iii (I. I. I! as?! otttfit vial‘f: I In» "3...ng id? . in 35‘: vl ) ft .41 5.24:... \r at.) g .i , 39);?!" rift. .. a Hafiz» In“: .1er in. ,5 2:! . 11.. .. .. e EL: .122. til. Kn.‘llnflfuér- ; . . .It Ptcllxrafitvflxa Do. {7323‘}!!! fir!!! uF. a Ill‘r. 1. If!» a i. u x: it!‘ re ‘I‘lvl (.1165), PVI‘E‘ In)» "Maxi-a a}: {pas . t...) .). . (0:3 Q3. . . lulu. 41 {filtpl z 5):)! ( 1! Dell... , (r)?! I:‘\va:. 1i!!a.rft1.tr.v 130i 3.37532351“ ((-7. Dixig‘v 111.41.).‘Vlfvfii If]; «'1! III‘ I. '1 {Le t. I]?! .1! rfatllllrnb? I‘If. r5813 IIIIIIIIIIIIIIIIIIIIIIIIIIIIII llHllllllllllllllllllllllllllllllllllllllllllllllllllllllll 31293016914198 This is to certify that the thesis entitled A TWTATIVE DOUBLE—IPRAY MODEL FEA‘ Tum/e. SPRAY TflPORIi‘AT 105/ AND T HON COUNTEQFLOW gagged by FL mar MI mm; L YANCq has been accepted towards fulfillment of the requirements for MAf-TEQ 0F “(70165“; degree in MECHANICAL ENGI'INEEKIVQ 9M? 2. ‘ Major professor Date 22' AtguS‘, 1777 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE IN RETURN BOX , to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 1/98 c:/ClFlC/DatoDm.p65—p.14 A TENTATIVE DOUBLE-SPRAY MODEL FEATURING SPRAY VAPORIZATION AND COUNTERFLOW DIFFUSION FLAMES By Michael Yang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1997 ABSTRACT A TENTATIVE DOUBLE-SPRAY MODEL FEATURING SPRAY VAPORIZATION AND COUNTERFLOW DIFFUSION FLAMES By Michael Yang The word “tentative” means purely theoretical. In this model we consider two gas- eous streams (e.g., helium and nitrogen) approaching each other from opposite directions in a counterflow. The two opposed streams each carry a distribution of liquid droplets. Hence, we are examining the problem of a dual counterflow spray flame. We consider that the sprays consist of liquid fuel (e. g., hydrogen) and liquid oxidizer (e.g., oxygen), respec- tively. The sprays vaporize, and their subsequent gaseous phases diffuse toward a chemical reaction region near the stagnation plane, at which reactants burn. At the outset, with the proper thermodynamic assumptions, we derive a set of steady-state, ordinary differential equations to describe the temperature of the gas flow and the mass fractions of each reactant. Then, we solve numerically the differential equations in three consequent cases, each more complicated than the one previous: (1) fast vaporiza- tion and fast chemistry, (2) finite-rate vaporization and fast chemistry, and (3) finite-rate vaporization and finite-rate chemistry. We compare the numerical results of our model to the fuel-spray-only and the purely-gaseous counterflow diffusion flame models. Dedicated to my beloved Mother, Father, and Sister. iii ACKNOWLEDGMENTS This thesis strictly resembles the paper, “Counterflow Spray Diffusion Flames: Com- parison between Asymptotic, Numerical, and Experimental Results,” a masterpiece by Mr. Philippe Versaevel, Ecole Centrale de Paris, France, 1993. In other words, Mr. Versaevel’s work is so excellent as well as promising and extendable, that it enables me to construct a similar, extended version upon itself. However, due to logistic limitations, the asymptotic and experimental methods applied by Mr. Versaevel are not covered in this thesis. I shall be full-heartedly grateful to my academic advisor, Dr. Indrek S. Wichman, for his four-year-long guidance, encouragement, and... tolerance! He changed my life. TO CHANGE A LIFE IS TO CHANGE A WORLD ENTIRE. iv Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 TABLE OF CONTENTS LIST OF FIGURES NOMENCLATURE INTRODUCTION MAIN ASSUMPTIONS GENERAL FORMULATION 2.]. Governing Equations 2.2. Nondimensionalization FAST VAPORIZATION AND FAST CHEMISTRY 3.1 Formulation by Coupling 3.2 Formulation by Other Manipulations 3.3 Locations of the Vaporization Fronts 3.4 Location of the Flame Front 3.5 Procedure to Determine the Profile FINITE-RATE VAPORIZATION AND FAST CHEMISTRY 4.1 Preliminary Relations 4.2 Procedure to Determine the Profile FINITE-RATE VAPORIZATION AND FINITE-RATE CHEM- ISTRY 5.1 Procedure to Determine the Profile NUMERICAL RESULTS OF THE DOUBLE-SPRAY MODEL 6.1 Fast Vaporization and Fast Chemistry 6.2 Finite-Rate Vaporization and Fast Chemistry 6.3 Finite-Rate Vaporization and Finite-Rate Chemistry COMPARISON BETWEEN DOUBLE—SPRAY, FUEL-SPRAY- ONLY, AND PURELY-GASEOUS MODELS vii xii \O\O 14 15 18 23 26 3O 35 36 38 45 45 55 56 57 61 92 CONCLUSIONS BIBLIOGRAPHY vi 94 95 g. Figure I Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 LIST OF FIGURES A representation of the tentative double-spray model Five functions versus X: F _ RX) 1 —X and 1 X over [0, 0.5], X and X over [0.5, l], and F(X) over [0, l]. A result of fast vaporization and fast chemistry in X-coordinate, with Yoo = 0.15 , YFF = 0.03 (i.e., ¢ = 1.6 ). Upper diagram: T [K]. Lower diagram: ng, Yog (_), sz , Yo, (-.), Yp (--). A result of fast vaporization and fast chemistry in 32 -coordinate, with YOO = 0.15 , YFF = 0.03 (i.e., ¢ = 1.6 ). Upper diagram: T [K]. Lower diagram: ng, Yog (_), YFI , Y0] (-.), Yp (--). Results of fast vaporization and fast chemistry in X-coordinate, with Yoo = 0.15 , YFF = 0.015 , 0.03, 0.045, 0.06 (i.e., (l) = 0.8 , 1.6, 2.4, 3.2, respectively). Upper diagram: T [K]. Lower diagram: irpg. iog (_), Yp (--). Results of fast vaporization and fast chemistry in SC -coordinate, with Y00 = 0.15 , YFF = 0.015 , 0.03, 0.045, 0.06 (i.e., (1) = 0.8 , 1.6, 2.4, 3.2, respectively). Upper diagram: T [K]. Lower diagram: YFg, YOg (_), YP (u). Results of fast vaporization and fast chemistry in X-coordinate, with 6 sets of inputs: Yoo = 0.15 , YFF = 0.015 and 0.06 (i.e., q) = 0.8 and 3.2); YOO = 0.1 , YFF = 0.01 and 004(4) = 0.8 and 3.2); YOO = 0.05, YFF = 0.005 and 0.02 (0 = 0.8 and 3.2). vii 34 67 68 Figure 8 Figure 9 Figure 10 Figure l 1 Figure 12 Figure 13 Figure l4 Upper diagram: T [K]. Lower diagram: ng , Yog (_), Yp (--). Results of fast vaporization and fast chemistry in SE -coordinate, with 6 sets of inputs: Y00 = 0.15 , YFF = 0.015 and 0.06 (i.e., 4) = 0.8 and 3.2); YOO = 0.1, YFF = 0.01 and 004(4) = 0.8 and 3.2); YOO = 0.05, YFF = 0.005 and 0.02 (q) = 0.8 and 3.2). Upper diagram: T [K]. Lower diagram: ng, Yog (_), Yp (--). A result of finite-rate vaporization and fast chemistry in X-coor- dinate, with Yoo = 0.15, YFF = 0.03, l/DOv = 0.1 . Upper diagram: T [K]. Lower diagram: ng, Yog (_), Y“, Yo] (-.), Yp (--). - A result of finite-rate vaporization and fast chemistry in SE -coor- dinate, with Yoo = 0.15, YFF = 0.03, 1/DOV = 0.1 . Upper diagram: T [K]. Lower diagram: ng , Yog (_), YFI, Yo; (-.), Yp (--)- Results of finite-rate vaporization and fast chemistry in X-coor- dinate, with Y00 = 0.15, YFF = 0.03, l/Dov = 0.0001 ,0.25, 0.5, 0.75, 1.0. Upper diagram: T [K]. Lower diagram: ng , Yog (_), §FI , ?O[ ('3), QP (")- Results of finite-rate vaporization and fast chemistry in it -coor- dinate, with Yoo = 0.15, YFF = 0.03, l/DOv = 0.0001,0.25, 0.5, 0.75, 1.0. Upper diagram: T [K]. Lower diagram: ng, Yog (_), YFI. {for (--), YP (--). Results of finite-rate vaporization and fast chemistry shown as X-locations and temperatures versus a broad range of l/D0v , with YOO = 0.15. YFF = 0.03. Upper diagram: XF‘v , XF: , xeq, X01, , Xo+v . Lower diagram: TH, TFH, T0,, TOW , T [K]. eq Results of finite-rate vaporization and fast chemistry shown as X-locations and temperatures versus a broad range of 1/DOv , with viii 71 72 73 74 75 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Yoo = 0.15 , YFF = 0.015 , 0.03, 0.045, 0.06. Upper diagram: X0; (_), Xeq (-.). Lower diagram: TOW (_), Teq (-.) [K]. Results of finite-rate vaporization and finite-rate chemistry shown as temperature at the fast-chemistry flame front, T f [K], ver- sus strain rate, K [1/sec], with Yoo = 0.15 , YFF = 0.025 , 0.0] and the original diameter of 02 = 1 pm, 10 um, 20 um. Results of finite-rate vaporization and finite-rate chemistry shown as temperature at the fast-chemistry flame front, Tf [K], ver- sus strain rate, K[1/sec], with Y00 = 0.15 , YFF = 0.025 , 0.026, 0.027, 0.028, 0.029, 0.03, 0.031, 0.032, 0.033, 0.034, 0.035, and the original diameter of 02 = 1 um, 10 um, 20 mm. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=150.88 /sec, and the original diameter of 02 = 1.0 pm. Upper diagram: T [K]. Lower diagram: ng, Yog (_), IF! , Yo: (-.), Yp (--). A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=150.88 /sec, and the original diameter of 02 = 1.0 um. Upper diagram: T [K]. Lower diagram: ng, Yog (_), sz , Yo] (-.), Yp (--). A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.035, K=97.27 /sec, and the original diameter of 02 = 10 um. Upper diagram: T [K]. Lower diagram: ng, Yog (_), Yp, , Yo, (—.), Yp (")- A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=97.27 /sec, and the original diameter of 02 = 10 p. m. Upper diagram: T [K]. Lower diagram: ng, Yog (_), sz , Yo) (-.), Yp (--)- ix 77 78 80 81 82 Figure 28 Figure 29 Figure 30 extinction in X-coordinate, with YOO = 0.15 , YFF = 0.025 , K=l7.02 /sec, and the original diameter of 02 = 20 p. m. Upper diagram: T [K]. Lower diagram: ng, Yog (_), Yp; , Yo) (-.), Yp (--)- A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.025 , K=17.02 /sec, and the original diameter of 02 = 20 u m. Upper diagram: T [K]. Lower diagram: ng, Yog (_), sz , Yo; (-.), Yp (--)- Results of finite-rate vaporization and finite-rate chemistry shown as (upper) the global equivalence ratio 0 versus strain rate at extinction, Kext [1/sec], and (lower) 0 versus the highest flame temperature at extinction, Tmax, ext [K], with Y00 = 0.15 , and the original diameter of 02 = l um, 10 um, 20 um. Comparison between the Double-spray, Fuel-Spray-Only, and Purely-Gaseous Models xi 89 90 91 93 NOMENCLATURE frequency factor for the chemical reaction specific heat per unit mass of the gas mixture original droplet diameter diffusion coefficient classical Damkohler number vaporization Damkohler number for the fuel vaporization Damkohler number for the oxidizer strain rate of the counterflow heat of vaporization per unit mass of fuel heat of vaporization per unit mass of oxidizer droplet number density heat released by combustion per unit mass of gaseous fuel droplet radius temperature of gas activation temperature for the chemical reaction xii Fl Fv O] 00 CV boiling temperature of the fuel boiling temperature of the oxidizer axial velocity axial coordinate mass fraction Greek Symbols thermal conductivity of gas stoichiometric coefficient global equivalence ratio (g.e.r.) density gas density Subscripts and Superscripts relating to the origin of the fuel relating to gaseous fuel relating to liquid fuel relating to the fuel vaporization relating to gaseous oxidizer relating to liquid oxidizer relating to the origin of the oxidizer relating to the oxidizer vaporization xiii INTRODUCTION Liquid fuel, especially when injected as a spray, is popular to many types of combustion engines. To investigate how the existence of the liquid phase of a reactant influences the combustion phenomenon, Philippe Versaevel [1] developed a simplified model by installing two nozzles injecting two gaseous nitrogen streams which approach each other from opposite directions in a counterflow. One of the opposed streams carries a distribution of liquid hydrocarbon droplets, considered as the fuel spray, and the other stream, which is purely gaseous, consists of an inert gas mixed with a certain amount of gaseous oxygen as the oxidizer. The fuel spray vaporizes, and then both the subsequent gaseous phase of fuel and the gaseous oxidizer diffuse and convect toward a chemical reaction region (denoted as the “flame zone”) near the stagnation plane, at which reactants bum. This model, later referred to as the “fuel-spray-only” model in this thesis, is described as one-dimensional, steady-state, isobaric (under one atmosphere), constant- strain, fuel-lean, diffusional-flaming, and the distance between the origins of the fuel and oxidizer, which we regard as the two boundaries of the entire flow field being considered, is infinitely long compared to the thickness of the flame zone. The spacecraft liquid rocket engines inspire us to consider the oxidizer as well as the fuel as originally in liquid phase. However, in our research model, we do not intend to imitate a liquid rocket engine, say, the Space Shuttle Main Engine, where pure hydrogen and pure oxygen are originally compressed and stored in the liquid phase and are afterwards released and vaporized due to pressure difference [2]. Yes, we do hope that our current research will lead to better understanding of a liquid-fuel-liquid-oxidizer com- bustion mechanism, but, to make it theoretically derivable as well as experimentally con- trollable, we, instead, establish our model through a strict extension of Versaevel’s model, so that the governing Navier-Stokes Equations can be largely simplified because of the isobaric condition and that the flame temperature can be easily controlled by changing the original mass fractions of the reactants (the fuel and oxidizer) within the non-reacting, background gases. After the extension, our model is identical to Versaevel’s except that both the fuel and oxidizer are originally in liquid droplets, and hence we call ours the “double-spray” model. Chapter One employs all of Versaevel’s assumptions, which lead to our governing equations in Chapter Two. It is worth noting that these assumptions treat, at any given horizontal location (i.e. the x -coordinate), the gas flow as homogeneous and station- ary relative to the reactants’ droplets, and these droplets as uniform-sized, uniformly vaporized, in a vaporization mechanism macroscopically described and derived by F. A. Williams via the “(12 law” [3], which constitutes the vaporization terms for both the fuel and oxidizer in the governing equations. In Chapter Two we shall extend Versaevel’s fuel-spray-only model into our double-spray model by adding the vaporization term for the oxidizer to our governing equations. It is worth noting that this term can hardly be balanced in the overall coupling function (see Section 3.1) by the already introduced quantities; therefore, we introduce a quantity, the mass fraction of the product, denoted as Yp , which plays a significant role in the coupling functions of Chapter Three, and in determining the location of the fast-chem- istry flame front in Chapter Four. Now we are able to investigate the key features of our model in Chapter Three by starting with the assumptions that both the chemical reaction rate and vaporiza- tion rates (for the fuel and oxidizer) are infinitely large, which implies that both the chem- ical reaction and the vaporization of sprays will occur at infinitesimally thin fronts. We denote this most special case as “fast vaporization and fast chemistry.” The fast—rate assumptions enable us to manipulate and transform the comprehensive differential equa- tions into algebraic expressions, which henceforth allow us to examine certain key factors theoretically and to obtain numerical solutions through a two-parameter algorithm which repeats merely algebraic calculations, as sketched in Section 3.4. It is also worth noting that, when we numerically locate the oxidizer’s vaporization front at it own origin, all the algebraic formulas derived in our double-spray model will reduce to those in the fuel- spray-only model. In Chapter Four we continue our analysis from the previous chapter by allowing the vaporization rates of sprays to be finite, which is more realistic. The two vaporization fronts for the fuel and oxidizer in Chapter Three are now released into two broader regions, denoted as the “vaporization zones.” We denote this case as “finite-rate vaporization and fast chemistry.” The associated solution to the governing equations may be numerically acquired through a two-parameter algorithm which repeats integration over the two vaporization zones, as sketched in Section 4.2. To accomplish the most realistic case in Chapter Five, we continue the analysis from the previous chapter by setting the chemical reaction rate to be finite, hence stretching the flame front onto a broad domain known as the flame zone. We denote this case as “finite-rate vaporization and finite-rate chemistry.” The associated solution to the governing equations may be numerically acquired through an eight-parameter algorithm (containing two more parameters than that for the fuel—spray-only model) which repeats an overall integration throughout the entire flow field, and which is thoroughly sketched in Section 5.1. Next, we proceed to demonstrate our double-spray model by selecting liq- uid hydrogen as the fuel and liquid oxygen as the oxidizer, and we employ certain refer- ence for their thermodynamic properties [4,5]. Since the liquid phase of hydrogen under one atmosphere requires an ambient temperature as low as 20.38 Kelvin, we tend to select helium, which has the lowest boiling point in the Universe [6], as the non-reacting, back- ground gas to carry both the fuel and oxidizer liquid phases. However, the subsequent computations show trouble. Basically, to prevent droplets from deterring the momentum of the gas flow, the original mass fraction of each liquid phase must be as low as 15% of the mass of the background gas. But, due to oxygen’s large molecular weight compared to helium’s, a low oxygen mass fraction compared to the mass of helium will lead to a even lower molecular fraction which results in low concentration, hence weak chemical reac- tion, and low flame temperature. Therefore, we finally employ gaseous nitrogen, which has a similar molecular weight and slightly lower boiling temperature than that of oxygen, as the background gas carrying liquid oxygen droplets, and gaseous helium to carry the liquid hydrogen droplets only (see Figure 1). Nevertheless, we shall regard the gas proper- ties such as p g , 7‘. , Cp , and D as constant throughout the flow field, and their values are based upon the mixture of equal numbers of moles of helium and nitrogen. With our final selection for the reactants and background gases, we input the values of their associated properties into the computing algorithms for the three cases described in Chapters 3, 4, and 5, respectively. The numerical results of these cases, illus- trated by diagrams, are placed in the three corresponding sections of Chapter Six. Finally, for comparison, we transform our model into the fuel-spray—only model by employing the same reactants and background gases from Chapter 5 but slightly raising the ambient temperature on the oxidizer side so that the oxidizer is originally gas- eous; consequently, we transform that fuel-spray-only model into the purely gaseous model by slightly raising the ambient temperature on the fuel side so that the fuel is origi- nally gaseous, too. We compare the numerical results obtained in Chapter 5 to those of the two models here in Chapter Seven, which gives us a comprehensive understanding of counterflow spray flame problems. Stagnation plane Diffusion flame front Vaporization front for the oxidizer Vaporization front for fuel Ya: Yr? 01=Yoo X: 0.5 Liquid hydrogen droplets Liquid oxygen droplets and and helium gas mtrogen gas Figure l. A Representation of the Tentative Double-Spray Model CHAPTER ONE MAIN ASSUMPTIONS In this chapter, the six main assumptions made in this thesis are described in detail: 1. Both the fuel and oxidizer droplets are so scarce that the mixture density is equal to the gas density p g , and all gas properties are constituted of those of the back- ground gases only. 2. The density, thermal conductivity, and specific heat of the gas, denoted by p g , A , and CI) respectively, are constant. The diffusion coefficients of gaseous species (say, i), denoted by D,- ,are equal to a constant value D. The gas phase Lewis num- ber is unity. 3. Time rates of vaporized mass per unit volume per second for the fuel and oxidizer droplets, respectively, are as follows: . 1. Cp p F . C p O O,t<0, whereut 5' <> 1_. The axial velocity is proportional in magnitude and opposite in direction to the axial coordinate, which is expressed as: v = - K x . The time rate of fuel mass consumption per unit volume is expressed as: . T WC = B pg Yng Yqu exp(— _?a) . The product is in the gaseous phase only. CHAPTER TWO GENERAL FORMULATION In this chapter, the governing equations are first written, then nondimen- sionalized. These equations, subject to various simplifications and alterations, are solved in Chapters 3, 4, and 5. 2-1. Governing Equations The equations for conservation of energy [(1)], gaseous species [(2), (3), (4)] and liquid species [(5), (6)] are: pvdr pvd2T+vi LFW Low. (1) — _ ‘ _ F ‘— Ov’ 8 dx ‘1 dx2 Cp ° Cp " Cp 2 Pg v dxg = pg D (1ng — wc + va, (2) 2 dYO dYO n . Pg dxg =pgD———dzg ‘VWc+Wov» (3) x de Pg dx dYFl p8 V dx dYOl p3 V dx- 10 szP . pgD dx2 + (1+v)wc, (4) - WFv , (5) — w0v . (6) These equations are subject to the relations: 4 PF! YFI=_3-annF—6-g_’ Po: YO,——§—1cr0n0 Pg, and the boundary conditions: atxz—oo T=TFF, YF8=0, Yog=0, YP=0, YO,=0, PF! YFI=YFF=—T”FF"F—§ 3 pg atx=+oo T=TOO, YFg=0, Yog=0, YP=O. YF,=0, 4 901 Yor=Yoo="3‘7”oo"o p g 1 11 2-2. Nondimensionalization In order to nondimensionalize these equations, we introduce: ~ K 1 ~ “' C T ~ YF x2 ’—x XE—[er( )+l] T_=. p YFE 8, D 2 f af— , Q YFF . g YFF " Yo “ YP ~ YFI “ Yoz Y E = Y a Y E 08 V YFF P (1 +V) YFF Fl FF , V YFF and then Equations (1) to (6) become, respectively: d2'1‘ dX 2 ~ ~ i. — = - D Y Y - —..— dxz ( d} J c F8 08 exp T L ~ Y ~ ~ - - + —F- DF, rF ln[l + —Q——FF— (T—TF1)-u(T—TF1)] Q LF v L ~ Y ~ ~ ~ ~ + J— 0 DOv r0 1n[l 4.3—3 (T—T01)-u(T—Tol)], (7) (D Q Lo dng dx 2 " p “ q T 5% .. ) _ DCYFg Yo, exp — -3 dX dx T .. QY .. .. - - ~ - DP. rFln[1+ LFF (T-TFI) - u(T-sz)], (8) F 12 2.. .. d Yo dX 2 “ p " q T 28 (di ) = Dc ng Yog exp[— 4] dX T 1 ~ Q Y ~ ~ ~ ~ — "‘ Dov rO 1:1[1 + L FF (T-TOI) ° “(T-TOD] . (9) ¢ 0 2.. .. d Y dX 2 ~ ~ T (1sz ( d} ) = — DC Yng Yo: exp[- if], (10) d§Fl (dX) ~ "' Q YFF ~ ~ ~ ~ .. (-x) = - D v n: ln|:l + (T—sz) - u(T—Tp1):] , (ll) dX dx F LF d§01(dX)~ 1 " QYFF ~ ~ ~ ~ .. = — D v r0 ln|:l+——— (T-T01)-u(T-T01)], (12) dX dx (1) 0 Lo subject to the boundary conditions: atX=02 T=TFF, ng=0, Yog=0, YP=O, Ypl=l, Y01=0, atX= l: T=TQQ, ng=0, Yog=0, Yp=0, YF1=O, Y01=-;T, where v YFF YOO 13 = B vq YFILW" c" K ’ DFV 3-D— 12—pg , K rFF PF! D 1 Pg DOV __ 2 -_ 9 roo pg, _1_ .. r - 3 rF=-F— = YF! , rFF I i .. r .. ms 0 =¢ Ym , roo T = Cp TF1 if = CpTO, 1.; = cpTFF FF— Q YFF ~ C T T E p 00 oo QYFF These nondimensional parameters are defined in the Nomenclature. CHAPTER THREE FAST VAPORIZATION AND FAST CHEMISTRY Fast chemistry assumes that the reaction terms in Equations (7) to (10) exists only at an infinitesimally thin flame front, i.e. X = X and that the reaction at the eq’ flame is complete, i.e.: ~< J.” n 0 , over [Xeq, 1] , Yog = 0 , over [0, Xeq] . In other words. the fuel and oxidizer cannot coexist. Fast vaporization assumes that the vaporization terms for the fuel and oxi- dizer in Equations (7) to (12) exist only at two infinitesimally thin vaporization fronts: X = XFv at the fuel vaporization front and X = XOV at the oxidizer vaporization front and that the vaporization at each of the two fronts is complete. Therefore, 14 15 YFF , over [0, XFV] , Y = H 0 , over [XFW 1] , O , over [0, XOV] , Y = 0’ Yoo , over [XOW 1] . The relative positions among these three fronts are: 0.5 . 0 0 < XFV < < X0v < 1 , i.e. va < < xov . Xeq xeq where the brackets denote that X eq may be greater than, equal to, or smaller than 0.5 (i.e., xeq may be greater than, equal to, or smaller than zero). 3-1. Formulation by Coupling By proper combinations of Equations (7) to (10), we create coupling func- tions whose second derivatives with respect to X are zero and which are thus linear func- tions of X over certain applicable domains: Over [XFW l] we have ng + Yp 2 cl X + dl . The enforcement of boundary conditions at X = Xeq and at X = 1 leads to: 16 ~ ~ ~ 1—x YFg + YP = YReq —1-T . (13) eq Over [0, XOV] we have Yog + Yp = c2 X + (12 . The enforcement of boundary conditions at X20 and at X = Xeq leads to: X . (14) Xeq Over [va’ XOV] we combine Equations (13) and (14), and obtain: xm—x YFg—Yog = YP’eq Xeq(1_xeq) . (15) At the vaporization fronts X = XFV and X = X0V , respectively, Equation (15) becomes: §Fg, Fv = §P, eq Xeq — XFV , (16) ch (l - Xeq) . ~ X — X Yog,0v = YP,eq X 0" eq (17) eq (1_Xeq) . l7 Throughout the entire flow field [0, 1], without any fast-chemistry or fast- vaporization assumptions, we always have: .. L ~ vL ~ L vL ~ T+—Q—F-ng+ Q0 Yog-(l- F— O)Yp=c3X+d3, which, with boundary conditions at X =0 and at X = 1, is rewritten as: % % LP? ““0? (1 1 )x (1 LF ”0)? - FF '1' — F + 0 = 00- FF + -—- P- Q g Q g Q Q (18) By applying the fast-chemistry assumption and setting X = Xeq , Equation (18) becomes: ~ ~ ~ ~ LF v LO ~ ch = TFF + (TOO-’TFF) Xeq + (I — Q — Q )Ypyeq. (19) By applying both fast-chemistry and fast-vaporization assumptions and set- ting X : XFv and X = XOV, respectively, Equation(18) becomes: LF Q TF1 - TFF + ~ ~ ~ LF v L0 ~ YFg,Fv = (Too-TFF) X12. + (1- - j Y v, 20 Q Q P,F () 18 vLO Q T0! - TFF + YOg,Ov = (Too-TFF) XOV + (1- - )chv. 3.2 Formulation by Other Manipulations According to the fast-chemistry and fast-vaporization assumptions, the chemical reaction term and vaporization terms of Equations (7) to (12) are zero over the entire field excluding the flame front and two vaporization fronts which are infinitesimally thin; in other words, the second derivatives of the mass fractions of fuel, the oxidizer, and the product with respect to X are zero over the segmental domains divided by these three fronts. Therefore, without any coupling and simply by applying conditions at X = 0, X = XFV , X = X cq, X = XOv , and X = 1, the mass fractions may be determined as: vpgfl 35— , over [0, XFV] , XFv §Fg - " xeq _ X (22) YF,Fv———'—, over[X ,X ], i 8 Xeq __ XFv Fv eq 1 0 , over [Xeq, l] , r O , over [0, ch] , ~ X -— X ~ Yo, 0, ———°—“— , over [Xe ,XOV] , Yog = 1 ’ X0v — Xeq q _ (23) " - X [ YOg,Ov 1_ X0 , over 1X0. 1] , V 19 r " X Yp,eq T , over [0, Xeq] , Y1» = 1 __ 1°“ X (24) Yp,eq .1-_X_ , OVCI‘ [Xeq, 1] . 6C1 The discontinuities of the mass fraction gradients at the fronts X = Xeq, X = XFV , and X = XOV may cause unequal diffusion rates for each species across these fronts. This inspires us to investigate how the mass fluxes are balanced. Here we use XF; and X0: to denote the upstream sides (which contain dr0plets) of the vaporization fronts for the fuel and oxidizer, respectively, XF: and XO-v the downstream sides of the vapor- ization fronts, and Xe; and Xe; the two sides of the flame front, facing the fuel and oxi- . . .. + dIZer mass fluxes, respectively. Also note that Xeq = Xeq = Xeq, x‘-x*—x dx‘—x*—x ~ ‘— *- Fv — Fv - Fwa" Ov — Ov - Ov’l'e"xeq “xeq _xcq’ — _ + _ ' d — _ + _ va - va - wa‘m xOv - XOV ‘ XOV' Under the fast chemistry and fast vaporization assumptions, the mass con- servation law describing mass across the two vaporization fronts, X = XFv and X = Xov , respectively, is stated as: 20 Y K ‘ Y K ‘ D dYFg Pg 1:1:(- va) + Pg FFFV (" va) ‘ pg 1 dx 1;. = x15. ~ ~ dY _ _ + _ Pg _ pg YFg,Fv( Kva) pgD( dx )x : XFT, , (25) Y K ‘ D dYOg pg OngV (_ xOV) — pg ( dx )x = x0_v ~ dY x = xOV . (26) Under the fast chemistry assumption, the mass conservation law describing mass conservation across the flame front is stated as: D dYFg Y K ‘ D W" 1—1... (W) dYOg - + dYP = —pgD(—d—;_)x—x+ + ngP,eq(_Kxcq ) _ pgD(—ZI-x_)x:xc+ cc1 q ' (27) S. -— _ + _ — _ + _ d - _ + _ . h mce xeq — xeq —- xeq, va — va — va,an x0v — x0v — .chv (or sincet ere are no velocity jumps), all of the gaseous convection terms above are cancelled out, and Equations (25) to (27) henceforth become: 21 p Y (-Kx) p D(dYF8) = —p D(dYF8) (28) g FF H 8 dx x = x13; 8 dx x — x1312, dYOg dYOg —pg (dx )xsz—v ”ngoo(-KxOv) PgD( dx jx=x0v, (29) —p (dYFg) -p D(dYP) 8 dx x-xe; g dx x=xe; dYOg dYP _ —pgD( dx )X=XC; —pgD( dx )x=xe;. (30) Then, in terms of absolute values, equations (28) to (30) may be rewritten as: dYF dY . : ' —£‘ -——F—g— ngFFK levI pgD H dx xzxp; + dx x=xF:], (31) dYO dY V V 22 dY dY Pg D . I F8 _ + __08_ + dx x = xeq dx x = xeq D l de de (33) = . + . Pg dx x = x6; dx x = x6; Equations (31) and (32) show that, for either the fuel or oxidizer, the total amount of mass diffusion at the vaporization front is equal to the mass convection of the liquid phase from upstream; moreover, Equation (23) shows that the total amount of mass diffusion of the product at the flame front is equal to the total amount of mass diffusion of the gaseous fuel from the fuel side plus the gaseous oxidizer from the oxidizer side. In addition, Equations (31) and (32) may be further transformed through nondimensionalization and then be substituted into by the derivative forms of Equations (22) and (23), respectively; accordingly, relations concerning the locations of the vapor- ization fronts, EFV and firm, respectively, are obtained as follows: 1 ~ X i 2 —iFv = YFg,pv 6" ) exp[— F” J (34) ~ 2 ‘r' _—_ 4M“) 0 l—Xeq exp—£2:— ' 0v «I 2 113 g, v (XOv—Xeq) (1_X0v) 2 . 23 3-3. Locations of the Vaporization Fronts Rearrangement of Equations (34) and (35), respectively, gives the follow- ing relations: 3c 2 - x (1—x ) J21: (—iFv)va(1—XF,) exp 2" = ng,p,, 3): XF” , (36) eq_ Fv 3e 2 .. (1—x )x J27: 3E0, XOVU—XOV) exp ‘2’” = ¢Yog,0v x “'x 0" . (37) Ov— eq ~2 Then, we define F(X) _=. A/ 2 1t - I St I . X - (1 - X) - exp(xT) , where 3c is related to X 3c 2 )+ I] . Hence Equations (36) and (37), respectively, by the definition X a g. [erf( become: - x —XF Ymv = th ,)- °“ V , (38) g F ch(l—XFV) - 1 X —X Yog,ov = —-F(x0,>- 0” e“ (39) (I) (1_xeq) xOv - 23 3-3. Locations of the Vaporization Fronts Rearrangement of Equations (34) and (35), respectively, gives the follow- ing relations: 32 2 ~ x (1 —x ) V 2 TC (_ iFv) XFv (l —XFV) exp[ 2" J = YF8.FV 2;: -X Fv , (36) eq Fv 32 2 ~ (1—x )x 0v“ eq ~2 x/27t ~II{‘I~X-(1—X)-exp(i—), where} isrelatedtoX Then, we define F(X) 2 52 f5: by the definition X a —5— [erf( )+ I] . Hence Equations (36) and (37), respectively, become: - x — xF YF ,F = W ). 2" V , (38) 8 V Fv ch ( l _ XFV) " _ l XOv — Xeq 24 Next, by combining Equations (16) with (38), (17) with (39), to eliminate ing, FV, Yog, 0v , respectively, we have: it F(X > P, eq ___ Fv , (40) 1—x,q l—XFV v F(X P, eq = _1_ CV) . (41) xeq ‘12 XOv Also, by setting X = XFV and X = X0v , respectively, Equation (24) becomes: 2' 2' XFv YP, Fv = YP, eq X , (42) eq 2’ 2' 1 _ XOv YP 0v — YP eq 1 _ X (43) Now, to determine the locations of the vaporization fronts, i.e., iFv and 320v, we substitute Equations (16) and (42) into (20) to replace ng, Fv and Yp, pv , and we substitute Equations (17) and (43) into (21) to replace X081” and Xp, 0v. Then, ~ ~ YP, eq and YP, eq i—xcq xeq , the former of within either Equation (20) or (21), there coexist 25 F X which can be replaced by —1_(_):—V)_ from Equation (40), and the latter of which by _ Fv F fl from Equation (41). As a result, Equations (20) and (21) become, respec- 1 4’ xOv " - - - LF v = 1 v Lo F(Xov) {TFI — TFF - (TOO-TFF) XF, + 7)— F(XFV) }, (44) .. ~ ~ ~ 1 V L0 = 1 LP F(XFV) {T01 - TFF - (Too-TFF) X0V + 71; Q I2(Xov) } (45) Figure 2 shows five functions with respect to X: F(X) over [0, l], 22%)- and l-X over [0, 0.5], and f—(XEZ- and X over [0.5, 1]. Since, as mentioned at the begin- ning of this chapter, 0 < XFv < 0.5 < X0v < 1 , the diagram of Figure 2 informs us that . Ov . . Fv . . . . neither the term in E uation 44 nor —— inE nation 45 is like] to be 26 LF Q zero; therefore, XFV —> 0 only when Tp1—> TF1: and —> O, and X0V -—> 1 only when ~ ~ L To) —-> T00 and —9- —9 0 ; that is to say, XFV = 0 only when the fuel is originally gas- Q eous, and X0" = 1 only when the oxidizer is originally gaseous, both of which are con- sistent with physical reality. 3-4. Location of the Flame Front Equations (40) and (41) give: x + 1.2532. . 9,6 = 1. sq F(XFV) ' q i ‘12 X0v " L ch + F(XOV) ' YP’ eq — O . Then, by solving the simultaneous equations above, we are able to determine the location of the flame front, Xeq , and the rescaled mass fraction of the product at the flame front, Xp, eq , as follows: q) XOv (1) F(XOv) xeq = = , (46) l " XFv q) x0v 1 - XFV F(XOV) F(XFV) F(XOV) F(XFV) XOv 27 Yp,eq = 1 _ XFV ¢ XOV . (47) F(XFV) + F(XOV) From Equation (46) we observe that the variation in either XFV or X0v may influence the value of Xeq , and hence may shift the flame front either toward the fuel side or toward the oxidizer side. For convenience of discussion, here we introduce some new indices which are used in this section only: 0g: refers to the fuel-spray-only model, in which the oxidizer is originally gaseous, i.e. V F g: refers to the model that the fuel is originally gaseous, i.e. XFv = O; OFg: refers to the purely-gaseous model, where both X0v = 1 and XFv = 0. Note that OFg is a special situation of either 0g or Fg , and all these three models are special situations of our generalized “double-spray” model. Then we may express the location of the flame front in each of the four models. First, we recall Equation (46): 28 ¢ X.., = I-Xev F(xoa ’ F(XFV) XOv + (22 which generally refers to the double-spray model and, when applied to models 0g , F g , and OFg, respectively, reduces to: <1) xeq, 08 = 1 _ XFV ¢ 3 —— + F(va) ¢ qu, Fg = F(XOV) v —_ + xOv <1) Xeq, OFg 1 + (1) For comparison, let the value in the global equivalence ratio (1) be fixed throughout these four models. Then, look at XFV and XOV. Although Equations (44) and (45) indicate that the variation in the value of XOv may influence that of XFV , and vice- versa, the influences may be negligible since, numerically speaking, X0" is usually close 29 (but not equal) to 1, and XFv usually close (but not equal) to zero, so that the values in XFv and in XOV throughout the four expressions above may be considered as fixed, which enables us to make the following comparison: xeq O < Xm,Og < X < XW:g < 1 , (the fuel side) “1' OFS (the oxidizer side) where Xeq may be greater than, equal to, or smaller than ch, 01:8. The inequality relation above may be interpreted in a few ways, and here we take what we are most interested in: Xeq’ 0g < Xeq : means that the existence of the oxidizer spray (Fuel spray only) (Double spray) shifts the flame toward the oxidizer side; Xeq, 08 < Xeq, Opg : means that the existence of the fuel spray shifts (Fuel spray only) (Purely gaseous) the flame toward the fuel side. To sum up: The flame moves toward the side where droplets originate. 30 3.5 Procedure to Determine the Profile The entire fast-vaporization-fast-chemistry profile is linear with the physi- cal coordinate X and can thus be explicitly determined by determining the locations of the vaporization and flame fronts (XFV , X0V , and ch ), the former two of which may be found through mutual iterations into Equations (44) and (45), and the latter by substituting the final values of the former two into Equation (46). The procedure is sketched below: 1. Setting up: Starting values: XFV’ i, XFV’ j; XOV’ i , XOv’j. . (old) ld Iteration parameters: 0’1 , 0'1 , ofinew) (for XFV); 0'30 ), 0‘2, oénew) (for XOv)° . . (old) 1 Evaluation variables: Tl] , m ; 112021) , n2. Equat1on (45) => -—————- = f 1(X0v) . (f 1 : function) 1 — XFv . F(XOv) . Equation (44) 2 7— = f 2(va) . ( f 2: function) Ov Preliminary inputs: Yoo , YFF. 2. Algorithm for finding XFV and XOV: I. Set: 62 = XOV,1 . II. i. Set: 0’1 = XFV,i . ii. iii. iv. V. 111. Set: IV. Do: ii. 31 Set° - F(o‘) f (o) - 111 1_O.1 1 2 - ld .. Set: C(IOId) = O] , 11(10 ) = “I, 01 = XFV,j . Repeat (11). Do: ‘ (new) (old) ‘11 , a. Set: (51 = 61—(61-0'1 )--—. (Newton 3) (old) Tl] -m ld .. . -6 Until m < 10 . F(Oz) Set: n2= —f2(ol). 0'2 Id C(22)”) = oz, n30 2 = 112,02 = X0“. .Repeat(II). S , (neW) _ (old) T12 , 61. 0'2 — 0'2 - (0'2— 0'2 )' W . (NCWtOI’l S) ‘12—‘12 Set: 0:0”) = 0'2, ngfld) = 112, 0'2 = 65mm. Repeat (11). Until 112 < 10”. V. Final values: XFv 2 0'1; XOV = 0'2 . The profile: Once the final values of XFV and X0" are obtained, the profiles of all the 32 key variables can be subsequently determined by recalling equations from the previous sections: <1) X = ‘22 1"va F(XOV) F(XFV) XOv ~ 1 YP,eq = 1 —XFV (I) XOv F(va) F(XOV) ~ ~ X _ XF YFg, Fv = YP, eq eq v ’ xm(1—xm) - .. x0 -x Yo ,o = YP, v eq 8 V eq xeq(1—xeq) fi‘eq : r ~ x YF ,F . g V XFv YFg—49F F xeq_x , V a g xeq - XFv 0 , l 0 , ~ X — X .. Y eq , YOg = 08’ CV xOv _ Xeq ‘ 1 — X Yo ,o k g V 1 _ XOv 9 ~ ~ ~ LF v LO ~ TFF + (Too-TFF) Xeq + (1- - )YP,eq , Q Q over [0, XFV] , over [XFW ch] , over [ch, l] , over [0, Xeq] , over [Xeq, XOV] , over [XOW l] , (46) (47) (16) (17) (19) (22) (23) 33 ~ X YP,eq X 9 {r 3‘2 2 ‘~ 1-x Y“, ’17)?" en over [0, Xeq] over [Xeq, l] k r X XF ’ X - XFv Xeq _ XFv 2 x0v — X T01+ (Teq — TODX—X— , 0v _ eq ~ ~ ~ 1 — x Too + (T01 - Too)—1 X , k _ 0V TF1: + (TF1 - TFF) - V TF1 + (Teq "' TF1) r—Ji ll in = 11(va — X), over [0, 1] , v0, = u(X—XOV) , over [0, l] , 9|»— O,t<0, where u(t) E 1, t2 0 . 7 9 over [0, XFV] , over [Xth Xeq] , over [Xeq, XOV] , over [XOw l] , (24) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 34 X over [0.5, l], and F(X) over [0, l]. \ l I l I I I I I I / .\O /' \. /' \ /. .\ /, /. \ \. /' '- \ \. /' " \ \_ / / \ \ 1-X X ./ / - \ \. .’ / a \ \. _/ / \ / \ . ~ / \ / - \ '\ /' / .. \ /' l \ /' / \ /' / _ \ .. ./ _‘ / \ F(X) / (1 —X) F(X) / X / _ \ / _ \ / \ / \ / " \ / — F(X) \ / F(X) \ / " \ / ‘ \ / \ / .. \ / _ \ / \ / \ / l L 1 1 L l l l ‘ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X . . . F X F X Figure 2. Five functions versus X: l ( )1 and l-X over [0, 0.5], (X) and CHAPTER FOUR FINITE-RATE VAPORIZATION AND FAST CHEMISTRY An obvious distinction between fast vaporization and finite-rate one is that the former features XF—v = XF: = XFv and X0} = X02; = XOV, and the latter XF v 7: XF: and X0: at XO-V . For finite-rate vaporization, intervals [XFZ , XF: ] and [XO_v , X022, ] are the so-called vaporization zones, and we have: r l , over [0, X12; ] , in, = l l +j x: (2%?)(11, forXe [xF'v , xF’; ], 1 O , over [XF: , l] , O , over [0, XO_v] , 901: T71,714X2;(d-—2;£’)dt, forXe [xo'v, x02], . 7:)— , over [x0*,, 1] , 35 36 XF; : beginning of vaporization, X0: : beginning of vaporization. where and XF: : end of vaporization, XO-v : end of vaporization. For expressional convenience in the following sections, we define: TFH E212(X = X12: ), Xian». E YP(X = XF: ), YFg, Fv+ E YFg(X = XF: ), "I‘m- a fix = xo‘.), inov- -=- Ynx = X01). Yogov- _=. Yog(x = X01). 4.1 Preliminary Relations By eliminating the chemical reaction term from the Governing Equations (7) to (10), we are able to derive the following coupling functions: T + F128 = CF X + dF , over [XF: , X01] . Applying the boundary condi- tions, we have: + + 7 " at X : XFV , CF XFV + dF : TFv+ + YFg’ Fv+ , 31X = XOV , CF XOV + dF : TOV— , _ TOv- ‘ TFv+ - YFg, Fv+ 2 CF - _ + . (48) XOv _ XFv 37 Similarly, T + {fog = 00 X + do , over [Xpt , XO-v] . With the boundary conditions we have: + + " at X = XFV , CO XFV + do = TFv+ , at X = XO-V , CO XO_V '1' do = TOV— + YOg’OV_ , :> CO = TOV- + YOg,Ov- “ TFv+ _ + (49) XOV _ XFV 2 2 These two coupling functions, T + ng and T + Yog , can be reduced respectively by applying the following relations: {(Fg = 0 , over [xeq, 1] , ‘r = cF x + tip, over [xeq, xo‘v] , (50) X0820, over[0,X T=cOX+dO, over[XF:,X eql , => ml, (51) of which the derivative forms lead to the following relations. respectively: _di _ = fl. _ (52) dx Xe [xcq,x0,] dX x xO . _ di‘ ll X '11 38 Finally, we apply the boundary conditions of equation (50) and of equation (51), respectively, and obtain: atx=xeq, i‘Cq=CFXeq+dF’ ~ .. _ _ ~ _ => Teq = TOv- — CF (XOV — Xeq) , (54) at X = XOV , TOV- = CF XOV + (1!: , at X = XF+ , fFv+ = C0 XF+ + do , ~ ~ .. V .. V = Teq = TFv+ + Co (Xeq _ X12: ) a (55) atx=Teq, Teq=COXeq+do, the two of which imply: X - i‘Ov_ + CO XF: “’f‘i‘Fv... _CF XO—V 56 eq _ C0—CF v ( ) - + .. _ TOv- + va ' " TFv+ _ XOV ' a" 2i dxx=xF‘; arxx=x0‘v , .di _ é dxx=xpj crxx=x0‘v 4.2 Procedure to Determine the Profile 1. Setting up: . — — + + Starting values. XFV’i , XFV'J- , XOv,i , XOV’J- . 39 . (old) _ 1 Iteration parameters: 0' 1 , 0'1, ofinew) (for XFV ); 0'? d), 0'2, can”) (for X0+v ). (old) . . (old) Evaluation variables: Th , n1 ; n2 , n2. Dependent variables of 61 only: XF: , c0, TFH. Dependent variables of 0'2 only: XO-v , CF, Toy- . Dependent variables 0f(61 , 62 ): Xeq , feq , QP, eq ; QP’ Fv+ , §P, 0v- ; YFg, Fv+ , YOg, Ov— - Governing differential equations: on the fuel side: 2" 2 L ~ Y " " " ' H (6“): .11)“ ern[1+ Q FF (T—Tm>-u(T-Tr=z>] d§Fl dX ~ _ ~ QYFF " ~ " " dx ((132)-20- — DFVrFIn[1+—EF——(T-Tm-ud—Tml on the oxidizer side: dzi (2131 dx2 di ‘ 1 VLO ~ QYFF ‘ ' ” " E—Q— DOv r0 ln[l + T (T-Toz) “(T-T00] , d90,dx-_1 ~ QYFF“" ”“ (IX ((12)): .. ¢ DOvroln[l+ L0 (T—TOI) “(T-TOD] Preliminary inputs: YOO , YFF , DOV , DFv . 40 Algorithm for finding XF; and X0: : II. III. - x ‘ an? i. Find Xov , so that: —1- +J 0" (-—OI)dX = O . (Kutta’s) ¢ (,2 dx ii. Applying equation (52) and the definition for Toy- , set: ~ ~ _ i. X " 2" cF = fl _ = M +J 0" d—T‘; dX , (Kutta’s) (IX X = XOV l _ 62 02 (1X .. - _ - x ‘ " Toy- E T(X = XOv) = T01 +1 0v (fl )dx . (Kutta’s) 0'2 dX i. Set: 0'1 = XFV:i .. . + XF: d?!” 11. a. Find XFv , so that: 1 +1 (_)dX = O . (Kutta’s) 0'1 dX b. Applying equation (53) and the definition for TFH , set: .. ~ _ ~ X + 2" .0 = a -121... j Fv [cl—3].“, (Km...) (IX X = XFV Cl 6] dX ~ ~ ~ X + d"? Tpv+ ET(X = XF: ) = TF1 +J. Fv (d—i )dX . (Kutta’s) o c. Applying equations (56), (55), (I9), (24), and (18), set: ~ + ~ _ X TOV- + CO XFV - TFv+ "’ CF XOV eq Co— CF , 41 Teq = TFv+ + CO (ch — XF: ) , ~ Teq - TFF - (Too - TFF)Xeq Y = , Re“ LF v LO Q Q ' " " XF: " " l _ XOhv YP,F + = YP,eq YP,O - = YP, _— , v xeq , v eq l-Xeq YFg, Fv+ = .. .. ~ - L v L ~ —LQ— {TFF—TFV+ + (T00 —TFF)XF: +(1 — F - O )YP,FV+} , F YOg, Ov— 7- Q VLO - ~ - ~ _ LF vLO ~ TFF-TOv—+(TOO—TFF)XQV+(1— Q — Q )YP,ov_ . d. Applying (49), SCI: T1] = (30— TOv- + YOg,Ov—- _TFV+ . — + XOV _va ... (old) (old) .. 111. Set: 0'l = 0", 1]] = 1]], 0'1 = XFM- . Repeat (11). iv. Do: (new) (Old) 111 , a. Set: 61 = 01"(01‘ 0'1 )°——_(6F)' (Newton 5) 111 ‘ni ld ld .. b. Set: 0(10 ) = 01,1120 ) = 111,01 = cam”). Repeat (11). Until 111 < 10—6. 42 TOv- _ TFV+ " YFg, Fv+ .. ... ' X0v _ XFV IV. Applying equation (48), set: 112 = cF — v. Set: 55°”) = oz, 11‘2“” = 112,02 = x0“. Repeat (11) to (IV). VI. Do: i Set: om”) - o — (o o(°ld)) - —n2—— N t ’ . . 2 — 2 2— 2 (old). (ewons) I12-le ii. Set: oéom) = 02,119“) = 112,02 = oémw). Repeat (II) to (IV). . —6 Until 112 < 10 . VII. Final values: XF; = 0'1; X0: = 0'2 . 3. The profile: ~ ~ .... x - TFF + (TF1 _ TFF) _ a over [0,XFV ] 9 XFV X .. " dT — + TF1+JX_(E)(II, ‘fOl‘XE[XFv,XFv], Fv " " ” X — XF: + TFW + (ch — TFV+)———+— , over [XFv ,ch] , ~ Xeq — XFV T = < .. .. ~ X0; - X _ TOv— + (Teq —TOV- )——:—_ 9 over [xeq’XOv] , x0v _ Xeq X .. " dT _ + 0v ‘ "' " l — X + Too + (Tor - Too)-—-—+- . over [X0v ,1] , 1 — X0v 43 ' ~ x Yp,eq X , over [0,Xeq] , .. eq Y? = i (24) ~ 1 — X Yp'eq-l—T , OVCI’ [Xeq’l] , L Cq then, derived from equation (18), there are: r Q " ~ “' " LF V LO "’ — TFF— T + (Too — TFF)X +(1 - — )Yp , over [0,X ] , .. L Q Q eq F YFg : < L 0 , over [Xeq,l] , ' O , over [0,ch] , 90, = t QiiiiX1LFVL0? X1 iVLo FF- +( 00- FF) +( -Q- Q j? , 0V6r[ eq.1. Finally, recalling from the beginning of this chapter, we have the following: 1 , over [0, XF; ] , ~ X d? Fl _ + YF1= j1+JXF_(-—d;—)dt, for XE [XFV , XFV ] , v l O , over [XF:, l] , Y01=< + X( 0, ~ (”01 dt (1) , over [0, X01] , )dz, for x e [X01103] , over [XOJ'W 1] . CHAPTER FIVE FINITE-RATE VAPORIZATION AND FINITE-RAT E CHENIISTRY In this chapter the reaction rate is no longer infinitely large and the reaction can not be completed merely at an infinitesimally thin interface. Therefore, integration over the broad flame zone as well as the overall domain [0,1] is required, and all the gov- erning Equations (7) to (12) must be taken into account. Then, the whole computation pro- cess is sketched as follows: 5.1 Procedure to Determine the Profile 1. Setting up: Starting values: t£~ HE" HQ" HE" ): dxx=01’dxx=0j’dxx=iidxx=ij 45 46 Iteration parameters: ld) ~ ~ 6(10 , ol , can”) (for Q ); 6?“), 02, 6‘2"”) (for g ); dX X=O dX X=l (old) _ (new) 51?]: (old) (new) d? o ,o,o (for—8 );o ,o,()' (for—DE ; 3 3 3 dX X=O 4 4 4 dX le) (old) (new) (1&0 . (old) (new) d§0 o ,o ,o (for 8 ), o ,o ,o (for 8 ); 5 5 5 (1X X=O 6 6 6 le 69’“), 07, 6‘7“”) (for {(1:8(X = Xeq) ); 6?“), 68, 63‘”) (for K). Evaluation variables: ( ld) ( ld) ( ld) ( Id) Tho ’Tll;n20 »Tl2;n3o ’Tl3in4o ”l4; ( ld) ( ld) ( ld) ( ld) 115° ,ns: 116° .716; 117° .117; 118° ,ng. Dependent variables of 08 only: DC, DOV, DFV, Xeq. Dependent variables of (0'I , 0'3 , 0'5, 0'8 ): El _ dfig _ dYOs _ Yog(x = x6; ). dXX=ch,dXX=ch,dX X: eq, Dependent variables of (0’2, 04, 0'6, 0'8 ): ail , £515 + 4h + dXX=Xeq.dXX=ch,dX X=Xeq Preliminary inputs: YOO, YFF, r00, rFF, Tr. 47 2. Algorithm for determining the iteration parameters: 1. Set: 0'8 = Ki . ' B vq YFgfl" c _ 08 D l P 11. i. Set: (Dov = 3 2 g . ‘ 68 r00 901 D l P DFV = G 2 g . ( 8 rFF pFl ii. Applying the algorithm in Section 4.2, find Xeq. 111. i. Set: 07 = vegx = Xeqh . ii. Set: 0'5 = (dfl: ) . dX X=O 1 iii. a. Set: 61 = (2T. ). dX X=O 1 b. Set: 0'3 = (d_Y_Fg ) . dX X=0 1 X :1)? . St: = cq( F‘ijth— . Ktt’ c e 113 I 0 (IX 07 ( u as) . (old) (old) din: d. Set. 0 =6 ,1] =1] ,o=(_& ). 3 3 3 3 3 dx :0 J Repeat (c). 6. Do: Tl Set: oénew) = 0'3 — (0'3 — 6501c”) - —-—3m . (Newton’s) ‘13-‘13 48 . (old) (old) (new) Repeat (c). Until 113 < 10*. .. X ~ .. Set: 111 = TF1: +J Oeq (:7: )dX — Tf . (Kutta’s) . (old) (old) (1} Set. 0 =6, = ,o=(_ ). ‘ 1 m m ‘ dxl X=0 j Repeat from (b) to (f). Do: . (new) (old) ‘11 , Set. 0'l = o] — (0'l — 0'1 )~ (old). (Newton S) '11 “Ill Set: 0'30”) = 61,1](101d) = m, 61 = Gin“). Repeat from (b) to (f). . —5 Until m < 10 . f ~ X 2~ id: _ =ol+J 6" {—1—1; dX dX X = Xeq 0 ~ X 2" dz)? x x‘ =G3+J dqld Yggjdx Set: < _ 6“ dX ~ X 2" dYOg -___0-5+J eq dYSde (IX X - Xeq 0 dX ~ d? _ eq 03 . Yoga—X”) [0 (dX )dX (Kutta’s) iv. 49 dX X=1 1 Set: o4=(d_Yi9 ). dX X=1 1 Set: o6=(d_Y_08 ). dX X=1 1 X ~ _ Set: T16 = J leq (digng- Yog(X = Xeq ) . (Kutta’s) (old) (old) _ d‘h) ) S t: 0' = o , = , 0' — (_8 . . e 6 6 n6 n6 6 (IX leJ Repeat ((1). Do: (new) (old) “6 , 1l6‘Tl6 Set: 020w) = 06,1120”) = 116,06 = 02mm. Repeat (d). . -5 Until n6< 10 . Xe flF ' = q g dX-O' . K tta’ SCl.n4J1(dX) 7 (U 5) (old) (old) din: ) Stzo =0, = ,o=(___8 .- e 4 4 n4 n4 4 dX X=lJ Repeat from (c) to (g). Do: 50 1] Set: can”): 0'4—(0'4-0'201d))- 4( . (Newton’s) . old) 114'114 Set: 0(told) : 0'4, T12pm) = 114’ 0'4 = 0'2“”) Repeat from (c) to (g). . —5 Untll n4< lO . .. X d'iv . Set: =T + eq(—) — . ’ 712 00 I 1 dX dX Tf (Kutta s) , (old) (old) di‘ Set.o =o,n =n,o=(_ ). Repeat from (b) to (j). Do: . (new) (old) "2 Set. 62 = 02 "' (62— 02 ) ‘ ——(_ . (NCWIOIYS) old) Tlz‘l'lz Set: O_(zold) : 0'2, T1(zold) = 112’ 02 = 6(2new). Repeat from (b) to (i). . —5 Until 112 < 10 . ' di: Xeq dzT dX x = x,q 1 dX ... X 2" Set: H20, where the stoichiometric ratio is: v = 8, and, according to Strahle [7], the indices p and q in the ~ ~ i“ . . chemical reaction term, DC ngp Y0;I exp[-— 4], 1n Equations (7) to (10), are deter- T mined as: p=1.5, and q=O_.5. 55 56 Then, our computations ensue, and their subsequent results are plotted onto diagrams. Now we are going to present and explain these diagrams which consist of the three cases corresponding to our theoretical analysis in Chapter 3, 4, and 5, respectively: 6.1 Fast Vaporization and Fast Chemistry Figure 3 features the entire profile, which is linear with X. The dotted lines indicate the locations of the vaporization and flame fronts, where the inflection points exist. Figure 4 is an image of Figure 3 by transforming the horizontal axis from coordinate X into it , a nondimensionalized but much realistic coordinate directly proportional to the exact spacial coordinate x. From Figures 5 and 6 we observe that, the higher YFF (i.e. the higher glo- bal equivalence ratio (1) , since YOO is fixed), then the farther the flame front is pushed away from the fuel side, as well as the higher the flame temperature is raised (due to the fuel-richer chemical reaction), both of which are consistent with physical reality. Figures 7 and 8 feature six combinations of (Yoo’ YFF) which construct two different values for the global equivalence ratios (1) (0.8 and 3.2) versus three different values of Yoo (0.15, 0.1, and 0.05). Amazingly, the location of the flame front, indicated by dotted lines, remains almost the same with the same value of the global equivalent ratio, despite the variety of Yoo; in other words, the global equivalent ratio is the key 57 parameter controlling the location of the flame front. In addition, we notice that the flame temperature shifts tremendously with the variety of Y00 rather than with that of the glo- bal equivalent ratio. Therefore, in order to create a flame temperature high enough to keep the product of the reaction, i.e. water, from condensing, we decide to pick up a sufficiently high value, 0.15, for Yoo throughout the rest of our report. 6.2 Finite-Rate Vaporization and Fast Chemistry In Chapter 3 and in Section 6.1, fast vaporization assumes that the vapor- ization Damkohler numbers of fuel and the oxidizer, denoted DFV and D0" respectively, are infinitely large, and the entire profile may be addressed without involving any specific value for Damkohler numbers. Finite-rate vaporization in this section as well as in Chap- ter 4, in contrast, requires inputs of specific values for both DFV and DOV. Instead of arbi- trarily generating a variety of values for DFV versus a variety of values for DOV, we try to make our investigations simple and efficient by seeking the correlation, if any, between these two numbers, prior to our numerical work. Quoting Equation (2.9) from “Atomization and Sprays,” by Lefebvre [8], we have: 86 CD PA URZ, maximum stable drOpIet size: Dmax = 58 where 0' is the surface tension of the droplet, CD the drag coefficient, p A and UR the density and relative velocity of the surrounding air, respectively. With the relation above, we may estimate the ratio of the original droplet size on the oxidizer side to that on the fuel side, by assuming the drag coefficient and flow velocity to be identical on both sides, and by quoting data for surface tensions of liquid oxygen and liquid hydrogen [5] and data for densities of gaseous nitrogen and helium [4]: 0' r (—) 'd‘ 'd 00 z PA oxr izer 51 e 24.83. rFF (3) PA fuel side Then, by applying the mathematical definitions of these two Damkohler numbers, as expressed on Page 12, and by quoting data for densities of liquid oxygen and liquid hydro- gen [4], we obtain a rough yet usable relation between the vaporization Damkohler num- bers: which, in our belief, is realistic. Therefore, in this section and next one, when we pick up a value for DOV, we automatically set the value of DFV to be 380.4 times Dov- 59 Figures 9 and 10 feature a typical finite-rate-vaporization-fast—chemistry profile, with dotted lines showing the ranges of two vaporization zones and the location of the fast-chemistry flame front. We shall keep in mind that the input value of the vaporiza- tion Damkohler number for fuel is 380 times that for the oxidizer. As the physical defini- tion shows, a larger vaporization Damkohler number means a larger mechanical-time-to- vaporization-time ratio, i.e. faster vaporization with respect to a time frame attached to a specific group of moving droplets, i.e. the droplets vanish completely while travelling for a shorter distance; that is to say, a larger vaporization Damkohler number should physically lead to a thinner vaporization zone, and vice-versa. Our numerical result, as plotted on Figures 9 and 10, shows the fuel vaporization zone obviously thinner than the oxidizer’s, which implies that our calculations are trustable. Figures 11 and 12 present different profiles with the inverse of D0‘, rang- ing from 0.0001 to 1.0, i.e. DOv ranging from 10000 to l and DFV decreasing accord- ingly. The results show that, as the two vaporization Damkohler numbers decrease, both the oxidizer and fuel vaporization zones stretch toward the flame front, which is consistent to our physical predictions. The oxidizer vaporization zone may ultimately coincide with the flame front, which is demonstrated later in Figure 15. It is worth noticing that, while both vaporization zones stretch toward the flame front, the flame front itself shifts toward the oxidizer side instead of toward the fuel side, implying that the oxidizer in our model, whose droplet size is estimated at 4.83 times the fuel’s, has a dominant influence rather than fuel does. 60 Figure 13 is corresponding to figure 11. The end of oxidizer vaporization zone and the flame front meet at 1/D0v = 1.06 (i.e. D0" = 0.943), beyond which are indi- cated by the dashed lines to represent the unrealistic, merely numerical result. The unreal- ity can be demonstrated by following exactly the same derivation procedure for Equation (15) except over [XFT/ , X01] , which implies: where the value of Yog, 0v- turns negative whenever XO—v < Xeq. If we replace the input value, 0.03, of YFF in Figure 13 with a variety of values, the X-versus- l/ Dov and T-versus-l/D0V curve patterns will remain similar, but the l/DOv value at which X0; = ch may differ, as shown in Figure 14. This figure fea- tures YFF = 0.015, 0.03, 0.045, and 0.06 (i.e., the global equivalence ratio = 0.8, 1.6, 2.4, and 3.2, respectively), whose corresponding maximum values of l/DOv that can exist in reality are: 1.37, 1.06, 0.91, 0.83, respectively, at which X02, = Xcq = 0.577, 0.682, 0.746, 0.787, respectively. 61 6.3 Finite-Rate Vaporization and Finite-Rate Chemistry 1. Tf versus K The computation procedure in Section 5.1 requires the input of Tf, which is obtained through the following definition: .553. QYFi, where Tf is our presumed value for the temperature measured at the location of the fast- chemistry flame front, Xeq , which is determined through the algorithm in Section 4.2. Each input value of Tf leads to a solution set which includes the strain rate K: hence, by inputting a variety of Tf values, we may create a Tf-versus-K curve, for any fixed (YOO , YFF , r00) combination. Figure 15 presents six such curves corresponding to six input combinations which represent three different original droplet diameters of oxygen being 1 u m, 10 u m. and 20 u m (i.e., the original droplet diameters of hydrogen being 0.2] u m, 2.1 u m, and 4.1 u m. respectively) versus two different YFF values being 0.025 and 0.035 (i.e., the global equivalence ratios being 1.33 and 1.87, respectively). The diagram informs that, as the fuel becomes richer, the effect of droplet size becomes more distinctive. Figure 16 is a rearrangement of Figure 15 by dividing the curves represent- ing three different droplet sizes into three separate diagrams and by inputting YFF values 62 between 0.025 and 0.035. The significance of these three diagrams is explained as follows: First, each of the diagrams constructs a good data base for relations among Tf , K, the glo- bal equivalence ratio, and droplet size, any of which, if given the other Mo quantities, can be easily anticipated by interpolation of the diagrams. Second, the higher the global equiv- alence ratio, the higher upper limit of strain rate there exists to maintain the flame from extinction. Third, given the same global equivalence ratio, the smaller droplet size will allow a higher upper limit of strain rate to maintain the flame from extinction. The second and the third will be discussed later. 2. The Profile (T, Tpg, Tog, Tp, T'Fz, To; versus X, SE) at Extinction The largest value of strain rate on each Tf—versus—K curves, beyond which the flame can no longer exist, is exactly where extinction occurs. Figures 17 to 28 present the temperature and mass fraction profiles (T, Tpg , Tog , Tp , TF1, To, versus X, it) cor- responding to the largest values of strain rate ever obtained on each of the selected Tf-ver- sus-K curves featuring the original droplet diameter of oxygen at 1 u m, 10 u m, and 20 u m (i.e. the original droplet diameter of hydrogen at 0.21 11 m, 2.1 u m, and 4.1 u m, respectively), with YFF = 0.035 and 0.025 (i.e., the global equivalence ratio (1) = 1.87 and 1.33, respectively). The dotted lines indicate the location of the fast-chemistry flame front, Xeq or icq. From these twelve figures, we may observe qualitatively that either a higher global equivalence ratio or a smaller droplet size will lead to a sharper temperature profile 63 at the flame area, which implies faster chemistry. Moreover, in every diagram from Figures 17 to 28, we are happy to observe that the Tpg and Tog profiles intersect with each other at roughly the location of the fast- chemistry flame front, which may be analyzed by the following derivation: ng(x = xeq) z Yog(x = xeq) , YFi V YFi ’ Y0g(X = xeq) ~ which means that the location of the fast-chemistry flame front, Xeq , stoichiometric ratio v occurs! 3. The Global Equivalence Ratio versus Kext and T max, CXI. is roughly whe’.‘ * the We denote the strain rate at extinction as Km and the highest flame tem- perature on the profile corresponding to Ken as Tmax, ex, , both of which are associated with a global equivalence ratio and a droplet size, as presented earlier in Figures 15 and 16. In Figure 29 we present, with the original dr0plet diameter of oxygen at 1 u m, 10 u m, 64 and 20 u m (and the original droplet diameter of hydrogen at 0.21 11 m, 2.1 p. m, and 4.1 u m, respectively), the global equivalence ratio (1) versus Kext and (1) versus T max, ext ° From this figure we observe the following: (1) (2) (3) The curves of the global equivalence ratio versus Kext are concave downward and those versus Tmax, ext concave upward, but both converge as the global equivalence ratio reduces toward zero and both diverge as that increases, which indicates that the variety in the droplet size has a significant effect upon extinction when the glo- bal equivalence ratio is large (i.e., when the chemical reaction strong), but that it makes little difference when the chemical reaction is weak. The strain rate at extinction approaches zero just as the global equivalence ratio does, which implies that even a slow fluid velocity can extinguish the flame if the chemical reaction is weak. The curves demonstrate that either raising the global equivalence ratio or reducing the droplet size will increase Tmax ext and, we believe, will more or less increase the overall flame temperature. 65 FAST VAPORIZATION AND FAST CHEMISTRY 1000 , r l I 800 600 .2 400 € 200 Figure 3. A result of fast vaporization and fast chemistry in X—coordinate, with Yoo = 0.15 , YFF = 0.03 (i.e., ¢ = 1.6 ). Upper diagram: T [K]. Lower diagram: TF8, Tog (_), TF1, To; (-.), Tp (--). 66 FAST VAPORIZATION AND FAST CHEMISTRY . I I 1 000 . I _ I 800 600 400 200 0.8 0.6 0.4 0.2 Figure 4. A result of fast vaporization and fast chemistry in EC -coordinate, with Yoo = 0.15, YFF = 0.03 (i.e., q) = 1.6). Upper diagram:T [K]. Lower diagram: TF3, Tog (_), TF1, To; (—.), Tp (--). 67 FAST VAPORIZATION AND FAST CHEMISTRY 1000’- l I I ..-‘ ........ I ........ r. I I .4 Y =0.060 ................................... goo — FF ................... ‘ YFF=0.O45 600- _ - YFF-0.030 400 _ YFF=0.015. ' . _ 200 r .. 0 I I L I I i I I I 0 01 02 0.3 04 0.5 06 07 08 09 1 X Figure 5. Results of fast vaporization and fast chemistry in X-coordinate, with Yoo = 0.15 , YFF = 0.015 , 0.03, 0.045.006 (i.e., d) = 0.8 , 1.6, 2.4, 3.2, respectively). Upper diagram: T [K]. Lower diagram: 9F8, 90g (_), i'I’ (")° 68 FAST VAPORIZATION AND FAST CHEMISTRY 1 000 800 - 600 - 400 - 200 0.8 - 0.6 0.4 - 0.2 - Figure 6. Results of fast vaporization and fast chemistry in SE -coordinate, with Y00 = 0.15 , YFF = 0.015 , 0.03, 0.045, 0.06 (i.e., (1) = 0.8 , 1.6, 2.4, 3.2, respectively). Upper diagram: T [K]. Lower diagram: YFg. YOg (_), YP (--). 69 FAST VAPORIZATION AND FAST CHEMISTRY 1000 I I I | '..---"l ........ '.. I 1 YOO_0.15,g.e.r.=3.2- - - 3 800 - Yoo=o.1o,g.e.r.=a.2- - . . . , YOO=0.05,g.e.r.=3.2'- , 600 .. YOO=O.15,g.e.r.=O.8 ............... _ YOO=0.10,g.e.r.=0.8 ----- . . 400 ' YOO=0.05,g.e.r.=O.8'- . ‘ 200 - - 0 I I I I I I I I I 0 O 1 O 2 0 3 0 4 0.5 0 6 0 7 0 8 0 9 1 X 1 _ I I F I I I T . .1 ........ I ..... YOO=0.15,g.e.r.=0.8. . . . .. . 0 8 \ YOO=O.1O,g.e.r.=O.8- ' . . , . .- - YOO=O.05,g.e.r.=0.8 0.6 - H YOO=O.15,ger=3.21 0-4 ~ YOO=O.10,ger=3.2'- - ~ ‘ Y =0.05,ger=3.2 5 0.2 __ .. - —‘- 0° 0 0 1 0 2 0 3 0 4 0.5 0 6 0 7 0 8 0 9 1 X Figure 7. Results of fast vaporization and fast chemistry in X-coordinate, with 6 sets of inputs: Y00 = 0.15, YFF = 0.015 and 0.06 (i.e., ¢ = 0.8 and 3.2); Yoo = 0.1, YFF = 0.01 and 0.04 ((1) = 0.8 and 3.2); Yoo = 0.05, YFF = 0.005 and 0.02(¢ = 0.8 and 3.2). Upper diagram: T [K]. Lower diagram: ng , Yog (_), Yp (--). 70 FAST VAPORIZATION AND FAST CHEMISTRY 1000 ' ' ' r F ' ° Yoo=0.15,g.e.r.=3.2 800 - . . _ _ _ --------- Yoo-O.10,g.e.r._3.2 600 - ~ ........ sYOO=O.OS,g.e.r.=3.2 ‘ YOO=0.15,g.e.r.=0.8' " 400 - ............... . YOO—0.10,g.e.r.-0.8 200- YOO=O.05,g.e.r,=o.8 ............ 0 l 1 I x—tilt 1 - l I f I. ' I -1 . - . . . . YOO=O.15,g.e.r.=O.8 0.8 - YOO=O.10,g.e.r.=O.8 - 0. 6 _ YOO=0.05,g.e.r.=0.8 .. Q4 _ YOO=O.15,g.e.r.=3.2 _ {9 YOO=0.1O,g.e.r.=3.2 0.2 - \OO=O.05,g.e.r.=3.2 ‘ o _ I -6 -4 4 6 Figure 8. Results of fast vaporization and fast chemistry in 3: ~coordinate, with 6 sets of inputs: Y00 = 0.15 , YFF = 0.015 and 0.06 (i.e., 0 = 0.8 and 3.2); Y00 = 0.1, YFF = 0.01 and 0.04 ((1) = 0.8 and 3.2); Yoo = 0.05, YFF = 0.005 and 0.02 (q) = 0.8 and 3.2). Upper diagram: T [K]. Lower diagram: ng, Tog (_), Yp (--). 1000 ~33 ' ' r ' r ' ' ' I I ‘1 800 -13 4 600 ~23 6 400 200 -§§ «f 0 ' 4 l 1 4 1 1 ' L I l ' ' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 .. I I I f I I I I I I 1. 0.8 ~ 0.6 ~ 0.4 71 FINITE-RATE VAPORIZATION AND FAST CHEMISTRY ~1—h ' 3'". 1 :0~Q'U*v 0V1 0.2 $1; '5 . 0 . ’ .. . .. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X Figure 9. A result of finite-rate vaporization and fast chemistry in X-coor- dinate, with Yoo = 0.15, YFF = 0.03, l/DOv = 0.1 . Upper diagram: T [K]. Lower diagram: {0:8, {(08 (_), Tm, Yo, (-.), Yp (--). 72 FINITE-RATE VAPORIZATION AND FAST CHEMISTRY 1000 ' : : ' ' : : ' : T 800 600 - 400 - 200 - 0.8 I 0.6 '- 0.4 - 0.2 - Figure 10. A result of finite-rate vaporization and fast chemistry in SE -coor- dinate, with Y00 = 0.15, YFF = 0.03 , l/Dov = 0.1 . Upper diagram: T [K]. Lower diagram: ng, Tog (_), TF1, Yo; (-.), Yp (--)- 73 FINITE—RATE VAPORIZATION AND FAST CHEMISTRY 100°" ._,...--1/DOV=O.0001 ...... 1/0 =0.25 800- ............. °"_ - - «3;: ....... Ijgov-g-gg 600- ""--...1/D°V:1.00 ‘ . 0V- ' 400— ~ 200— - O L I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1" I I I I I I I I I _ 000500001 0.8 00050.25 - Figure 1 1. Results of finite-rate vaporization and fast chemistry in X-coor- dinate, With YOO = 0.15 , YFF = 0.03 , l/DOV = 0.0001 , 0.25, 0.5, 0.75, 1.0. Upper diagram: T [K]. Lower diagram: ng , Tog (_), 9F] 9 Q0’ (")9 ?P (--)' 74 FINITE-RATE VAPORIZATION AND FAST CHEMISTRY 1000 - .,_....-1/DOV=0.0001 ........ 1/D =0.25 800 ' ---- ' ' f .' ......... TIDOV-o 50 ~ ' :_:'2 I i ............ 1 lDOv‘0'75 600- ""'~-..._.1/D°V:1.00 ‘ Ov- o 400 - _ 200 - _ 0 1 -5 —4 -2 O 2 4 5 x—tilt 1 ........ L ....... T _ ' f I , 1/oOv_0.0001 ’ 1/DO =O.25 V 0'8 ' 1/00 =0.50 . v . 1 |0v=0.75 _ 0.6- 1/-=1.00 - 0.4 - fl .. ’a 0.2 - " _ a? \ fl \ / 0 1 . *‘6 -4 O 6 x-tilt Figure 12. Results of finite-rate vaporization and fast chemistry in i -coor- dinate, with Yoo = 0.15, YFF = 0.03, 1/D0v = 0.0001,0.25, 0.5, 0.75, 1.0. Upper diagram: T [K]. Lower diagram: ng , Yog (...)9 9F, ’ §OI (‘.), YP (--)' 75 FINITE-RATE VAPORIZATION AND FAST CHEMISTRY 1 I *f I ' '__L__;__;__;- Ov+ 0.8 ' x0v- 1 0.6" x _~“—‘_~'—"-——_— 80 0.4 - - 0.2 - _ X X Fv+ F .- 0 ‘ —====:——f—’4:ZZZIZZZZZZ 0 0.2 04 06 08 1 1.2 14 1.6 1.8 2 1 / D CV 1000 " I r I I I I I I I .4 800 - T ,_ —————— - eq .. ‘_ .. -—- " 600 - _ TOV- 400 - . 200 TFv+ TOI /TF:l .- 0 x I 4 I ,I=,:=:;::::;:::1::: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 / DO V Figure 13. Results of finite-rate vaporization and fast chemistry shown as X-locations and temperatures versus a broad range of l/DOv , with YOO = 0.15 , YFF = 0.03 . Upper diagram: XF; , x13; , xeq, x0; , x03 . Lower diagram: TH, Tm, T0,, T0,_ r , ,q 1K1. 76 FINITE—RATE VAPORIZATION AND FAST CHEMISTRY 1 I I I I I I I I I YFF=O'060 0.8 _______ _ YFF=O'045 _ ::::::: ................ v..=o.oso 0.6 - ______________________ YFF=0.015 . .. 0.4 - - 0.2 - .. 0 I L I I I I l I I 0 0 2 0.4 0 6 0.8 1 1 2 1 4 1 6 1 8 2 1 / DOV 1000”‘--.;'_. I I I I r I I I a _ ~ . _ . :1 ___________ YFF—0.060 800 —~ - a ..... -. """" - — - — YFF=O.O45 _ T ' ------- YFF=0'030 600 r- . ~~~~~~ - ____________________ Y =0.015 400 - FF ‘ 200 - - 0 I I L I__ I I I l I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 / DOV Figure 14. Results of finite-rate vaporization and fast chemistry shown as X-locations and temperatures versus a broad range of 1/ Dov , with Yoo = 0.15 , YFF = 0.015 , 0.03, 0.045, 0.06. Upper diagram: X01, (_), ch (-.). Lower diagram: To“ (_), Teq (-.) [K]. 77 FlNITE-RATE VAPORIZATION AND FlNITE-RATE CHEMISTRY 700 I I I I T I r I 680 - _ _ ..... YFF-O.025 660 - d 640 - - 620 - - 600 - _ 580 - - 560 - - 540 - do = 1um _ 2 d =1 Oum 520 — 02 ~ doz =20um 500 g 1 I I 1 1 1 l 0 20 40 60 80 100 120 1 40 160 1 80 Strain Rate : K [1/sec] Figure 15. Results of finite-rate vaporization and finite-rate chemistry shown as temperature at the fast-chemistry flame front, Tf [K], ver- sus strain rate, K [l/sec], with Yoo = 0.15 , YFF = 0.025 , 0.035, and the original diameter of 02 = 1 um, 10 1.1m, 20 um. 78 FINITE—RATE VAPORIZATION AND IFlNITE-RATE CHEMISTRY 700 r 2:2: 1W4 —o 500 ‘ 0 20 410 60 80 1 00 1 20 1 40 160 1 80 700 I I I T I I I ....................... do; 10um 650 - - 600 025 YFF=O.035 ‘ ‘ 550 _ 500 1 l I l _L 0 80 100 120 140 160 1 80 700 I I I T F I I T .............. do =20um 650 - 2 - 600 "Yr? - 0.025 YFF=0'035 550 - - 500 I I I I 4% l I I 0 2O 40 60 80 100 120 1 40 1 60 1 80 Strain Rate : K [1 /sec] Figure 16. Results of finite-rate vaporization and finite-rate chemistry shown as temperature at the fast-chemistry flame front, Tf [K], ver- sus strain rate, K[1/sec], with Yoo = 0.15 , YFF = 0.025 , 0.026, 0.027, 0.028, 0.029, 0.03, 0.031, 0.032, 0.033, 0.034, 0.035, and the original diameter of 02 = 1 um, 10 um, 20 um. 79 FlNITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 1 000 r u r r r 1 I. r r j 800 - d 600 - - .... - 200 - .. C)0 011 012 0:3 014 0:5 016 017 0L8 0L9 1 X 1 '1 I f 1 I I I I I I '1 Figure 17. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.035, K=150.88 /sec, and the original diameter of O2 =.1.0 11 m. Upper diagram: T [K]. Lower diagram: Tag, {(03 (_), TF1, Yo; (-.), Yp (--)- 80 1000 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY I T , I 800 - 600 - 400 '- 200 0.8 - 0.6 0.4 0.2 - Figure 18. A result of finite-rate vaporization and finite-rate chemistry at extinction in STE-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=150.88 lsec, and the original diameter of 02 = 1.0 [.1 m. Upper diagram: T [K]. Lower diagram: TF8, Tog (_), Yp; , To; (-.), Yp (--). 81 FINITE—RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 1 000 i r r I I I , I I I 800 '- 600 - 400 - 200 I 0.6 ~" 0.4 ~- 02” .- " “'5 . O Figure 19. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.035, =97.27 lsec: and the original diameter of 02 = 10 ttm. Upper diagram: T [K]. Lower diagram: Sips, r0, (_), In, {(0, (-.), 1?.» (--). 82 o FINITE-RATE VAPORIZATION AND FlNITE-RATE CHEMISTRY 1 00 r r I . I I 800 - 600 - I 400 200 - 0.8 _ 0.6 - 0.4 - 0.2 Figure 20. A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=97.27 /sec: and the original diameter of 02 = 10 um. Upper diagram: '1‘ [K]. Lower diagram: in, To, (_), 9... r0, (-.), vp (--). 83 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 1000 I r I r I I , I I I 800 - 600 ~ 400 I 200- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 0.4 ~ 0.2 - . q ‘ ’ . \ Figure 21. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.035 , =66.53 /sec, and the original diameter of 02 = 20 u m. Upper diagram: T [K]. Lower diagram: ng , Yog (_), sz , Yo; (-.), Yp (--). I 1 1 84 100 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 0 . r I . I I 800 - 600 - 400 - 200 - 0.8 - 0.6 - 0.4 - Figure 22. A result of finite-rate vaporization and finite-rate chemistry at extinction in S'c-coordinate, with Yoo = 0.15 , YFF = 0.035 , K=66.53 lsec,‘ and the original diameter of 02, = 20 It m. Upper diagram: T [K]. Lower diagram: ng, Tog (_), Ypl , To; (-.), Tp (--). 85 1000 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY I I . I I I I 800 — 600 400 200 Figure 23. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Y00 = 0.15 , YFF = 0.025 , K=26.50 /sec,' and the original diameter of 02 = 1.0 um. Upper diagram: T [K]. Lower diagram: ng, Tog (_), sz , Yo, (-.), Yp (--). 1 000 86 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 800 '- 600 '- 400 '- 200 - I I I I . I 0.8 r 0.6 I 0.2 Figure 24. A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.025 , K=26.50 lsec: and the original diameter of 02 = 1.0 1.1m. Upper diagram: T [K]. Lower diagram: ng, Tog (_), 171:1 , Yo; (-.), Yp (--)- 87 1000 FlNITE-RATE VAPORIZATION AND FlNITE-RATE CHEMISTRY 800- . 600- ~ 400- ~ 200 - - O I 1 1 I 1 ' l I I 1 0 OA (12 113 0A1 (15 (L6 03' 013 (19 1 X 1 Ii I I I T I I I I I (18 - .’ ()15 ’ ~ 1 i 0.4!I ; ‘ - i ’v--+- ' 0.2 I , , , " ’ ' \ ~ I ' 1,,le” “~\-' 0 L I 1 1 I I I 2 f‘ 0 01 02 03 04 05 06 O7 08 09 1 X Figure 25. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.025 , =21.39 /sec, and the original diameter of 02 = 10 um. Upper diagram: T [K]. Lower diagram: TF8, Tog (_), TF1, To; (—.), Tp (--)- 88 FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 1 000 800 - 600 - I 400 200 - I I I . [ ‘ 0.8 I I 0.6 Figure 26. A result of finite-rate vaporization and finite-rate chemistry at extinction in SE-coordinate, with Yoo = 0.15 , YFF = 0.025 , =21.39 /sec, and the original diameter of 02 = 10 um. Upper diagram: T [K]. Lower diagram: ng, Tog (_), TF1, Yo: (-.), Yp (--)- 89 FINIT—RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 1000 I f I 800 - 600 - 400 - 200 - Figure 27. A result of finite-rate vaporization and finite-rate chemistry at extinction in X-coordinate, with Yoo = 0.15 , YFF = 0.025 , K=17.02 lsec,' and the original diameter of 02 = 20 um. Upper diagram: T [K]. Lower diagram: ng , Yog (_), Yp, , Yo, (-.), Yp (--)- 90 1o FINIT-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 00 I I I , r I 800 '- 600 - 400 - I 200 0.6 I 0.4 - 0.2 - O Figure 28. A result of finite-rate vaporization and finite-rate chemistry at extinction in i-coordinate, with Yoo = 0.15 , YFF = 0.025 , K=17.02 lsec: and the original diameter of 02 = 20 p. m. Upper diagram: T [K]. Lower diagram: ng, Tog (_), Y5; , To; (-.), Tp (--)- FINITE-RATE VAPORIZATION AND FINITE-RATE CHEMISTRY 2.2 I I . I I r I T 1 g 2.. do =20um do =10um do =1um , a: 2 2 2 8 1.8 - - C .92 316- - 3 O' E 1.4 r - m D O (‘5' 1.2 - ~ 1 1 1 I I I ' I I 0 20 40 60 80 100 120 140 160 Strain Rate at Extinction: Kext [1/sec] 2.2 I I F I I I Global Equivalence Ratio 0) do =20um do =10um d =1um,‘ 2 2 2 I I I I I I I Figure 29. 580 600 620 - 640 660 680 700 Highest Flame Temperature at Extinction: Tmax ext [K] Results of finite-rate vaporization and finite-rate chemistry shown as (upper) the global equivalence ratio 4) versus strain rate at extinction. Ke'xt [1/sec], and (lower) ‘1’ versus the highest flame temperature at extinction, Tm“, ext [K], with Yoo = 0.15 , and the original diameter of 02 = 1 um, 10 um, 20 um. CHAPTER SEVEN COMPARISON BETWEEN THE DOUBLE-SPRAY, FUEL-SPRAY- ONLY, AND PURELY-GASEOUS MODELS Figure 30 compares Figure 29 with the fuel-spray-only case, featuring oxy- gen in the gaseous phase and the original droplet diameter of hydrogen at 0.21 p. m, 2.1 1,1. m, and 4.1 u m, and with the purely-gaseous case, featuring both hydrogen and oxygen in gaseous phases. Either in the global equivalence ratio versus Kext or in that versus Tmax. ext , we may conclude the following: (1) First look at the purely-gaseous curve, and then the existence of fuel spray shifts the curve leftward (meaning more extinguishable and lower flame temperature), and the existence of “double-spray” shifts the curves further leftward. (2) The variety in the droplet size causes divergence of the curves, and the variety in the oxidizer dr0plet size exerts a dominant influence upon such divergence rather than that in the fuel droplet size does. 92 93 DOUBLE-SPRAY, FUEL-SPRAY-ONLY, AND PURELY-GASEOUS MODELS 2.2 1— I I I I I I I I DOUBLE-SPRAY FUEL-SPRAY-ONLY PURELY-GASEOUS Global Equivalence Ratio I I I 0 100 200 300 400 500 600 700 800 900 1000 Strain Rate at Extinction: Km [1/sec] 2.2 I I I I DOUBLE-SPRAY FUEL-SPRAY— PURELY- ONLY GASEOUS'1 1.8- ~ 1.6- 4 Global Equivalence Ratio 1 l I I I 600 650 700 750 800 Highest Flame Temperature at Extinction: Tmax m [K] Figure 30. Comparison between the Double-spray, Fuel-Spray-Only, and Purely-Gaseous Models 0 CONCLUSION We have investigated our model both theoretically and numerically: we make a few thermodynamic assumptions in Chapter 1, develop governing differential equations in Chapter 2, and set up algebraic analysis as well as computation routines for three typical cases in Chapters 3, 4, and 5, respectively, present and discuss the numerical results of these three cases in Chapter 6, and finally, compare the results in our model to those in the fuel-spray-only and purely-gaseous models. The numbers we obtain are con- sistent with physical reality, which means that our theoretical derivations are correct. One of the recommended directions to extend our work is to change the double-spray model toward the liquid rocket engine, where the reactants’ vaporization is driven by pressure difference, and where the reactants are pure at their origins (i.e., no background gases). We anticipate this as a significant approach in the combustion studies. MORAL OF THE STORY: EACH NUMBER CARRIES A PHYSICAL REALITY! 94 BIBLIOGRAPHY BIBLIOGRAPHY Versaevel, P. (1993), “Counterflow Spray Diffusion Flames: Comparison between Asymptotic, Numerical, and Experimental Results,” Ecole Centrale de Paris, France. Unpublished. Doyle, S. E. and Hall, R. C. (1992), “History of Liquid Rocket Engine developed in the United States 1955-1980,” American Astronautical Society. Williams, F. A. (1985), “Combustion Theory,” Benjamin/Cummings Publishing Co. Reid, R. C., Prausnitz J. M., and Poling, B. E. (1987), “The PrOperties of Gases and Liquids,” McGraw-Hill Publishing Co. Verkin, B. I. (1991), “Handbook of Properties of Condensed Phases of Hydrogen and Oxygen,” Hemisphere Publishing Co. Sychev, V. V., Vasserman, A. A., Kozlov, A. D., Spiridonov, G. A., and Tsymamy, V. A. (1987), “Thermodynamic Properties of Helium,” Hemisphere Publishing Co. Strahle, W. C. (1993), “An Introduction to Combustion,” Chapter 3, Gordon and Breach, Overseas Publishers Association, Amsterdam (2nd printing 1996). Lefebvre, A. H. (1989), “Atomization and Sprays,” Hemisphere Publishing Co. 95 HICHI N S RIES will 11111111111111 31