Biomedical Engineering Reference
In-Depth Information
3.1
Mass Balance in Immobilized Enzyme Particle
The effectiveness factor in Eqs.(1)-(3) is defined as
v
obs
h
=
(6)
7
v
kin
v
kin
represents the reaction rate in the kinetic region,it means the rate that would
be attained if the substrate concentration in the whole particle were the same as
that in the bulk phase, c
Sb
. In order to calculate the observed reaction rate, v
obs
,
the particle mass balance equations have to be solved:
D
s
d dc
s
64
r
n
= v(c
s
,c
P
,
P
)
(7)
6
r
n
dr dr
D
P
d dc
P
64
r
n
= -v(c
s
,c
P
,
P
)
(8)
6
r
n
dr dr
while required boundary conditions are
dc
S
dc
P
r = 0:
=
= 0
(9)
6
6
dr dr
r = R: c
S
= c
Sb
c
P
= c
Pb
.
(10)
The kinetic term,v,is a function of the kinetic parameters: vector
P
and the par-
ticle substrate and product concentrations,c
S
and c
P
,respectively.D
S
and D
P
are
the corresponding effective diffusion coefficients and r is the particle coordinate
(in the case of spherical geometry it is the radial distance).Parameter n depends
on the geometry of the biocatalyst particle and is 0,1,2 for a plate,a cylinder and
a sphere, respectively. Since concentrations on the particle surface are assumed
to be identical with bulk concentrations, boundary conditions do not include
the influence of external mass transfer.Solving the above differential equations,
the observed reaction rate in the packed bed is evaluated from the rate of sub-
strate flux to the particle or of product flux from the particle
A
◊
D
S
dc
S
A
◊
D
P
dc
P
6
6
v
obs
=
= -
(11)
631
631
V
L
dr
r=R
V
L
dr
r=R
where A is the total external surface area of the particle and V
L
is the volume of
the bulk phase in the packed bed. For spherical geometry, Eq. (11) can be sim-
plified to
(1-
e
) dc
S
(1-
e
)dc
P
6
6
v
obs
=
= -
(12)
63
63
e
R dr
r=R
e
R
dr
r=R
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