r- exchange coefficient

Transcription

1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y IGT-29 The Society shall not be responsible for statements or opinions advanced in papers or in dis cussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available from ASME for fifteen months after the meeting. Printed in USA. Copyright 1985 by ASME A Preliminary Calculation of the Combustion Recirculating Flow Behind Flame-Holders C. F. ZENG*, J. X. ZHAO**, Z. T. DING*** Nanjing Aeronautical Institute Nanjing, People's Republic of China ABSTRACT In this paper the numerical method is adopted to predict the combustion recirculating flow behind flameholders. A modified k- &turbulence model and Magnussen's combustion model are employed. Becouse of the large variation of density in combustion flow field,the influences of variable density have been considered. The results predicted include: the combustion recirculating flow field and the flame spreading; the variation of recirculation length with equivalence ratio, and inlet velocity, etc. All of these are qualitatively good comparing with experimental results. NOMENCLATURE turbulence dissipation rate viscosity density Prandtl/Schmidt number dependent variable r- exchange coefficient SUPERSCRIPT SUBSCRIPT time average value density-weighted average value time average fluctuation density-weighted average fluctuation A CI,C2,CA C P C p D d EQ f H k P' r R R d R S T u,v u b x GREEK 6ij constant of combustion model constants of turbulence model constant-pressure specific heat and static pressure coefficient flame-holder diameter equivalence ratio mixture fraction (f mox-m ft; s) stagnation enthalpy turbulence kinetic energy mass fraction molecular weight and mass flow rate pressure pressure corretion radial coordinate universal gas constant and volumetric reaction rate flameholder radius pipe radius stoichiometry mass ratio and Strouhal number absolute temperature axial, radial velocities maximum of axial velocity at certain section axial coordinate Kroneker delta i,j,k tensor index species fu fuel ox oxygen pr product 0 dependent variable INTRODUCTION The main object of this paper is to study the possibility of predicting the combustion recirculating flow behind flameholders. This work is a preparation for the further calculation of the flame-holder flamestability (lean) of jet-engine's afterburner. The calculation of this paper includes the combustion recirculation zone and the flame spreading starting from this zone. The predicted results are compared with the experimental data of Refs. (1), (2), (I). Because of the large variation of density in combustion flow field, the influences of density variation have been considered and the density-weighted average has been used in the conservation equations. For simplification, a thin disk is used as a flameholder inside the pipe, with a blockage ratio of The schematic calculation domain is shown in Fig. 1. The inlet flow is assumed to be fully developed turbulent and axi-symmetrical. The inlet mixture is well premixed (propane{ air), and the inlet variables are uniformly distributed. Several cases with different Presented at the 1985 Beijing International Gas Turbine Symposium and Exposition Beijing, People's Republic of China September 1-7, 1985

2 Renolds-stress: in 04 Mt 0.2m r R ==0.1m R d ==0.05m \VA\ \ Mean Flame Spreading L r 1. 5m Recirculation zone FIG.1 SCHEMATIC CALCULATION DOMAIN equivalence ratios, and inlet velocities,includiqgcold flow case, have been calculated, and the predictions are qualitatively good. FUNDAMENTAL EQUATIONS AND MODELS The Conservation Equations The general form of two-dimensional, differential equations for the turbulent combustion flow with recirculation may be expressed as: ribeff ) :C. (1 ) where represents respectively the density-weighted average values, i.e. u, v, lc, E, H, m and f, is the time average value of density, u S is the source term. roe,f is the effective turbulent exchange coefficient of 4. In combustion flow field the variation of temperature and density is rather large, so it is believed that the consideration of variable density and the densityweighted average may be more resonable, although this problem remains unsolved clearly in theoretical aspects at present. In this paper, the influences of variable density are considered and the density-weighted average is employed, so that the source terms in eq. (1) differ from those of equations of constant density and time average, because of the addition of variable density terms (See Table. 1). The Renolds-stress and turbulent fluxes of scalar values are modeled as follows, (4). X fuiui ai(e10-atplixdgij itit(fx +al ax.) (2) J turbulent fluxes of scalar values: a'43 P uy of arc; whezt813.. is Kronecker delta, For k and E equations the added terms from density variation, the last density-fluctuation corelation in k, E equations' source terms shown in Table. 1, are somewhat difficult to be modeled, hence they are neglected in this paper for simpification. Therefore the k, E equations in this paper have the same forms as those of constant density and conventional average,(5). Temperature is determined from the expression of the mixture specific stagnation enthalpy: irifu.hfut (112+v) (4) where Cis determined from six-order polynominals P,J in temperature given in (6). Density is determined from the state equation of ideal gas: (3) F-13/(RT. ) ( 5) where M. is the molecular weight of species j, and R is the universal gas constant. The Turbulence Model The two-equation model of turbulence is employed. The turbulent viscosity Alt is given by: Jilt= CA. P. R2/ E (6) where C AL and Cf, C2 in the source term of E's equation are turbulent model constants. These constants are assumed to have the same values as those of the conventional turbulence model, i.e. CAA. = 0.09; C1=1.44; except that C2 is adjustable, C2 is 1.92 for cold flow and 1.73 for reacting flow. The Combustion Model A simple combustion model, which neglects the fluctuations of species and temperature, is employed in this paper. The reaction rate Rfu is given by TABLE. 1 dp r$eff '--(p EC..uhf _ Ril, 4_ a r 1.11 _2_ 1,,,, c )c..-, axi ' K-t-Ateft ) utj.1 DX; 1--"'" e il axi 3,1V axic i-i Aeff/Crk dilyc, -1-- (P k -frjtteff a l V e(itif.?_,4. + E i: )3 - g- P k g. ST((-4e1J F-Leff)A (i (- Elf- 2)) 1 (- uu. ap Aeff Ark d xi ' p DXt E /Lief f ATL Lii".U} I -a----ki C2 E ) - - E/ k c i 'DP (Cr t (5-i. _e_lu" ax) p ax. 114, Alef Al, k;, -f,tieff /07i 0 2

3 Magnussen & Hjertager's formula (7) or by Arrhenius' formula. The formula proposed by Magnussen, etal is expressed as: f, (3A E / KJ. minimum Rods 0511pr/(1 + 5)) (7) where A is an empirical constant having the value of about 8.0in this paper. The Arrhenius formula is: _ Rfu,2 = *5 ' 771fu. mox. exp (-19000/T ) (8) RESULTS AND DISCUSSIONS The distribution of streamlines is shown in Fig.2, from which it can be seen that combustion makes the eye of vortex relatively more close to the flame-holder. The ratio of reverse to total mass flow rate, Mr/Mt, which is indicative of the recirculation strength,is platted in Fig.3. The maximum value of reverse flow rate, because of lower density, decreases in combustion cases. These predictions are well conform with the experiments of Ref.(2). The minimum one is taken to be Tffn, namely: - Riu minimum RfU, i 5 krel4,2} (9) Finite-difference Equations The partial differential equations, eq. (1), can be transformed into the finite difference ones by approximate integration over control volumes surrounding each intersection point. The finite difference equations for each node P have the form: 0. Eye of Vortex -41-1tr-- -, -2.0, 0.0% an(1) + (l o) where the a's are the coefficients expressing the combined influence of convection and diffusion, Sv is the integrated version of source term -S-4, in eq. (1), andql, denotes the summation over the four neihouring nodes x/dd FIG.2 MEAN STREAMLINES DISTRIBUTION (uin= 40mis, T in = 600K, EQ =0 7 ) THE BOUNDARY CONDITIONS AND SOLUTION PROCEDURE Mr/Mt uin.40mis The Boundary Conditions =600K Tin The flow variables at inlet are assumed to be uni / form, i.e. u=, T=Tir,. The exit section is located far away from the recirculation zone, that is to say, about five times as long as the recirculation length, Lr, so that the longitudinal gradients of all variables at exit can be assumed to \ Cold EQ=0.7 \\ be vanish. Also, at exit the continuity correction of mass flow should be made. On the axis of symmetry, transverse gradients of all variables are zero and radial velocity, v, is zero. 0.0 At the walls of pipe and flame-holder the values of 0.5 x/lr u,v,k,e are given to be zero. Because of the large variation of turbulent transport coefficients in the region FIG.3 REVERSE FLOW RATE RATIO DISTRIBUTION near wall, "wall functions" are used for u,v,k,e, (8). The walls of pipe and flame-holder are assumed to be adiabatic and impermeable, so that the normal gradients of stagnation enthalpy, H, and fuel concentration, me., are zero. Solution Procedure The SIMPLE algorithm embodied in (8) is used to solve the finite-difference equations for u,v,p',k,e,h, m +, and f. Convergence criteria is given by the following: a.,(1).-54,1/(m-cpref) < (11) where M is mass flow rate. Under-relaxation has to be used to ensure convergence. About 300 iterations are needed to achieve a converged solution. The computational grids employed are 35x26 (non-uniformly distributed). The CPU time on IBM 4341 is about 30 min. FIG.4 AXIAL-VELOCITY DISTRIBUTIONS ALONG THE AXIS OF SYMMETRY (u in = 4Ornis, T in = 600K) 3

4 Fig. 4 shows the axial-velocity variation along the axis of symmetry and shows how the velocity distribution is changed with different equivalence ratios. From Fig. 4, the dimensionless recirculation length, Lr/Dd, could be determined by zero axial velocity at axis of symmetry. The effects of equivalence ratio and inlet velocity on axial extents of recirculation zone are summarized in Fig. 5, and these are good in general trend with the experimental data of Refs. (1), (2). corresponds with the experimental findings in (1). Fig. 6 shows the radial profiles of temperature and fuel concentration.there exists a certain region where both the temperature gradient and fuel concentration gradient are very large. The flame front can be regarded to be located in this region. The positions of flame front for each section are slightly different,thus the so-called flame spreading is formed. The predicted mean angles of flame spreading are about 2. The angles iceepnearlyconstantfordifferenteqandu.,which is similar to that results of (10), (11). in (a) 2000K I x D d Fuel Concentration (b) 0. 5 (b) x/dd FIG.6 RADIAL PROFILES OF TEMPERATURE AND FUEL CONCENTRATION (uin, 40m/s, Tin = 600K, EQ =0.7) x/dd Cold FIG.5 EFFECTS OF EQUIVALENCE RATIO AND INLET VELOCITY ON THE LENGTH OF RECIRCULATION -2 EQ=0.7 Following J. E. C, Topps (9), propably the effects of equivalence ratio and inlet velocityon the reversal length can be explained by the Strouhal number which is defined by the following formula: S = n T d/uin C FIG.7 PRESSURE COEFFICIENT VARIATION ON THE AXIS OF SYMMETRY (l in ==40m/s, T in =600K) where n is the flame frequency, and n is the reciprocity of the ignition time. The larger the Strouhal number is, the smaller the recirculation zone would be. As a rule, the ignition time increases with the decrease or increase of the equivalence ratio, hence n will be reduced and S will become smaller. Therefore the Lr/Dd would be larger. In hot cases, the effect of the inlet velocity on the Lr/Dd could be explained in a similar way. This explanation seems resonable and The axial distribution of the static pressure coefficient, C p, which is (p - P in )/(ip e in ), are shown in Fig.7. It can be seen that the Cp varies larger in isothermol flow while smaller in reacting flow, and the minimum values of these two cases are nearly of the same value. The turbulent flow properties are summarized in Fig.8. They comprise: the normalized turbulent kinetic 4

5 energy distribution A/a/ub ; the dissipation rate of turbulent kinetic energy E. The distributions of these properties for cold flow are well agreable to relevant experimental data (3). These for reacting flow are found to be similar to those for cold flow (/ub FIG. 8 RADIAL DISTRIBUTIONS OF TURBULENT PROPERTIES (l in =40m/s, Tin =600K) CONCLUDING REMARKS _-- Cold I/u b ; EQ=0.7 A/-21/ub Cold 6 ; EQ=0.7 6 ex io4 Preliminary calculations for flameholder's recirculating flow with flame have been done with considering the influence of variable density and using the density-weighted average in conservation equations. The "k-e," turbulence model, with adjustable model constant C2, is employed. Also a simple combustion model combining Magnussen's formula and Arrhenius formula is used to determine the reaction rate. The comparisons show that the results obtained with the present procedure are in agreement with the experimental results qualitatively. The distributions of temperature and fuel concentration seem to be not good enough and the angle of flame spreading is smaller. However, the work presented in this paper seems encouraging No doubt a lot of research work, for example, to adopt other compustion models, could be done further to improve the precision and to make the procedure and program more flexible. REFERENCES 1 Winterfeld, G. : "On Processes of Turbulent Exchange Behind Flame-holders" 10th Symposium (Int) on Combustion, 1965, p Fujii, S., etal : "A Comparison of Cold and Reacting Flows Around a Bluff-Body Flame Stabilizer" ASME J. Fluids Engineering, Vol. 103, June. 1981, p Fujii, S., etal : "Cold Flow Tests of a Bluff- Body Flame Stabilizer" ASME J. Fluids Engineering. Vol. 100, Sept. 1978, p Jones, W.P : "Models for Turbulent Flows with Variable Density and Combustion", VKI Lecture Series, , Jan , Bradshow, P., etal : "Engineering Calculation Methods for Turbulent Flow", ch. 15, Academic Press Inc. (London), Prothro, A. : "Computing With Thermochemical Data" Combustion and Flame, Vol. 13, 1969, p Magnussen, B.F, and Hjertager, B.H. : "On Methmatical Modeling of Turbulent Combustion with special Emphasis on Soot Formation and Combustion" 16th Symposium (Int) on Combustion, 1976, p Gosman, A.D. : "TEACH - T General Program", Topps, J. E. C, " An Optical Technique for the Investigation of Flow in Gas Turbine Combustors " 17th Symposium (Int) on Combustion, 1978, p Wright, F. H & Zukoski, E. E. 8th Symposium (Int) on Combustion, 1960, p Howe, J. H., etal: 9th Symposium (Int) on Combustion, 1963, p.36. N.B. The Authors: * Professor a* Lecturer NM* Research Assistant 5