new 2012 04 rabibrata bio transport
TRANSCRIPT
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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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Material or Substantial Derivative:
= ax
Local or
Temporal
Acceleration
Convective Acceleration
Fluid Acceleration has two
Components:
Temporal Acceleration and
Convective Acceleration
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Material or Substantial Derivative:
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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Possible Movement/ Deformation modes of a
Fluid ParticleTranslation with Linear and Angular Deformation
(Rate of Angular Deformation) =
Sign Convention:
ACW is + ve
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Possible Movement/ Deformation modes of a
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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Possible Movement/ Deformation modes of a
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
=
Equation of Continuity
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
Equation of Continuity
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
Nusselt number = convective to conductive heat transfer = hL/k
Sherwood number = convective to diffusive mass transfer = kmL/D
ud
/pc k
/
T
c
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Mass Transfer
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Mass Transfer operations
Pris replaced by Sc
Nusselt number = convective to conductive heat transfer = hL/k
Sherwood number = convective to diffusive mass transfer = kmL/D
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