AME 513 Principles of Combustion Lecture 10 Premixed flames III: Turbulence effects

2 AME Fall Lecture 10 - Premixed flames III Motivation  Study of premixed turbulent combustion important because  Turbulence increases mean flame propagation rate (S T ) and thus mass burning rate (=  S T A projected )  If this trend increased ad infinitum, arbitrarily lean mixtures (low S L ) could be burned arbitrarily fast by using sufficiently high u’...but too high u' leads to extinction - nixes that idea  Even without forced turbulence, if the Grashof number gd 3 / 2 is larger than about 10 6 (g = 10 3 cm/s 2, ≈ 1 cm 2 /s  d > 10 cm), turbulent flow will exist due to buoyancy  Examples  Premixed turbulent flames »Gasoline-type (spark ignition, premixed-charge) internal combustion engines »Stationary gas turbines (used for power generation, not propulsion)  Nonpremixed flames »Diesel-type (compression ignition, nonpremixed-charge) internal combustion engines »Gas turbines »Most industrial boilers and furnaces

3 AME Fall Lecture 10 - Premixed flames III Turbulent burning velocity  Models of premixed turbulent combustion don’t agree with experiments nor each other!

4 AME Fall Lecture 10 - Premixed flames III Basics of turbulence  Good reference: Tennekes: “A First Course in Turbulence”  Job 1: need a measure of the strength of turbulence  Define turbulence intensity (u’) as rms fluctuation of instantaneous velocity u(t) about mean velocity ( )

5 AME Fall Lecture 10 - Premixed flames III Basics of turbulence  Job 2: need a measure of the length scale of turbulence  Define integral length scale (L I ) as  A measure of size of largest eddies  Largest scale over which velocities are correlated  Typically related to size of system (tube or jet diameter, grid spacing, …) Here the overbars denote spatial (not temporal) averages  A(r) is the autocorrelation function at some time t  Note A(0) = 1 (fluctuations around the mean are perfectly correlated at a point)  Note A(∞) = 0 (fluctuations around the mean are perfectly uncorrelated if the two points are very distant)  For truly random process, A(r) is an exponentially decaying function A(r) = exp(-r/L I )

6 AME Fall Lecture 10 - Premixed flames III Basics of turbulence  In real experiments, generally know u(t) not u(x) - can define time autocorrelation function A(x,  ) and integral time scale  I at a point x Here the overbars denote temporal (not spatial) averages  With suitable assumptions L I = (8/π) 1/2 u’  I  Define integral scale Reynolds number Re L  u’L I / (recall = kinematic viscosity)  Note generally Re L ≠ Re flow = Ud/ ; typically u’ ≈ 0.1U, L I ≈ 0.5d, thus Re L ≈ 0.05 Re flow  Turbulent viscosity T  Molecular gas dynamics: ~ (velocity of particles)(length particles travel before changing direction)  By analogy T ~ u’L I or T / = C Re L ; C ≈  Similarly, turbulent thermal diffusivity  T /  ≈ Re L

7 AME Fall Lecture 10 - Premixed flames III Turbulent burning velocity  Experimental results shown in Bradley et al. (1992) smoothed data from many sources, e.g. fan-stirred bomb

8 AME Fall Lecture 10 - Premixed flames III = S T /S L Bradley et al. (1992)  Compilation of data from many sources = u’/S L

9 AME Fall Lecture 10 - Premixed flames III Characteristics of turbulent flames  Most important property: turbulent flame speed (S T )  Most models based on physical models of Damköhler (1940)  Behavior depends on Karlovitz number (Ka)  Low Ka: “Huygens propagation,” thin fronts that are wrinkled by turbulence but internal structure is unchanged  High Ka: Distributed reaction zones, broad fronts Defined using cold- gas viscosity

10 AME Fall Lecture 10 - Premixed flames III Characteristics of turbulent flames

11 AME Fall Lecture 10 - Premixed flames III Turbulent combustion regimes  Comparison of flamelet and distributed combustion (Yoshida, 1988) Flamelet: temperature is either T ∞ or T ad, never between, and probability of product increases through the flame Distributed: significant probability of temperatures between T ∞ or T ad, probability of intermediate T peaks in middle of flame

12 AME Fall Lecture 10 - Premixed flames III Estimates of S T in flamelet regime  Damköhler (1940): in Huygens propagation regime, flame front is wrinkled by turbulence but internal structure and S L are unchanged  Propagation rate S T due only to area increase via wrinkling: S T /S L = A T /A L S T /S L = A T /A L

13 AME Fall Lecture 10 - Premixed flames III Estimates of S T in flamelet regime  Low u’/S L : weakly wrinkled flames  S T /S L = 1 + (u’/S L ) 2 (Clavin & Williams, 1979) - standard for many years  Actually Kerstein and Ashurst (1994) showed this is valid only for periodic flows - for random flows S T /S L - 1 ~ (u’/S L ) 4/3  Higher u’/S L : strongly wrinkled flames  Schelkin (1947) - A T /A L estimated from ratio of cone surface area to base area; height of cone ~ u’/S L ; result  Other models based on fractals, probability-density functions, etc., but mostly predict S T /S L ~ u’/S L at high u’/S L with the possibility of “bending” or quenching at sufficiently high Ka ~ (u’/S L ) 2, e.g. Yakhot (1988):

14 AME Fall Lecture 10 - Premixed flames III Effects of thermal expansion  Byckov (2000):  Same as Yakhot (1988) if no thermal expansion (  = 1)  Also says for any , if u’/S L = 0 then S T /S L = 1; probably not true

15 AME Fall Lecture 10 - Premixed flames III S T in distributed combustion regime  Much less studied than flamelet combustion  Damköhler (1940): A ≈ 0.25 (gas); A ≈ 6.5 (liquid)  Assumption  T ≈  L probably not valid for high  ; recall …but probably ok for small   Example: 2 equal volumes of combustible gas with E = 40 kcal/mole, 1 volume at 1900K, another at 2100K  (1900) ~ exp(-40000/(1.987*1900)) = 3.73 x 10 4  (2100) ~ exp(-40000/(1.987*2100)) = 1.34 x 10 4 Average = 2.55 x 10 4, whereas  (2000) = 2.2 x 10 4 (16% difference)!  Averaging over ±5% T range gives 16% error!