Analogies

Ajay Pal SharmaAjay Pal Sharma

Assistant ProfessorAssistant Professor

Department of Chemical EngineeringDepartment of Chemical Engineering

Seth Jai Parkash Mukund Lal Institute of Seth Jai Parkash Mukund Lal Institute of Engineering & Technology, Radaur.Engineering & Technology, Radaur.

Yamunanagar .(Haryana)Yamunanagar .(Haryana)

INDIAINDIA

Heat ,mass and momentum Heat ,mass and momentum transfertransfer

Unit process can be classified into three fundamental Unit process can be classified into three fundamental transfer processes.transfer processes.

Momentum transferMomentum transfer

Heat transferHeat transfer

Mass transferMass transferdz

dcDj 1

1

dz

dTkq

dz

du

AnalogiesAnalogies

General molecular transport equation:General molecular transport equation:

All three of the molecular transport process of All three of the molecular transport process of momentum, mass and heat are characterized momentum, mass and heat are characterized by the same general type of equation.by the same general type of equation.

sistanceRe

force Drivingprocess transfer of Rate

AnalogiesAnalogies

Molecular diffusion equation for momentumMolecular diffusion equation for momentum::

momentum transferd/s.m=momentum transferd/s.m22= حح

μμ//ρρ is kinematic viscosity in m is kinematic viscosity in m22/s/s

z is the distance in mz is the distance in m

vvx x ρρ is momentum/m is momentum/m3 3

where the momentum has the units of kg. m /s.where the momentum has the units of kg. m /s.

dz

vd x )(

AnalogiesAnalogies

Molecular diffusion for heat conduction for Molecular diffusion for heat conduction for constant cconstant cp p and and ρρ

q/A is heat flux in W/mq/A is heat flux in W/m22

αα is thermal diffusivity in m is thermal diffusivity in m22/s /s

ρρ c cp p T is J/mT is J/m33

dz

Tcd

A

q p )(

AnalogiesAnalogies

Molecular diffusion equation for mass Molecular diffusion equation for mass transfer transfer

jjAZAZ is molar flux of component A due to molecular diffusion in is molar flux of component A due to molecular diffusion in kg mol A/ s.mkg mol A/ s.m22

DDABAB is the molecular diffusivity of molecules A in B in m is the molecular diffusivity of molecules A in B in m22/s/s

CCA A is the concentration of A in kg mol/mis the concentration of A in kg mol/m33

z is the distance o f diffusion in metersz is the distance o f diffusion in meters

dz

dCDj

A

ABA

AnalogiesAnalogiesAll three molecular transport equations are All three molecular transport equations are identical. There is mathematical analogy identical. There is mathematical analogy between these equation but the actual between these equation but the actual physical mechanism occurring is totally physical mechanism occurring is totally different .E.g. In the mass transport two different .E.g. In the mass transport two components are being transported by components are being transported by relative motion .In heat transfer, molecules relative motion .In heat transfer, molecules are relatively stationary and transport is are relatively stationary and transport is taken by electrons. Transport of momentum taken by electrons. Transport of momentum is occurred by several types of mechanism.is occurred by several types of mechanism.

AnalogiesAnalogiesAll the fluxes are on the left hand side ofAll the fluxes are on the left hand side of

three equations. All fluxes have the samethree equations. All fluxes have the same

units i.e. quantity transferred /per unit timeunits i.e. quantity transferred /per unit time

per unit area .per unit area .

The transport properties The transport properties μμ//ρρ,,αα and D and DABAB have the units of m2/s.All concentration are represented as momentum/m3, j/m3, kgmol/m33.

AnalogiesAnalogies Since basic mechanism of heat, mass and Since basic mechanism of heat, mass and

momentum transport is essentially same, it is momentum transport is essentially same, it is some times possible to directly relate heat some times possible to directly relate heat transfer coefficients, mass transfer coefficients transfer coefficients, mass transfer coefficients and friction factors by means of analogies.and friction factors by means of analogies.

Analogy involving momentum transfer is only Analogy involving momentum transfer is only valid if there is no form drag, hence these are valid if there is no form drag, hence these are limited to flow over flat plates and inside limited to flow over flat plates and inside conduitsconduits..

AnalogiesAnalogiesTurbulent diffusion equation for momentum, Turbulent diffusion equation for momentum, heat and mass transferheat and mass transfer

dz

dcDj

dz

Tcd

A

q

dz

vd

A

MABAZ

p

t

z

x

tzx

)(

)(

)(

AnalogiesAnalogies

The most simple and crude analogy in The most simple and crude analogy in turbulent diffusion equations is that turbulent turbulent diffusion equations is that turbulent eddies transport or eddies diffusivities are eddies transport or eddies diffusivities are same for all modes of transport.same for all modes of transport.

mtt Momentum eddy diffusivity

Thermal eddy diffusivity

Mass eddy diffusivity

AnalogiesAnalogiesAnother analogy probably the oldest is the Another analogy probably the oldest is the

““Reynolds Analogy”. This relates fanning friction Reynolds Analogy”. This relates fanning friction factor for fluid flow to heat transfer.factor for fluid flow to heat transfer.

Fanning friction factor can be defined as shear Fanning friction factor can be defined as shear stress at the surface divided by the product of stress at the surface divided by the product of density times velocity head.density times velocity head.

2/2vf

s

Fanning friction factor

AnalogiesAnalogies

For momentum transfer For momentum transfer

For fluid flow in a pipe, heat transfer equation from For fluid flow in a pipe, heat transfer equation from fluid to wall can be written asfluid to wall can be written as

dz

dvt)(

dz

dttCp

A

q)(

AnalogiesAnalogies

After dividing the momentum transfer equation After dividing the momentum transfer equation with heat transfer equation and by assuming with heat transfer equation and by assuming thermal diffusivity and momentum diffusivity thermal diffusivity and momentum diffusivity negligible and equal eddy diffusivities we get a negligible and equal eddy diffusivities we get a equationequation

dvCpdTAq

)/

(

AnalogiesAnalogiesIntegrating between the T=Ti and v=0 to Integrating between the T=Ti and v=0 to some point where T is the same as the some point where T is the same as the bulk T and assume that velocity at this bulk T and assume that velocity at this point is same as average velocity.point is same as average velocity.

2/2

)(/

0)(/

avs

i

avi

s

fv

TThAq

vTTCpAq

AnalogiesAnalogies

This become Reynolds Analogy. It This become Reynolds Analogy. It postulates direct interaction postulates direct interaction between the turbulent core of the between the turbulent core of the flow and the walls.flow and the walls.

Gc

h

vc

hf

pavp

2

AnalogiesAnalogiesRight hand side term in the Reynolds analogy is Right hand side term in the Reynolds analogy is the Stanton number. Stanton number is a the Stanton number. Stanton number is a dimensionless group made up of other more dimensionless group made up of other more familiar groups. Reynolds analogy gives familiar groups. Reynolds analogy gives reasonable values for gases where Prandtl reasonable values for gases where Prandtl number is roughly one.number is roughly one.

u

kD

LuD

Lk

Sc

ShSt

Gc

h

c

k

dGk

hdNuSt

cc

M

pp

H

Re

PrRe

AnalogiesAnalogies

Nusslet number:Nusslet number: It establishes the It establishes the relation between convective film coefficient relation between convective film coefficient h, thermal conductivity of fluid k and length h, thermal conductivity of fluid k and length parameter d of a physical system.parameter d of a physical system.

It is also interpreted as the ratio of It is also interpreted as the ratio of temperature gradient to an overall temperature gradient to an overall reference temperature gradient. reference temperature gradient.

AnalogiesAnalogies

Reynold number: Reynold number: It is ratio of inertial It is ratio of inertial force to viscous forces in fluid motion. At force to viscous forces in fluid motion. At low Reynold numbers viscous effects low Reynold numbers viscous effects dominate and fluid motion is laminar. At dominate and fluid motion is laminar. At high Reynold numbers inertial effects high Reynold numbers inertial effects leads to turbulent flow.leads to turbulent flow.

NNReRe< 2100< 2100

NNRe Re > 4000> 4000

Laminar flow

Turbulent flow

AnalogiesAnalogies

Prandtl number:Prandtl number: It is ratio of kinematic It is ratio of kinematic viscosity to thermal diffusivity of a fluid. It viscosity to thermal diffusivity of a fluid. It indicates the relative ability of the fluid to indicates the relative ability of the fluid to diffuse momentum and internal energy by diffuse momentum and internal energy by molecular mechanism. Prandtl number is molecular mechanism. Prandtl number is connecting link between velocity field field connecting link between velocity field field and the temperature field and its value and the temperature field and its value strongly influences relative growth of strongly influences relative growth of velocity and thermal boundary layers. velocity and thermal boundary layers.

AnalogiesAnalogies

Schmidt number: Schmidt number: It is the ratio of shear It is the ratio of shear component of diffusivity component of diffusivity μμ//ρρ to the to the diffusivity of mass transfer. Values of diffusivity of mass transfer. Values of Schmidt number for gases range from Schmidt number for gases range from about 0.5 to 2. For liquids Schmidt about 0.5 to 2. For liquids Schmidt numbers range from about 100 to over numbers range from about 100 to over 10000 for viscous liquids.10000 for viscous liquids.

AnalogiesAnalogies

Reynolds analogy breakdown if viscous Reynolds analogy breakdown if viscous sub layer become important, because sub layer become important, because eddy diffusivities diminish to zero and eddy diffusivities diminish to zero and molecular diffusivities becomes important.molecular diffusivities becomes important.

Prandtl modified the Reynolds analogy by Prandtl modified the Reynolds analogy by writing regular diffusion equations for writing regular diffusion equations for viscous sub layer and Reynolds analogy viscous sub layer and Reynolds analogy equation for turbulent core region.equation for turbulent core region.

AnalogiesAnalogies

Viscous sub layer is the region where the Viscous sub layer is the region where the velocity is proportional to distance from the velocity is proportional to distance from the wall. The second region called buffer layer wall. The second region called buffer layer is the region of transition between the is the region of transition between the viscous sub layer with practically no eddy viscous sub layer with practically no eddy activity. Third region called turbulent core activity. Third region called turbulent core has violent eddy activities has violent eddy activities

Boundary layer in pipeBoundary layer in pipe

Viscous sub layer

Buffer zone

Turbulent core

Viscous sub layer Buffer layer

Turbulent core

y+

V+

Dim

ensi

onle

ss v

eloc

ity r

atio

Dimensionless number

AnalogiesAnalogiesIf the laminar sub layer is included in Reynolds If the laminar sub layer is included in Reynolds analogy, the Prandtl analogy applies. Prandtl analogy, the Prandtl analogy applies. Prandtl analogy included the ratio of mean velocities in analogy included the ratio of mean velocities in laminar sub layer and core as well as the laminar sub layer and core as well as the Prandtl number for heat transfer. If the Prandtl Prandtl number for heat transfer. If the Prandtl number is reduces to one it becomes Reynolds number is reduces to one it becomes Reynolds analogy.analogy.

)1(/1

2/

prxmxδ

st

Nuu

fN

Analogies Analogies

Von korman modified the Prandtl analogy Von korman modified the Prandtl analogy by considering buffer layer in addition to by considering buffer layer in addition to viscous sub layer and turbulent core.viscous sub layer and turbulent core.

Chilton and Colburn J-factor analogy is Chilton and Colburn J-factor analogy is most successful and widely used analogy. most successful and widely used analogy. This analogy is based on correlations and This analogy is based on correlations and data rather than assumptions about data rather than assumptions about transport mechanism.transport mechanism.

AnalogiesAnalogies

Chilton Colburn j-factor analogy is simple Chilton Colburn j-factor analogy is simple analogy. This analogy is valid for turbulent flow analogy. This analogy is valid for turbulent flow in conduits with N in conduits with N ReRe > 10000, 0.7< N > 10000, 0.7< N Pr Pr > 160> 160

and tubes where L/d > 60and tubes where L/d > 60

MH

smooth

jjf

2

Thanks.Thanks.