OCR Text |
Show f e e } Rs = Rs./ ~ Rs.c = mp(a-P Ns)n - min ACs~, ^ - ^^ V K Rs K Csrs + CrrF J (11) where Rsf and Rsc are the rate of soot formation and rate of soot combustion, and mp, and Ns, are the mass and particle number density of soot particle, n is radical nuclei concentration. The a and (3 are empirical constants with assigned values of 1 x IO5 /s and 8xl0"14 (number/n^s)"1, A = 4, and c0, rs,cF, and rF are the concentration of oxidizer species, mass stoichiometry for combustion of soot, concentration of fuel, and mass stoichiometry for combustion of fuel, respectively. The net rate of nuclei generation in the Tesner model is given by the balance of the nuclei formation rate and the nuclei combustion rate: Rn = Rnj Rn,c = aocFexp(-E/RT) + (f -g)n-g0nNs- Rs.c- (12) cs where R^f and R^c are the rate of nuclei formation and rate of nuclei combustion, ao is a pre-exponential rate constant, E/R, f, g, and g0 are the activation energy term, linear branching coefficient, linear termination coefficient, and linear termination on soot particles, respectively; and r\0 is the kinetic rate. Soot formation and the combustion of soot particles are largely determined by the local C/O ratio, the local hydrocarbon concentrations, temperature, and residence times of the particles both in soot forming and burning region. A n mcrease in the C/O ratio essentially increases the particle size, and it has little effect on the final number and shape of the size distribution. The rate of growth of soot is proportional to the hydrocarbon concentrations and is greatly retarded by the hydrogen formed during the reaction. 2.3 Dispersed Phase In addition to solvmg transport equations for the continuous phase, a discrete second phase for oil droplets is solved in the Lagrangian frame of reference. The trajectories and heat and mass transfer from/to these discrete particles are coupled with the continuous phase. The trajectory of the discrete phase particle is calculated by integrating the force balance on the particle, which is given in Eq (9). -l = FD(u-uP) + gx(pp-p)/pP + Ft ^^) where F D (u -Up) is the drag force per unit particle mass and 18/i CoRe (14) ppDp1 24 Here, u is the fluid phase velocity, up is the particle velocity, fi is the molecular viscosity of the fluid, the fluid density, pp is the particle density, D p is the particle diameter, and Re is the Reynolds number, which is defined based on the relative velocity of the fluid and particle velocities. The drag coefficient C D is defined from Morsi and Alexander's expression (1972): |