Problem 6.67
(Difficulty 2)
6.67 Water is being pumped from the lower reservoir through a nozzle into the upper reservoir. If the
vacuum gage at 𝐴 reads 2.4 𝑝𝑝𝑝 vacuum,
(a) find the flow velocity through the nozzle.
(b) find the horsepower the pump must add to the water.
(c) draw the energy line and the hydraulic grade line.
Assumption: The flow is steady, incompressible, uniform, and frictionless.
Solution: Apply the continuity and Bernoulli equations to find the velocity and power.
The continuity equation is:
𝑄 = 𝑉1 𝐴1 = 𝑉2 𝐴2
The Bernoulli equation along a streamline is
The gage pressure at A is:
𝑝 𝑉2
+
+ 𝑔𝑔 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝜌 2
𝑝𝐴 = −2.4 𝑝𝑝𝑝 = −345.6
𝑙𝑙𝑙
𝑓𝑓 2
Applying the Bernoulli equation from station 1 to station A:
𝑝𝐴 𝑉𝐴2
𝑝1 𝑉12
+
+ 𝑔𝑧1 =
+
+ 𝑔𝑧𝐴
𝜌
2
𝜌
2
Where p1 = 0 and V1 = 0
𝑔𝑧1 =
The water density is
𝑝𝐴 𝑉𝐴2
+
+ 𝑔𝑧𝐴
𝜌
2
𝜌 = 1.938
The velocity at A is then
𝑙𝑙𝑙 ∙ 𝑠2
𝑠𝑠𝑠𝑠
=
1.938
𝑓𝑓 3
𝑓𝑓 4
𝑙𝑙𝑙
2 × −345.6 2
2𝑝𝐴
𝑓𝑓
𝑓𝑓
𝑓𝑓
− 2𝑔𝑧𝐴 = �2 × 32.2 2 × (20 𝑓𝑓 − 25 𝑓𝑓) −
= 5.90
𝑉𝐴 = �2𝑔𝑧1 −
2
𝑙𝑙𝑙 ∙ 𝑠
𝑠
𝑠
𝜌
1.938
𝑓𝑓 4
The volumetric flow rate is:
𝑄 = 𝑉𝐴 𝐴𝐴 = 5.90
2
𝑓𝑓 𝜋
12
𝑓𝑓 3
× × � 𝑓𝑓� = 4.63
𝑠 4
12
𝑠
Applying the continuity equation, the volumetric flow rate at location 2 is the same as at A:
𝑓𝑓 3
4.63
𝑄
𝑓𝑓
𝑠
=
= 53.1
𝑉2 =
2
𝐴2 𝜋
𝑠
4
× � 𝑓𝑓�
12
4
Applying the energy equation from station 1 to station 2:
𝑝2 𝑉22
𝑝1 𝑉12
+
+ 𝑧1 + 𝐸𝑝 =
+
+ 𝑧2
𝜌𝜌 2𝑔
𝜌𝜌 2𝑔
Where Ep is the head provided by the pump
The pump power is then
𝐸𝑝 = (𝑧2 − 𝑧1 ) +
𝑊̇𝑝 =
𝑉22
= 148.8 𝑓𝑓
2𝑔
𝑄𝑄𝑄𝐸𝑝
= 70 ℎ𝑝
550
The energy line and hydraulic grade lines are