Dimensionless Numbers & Their Application.pptx

8/14/2019 Dimensionless Numbers & Their Application.pptx

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By: Adeel-ur-RehmanAmir Bashir

Hamood Ahmad

Safdar Abbas


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What are DIMENSIONLESS NUMBERS?

Some examples of dimensionless numbers.


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Why are they used?


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The concept was introduced by Gabriel Stokesin 1851. But Reynolds number is named afterOsborne Reynolds who popularized its use in1883.


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Definition Reynolds Number is a dimensionless number that

gives a measure of the ratio of inertial forces toviscous forces.

cesviscousfor

rcesinertialfoRe


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Mathematical form:

D= diameter of the pipe

u= velocity of the fluid

p= density of the fluid

= viscosity of the fluid

DupRe


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It is primarily used to analyze different flowregimes namely laminar, turbulent or both.

If Re


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Named after the ancient Greek scientistArchimedes Definition

It is defined as the ratio of gravitational forces toviscous forces.


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The mathematical form is given by:

Where g=gravitational acceleration (9.8 m/s2).

l= density of the fluid.

= density of the body. = dynamic viscosity.

L = characteristic length of body.

3

3 )(

ll pppgLAr


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It is used to determine the motion of fluiddue to density differences

When analyzing potentially mixed convectionof a liquid, the Archimedes numberparameterizes the relative strength of freeand forced convection.

When Ar>> 1 natural convection dominates, i.e.less dense bodies rise and denser bodies sink,

when Ar


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Named after German Engineer Franz Grashof Definition

The Grashof number (Gr) is a dimensionlessnumber in fluid dynamics which approximates the

ratio of the buoyancy to viscous force acting on afluid.


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For vertical flat plate

Where g= acceleration due to Earths gravity

= volumetric thermal expansion coefficient

Ts= surface temperature

T= bulk temperature L= length ofchoosen scale

v= Kinematic viscosity

2

3)(

v

LTTgGr sL


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For Pipe

Where g= acceleration due to Earths gravity.

= volumetric thermal expansion coefficient.

Ts= surface temperature.

T= bulk temperature. D= diameter of chosen scale.

v = Kinematic Viscosity

2

3)(

v

DTTgGr SD


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It used in analyzing the velocity distribution infree convection systems.

When Gr >> 1, the viscous force is negligible

compared to the buoyancy and inertial forces.When buoyant forces overcome the viscousforces, the flow starts a transition to theturbulent.

At higher Grashof numbers, the boundary layer isturbulent; at lower Grashof numbers, theboundary layer is laminar.


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Named after Austrian physicist andphilosopher Ernst Mach.

Definition In fluid mechanics, Mach number( or ) is a

dimensionless quantity representing the ratio ofspeed of an object moving through a fluid and thelocal speed of sound.


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Its mathematical form is given as:

Where M is the Mach number,

v is the velocity of the source relative to themedium.

vsound

is the speed of sound in the medium.

soundv

vM


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The Mach number can be used to determine if aflow can be treated as an incompressible flow.

If M < 0.20.3 and the flow is steady,

compressibility effects will be small and fluid isincompressible.

The Mach number is commonly used both with

objects traveling at high speed in a fluid, andwith high-speed fluid flows inside channels suchas nozzles


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If Mach no =1 then condition is sonic

Mach no


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Named after German physicist LudwigPrandtl Definition

It is the ratio of kinematic viscosity to thermal

diffusivity.

Pr = Kinematic Viscosity/Thermal Diffusivity


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Its mathematical form is given by:

Where Cp= Specific heat

= Dynamic viscosity k = Thermal conductivity

kCpPr


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The prandtl number controls the relativethickness of the momentum and thermalboundary layer.

When Pr is small, it means that the heatdiffuses very quickly compared to thevelocity.


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Dimensionless Numbers & Their Application.pptx

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