Water at 20^∘ C flws through a 5 -cm-diameter pipe that has a 180^∘ vertical bend, as in Fig. P

step 1 Let's see here. My bad. Water at 20 degrees flows through a 5 -centimeter diameter pipe that has

Water at 20^∘ C flws through a 5 -cm-diameter pipe that has...

Water at $20^{\circ} \mathrm{C}$ flws through a 5 -cm-diameter pipe that has a $180^{\circ}$ vertical bend, as in Fig. $\mathrm{P} 3.43 .$ The total length of pipe between flanges 1 and 2 is $75 \mathrm{cm}$. When the weight flow rate is $230 \mathrm{N} / \mathrm{s}, p_{1}=165 \mathrm{kPa}$ and $p_{2}=134 \mathrm{kPa}$ Neglecting pipe weight, determine the total force that the flanges must withstand for this flow.

Key Concepts

Fluid Flow Through Curved Pipes

Fluid flow in pipes with bends or curves involves changes in the direction of momentum, which in turn produce additional reaction forces on the pipe walls and supports. Understanding these forces is vital in ensuring that components like flanges are designed to withstand the forces resulting from such directional changes in fluid flow.

Control Volume Analysis

Control volume analysis is a method in fluid mechanics where a specific region (or volume) is defined to apply conservation laws. It simplifies the complex behavior of fluid flow by focusing on the net effects of mass, momentum, and energy entering and leaving the volume, which is critical in determining forces in flowing systems.

Pressure Forces

Pressure forces are generated by the pressure exerted by a fluid on surfaces. In fluid mechanics, analyzing the resultant force on structures such as flanges requires understanding how pressure differences at various points in the system contribute to the overall force that must be resisted.

Conservation of Momentum in Fluid Flow

This concept involves applying Newton’s second law to a flowing fluid by considering a control volume. The net force acting on the fluid is equal to the difference in momentum flux between the inlet and outlet, which is essential when analyzing changes in fluid direction and speed in systems like pipe bends.

Transcript

00:01 Let's see here.

00:04 My bad.

00:07 Water at 20 degrees flows through a 5 -centimeter diameter pipe that has a 180 vertical bend, and i've shown that.

00:15 The total of the pipe between the flanges 1 and 2 is 75 centimeters.

00:21 When the weight flow rate is 230 newton per second, p1 equals 165 kilopascals, and p2 equals 134 kilos.

00:30 Back pascales.

00:32 Neglecting the pipe weight, determine the total force that the flanges must withstand.

00:39 Okay, so let's figure this out.

00:43 Where's my pen? there we go.

00:48 So you can go reference that if you need to.

00:50 And we're going to give 20 degrees.

01:00 So we're going to be at 20 degrees c.

01:04 And we know that our p1, p2, and our vertical bent.

01:09 Our weight flow rate and the total length.

01:14 We're going to use the momentum balance, and this will be for us, m times vq minus v1.

01:46 Okay, fx is the force on the flange, force on flange, and vq equals negative v1.

02:04 Then we'll have p1 minus p -aa plus p2 minus p -aa -a minus f -x, that will equal negative m.

02:22 2b1.

02:24 And then we can find the mass flow rate and that will equal 230 divided by 9 .81.

02:35 Whoops, what did i do there? 230 divided by 9 .81.

02:42 That'll be, and that's kilograms per second.

02:48 That'll equal 23 .445 kilograms per second.

02:54 Okay, we already know that m equals density of, let's do my density a little bit nicer, nicer.

03:05 And this is going to be 23 .445 equals, let me see here.

03:28 This will equal pi over four times 0 .05 squared times v1.

03:38 And i believe this will be 995...

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