Relative Equilibrium of Fluids
Rectilinear Motion (Moving Vessel)
• Horizontal Motion
𝜃
at rest vehicle/truck
moving vehicle
•
the fluid is disheveled inside
•
the rear side is experiencing
max Fh (hydrostatic force)
Relative Equilibrium of Fluids
Rectilinear Motion (Moving Vessel)
•
W
Thus:
REF
𝜃
𝜃
N
𝜃
𝑡𝑎𝑛𝜃 =
𝑅𝐸𝐹
𝑀𝑎
=
𝑊
𝑀𝑔
𝑎
𝑡𝑎𝑛𝜃 =
𝑔
REF
W
Relative Equilibrium of Fluids
Rectilinear Motion (Moving Vessel)
Inclined Motion
•
𝑎𝐻
𝑡𝑎𝑛𝜃 =
𝑔 ± 𝑎𝑣
•
•
•
Vertical Motion
𝑃𝐵𝑂𝑇𝑇𝑂𝑀
𝑎
= 𝛾ℎ 1 ±
𝑔
Use (+) upward motion
Use (- )downward motion
Relative Equilibrium of Fluids
Rotating Vessels
Volume of
paraboloid
air
liquid
Relative Equilibrium of Fluids
Rotating Vessels
𝜔2 𝑥
𝑡𝑎𝑛𝜃 =
𝑔
𝝎
𝑡𝑎𝑛𝜃
is the slope of the paraboloid at (x,y)
𝜔2 𝑥 2
𝑦=
2𝑔
𝜔 = angular speed in radians/s
𝑟𝑒𝑣
1
𝑚𝑖𝑛
=
𝜋
30
rad/s
Relative Equilibrium of Fluids
𝝎
Rotating Vessels
𝒙
𝑃𝑎𝑟𝑎𝑏𝑜𝑙𝑜𝑖𝑑:
𝑦
𝒙𝟏
𝒚𝟏
𝑥12
𝑥2
=
𝑦1 `
𝑦`
𝑦 = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑜𝑖𝑑
1 2
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 𝑦
2
𝒚
𝝎
If 𝟐 < 𝑫
𝑦
open
𝜔2 𝑥 2
𝑦=
2𝑔
𝒚
𝟐
If = 𝑫
𝐷
𝝎
air
𝑥
𝑦
If > 𝑫
air
𝝎
𝑥
4
If 𝒚 = 𝑯
𝑦
NO SPILL
1
air
𝑥
5
𝝎
If y > 𝑯
𝝎
𝑥
IN THE VERGE
OF SPILLING
2
air
SOME SPILLED,
VORTEX NOT
TOUCHES YET
THE BOTTOM
3
STAGES OF PARABOLOID AT INCREASING
ANGULAR VELOCITY, 𝝎 (rad/s)
𝐻
LIQUID
𝒚
𝟐
𝑦
air
𝑦
SOME SPILLED,
VORTEX NOW
TOUCHES THE
BOTTOM
CENTER
SOME SPILLED,
VORTEX
BEYOND THE
BOTTOM
Examples
An open rectangular tank mounted on a truck is 5 m
long, 2 m wide and 2.5 m high is filled with water to a
depth of 2 m.
a. What is the maximum horizontal acceleration can be
imposed on the tank without spilling any water
b. Determine the accelerating force on the liquid mass.
c. If the acceleration is increased to 6 m/s^2, how
much water is spilled out?
Ans: a=1.962 m/s^2; F = 39.24 kN; Vspilled = 9.78 cu.m
Examples
A closed horizontal cylindrical tank 1.5 m in diameter
and 4 m long is completely filled with gasoline (s.g. =
0.82) and accelerated horizontally at 3 m/s^2.
a. Find the total force acting on the rear wall of the
tank.
b. Find also the accelerating force on the fluid mass.
Ans: Frear = 28.05 kN; Fnet = 17.36 kN
Examples
A closed rectangular tank 4 m long, 2 m wide, and 2 m
high is filled with water to a depth of 1.8 m. If the
allowable force at the rear wall of the tank is 200 kN,
how fast can it be accelerated horizontally?
Ans: 13.72 m/s^2
Examples
A vessel 3 m in diameter containing 2.4 m of water is
being raised.
a. Find the pressure at the bottom of the vessel in kPa
when the velocity is constant.
b. Find the pressure at the bottom of the vessel when it
is accelerating 0.6 m/s^2 upwards.
Ans: 23.54 kPa; 24.98 kPa
Examples
A vessel containing oil is accelerated on a plane inclined
15 degrees with the horizontal at 1.2 m/s^2. Determine
the inclination of the oil surface when the motion is:
a. Upwards
b. Downwards
Ans: 6.53 degrees; 6.96 degrees
Examples
Rotating Vessels
An open cylindrical vessel having a height equal to its
diameter is half-filled with water and revolved about its
own vertical axis with a constant angular speed of 120
rpm. Find its minimum diameter so there can no liquid
spilled.
Ans: 497 mm
Examples
Rotating Vessels
An open cylindrical tank 1.6 m in diameter and 2 m high
is full of water. When rotated about its vertical axis at
30 rpm, what would the slope of the water surface at
the rim of the tank?
Ans: 0.81
Examples
Rotating Vessels
An open cylindrical tank 1.2 m in diameter and 2.1 m
high is 2/3 full of water. Compute the amount of water
in liters that will be spilled out if the vessel is rotated
about its vertical axis at a constant angular speed of 90
rpm.
Ans: 130 L
Examples
Rotating Vessels
An open cylindrical tank, 2 m in diameter and 4 m high
contains water to a depth of 3 m. It is rotated about its
own vertical axis with a constant angular speed 𝜔.
a. If 𝜔 = 3 rad/sec, is there any liquid spilled?
b. What maximum value of 𝜔 (in rpm) can be imposed
without spilling any liquid?
c. If 𝜔 = 8 rad/sec, how much water is spilled out?
d. What angular speed 𝜔 (in rpm) will just zero the
depth of water at the center of the tank?
e. If 𝜔 = 100 rpm, how much area at the bottom will be
uncovered?
Ans: no spill; 59.8 rpm; 1.98 cu.m; 84.6 rpm; 0.889 sq.m
Examples
Rotating Vessels
A closed cylindrical vessel, 2 m in diameter and 4 m
high is filled with water to a depth of 3 m and rotated
about its own vertical axis at a constant angular speed
𝜔. The air inside the vessel is under a pressure of 120
kPa.
a. If 𝜔 = 12 rad/sec, what is the pressure at the center
and the circumference at the bottom of the tank?
b. What angular speed 𝜔 will just zero the depth of
water at the center?
c. If 𝜔 = 20 rad/sec, how much area of the bottom is
uncovered?
Ans: 121.66 kPa and 193.64 kPa; 119.6 rpm; 0.48 sq.m
Examples
Rotating Vessels
A closed cylindrical vessel, 1.5 m in diameter and 3.6 m
high is ¾ full of brine (s=1.3) and is revolved about its
own vertical axis with a constant angular speed. The
vessel is made up of steel 9 mm thick with an allowable
tensile stress of 85 MPa and has a small opening at the
center of the top cover.
a. If the angular speed is 210 rpm, what is the
maximum stress in the walls?
b. To what maximum angular speed can the vessel be
revolved?
Ans: 12.96 MPa; 576 rpm