What is the average radius of the pipe within the hill?

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SUMMARY

The average radius of the pipe within the hill can be calculated using the principle of conservation of mass, specifically the equation AoVo = AV, where Ao is the cross-sectional area of the smaller pipe and A is the area of the larger pipe. Given the flow velocity of 2.50 m/s and the dye travel time of 88.8 seconds, the correct approach involves using the total distance of 110m to determine the effective distance the dye travels through the hill. The calculations indicate that the average radius of the pipe within the hill is approximately 0.03m, derived from the correct application of the flow equations and understanding of the system's geometry.

PREREQUISITES
  • Understanding of fluid dynamics principles, specifically the continuity equation.
  • Familiarity with basic kinematics, particularly the relationship between distance, speed, and time.
  • Knowledge of cross-sectional area calculations for circular pipes.
  • Ability to manipulate algebraic equations to solve for unknown variables.
NEXT STEPS
  • Study the continuity equation in fluid dynamics to deepen understanding of flow rates.
  • Learn about the effects of pipe diameter on flow velocity and pressure loss.
  • Explore practical applications of dye tracing in hydraulic engineering.
  • Investigate the impact of pipe material and shape on fluid dynamics in various systems.
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Hydraulic engineers, civil engineering students, and anyone involved in fluid mechanics or pipe system design will benefit from this discussion.

jinman
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Homework Statement



An old two pipe system runs through a hill, with distances D_a=D_b=30m and the total Distance is D=110m. On each side of the hill, the pipe radius is .0200m. However, the radius of the pipe inside the hill is no longer known. To determine it, hydraulic engineers first establish that the water flows through the left and right sections at 2.50m/s. Then they release a dye in the water at point A and find that it takes 88.8s to reach point B. What is the average radius of the pipe within the hill (sorry i don't have a diagram)?


Homework Equations


AoVo=AV

v=d/t


The Attempt at a Solution


I cut this problem in half to try and make it work. I used AoVo=AV where Ao is the little pipe, and A is the the middle pipe.

AoVo=AV>>>>(pi_r^2)(2.50m/s)=(pi_r^2)(V)

solve for r^2 on right

(pi_r^2)(2.50m/s)=(pi_r^2)(d/t)>>>>sq. root(.02^2)(2.50m/s)(44.4s) / 55m= .03m

i know i am going wrong in many places. Any suggestions?
 
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The method used seems Ok, but I can't tell where the distance of 55m you used comes from, nor do I understand why you use 44.5s instead of 88.8s. I thought d was the lenth of the pipe in the hill and t the time it for the dye to go through this pipe?
 
I used 55m because D=110m is the the whole system, so i looked at the first half. which is why i used 44.4 seconds. Is this the right to do the problem?
 

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