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At what flow rate might turbulence begin to develop in a water main with a 0.200-m diameter? Assume a 20o C temperature.

$Q=3.12 \times 10^{-4} \mathrm{m}^{3} \mathrm{s}^{-1}$

Physics 101 Mechanics

Chapter 12

Fluid Dynamics and Its Biological and Medical Applications

Fluid Mechanics

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to hear, Um, it's first. It's we have to say first that the flow is said to be turbulent when the Reynolds number is greater than 3000. However, when the Reynolds number is between 5000 the flow can be a mixture of land, lamb inner and turbulent. So here for the surf, for the sake of this problem, we can say that the Reynolds number has to be less than or equal to 2000 in order in order for the flow to be lamb in there. And so we can first say that the volumetric florid is equaling the cross sectional area of the pipe or the hose of the tube, whatever it may be. In this case, uh, in this case, we have a water main, so essentially just a pipe. And so we have than the velocity is equaling the volumetric floor eight divided by the cross sectional area of the pipe pi r squared. Now we know that the formula for the Reynolds number is equaling two times the density of water times the velocity times, the radius of the pipe divided by new, which would, of course, be the coefficient of viscosity and we're going to substitute this equation in for the velocity V. So then we have. This would be equal to two times row times are over the viscosity new, and this would be multiplied by the volumetric flow rate divided by pi r squared. And so we have that the Reynolds number is going to be equaling two times the density times the volumetric Fleury divided by pi times the radius times and knew the coefficient of viscosity. And so we can solve for the volumetric flow rate. This would be the Reynolds number multiplied by pi. Times are the radius times new divided by two times three density. And so now we can solve. So the volumetric floor, it would be equally here. We're going to use a Reynolds number of of course, of course, 2000. So this would be the maximum Ah, the maximum volumetric flow rate where the flow can still be laminated. So 2000 times pi times 0.100 meters so that the diameter divided by two multiplied by the viscosity of 1.5 times 10 to the negative third pascal seconds. And then this would be divided by two times the density of 1000 kilograms per cubic meter. And we find that the volumetric flow rate is gonna be equaling 3.16 times 10 to the negative forth meters cubed per second. This would be our final answer again. This would be the maximum boy metric Florida, where the flow is still lamb in there. That is the end of the solution. Thank you. For what?

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