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A pond drains through a pipe as shown in Fig. P22.21. Under a number of simplifying
Chapter 22, Problem 22.21(choose chapter or problem)
A pond drains through a pipe as shown in Fig. P22.21. Under a number of simplifying assumptions, the following differential equation describes how depth changes with time: dh dt = d2 4A(h) 2g(h + e) where h = depth (m), t = time (s), d = pipe diameter (m), A(h) = pond surface area as a function of depth (m2 ), g = gravitational constant (= 9.81 m/s2 ), and e = depth of pipe outlet below the pond bottom (m). Based on the following area-depth table, solve this differential equation to determine how long it takes for the pond to empty, given that h(0) = 6 m, d = 0.25 m, e = 1 m.
Questions & Answers
QUESTION:
A pond drains through a pipe as shown in Fig. P22.21. Under a number of simplifying assumptions, the following differential equation describes how depth changes with time: dh dt = d2 4A(h) 2g(h + e) where h = depth (m), t = time (s), d = pipe diameter (m), A(h) = pond surface area as a function of depth (m2 ), g = gravitational constant (= 9.81 m/s2 ), and e = depth of pipe outlet below the pond bottom (m). Based on the following area-depth table, solve this differential equation to determine how long it takes for the pond to empty, given that h(0) = 6 m, d = 0.25 m, e = 1 m.
ANSWER:Depth (m)|Area (m2 )
0.25 |5.5
0.75 |7.5
1.25 |9.5
1.75 |11.5
2.25 |13.5
2.75 |15.5
3.25 |17.5
3.75 |19.5
Assuming the flow is non-uniform, the solution to this differential equation is
T = ?h0dh d2 4A(h) 2g(h + e)
Integrating with respect to h,