Problem 4.62
(Difficulty: 2)
4.62 The lower tank weighs 224 𝑁, and the water in it weighs 897 𝑁. If this tank is on a platform scale,
what weight will register on the scale beam?
Given: Tank weight: 𝐹𝑡𝑡𝑡𝑡 = 224 𝑁. Water weight: 𝐹𝑤𝑤𝑤𝑤𝑤 = 897 𝑁. All the other parameters are
shown in the figure.
Find: The weight on the scale beam.
Assumptions: Flow is steady
Density is constant
Solution:
Basic equation:
Continuity
Bernoulli equation;
0=
𝜕
� 𝜌𝜌∀ + � 𝜌𝑉� ∙ 𝑑𝐴̅
𝜕𝜕 𝐶𝐶
𝐶𝐶
𝑝 𝑉2
+
+ 𝑔𝑔 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
2
𝜌
Momentum equation for the y-direction
𝐹𝑠𝑠 + 𝐹𝐵𝐵 =
𝜕
� 𝑣𝑣𝑣∀ + � 𝑣𝑣𝑉� ∙ 𝑑𝐴̅
𝜕𝜕 𝐶𝐶
𝐶𝐶
For the upper surface of the lower tank, from Bernoulli equation we have:
𝑉12
− 𝑔ℎ1 = 0
2
ℎ1 = 7.8 𝑚
𝑉1 = �2𝑔ℎ1 = �2 × 9.81
For the bottom of the lower tank, we have:
𝑚
𝑚
× 7.8 𝑚 = 12.36
2
𝑠
𝑠
𝑉22
− 𝑔ℎ2 = 0
2
ℎ2 = 1.8 𝑚
𝑉2 = �2𝑔ℎ2 = �2 × 9.81
The mass flow rate of the lower tank is:
𝑚
𝑚
×
1.8
𝑚
=
5.94
𝑠2
𝑠
𝑘𝑘
𝑚 𝜋
𝑘𝑘
𝜋
𝑚̇ = 𝜌𝑉2 𝐴2 = 𝜌𝑉2 𝐷22 = 999 3 × 5.94 × × (0.075 𝑚)2 = 26.22
𝑚
𝑠 4
𝑠
4
Force on scale:
𝐹𝑦 = −𝑉1 𝑚̇ + 𝑉2 𝑚̇ = 26.22
Direction is going down.
𝑘𝑘
𝑚
𝑚
× �5.94 − 12.36 � = −168.3 𝑁
𝑠
𝑠
𝑠
Weight on the scale beam:
𝐹𝑤 = 𝐹𝑦 + 𝐹𝑡𝑡𝑡𝑡 + 𝐹𝑤𝑤𝑤𝑤𝑤 = 168.3 𝑁 + 224 𝑁 + 897 𝑁 = 1289 𝑁