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chapter 5 turbulent diffusion flames - FedOA

chapter 5 turbulent diffusion flames - FedOA

The energy expended in

The energy expended in vaporization of carbon is written in terms of heat of vaporization of carbon, ΔHv, molecular weight of solid carbon, M = 12 g/mol and rate of mass vaporization, dm/dt, obtained solving mass balance: dm dt 2 da 2 = 4πρs a = −4πa ρνU dt where ρs and ρv are the density of soot in vapor and solid phase respectively, Uv is the velocity with which the vapor leaves particles normally given by the relation of Langmuir: U v ⎛ RT = ⎜ ⎝ 2M in which Ts is the surface temperature of the particle, that for the simplification done is equal to that inner T, R is the universal gas constant and Mv is the molecular weight of carbon in vapor phase. The energy loss by blackbody radiation, the radiative transfer expression, is simply given by the Stefan-Boltzmann law: 40 s v ⎞ ⎟ ⎟ ⎠ 1 2 2 4 4 ( 4π a ) ( T − T ) Qb = εσ SB where σSB is the Stefan Boltzmann constant and ε is the soot emissivity that can be taken equal to the absorption coefficient Kabs. 0 ν

Solving the mass and energy balances we obtain the time dependant particle size, a(t) and temperature, T(t), for a particular choice of the excitation wavelength, λexc, flame temperature T0 and initial particle diameter, a0. Finally the LII signal must be calculated taking into consideration the density of primary particles, Np = N np, and the spectral bandwidth of detection, Δλ, around a central wavelength λ0: C1 ⎡ C2 ⎤ 2 ( λ0, t) = ⎢exp( − −1⎥ N 4πa ( t) ε ( t) Δλ λ0 ⎣ λ0T ( t) ⎦ 41 −1 LII p where C1 and C2 are the first and second Planck constant. By integrating the above equations Melton showed that the LII signal at the maximum temperature (dT/dt), also called “prompt LII”, is proportional to: x Pr omptLII ∝ Npdp where dp is the particle diameter and the exponent x is x= 3 + 154 nm/ λdet This simple finding and the easiness of the experimental set-up promoted the development of the LII technique as the major diagnostic tool for soot detection in practical combustion systems. Based on the Melton interpretations of the LII signal a great number of models have been developed to describe the heating and the cooling of the particles by solving the energy and mass balance equations for temperature and primary particle size. Differences and comparisons from the models are well reported in the review work of Schulz et al. [50], where a briefly description of the major models, nine in the work of Schulz, and their reference are given.

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