Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
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Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
Advanced Transport PhenomenaModule 7 Lecture 31
1
Similitude Analysis: Full & Partial
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“Inspectional Analysis”– Becker (1976)
Based on governing constitutive equations, conservation
principles, initial/ boundary conditions
Similitude conditions extracted without actually solving
resulting set of dimensionless equations
SIMILITUDE ANALYSIS
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More powerful than dimensional analysisRemoves guesswork/ intuition regarding relevant
variablesDemonstrates physical significance of each
dimensionless group Suggests when certain groups will be irrelevant based
on competing effectsEnables a significant reduction in # of relevant
dimensionless groupsSuggests existence & use of analogies
SIMILITUDE ANALYSIS
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Example: Convective heat flow
Steady heat flow from isothermal horizontal cylinder of
length L, in Newtonian fluid in natural convective flow
induced by body force field g
Dimensional interrelation:
SIMILITUDE ANALYSIS
1
'w
T w pq fct L,g , ,T ,T ,k , ,c , ,shape,orientationL
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total rate of heat loss per unit axial length of cylinder
L proportional to cylinder surface area per unit axial
length
T thermal expansion coefficient of fluid
SIMILITUDE ANALYSIS
'wq
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Example: Convective heat flow
By dimensional analysis (-theorem), “only” 6
independent dimensionless groups:
SIMILITUDE ANALYSIS
23
2 2
'w w
T wp ww
q / L v / LTgL vfct , T T , , , ,shape,orientationv T c T Tk T T / L
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By similitude analysis, only 2 (Pr, Rah):
SIMILITUDE ANALYSIS
'w
h hw
q / Lconst shape .Nu Ra ,Pr,shape,orientation )
k T T / L
3
2T w
h h
g T T L vRa . Gr .Prv
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Example: Convective heat flow
Nondimensionalizing equations & bc’s for velocity &
temperature fields:
SIMILITUDE ANALYSIS
ref
wref
ref
L L
T T T T
U v / L
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Example: Convective heat flow
Solutions of the PDE-system, v* and T*:
SIMILITUDE ANALYSIS
1
0vv* grad*v* grad v* g
v* grad* *= grad* T*h
div* * (mass ). div* Gr . / g .T * (momentum )
. T Pr div* ( energy )
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Example: Convective heat flowDimensionless groups have physical significance, e.g.:
Grh measure of relative magnitudes of buoyancy and viscous forces
SIMILITUDE ANALYSIS
h
*local buoyancy force / mass Gr .local viscous force / mass div*
Tgrad v*
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Example: Convective heat flow
Mass-transfer analog of heat-transfer problem:
Example: slowly subliming (or dissolving) solid cylinder
of same shape & orientation, with solute mass fraction
A,w = constant (<< 1) and A,∞(also << 1) specified
Local buoyancy force/ mass = g(A-A,∞)
SIMILITUDE ANALYSIS
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Example: Convective heat flow
Composition variable
Satisfies:
(neglecting homogeneous chemical reaction & assuming local
validity of Fick’s law for dilute species A diffusion through
Newtonian fluid)
SIMILITUDE ANALYSIS
A A,
A,w A,
*
1* Sc div* * v*.grad* grad*
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Example: Convective heat flow
v* satisfies nonlinear PDE:
Transport property (diffusivity) ratio:
Grashof number for mass transport:
SIMILITUDE ANALYSIS
v*.grad*v gradv gm* div* * Gr / g *
A
vSc Schmidt numberD
3
2A,w A, m
m
g L RaGrv Sc
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Example: Convective heat flow
By inspection & comparison:
Functions on RHS are same for mass & heat transfer
Can be obtained by heat- or mass-transfer experiments,
whichever is more convenient
Dimensional analysis could not have led to this prediction &
conclusion
SIMILITUDE ANALYSIS
'A,w
m mA A,w A,
j / Lconst shape .Nu Ra ,Sc,shape,orientation )
D / L
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SIMILITUDE ANALYSIS
Correlation of perimeter-averaged “natural convection” heat transfer from/toa horizontal circular cylinder in a Newtonian fluid (adapted from McAdams (1954))
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Laminar Flame Speed:
Simplest problem involving transport by convection & diffusion, along with simultaneous homogeneous chemical reaction: prediction of steady propagation of the “wave” of chemical reaction observed subsequent to local ignition in an initially premixed, quiescent, nonturbulent gas Heat & reaction intermediaries diffusing from initial zone
of intense chemical reaction prepare adjacent layer of gas, which prepares next layer, etc.
SIMILITUDE ANALYSIS
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Laminar Flame Speed:
Su steady propagation speed relative to unburned gas
Simple to measure
Not trivial to interpret
Transport laws can be approximated
But, combustion reactions occur via a complex network
Problem lends itself to SA
SIMILITUDE ANALYSIS
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Laminar Flame Speed:Assumptions: Single, stoichiometric, irreversible chemical reaction Simple “gradient” diffusion Equality of effective diffusivities (eff =eff =Dieff)
Constant heat capacity (w.r.t. temperature & mixture composition)
Deflagration waves propagate slowly enough to neglect relative change of pressure across them, (pu – pb)/pu
SIMILITUDE ANALYSIS
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Laminar Flame Speed:
Stoichiometric fuel + oxidizer vapor reaction assumed
to occur at local rate:
n ≡ O + F overall reaction order Generalization of bimolecular (n = 2) form
necessary to describe overall effect to many elementary
steps of different reaction orders
SIMILITUDE ANALYSIS
1
1F
nv''' vo
F O Fvo VFO F
pM Er .Aexp .M M RT RT
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Laminar Flame Speed:
Normalized temperature variable
Characteristic length: /Su
mixture thermal diffusivity
Dimensionless distance variable
SIMILITUDE ANALYSIS
u
F ,u p
T TQ / c
uS z
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Laminar Flame Speed:
maximum reaction rate, occurs at
Normalized reaction rate function:
Problem now reduces to finding eigen-value, ,
corresponding to solution of BVP:
SIMILITUDE ANALYSIS
'''
'''F O F
'''F ,max r max
r T , T ,TR
r T
2
2 2
1d d .Rd d
'''F ,maxr '''
F maxrT
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SIMILITUDE ANALYSIS
2
2
01
u u F ,u'''F ,max
at ,at ,
Sr
where
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Laminar Flame Speed:
where
SIMILITUDE ANALYSIS
1
1
F ,u O,u
b
F ,uF ,u
p b p u
/ mixtureratiof
EArr ArrheniusRT
Q chemicalQenergy releasec T c T
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Laminar Flame Speed:
Therefore, at most:
Or flame speed must be given by:
fct evaluated by numerical or analytical methods
SIMILITUDE ANALYSIS
O Ffct Arr, , ,v ,v
1 2/'''
F ,maxu O F
u F ,u
rS . fct Arr, , ,v ,v
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Laminar Flame Speed:
Above similitude result contains pressure-dependence
of Su
since ̴p-1, ̴pn, u ̴p+1
Effective overall reaction order
SIMILITUDE ANALYSIS
2 1n /uS p
2 1 ueff
d ln Snd ln p
~
'''F maxr
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Include many additional parameters
Many reference quantities, e.g., for a combustor:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
212
ref
ref
ref
ref
L L
U U , ( forced convection )
LtU
p U
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Can true similarity ever be achieved except in the
trivial case of Lp = Lm?
Alternative: allow “approximate similarity”, or “partial
modeling”
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
adiabrefT T T T ,etc.
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Gas-Turbine Combustor Efficiency:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
Aircraft gas turbine GT combustor (schematic)
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Gas-Turbine Combustor Efficiency:
Complex geometry
Liquid fuel introduced into enclosure as a spray
Each spray characterized by a spray angle, spray
momentum flux, droplet size distribution, etc.
Two-phase effects
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
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Simpler limiting case: fuel droplets sufficiently small so
that their penetration is small
Vaporization rapid enough to not limit overall
chemical heat release rate
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
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Gas-Turbine Combustor Efficiency:
Performance criterion: combustion efficiency
Similarity criteria:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
0 0
0 0
,b ,ucomb
,b;adiab ,u
T TT T
u u
u u
u F ,u
p v u
u
shapeRe U L / vPr v /Sc v / D
c / c , and
Ma U / a
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Gas-Turbine Combustor Efficiency:Additional factors:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
stoich
F ,u
p ,u b,adiab
fuel / air mass flow ratiofuel / air
Qc T
b,ad
flow
chem,ref
EArr dimensionless Arrhenius activation energyRT
t Damkohler ratio of characteristic flow time toDam
chemical oxidation timet
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Gas-Turbine Combustor Efficiency:
If combustion efficiency comb exhibits functional
dependencies:
We can conclude: m = p
if each nondimensional parameter is same for model &
prototype
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
comb Re,Pr,Sc, ,Ma, , ,Arr ,Dam,shape
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Gas-Turbine Combustor Efficiency:
If scale model is run with same fuel, at same inlet
temperature (Tu) & same mixture ratio () as
prototype, nondimensional parameters will be same if:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
m p
m p
m p
Re Re
Ma Ma
Dam Dam
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Gas-Turbine Combustor Efficiency:
Is there a combination of model pressure, velocity &
scale (pm, Um, Lm) such that remaining similarity
conditions can be met?
Answer requires specification of p, U, L-dependence of
each parameter
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
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PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
Gas-Turbine Combustor Efficiency:
-for a perfect gas, Re-equivalence implies:
-Ma-equivalence implies:
m ppUL pUL
m pU U
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Gas-Turbine Combustor Efficiency:
Therefore, model pressure
This conflicts with Dam-equivalence!
For example, in case of a simple nth-order
homogeneous fuel-consumption reaction:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
pm p
m
Lp p
L
1F ,ref n
chem n'''F ref
pt ppr
~ ~
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Gas-Turbine Combustor Efficiency:
Since tflow L/Uu, Dam-equivalence requires:
In light of Ma-equivalence requirement:
Differs from earlier expression for pm when n≠ 2
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
1 1n n
m p
Lp LpU U
1 1/ np
m pm
Lp p
L
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Gas-Turbine Combustor Efficiency:
Thus, even in simple combustor applications, strict scale-model
similarity is unattainable
comb is much more sensitive to Dam than to Re
Especially at high (fully turbulent) Re
Hence, for sufficiently large Re, Re-dependence of comb can be
neglected
“approximate similitude”
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
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Gas-Turbine Combustor Efficiency:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
Dependence of GT combustor efficiency on Re at constant (inverse) DamkohlerNumber (schematic, adapted from S. Way (1956))
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Gas-Turbine Combustor Efficiency:
Under “approximate similitude”, scale-model
combustor tests should be run with:
and
Apparent reaction order, n: 1.3-1.6 (depending on fuel)
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
m pU U 1 1/ n
pm p
m
Lp p
L
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Gas-Turbine Combustor Efficiency:
Efficiency & stability data on combustors should appr
correlate with a parameter proportional to Dam (or to
Dam-1):
Examples: efficiency, stability-limits
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
1 3air
n n
mU orp L p L
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Gas-Turbine Combustor Efficiency:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
Correlation for the GT combustor efficiency vs parameter proportional to (inverse) Damkohler number (adapted from S. Way (1956)) 43
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Gas-Turbine Combustor Efficiency:
PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS
Correlation of the GT combustor stability limits vs parameter proportional to (inverse) Damkohler number (after D.Stewart (1956)) 44