New 2012 04 Rabibrata Bio Transport

7/31/2019 New 2012 04 Rabibrata Bio Transport

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RABIBRATA MUKHERJEE

Department of Chemical Engineering

Indian Institute of Technology Kharagpur

E-mail: [email protected]

Basic Pr inciples of TransportPhenomena:

w it h relevance t o Living Syst ems

BS20001 Science of Living Systems


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Lecture Description:This lecture presents an introduction to the principles of heat,mass and momentum transfer and their relevance in living

systems.

Lecture Objective:

Learn the fundamental conservation principles and constitutivelaws that govern heat, mass and momentum transport

processes in fluids;

The key constitutive properties

Text:

Fundamentals of Heat and Mass Transfer by F. P.

Incropera and D. P. Dewitt, Fifth Edition, Wiley India


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Transport Phenomena:

The classical transport phenomena involves thermal

transport and diffusion mass transfer in conjugation with

momentum transfer (also identified as fluid flow).

Glossary : Fluid, Fluid Flow, Momentum Transfer

Examples:

Fluid Flow: Flow through a tube/ pipe/ open channel flow ofriver etc.

Heat Transfer: Heating of a Block of Solid or a Can of Liquid

or Feeling warm under the Sun.

Mass Transfer: Salt Dissolving in water, distillation,

absorption, adsorption, leaching etc.


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Examples of Transport

Phenomena in Biological

Systems:

- CARDIOVASCULAR SYSTEM

- RESPIRATORY SYSTEM

- LYMPHATIC SYSTEM

- OTHER SMALLERCANALIZATIONS WITH FLUID

MOTION


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Constitution of Blood:Its not a simple liquid like Water. It contains variety of Cells, most notably

the RBC and WBC.

Flow of Blood

Supplying Dissolved Oxygen to theCells/ or removing Carbon dioxide.

Blood Cooling:

When blood flows through

tissues or organs, it

functions not only as a

carrier of nutrients and

metabolic wastes but also asa coolant to remove the heat

produced by metabolism.

Blood gains heat which is

transferred by circulation tothe skin where it is dissipated

to the environment

The interaction between these particles is critical


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Basic Concepts in Fluid Flows

Different types of Flow:

(1) Steady and Unsteady

(2) Uniform and Non Uniform

(3) Internal and external Flow

(4) Compressible and incomprissible

(5) Inviscid and Viscous

(6) Laminar and Turbulent

(7) Single phase flow vs. 2 phase flow.

In order to understand the Science of Bio Transport

Processes, we need to understand the Basics of Fluid Flow

The Basic Governing Equations : Continuity and Conservation of

Momentum

Boundary Layers


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Flu id :A material that flows

Flow:Bounded f low (Flow through a conduit): Internal FlowUnbounded flow (free surface flow): External Flow

Flow Characterizat ion: To obtain velocity profile i.e. velocity components in x, y,

z and t-coordinates/ t, r, , z co-ordinates.

Temperature profile as

a function of time and

space.

Concentration profile

as a function of time

and space.


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Physical properties associated

Density (), Viscosity (), Specific Heat (CP), Thermal

conductivity (K), diffusivity (DAB

), Surface Tension ( )etc

Governing Equations:

Overall mass balance equation known as Equation ofContinuity. (1)

Momentum balance equations (in three directions). (2)

Overall Energy Equation. (3)Species conservation equation/mass balance equation (4)

(1) + (2) Velocity profile (u, v, w)

(1) + (2) + (3) Temperature profile(1) + (2) + (4) Concentration profile

Coupled PDEs which may be decoupled for some simple

cases.


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Frame of references:Before solving a fluid flow problem fix up the co-ordinate system.

Lagrangian Approach:

Moving frame of reference, where the kinematic behavior of each particle

is identified by its initial position ( ).

Eulerian Approach:

Fixed frame of reference, it seeks the velocity and its variation at each and

every location in the flow field.

We deal with mostly Eulerian approach.

Kinematics: Geometry of Motion


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Types of Flow:

Steady FlowUnsteady Flow

However, whether a flow is steady or Not largelydepends on the Frame of reference.

Uniform and Non Uniform Flow.

(When velocity and other hydrodynamic parameters donot change from point to point within the flow field)

Compressible and Incompressible Flow.

Internal and External Flow.


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Material or Substantial Derivative:

Position of a Particle given in the flow field by space co-ordinates as

u, v, w are the three components of velocity

After time t, let the particle move to position (x + x, y + y and z + z)

Corresponding velocity components are (u + u, v + v and w + w)

u + u = u (x + x, y + y, z + z, t + t)

v + v = v (x + x, y + y, z + z, t + t)

w + w = w (x + x, y + y, z + z, t + t)

x = u t

y = v t

z = w t


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= ax

Local or

Temporal

Acceleration

Convective Acceleration

Fluid Acceleration has two

Components:

Temporal Acceleration and

Convective Acceleration


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Multiplicity of TubeBranching The branched

networks of tubes from

the cardiovascular system

and lungs are extremely

intricate and complex.

Every time Blood/ Fluid

enters a narrower tube,

there is some convective

acceleration


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Geometric description of the flow field

Flow Field: An area over a liquid/ fluid flow is

occurring.

Streamlines:

An imaginary line in the flow field suchthat tangent at every point gives the

direction or velocity vector.

Pathline:Trajectory of a particular fluid particle in the

flow field. Identity of a particle, Tracer

experiment.

Streakline:A streakline at any given instant of time in the

locus of the temporary location of all particles

who have passed through a fixed point earlier

in the flow field.


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St ream funct ion

As such the flow or the particles that movealong the streamlines.

u = /y v = /xas follows from a consideration of =constant and take the differential d = 0.

Analytically, the stream function is amathematical device to satisfy thecontinuity equation identically (note thatux + vy = 0 automatically)


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Possible Movement/ Deformation modes of a

Fluid Particle

Translation

Translation and Rotation

without deformation

Represents Rigid BodyDisplacement

Translation with Linear Deformation

Strain


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Fluid ParticleTranslation with Linear and Angular Deformation

(Rate of Angular Deformation) =

Sign Convention:

ACW is + ve


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Fluid ParticleTranslation with Linear and Angular Deformation

Under the specific Condition

The Line segments AB and AD are moving with the same angular velocity and

therefore, this is a case of PURE ROTATION


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Fluid Particle

Rotation of a Fluid Element is defined as the arithmetic

mean of angular velocities of two perpendicular linear

segments meeting at that point

Under the specific ConditionIrotational Flow Field


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Rotational Flow:In a 3-D flow field:

Similarly,

Rotation Vector:

angular velocity (x-component)

1

2

x

yzvv

y z

=

=

1

2

1

2

x zy

y xz

v v

z x

v v

x y

=

=

( )1

2v =

rr r( ) Curl of

x y z

i j k

v vx y z

v v v

= =

r r r

Vorticity: Twice of Rotation Vector

u and vx same


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When components of rotation vector ofeach point of flow field is equal to zero,

flow is termed as Irrotational flow.

So, for irrotational flow.2 v = =

r r

0 =


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Equation of Continuity

Consider a parallelepiped (control volume)

Let a fluid enters face ABCD with velocity vx and density .

Fluid leaves face EFGH, with velocity

So, rate of mass entering the CV through ABCD =

and densityxxv

v dx dxx x

+ +

xv dydz

z

x

dx

H

G

F

E

D

C

B

Ady

dz

y


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Rate of mass leaving through EFGH:

Hence net rate of mass efflux (out - in) in x direction

Similarly, the net rate of mass efflux in,

y-direction is

xx

vdx v dx dydz

x x

= + +

( )x xv v dx dydz

x

= +

( )x x xv v dx dydz v dydz x

= +

( )xv dxdydzx

=

( )yv dxdydzx

=


z

x

dx

H

G

F

E

D

C

B

Ady

dz

y


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net rate of mass efflux in,z-direction is

Net rate of accumulation in CV is

So, mass conservation equation is:

For an incompressible flow,

For a steady state flow,

For an incompressible, steady state flow:

( )zv dxdydzx

=

dxdydzt

=

() 0t

=

( ), ,x y z


z

x

dx

H

G

F

E

D

C

B

Ady

dz

y


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Conservation of momentum (EOM): for an Fluid

Direct Consequence of Newtons Second Law

where and from definition

Now for a system with infinitesimal mass dm, Newtons Second Law can be written as

For a fluid we know that gets replaced with

Which implies

Fx is the TotalForce Acting in X

direction.

Inertial Terms


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Fx is the Total Force Acting in X direction.

The constituents of force in any particular direction

are the Surface Forces and the Body Forcesz

x

dx

H

G

F

E

D

C

B

Ady

dz

y

Body Force: Gravity

Electro Magnetic Forces

Surface Force:

Normal and Shear Stress,

Surface Tension etc.

gx dx dy dz

Pressure Gradient can be

handled as a part of NormalStress

Final Equation for an incompressible fluid is:


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There is Angular Deformation of the Liquid

The layer of the Liquid right adjacent to the solid surface attains the velocity

of the surface itself.

And, a stagnant layer tries to oppose the flow of the next adjacent layer.

This resistance to flow is an intrinsic property of the fluid, which in Simple

Terms is Known as viscosity.

No Slip Boundary Condition

Shear Stress and Viscosity

Liq

Plate 1

Plate 2

A fluid which has No viscosity is Known as

an Inviscid Fluid.


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A velocity gradient results in Shear Stress, which is imparted by the layer of

liquid on the next adjacent layer.

yx = Fx/AyRate of Angular Deformation = d/dt

Now tan d = dl/dyFor small d , tan d = d

Further: dl = du. dt

d = du. dt / dy

or (d /dt) = (du/dy)The Angular Deformation is caused

due because of the applied force,

which results in the shear stressyx

yx

(d/dt)

yx (du/dy)

yx = (du/dy)

Newtonian Fluid


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Conservation of momentum (EOM): for a Newtonian Fluid


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Different Rheological Behaviors


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Rheology of Blood

For a biologicalSystem like blood

the assumption of a

Newtonian Fluid isHARDLY valid.

E. W. Merril. Philosophical Rev. 49, 863, 1969


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Typical boundary condit ions for f luid f low :

5 types of boundary conditions for may appear in

fluid flow (based on the Physical condition)

They are:

1. A solid surface (may be porous)2. A free liquid surface

3. A vapor-liquid interface

4. A liquid-liquid interface

5. An inlet/outlet section


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Condition at solid surface:

If it is a stationary/impervious wall then,

If it is a moving surface with velocity u0 inx-direction whichis known as NO-slipboundary condition,

Constant wall temperature (CWT): T=Tw as the surface

Constant wall flux (CHF):

Insulated Surface:

0x y zv v v= = =

0, 0x y zv u v v= = =

0 constantT

k qy

= =

0 or 0 at the wall

T T

k y y

= =


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Cooling/heating at the wall:

Permeable wall:

at the wall,

( ) at the wallcT

k h T T y

=

tangential 0xv v= =

normal 0yv v=

Due to No slip


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2. Condition at liq-liq/liq-vapor interface:

At interface of two immiscible liquids:

At Liquid-vapor interface:

if 1 represents vapor

3. Inlet/outlet condition: May be specified

4. Physical B.C.:

1 2 1 2 1 2; ;v v T T = = =

1 2


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Mathematical Types of the Boundary Conditions:

1. Dirichlet B.C.:

Constant valued B.C.

2. Neumann B.C.:

Derivative of dependent variable is specified.

3. Robin-mixed B.C.:

Dependent variable & its derivative are

specified through an algebraic equation.

0 constantT

k qy

= =

( )at the wall

c

Tk h T T

y

=


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Non-dimensional NumbersRe=Reynolds number= Inertial forces/viscous forces=

Pr=Prandtl number=momentum diffusivity/thermal diffusivity=

Sc=Schmidt number=momentum diffusivity/mass diffusivity=

Heat transfer coefficient: h

Q=Heat flow rate=h*A*

Mass transfer coefficient: k

M= mass flow rate=kA

Nusselt number = convective to conductive heat transfer = hL/k

Sherwood number = convective to diffusive mass transfer = kmL/D

ud

/pc k

/

Tc


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Conservation of momentum (EOM): for an Fluid

Direct Consequence of Newtons Second Law

where and from definition

Now for a system with infinitesimal mass dm, Newtons Second Law can be written as

For a fluid we know that gets replaced with

Which implies

Fx is the TotalForce Acting in X

direction.

Inertial Terms


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VISCOUS FLOW

Equation of motion for viscous, incompressible flow:

For non-viscous (inviscid flow) flow:

Inertial term = pressure force term + body force term

For viscous flow:

Inertial term = pressure force term + body force term + viscous orshear force terms

In terms of Velocity gradient:

X-Comp.: 2 2 2

2 2 2

x x x x x x x

x y z

v v v v p v v vv v v

t x y z x x y z

+ + + = + + +


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Y-Component:

Z-Component:

Developing and fully developed flow:

Consider flow through a pipe. At entrance the uniform velocity u0.

As the fluid enters the pipe, the velocity of fluid at the wall is zerobecause no-slip boundary.

The solid surface exerts retarding shear force on the flow. Thus, thespeed of fluid close to wall is reduced.

2 2 2

2 2 2

y y y y y y y

x y z

v v v v v v vpv v v

t x y z y x y z

+ + + = + + +

2 2 2

2 2 2z z z z z z z

x y zv v v v p v v vv v vt x y z z x y z

+ + + = + + +


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At successive sections, effects of solid wall is felt further into the

flow.

A boundary layer develops from both sides of the wall

After a certain length, boundary layers from both surfaces meet at

the center and the flow becomes fully viscous. This length is

Entrance length.

For laminar flow: 0.06ReL

D= here, Re=

vD


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Beyond entrance length,

velocity profile does not change in shape and flow is termed as

Fully developed flow.

If flow is fully developed in x-direction, mathematically it is describedas,

For laminar flow, typically entrance length (L) is about a few cm.

0xv

x

=


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By Considering the Energy Balance about the Control Volume, it becomes

Possible to obtain the Energy Conservation Equation

Heat Transfer by Conduction/

Molecular Level Motion

Heat Transfer by Convection/

Due to Bulk Flow of the Liquid

A similar species about the Control Volume, one obtains the Species Transport

Equation


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Definition:

for flow over a flat surface, the boundary layer is defined as

locus of all points in the flow field such that velocity at each point is

99% of the free stream velocity.

Laminar Boundary Layers

u

u

u(x,y)(x)

L

y

x


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Laminar Boundary Layers

u

u

u(x,y)(x)

L

y

x

For an open Channel Flow

itself, you can cancel several

terms and you are eventually

left with:

Continuity equation:

If we regard order of u

Then comparing the two terms in the continuity equation

u


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Laminar Boundary Layersu

u

u(x,y)(x)

L

y

x

Look at the order of the Terms in the Eqn. of motionLHS 1: O(u). O(u)/O(L) ---- > O(u2/L)

LHS 2: O(v). O(u)/O() ---- > O(u/L). O(u)/O() ------ > O(u2/L)

RHS 1: O(u) / O(L). O(L) ---- > O(u/L2)

RHS 2: O(u) / O(). O() ---- > O(u/ 2)

> RHS 1

T b l f l


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Turbulent f low

Characterization of turbulent flow:

Irregular motion

Random fluctuation

Fluctuations due to disturbances, e.g., roughness of solidsurface

Fluctuations may be damped by viscous forces / may grow

by drawing energy from free stream

Re = DV/


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Critical Reynolds number:

Re < Recr:

Kinetic energy is not enough to sustain random fluctuations

against the viscous dampening. So, laminar flow continues

Re > Recr:

Kinetic energy of flow supports the growth of fluctuations and

transition to turbulence occurs.

Origin of Turbulence:

Frictional forces at the confining solid walls Wall turbulence

Different velocities of adjacent fluid layers Free turbulence


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Turbulence results in better mixing of fluid and produces additional

diffusive effects Eddy diffusivity.

Velocity Profile:

The mean motion and fluctuations:

Axial velocity is written as,

Here in RHS the first term is time averaged component

second term is time dependent fluctuations.

( ) ( ) ( )', ,u y t u y u y t = +

Fully developed

LaminarFully developed

TurbulentPlug flow

Reynolds Decomposition of Turbulence


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0yx zvv v

x y z

+ + =

Equation of Continuity for Turbulence Flow:

Laminar Flow

Intensity of Turbulence

Isotropic Turbulence

X-component EOM:


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2 ' '2 '2 2 ' '

2

2 2 2

x yx x x x x x z

x y z x

v vv v v v v v vpv v v v

t x y z x x y z

+ + + = + + +

Enhanced Momentum diffusivity:

molecular leve transport is favored

X-component EOM:

The last three terms are the additional terms known as

Reynolds stress terms.

Where

Is NOT a fluid property but is a property of the fluctuation

Semi empirical expressions for Reynolds stresses:

Boussinesqs eddy viscosity:( ) ( )t t x

yx

dv

dy =


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2. Prandtls mixing length:

Assuming eddies move around like gas molecules,

analogous to mean free path of gas in kinetic theory:

where,

( ) 2t x x

yx

d v d vl

dy dy

=

; y is distance from solid and 0.4l ky k = =

( ) 2eddy viscosityt xdv

ldy

= =

K is von Karman

Constant

Th l C ti


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Thermal Convection

Thermal Convection can be of two types:1. Forced Convection: The flow is triggered by an external pressure or

other driving force, in course of the flow it takes away heat.

2. Natural convection, where a change in temperature leads to variation in

density and that in turn triggers a flow.

Th l B d L


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Thermal Boundary Layer

Thermal Boundary Layer:

If entry temperature,

the convection of heat occurs.

Wall condition:CWT = constant wall temperature TS= constant

CHT = constant Heat flux qS= constant

In both cases fluid temperature changes compared to inlet temperature


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In both cases, fluid temperature changes compared to inlet temperature.

If Pr > 1:

xt> xh hydrodynamic BLgrows earlier than thermal BL

If Pr < 1:

xt< xh thermal BL grows faster.

0.05Re Thermal entry length

0.05Re Hydrodynamic entry length

t

D t

Lam

hD h

Lam

xPr x

D

x xD

= =

= =

t

h

xPr

x=

/pc kPr =

Free Convection or Natural Convection


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Free Convection or Natural Convection

Forced convection:

flow is induced by external source, pump/ compressor.

Free Convection:

No forced fluid velocity.

Ex: Heat transfer from pipes/ steam radiators/ coil of

refrigerator to surrounding air

Consider, two plates at different temperatures, T1 & T2 and T2 > T1

2 < 1 means Density decreases in the direction of gravity(Buyoant force)


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If Buyoant force overcomes the viscous forces, instability occurs

and fluid particles start moving from bottom to top.

Gravitational force on upper layer exceeds that at the lower one and

fluid starts circulating.

Heavier fluid comes down from top, warms up and becomes lighter

and moves up.

In the case, T1 > T2;

Density no longer decreases in the direction of gravity and there is

no bulk motion of fluid.

0 & 0dT d

dx dx

< >


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Boundary layerdevelopment on a heatedvertical plate:

Fluid close to the plate isheated and becomes lessdense.

Buoyant force induces afree convection BL inwhich heated fluid rises atvertically entraining thefluids from surroundings

Velocity is zero at the wallandy = .

G h f N b


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Grashof Number:

Expected:

Both free and forced convection are important if

if

if

( )

( )

2

0

2

0

3

2

Grashof number

Buoyancy force

Viscous force

s

s

g T T L u L

u

g T T L

=

= =

( )Re , ,Pr L L L LNu Nu Gr=

21

Re

L

L

Gr

( )2

1 free convection is small, Re,PrRe

L

L L

L

GrNu Nu< =

( )2 1 forced convection is small, ,PrReL

L L L

L

Gr Nu Nu Gr> =


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Non-dimensional NumbersRe=Reynolds number= Inertial forces/viscous forces=

Pr=Prandtl number=momentum diffusivity/thermal diffusivity=

Sc=Schmidt number=momentum diffusivity/mass diffusivity=

Heat transfer coefficient: h

Q=Heat flow rate=h*A*

Mass transfer coefficient: k

M= mass flow rate=kA



ud

/pc k

/

T

c


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Mass Transfer


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Mass Transfer operations

Pris replaced by Sc



Pr: Ratio of momentum and thermal diffusivity

Sc: Ratio of momentum and mass diffusivity

Lewis Number: Le = Ratio of thermal and mass diffusivity = (k/CP)/DAB

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New 2012 04 Rabibrata Bio Transport