This section gives information on important boundary conditions not presented in the modeling sections.
Boundary conditions of the variables at walls, inlets, outlets and symmetry boundaries are either of Dirichlet or Neumann i.e. of fixed or gradient type, respectively. Inlet boundary conditions are set as fixed values for all variables except the pressure. Neumann type boundary conditions are applied to all other boundaries, with first-order or second-order gradients. The pressure at the outlet, on the other hand, is linearly extrapolated. For more detailed information, please refer to the literature [37, 125].
Discretization of the particle size distribution
It was stated in Section 4.1.5 that the particle-laden fluid is modeled by a quasi-one-phase Eulerian approach without resolving the particle phase. However, chemical reactions depend on the external surface of the particles and thus on the particle size, as will be pointed out in Section 4.5. The particle size distribution is obtained from mass-based or volume-based methods, such as sieving or laser diffraction. It can be approximated by the Rosin-Rammler distribution, also known as the Weibull distribution. This cumulative density function is given in equation (4.33). The shape parameter λRR can be seen as a measure of the spread of the particle sizes, whereas the scale parameter kRR refers to a mean diameter of the distribution.
Their derivation is described in [172] and they are obtained by transposing equation (4.33), leading to equation (4.34).
xcum = 1 − exp
− D kRR
λRR
(4.33) ln (− ln (1 − xcum)) = λRR ln(D) + ln
1 kRR
λRR
(4.34) Equation (4.34) can be seen as a line equation, i.e. as a function of the particle diameter ln(D). The results from particle size characterization can be transferred to a diagram with ln(D) on the x-axis and ln (− ln (1 − xcum)) on the y-axis. The shape parameter λRR can be thus derived from the slope of a line fitted through the results of experimental particle size characterization. The scale parameterkRR instead can be obtained by means of the intersection of the fitted line with they-axis, i.e. if ln(D) equals 0.
Mass-based methods, such as sieving, and volume-based methods, as laser diffraction is, can be used to obtain the required particle size distribution. Both methods were used, but the sieving is performed only with coal samples from the PT1 experiments (cf. Section 3.4).
The sieving result is given in Table 4.1. It was concluded from the laser diffraction results that particle size distribution for coal samples from both the PT1 and PT2 experiments are very similar (cf. Table 3.4). It is assumed that the mass-based sieving analyses would be very similar, too. The mass-based sieving method is regarded more suitable for AIOLOS, as the program is mass-based, too. The particle size distribution and the Rosin-Rammler parameters derived from sieving analysis are used for all simulations, see Table 4.1.
Table 4.1: Mass based particle size distribution properties of Calentur coal Particle size distribution Rosin-Rammler parameters
D10 in µm 44.53 λRR in — 3.80
D50 in µm 73.10 kRR in µm 80.5
D90 in µm 100.3
The conduction of the CFD simulation on the basis of a continuous particle size distribu-tion would result in very high computadistribu-tional costs. The continuous particle size distribudistribu-tion is discretized using an equal spacing of the diameters of the particle size range. Consequently, the mass fraction represented by each discretized mean diameter is calculated using the Rosin-Rammler function and its shape and scale parameters λRR and kRR, respectively. These pa-rameters are fitted to the continuous particle size disctribution, as was described before.
The number of intervals i.e. particle size classes, depends on the targeted accuracy. It is noted that a finer discretization increases the computational effort, since more particle size classes have to be computed. Five particle size classes have been found to be a good compromise between accuracy and speed of computation. Finer particle size discretizations were tested, but without a significant impact on the simulation results.
Figure 4.3: Particle size distribution of Calentur coal
The particle size distribution is shown in Figure 4.3: approximation of particle distribution by the Rosin-Rammler function (RR), sieving results (Dsieving) and the five representative particle diameters Drep as applied in the simulations. The five representative particle size classes are given with their diameter and mass fraction in Table 4.2.
Table 4.2: Representative particle size distribution of Calentur coal used in CFD simulations
D in µm x in %
34.00 5.89
59.20 23.82
84.40 38.06
110.0 25.79
135.0 6.98
Furnace wall temperatures
The furnace wall temperature ϑW is derived from thermocouples located in the wall close to the surface. This measured data was averaged over the measurement period of each operating point. The furnace wall temperature as applied in AIOLOS is discretized along the axial, i.e.
z-direction of the furnace. The givenz values in Table 4.3 represent the axial distance of each wall section up to which the corresponding temperature prevails.
The discretized temperatures and wall sections of the FLOXCO2 case are compiled in Table 4.3, as this case is discussed in the following Section 4.7. The furnace wall temperatures of the other cases depicted in Table 3.2 can be found in Appendix C.
Table 4.3: Furnace wall temperatures of the FLOXCO2 case z in mm ϑW in °C
155 1031
380 1050
710 1064
1060 1074
1400 1064
1730 1024
2060 1013
2400 970
5900 912
Inlet conditions
The turbulence intensity at the inlet must be set as a boundary condition and is calculated as displayed in equation (3.4), Chapter 3. The root-mean-square velocity fluctuations are usually unknown, thus the turbulence intensity can be estimated from the Reynolds number for a fully developed pipe flow, see equation (4.35). Therein, the Reynolds number Re is based on the characteristic length of a pipe, the hydraulic diameter Dhyd. Dhyd equals the pipe diameter D for a conventional pipe as the combustion-air nozzle or the difference of outer and inner diameter for a coaxial pipe as the coal-carrier-gas annulus.
Iturb = 0.16Re−18 (4.35)
With equation (4.35), one obtains turbulence intensities of 5.5 % and 3.8 % at the coal-carrier-gas annulus and the combustion-air nozzle, respectively. This is close to the turbulence intensity of 5 % anticipated before the first experimental campaign.