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 32

1

Similitude Analysis: Flame Flashback,

Blowoff & Height

Flashback of a Flame in a Duct:

Depends on existence of region near duct where local

streamwise velocity < prevailing laminar flame speed, Su

No flame can propagate closer to wall than “quenching

distance” q, given by:

mixture thermal diffusivity

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

q u uconst. / S

2

Flashback of a Flame in a Duct:

Critical condition for flashback is of “gradient” form, i.e.,

U/d :Su/q, or:

Multiplying both sides by d2/u leads to correlation law

of Peclet form:

Basis for accurate flashback predictions in similar

systems

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

2

u

u ufb

S dUconst.

2u

fb u

SUconst.

d

3

Flashback of a Flame in a Duct:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

Correlation of flashback limits for premixed combustible gases in tubes (after Putnam and Jensen (1949))

4

“Blow-Off” from Premixed Gas Flame “Holders”:

Oblique premixed gas flames can be stabilized in ducts even

at feed-flow velocities >> Su

Anchor is well-mixed zone of recirculating reaction products

e.g., found immediately downstream of bluff objects (rods,

disks, gutters), in steps of ducts

Sharp upper limit to feed-flow velocity above which blow-out

or extinction occurs Ubo

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

5

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

Schematic of an oblique flame “ anchored” to a gutter-type (2 dimensional) flame-holder (stabilizer) in a uniform stream of premixed combustible gas (Su<U<Ubo)

6

“Blow-Off” from Premixed Gas Flame “Holders”:

Su measure of reaction kinetics

Similitude to GT combustor efficiency example yields:

where

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

1boRe fct Dam, Arr, , ,Pr,Sc,geometry

bobo

u

U LRe

v

7

“Blow-Off” from Premixed Gas Flame “Holders”:

Based on SA of Su data:

Solving for Peclet number at blow-off:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

22

boO F

u u

L / UDam . fct Arr, , ,v ,v

/ S

3bo u

O Fu u

U L S Lfct , Arr, , ,v ,v ,Pr,Sc,geometry

8

“Blow-Off” from Premixed Gas Flame “Holders”:

Experimentally:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

4bo u

u u

U L S Lfct ,geometry

9

“Blow-Off” from Premixed Gas Flame “Holders”:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

Test of proposed correlation of the

dimensionless "blow-off” velocity,

for a flame

stabilized by a bluff body of transverse

dimension L in a uniform, premixed gas

stream ( adapted from Spalding (1955))

bo u u uU L / vs.S L /

10

“Blow-Off” from Premixed Gas Flame “Holders”:

Alternative approach: recirculation zone likened to

WSR

3D stabilizer of transverse dimension L exhibits

recirculation zone with effective volume given by:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

35WSR eff

V L . fct Re,Pr,Sc,geometry

11

“Blow-Off” from Premixed Gas Flame “Holders”:

Fuel-flow rate into recirc/ reaction zone can be written

as:

Blow-out occurs when corresponding volumetric fuel

consumption rate is near

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

26F u F ,um U . L . fct Re,Pr,Sc,geometry

'''F F ,max WSRbo eff

m r . V

'''F ,maxr

12

“Blow-Off” from Premixed Gas Flame “Holders”:

Rearranging:

Additional insight: blow—off velocity scales linearly with

transverse dimension of flame stabilizer, at sufficiently

high Re

Experimentally verified

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

2

7 2bo uO F

u u

U L S L. fct Arr, , ,v ,v ,Re,Pr,Sc,geometry

13

Laminar Diffusion Flame Height:

Buoyancy and fuel-jet momentum contribute to height,

Lf, of fuel-jet diffusion flame

Simple model: relevant groupings of variables

Beyond realm of ordinary dimensional analysis

Treat hot “flame sheet” region as cause of natural

convective inflow of ambient oxidizer

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

14

Laminar Diffusion Flame Height:

For any fuel/ oxidizer pair, when buoyancy dominates:

If fuel-jet momentum dominates, at constant Re:

Rj = ½ dj

Fr Froude number:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

1 32 /f

j

L. Re Fr const

R

1 3 1 32 / /f

j

L. Re Fr const' . Fr

R

2j jFr U / gR

15

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

Correlation of laminar jet diffusion flame lengths ( adapted from Altenkirch et al. (1977))16

Fuel Droplet Combustion at High Pressures:

Burning time, tcomb, depends on pressure

In rockets, aircraft gas turbines, etc., pressures > 20 atm may be

reached

Based on diffusion-limited burning-rate theory & available

experimental data at p ≥ 1 atm:

K burning rate constant

Dependent on fuel type & environmental conditions

Independent of droplet diameter

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

20comb p ,t d / K

17

Fuel Droplet Combustion at High Pressures:

Each fuel also has a thermodynamic critical pressure, pc, to

which prevailing pressure, p, may be compared:

Hence, following correlation may be obtained:

Corresponding-states analysis for high-pressure droplet

combustion is reasonably successful

Allows estimation of burning times where data

are not available

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

1

1c c

K p, fuel type,... pvs

K p ,same fuel ,... p same fuel

18

Fuel Droplet Combustion at High Pressures:

PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS

Approximate correlation of fuel-droplet burning rate constants at elevated pressures ( based on data of Kadota and Hiroyasu (1981))

19

Configurational Analysis (Becker, 1976):Establishing conditions of similarity by forming a

sufficient set of eigen-ratiosLeads to similitude criteria in the form of dimensionless

ratios of: Inventories Source strengths Fluxes Lengths Time, etc.

Can start analysis from macroscopic or microscopic CVs

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

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But, interpretation not unique:

All may be regarded as ratios of same type, e.g.,

characteristic times

Relevant even to SS problems

e.g., length ratio in forced convection system fluid

transit-time ratio

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

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e.g., momentum-flux ratio (Re) ratio of times

required for momentum diffusion & convection through

common area, L2

e.g., Pr ratio of times governing diffusive decay of

nonuniformities of energy (L2/) and momentum (L2/)

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

22

Characteristic Times:

e.g., fluid-phase, homogeneous-reaction Damkohler

number ratio of characteristic flow time to

characteristic chemical reaction time

e.g., surface (heterogeneous) Damkohler number

ratio of characteristic reactant diffusion time across

boundary layer to characteristic consumption time on

surface

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

2m diffw m

chem,we m

e w

/ D tkC

D t

k

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Characteristic Times:

e.g., Stokes number (governing dynamical non-equilibrium

in two-phase flows) ratio of particle stopping time to

fluid transit time (L/U)

Attractive way of dealing with complex physicochemical

problems, e.g.:

Gas-turbine spray combustor performance

Coal-particle devolatilization

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

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Further simplifications possible through:

rational parameter groupings

Dropping parameters via sensitivity analysis

Make maximum use of available insight & data–

analytical & experimental

ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS

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Integrates results of:

Scale-model testing

Full-scale testing

Mathematical modeling

Allows judicious blend of all three

Based on fundamental conservation principles & constitutive

laws

Maximum insight with minimum effort

BENEFITS OF SIMILITUDE ANALYSIS

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Yields set-up rules for designing scale-model experiments

amenable to quantitative use

Reflects relative importance of competing transport

phenomena

Leads to smaller set of relevant parameters compared to

“dimensional analysis”

Sensitivity-analysis can enable “approximate” or “partial”

similarity analysis

BENEFITS OF SIMILITUDE ANALYSIS

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OUTLINE OF PROCEDURE FOR SA

Write necessary & sufficient equations to determine QOI in most appropriate coordinate system PDEs bc’s ic’s Constitutive equations

Introduce nondimensional variables Use appropriate reference lengths, times, temperature

differences, etc. Normalize variables to range from 0 to 1

Make suitable (defensible) approximations Drop negligible terms

28

Express dimensionless QOI in terms of dimensionless

variables & parammeters

Inspect result for implied parametric dependence

This constitutes “similitude relation” sought

Stronger than conventional dimensional analysis

Fewer extraneous criteria

OUTLINE OF PROCEDURE FOR SA

29

Apparently dissimilar physico-chemical problems lead to

identical dimensionless equations

Establishing useful analogies (e.g., between heat &

mass transfer)

Dimensionless parameters will represent ratios between

characteristic times in governing equations

Problem may simplify considerably when such ratios

0 or ∞

SIMILITUDE ANALYSIS: REMARKS

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Approximate similitudes possible for complicated

problems

Some conditions may be “escapable”

Resulting simplified equations may have invariance

properties

Allow further reduction in number of governing

dimensionless parameters

Allow extraction of functional dependencies, simplify

design of experiments

SIMILITUDE ANALYSIS: REMARKS

31