16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 20: Combustion of Liquid Propellants Simplified Drop-Wise Liquid Combustion Theory 1. Introduction In this lecture we examine in more depth the fluid phenomena which dominate the operation, and hence the design, of a liquid rocket. We start by considering the two phase aspects of combustion, which usually determine the combustor size. Next we examine the stability of combustor operation and the design modification required to achieve it. Heat transfer to walls, wall compatibility and wall cooling form the next topic. We conclude with some notions in nozzle aerodynamic design and nozzle optimization. 2. Drop Vaporization and Spray Combustion Returning to Fig. 12.1 (lecture 12), we note again the existence of a zone AC where the propellants go from a series of liquid jets issuing through a multiplicity of small injector holes, through breakup of these jets into droplets, impingement (in some designs) of jets or droplet streams on each other, dispersion of the droplets into a recirculating mass of combustion products, evaporation of the droplets, interdiffusion of the vapors and kinetically controlled combustion. These are obviously complicated processes, and a comprehensive analysis good enough for first principles design requires large-scale computation [18, 19, 20]. In fact, the largest number of existing liquid rocket combustors, those dating from before 1970, were developed mainly through empirical methods, supplemented by very extensive testing. Improved modeling and computational capabilities have more recently permitted a more direct approach, with fewer hardware iterations, but theory is still far from completely developed in this area, and serves at this point mainly to ascertain trends and verify mechanisms. For an in-depth discussion of liquid propellant combustion, see Refs. 21 and 22. Here we will only review the fundamental concepts which underlie current spray combustion models. 2.1 Single-Drop Combustion Even when liquids are originally injected into the chamber, actual combustion takes place in the gas phase, following vaporization of both, oxidizer and fuel. In fact, vaporization is usually the rate–controlling step in the whole process, although its rate is itself affected by the reactions occurring near each droplet. In a few instances, both propellants enter the chamber as gases or easily vaporized liquids, and then gas phase mixing is the limiting step. (An example is the Shuttle SSME, where the hydrogen is vaporized in the cooling circuit, and the oxygen is partially vaporized by the hydrogen in a co-flowing heat exchanger arrangement ahead of the injectors, the rest being atomized and evaporated shortly after injection). In a fuel- rich environment, vapor from a fuel droplet diffuses without reacting into the surrounding gas, while heat diffuses inwards to supply the latent heat of evaporation. In the same environment, vapor from an oxidizer drop comes in contact, and reacts with the fuel-rich gas at some distance from the droplet, forming a spherical flame, from which heat diffuses both ways, while reaction products diffuse outwards and also stagnate in layer around the droplet. A similar description applies to a locally oxydizing environment. It will be sufficient to analyze, for example the case of a fuel drop in an oxydizing environment, from which we can easily derive other cases by suitable reinterpretations. As a preliminary, some relationships will be derived between the overall stoichiometry (oxidizer/fuel ratio, or OF, by mass), and the mass fractions of products and excess oxidizer or fuel to be found in the burnt gas far from the droplets. Let the actual mass ratio be OF, and the stoichiometric mass fraction be OFs (later called i for short), Assuming excess oxidizer, and complete combustion (a rough assumption), the reaction can be mass balanced as 1 (Fuel) + OF (Oxidizer) → (1+OFs) (Products) + (OF-OFs) (Oxidizer) and so the mass fraction of excess oxidization at infinity is Yox,∞ = OF − OFs OF − OFs = (1 + OFs ) + (OF − OFs ) 1 + OF Similarly, for fuel-rich operation, ⎛ ⎛ 1 1 ⎞ 1 ⎞ 1 (Fuel) + I (oxidizer) → ⎜1 + − ⎟ (Pr oducts ) + ⎜ ⎟ (Fuel) OF OFs ⎠ ⎝ ⎝ OF OFs ⎠ and the mass fraction of excess fuel at infinity is YF,∞ = 1 1 − OF OFs ⎛ 1 ⎞ ⎛ 1 1 ⎞ − ⎟ ⎜1 + ⎟+⎜ ⎜ ⎜ ⎟ OFs ⎠ ⎝ OF OFs ⎟⎠ ⎝ = OFs − OF 1 ( + OF ) OFs which could be also formally obtained from YOX, ∞ by reversing the roles of fuel and oxidizer (including OF → 1 ) OF Consider a spherical fuel droplet of radius R and density ρl . Its rate of vaporization is Γ = −4 π R 2 ρρ dR dt and the main purpose of the analysis is the calculation of Γ from given R and gas properties. Let Yf (r ) , Ypr (r ) by the mass fractions of fuel and of combustion products in the region R < r < rFl between the drop and the flame location (no oxidizer is to be found there, so Yf + Ypr = 1). These two constituents are both convected by the mean outwards flow, and diffuse according to their concentration gradients. Since the products do not penetrate the drop surface, the net outflow of fuel at radius r is equal to the total mass flow, and is given by Γ Yf − 4 π r2 ρ Df dYf =Γ dr where Df is the diffusivity in the products. The net flux of products remain zero in this region (but not their concentration Ypr = 1 − Yf ). Imposing that Yf approaches zero at the flame radius, the above equation integrates to Yf = 1 − e ⎛1 1 ⎞ −λ f ⎜⎜ − ⎟⎟ ⎝ r rFl ⎠ (R < r < rFl) where λf = Γ 4 π ( ρ Df ) and the product ρ Df is taken to be constant. While this gives information about the composition of the gas near the droplet, the more important results come from a similar mass balance outside the flame radius (r > rF). Assume cp f cp OX cpprods. k+ , k− k , ρg D k cP For r > rF , the oxidizer mass balance is Γ Yox = 4 π r 2 ρ D dYox = -i Γ dr (1) or dYox Γ iΓ + Yox = − d (1 r ) 4 π ρ D 4 π ρD Define λ= Γ 4 π ρD Γ Cp 4 πk and impose Yox (r = ∞ ) = Yox,∞ Yo x = (i + Yox, ∞ ) e − λ r (2) (3) (4) −i (5) Impose also Yox (r = rF ) = 0 , solve for λ rF : Y ⎛ ⎞ λ = ln ⎜ 1 + ox,∞ ⎟ rF i ⎠ ⎝ (6) which is a relationship between Γ and rF (rF turns out to be proportional to Γ , i.e., a “blowing” effect) The distribution of both, mass fluxes and mass fractions are displayed below: We now need the heat flux balance, both inside and outside the flame: R < r rF to complete the formulation. For this purpose, we now move the enthalpy zero to T=TF, the flame temperature. The total heat flow (convection plus diffusive) is now Γ cp ( T − TF ) − 4 π r2 k dT dr and this must be the heat released at the flame front, minus the heat sent inwards from the flame to the liquid. The former is Γ Q (Q= heat of combustion per unit fuel mass). The latter would be Γ L , except for the extra Γ cp ( TF − Tv ) due to the change of enthalpy references. Altogether, dT = Γ ⎡⎣Q − L − cp ( TF − Tv ) ⎤⎦ dr (11) Γ cp dT Γ ⎡Q − L − cp ( TF − Tv ) ⎤⎦ + ( T − TF ) = d (1 r ) 4 π k 4 πk ⎣ (12) Γ cp ( T − TF ) − 4 π r 2 k or Integrating with T ( ∞ ) = T∞ as a boundary condition, T − TF = λ λ Q − L − cp ( TF − Tv ) ⎛ − ⎞ − ⎜1 − e r ⎟ + ( T∞ − TF ) e r cp ⎝ ⎠ and particularizing at r = rF (T=TF), (13) TF − T∞ = ⎞ Q − L − cp ( TF − Tv ) ⎛ − rλF ⎜e − 1⎟ ⎜ ⎟ cp ⎝ ⎠ (14) which is the needed extra relationship to complement (6) and (10) Eqs. (10) and (14) can be simplified if λ is substituted from (6): rF ⎛ ⎞ λ ⎟ L ⎜ eR ⎜ − 1⎟ TF − Tv = Yox,∞ cp ⎜ ⎟ ⎜1 + ⎟ ⎝ ⎠ i TF − T∞ = (10’) Q − L − cp ( TF − Tv ) Yox,∞ cp i (14’) Eliminate TF between these two equations: ⎡ ⎛ ⎞⎤ ⎛ ⎞ λ λ ⎢Q − L ⎟⎥ Y ⎟ L ⎜ eR L ⎜ eR ⎜ ⎜ Tv + − − 1 ⎟ ⎥ ox,∞ − 1 ⎟ − T∞ = ⎢ Yox,∞ Yox,∞ cp ⎜ cp ⎜ ⎢ cp ⎟⎥ i ⎟ ⎜1 + ⎟⎥ ⎜1 + ⎟ ⎢ ⎝ ⎠⎦ i i ⎝ ⎠ ⎣ L cp e 1+ Solve for λ R Yox,∞ Yox,∞ ⎞ ⎛ Q Yox,∞ L + + T∞ − Tv ⎜1 + ⎟ = i ⎠ cp i cp ⎝ (15) i Γ cp λ , and use λ = : R 4 πk Γ= Q Yox,∞ i + cp (T∞ − Tv ) ⎤ ⎡ 4 πk R ln ⎢1 + ⎥ cp L ⎣ ⎦ (16) In the literature, this is written as Γ= with B ≡ 4 πk R ln ⎡⎣1 + B⎤⎦ cp Q Yox,∞ i + cp ( T∞ − Tv ) L (17) (18) (the Spaulding parameter). We have solved for Γ , and can go back to (6) to get the flame radius rF = R ln (1 + B ) Y ⎛ ⎞ ln ⎜1 + ox,∞ ⎟ i ⎠ ⎝ (19) and to (10’) to get TF. From (15), (18), lλ R = 1 + B , and then, from (10’), TF = Tv + L B − Yox ,∞ i cp 1 + Yox ,∞ i (20) Order of Magnitude: Assume an oxidizer–rich LOX-kerosene burner, with (O/F)s=i=2.4, and O/F =3. 3 − 2.4 This gives Yox,∞ = = 0.15 . The heat of reaction is roughly Q=4.3 x 107 1+3 J/Kg/fuel and the fuel latent heat is L 2.5 × 105 J / Kg . Assume the surrounding gas is at T∞ − Tv = 2000K , and take Cp Yox,∞ i B= = 2000J / Kg / K . We calculate 0.15 = 0.0625 2.4 4.3 × 107 × 0.0625 + 2 × 103 × 2 × 103 = 26.8 2.5 × 105 Yox,∞ ⎞ ⎛ In view of these numbers, ln ⎜1 + ⎟ i ⎠ ⎝ can simplify further to rF TF R Tv + i Yox,∞ Yox,∞ i 1 , but B 1 , so (19) and (20) ln (1 + B ) (19’) Q Yox, ∞ i L B = T∞ + cp cp (20’) Vaporization time - The droplet is actually being consumed, so its radius varies (on a slower time than the gas transit time), according to ρl 4 π R 2 dR = −Γ dt (21) and since Γ is given by (17) ρe 4 π R 2 ρl dR k = −4π R ln (1 + B ) dt cp R 02 − R 2 k ln (1 + B ) t = 2 cp R = R0 1 − 2 k ln (1 + B ) ρl cp R 20 t or R t = 1− R0 τv ; τv = ρl cp R 20 2 k ln (1 + B ) (22) τv is the deep vaporization time For an order of magnitude, assume R 0 = 50 µm = 5 × 10−5 m, ρl = 800 Kg / m3 , K 0.1 W / m / k, plus the numbers used in the previous examples. We calculate Γv = ( 800 × 2000 × 5 × 10−5 ) 2 × 0.1 × ln (1 + 26.8 ) On the other hand, since 2 = 6.0 × 10−3 s = 6 ms. τv ∼ R20 , if R0 were instead 10 µm , τv would be 25 times shorter, or 2.4 × 10−4 . To see the significance of this, assume the gas moves in the combustion chamber at M 0.2 , or ug ∼ 240 m / s . A droplet of radius 50 µm would then require a combustor length no less than 240 × 0.006 = 1.44 m , which is excessive from the mass point of view. A drop of R 0 = 10µm, on the other hand, require only 6 cm to evaporate. This brings out clearly the importance of good atomization. Other cases For a fuel drop in a fuel-rich background, everything is the same if we make Q=0 (no flame), and, of course, ignore rF and TF . In fact, the only equation needed is the r < rF heat flow balance (Eq. 9), which, using T(∞) = T ∞ , gives T∞ − Tv = L ⎡eλ R − 1⎤⎦ cp ⎣ and so, finally ⎛ cp ( T∞ − Tv ) ⎞ λ = ln ⎜ 1 + ⎟⎟ ⎜ R L ⎝ ⎠ or Γ= 4 πk R ln (1 + B ) cp (as in (17)), with B= cp (T∞ − Tv ) L (F / F Case) (23) Again, this is as in (18) if Q=0. So, the fuel drop evaporates faster if it is in an oxidizing environment (larger B). The reason, clearly, is the extra heating from the nearly flame in that case. For an oxidizer drop in an oxidizing background (0/0), we have a very similar situation (no flame), so, once again, Γ ox is given by (17), with B given by (23), but now Tv and L must be reinterpreted to be those for the oxidizer liquid. Finally, for an oxidizer drop in a fuel-rich background, we return to the first case (fuel drop in oxidizing gas), and switch the roles of fuel and oxidizer throughout. For, example, Q must be the heat of reaction per unit oxidizer mass, Yox,∞ must be replaced by Yf,∞ , i = (O / F)st. becomes 1 i , and (Tv , L) refer to the oxidizing liquid. CALCULATION OF CHAMBER LENGTH (Generalizing Spaulding’s ) The drop vaporization time Tv = (2000 version) ρl cp R 20 2 k ln (1 + B ) depend on B. For a fuel-rich background, as normally seen in rocket, we have two types of droplets: Fuel drop in fuel-rich medium: BF,F = Oxidizer drop in fuel-rich medium: BO,F = cp ( T∞ − Tv,f ) Lf Q Yf,∞ + cp ( T∞ − Tv,ox ) L ox (23) (24) (notice Q is still the heat of reaction per unit fuel mass). In many cases L ox < L f (especially for Lox), and also Q is a large quantity appearing in BO,F , but not in BF,F . So, BO,F BF,F , meaning the oxidizer drops are likely to evaporate much earlier than the fuel drops. For chamber length calculations, then, we focus on the fuel drops mostly. Assume fuel drops are injected into the chamber at a velocity uDo. Oxidizer drops are assumed to vaporize very fast (or to be fully evaporated at injection, as when Lox is used). At any section prior to fuel vaporization, the fuel liquid fraction (by flux) is γ= 4 A π R 3ρl nD uD • 3 mF (25) The number-flux of drops nDuD A is conserved, so γ ∼ R3 and from (22), since the initial γ is unity, 32 ⎛ t ⎞ γ = ⎜1 − ⎟ τv ⎠ ⎝ ( τv = fuel drop vap. time) (26) The vaporized fuel fraction is 1- γ , so the gas mean velocity is • ug • (1 − γ ) mF + mox = ρg A (27) and, at the end of the vaporization, this becomes • m u2 = ρg 2 A (28) If the molecular mass difference between fuel and oxidizer fuel and oxidizer vapors is ignored (consistent with earlier approximations), and ρg , Tg are nearly uniform, then we treat ρg as a constant, and so or • ug mF u2 • ug • (1 − γ) + m mox • = m 1 − γ + O /F 1 + O /F (29) 32 ⎡ ⎛ u2 t ⎞ ⎤ ⎢1 + O / F − ⎜1 − ⎟ ⎥ τv ⎠ ⎥⎦ 1 + O /F ⎢ ⎝ ⎣ (30) Notice that (29) implies an initial gas mean velocity ug ( 0 ) O /F u2 , which is 1 + O /F due to the vaporized oxidizer flow. The particle velocity, uD is different from ug in general, so there is slip, and therefore drag force. Assuming small Reynolds number around each drop, Stokes flow occurs, so duD 4 π R 3 ρl = 6 π R µ g (ug − uD ) 3 dt (31) or, defining a velocity relaxation time τRe l = 2 R 20 ρl 9 µg (32) 2 ⎛ R ⎞ duD τRe l ⎜ ⎟ = ug − uD ⎝ R 0 ⎠ dt (33) We now introduce non dimensional variable τv =b τRu ; 1− t τv =θ ; uD =υ u2 (34) and write (33) as θ dυ 1 + O / F − θ3 2 =− +υ b dθ 1 + O /F (35) This is a linear, first-order equation for υ(θ) , although with non-constant coefficients. It can be integrated by standard methods, like parameter variation. uD Imposing uD ( t = 0 ) = uDo , or υ ( θ = 1) = 0 , the solution is found to be u2 ⎡u ⎤ b 2b 2b 32 ν = 1 + ⎢ Do − 1 − ⎥ θ + (3 − 2b)(1 + O / F) θ 2 (3 − 2b)(1 + O / F) ⎣ ⎦ (36) One more integration will yield the vaporization length: L = τv 1 ∫ u dt = u τ ∫ υ ( θ ) dθ D 0 (37) 2 v 0 After simplification, this gives the result L 1 ⎡ uDo ⎛ 25 ⎞ ⎤ = + ⎜1 − b⎥ ⎢ u2 τv 1 + b ⎣ u2 1 + O / F ⎟⎠ ⎦ ⎝ (38) Notice that, from (22) and (32), the parameter b can be written as b= or ρl cp R 20 τv = τRe l 2 k ln (1 + B ) b= 1 2 ρl R 20 9 µg Pr 9 4 ln(1 + B) = 9 ( µgcp / k ) 4 ln (1 + B ) (39) For B=30 and Pr=0.8, this gives b=0.52. Spaulding defined a related parameter ξ = 2b and presented results for the monopropellant case. We can reduce to that case by setting O/F=0. Result are uD0 , and shown in the next figure for ξ = 0.5 (b = 0.25). The parameter χ0 is u2 ξ= x 2 u2 τv . As an example, for the middle graph χ0 = 0.5 , so using O/F = 0 in Eq. (38) we calculate ξ2 = L 1 = (0.5 × 0.6 × 0.5) or ξ2 = 0.267 . This coincides 2 u2 τv 2 × 1.5 with the result in the figure. Notice how the droplets are initially slowed down by gas drag, but later, as the gas evolves and accelerate, they are pulled along. Near the end of vaporization, the rates accelerate, because the droplets are so light. Evolution of evaporating droplet size and velocity. Here the reference vref for velocity is that of the gas after complete evaporation, and for length, lref = 2 vref τv References-Cont. 18. R.D. Sutton, W.S. Hines and L.P. Combs, “Development and Application of a Comprehensive Analysis of Liquid Rocket Combustion”, AIAA JI., Vol. 10, no. 2, Feb. 1972, pp. 194–203. 19. P-Y Liang, S. Fisher and Y.M. Chang, “Comprhensive Modelling of Liquid Rocket Combustion Chamber”. J. of Propulsion, Vol. 2, no.2, March-April 1986, pp 97-104. 20. M.S. Raju and W. A Sirignano, “Multicomponent Spray Computations in a Modified Centerbody Combustor”, J. Propulsion Vol. 6, No. 2, pp 97-105, March-April 1990. 21. F.A. Williams, Combustion Theory, Benjamin/Cummings Publishing Co., Inc., 2nd Ed., 1985. 22. W.A. Sirignano, “Fuel Droplet Vaporization and Spray Combustion Theory”, Progress in Energy and Combustion Science, Vol. 9, 1983, pp.291-322. 23. D.B. Spaulding, Aero. Quarterly, 10, 1. (1959). See also Ref. 21, Sec. 11.15. 24. H.H. Chiu, H.Y. Kim and E.J. Croke,”Internal Group Combustion of liquid Droplets”, Proc. 19th International Symposium on Combustion, 1983. 25. P.Y. Liang, R.J. Jensen and Y.M. Chang, “Numerical Analysis of SSME Preburner Injector Atomization and Combustion Process”. J. of Propulsion, Vol. 3, No. , Nov-Dec.1987, pp. 508-514. 26. D.T. Harrje and F.H. Reardon (Editors). Liquid Propellant Rocket Combustion Instability, NASA SP-194, 1972. 27. R. Bhatia and W.A. Sirignano, “One dimensional Analysis of Liquid-Fueled Combustion Instability”, J. of Propulsion, Vol 7, No. 6, Nov-Dec. 1991, pp 953-961. 28. G.A. Flandro, ”Vortex Driving Mechanism in Oscillatory Rocket Flows”, J. of Propulsion, Vol. 2, No. , May-June 1986, pp. 206-214. 29. R.J. Priem and D.C. Guentert, “Combustion Instability Limits Determined by a Non linear Theory and a One-dimensional Model”. NASA TND-1409, Oct. 1962. 30. M. Habiballa, D. Lourme and F. Pit. “PHEDRE-Numerical Model for Combustion Stability Studies Applied to the Ariane Viking Engine”. J. Propulsion, Vol. 7, no. 3, May-June 1991, 322-329. 31. T. Poinsot, F. Bourienne, S. Candel and E. Esposito, “Suppression of Combustion Instabilities by Active Control”. J. Propulsion, Vol.5, No. 1, JanFeb. 1989, pp. 14-19. 32. NASA SP-8113, Liquid Rocket Combustion Stabilization Devices, Nov. 1974.

Lecture 20: Combustion of Liquid Propellants

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