Transcript Advanced Transport Phenomena

Advanced Transport Phenomena
Module 3 Lecture 9
Constitutive Laws: Momentum Transfer
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
CONSTITUTIVE LAWS
Conservation equations are necessary but not
sufficient for predictive purposes, since they lack
closure on:
 Local state functions (thermodynamic)
 Local diffusion fluxes (mass, momentum,
energy)
 Reaction-rate laws
All must be explicitly related to “field densities”.
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EQUATIONS OF STATE
Appropriate laws must be used for fluid mixture
under consideration
Come
from
equilibrium
chemical
thermodynamics.
Mixture assumed to be describable in terms of
state variables
p, T, composition ( 1 , 1 ,...,  N ), e, h, s, f (= hTs)
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EQUATIONS OF STATE
Nature of relationship may differ from fluid to fluid
Perfect gases
Liquid solutions
Dense vapors, etc.
Appropriate laws must be used for fluid mixture
under consideration
Come
from
thermodynamics
equilibrium
chemical
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CHEMICAL KINETICS
Individual net chemical species source strengths
ri ''' (and ri'' for the relevant boundaries ) must be related
to local state variables (p, T, composition, etc.)
To define reacting mixture
Info comes from chemical kinetics
Comprehensive
expression based on all
relevant (molecular-level) elementary steps, or
Global expressions empirically derived
Needed to size chemical reactors
Rate laws can be simple or quite complicated
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CHEMICAL KINETICS
Rate laws must satisfy following general
constraints:
No net mass production
N
'''
r
 i  0(i  1, 2,..., N )(corollary of conservation of each
i 1
chemical element : r'''k   0 for k  1, 2,..., N elem ),
No net charge production
'''
z
F
r
 i  i / mi   0
N
i 1
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CHEMICAL KINETICS
Vanishing
of net production rate of each
chemical species at local thermo chemical
equilibrium (LTCE)
ri''' T , p; 1LTCE ,  2 LTCE ,....,  NLTCE   0
where
iLTCE (i=1,2, …, N) are calculated at
prevailing T, P, chemical-element ratios
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DIFFUSION FLUX– DRIVING FORCE
LAWS/ COEFFICIENTS
Simplest laws :
Fluxes linearly proportional to driving forces,
i.e., local spatial gradients of field densities
Valid for chemically reacting gas mixtures
provided state variables do not undergo an
appreciable fractional change in:
A spatial region of the dimension ca. one
molecular mean-free-path
A time interval of the order of mean time
between molecular collisions
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DIFFUSION FLUX LAWS– GENERAL
CONSTRAINTS
Positive energy production
in the presence of diffusion, irrespective of
direction of diffusion fluxes
Material frame invariance
same form irrespective of changes in vantage
point
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DIFFUSION FLUX LAWS– GENERAL
CONSTRAINTS
Local action (space & time)
Isotropy
transport
properties
are
not
direction-
dependent
Linearity
Laws linear in local field variables and/ or their
spatial gradients
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LINEAR MOMENTUM DIFFUSION VS RATE
OF FLUID-PARCEL DEFORMATION
where T = “extra” stress associated with fluid
motion;
viscous stress
Π = total local stress
p = thermodynamic (normal) scalar
pressure
I = unit tensor
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VISCOUS STRESS
  xx  xy  xz 


T   yx  yy  yz 




zx
zy
zz


only 6 of the 9 components are independent
because of symmetry  xy   yx 
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VISCOUS STRESS
Each component a force per unit area
First subscript: surface on which force acts
(e.g., x = constant)
Second subscript: direction of force
 xx : normal (tensile) stress
  xy : shear stress in y direction on x = constant
surface
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
-T  rate of linear momentum diffusion
JC Maxwell: For gases, fluid velocity gradients
result in corresponding flux of linear momentum
Proportionality constant: viscosity coefficient
For a low-density gas in simple shear flow:
 yx
vx
  (T ).
.
y
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
Stokes: For more general flows, T is linearly
proportional to local rate of deformation of fluid
parcel, which has two components:
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
 Rate of angular deformation in x-y plane:
 vx vy 


;
 y x 
 Rate of volumetric deformation:
vx v y vz


x y z
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
Stokes’ constitutive law for local extra (viscous) stress:
2 

T  2.Def v       .div(v ) Ι
3 

where, in general:
1
†
Def v   grad v    grad v  

2

Applicable for Newtonian fluid
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
 dynamic viscosity
  /   kinematic viscosity (diffusivity; cm2/s)
   bulk viscosity, neglected for simple fluids

When momentum-flux law is inserted into PDE
governing linear momentum conservation,
Navier-Stokes equation is obtained
 Basis for most analyses of viscous fluid
mechanics
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STOKES’ EXTRA STRESS VS RATE OF
DEFORMATION RELATION
grad v may be decomposed into a symmetrical & antisymmetrical part:
1
grad v=  grad v  +(grad v)† 
2
Def v
1
+  grad v  -(grad v)† 
2
Rot v
Symmetric portion: Def v, associated with local
deformation rate of fluid parcel
Anti-symmetric portion: Rot v, “spin rate”, defines local
rotational motion of fluid parcel
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ENERGY EQUATION IN TERMS OF WORK
DONE BY FLUID AGAINST EXTRA STRESS
 v

 e 

t V 
2
2

 v
 dV  S   h 
2


2

 v.n dA   s q ''.n dA

N
  q '''.dV   v.  T.n dA    m .g i dV
V
S
where specific enthalpy,
i 1
V
''
i
h  e  ( p / )
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ENERGY EQUATION IN TERMS OF WORK
DONE BY FLUID AGAINST EXTRA STRESS
Subtracting
mechanical-energy
equation from above gives:
conservation


 e dV    ev . ndA    q ''. ndA   q '''.dV

S
s
V
t V

  T:grad v dV
V
Both rate of heat addition (term on RHS in
parentheses) and rate of viscous dissipation
contribute to accumulation rate and/ or net
outflow rate of thermodynamic internal energy 22
VISCOUS DISSIPATION
T . n dA  surface force on differential area n dA
associated with all contact stresses other than
local thermodynamic pressure
div T  local net contact force per unit volume in
the limit of vanishing volume
div  T.v   v. div T  T:grad v
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VISCOUS DISSIPATION
In Cartesian coordinates:
v y
 vx
vz
 xx x   yx x   zx x

v y
vz
 vx
T:grad v   xy
  yy
  zy
y
y
 y
 v
v y
vz
x
  yz
  zz
 xz
z
z
 z
 Local viscous
 dissipation

 rate per unit

 volume
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VISCOUS DISSIPATION
Angular momentum conservation at local level
leads to conclusion that T is symmetric, and
T : grad v = T : Def v
where local fluid parcel deformation rate
1
†

Def v   grad v  +(grad v) 
2
is also symmetric.
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VISCOUS DISSIPATION
Rate of entropy production due to linearmomentum diffusion  T : Def v
T Def v => positive entropy production for
positive 
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VISCOUS DISSIPATION IN TURBULENT
FLOWS
Velocities fluctuate
Contribute additive “correlation terms” to time-
averaged energy equations
For incompressible turbulent flow of a constant-
property Newtonian fluid:
T:Def v



 2 Def v:Def v  2 Def v ' :Def v '


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VISCOUS DISSIPATION IN TURBULENT
FLOWS
2nd set of terms: viscous dissipation rate per unit
mass associated with turbulent motion of fluid.
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VISCOUS DISSIPATION IN TURBULENT
FLOWS
For steady turbulent flows (e.g., through straight
ducts,
elbows,
valves),
viscous
dissipation
associated with both time-mean & fluctuating
velocity fields contributes to:
Net inflow rate of
 p /    v
2
/ 2    per unit
mass flow,
Corresponding rise in internal energy per unit
mass of fluid.
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VISCOUS DISSIPATION IN TURBULENT
FLOWS
Heating associated with local viscous dissipation
can strongly modify:
Local temperature field,
All temperature-dependent properties, including
m itself
Especially important in high-Ma
viscous flows
(e.g., rocket exhausts); low-Re, low-Ma flows in
restricted passages (e.g., packed beds in HPLC
columns)
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DYNAMIC VISCOUS COEFFICIENT
Experimentally obtained by establishing a simple
flow (e.g., steady laminar flow in a pipe) & fitting
observations (e.g., pressure-drop for given flow
rate) to predictions based on mass & linearmomentum conservation laws & constitutive
relations
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DYNAMIC VISCOUS COEFFICIENT
Unit: cp (centi-Poise) in cgs; kg/ (m s)
  /    , momentum diffusivity, or kinematic
viscosity; m2/s
For low-density gases,  independent of P, ~ T0.5-1
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 FROM KINETIC THEORY OF GASES
Chapman – Enskog Expression:
1
2
 mk BT 

5
 .
16  2  .  k BT /  
  kBT /   intermolecular potential function
T /  / kB   dimensionless temperature
 depth of potential energy “well”
  intermolecular spacing at which
potential crosses 0
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INTERMOLECULAR POTENTIAL
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 FROM KINETIC THEORY OF GASES
1/3
  1.16V
1/3
nbp

kB
 0.841V
1/3
c
 Tc 
 2.44  
 pc 
 1.22 V
1/3
s ,mp
 1.18Tnbp  0.77Tc  1.92Tmp
(Volumes  cm3/g-mole, T  K, critical pressure
 atm)
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MIXTURE & LIQUID VISCOSITY
Square-root rule:
N
 mix 
M
i 1
N
1/2
i
M
i 1
yi i
1/2
i
yi
For liquids, viscosity decreases with increasing
temperature
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MIXTURE & LIQUID VISCOSITY
Andrade-Eyring two-parameter law:
 E 
   .exp 

 RT 
E   activation energy for fluidity (inverse
viscosity)
R  universal gas constant
  (hypothetical) dynamic viscosity at infinite
temperature
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CORRESPONDING STATES
CORRELATION FOR VISCOSITY OF
SIMPLE FLUIDS
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VISCOSITY OF LIQUID SOLUTIONS,
TURBULENT VISCOSITY
No
simple relations for viscosity of liquid
solutions
Empirical relations specific to mixture classes
employed
e.g., glass, slags, etc.
Gases & liquids in turbulent motion display
augmented viscosities
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