A Primary Reaction Due to Engineering Creation using Fluid Flows…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Force System Generated/Needed by Fluid Flows

Application of RTT on Conservation of Linear Momentum

• Mathematical model for Reynolds Transport Theorem

tV

dVvtxft

txf

Dt

tDF

,.,

• Reynolds field variable for this purpose is momentum flux or mass velocity and defined as:

vtxf ,

GSmESED

tV

FFFFFdVvtxft

txf

,.,

• Newton’s Second Law:

n

iiF

Dt

tDF

1

GSmESED FFFFF

Fluid Flows using a selected combination of Forces

• Systems only due to Body Forces.

• Systems due to only normal surface Forces.

• Systems due to both normal and tangential surface Forces.

– Thermo-dynamic Effects (Buoyancy forces/surface)…..

– Physico-Chemical/concentration based forces (Environmental /Bio Fluid Mechanics)

The System of Body Forces in Fluid Flows

• Gravitational Forces

• Magnetic Forces (Magneto Hydrodynamic)

– Lorentz force

• Electrical forces.

– Coulomb force

– Dielectric force

– Electrostriction

Flows Driven by Gravitational Force

tV

nalGravitatioFdVvvt

v

.

tV

nalGravitatio gdVF

tV tV

gdVdVvvt

v

.

Magneto Hydrodynamics

• Magneto-fluid dynamics

• Fluid Dynamics of electrically-conducting fluids and their interactions with magnetic fields.

• Interaction among Magnetic field, Electric field and Plasma flow

• Magnetic Field flow produce Electric field.

• Interacting Magnetic& Electrical fields produce Flow Lorentz Force

Faradays EMF

BV

Lorenz Force

BJ

Turbo Generator Vs MHD Generator

Important aspects of MDH

1. Accelerate fluids through Lorentz force, which refers to

MHD accelerators.

2. Convert the kinetic energy of a fluid and its enthalpy into

electric energy, which refers to MHD generators.

Principle of Operation

• The underlying principle of MHD power generation is elegantly simple.

• An electrically conducting fluid is driven by a primary energy source (e.g., combustion of coal or a gas) through a magnetic field.

• This results in the establishment of an electromotive force within the conductor in accordance with the principle established by Faraday.

• If the conductor is an electrically conducting gas, it will expand, and so the MHD system constitutes a heat engine involving an expansion from high to low pressure in a manner similar to that of a gas turbine.

• The gas turbine operates through the gas interaction with the surfaces of a rotating blade system.

• The MHD system, however, involves a volume interaction between a gas and the magnetic field through which it is passing.

• It is a system that depends on volume rather than surface interaction.

General Characteristics

• The MHD generator can properly be viewed as an electromagnetic turbine .

• Output is obtained from the conducting gas-magnetic field interaction directly in electrical form rather than in mechanical form.

• Electrical conduction in gases occurs when electrons are available to be organized into an electric current in response to an applied or induced electric field.

• The electrons may be either injected or generated internally, and, because of the electrostatic forces involved, they require the presence of corresponding positive charge from ions to maintain electrical neutrality.

• An electrically conducting gas consists in general of electrons, ions to balance the electric charge, and neutral atoms or molecules.

• Such a gas is termed a plasma.

Principle of MHD Generator

x

yz

Principle of MHD Accelerator

Development of Micro Pumps

Superiority of The Lorenz Force

• Simple Structure

• Only MHD channel (electrodes, insulator) and Magnet

• High Power density ---- high electric field, current density

• Compact machine

• Small output applications

• High temperature operation ---- looking for suitable material

• No turbine, and no rotating machine

• Ceramic material can be used

• No moving parts

• No turbine and no rotating generator

• Good for space applications

Electro Static Forces

Principles of Electrostatic Precipitator

Supplying high voltage between the Collecting Electrode and Discharge Electrode generates a Corona Discharge that produces minus ions.The electrically charged dusts are attracted towards the Collecting Electrodes by an electrostatic force

The accumulated dusts, due to the impact strength of hammer rapping to the Collecting Electrodes, are dropped and collected in the hopper.

Electro Static Forces

The Surface Forces on A Fluid Flow

• The Surface forces are characterized by length scales relevant to the microscopic dynamics.

• This would mean that such forces decay extremely rapidly with distance on the macroscopic scale.

• The short-ranged forces will decay on distances of the order of a mean free path.

• Any infinitesimal element in a continuum description is, by definition, must be much larger than all microscopic scales.

• The effect of such forces would be negligible unless two interacting elements are directly in contact.

• These forces are expected to act across the contact surface.

• In other words, the short-ranged forces manifest as surface forces, in a continuum description.

Concept of Surface & Fluid Particle Interaction

Diffuse reflection

U2

U U

Φ

U2

Φ

U1

U1

Φ

U2

Specular reflection

• Perfectly smooth surface (ideal surface) Real surface

• The surface forces are defined for a combined solid and fluid system.

• The fluid packets close to a solid wall tend to reach mechanical equilibrium with the wall.

• The fluid particles will exchange maximum possible momentum flux with the solid wall.

• A small layer of fluid particles close the the wall come to Mechanical, Thermal and Chemical Equilibrium With solid wall.

• Fundamentally this fluid layer is in Thermodynamic Equilibrium with the solid wall.

Flux Nature of Surface Forces

• The surface forces act to transport momentum across the boundaries of an infinitesimal element.

• In dilute gases, the momentum transport occurs due to molecules randomly crossing the boundary.

• Carry momentum across in the appropriate direction.

• Often referred to as the kinetic contribution to the stress.

• In liquids, the transport of momentum can occur without physical translation of molecules via short-ranged forces acting between pairs of molecules on either side of the boundary.

• Separated by a distance comparable to the range of the inter-molecular potential.

• Considered as the potential contribution to the stress.

• Clearly, the total effect of short-ranged forces acting on a differential element is decided by its surface area rather than the volume.

The System of Surface Forces in Fluid Flows

• Mechanical forces.

• Electro-kinetic forces.

• Electroosomosic Forces

• Electrophoresic Forces

Surface Forces

tV

SB FFdVvvt

v

.

The Second term on the right-hand side of above equation represents the resultant surface force acting on the entire control surface. As discussed before, force is an extensive quantity and hence surface force is proportional to surface area.

CS

ss dFF

CS

s dSnF

ˆ

Depending on the global surface under study, this gives a force vector which can be decomposed into a normal and a shear forces.

dFs is written as a scalar product of the stress tensor and area vector acting on the surface element ds:

Components of Forces

.ˆ.ˆˆ tpnn

n is the normal unit vector that points away from the surface.t is the tangential unit vector. The negative signs of n and t have been chosen to indicate that the pressure p and the shear stress are exerted by the surroundings on the surface S. Thus, the surface force acting on a differential surface is:

dStdSpndFs

.ˆ.ˆ

Dealing with Surface Forces

B

SSV

FdStpndSvvndVt

v

CCC

.ˆ.ˆ.ˆ

In above equation, the integration must be carried out over the entire control surface. Fora control surface consisting of inlet, exit, and wall surfaces, the second integral on the left-hand side gives:

CWCoutCinC SSSS

dSvvndSvvndSvvndSvvn .ˆ.ˆ.ˆ.ˆ

tV S

B

C

dStpnFdVvvt

v

.ˆ.ˆ.

Momentum Conservation Equations for Fluid Flows

tV

SB FFdVvvt

v

.

The surface forces are those applied by external stresses on the sides of the element.

3

333

3

322

3

311

2

233

2

222

2

211

1

133

1

122

1

111

xe

xe

xe

xe

xe

xe

xe

xe

xe

dV

Fd

Thus the total vector surface force is defined as surface force per unit volume

j

ijijsurface x

f

Differential Form of Momentum Conservation Equations for Fluid Flows

tV

ij

tVtV

dVdVgdVvvt

v

.

ijgvvt

v

.

From the definition of a fluid, the stresses parallel to planes must vanish if the fluid is at rest. Thus the shear stresses are zero, and the normal stresses become equal to the hydrostatic pressure:

p 332211

ij as drepresente is for jiij

We must ensure that the stresses in general flow reduce to this special case when the velocity is zero.

The Fluid at Rest : Clue From Pascal’s First Law

0 ij