DIMENSIONLESS NUMBERS CLASS NOTES FOR ENGINEERS
DIMENSIONLESS NUMBERS
CLASS NOTES FOR ENGINEERS
Convention equations in terms of
non-dimensionless numbers reduce
the number of variables. This makes the
equations convenient and easy to handle.
It reduces the time and level of difficulty.
Dimensionless numbers do not have
any units. Use Reynolds number
to know laminar, turbulent or transition
flow. In every dimensionless number,
Numerator is always inertia force and
Denominator is some other force.
NUSSELT NUMBER
Nusselt number describes the mechanism of convection. Convective heat transfer is greater than conduction across the same fluid stationary layer.
Nu=Convective Heat Flux in a moving fluid/conductive Heat Flux in the same stationary fluid
Nu= hl/k_{f}
The higher the Nu number, the more is the HT by convection. Nusselt number is higher for a turbulent flow than a laminar flow.
REYNOLD NUMBER
It describes the type of flow/laminar/transition/turbulent.
Re=Inertia force / viscous force
Procedure to find Reynolds number for a plate and pipe is
Plate Re=ρvl/µ
Pipe Re= ρvd/ µ
(i) When Re is low, viscous forces are more significant and the flow is laminar.
(ii)As Re increases, inertia forces are more predominant and the flow is turbulent.
CRITICAL REYNOLDS NUMBER
Reynolds number where the laminar region ends. Its value is 5 x 10^{5}for a horizontal plate. Otherwise its value is different for different geometries.
GRASHOFF’S NUMBER
Gr= Buoyant force x inertia force/ (viscous force)^{2}
Gr= L^{3}g β ΔT/ν^{2}
Where
L is the length of the plate
g is acceleration due to gravity
β = 1/T_{av}
ΔT is the temperature difference
ν is the kinematic viscosity
PRANDTL NUMBER
Pr= momentum diffusivity/Thermal diffusivity
Pr =µc_{p}/k_{f}
Take Pr=0.7 for gases if not given
Take Pr = 10 for water if not given
Data on Prandtl number
Fluid Range of Pr Number
Liquid metals 0.004 to 0.030
Gases 0.7 to 1.0
Water 1.7 to 13.7
Light organic fluids 5 to 50
Oils 50 to 100000
Glycerin 2000 to 100000
Prandtl number best describes the relation between velocity boundary layer thicknesses to thermal boundary layer thickness. (δ_{th}=δ Pr ^{-1/3}). When Pr=1 means rate of momentum transfer and heat transfer are same.
RAYLEIGH NUMBER
Ra=Re Pr
BIOT NUMBER
Bi= convective HT in moving fluid/conductive HT in a solid surface
Bi = hl/k_{s}
Use of Biot number exists in the study of
(i) 1- D unsteady conductive HT
(ii) In fins of varying Cross section
FOURIER NUMBER
Fo= αt/L_{c}^{2}
Where L_{c} is characteristic length = surface area/ volume
Firstly L_{c} = δ/2 for a plane wall
Secondly L_{c}= r/4 for a cylinder
Thirdly L_{c}= r_{0}/3 for a sphere
Fourthly L_{c}= a/6 for a cube
Use of Fourier number exists in the analysis of
(i) 1- D unsteady state conduction
(ii) In fins of varying cross section
STANTON NUMBER
St=Nu/ (Re Pr)
Reynolds, Colburn and Prandtl Analogies use Stanton number in convection.
FROUDE NUMBER
The Froude Number is a dimensionless number. It is the ratio of inertia force on an element of fluid to the weight of the fluid element. Mathematically
Fr= (Inertia force/gravitational force)^{0.5}_{ }
Fr=v/ (gl)^{0.5}
^{WEBER NUMBER}
W_{b}= Inertia force/ Surface tension
W_{b} =ρv^{2}l/σ
EULER NUMBER
The Euler Number (Eu) is a dimensionless number. It gives a relationship between a local pressure drop & the kinetic energy of flow per unit volume. Eu is zero for a perfect frictionless flow. It gives the energy losses in the flow.
Eu = pressure force/inertia force
Eu=ρv^{2}/p
PHYSICAL SIGNIFICANCE OF PRANDTL NUMBER
Prandtl number signifies the thickness of thermal boundary layer. It also connects with the thickness of hydro boundary layer.
firstly if Pr =1, δ =δ_{th}
Secondly if Pr <1 then δ < δ_{th}
Thirdly if Pr >1 then δ > δ_{th}
PHYSICAL SIGNIFICANCE OF ‘GRASHOFF’S NUMBER
Grashoff’s number (Gr) is the ratio of the buoyancy to viscous force acting on a fluid layer. Used in free convection. Its value indicates whether the Buoyancy force is more predominant or the viscous force. Used where Buoyancy force is more predominant. Grashoff’s number is similar to to the Reynolds number in forced convection. Used in fluid dynamics and heat transfer.
Gr = βg Δt l^{3}/ ϑ^{2}
Buoyancy force is due to density difference. The temperature difference causes the density difference. Greater is the temperature difference, greater will be density difference. Greater will be the Grashoff’s number.
Physical Significance of Nusselt number
More is the Nusselt number, more is the heat transfer by convection than conduction. Value of Nusselt number as one represents a case of heat transfer by pure conduction. Nusselt number increases as velocity of flow increases. Nusselt number for a turbulent flow is much higher than that for a laminar flow.
Physical Significance of Reynolds Number
Reynolds number classify the type of flow as laminar, transition and turbulent. It gives the effect of viscosity on velocity of flow. More is the viscosity, lesser is the velocity of flow and vice-versa.
Physical Significance of Rayleigh’s number
This depends on the Buoyancy driven flow. It is equal to the product of Grashoff’s number and Prandtl number. This governs free or natural convection. Its value is large, normally 10^{6} to 10^{8}. Rayleigh critical value is 1708. Below this value, there is no flow and hence no convection.
Physical significance of Biot Number
It is the ratio of internal resistance to external resistance. It means for small value of Biot number, internal resistance is less. The temperature gradient is less.
Develop the model of a prototype with the help of these numbers. A prototype means an actual object. A prototype can be a big object like an airplane, a dam and a turbine. A model in case of a big object is small. Make the model and test it for geometric, kinematic and dynamic similarity in terms of dimensionless numbers. These numbers help to achieve predicted performance for the actual prototype. Then make the prototype and put to practical use. Hence a model can be smaller or bigger than the actual object (Prototype).
TABLE: Dimensionless numbers
Sr. No. |
Dimensionless number |
Definition |
Symbol |
Formula |
Practical application |
1. |
Reynoldsnumber |
Inertia force/viscous force |
R_{e} |
ρ v L / µ |
Flow of viscousfluids in pipesand over a flatplate |
2. |
Froudenumber |
(Inertia force/Gravity force)^{0.5} |
Fr |
v/(Lg)^{0.5} |
Flow in openchannels |
3. |
Eulernumber |
(Inertia force/pressure force)^{0.5} |
Eu |
v/(P/ρ)^{0.5} |
Flow in pipes/tubes |
4. |
Webernumber |
(Inertia force/Surface tension force)^{0.5} |
We |
v/(σ/ρL)^{0.5} |
Flow throughcapillary tube |
5. |
Machnumber |
(Inertia force/Elastic force)^{0.5} |
M |
(v/(σ/ρ))^{0.5} |
High speedflow of fluids |
6. |
Schmidtnumber |
momentum diffusivity/mass diffusivity |
Sc |
μ / ρD |
Schmidt numberis analogous toPrandtl number |
Every dimensionless number in fluid flow is the ratio of inertia force to some other force as described below. There are in all six forces. Accordingly, there are five dimensionless numbers in fluid mechanics.
1. Reynolds number
It is a ratio of inertia force to viscous force.
Inertia force = mass x acceleration
= ρ Vol v/t = ρ (Vol/t) v
= ρ A v v= ρ A v^{2}
Viscous force = Shear stress x area
= τ x A = µdu/dy A = µ (v/L) A
Re = ρA v^{2} / µ (v/L) A = ρ v L / µ
Reynolds number signifies that viscous forces play a significant role or the fluid is viscous.
It analyzes different types of flow namely Laminar, Turbulent, or Transition.
(i) When Viscous forces are dominant, it is a laminar flow
(ii) If Inertial forces are dominant, it is a Turbulent flow.
Practical applications of viscous flow are
(a) Low velocity fluid flow around the airplanes and automobiles
(b) Motion of fluid around a completely submerged submarine
(c) Flow in a high speed centrifugal compressor
(d) In-compressible fluid flow in a small diameter pipe (capillary tube)
2. Froude number
It is square root of the ratio of inertia force to gravity force.
Fr =(Fi / Fg) ^{0.5} = ( ρA v^{2}/m g)^{0.5} = v/ (Lg)^{0.5}
Froude number is applicable where gravity forces play a significant role.
Practical applications where gravity force play a significant role are
(a) Spillway of a dam
(b) Notches and weirs
(c) Fluid flow in open channels
(d) Motion of a fluid around a ship in the ocean
3. Euler number
It is square root of ratio of inertia force to pressure force.
Eu =(Fi / Fp )^{0.5} = (ρA v^{2}/p A)^{0.5} = v/(P/ρ)^{0.5}
It is applicable where pressure forces are predominant.
Such practical applications are
(a) Flow through pipes, orifices, mouthpieces and sluice gates
(b) Water hammer phenomenon in a pen-stock
(c) Pressure rise due to sudden closure of valves
4. Weber number
It is the square root of the ratio of inertia force to surface tension force.
We = (Fi / Fs)^{0.5} = (( ρA v^{2}/σ L)^{0.5} = v/(σ/ρL)^{0.5}
Its practical applications are
(a) Flow in capillary tube
(b) Blood flow in veins
5. Mach number
Square root of ratio of inertia force to the elastic force.
M = (Fi /Fe)^{0.5} = ((ρA v^{2}/σ x A)^{0.5} = v/( σ/ρ)^{0.5}
σ/ρ = velocity of sound in the fluid
Practical applications for elastic force are
(a) Compressible fluid flow problems at high velocities
(b) High velocity fluid flow in pipes
(c) Fluid motion across high speed projectiles and missiles
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