Manometry and Buoyancy
- The pressure is proportional to the height of a column of fluid.
- Manometry is the field of science which deals with the evaluation of the pressure of the fluid.
- The instrument used to carry out the complete process is termed a Manometer.
- Types of Manometers: Barometer, Piezometer and U-tube Manometer.
Manometers use the relationship between pressure and head to measure pressure.
Relation between Hydrostatic pressure & Head
We have the vertical pressure relationship p = ρgz + constant measuring z from the free surface so that z = -h and surface pressure is atmospheric,patm
We generally assume atmospheric pressure as the datum,
Gauge pressure, pg = ρgh
The lower limit of any pressure is the pressure in a perfect vacuum. Pressure measured above a perfect vacuum (zero) is known as absolute pressure.
Absolute pressure, pa = ρgh+ patmospheric
Absolute pressure = Gauge pressure + Atmospheric
Piezometer Tube Manometer
- The simplest manometer is an open tube. This is attached to the top of a container with liquid at pressure. containing liquid at a pressure.
- The tube is open to the atmosphere, The pressure measured is relative to atmospheric so it measures gauge pressure.
- Pressure at A = pressure due to column of liquid h1
pa = ρgh1
- Pressure at B = pressure due to column of liquid h2
Pb = ρgh2
Limitations of Piezometer:
- Can only be used for liquids
- Pressure must above atmospheric
- Liquid height must be convenient i.e. not be too small or too large
- It consists of a U shaped bend whose one end is attached to the gauge point ‘A’ and the other end is open to the atmosphere.
- It can measure both positive and negative (suction) pressures.
- “U”-Tube enables the pressure of both liquids and gases to be measured “U” is connected as shown and filled with manometric fluid.
- The manometric fluid density should be greater than of the fluid measured, ρman > ρ
- The two fluids should not be able to mix they must be immiscible.
- Pressure in a continuous static fluid is the same at any horizontal level, pressure at B = pressure at C
PB = PC
- For the left-hand arm pressure at B
- pressure at A + pressure of height of liquid being measured
PB = PA + ρgh1
- For the right-hand arm pressure at C =
- pressure at D + pressure of height of manometric liquid
PC = P + ρmanogh2
We are measuring gauge pressure we can subtract patmospheric giving
PB = PC
PA = P +ρmanogh2- ρgh1
Differential U-Tube Manometer
- A U-Tube manometric liquid is heavier than the liquid for which the pressure difference is to be measured and is not immiscible with it.
- The pressure difference between A and B is given by equation
PA – PB = ρ2h2 + ρ3h3 – ρ1h1
Inverted U-Tube Manometer
- Inverted U-Tube manometer consists of an inverted U Tube containing a light liquid.
- This is used to measure the differences of low pressures between two points where better accuracy is required.
- It generally consists of an air cock at top of the manometric fluid type.
The pressure difference can be calculated from the equation:
P1 – ρ1gH2 – ρmg(H1– H2)=P2 – ρ2gH1
- Micro Manometer is the modified form of a simple manometer whose one limb is made of a larger cross-sectional area.
- It measures very small pressure differences with high precision.
Let ‘a’ = area of the tube, A = area of the reservoir, h3 = Falling liquid level reservoir,
h2 = Rise of the liquid in the tube,
- By Volume Equality, Ah3 = ah2
- Equating pressure heads at datum,
P1 = (ρm – ρ1)gh3 + ρmgh2 – ρ1gh1
- An inclined manometer is used for the measurement of small pressures and is to measure more accurately than the vertical tube type manometer.
- Due to inclination, the distance moved by the fluid in the manometer is more.
- The pressure difference between A and B is given by equation
PA – PB = ρ2LsinΘ + ρ3h2 – ρ1h1
Buoyancy is also known as buoyant force. It is the force exerted on an object that is wholly or partly immersed in a fluid.
Concept of Buoyancy: When a body is immersed in a fluid, an upward force is exerted by the fluid on the body which is equal to the weight of the fluid displaced by the body. This acts as upward.
Archimedes’ Principle: It states, when a body is immersed completely or partially in a fluid, it is lifted up by a force equal to the weight of the fluid displaced by the body.
Buoyant force = Weight of fluid displaced by the body
The buoyant force on cylinder =Weight of fluid displaced by cylinder
Value of immersed part of solid or Volume of fluid displaced
= Volume of cylinder immersed inside the water
Principle of Flotation: According to this principle, if the weight of the body is equal to buoyant force then, the body will float.
- The factors that affect buoyancy are: the density of the fluid, the volume of the fluid displaced, and the local acceleration due to gravity.
- The buoyant force is not affected by the mass of the immersed object or the density of the immersed object.
Centre of Buoyancy: The point at which the force of buoyancy acts is called centre of buoyancy. It lies on the centre of gravity of the volume of fluid displaced or the centre of gravity of the part of the body which is inside the water. Point B is the centre of buoyancy.
Buoyancy on a submerged body:
- The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of the liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.
- Thus the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.
Buoyancy on a partially immersed body:
- According to Archimedes principle, the buoyant force of a partially immersed body is equal to the weight of the displaced liquid.
- Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body.
- For a floating body in static equilibrium and in the absence of any other external force, the buoyant force must balance the weight of the body.
Metacentre of a Floating Body: If a body that is floating in liquid is given a small angular displacement, it starts oscillating about some point M. This point is called the metacentre.
The equilibrium of a submerged body in a liquid requires that the weight of the body acting through its centre of gravity should be colinear with equal hydrostatic lift acting through the centre of buoyancy. Let us suppose that a body is given a small angular displacement and then released. Then it will be said to be in distance MG is called metacentric height (it is the distance between gravity centre and meta centre)
Stability of Submerged Body: It is classified into three groups.
- Stable Equilibrium: When the centre of buoyancy lies above the centre of gravity, the submerged body is stable.
- Unstable Equilibrium: When B lies below G, then the body is in unstable equilibrium.
- Neutral Equilibrium: When B and G coincide then, the body is in neutral equilibrium.
Stability of Floating Bodies: When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.
- Stale Equilibrium: When a body is given a small angular displacement by external means and if the body comes to its original position due to internal forces then, it is called stable equilibrium.
It occurs when the metacentre lies above the centre of gravity.
- Unstable Equilibrium: In the above case, if the body does not come in its original position and moves further away then, it is known as unstable equilibrium. M lies below the centre of gravity.
- Neutral equilibrium: When a body is given a small angular displacement and it sets on new position then, body is called in neutral equilibrium. In this, M and G coincide.
- Relation between B,G and M is
GM = I/V - BG
Here, l = Least moment of inertia of plane of the body at the water surface
G = Centre of gravity
B = Centre of buoyancy
M = Metacentre
V is volume submerged inside the water can be given as
Where b,d and x are the length, width and depth of the section or body.
- BG is the distance between the centre of gravity and the centre of buoyancy. (In other words, BG=distance between the centre of gravity of whole body and centre of gravity of submerged part of the body)
- When we find out GM then, we can determine the status of body as
- GM > 0 (stable equilibrium),
- GM < 0 (unstable equilibrium),
- GM = 0 (neutral equilibrium).
If you aim to crack ESE, GATE & Other PSU Exams then you must try BYJU’S Exam Prep Comprehensive Preparation for ESE and GATE CE Exams
All the Best