# Can the continuity equation and Bernoulli contradict each other?

Please bear with me - I'm a lapsed mathematician and I'm self-studying these concepts.

Water flows in a pipe of diameter 5 m at a velocity of 10 m/s. It then flows down into a smaller pipe of diameter 2 m. The height between the centre of pipe sections is 5 m. The density is assumed to be uniform over the cross sections. The gauge pressure at Boundary 1 is 120 kPa. Calculate the velocity at the smaller pipe section.

There is no reason I can see to assume mass flow continuity doesn't apply, and using $v_1A_1 = v_2A_2$ one obtains $v_2 = 62.5$ m/s. However, using the Bernoulli equation while assuming atmospheric pressure at the smaller section, one gets $$\frac{p_1}{\rho} + \frac{1}{2}v_1^2 + + gz_1 = \frac{p_2}{\rho}+\frac{1}{2}v_2^2 + gz_2$$ $$120+ 50 + 5g = 0 + \frac{1}{2}v_2^2 + 0$$ $$v_2 = 20.93,$$ and in fact this is what the textbook answer gives. I am quite confused as to why mass continuity applies in other situations, even with changes in pressure, but doesn't seem to apply here.

My question is: is this textbook question then ill-posed? I feel as though by providing too much information about the pipe section without checking the calculations, the question is bound to create a contradiction. The 5m/2m diameters don't actually make it into the final answer.

• Consider a fixed volume system. In that, mass entered - mass exited = mass stored. Commented Jun 3, 2018 at 11:24
• Also, in bernouli equation, avoid using gauge pressures. Use absolute pressure. At first time reading question i saw 0 pressure on RHS that was fishy. Commented Jun 3, 2018 at 11:29
• I'm curious about the textbook used. I have seen similar issues before. Commented Jun 9, 2018 at 18:17
• Hi @SalomonTurgman - it's from Renewable Energy Engineering by Jenkins, Ekanayake. It's one of a few issues I've found - not all massive, but certainly in the problems and exercises, one would hope there weren't glaring contradictions... Commented Jun 11, 2018 at 20:14