CFD Study of Industrial FCC Risers: The Effect of Outlet Configurations on Hydrodynamics and Reactions

Table 4

Summary of Lagrangian equations.

Lagrangian discrete phase model
Discrete phase trajectory:	𝑑𝑥𝑑𝑑𝑡=𝑢𝑑	(A)
Discrete phase velocity:	𝑑𝑢𝑑=𝐠𝜌𝑑𝑡^𝑑−𝜌𝑔𝜌𝑑+𝐹𝐷𝑢𝑔−𝑢𝑑	(B)
Drag force:	𝐹𝐷=18𝜇𝑔𝜌𝑑𝑑^2𝑑𝐶^𝑑𝐷Re𝑑24	(C)
Droplet Reynolds number:	Re𝑑=𝜌𝑔𝑑𝑑𝑢𝑑−𝑢𝑔𝜇𝑔	(D)
Drag coefficient:	𝐶^𝑑𝐷=𝑎1+𝑎2+𝑎ReRe^32	(E)
Heat balances:
𝑇𝑑<𝑇vap:	𝑚𝑑𝐶𝑝𝑑𝑇𝑑𝑑𝑡=ℎ𝐴𝑑(𝑇∞−𝑇𝑑)	(F)
𝑇vap≤𝑇𝑑<𝑇bp:	𝑚𝑑𝐶𝑝𝑑𝑇𝑑𝑑𝑡=ℎ𝐴𝑑(𝑇∞−𝑇𝑑)+𝑑𝑚𝑑𝐿𝑑𝑡	(G)
Heat transfer coefficient:	𝜆ℎ=𝑑𝑑(2.0+0.6Re𝑑^1/2Pr^1/3)	(H)
Mass transfer:	𝑑𝑚𝑑𝑑𝑡=−𝑘𝑐(𝐶𝑖,surf−𝐶𝑖,∞)𝐴𝑑𝑀𝑤,𝑖	(I)
Mass transfer coefficient:	𝑘𝑐=𝐷𝑚,𝑖𝑑𝑑(2.0+0.6Re𝑑^1/2𝑆𝑐^1/3)	(J)
Droplet diameter variation:	−𝑑(𝑑𝑑)=𝑑𝑡4𝑘∞(1+0.23Re𝑑)𝜌𝑑𝐶𝑝,∞𝑑𝑑𝑇ln1+^∞−𝑇𝑑𝐿	(K)