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.
of radiation represented by the spectral radiative intensity Iν along the direction~s is described by the radiative transfer equation (RTE), given in equation (4.36).
dIν(~s)
ds = −βνIν(~s) + κabs,νIb,ν + σscat,ν
4π Z
4π
Iν(~sscat,in)Φν (~sscat,in → ~sscat,out) dΩ (4.36)
Therein, the spectral extinction coefficient βν is the result of summation of the spec-tral absorption coefficient κν and the spectral scattering coefficient σscat,ν. The spectral blackbody intensity Ib,ν is determined by Planck’s spectral distribution of emissive power.
Φν (~sscat,in → ~sscat,out) denotes the scattering phase function from the incoming direction~sscat,in
to the outgoing direction~sscat,out. It is integrated over the solid angle sector dΩ.
According to Leiser [80], the number of radiative transport equations to be resolved can be seen from equation (4.37). It depends on the discretization of the radiative space on the one hand and on the spectral discretization of the wave length range on the other hand. For the spatial discretization, different methods have been developed that can be classified in the three groups zonal, ray tracing and differential methods. A comprehensive review can be found in Str¨ohle [140].
NRTE = N~s · N∆ν (4.37)
In this work, a member of the differential methods group denoted as the discrete ordinates method is employed. The radiative space is discretized by its total solid angle, resulting in a discrete number of directions N~s. The number of directions can be increased to infinity and in that case, the solution of the discrete ordinates method equals the exact solution of the radiative transfer equation. It is limited to 24 in this work, and they are weighted by means of a S4 quadrature scheme.
In order to restrict NRTE, the spectral discretization N∆ν can be kept low. Scattering, accounted for by scattering coefficient σscat,ν, significantly occurs by particles only, and thus it is neglected for the gas phase. Absorption, accounted for by absorption coefficient κabs,ν, however, is of importance in the gas and particle phase, and is therefore modeled for both phases by separate sub-models.
Gas absorption
The easiest way to achieve N∆ν being low is to assume the gas phase as a gray medium. The spectral dependency is thus neglected. However, the inherited error can become considerably high, if the atmosphere contains significant portions of CO2 or H2O. Spectral and global models instead discretize the spectral range, i.e. N∆ν depends on the chosen discretization.
Spectral models incorporate either each spectral line or spectral line bands. The first
ap-proach, known as line-by-line apap-proach, is the computationally most expensive, but most ac-curate one of the spectral models. It is usually applied for validation of less acac-curate spectral models. Narrow-band models exhibit a coarser resolution of spectral lines, but they are still too expensive to be used in furnace simulations and serve as a reference, as well. Exponential wide-band models have been used in coal furnace simulations for instance by Str¨ohle [140] and Erfurth [35]. They assume that absorption and emission concentrates to distinct regions of the spectral range, called bands. It neglects the position of spectral lines within a band and assumes an equal spacing between the spectral lines. Absorptivity and emissivity is highest in the band center and decreases exponentially towards the band limits. Str¨ohle [140] proposed the use of three gray gases derived from the exponential wide-band model and one clear gas, resulting in N∆ν = 4. An exponential wide-band model was tested in the course of the FLOX-COAL-II project. The obtained simulation results, however, can not justify the increased computational effort [55]. Additionally, the computational stability strongly decreased with application of the exponential wide-band model.
Global models do not account for the spectral nature of radiation and provide the optical property for one gray gas. A widely used model in CFD combustion simulations is the weighted sum of gray gases model (WSGGM), developed by Hottel and Sarofim [66]. It obtains the absorption coefficientsκabs,G,i for a predefined mixture of i gray gases which can be seen as a pseudo-spectral discretization. It finally derives one absorption coefficientκabs,G by a weighted summation ofκabs,G,i, that is valid for a narrow range of the supposed gas phase composition.
This limited validity is one of the major drawbacks of the WSGGM, and requires careful use.
Particle absorption and scattering
As the fluid in pulverized coal combustion simulations contains particulate matter such as soot, char and ash, their contribution to the radiative heat transfer has to be considered. These solid components of the fluid are assumed as gray, i.e. with no spectral dependency and N∆ν = 1.
The optical properties of the dispersed particle cloud, the absorption coefficient κabs,P and the scattering coefficient σscat,P, are obtained in equations (4.38) and (4.39) by means of the absorption and scattering efficiency factorsηabs and ηscat, the particle load NP and the particle diameter DP [63]. The applied absorption and scattering efficiency factors differ for coal, char and ash and are given in Table 4.4.
κabs,P = π
4 · ηabs · DP2 · NP (4.38)
σscat,P = π
4 · ηscat · DP2 · NP (4.39)
Table 4.4: Absorption and scattering efficiency factors for coal, char and ash ηabs ηscat
Coal and char 0.85 1.30
Ash 0.85 1.70
The probability of radiation scattering occurring at a particle is given by the scattering phase function Φν (~sscat,in → ~sscat,out) in equation (4.36) [140]. It can be derived by the Delta-Eddington approximation [70], and thus the effective scattering coefficient is obtained for each direction~s in the DOM [80, 140].
The absorption coefficient of soot has to be calculated in a different manner, see Leiser [80].
Since the presence of soot is neglected in this work, it is not detailed here.
Combined gas and particle absorption
The total absorption coefficientκabs,tot can now be obtained by the sum of gas and particulate matter absorption coefficients, see equation (4.40).
κabs,tot = κabs,G + κabs,P (4.40)
The influence of gaseous radiation on the combustion process is of minor importance if particulate matter is present, as results of Gronarz et al. [55] indicate. This was confirmed in this work in simulations comparing the κabs,G obtained from the WSGGM and κabs,G set constant. The total absorption coefficient κabs,tot showed the very similar results regardless of the κabs,G origin. Consequently, temperature remains unaffected [165].
Wall properties
Similar to gas phase and particulate matter, the furnace walls absorb and emit radiation, depending on their optical properties and temperature. The furnace walls are supposed to be a gray body without spectral dependency and they are expected to be diffuse, i.e. they emit or reflect the radiation isotropically [125]. The absorption coefficient κabs,W is set to 0.65.
The convective heat transfer from the fluid to the wall is defined by the convective heat transfer coefficientαW. It is set to 25 W (m2K)−1.