The
substrate concentration gradient in a homogeneous bead with cells or
enzyme consuming the substrate can be described with a second order
differential equation that, after conversion to dimensionless
form, can be solved numerically.
This
SIMSPEC file calculates the dimensionless relative susbtrate
concentration in a homogeneous bead with a substrate consuming
biocatalyst, assuming Monod kinetics. The calculation starts with an
arbitrary concentration (s)
at the bead center (r = 0) and steps gradually towards r=1. If s ≠ 1 when r = 1 is reached, the calculation starts again with a
larger/smaller
s at the center, and this procedure is repeated until s = 1 at r = 1 (see left hand graph). Then all incorrect gradients in the graph are eliminated (right hand graph), and the
effectiveness factor is
calculated. The effectiveness factor can be used to calculate the total reaction rate of the
bead and from the bead concentration in the reactor the total reactor reaction rate can be obtained.
The
theory
of substrate diffusion limitation and its effect of the effectiveness
factor (diffusion limited/ nonlimited reaction rate) is included in
Fermentation Process
Engineering, 2nd ed. (will become available for downloading from
www.enfors.eu).
