U-tube bubble reactor—M. Fig. P10.14 shows a bubble reactor of height H, in which volumetric flow rates G1 and G2 of gas are injected at the top of the downcomer and at the base of the riser,...

U-tube bubble reactor—M. Fig. P10.14 shows a bubble reactor of height H, in which volumetric flow rates G1 and G2 of gas are injected at the top of the downcomer and at the base of the riser, respectively. The cross-sectional areas, void fractions, and absolute gas-bubble velocities are denoted by A, ε, and uc, with subscripts 1 and 2 referring to the downcomer and riser, respectively. The volumetric liquid circulation rate is L. All injected bubbles leave at the top of the riser. Prove that the void fraction in the downcomer is: ε1 = G1 G1 + L − ubA1 , in which ub is the rise velocity of a discrete swarm of bubbles relative to stagnant liquid. Obtain the corresponding expression for ε2 in the riser.

The frictional head loss per unit depth is found experimentally to be c times the square of the corresponding superficial liquid velocity, where c = 0.0452 s2/m2. If A1 = 0.02, A2 = 0.1 m2, G1 = 0.01, G2 = 0.05 m3/s, and ub = 0.25 m/s, calculate ε1, ε2, and L (m3/s).