with vwp the slip-velocity between water and gel bead, and vwo the slip-velocity between
water and organic phase. The droplet diameter is calculated with the equation of Kumar
and Hartland
with , and
The rise velocity of a single droplet is calculated with Vignes’ equation:
with
The mass transfer coefficient is calculated with the equation of Newman (1931).t represents the contact time and is equal to: t=Htε 0 /Ud
The mass balances are made dimensionless by dividing with the initial substrate
concentration, the bead diameter, and with DS/R^2. The resulting dimensionless parameters
are: dimensionless length inside the bead z=r/R, dimensionless concentration ,
and a dimensionless time Fo=Dst/R^2.
The resulting mass balances with the implementation of the transfer flux relations are:
Multiphase bioreactor design 384