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

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

2. 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

3. SIMILITUDE ANALYSIS • 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

4. 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:

5. SIMILITUDE ANALYSIS total rate of heat loss per unit axial length of cylinder pL proportional to cylinder surface area per unit axial length bT  thermal expansion coefficient of fluid

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

7. SIMILITUDE ANALYSIS • By similitude analysis, only 2 (Pr, Rah):

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

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

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

11. 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 = gbw(wA-wA,∞)

12. SIMILITUDE ANALYSIS • 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)

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

14. SIMILITUDE ANALYSIS • 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

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))

16. 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.

17. 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

18. SIMILITUDE ANALYSIS • Laminar Flame Speed: • Assumptions: • Single, stoichiometric, irreversible chemical reaction • Simple “gradient” diffusion • Equality of effective diffusivities (neff = aeff =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

19. SIMILITUDE ANALYSIS • Laminar Flame Speed: • Stoichiometric fuel + oxidizer vapor reaction assumed to occur at local rate: • 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

20. SIMILITUDE ANALYSIS • Laminar Flame Speed: • Normalized temperature variable • Characteristic length: a/Su • a mixture thermal diffusivity • Dimensionless distance variable

21. SIMILITUDE ANALYSIS • Laminar Flame Speed: • maximum reaction rate, occurs at • Normalized reaction rate function: • Problem now reduces to finding eigen-value, Y, corresponding to solution of BVP:

22. SIMILITUDE ANALYSIS where

23. SIMILITUDE ANALYSIS • Laminar Flame Speed: • where

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

25. SIMILITUDE ANALYSIS • Laminar Flame Speed: • Above similitude result contains pressure-dependence of Su • since a ̴p-1, ̴pn, ru̴p+1 • Effective overall reaction order ~

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

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

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

29. 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

30. 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

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

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

33. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Gas-Turbine Combustor Efficiency: • If combustion efficiency hcomb exhibits functional dependencies: • We can conclude: hm= hp • if each nondimensional parameter is same for model & prototype

34. 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 (F) as prototype, nondimensional parameters will be same if:

35. 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

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

37. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • 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: ~ ~

38. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • 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

39. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Gas-Turbine Combustor Efficiency: • Thus, even in simple combustor applications, strict scale-model similarity is unattainable • hcomb is much more sensitive to Dam than to Re • Especially at high (fully turbulent) Re • Hence, for sufficiently large Re, Re-dependence of hcomb can be neglected • “approximate similitude”

40. 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))

41. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • 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)

42. 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): • Examples: efficiency, stability-limits

43. 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))

44. 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))