Practice Problem 13.9 SOLUTION SET UP Let point 1 be at the inlet pipe and point

Question

Practice Problem 13.9

SOLUTION

SET UP Let point 1 be at the inlet pipe and point 2 at the bathroom. The speed v2 of the water at the bathroom is obtained from the continuity equation, A1v1=A2v2. We take y1=0 at the inlet and y2=5.0m at the bathroom. We are given p1 and v1; we can find p2 from Bernoulli's equation. To find the time required to fill the bathtub, we use the volume flow-rate relationship V/t=Av. The bathtub's volume is 100L=100×103m3.

SOLVE From the continuity equation, the flow speed v2 in the bathroom is

v2=A1A2v1=(1.0cm)2(0.50cm)2(2.0m/s)=8.0m/s

From Bernoulli's equation, the gauge pressure p2 in the bathroom is

p2====p112(v22v12)g(y2y1)4.0×105Pa12(1.0×103kg/m3)(64m2/s24.0m2/s2)(1.0×103kg/m3)(9.80m/s2)(5.0m)3.2×105Pa3.2atm=47lb/in.2(gaugepressure)

The volume flow rate is

V/t==A2v2=(0.50×102m)2(8.0m/s)6.3×104m3/s=0.63L/s

The time needed to fill the tub is 100L/(0.63L/s)=160s.

REFLECT Note that when the water is turned off, the second term on the right of the pressure equation vanishes, and the pressure rises to 3.5×105Pa. In fact, when the fluid is not moving, Bernoulli's equation reduces to the pressure relationship that we derived for a fluid at rest.

Part A - Practice Problem:

Determine the gauge pressure for a 2-cm-diameter faucet located next to the washing machine in the basement, 2.0 m below where the pipe enters the house.

Express your answer to two significant figures and include the appropriate units.

SubmitMy AnswersGive Up

Provide FeedbackContinue

Practice Problem 13.9

Let's begin by looking at a flow problem in which the diameter of a pipe changes along the flow path. We will need to use both the continuity equation and Bernoulli's equation for this problem. Water enters a house through a pipe with an inside diameter of 2.0 cm at a gauge pressure of 4.0×105Pa (about 4 atm, or 60 lb/in.2). The cold-water pipe leading to the second-floor bathroom 5.0 m above is 1.0 cm in diameter(Figure 1) . Find the flow speed and gauge pressure in the bathroom when the flow speed at the inlet pipe is 2.0 m/s. How much time would be required to fill a 100 L bathtub with cold water?

SOLUTION

SET UP Let point 1 be at the inlet pipe and point 2 at the bathroom. The speed v2 of the water at the bathroom is obtained from the continuity equation, A1v1=A2v2. We take y1=0 at the inlet and y2=5.0m at the bathroom. We are given p1 and v1; we can find p2 from Bernoulli's equation. To find the time required to fill the bathtub, we use the volume flow-rate relationship V/t=Av. The bathtub's volume is 100L=100×103m3.

SOLVE From the continuity equation, the flow speed v2 in the bathroom is

v2=A1A2v1=(1.0cm)2(0.50cm)2(2.0m/s)=8.0m/s

From Bernoulli's equation, the gauge pressure p2 in the bathroom is

p2====p112(v22v12)g(y2y1)4.0×105Pa12(1.0×103kg/m3)(64m2/s24.0m2/s2)(1.0×103kg/m3)(9.80m/s2)(5.0m)3.2×105Pa3.2atm=47lb/in.2(gaugepressure)

The volume flow rate is

V/t==A2v2=(0.50×102m)2(8.0m/s)6.3×104m3/s=0.63L/s

The time needed to fill the tub is 100L/(0.63L/s)=160s.

REFLECT Note that when the water is turned off, the second term on the right of the pressure equation vanishes, and the pressure rises to 3.5×105Pa. In fact, when the fluid is not moving, Bernoulli's equation reduces to the pressure relationship that we derived for a fluid at rest.

Part A - Practice Problem:

Determine the gauge pressure for a 2-cm-diameter faucet located next to the washing machine in the basement, 2.0 m below where the pipe enters the house.

Express your answer to two significant figures and include the appropriate units.

SubmitMy AnswersGive Up

Provide FeedbackContinue

Explanation / Answer

by contunuity equation

area1 * velocity1 = area2 * velocity2

pi * 0.02^2 * 2 = pi * 0.01^2 * velocity2

velocity2 = 8 m/s

speed of the flow in the bathroom = 8 m/s

by bernoulli's equation

P1 + density * g * h1 + 0.5 * density * velocity1^2 = P2 + density * g * h2 + 0.5 * density * velocity2^2

P1 + density * g * (h1 - h2) + 0.5 * density * velocity1^2 = P2 + 0.5 * density * velocity2^2

4 * 10^5 + 1000 * 9.8 * 5 + 0.5 * 1000 * 2 = P2 + 0.5 * 1000 * 8^2

P2 = 418000 Pa

absolute pressure at bathroom = 418000 Pa

gauge pressure at bathroom = 418000 - 400000

gauge pressure at bathroom = 18000 Pa

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.