Buoyancy Study notes for Chemical Engineering

For floating or submerged body:-

W = [equilibrium equation]

: Volume of body [Submerged portion]

Density of liquid/ fluid.

From equilibrium equation

Density of body.

Submerged volume fraction of body depends upon the density of liquid and body.

Stability of floating or submerged body:

Ø Rotational stability or stability of floating or submerged body, depends on Relative position of centre of gravity and centre of buoyancy.

Stability of Floating body:

Ø Stability of floating body depends on the position of metacentre or metacentric height.

Metacentre:-

Intersection point of , line of action of buoyancy force before and after rotation.

Ø Metacentric height equal to the distance between metacentre and centroid of body.

M G

G M

Ø M Above G

Ø MG +ve

Ø Stable condition

Ø GM (metacentric height) is the property of cross-section for a given weight.

Metacentric height

Metacentric –MG GB distance between C and B

C centroid of full body

Example: -

Given: A rectangular block having dimension 3

Find 1. Depth of immersion

2. Level of water after immersion

3 Metacentricheights

· Weight of body = weight of displaced water

36 1000

depth of immersion

· Volume of water is constant so.

· Metacentric Height :-

So. Floating block is in stable condition

Rigid Body Motion: -

Assumption: - incompressible fluid.

· Rigid body motion is the motion of fluid like a solid.

· In Rigid body motion, there is no relative motion between adjacent layers.

· For Rigid body motion consider Newton’s second law of motion –

F = surface force +body force

Surface force = pressure force

Body force = Gravity force or weight

Consider a Fluid element having dx, dy, dz and density and pressure at its centre P.

Apply equillbrium equation in Z- direction (

Similarly in x, and y direction

Body force = - ρgdxdydz

Equation for Rigid body motion:-

In x- direction: ()

Body force zero in x direction acceleration in x direction

Similarly in y –direction

In z direction

Surface of constant pressure in horizontal motion (2-D motion)

In horizontal motion

Than …………(1)

And equation of motion

So put the values of in eq. (1)

dp =

Surface of constant pressure:-

Change in pressure

From previous equation

………………equation of “Surface of constant pressure”.

Rotation in a Cylindrical Container :- (Rigid Body Motion)

Force involves: (i) Centripetal force

(ii) Gravitational force [pressure force]

Consider an element, having position (r,z) : r=radial distance from z-z axis and Z= verical distance

from cetre of cylinder (of bottom as shown in figure)

From Newton’s second law of motion

dm elemental mass

………….(a)

And pressure change in vertical direction ………….(b)

So. From eq. (a) and (b) we can say that

By differenting p with respect to r and z

………..(c)

Put values of from eq. (a) and (b) in eq. (c)

This is the fundamental equation of flow in cylinder or for forced vortex flow.

Equation of free surface:-

For free surfaceb/w two point on that surface

So. Form fundamental equation: -

………(d) here represent the slope of free surface

by integration equation (d) with respect to r

Put Boundary condition r=0, z=H

So, Z (Equation of parabola)

NOTE: If we assume origin at apex of parabola than

Volume of parabolic

Volume of parabolic = Volume of shaded portion

NOTE:-

Example:-

Find the maximum angular velocity for which water is not spilled form tank.

Equation of surface

Example:-

Find the volume of spilled water if w=6 rad per second and w=7 rad/sec.

Assume cylinder, and quantity of water same as in example (1)

For Case –I

given w=6 rad/sec.

So. (Assume apex as origin)

Remaining Volume inside the cylinder =

Spilled Volume =

20.979

Case -2

And

So. Apply free surface equation at point A

Volume of shaded portion =

Final volume of water in cylinder

Example:-

Given conditions

I. Closed container

II. Fully filled with fluid without any pressure.

III. No free space means fully filled with liquid.

Calculate force on bottom and top surface.

Solution:

Force on bottom surface Hydrostatic force Force due to Rotation

Hydrostatic force

Pressure force due rotation

Pressure

Pressure force an elemental Ring (dr thickness)

Total force on bottom

Total force on top