Question 8..8: Under what conditions can you compute the pressure in the sy...

For flow along a streamline between points 2 and 4,

From overall mass balance,   \nu _{2}A_{2}=\nu _{4}A_{4}. For the pipe from point 2 to point 4,  A_{2}=A_{4};  therefore,   \nu _{2}=\nu _{4}.   With \nu _{2}=\nu _{4} and z_{2}=z_{4}, Bernoulli’s equation FIGURE 8.15 Flow from a reservoir through a segment of rigid tubing. suggests that the pressures p_{2} and p_{4} are equal. Because we discharge to atmospheric pressure at an assumed subsonic velocity, p_{4}=0 (gauge) and the pressure at point 2 is also predicted to be zero. Would we expect this to be the case particularly given that we are driving the flow only via a pressure gradient? Recall that Bernoulli should not be used over long distances. For flow along a streamline between points 1 and 2, assuming a rounded entrance at the chamber-tube interface,

where the pressure at point 1 is zero (gauge) and v_{1}\ll \nu _{2}  if    A_{1}\gg A_{2}; thus,

where \nu _{2} is nonzero and equal to \nu _{3} (which is measured via the flowmeter) by mass balance. We observe, therefore, that one application of Bernoulli suggests that p_{2}=0, whereas another yields P_{2}=\rho gH-½\rho \nu ^{2}_{2}. Bernoulli is applicable across contractions and over short distances.