Hydrodynamics of a Novel Design Circulating Fluidized Bed Steam Reformer Operating in the Dense Suspension Upflow Regime

Table 3

Summary of the used Eulerian-Eulerian model.

Conservation equations
Continuity
𝜕𝛼𝜕𝑡^𝑞𝜌𝑞+∇⋅𝛼𝑞𝜌𝑞𝑣𝑞=0
𝛼𝑔+𝛼𝑠=1
Species equation:  𝜕𝛼𝜕𝑡^𝑞𝜌𝑞𝑥^𝑖𝑞+∇⋅𝛼𝑞𝜌𝑞𝑣𝑞𝑥^𝑖𝑞=∇⋅𝛼𝑞𝜌𝑞𝐷^𝑖𝑘∇𝑥^𝑖𝑞+𝑆𝑞
Momentum
𝜕𝛼𝜕𝑡^𝑞𝜌𝑞𝑣𝑞+∇⋅𝛼𝑞𝜌𝑞𝑣−𝑣𝑞=∇⋅𝜏𝑞−𝛼𝑞∇𝑝+𝛼𝑞𝜌𝑞𝑔+𝐹𝑞
For gas phase
𝐹𝑞=K𝑔𝑠𝑣𝑠−𝑣𝑔
𝜏𝑔=𝛼𝑔𝜇𝑔,eff∇𝑣𝑔+∇𝑣^𝑇𝑔+𝛼𝑔𝜆𝑔−23𝜇𝑔∇⋅𝑣𝑔𝐼
𝜇𝑔,eff=𝜇𝑔+𝜇𝑔,𝑡
For solid phase
𝐹𝑞=K𝑠𝑔𝑣𝑔−𝑣𝑠
𝜏𝑠=−𝑃𝑠𝐼+𝛼𝑠𝜇𝑠∇𝑣𝑠+∇𝑣^𝑇𝑠+𝛼𝑠𝜆𝑠−23𝜇𝑠∇⋅𝑣𝑠𝐼
𝜆𝑠=43𝛼𝑠𝜌𝑠𝑑𝑠𝑔0,𝑠𝑠1+𝑒𝑠𝑠Θ𝑠𝜋^1/2
Energy
𝜕𝛼𝜕𝑡^𝑞𝜌𝑞ℎ𝑞+∇⋅𝛼𝑞𝜌𝑞𝑣𝑞ℎ𝑞=−𝛼𝑞𝜕𝑝𝑞+𝜕𝑡𝜏𝑞∶∇𝑣−𝑞−∇⋅𝑞𝑞+𝑄𝑃𝑄+𝑆𝑞
𝑄𝑔𝑠=6𝑘𝑔𝛼𝑔𝛼𝑠𝑁𝑢𝑔𝑑^2𝑠𝑇𝑔−𝑇𝑠
Kinetic fluctuation energy (granular temperature)
32𝜕𝜌𝜕𝑡^𝑠𝛼𝑠Θ𝑠+∇⋅𝜌𝑠𝛼𝑠𝑣𝑠Θ𝑠=−𝑃𝑠𝐼+𝜏𝑠∶∇𝑣𝑠+∇⋅𝑘Θ𝑠∇Θ𝑠−𝛾Θ𝑠+𝜙𝑔𝑠