ANALYTICAL SOLUTION FOR THE DROPLET COMBUSTION IN PREMIXED FLAME REGIME

Texto completo

(1)ANALYTICAL SOLUTION FOR THE DROPLET COMBUSTION IN PREMIXED-FLAME REGIME. Jaime Bortoli Filho 1 Jaime Filho 2 Andriel Paz Reis 3 Fernando Fachini Filho 4. Resumo: The present work presents an analytical model for the combustion of droplets in the premixed-flame regime. In the premixed-flame regime either fuel or oxidizer leaks through the flame sheet. In the present model, only fuel leakage is considered. The present model is able to describe the main characteristics of the flame, such as, temperature and flame position. The results show the same behavior as those found in experimentally works. However, due to the simplicity of the model, it is only possible to make parametric estimates with respect to the behavior of the flame assuming the fuel leakage as a known parameter.. Palavras-chave: DROPLET COMBUSTION, FUEL DROPLET. Modalidade de Participação: Pesquisador. ANALYTICAL SOLUTION FOR THE DROPLET COMBUSTION IN PREMIXED-FLAME REGIME 1 Aluno de graduação. jaime.paaz@gmail.com. Autor principal 2 Aluno de graduação. jaime.paaz@gmail.com. Apresentador 3 Aluno de graduação. andrielpr@gmail.com. Co-autor 4 Outro. fachini@lcp.inpe.br. Co-autor. Anais do 9º SALÃO INTERNACIONAL DE ENSINO, PESQUISA E EXTENSÃO - SIEPE Universidade Federal do Pampa | Santana do Livramento, 21 a 23 de novembro de 2017.

(2)

(3) METHODOLOGY Formulation of the quasi-steady droplet combustion problem are presented elsewhere (Fachini, 1999). Thus, because of the problem to be well known, only the parts of the formulation essential for the understanding will be explicitly presented. Considering the ambient conditions to be characterized by the temperature T∞ , density ρ∞ , oxygen mass fraction YO∞ . Without loosing any important feature of the problem, the transport coefficients (thermal conductivity and diffusion coefficient, respectively) are dependent only on temperature according to k/k∞ = Di /Di ∞ = (T /T )n = θn . The non-dimensional quasi-steady conservation equations, describing the gas phase around the droplet with radius a at time t (a0 at the time t = 0), are expressed by x2 ̺v = λ(τ )        LeF yF  yF  −1     λ ∂ 1 ∂  2 n ∂ −S Le y yO  = ω̇ xθ − O O    x2 ∂x  x2 ∂x  ∂x  Q θ θ. (1) (2). The definition of the non-dimensional independent variables are as following: the time τ ≡ t/tc , where tc is the vaporisation time tc ≡ ε(a20 /α∞ ) and the radial coordinate x ≡ r/a0 ; ε ≡ ρ∞ /ρl is the ratio of the gas density to the liquid density and α∞ ≡ k∞ /(ρ∞ cp ) is the thermal diffusivity. The definition of the nondimensional dependent variables (temperature, density, oxygen mass fraction, fuel mass fraction and velocity) are as following: θ ≡ T /T∞ , ̺ ≡ ρ/ρ∞ , yO ≡ YO /YO∞ , yF ≡ YF and v ≡ V a0 /α∞ , respectively. The parameters appearing in Eqs. (2) are defined as: Lewis number Lei ≡ α∞ /Di ∞ , S ≡ LeO ν/(LeF YO∞ ) and heat of combustion Q ≡ q/(LeF cp T∞ ). Note that ν, νC and νH are the stoichiometric coefficients in terms of mass that satisfy F + νO2 → (1 + ν)P ; ν/YO∞ is the quantity of air necessary to burn stoichiometrically a unity mass of fuel. The fuel oxidation performing by one-step kinetic mechanism is of the Arrhenius type, ω̇ =. LeF Ba20 ρn∞1 +n2 −1 n1 n2 y y exp(−θa /θ) α∞ WOn1 WFn2 O F. (3). where Wi is the molecular weight of species i and the nondimensional activation energy is defined by θa ≡ E/RT∞ ; The non-dimensional vaporisation rate is λ = ṁ/(4πa0 k∞ /cp ) and the non-dimensional droplet radius, a = a/a0 . According to d2 law, the ratio λ/a, known as vaporisation constant and defined as β, depends on the heat flux to the droplet imposed by the flame and is a constant value. Equations (2) are integrated from the droplet surface x = a to the ambient atmosphere x → ∞, the flame is at a position between these two boundaries. To proceed the integration, the boundaries conditions must be specified; at x = a ≡ a/a0 : ∂θ = λl + q − = λl′ , ∂x and for x → ∞:. θ = θs ,. x2 θ n. yF = yF s = exp[γ(1 − θb /θs )],. θ − 1 = yO − 1 = yF = 0. x2 θ n. ∂yF = −λLeF (1 − yF s ), (4) ∂x (5). The subscript s represents the droplet surface condition. The nondimensional latent heat l is expressed by L/(cp T∞ ), q − is the heat to inside the droplet. The parameter γ is defined as γ = L/RTb . The solution of the problem demands knowing the jump condition for the gradients of fuel, oxidant and temperature through the flame, which is determined integrating Eqs. (2) around the the flame x2 θn ∂YF LeF ∂x. x+ p x− p. x2 θn ∂YO = SLeO ∂x. x+ p x− p. x2 θn ∂θ =− Q ∂x. x+ p. (6) x− p. 2.

(4) in which x2 θn ∂YO SLeO ∂x. =0 x− p. In this work, it is admitted uniform temperature profile inside the droplet and close to the boiling value, θ = θs < θb . Thereby, the mass conservation equation for the liquid phase leads to da2 λ = −2 dτ a. (7). The closure for the system of equations is provided by the dimensionless equation of state of the gas, ̺θ = 1. The system of Eqs. (1) and (2) is characterized by the boundary conditions Eqs. (6) and (7). According to the type of the problem, at the flame x = xp , the properties are θ − θp = y F − y F p = y 0 = 0. (8). Equation (7) expresses the droplet problem in a general form, either fuel and oxygen leak by the flame. Under reactants leakage condition, the flow field analysis does not provide a closed solution. The flame properties (position and temperature of the flame) and the droplet properties (droplet temperature and vaporisation rate) are determined as function of the fuel leakage quantity. As can be seen, the system of equation (2) has a first integration if it is performed in the regions a ≤ x ≤ xp for condition θn = 1, thus. LeF λ(yF − c1 ) = x2. λ(θ − c2 ) = x2. ∂yF ∂x. (9). ∂θ ∂x. (10). applying the boundary condition Eq. (4) in Eq. (9) the constant c1 = 1 is obtained. Thus the Eq. (9) is integrated again and applying the boundary condition (8) results yF = 1 − e. λLeF. . 1 − x1 xp. . (1 − yF p ). similarly, the same procedure is repeated in Eq. (10) with the boundary conditions Eqs. (4) and (9). θ=. . Q− θs − L − λ. . 1−e. λLeF. . 1 − x1 xp. . + θp e. λLeF. . 1 − x1 xp. . For region xp ≤ x ≤ x∞ the Eqs. (9) and (10) are integrated with the boundary condition in a region far from the droplet Eq. (5) resulting in. y F = y F∞ + e. θ = θ∞ + e. λLeF. λLeF. . . 1 − x1 xp. 1 − x1 x p. . . (yF p − yF∞ ). (θp − θ∞ ) 3.

(5) Thus the flame position xp , flame temperature θp and vaporization rate λ are obtained 1+s xp = s(1 − YF s ) . . . 1+s ln s(1 − YF p ). θp = Q − (L + 1) +. λ=. . 1+s 1 + sYF p. . . (1 − QYF p ). . 1+s −1 1 + sYF p. . 1+s s(1 − YF s ). RESULTS The cases analyzed here consider n-heptane droplet in a quiescent atmosphere at 298K. The fuel leakage (yF p ) is considered as a known parameter. Figure (1-a) exhibits the temperature profile of the flame for different fuel leakage. The results shown that the flame temperature decrease with the increase of fuel leakage. This behavior is expected because the flame loss heat for fuel unburned by the flame according to the behavior of the Fig. (1-b). Figure (2) shows the flame position as a 10. yfp = 0.1 yfp = 0.2 yfp = 0.3 yfp = 0.4 yfp = 0.5. 9 8 7. θ. 6 5 4 3 2 1 0 0. 10. 20. 30. 40. 50. 60. x 10 9 8 7. θp. 6 5 4 3 2 1 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. yFp. Figure 1: a) temperature profile, b) flame temperature as a function of fuel leakage, 4.

(6) 90 80 70 60. xp. 50 40 30 20 10 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. yFp. Figure 2: flame position as a function of fuel leakage function of fuel leakage, as a result, the flame is attracted to the surface of the droplet when the fuel leakage increases. CONCLUSION The present formulation, based on quasi-steady model for the droplet combustion considering fuel leakage by the flame, was used to address the droplet combustion in premixed-flame regime. The flame temperature decreases with the fuel leakage but flame position comes close to the droplet even with small leakage. Also, the analysis is able to capture the unexpected small reduction on the vaporization rate even with large fuel leakage, a feature observed experimentally. However, the model over-predicts the value for flame position. Therefore, the analytical model is only able to see the features of droplet combustion in premixed-flame regime. In addition, for a more detailed analysis, a chemical kinetic model is more appropriate. REFERENCES FACHINI F. F., Combustion and Flame, 116 (1999) 302-306. GODSAVE G. A, Proc. Comb. Inst., 4 (1953) 818-830. KANURY A.M., Introduction to Combustion Phenomena, Vol. 2, CRC Press, Boca Raton, FL, (1975), 148215. KUO, K, Principles of Combustion, Ed. Wiley, (2005). LIAN A., Asymptotic structure of counterflow diffusion flames for large activation energies, Acta Astronautica 1 (1974), pp. 10071039. NAYAGAM, V., Dietrich, D., Williams F., A BurkeSchumann analysis of diffusion-flame structures supported by a burning droplet, Combustion Theory and Modelling, (2017) 1-12. POINSOT, T., Veynant D., Theoretical and Numerical Combustion, Ed. R.T. Edwards, (2005). TURNS, S. R, An Introduction to Combustion: Concepts and Applications, Ed. McGraw-Hill, (2011).. 5.

(7)