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

Table 3

Semiempirical CFB mathematical model.

	Riser
	Height (H), Kunii and Levenspiel [30]
	𝜀𝑔,𝑑−𝜀𝑔,𝐻𝜀𝑔,𝑑−𝜀𝑔,𝑎=exp−𝑎𝐻−ℎ𝑖
	Decay constant, Kunii and Levenspiel [31]
	𝑑𝑎=4.1011ln𝑠⋅10^6−15.181𝑈
	Choking voidage, Yang [32]
	𝑈ch𝜀𝑔,ch=𝑈𝑡,ns+2𝑔𝐷𝜀^−4.7𝑔,ch−1𝜌𝑠^2.26.81×10^5𝜌𝑔^2.2
	Solids circulation flux, Yang [32]
	𝐺𝑠=𝑈ch−𝑈𝑡1−𝜀𝑔,ch𝜌𝑠
	Terminal velocity of particles, Basu [2]
	𝑈𝑡,ns=𝐾𝑡,ns𝑈𝑡
	where 𝑈𝑡 and 𝐾𝑡,ns are given by, Basu [2]
	𝑑𝑠𝑈𝑡𝜌𝑔𝜇=Ar18ifRe<0.4
	𝑑𝑠𝑈𝑡𝜌𝑔𝜇=Ar7.5^0.666if0.4≤Re≤500
	𝑑𝑠𝑈𝑡𝜌𝑔𝜇=Ar0.33^0.5if500<Re
	𝐾𝑡,ns=0.843log10𝜙𝑠𝑑0.065ifRe=^𝑠𝑈𝑡𝜌𝑔𝜇<0.2
	𝐾𝑡,ns=4.89𝜌𝑠−𝜌𝑔𝑔𝑑𝑠3𝜌𝑔5.31−4.88𝜙𝑠^0.5𝑑ifRe=^𝑠𝑈𝑡𝜌𝑔𝜇>1000
	Superficial gas velocity, Perales et al. [33]
	𝑈tr=𝑈ch𝜇=𝑈=1.45𝜌𝑔𝑑𝑠Ar^0.484
	Axial voidage fraction, Davidson [5]
	1−𝜀𝑔,𝐻𝜌𝑠=𝐸𝑈+𝑤𝑈𝑓
	Internal solids reflux, Davidson [5]
	𝐺𝑠𝑅=𝐸−^𝑠𝐸
	Solids inventory, Behainne and Martins [34]
	𝐼𝑠,𝑟=𝜋𝐷^2𝜌𝑠4𝜀𝑔,𝐻−𝜀𝑔,𝑎𝑎+𝐻1−𝜀𝑔,𝑎
	−𝐻−ℎ𝑖𝜀𝑔,ch−𝜀𝑔,𝑎
	Pressure drop, Behainne and Martins [34]
	Δ𝑃𝑟=4𝑔𝐼𝑠,𝑟𝜋𝐷^2+𝐺^2𝑠𝜌𝑠1−𝜀𝑔,𝐻
	Cyclone
	Swift geometrical configuration, Basu [2]
	𝐴=0.44𝐷𝑐;𝐶=0.21𝐷𝑐;𝑀=0.4𝐷𝑐;𝐹=0.5𝐷𝑐;𝑆=1.4𝐷𝑐;
	𝐵=3.9𝐷𝑐;𝑁=0.4𝐷𝑐
	Pressure drop, Muschelknautz and Greif [35]
	Δ𝑃𝑐=𝑓𝑤𝐴𝑅𝑉𝑏𝜌𝑔2𝑢𝑎𝑢𝑜^1.5+𝑢2+3^𝑜𝑣𝑜^4/3+𝑢𝑜𝑣𝑜^2𝜌𝑔2𝑣^2𝑜
	Wall friction coefficient, Basu [2]
	𝑓𝑤=𝑓01+2𝐶𝑒
	Tangential gas velocity, Muschelknautz and Greif [35]
	𝑢𝑜=𝑢𝑎𝑟𝑜𝑟𝑜+((𝑓𝑤/2)(4𝐴𝑅/(𝜋𝐷^2𝑈))𝑢𝑎√𝑟𝑎𝑟𝑜
	Gas exit tube radius, Muschelknautz and Greif [35]
	𝑢𝑎=𝑢𝑒𝑟𝑒𝑟𝑎𝛼1
	Parameters of cyclone design, Muschelknautz and Greif [35]
	𝛼1=1𝛽11−𝛽1−4^12−𝛽12^21−1−𝛽1^21+𝐶𝑒2𝛽1−𝛽1^2
	𝛽1=𝐶𝑟𝑎
	Velocity at exit
	𝑣𝑜=𝑄𝜋𝑟^20
	Standpipe/solids recycle valve
	Height above the aeration point, Knowlton [36]
	𝐿𝑣,lv=Δ𝑃lv+Δ𝑃𝑟,sr+Δ𝑃𝑐(Δ𝑃/𝐿)mf,sp
	Pressure drop in the vertical leg of the L-valve, Knowlton [36]
	Δ𝑃𝐿mf,sp=𝜌𝑠𝑔1−𝜀𝑔,mf
	Pressure drop in the L-valve, Geldart and Jones [37]
	Δ𝑃lv𝐺=216^0.17𝑠,lv𝐷^0.63lv𝑑𝑠^0.15
	Pressure drop in the riser above the solids return level,
	Behainne and Martins [34]
	Δ𝑃𝑟,sr=Δ𝑃𝑟−ℎsr1−𝜀𝑔,𝑎𝑔𝜌𝑠
	Aeration mass flow rate, Geldart and Jones [37]
	𝑚lv=𝑚mf𝐺𝑠,lv/𝐷lv+2,9653,354
	Solids inventory, Behainne and Martins [34]
	𝐼𝑠,sp-lv=𝐿𝑣,lv+𝐿ℎ,lv1−𝜀𝑔,mf𝜋𝐷^2lv𝜌𝑠4