Numerical Studies of the Gas-Solid Hydrodynamics at High Temperature in the Riser of a Bench-Scale Circulating Fluidized Bed

Table 4

Hydrodynamic and thermal models for the two-fluid gas-solid flow approach.

Continuity
𝜕𝜀𝜕𝑡^𝑖𝜌𝑖+∇⋅𝜀𝑖𝜌𝑖^→𝜐𝑖=0
Gas-phase momentum
𝜕𝜀𝜕𝑡^𝑔𝜌𝑔^→𝜐𝑔+∇⋅𝜀𝑔𝜌𝑔^→𝜐𝑔^→𝜐𝑔=−𝜀𝑔∇𝑃𝑔+∇^→𝜏𝑔+𝜀𝑔𝜌𝑔^→𝑔
+𝛽^→𝜐𝑠−^→𝜐𝑔
Gas-phase stress tensor
^→𝜏𝑔=𝜀𝑔𝜇𝑔∇^→𝜐𝑔+∇^→𝜐𝑔^𝑇−23∇⋅^→𝜐𝑔^→𝐼
Solid-phase momentum
𝜕𝜀𝜕𝑡^𝑠𝜌𝑠^→𝜐𝑠+∇⋅𝜀𝑠𝜌𝑠^→𝜐𝑠^→𝜐𝑠=∇^→𝑇𝑠+𝜀𝑠𝜌𝑠^→𝑔+𝛽^→𝜐𝑔−^→𝜐𝑠
Solid-phase stress tensor
^→𝑇𝑠=−𝑃𝑠+𝜆𝑠∇⋅^→𝜐𝑠^→𝐼+𝜇𝑠∇^→𝜐𝑠+∇^→𝜐𝑠^𝑇−23∇⋅^→𝜐𝑠^→𝐼
Solid-phase pressure, Ogawa et al. [38]
𝑃𝑠^𝑘+𝑡=𝜌𝑠𝜀𝑠1+2(1+𝑒)𝜀𝑠𝑔𝑜𝜃
Particle-particle restitution coefficient, Jiradilok et al. [21]
𝑒=0.9
Radial distribution function, Ogawa et al. [38]
𝑔𝑜=𝜀1−^𝑠𝜀𝑠,max^1/3−1
Solid-phase bulk viscosity, Lun et al. [39]
𝜆𝑠=43𝜀^2𝑠𝜌𝑠𝑑𝑠𝑔𝑜(1+𝑒)𝜃𝜋
Solid-phase shear viscosity, Gidaspow [17]
𝜇𝑠=45𝜀𝑠𝜌𝑠𝑑𝑠𝑔𝑜()𝜃1+𝑒𝜋^1/2
+𝜀𝑠𝜌𝑠𝑑𝑠√𝜃𝜋6()23−𝑒1+5(1+𝑒)(3𝑒−1)𝜀𝑠𝑔𝑜
Gas-solid drag, Gidaspow [17]
𝜀𝛽=150^2𝑠𝜇𝑔𝜀𝑔𝑑^2𝑠+1.75^→𝜐𝑠−^→𝜐𝑔𝜀𝑠𝜌𝑔𝑑𝑠for𝜀𝑔≤0.8
3𝛽=4𝐶𝐷^→𝜐𝑠−^→𝜐𝑔𝜀𝑠𝜌𝑔𝑑𝑠𝜀𝑔^−2.65for𝜀𝑔>0.8
𝐶𝐷=24Re1+0.15Re^0.687forRe<1,000
𝐶𝐷=0.44forRe≥1,000
Energy balance
𝜕𝜀𝜕𝑡^𝑖𝜌𝑖𝐻𝑖+∇⋅𝜀𝑖𝜌𝑖^→𝜐𝑖𝐻𝑖=∇𝛼𝑖∇𝑇𝑖+ℎ𝑖𝑗𝑇𝑖−𝑇𝑗+𝑆𝑖
Heat exchange, Ranz and Marshall [40]
ℎ𝑖𝑗=6𝑘𝑔𝜀𝑔𝜀𝑠Nu𝑑𝑠^2
Nu=2+0.6Re𝑠^0.5Pr^0.333