Computational Fluid Dynamics for Engineers by Klaus A. Hoffmann, Steve T. Chiang

By Klaus A. Hoffmann, Steve T. Chiang

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The mass diffusivities for all species are taken to be identical and are described by D. Higher-order terms in the Mach number are neglected and pressure is taken to be uniform. Continuity: r2 ∂ρ ∂ + (ρur 2 ) = 0, ∂t ∂r or, for the quasi-steady case, ˙ ρur 2 = constant = (m/4π). , λ/c p = ρ D, this becomes − ∂ ∂r ρ Dhmr 2 ∂ ∂h ∂ ∂ (ρr 2 h) + (ρur 2 h) − ρ Dr 2 ∂t ∂r ∂r ∂r = r2 ∂p − ρr 2 Qw˙ F . ∂t Vapor-Species Conservation: L(YF ) = ∂YF ∂ ∂ ∂ (ρr 2 YF ) + (ρur 2 YF ) − ρ Dr 2 ∂t ∂r ∂r ∂r = ρr 2 w˙ F .

In the special case of very fast chemistry, constant specific heat, and unitary Lewis number (ρ D = λ/cp ) for the gas phase, the quasi-steady gas-phase equations can be reduced to certain algebraic relations by means of the use of Eqs. 10) 16 Isolated Spherically Symmetric Droplet Vaporization and Heating where ∞ (r ) ≡ r B≡ dr ρ Dr 2 −1 , h∞ − hs + ν QYO∞ νYO∞ + YF s = . 11) When ρ D = constant, we obtain (r ) = ρ Dr and m˙ = 4πρ DR log(1 + B). The preceding analysis considers that the Lewis number is unity valued and the Spalding mass transfer number BM and heat transfer number BH are equal and identified simply as B.

Therefore heat and mass transfer in the liquid occur only because of diffusion with a moving boundary but without convection. Here, the spherically symmetric isolated droplet equations are presented. A quasi-steady assumption is often made for the gas phase because diffusion of heat and mass in the gas is usually relatively fast compared with that of the liquid; this assumption weakens as we approach the critical pressure. Overviews of general theoretical issues related to vaporization, heating, and burning of spherically symmetric droplets can be found in the reviews of Law (1982), Sirignano (1983, 1993b), and Williams (1985).

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