# 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.

A question asks the following:

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. 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. Jun 3, 2018 at 11:29
• I'm curious about the textbook used. I have seen similar issues before. 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... Jun 11, 2018 at 20:14

## 2 Answers

I think this question is ill posed.

If both pipes are flowing full, the correct answer is 62.5 m/s from continuity. If there is an abrupt transition from the large to the small pipe, there will be a total pressure loss and Bernoulli's equation will not apply without a loss factor.

I think @sam is correct. To provide some intuition here, imagine that we have a pipe that does what is described in the problem but that had constant diameter throughout. In this case, the pressure at point 2 is 169 kPa gauge. If we increase the diameter of the pipe at point 2, the pressure increases. You can think of this as some of the velocity head of the fluid at point 1 becoming pressure head at point 2. If the diameter decreases, the pressure must go down. That is, some of the head of the fluid at point 1 gets converted to velocity head at point 2. As the problem is posed, there is not enough head at point 1 to achieve the velocity at point 2.

To fix this we would either have to increase the pressure at point 1 or increase the height difference.