LII Theoretical Model

Melton (1984) first set up the mass and energy balance equation for quantitative measurement of soot. Two sources of model from Schulz, Kock et al. (2006), who reviewed LII techniques, and models and Michelsen, Liu et al. (2007) reviewed and compared LII models. Will, Schraml et al. (1998), Snelling, Liu et al. (2000), Hofeldt (1993), Bladh and Bengtsson (2004), Michelsen (2003), Dreier, Bougie et al. (2006) and Vander Wal (1994) are other modelling and theory sources. Dreier, Bougie et al.

(2006) reviewed models which are more specifically on LII in a high pressure spray combustion environment with time-resolved LII technique. The model described below is based on Kock, Tribalet et al. (2006) model as it applies to high pressure environment and his model is based on the work of Williams and Loyalka (1991). For other in-cylinder modelling, Boiarciuc, Foucher et al. (2007) used the high pressure LII model as proposed by Kock, Tribalet et al. (2006).

The basic equation for the rate of change in energy of a carbon particle undergoing laser heat up is shown in equation (1-6).

𝑄̇_𝑖𝑛𝑡 = 𝑄̇_𝑎𝑏𝑠− 𝑄̇_𝑠𝑢𝑏− 𝑄̇_{𝑐𝑜𝑛𝑑}− 𝑄̇_𝑟𝑎𝑑 (1-6) In equation (1-6), 𝑄̇_𝑖𝑛𝑡 is the rate of internal rate of energy change of the particle;

𝑄̇_𝑎𝑏𝑠 is the rate of laser energy absorbed by the particle; 𝑄̇_{𝑐𝑜𝑛𝑑} is the rate of heat loss through conduction; 𝑄̇_𝑠𝑢𝑏 is the rate of heat loss through sublimation; 𝑄̇_𝑟𝑎𝑑 is the rate of heat loss through radiation. Other heat loss mechanisms such as annealing and oxidation are omitted here. Annealing is when carbon particles melt

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and re-crystallise producing an annealed particle and oxidation of the carbon particle produces a compound such as carbon monoxide. More information about these heat loss mechanisms can be found in Michelsen (2003).

Absorption accounts for the temperature of the particle reaches after laser heating according to Mie theory. The maximum temperature reached is independent of particle size as the temperature of the particle in the Rayleigh limit is proportional to the particle volume.

𝑇_𝑃⁰ = 𝑇_𝑔− 6𝜋𝑅₀

𝜌_𝑝𝑐_𝑝 𝜆_{𝐿𝑎𝑠𝑒𝑟}𝐼𝑚 (𝑚²− 1

𝑚²+ 2) (1-7)

Equation (1-7) shows the particle temperature, 𝑇_𝑃⁰, after being heated up by the laser. R0 is the laser energy density, Tg is the gas temperature, ρp is the particle density, cp is the particle heat capacity and λlaser is the wavelength of the laser. m is the complex refractive index and Smyth and Shaddix (1996) has a review of soot refractive index. Sorensen (2001) reviewed the scattering and absorption of light by fractal aggregates which included different complex refractive indexes from a physical point of view. For LII application, Schulz, Kock et al. (2006) and Dreier, Bougie et al. (2006) included the complex refractive index in the models reviewed.

There is an issue with smaller particles as they have larger surface to volume ratio than larger particles and hence cool faster (Tropea, Yarin et al. (2007)). This has implication on detection gate setting as the longer or delayed detection gate produces an integrated signal which underestimates particle sizes (Ni, Pinson et al.

(1995), Vander Wal (1996)).

In order to avoid using m, the alternative is to apply two-color pyrometry to determine the particle temperature. The use of the two-color pyrometry allows the LII signal to be detected at two different wavelengths and the temperature can be calculated from the following equation (1-8).

79 temperature from black-body source during calibration and εp is the particle spectral emissivity. For small particles in the Rayleigh limit, the equation can be simplified as follows:

𝜀_𝑝(𝜆₂) 𝜀_𝑝(𝜆₁) ≈𝜆₁

𝜆₂ (1-9)

The rest of the energy equation relates to the heat loss through radiation and conduction. The cooling rate can be classified by the Knudsen number, Kn, which is the ratio of the particle radius, rp to the mean free path of the gas molecules, λg, in which the particle is suspended. For diesel engine conditions, it was shown that the Knudsen numbers are around 1, meaning that the radius of the particles and the mean free path of the gas molecules are about the same (Kock, Eckhardt et al.

(2002)). This means the system is in the transition regime. The conduction term can be expressed in equation (1-10) as:

80 𝑓_ℎ(𝐾𝑛_ℎ) = [1 + 𝐾𝑛_ℎ1.9234𝐴+1.3026

1.9234𝐾𝑛ℎ+1 ]⁻¹ with 𝐴 =_𝑄̇^{𝑄̇𝑐𝑜𝑛𝑑,𝑐}

𝑐𝑜𝑛𝑑,𝑓𝑚 = 5√𝜋 𝛼⁄ _𝑇(^𝛾−1_𝛾+1) 𝐾𝑛_ℎ (1-11) γ is the heat capacity ratio and αT is the translational energy accommodation coefficient which describes the efficiency of the energy transfer in molecular collision. The translational energy is in the range of 0.07 and 1.00 Kock (2005).

𝐾𝑛_ℎ =4𝜆_{𝑐𝑜𝑛𝑑}𝑇_𝑔 5𝑟_𝑝𝑝_𝑔 ( 𝑚_𝑔

2𝑘_𝐵𝑇_𝑔)

1⁄2

(1-12) pg and mg are the pressure and molecular mass of the ambient gas, λcond is the thermal conductivity of the gas. The high pressure environment has the effect of increasing the cooling by conduction, producing a shorter signal Hofmann, Bessler et al. (2003) and, conversely, a low pressure environment or a vacuum will reduce conduction and produce a longer signal (Beyer and Greenhalgh (2006), Liu, Daun et al. (2006)).

The sublimation heat loss is given by equation (1-13). This occurs as the carbon particle is heated up to above 2800 K causing the carbon to sublime into gas-phase carbon atom clusters (Michelsen (2003)).

𝑄̇_𝑠𝑢𝑏 = 𝑈̇_𝑠𝑢𝑏∆ℎ_𝑣 (1-13)

Δhv is the evaporation/sublimation enthalpy of the particle material, using the Clausius-Clapeyron equation, was found to be 21,946kJkg^-1 and 𝑈̇_𝑠𝑢𝑏 in transient condition can be expressed by equation (1-14).

𝑈̇_{𝑠𝑢𝑏,𝑡𝑟} = 𝑈̇_{𝑠𝑢𝑏,𝑐}𝑓_𝑐(𝐾𝑛_𝑐) = 2𝜋𝑑_𝑝𝐷(𝜌_𝑠− 𝜌_∞)𝑓_𝑐(𝐾𝑛_𝑐) (1-14) D is the diffusion coefficient of the vapour into the gas in m²s^-1, ρs and ρ∞ are the vapour density of the carbon at the particle surface and infinity respectively and ρ∞

is assumed to be negligible. ρs can be approximated to carbon vapour pressure ps

using the Clausius-Clapeyron equation as shown in equation (1-15).

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𝑝_𝑠 = 𝑝^∗𝑒𝑥𝑝 (−∆ℎ_𝑣𝑀_𝑣/𝑅_𝑚(1 𝑇⁄ − 1 𝑇_𝑝 ⁄ )) ^∗ (1-15) Mv is the molar mass of the vapour, Rm is the molar mass constant. p* and T* are referenced to the appropriate points on the vapour pressure curve taken at 61.5 Pa at 3000 K Leider, Krikorian et al. (1973).

The empirical function correcting the Knudsen affected continuum evaporation/

sublimation rate is defined in equation (1-16).

fc is the empirical function, αv is the accommodation coefficient for sublimation for non-equilibrium sublimation kinetics and real gas effects. This can be simplified to 1.

The empirical function of the Knudsen number, Knc is given in equation (1-17).

𝐾𝑛_𝑐 =2𝐷 Stefan-Boltzmann constant. Radiation heat loss is comparatively minor compared to conduction and sublimation heat loss.

𝑄̇_𝑟𝑎𝑑 = 𝜋𝑑_𝑝²𝜖_𝑝𝑡𝜎(𝑇_𝑝⁴− 𝑇_𝑔⁴) (1-18) The LII signal generated by each particle in the detection volume is given by equation (1-19). Using this equation, the signal can be found to be solved by integrating along the range of wavelength for a range of temperature from 1000K to 6000K for every 5K.

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constant, c is the speed of light, λ is the wavelength of light, kB is the Boltzmann constant and T is the temperature. Spectral intensity is the radiation in the interval dλ around a single wavelength and total intensity is the integral over all wavelengths.

𝜀_𝜆 = 4𝐶_𝑎𝑏𝑠

𝜋𝐷² (1-20)

The energy balance equation (1-6) can be used to calculate the theoretical temperature of the soot particle at any point of time after being energised by the laser. It also calculates the theoretical rate of temperature change and the contribution from each of the heat loss mechanisms and allows for the soot particle diameter to be calculated using the measured temperature and rate of temperature change.

Using equation (1-19), the LII signal can be calculated. The theoretical signal can be calculated for a range of temperatures using two set ranges of wavelengths. A ratio of the theoretical signal between the two wavelengths at a given temperature can be found and used with the measured signal to find the temperature in the measured volume. The LII signal equation can also be used to calculate the size of the particles as the diameter term in the equation cancels out when calculating the emissivity. Using the theoretical signal at a given temperature at either sets of wavelength, the emitting surface of the soot particle, and hence the size of the particle, can be calculated using the measured signal. Because this is a ratio of signals, the absolute value of the calculated signal and the measured signal is not important.

1.5.3 Laser

Most of the LII work done in engines, such as Miles, Collin et al. (2007), Boiarciuc (2006), Boiarciuc, Foucher et al. (2007) and Kock, Tribalet et al. (2006), Menkiel, Donkerbroek et al. (2012), used a Nd:YAG laser with the wavelength of 1064nm.

Boiarciuc (2006), Boiarciuc, Foucher et al. (2007) gave the reason for choosing 1064nm as being the avoidance of possible broadband PAH fluorescence interference and electronically excited C2 emission at 532nm or lower wavelength.

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This is a view supported by others (Vander Wal (1994), Bengtsson and Aldén (1995), Schoemaecker Moreau, Therssen et al. (2004)). Dec and Espey (1992), Dec, Zur Loye et al. (1992), Singh, Reitz et al. (2007) and Bougie, Ganippa et al. (2006), Bougie, Ganippa et al. (2007) used Nd:YAG lasers with the wavelength of 532nm. Aronsson, Sjöholm et al. (2010) used 4 double-pulse Nd:YAG lasers cluster for multiple measurements within 1 cycle. The laser was setup with a wavelength of 532nm and capable of 8 pulses per cycle with time separation between 5-440μs and up to 1.4J/pulse. A top-hat profile was used in Boiarciuc (2006), Boiarciuc, Foucher et al.

(2007), Aronsson, Sjöholm et al. (2010) and Kock, Tribalet et al. (2006) Menkiel, Donkerbroek et al. (2012) and one of the methods to generate this is by filtering the laser beam geometrically. Table 1-1 shows the collated information on laser used.

Author Laser

Wave-length Profile Energy Pulse width/

Frequency

specified 0.25 J/cm² 10ns (FWHM) Boiarciuc (2006), Boiarciuc,

specified 0.6 J/cm² Not specified Aronsson, Sjöholm et al.

(2010)

4

Nd:YAG 532nm Top-hat 0.2J/cm² Not specified Menkiel, Donkerbroek et al.

(2012) Nd:YAG 1064nm Top-hat 0.21 J/cm² 8ns

Table 1-3: Laser, laser power and profile used in various literature