Advanced Transport Phenomena
Module 7 Lecture 31
Similitude Analysis: Full & Partial
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
1
SIMILITUDE ANALYSIS
 “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
2
SIMILITUDE ANALYSIS
 More powerful than dimensional analysis
 Removes
guesswork/ intuition regarding relevant
variables
 Demonstrates
physical
dimensionless group

significance
of
each
Suggests when certain groups will be irrelevant
based on competing effects
 Enables
a significant reduction in # of relevant
dimensionless groups
 Suggests existence & use of analogies
3
SIMILITUDE ANALYSIS
 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:
q'w
 fct1  L,g,T ,Tw ,T ,k ,,c p ,,shape,orientation 
L
4
SIMILITUDE ANALYSIS
q'w 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
5
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 By
dimensional
analysis
(-theorem),
“only”
6
independent dimensionless groups:
2
 gL3

 q'w / L 
v Tw  v / L 
 fct2  2 ,T Tw  T  , , ,
,shape,orientation 


 T c p Tw  T 
 k Tw  T  / L 
 v

6
SIMILITUDE ANALYSIS
 By similitude analysis, only 2 (Pr, Rah):
 q /  L 
'
w
 k Tw  T  / L 
 const  shape  .Nu h  Rah ,Pr,shape,orientation )
gT Tw  T  L3 v
Rah 
.  Grh .Pr
2
v

7
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 Nondimensionalizing equations & bc’s for velocity &
temperature fields:
Lref  L
T  T ref  Tw  T 
U ref  v / L
8
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 Solutions of the PDE-system, v* and T*:
div* v*  0
( mass )
v*.grad*v*  div*  grad v*   Grh . g / g  .T * ( momentum )
v*.grad*T *=  Pr  div*  grad* T* 
1
( energy )
9
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 Dimensionless groups have physical significance, e.g.:
T*
local buoyancy force / mass
 Grh .
local viscous force / mass
div*  grad v*
Grh measure of relative magnitudes of buoyancy
and viscous forces
10
SIMILITUDE ANALYSIS
 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
wA,w = constant (<< 1) and wA,∞(also << 1) specified
 Local buoyancy force/ mass = gw(wA-wA,∞)
11
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 Composition variable
 Satisfies:
w A  w A,
w* 
w A,w  w A,
v*.grad*w*   Sc  div* grad*w*
1
(neglecting
homogeneous
chemical
reaction
&
assuming local validity of Fick’s law for dilute species
A diffusion through Newtonian fluid)
12
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 v* satisfies nonlinear PDE:
v*.grad*v*  div*  gradv*   Grm  g / g  w*
 Transport property (diffusivity) ratio:
v
Sc 
DA
 Schmidt number 
 Grashof number for mass transport:
Grm 
gw  wA,w  wA,  L3
v2
Ram

Sc
13
SIMILITUDE ANALYSIS
 Example: Convective heat flow
 By inspection & comparison:
 j'A,w /  L  
 const  shape  .Nu m  Ram ,Sc,shape,orientation )
 DA  wA,w  wA,  / L 
 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
14
SIMILITUDE ANALYSIS
Correlation of perimeter-averaged “natural convection” heat transfer from/to
a horizontal circular cylinder in a Newtonian fluid (adapted from McAdams (1954))
15
SIMILITUDE ANALYSIS
 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.
16
SIMILITUDE ANALYSIS
 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
17
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 Assumptions:

Single, stoichiometric, irreversible chemical reaction

Simple “gradient” diffusion

Equality of effective diffusivities (neff = eff = Di,eff)

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
18
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 Stoichiometric fuel + oxidizer vapor reaction assumed
to occur at local rate:
n
rF''' 
1
 pM 
 E  vo vF
.Aexp


 .wO wF
vo
VF 1 
M O M F  RT 
 RT 
n ≡ nO + nF overall reaction order
Generalization of bimolecular (n = 2) form


necessary to describe overall effect to many elementary
steps of different reaction orders
19
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 Normalized temperature variable
T  Tu

wF ,u Q / c p
 Characteristic length: /Su

 mixture thermal diffusivity
 Dimensionless distance variable
Su z


20
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
rF , maxmaximum reaction rate, occurs at
'''
Tr'''
F max
 Normalized reaction rate function:
R

rF'''  wO T  ,wF T  ,T 

rF''',max Tr'''  max

Problem now reduces to finding eigen-value, Y,
corresponding to solution of BVP:
d  d 2 1
 2  2 .R   
d d
w
21
SIMILITUDE ANALYSIS
where
0 at    ,

1 at    ,
w2 
u Su2 wF ,u
   rF''', max 
22
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 where
w


F ,u
E
Arr 
RTb
/ wO ,u 
f
 mixture 


ratio


 Arrhenius 
1
wF ,u Q   wF ,u Q    chemical


 1  
  




c pTb
  c pTu    energy release 
23
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 Therefore, at most:
w  fct  Arr, ,,vO ,vF 
 Or flame speed must be given by:

   r
Su  
u wF ,u


'''
F ,max
 . fct
1/ 2


 Arr, , ,vO ,vF 


 fct evaluated by numerical or analytical methods
24
SIMILITUDE ANALYSIS
 Laminar Flame Speed:
 Above similitude result contains pressure-dependence
of Su
 since 
̴p-1,
r
'''
F max
̴pn, u ̴p+1
Su p
~
 n / 21
 Effective overall reaction order
neff
 d ln Su 
 2 1 

d ln p 

25
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Include many additional parameters
 Many reference quantities, e.g., for a combustor:
Lref  L
U ref  U  ,
tref
pref
( forced convection )
L

U
1
 U 2
2
26
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
T  T ref  Tadiab  T  ,etc.
 Can true similarity ever be achieved except in the
trivial case of Lp = Lm?
 Alternative: allow “approximate similarity”, or “partial
modeling”
27
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
Aircraft gas turbine GT combustor (schematic)
28
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 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
29
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Simpler limiting case: fuel droplets sufficiently small so
that their penetration is small

Vaporization rapid enough to not limit overall
chemical heat release rate
30
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 Performance criterion: combustion efficiency
comb
 Similarity criteria:
T0 ,b  T0 ,u

T0 ,b;adiab  T0 ,u
shape
Re  U u L / vu
Pr  vu /  u
Sc  vu / DF ,u
   c p / cv  , and
u
Ma  U / a u
31
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 Additional factors:
fuel / air mass flow ratio 


 fuel / air stoich
wF ,u Q

c p ,uTb ,adiab
E
Arr 
RTb ,ad
Dam 
t flow
tchem ,ref
 dim ensionless Arrhenius activation energy 
 Damkohler  ratio of characteristic flow time to 


chemical
oxidation
time


32
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 If combustion efficiency comb exhibits functional
dependencies:
comb    Re,Pr,Sc,  ,Ma,, , Arr,Dam,shape 
We can conclude: m = p
 if each nondimensional parameter is same for model
& prototype
33
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 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:
 Re m   Re  p
 Ma m   Ma  p
 Dam m   Dam  p
34
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 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
35
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
-for a perfect gas, Re-equivalence implies:
 pUL m   pUL  p
-Ma-equivalence implies:
Um  U p
36
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 Therefore, model pressure
 Lp 
pm  p p  
 Lm 
 This conflicts with Dam-equivalence!
 For example, in case of a simple nth-order
homogeneous fuel-consumption reaction:
tchem 
F ,ref
'''

r
 F
ref
~
p
pn
~
p
 n 1
37
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 L/Uu, Dam-equivalence requires:
 Since tflow
 Lp n 1 
 Lp n 1 

 

 U m  U  p
 In light of Ma-equivalence requirement:
1/  n 1
 Lp 
pm  p p  
 Lm 
 Differs from earlier expression for pm when n≠ 2
38
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 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”
39
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
Gas-Turbine Combustor Efficiency:
Dependence of GT combustor efficiency on Re at constant (inverse) Damkohler
Number (schematic, adapted from S. Way (1956))
40
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 Under “approximate similitude”, scale-model
combustor tests should be run with:
Um  U p
and
1/  n 1
 Lp 
pm  p p  
 Lm 
 Apparent reaction order, n: 1.3-1.6 (depending on
fuel)
41
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
 Efficiency & stability data on combustors should appr
correlate with a parameter proportional to Dam (or to
Dam-1):
mair
U
or n 3
n 1
p L
p L
 Examples: efficiency, stability-limits
42
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
Correlation for the GT combustor efficiency vs parameter proportional to (inverse)
Damkohler number (adapted from S. Way (1956))
43
PARTIAL MODELING OF CHEMICALLY
REACTING SYSTEMS
 Gas-Turbine Combustor Efficiency:
Correlation of the GT combustor stability limits vs parameter proportional to (inverse)
Damkohler number (after D.Stewart (1956))
44