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U ̅_a=U_hexp[-0.038Z_c/SW+0.110] (3)
This value of average velocity, together with the height of contacting zone, is used directly for calculating gas residence time.
It is clear that the atomized liquid does not travel at the same velocity as the gas, and thus a ‘slip ratio’ is needed for mass transfer coefficient determinations. This ratio can be determined from
SR=∅_L V_c/qt_g (4)
where the value of liquid holdup ∅_L is to be determined. For the type of cocurrent contactor at hand, reference may be made to liquid holdup studies for gas-liquid mixtures owing vertically and cocurrently in large pipes. An excellent review of this area has been provided by Spedding and Chen, 19865 , who found that the classical work of Lockhart and Martinelli, 19496 is appropriate for vertical as well as horizontal two-phase flow. This confirms earlier work by one of the present authors (Fair, 1960)7 for the case of vertical thermosiphon reboilers where fractional liquid holdup is an important design parameter. The basic Martinelli flow parameter is
X_tt=〖(W_L/W_g)〗^0.9 〖(ρ_g/ρ_L)〗^0.5 〖(μ_L/μ_g)〗^0.1 (5)
The resemblance of this parameter to the flow parameter for determining flooding in tray columns is evident.
The liquid residence time on the tray is determined from the slip ratio and the gas residence time:
t_L=(SR)t_g (6)
The volumetric liquid holdup in the contacting zone is determined from a fit to the Martinelli plot:
∅_L=βX_tt^(2/3)/(3.5+X_tt^(2/3)) (7)
where the term b may be considered an adjustable parameter based on method of dispersion of the liquid into the flowing gas. For ‘perfect’ dispersion, β= 1.0. For dispersions involving liquid recirculation, as in the case of the concurrent tray, the value of β is greater than unity.
The effective interfacial area is based on free drops and is determined directly from the liquid holdup and the Sauter mean drop diameter:
a_e=6∅_L/D_32 (8)
The actual area should include that provided by wetted surfaces in the collector zone, but at present this is handled by the use of the adjustable parameter β.
The slip velocity is based on the average velocity of gas in the contacting zone and the slip ratio:
U_SL=U ̅_a(1-1/SR) (9)
A Reynolds number for the gas . lm is based on the slip velocity and the average drop size:
Re_g=D_32 U_SL ρ_g1000/μ_g (10)
Finally, the gas phase mass transfer coefficient is based on the early Froessling work with drops and bubbles, and in one form is represented as (Sherwood et al., 1975):
k_cg=D_g/D_32 [2+0.6(Re_g^0.5 Sc_g^0.333 )] (11)
Conventional mass transfer theory leads to the number of gas phase transfer units:
N_g=k_c a_e t_g (12)
The liquid phase mass transfer coefficient is based on the penetration model of Higbie, 19359 :
k_cL=2[D_L U_SL/(πD_32 )]0.5 (13)
This relationship is in general agreement with the work of Hughes and Gilliland, 195510 , who studied the rate of mass transfer inside liquid drops contacted by gas streams. The number of liquid transfer units is obtained from
N_L=k_cL a_e t_L (14)
Finally,
Nog = 1/[1/Ng +l/NL] (15)
where l is the ‘stripping factor,’ m/(L/V).On the basis of mass transfer theory, The overall efficiency for the cocurrent tray follows from equation (15):
Eog = 1 2 exp[2 Nog] (16)
This efficiency may be taken as the overall (Murphree) tray efficiency, since crossflow mixing effects are unlikely to be present. The efficiency may be corrected for entrainment as proposed by Colburn (and modified by one of the authors(Fair, 1963)11):