Water moves through a constricted pipe in steady ideal flow...

Water moves through a constricted pipe in steady, ideal flow. At the lower point shown in the figure below, the pressure is 1.70 ✕ 105 Pa and the pipe radius is 2.70 cm. At the higher point located at y = 2.50 m, the pressure is 1.26 ✕ 105 Pa and the pipe radius is 1.70 cm. (a) Find the speed of flow in the lower section. (b) Find the speed of flow in the upper section. (c) Find the volume flow rate through the pipe.

Transcript

00:01 In this question, we have water moving through a pipe in a steady flow.

00:06 The pipe changes elevation as well as radius, and we are given the pressure at the lower part and at the upper part, and we're asked to find the speed of the flow in the lower and upper sections, as well as the volume flow through the pipe.

00:22 And i want to point out that the question references a figure, but there was no figure provided.

00:29 So i've taken my best shot at figuring out what this should look like.

00:36 And the only thing that would be perhaps incongruent or inconsistent would be if the lower end of the pipe were at a height other than zero.

00:47 So just keep that in mind as we are solving.

00:50 So to get the speed of the fluid flow in the lower section, there are two basic relationships that we apply for fluids questions.

01:00 One is bernoulli's equation or principle, which says that it's basically a statement of conservation of energy.

01:11 So p1 plus one half times the density of the fluid times v1 squared plus rho gh1 equals p2 plus one half rho v2 squared plus rho g, and actually we're going to call it y instead just to be consistent with the symbols given in the question.

01:50 So we'll go back and change that h1 to y1.

01:53 And then the other one is the continuity equation, which helps tell us about having consistent volume flow that a1 times v1 equals a2 times v2, where a represents the area of the pipe.

02:12 So the area of the pipe multiplied by its velocity at one part equals the area of the pipe times its velocity at another part, which tells us why when the area gets smaller as it does in this pipe, the fluid will speed up.

02:27 Okay, so what i'm noticing is without having v1 or v2, we can't use the continuity equation.

02:37 So we're going to need bernoulli's equation for part a to find the speed of the flow in the lower section, and then we can use the continuity equation for part b.

02:48 So in fact, let's denote that we're going to do bernoulli, it's going to be part a, continuity is going to be part b.

02:58 Okay, so for bernoulli's equation though, you're going to notice that it still has v1 and v2 in it.

03:11 So we are going to need to express v2 in terms of v1, and we are going to need the continuity equation to do that, but it's also applicable for part b.

03:30 So here's what we can do...

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