest is $H=6.0\,\mathrm{m}$ above the free surface. Assume steady ideal flow, negligible reservoir speed and no losses. Take $p_{\mathrm{atm}}=100\,\mathrm{kPa}$, $p_v=2.0\,\mathrm{kPa}$, $\rho=1000\,\mathrm{kg/m^3}$ and $g=10\,\mathrm{m/s^2}$.

1. Find tube speed and crest absolute pressure.

2. Find the largest crest height permitted by the condition $p_{\mathrm{crest}}\ge p_v$. What happens to the assumed single-phase solution if the crest is raised to $8.0\,\mathrm{m}$?

Question 6 - Solution

A siphon is limited by absolute crest pressure

(a) Derivation and calculation

From reservoir surface to outlet,$$\frac{v^2}{2g}=d\Rightarrow\boxed{v=\sqrt{2gd}=\sqrt{40}=6.32456\,\mathrm{m/s}}.$$$$p_c+\tfrac12\rho v^2+\rho gH=p_{\mathrm{atm}}.$$$$\boxed{p_c=p_{\mathrm{atm}}-\rho g(H+d)=100000-1000(10)(8)=20000\,\mathrm{Pa}}.$$

(b) Result and interpretation

$$p_c\ge p_v\Rightarrow\boxed{H\le\frac{p_{\mathrm{atm}}-p_v}{\rho g}-d=7.80\,\mathrm m}.$$$$H=8.00:\quad p_c=100000-1000(10)(10)=0<p_v.$$The proposed continuous single-phase solution is inadmissible; vapor formation invalidates its assumptions. Losses can lower the usable limit.

Physical check

Absolute crest pressure must be checked; a negative gauge pressure alone does not imply failure.