1. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 7 Lecture 31 1 Similitude Analysis: Full & Partial

2.  “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

3.  More powerful than dimensional analysis  Removes guesswork/ intuition regarding relevant variables  Demonstrates physical significance of each dimensionless group  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 SIMILITUDE ANALYSIS 3

4.  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 4

5.  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 5

6.  Example: Convective heat flow  By dimensional analysis (  -theorem), “only” 6 independent dimensionless groups: SIMILITUDE ANALYSIS 6

7.  By similitude analysis, only 2 (Pr, Ra h ): SIMILITUDE ANALYSIS 7

8.  Example: Convective heat flow  Nondimensionalizing equations & bc’s for velocity & temperature fields: SIMILITUDE ANALYSIS 8

9.  Example: Convective heat flow  Solutions of the PDE-system, v* and T*: SIMILITUDE ANALYSIS 9

10.  Example: Convective heat flow  Dimensionless groups have physical significance, e.g.: Gr h  measure of relative magnitudes of buoyancy and viscous forces SIMILITUDE ANALYSIS 10

11.  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 11

12.  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 12

13.  Example: Convective heat flow  v* satisfies nonlinear PDE:  Transport property (diffusivity) ratio:  Grashof number for mass transport: SIMILITUDE ANALYSIS 13

14.  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 14

15. 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

16.  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

17.  Laminar Flame Speed:  S u  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

18.  Laminar Flame Speed:  Assumptions:  Single, stoichiometric, irreversible chemical reaction  Simple “gradient” diffusion  Equality of effective diffusivities ( eff =  eff =  D i  eff )  Constant heat capacity (w.r.t. temperature & mixture composition)  Deflagration waves propagate slowly enough to neglect relative change of pressure across them, (p u – p b )/p u SIMILITUDE ANALYSIS 18

19.  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 19

20.  Laminar Flame Speed:  Normalized temperature variable  Characteristic length:  /S u    mixture thermal diffusivity  Dimensionless distance variable SIMILITUDE ANALYSIS 20

21.  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 21

22. SIMILITUDE ANALYSIS where 22

23.  Laminar Flame Speed:  where SIMILITUDE ANALYSIS 23

24.  Laminar Flame Speed:  Therefore, at most:  Or flame speed must be given by:  fct evaluated by numerical or analytical methods SIMILITUDE ANALYSIS 24

25.  Laminar Flame Speed:  Above similitude result contains pressure-dependence of S u  since  ̴ p -1, ̴ p n,  u ̴ p +1  Effective overall reaction order SIMILITUDE ANALYSIS ~ 25

26.  Include many additional parameters  Many reference quantities, e.g., for a combustor: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 26

27.  Can true similarity ever be achieved except in the trivial case of L p = L m ?  Alternative: allow “approximate similarity”, or “partial modeling” PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 27

28.  Gas-Turbine Combustor Efficiency: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS Aircraft gas turbine GT combustor (schematic) 28

29.  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 29

30.  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 30

31.  Gas-Turbine Combustor Efficiency:  Performance criterion: combustion efficiency  Similarity criteria: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 31

32.  Gas-Turbine Combustor Efficiency:  Additional factors: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 32

33.  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 33

34.  Gas-Turbine Combustor Efficiency:  If scale model is run with same fuel, at same inlet temperature (T u ) & same mixture ratio (  ) as prototype, nondimensional parameters will be same if: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 34

35.  Gas-Turbine Combustor Efficiency:  Is there a combination of model pressure, velocity & scale (p m, U m, L m ) 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 35

36. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS  Gas-Turbine Combustor Efficiency: -for a perfect gas, Re-equivalence implies: -Ma-equivalence implies: 36

37.  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 ~~ 37

38.  Gas-Turbine Combustor Efficiency:  Since t flow L/U u, Dam-equivalence requires:  In light of Ma-equivalence requirement:  Differs from earlier expression for p m when n≠ 2 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 38

39.  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 39

40. Gas-Turbine Combustor Efficiency: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS Dependence of GT combustor efficiency on Re at constant (inverse) Damkohler Number (schematic, adapted from S. Way (1956)) 40

41.  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 41

42.  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 42

43. 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

44. 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