Enzyme reactor design
Mahesh Bule
Levenspiel’s four fundamental
questions
• In approaching the design of a reactor system the engineer has to
answer a number of important priliminary questions before
embarking on his detailed calculations. These questions are as
follows:
1. Do I have the right reactor type in mind: should it be plug flow,
mixed flow, recycle, multistage or what?
2. What temperature progression should I aim for: constant, rising,
falling etc. and should that require heat exchange, may be
multistage?
3. For a catalytic reaction what size of particle should be used? This
tells what type of reactor should be used: packed bed, fluidized
etc?
4. Does the catalyst deactivate and if so, does it deactivate rapidly or
slowly?
Platform to simulate and optimize the
enzyme reactor operation
BR – Batch reactor,
BRP – Batch reactor with intermittent addition of enzyme
SBR – Semi-batch reactor,
MACR – mechanically agitated continuous reactor,
FXBR – Fixed bed batch reactor
Reactor Types
• Ideal
– PFR
– CSTR
• Real
– Unique design geometries and therefore RTD
– Multiphase
– Various regimes of momentum, mass and heat
transfer
Reactor Cost
• Reactor is
– PFR
• Pressure vessel
– CSTR
• Storage tank with mixer
• Pressure vessel
– Hydrostatic head gives the pressure to design for
Reactor Cost
• PFR
– Reactor Volume (various L and D) from reactor kinetics
– hoop-stress formula for wall thickness:
–
t
PR
 tc
SE  0.6 P
• t= vessel wall thickness, in.
• P= design pressure difference between inside and outside of
vessel, psig
• R= inside radius of steel vessel, in.
• S= maximum allowable stress for the steel.
• E= joint efficiency (≈0.9)
• tc=corrosion allowance = 0.125 in.
Reactor Cost
• Pressure Vessel
– Material of Construction gives ρmetal
– Mass of vessel = ρmetal (VC+2VHead)
• Vc = πDL
• VHead – from tables that are based upon D
– Heat capacity Cp= FMCv(W)
Reactors in Process Simulators
• Stoichiometric Model
– Specify reactant conversion and extents of
reaction for one or more reactions
• Two Models for multiple phases in chemical
equilibrium
• Kinetic model for a CSTR
Used in early stages of design
• Kinetic model for a PFR
• Custom-made models (UDF)
Mass Balance on Reactive System
• In - out + gen - cons = accumulation
FA0
Rate of flow in
FA
Rate of flow out
System
GA
Rate of
generation/
consumption
• A mass balance for the system is
dN A
FA0  FA  G A 
dt
• NA is the mass of “A” inside the system.
• The reaction term can be written in more familiar
terms,
GA = rA V
• V is volume of the system.
• Note that the units for this relation are consistent:
mass
mass

 volume
time volume  time
• If GA (and hence rA) varies with position in the
system volume, we can take this into account by
evaluating this term at several locations. Then
DGA1 = rA1 DV1,
• Summing the reactions over the entire volume
yields:
k
k
i 1
i 1
G A   DG Ai   rAi DVi
• As k   (that is, as we decrease the size of these cubes
and increase their number)
• DV  0 which gives
V
GA   rA dV
Generalized Design Equation for
Reactors
• In - out + gen - cons = accumulation
V
dN A
FA0  FA   rA dV 
dt
Types of Reactors
• Batch
– No flow of material in or out of reactor
– Changes with time
• Fed- Batch
– Either an inflow or an outflow of material but not both
– Changes with time
• Continuous
– Flow in and out of reactor
– Continuous Stirred Tank Reactor (CSTR)
– Plug Flow Reactor (PFR)
– Steady State Operation
Batch Reactor
• Generalized Design Equation for
Reactors V
dN A
FA 0  FA   rA dV 
dt
• No flow into or out of the reactor,
then, FA = FA0 = 0
V
dN A
  rA dV
dt
• Good mixing, constant volume
dN A
 rAV
dt
or
d  N A V  dC A

 rA
dt
dt
Enzyme Batch Reactor
(constant volume, well mixed)
vmax S
dS
r

dt K M  S
• integrate from t = 0 to t = t, we obtain
Kmln (S0/S) + (S0 -S) = vmax t
• Batch reactors are often used in the early stage of
development due to their ease of operation and
analysis
Batch Enzyme Reactor
Determination of
M-M kinetic
parameters
Linear form becomes
(S0 – S)
ln(S0/S)
= - KM +
Vmax
t
ln(S0/S)
(S0 – S)
ln(S0/S)
Vmax
- KM
t
ln(S0/S)
Fed Batch Reactor
• Reactor Design Equation
V
FA0  FA  
dN A
rA dV 
dt
• No outflow FA = 0
• Good Mixing rA dV term
out of the integral
dN A d C A V 
FA0  rA  V 

dt
dt
Fed Batch Continued
• Convert the mass (NA) to concentration. Applying
integration by parts yields
• Since
• Then
dC A
dV
FA0  rAV  V
 CA
dt
dt
dV
 FA0
dt
dC A
FA0  rAV  V
 C A FA0
dt
• Rearranging
C A FA0
dC A FA0

 rA 
dt
V
V
Fed Batch Continued
• Or
dC A FA0
1  C A   rA

dt
V
• Used when there is substrate inhibition and
for bioreactors with cells.
Assumptions for a fed batch reactor
include
1. Only a feed in
2. Either a feed in or a
removal stream
3. Steady state
4. 2 and 3
5. All of the above
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Continuous Stirred Tank Reactor
• Assume rate of flow in = rate of flow out
• FA = v CA and FA0 = v CA0
• v = volumetric flow rate (volume/time)
CSTR - continued
• General Reactor Design Equation
V
FA0  FA  
• Assume Steady State
• Well Mixed
• So
V

dN A
rA dV 
dt
dN A
0
dt
rA dV  VrA
FA0  FA  VrA  0
or
FA0  FA
V
 rA
CSTR for Enzymes
(Enzyme remains inside)
• Input - output + generation - consump = accumulation
•
•
•
•
•
dS
FS0  FS  rV  v
dt
F - flow rate l/hr
S - substrate conc.
V- reactor volume
r - reaction rate
at Steady State dS/dt = 0
CSTR - enzymes
rV = F(S0 - S)
and
vmax S
r
KM  S
Introducing space-time θ ( = V/F) and r in above equation we get
S0
=S
VmaxS
+
KM + S
θ
Continuous Stirred Tank Enzyme Reactor at steady-state
Linear form becomes
CS
= - KM +
Vmax
Sθ
(42)
(S0 – S)
S
Vmax
- KM
Determination of
M-M kinetic
parameters
Sθ
(S0-S)
Plug Flow Reactor (PFR)
• Tubular Reactor
• Pipe through which fluid flows and reacts.
• Poor mixing
• Difficult to control temperature variations.
• An advantage is the simplicity of construction.
PFR Design Equation for Product
formation
• Design Equation
V
FA0  FA  
dN A
rA dV 
dt
• Examine a small volume element (DV) with length
Dy and the same radius as the entire pipe.
Flow of
A into
Element
Flow of
A out of
Element
• If the element is small, then spatial variations in rA
are negligible, and
Assumption of “good
V

rA dV  rA DV
mixing” applies only to
the small volume
element
• If volume element is very small, then assume steady
state with no changes in the concentration of A.
dN A
0
dt
• Simplify design equation to:
FA  y   FA  y  Dy   rADV  0
• rA is a function of position y, down the length of the
pipe and reactant concentration
• The volume of an element is the product of the
length and cross-sectional area,
DV = A Dy
• Design Equation becomes:
 FA  y  Dy   FA  y 
 ArA


Dy


• take the limit where the size of a volume
element becomes infinitesimally small
dFA
 ArA
lim
Dy 0 dy
• or because Dy A = V,
dFA
 rA
dV
• This is the Design Equation for a PFR
• Bioapplications - Sometimes hollow fiber
reactor analysis is simplified to a PFR
Plug-flow Enzyme Reactor at steady-state
F
S0
F
S
F
S+dS
F
Sf
dV
Mass balance for the substrate over dV:
FS = F(S + dS) + (-rS) dV
The above can be simplified to - FdS / dV = -rS
F for the steady flow rate through the reactor
S for concentration of the substrate
dV for small volume of the reacting mixture
(-rS) for substrate utilization rate in dV
Plug-flow Enzyme Reactor at steady-state
F
F
S
S0
F
S+dS
F
Sf
dV
Introducing space-time θ ( = V/F), we get
- dS / dθ = -rS
F for the steady flow rate through the reactor
S for concentration of the substrate
dV for small volume of the reacting mixture
(-rS) for substrate utilization rate in dV
Plug-flow Enzyme Reactor at steady-state
Substituting (-rS) for the simple enzyme reaction in, we get
-
VmaxS
dS
dθ
=
KM + S
Rearranging above equation we get
S
∫(
-
S0
KM + S
S
θ
) ∫
dS
=
Vmax dθ
0
Integrating above gives
()
KM ln
S0
S
+ (CS0 – CS)
= Vmax θ
Plug flow enzyme reactor
Determination of
M-M kinetic
parameters
Linear form becomes
(S0 – S)
ln(S0/S)
= - KM +
Vmax
θ
ln(S0/S)
(S0 – S)
ln(S0/S)
Vmax
- KM
θ
ln(S0/S)
Immobilized enzyme reactor (example)
Recycle packed column reactor
Advantages of immobilized enzymes:
- Easy separation from reaction mixture, providing the ability to control reaction
times and minimize the enzymes lost in the product
- Re-use of enzymes for many reaction cycles, lowering the total production
cost of enzyme mediated reactions
- Ability of enzymes to provide pure products
- Possible provision of a better environment for enzyme activity
Disadvantages of immobilized enzymes:
- Problem in diffusional mass transfer
- Enzyme leakage into solution
- Reduced enzyme activity and stability
- Lack of controls on micro environmental conditions
Methods of immobilization
1) Entrapment Immobilization
2) Surface Immobilization
3) Cross-linking
1) Entrapment Immobilization
It is the physical enclosure of enzymes in a small space.
-
Matrix entrapment (matrices used are polysaccharides, proteins,
polymeric materials, activated carbon, porous ceramic and so on)
-
Membrane entrapment (microcapsulation or trapped between thin, semipermeable membranes)
1) Entrapment Immobilization
Advantage is enzyme is retained.
Disadvantages are
- substrate need to diffuse in to access enzyme and
product need to diffuse out
- reduced enzyme activity and enzyme stability owing to the lack of control of
micro environmental conditions
2) Surface Immobilization
-
Physical adsorption (Carriers are silica, carbon nanotube,
cellulose, and so on; easily desorbed; simple and cheap;
enzyme activity unaffected )
-
Ionic binding (Carriers are polysaccharides and synthetic
polymers having ion-exchange centers)
-
Covalent binding (Carriers are polymers containing amino,
carboxyl, hydroxyl, or phenolic groups; loss of enzyme activity;
strong binding of enzymes)
Methods of immobilization
3) Cross linking
is to cross link enzyme molecules with each other using agents such
as glutaraldehyde.
Comparison between the methods
Adsorption
Covalent
coupling
Entrapment
Membrane
confinement
Simple
Difficult
Difficult
Simple
Low
High
Moderate
High
Variable
Strong
Weak
Strong
Yes
No
Yes
No
Applicability
Wide
Selective
Wide
Very wide
Running problems
High
Low
High
High
Matrix effects
Yes
Yes
Yes
No
Large diffusional
barriers
No
No
Yes
Yes
Microbial protection
No
No
Yes
Yes
Characteristics
Preparation
Cost
Binding force
Enzyme leakage
Immobilized enzyme reactor (example)
Recycle packed column reactor
- Allow the reactor to operate
at high fluid velocities
Immobilized enzyme reactor (example)
Fluidized bed reactor
- A high viscosity substrate solution
- A gaseous substrate or product in a
continuous reaction system
- Care must be taken to avoid the
destruction and decomposition of
immobilized enzymes
Immobilized enzyme reactor (example)
- An immobilized enzyme tends to
decompose upon physical stirring.
- The batch system is generally suitable for
the production of rather small amounts of
chemicals.
Continuous stirred
tank reactor
Effect of mass-transfer resistance in immobilized
enzyme systems:
Mass transfer resistance is present
- due to the large particle size of the immobilized enzymes
- due to the inclusion of enzymes in polymeric matrix
Effect of mass-transfer resistance in immobilized
enzyme systems:
Mass transfer resistance are divided into the following:
- External mass transfer resistance (during transfer of substrate
from the bulk liquid to the relatively unmixed liquid film
surrounding the immobilized enzyme and during diffusion
through the relatively unmixed liquid film)
- Intra-particle mass transfer resistance (during diffusion from
the surface of the particle to the active site of the enzyme in an
inert support)
External mass-transfer resistance:
Assumption:
- Enzymes are evenly distributed on the
surface of a nonporous support material.
- All enzyme molecules are equally
active.
- Substrate diffuses through a thin liquid
film surrounding the support surface to
reach the reactive surface.
- The process of immobilization has not
altered the enzyme structure and the MM kinetic parameters (rmax, KM) are
unaltered.
CSsSs
CSbSb
Enzyme
Liquid Liquid
Film Thickness,
L
film thickness,
L
External mass-transfer resistance:
Diffusional mass transfer across the liquid film:
CSsSs
CSbSb
JS = kL (CSb – CSs)
kL
liquid mass transfer
coefficient (cm/s)
CSb
substrate concentration in the bulk
solution (mol/cm3)
CSs
substrate concentration at the
immobilized enzyme surface (mol/cm3)
Enzyme
Liquid Liquid
Film Thickness,
L
film thickness,
L
External mass-transfer resistance:
At steady state, the reaction rate is equal to the
mass-transfer rate:
CSsSs
CSbSb
JS = kL (CSb – CSs)
Vmax CSs
=
KM + CSs
Vmax
maximum reaction rate per unit of
external surface area (e.g. mol/cm2.s)
KM
is the M-M kinetic constant (e.g.
mol/cm3)
Enzyme
Liquid Liquid
Film Thickness,
L
film thickness,
L
External mass-transfer resistance:
Vmax CSs
JS = kL (CSb – CSs)
=
KM + CSs
Non dimensionalizing the above equation, we get
1 - C’Ss
β C’Ss
=
NDa
1 + β C’Ss
where
C’Ss
=
CSs / CSb
NDa
=
Vmax / (kL CSb )
β
=
CSb / KM
is the Damköhler number
is the dimensionless substrate concentration
Damköhler number (NDa)
NDa =
Maximum rate of reaction
Vmax
=
Maximum rate of diffusion
If NDa >> 1, rate of diffusion is slow and therefore the limiting
mechanism
rp = JS = kL (CSb – CSs)
If NDa << 1, rate of reaction is slow and therefore the limiting
mechanism
rp =
Vmax CSs
KM + CSs
If NDa = 1, rates of diffusion and reaction are comparable.
kL CSb
Effectiveness factor (η)
actual reaction rate
η=
rate if not slowed by diffusion
η=
rmax CSs
β C’Ss
KM + CSs
1 + β C’Ss
rmax CSb
=
KM + CSb
Effectiveness factor is a function of β and C’Ss
β
1+β
Internal mass transfer resistance:
Assumption:
- Enzyme are uniformly distributed in
spherical support particle.
- Substrate diffuses through the
tortuous pathway among pores to
reach the enzyme
- Substrate reacts with enzyme on the
pore surface
-Diffusion and reaction are
simultaneous
- Reaction kinetics are M-M kinetics
CSs
CSr2
Diffusion effects in enzymes immobilized in a
porous matrix:
Under internal diffusion limitations, the rate per unit volume is
expressed in terms of the effectiveness factor as follows:
Vmax’ CSs
rS =
Vmax’
KM
CSs
η
η
KM + CSs
maximum reaction rate per volume of the support
M-M constant
substrate concentration on the surface of the support
effectiveness factor
Diffusion effects in enzymes immobilized in a
porous matrix:
Definition of the effectiveness factor
η=
η
reaction rate with intra-particle diffusion limitation
reaction rate without diffusion limitation
For η < 1, the conversion is diffusion limited
For η = 1, the conversion is limited by the reaction rate
Effectiveness factor is a function of β and C’Ss
Diffusion effects in enzymes immobilized in a
porous matrix:
β
η
φ
Theoretical relationship between the effectiveness factor (η) and firstorder Thiele’s modulus (φ) for a spherical porous immobilized particle for
various values of β, where β is the substrate concentration at the surface
divided by M-M constant.
Diffusion effects in enzymes immobilized in a
porous matrix:
Relationship of
effectiveness factor (η)
with the size of
immobilized enzyme
particle and enzyme
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