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Chapitre
Fluids Statics
1 Units – Dimensions
A unit is a meaure of a quantity
A dimension is a kind of quantity
1.1 Dimensions
There a four fundamental dimensions of importance to chemical Engineering
Length L, Mass M, Time θ, Temperature T.
Force F is sometimes used as basic dimension in place of mass.
From these basic ones others can be formed.
Velocity L/
, Acceleration L/
, Force ML/ , area L2, volume L3,
Density M/L3, energy ML2/ 2, power ML2/ , Viscosity M/L
1.2 Units
A) Absolute units
Basic dimensions are M, L,
, derived dimension is force
Force= M.L/
System
S.I (M.K.S)
Mass
Kilogram, Kg
Length
Meter, m
Metric (c.g.s)
Gram, gm
Centimeter, cm Second, S
Dyne
British, f.p.s
Pound, lb
Foot, ft
Second, S
Poundal
Time
Second, s
Force
Newton
1 Newton = 1kg.1m/sec2
1dyne= 1gm.1cm/ sec2
1 poundal = 1lb.ft/sec2
B) Gravitational Units
Basic Dimensions, F, L, Time
Derived Dimension Mass, M= F. 2/L
System
Metric
Force
Kilogram, Kgf
Length
Meter
Time
Second
Mass
Metric-Slug
British
Pound, Lbf
Foot
Second
Slug
1 Metric Slug = (1/9.81) Kg.Sec2/m
1Slug = (1/32.2) Lb.Sec2/ft
2. Force
- An important property of a static fluid is the pressure in the fluid.
- Pressure is equivalent to a force exerted by a fluid against the walls of its container.
- Pressure exists at any point in a volume of a fluid.
- According to Newton’s law, the equation for calculation of the force exerted by a
mass due to gravity is
In SI units,
F= Force in Newton (N)
M: Mass (Kg), g=acceleration of gravity 9.8066m/s2
-In c-g-s unit
F= force in dynes
M= mass (gm)
g=acceleration of gravity (980.66cm/s2)
In English units
F= Force in poundal
M: Mass in lb, g= acceleration of gravity (32.174ft/s2)
Or
F= force Lbf
M = mass (Lb)
g=32.174ft/sec2
gc = gravitational conversion factor= 32.174 lbm.ft/ lbf.s2
3. Pressure
-
Equations above give the force exerted by a mass under the influence of
gravity.
Pressure represents the force exerted by a mass of fluid on a supported area.
Pressure = Force/ unit area
-
This is the Pressure on A2 due to the mass of the fluid above it.
To get the total pressure P2 on A2, the pressure P0 on top of the fluid must be
added
-
This is the fundamental equation to calculate the pressure in a fluid at any
depth, to calculate P1
-
The pressure difference between points 2 and 1 is
P2- P1 =
-
Since it is the vertical height of a fluid that determines the pressure in fluid,
the shape of the vessel does not affect the pressure.
4. Head of fluid
- Pressures are given in many different sets of units ( Pascal, Psi, dynes/cm2 etc…)
- A common method of expressing pressure, is in terms of head in meter or of a
particular fluid.
5 Devices to measure pressures
- In chemical and other processing plants, it is important to measure and control the
pressure in a vessel or process.
- Also since many fluids are flowing in a pipe or conduct, it is necessary to measure
the rate at which the fluid is flowing.
- Many of these flow meters depend upon devices to measure a pressure or pressure
differences.
5.1 Simple U-tube manometer
-
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Pa is exerted on one arm of the U tube and Pb on the other arm. Both pressures
(Pa and Pb) could be pressure taps from a fluid meter or Pa (pressure tap) and
Pb (atmosphere pressure).
The top of the manometer is filled with liquid B ( b), and the bottom with a
more dense fluid A, ( a).
Liquid A is immiscible with B.
To derive the relationship between Pa(point1) and Pb (point 5)
P2= Pa + ( R+Z) b.g
(N/m2)
R= Reading of the manometer in (m)
-
A hydrostatic balance gives,
P3= P2
P3 = Pb + Z b g + R a g
= Pa + (Z + R) b g
Pa-Pb = R ( a - b)g
(N/m2)
In English units
Pa –Pb = R( a – b) g/gc lbf/ft2
-Z does not enter into the final result, nor do the tube dimensions.
5.2) Two fluids- U tube
-
Common surface is AA, level above common surface h1 and h2
Fluids at equilibrium h1g 1 = h2 g. 2
-
Pressure applied at right arm, fluids displaced and common surface descends
by amount y.
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New level A’A’
-
Levels above common surface x1 and x2.
At equilibrium
Pa + x1.g. 1 = Pb + x2.g. 2
Pb- Pa = x1.g. 1 - x2.g. 2
Mass of fluid displaced on one end = mass of fluid displaced on other end.
AZ = ay
Z = y A/a
x1 = h1 + y + Z
x2 = h2 – Z + y
Pb –Pa = 1.g [h1 + y +Z] - g. 2[h2 – Z + ]
Pb – Pa = g. 1[Z +y] – g. 2[y – Z]
Pb-Pa = y[g. 1(1 +a/A) – g. 2(1 – a/A)]
Pb – Pa= y.g [( 1 - 2) + ( 1 + 2) (a/A)]
As a/A tends to zero
Pb- Pa = y.g ( 1- 2).
5.3 Pressure measurement in vessel
At point 2 the pressure is
P2= Patm + h2. b.g (N/m2)
At point 1 the pressure is
P1= Pa + h1. a.g (N/m2)
Applying the principles of hydristatics, P1=P2, and arranging
Pa= Patm + h2. b.g – h1. a.g (N/m2)
5.4 Measurement of pressure differences between two vessels.
At point 2 the pressure is
P2= Pb + h2. c.g + h3. b.g (N/m2)
At point 1 the pressure is
P1= Pa + h1. a.g (N/m2)
Applying the principles of hydrostatics, P1=P2, and arranging
Pa-Pb = h2. c.g + h3. b.g – h1. a.g (N/m2)
5.5 Manometer with three fluids
-
considered, head upward negative, head downward positive, same liquid and
head=0
Pa –ha. a.g – hc. c g + hb b g = Pb
Pb= -hb b g + hc c g + a a g = Pa
5.6 Various Manometers
At Point 1 Pa+ h1 g= Pair= P1
At Point 2 P2= Pa + h1 g + h2 g= Pa + 2 g
At Point 3 P3 = Pa + h1 g + 2 g + h3 g
5.6 Gravity separator for two immiscible liquids
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To separate two immiscible liquids A (heavy liquid) and B (light liquid).
The feed mixture of the two liquids enters at one end of the separator vessel
and the liquids flow slowly to the other end and separate into two distinct
layers.
Each liquid flows through a separate overflow line.
Assuming the frictional resistance to the flow of the liquid is negligible,
principles of fluid static can be used to analyse the performance.
Ht= hA1 + hB is fixed by position of the overflow, ha1= height of fluid A,
hb= height of fluid B.
The heavy liquid A discharges through an overflow leg with height Ha2 above
the vessel bottom.
The vessel and the overflow lines are vented to the atmosphere.
An hydrostatic balance gives,
hB bg + HA1 ag = hA2 a g
hb=ht –hA1, and solving for hA1
hA1= (hA2- ht
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[ b/ a]/ (1 – [ b/ a])
This shows that the position of the interface or height Ha1 depends on the ratio
of the densities of the two liquids and on the elevations hA2 and ht of the two
overflow lines.
Usually the height hA2 is movable so that the interface level can be adjusted.
5.7 Bourdon Pressure gage
- A very common pressure measuring device.
- A coiled hollow tube in the gage tends to straighten out when subjected to
internal pressure, and the degree of straightening depends on the pressure
difference between the inside and outside pressures.
- The tube is connected to a pointer on a calibrated dial.
- Pressure is applied to the open end
- When P>Patm, the closed end will move as the tube tends to straighten out
(the reading is proportional to the displacement of the tube), causing the free
end to move proportional to the pressure.