8. Water Moves through a constricted pipe in steady, ideal flow. At the lower po

Question

8. Water Moves through a constricted pipe in steady, ideal flow. At the lower point shown in Figure, the pressure is P1=1.75x 104 Pa and the pipe area is 0.05m2. At another point y=0.25m higher, the pressure is P2=1.2x10%Pa and the pipe area is 0.2m2. Find the speed of low (a) in the lower section and (b) in the upper section. (c) Find the volume flow rate through the pipe. Density of the water is 1000kg/m3. Hint using both Continity equation and Bernoulli's Principle to solve for the speeds at 1 and 2 first. 9

Explanation / Answer

Bernoulli:

p1 + ½(v1)² = p2 + ½(v2)² + gy

(a) The flow rate at both ends must be the same, or

Q = v1*A1 = v2*A2, so

v2 = v1(A1/A2). So

p1 + ½(v1)² = p2 + ½(v1)²(A1/A2)2+ gy

Plug in values:

1.75e4Pa + ½(1000kg/m³)(v1)² = 1.2e4Pa + ½(1000kg/m³)(v1)²(0.05/0.02)^2 + 1000kg/m³*9.8m/s²*0.25m

3050 = (500kg/m³)(v1)²((0.05/0.02)2- 1) = 2625 * (v1)²

v1 = 1.078 m/s

(b) v2 = v1(A1/A2) = (1.078) )(0.05/0.02) = 2.69 m/s

(c) Q = A1* v1 = 0.05*1.078 = 0.0539 m³/s

[Check: Q = A2v2 = 0.02*2.69 = 0.0538m³/s, so its correct]

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