Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 7 Lecture 31

1

Similitude Analysis: Full & Partial

“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

2

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

3

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

4

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

5


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

6

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

7


Nondimensionalizing equations & bc’s for velocity &

temperature fields:

SIMILITUDE ANALYSIS

ref

wref

ref

L L

T T T T

U v / L

8


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 )

9

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*

10


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

11


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*

12


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

13


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

14

SIMILITUDE ANALYSIS

Correlation of perimeter-averaged “natural convection” heat transfer from/toa horizontal circular cylinder in a Newtonian fluid (adapted from McAdams (1954))

15

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

16


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

17

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

18


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

19


Normalized temperature variable

Characteristic length: /Su

mixture thermal diffusivity

Dimensionless distance variable

SIMILITUDE ANALYSIS

u

F ,u p

T TQ / c

uS z

20


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

21

SIMILITUDE ANALYSIS

2

2

01

u u F ,u'''F ,max

at ,at ,

Sr

where

22


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

23


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

24


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

25

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

26

Can true similarity ever be achieved except in the

trivial case of Lp = Lm?

Alternative: allow “approximate similarity”, or “partial

modeling”


adiabrefT T T T ,etc.

27

Gas-Turbine Combustor Efficiency:


Aircraft gas turbine GT combustor (schematic)

28


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

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


Performance criterion: combustion efficiency

Similarity criteria:


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

31

Gas-Turbine Combustor Efficiency:Additional factors:


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

32


If combustion efficiency comb exhibits functional

dependencies:

We can conclude: m = p

if each nondimensional parameter is same for model &

prototype


comb Re,Pr,Sc, ,Ma, , ,Arr ,Dam,shape

33


If scale model is run with same fuel, at same inlet

temperature (Tu) & same mixture ratio () as

prototype, nondimensional parameters will be same if:


m p

m p

m p

Re Re

Ma Ma

Dam Dam

34


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



-for a perfect gas, Re-equivalence implies:

-Ma-equivalence implies:

m ppUL pUL

m pU U

36


Therefore, model pressure

This conflicts with Dam-equivalence!

For example, in case of a simple nth-order

homogeneous fuel-consumption reaction:


pm p

m

Lp p

L

1F ,ref n

chem n'''F ref

pt ppr

~ ~

37


Since tflow L/Uu, Dam-equivalence requires:

In light of Ma-equivalence requirement:

Differs from earlier expression for pm when n≠ 2


1 1n n

m p

Lp LpU U

1 1/ np

m pm

Lp p

L

38


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



Dependence of GT combustor efficiency on Re at constant (inverse) DamkohlerNumber (schematic, adapted from S. Way (1956))

40


Under “approximate similitude”, scale-model

combustor tests should be run with:

and

Apparent reaction order, n: 1.3-1.6 (depending on fuel)


m pU U 1 1/ n

pm p

m

Lp p

L

41


Efficiency & stability data on combustors should appr

correlate with a parameter proportional to Dam (or to

Dam-1):

Examples: efficiency, stability-limits


1 3air

n n

mU orp L p L

42



Correlation for the GT combustor efficiency vs parameter proportional to (inverse) Damkohler number (adapted from S. Way (1956)) 43



Correlation of the GT combustor stability limits vs parameter proportional to (inverse) Damkohler number (after D.Stewart (1956)) 44

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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras