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Idealize Bioreactor CSTR vs. PFR .................................................................................. 3Analysis of a simple continuous stirred tank bioreactor ..................................................... 4

Residence time distribution................................................................................................. 4

F curve: ........................................................................................................................... 4

C curve: ........................................................................................................................... 4Residence time distribution or age distribution .............................................................. 4Residence time distribution and reaction kinetics .............................................................. 5

Well mixed continuous stirred tank reactor ........................................................................ 6

Plug Flow Bioreactor .......................................................................................................... 7

Example: ..................................................................................................................... 8

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Idealize Bioreactor CSTR vs. PFRWell mixed CSTR

Exit concentration the same as in reactor.

Instantaneous mixing upon addition of tracer to the reactor.

Plug flow reactor (PRFR)

Piston flow No back mixing No dispersion due to molecular diffusion

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Analysis of a simple continuous stirred tank bioreactor

Residence time distribution

F curve:

no traces t

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fraction of exit stream of age between t and t+dt is Edt

=1

1

0

1t

t

EdtEdt

In a closed vessel (no back mixing in the entrance and exit), C curve and E curve are

identical. Assume the flow into the reactor at 0t , red fluid and only red fluid in the exit stream has anage shorter then t.

(fraction of red fluid in the exit stream)=(fraction of exit steam younger then age t)

0

t

f Edt =

The mean residence time is

=0

tEdtt

Residence time distribution and reaction kinetics

In a batch reactor, the rate of reaction with a reactant concentration

dcr

dt=

If the concentration at time 0 is C0 at any given time

0 0

C t

Cc dc rdt = =

In a reactor, an element of fluid which has a residence time t will thus have a reactant

concentration of c . If the age distribution of that element is E, the distribution function

of the reactant concentration in that element of fluid is

C Edt

By integrating all elements through the age distribution curve one gets

0

( )oC C Exp kt Edt

=

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For first order reaction

dckc

dt=

( )oC C Exp kt =

The reactant concentration at the outlet of the reactor is

0

( )oC C Exp kt Edt

=

It can be see that the E function is multiplied by the expression of c for a batch reactor in

the integration over all age distribution.

Well mixed continuous stirred tank reactor

outin FFdt

dV=

in out if F F F = = 0=dt

dV

( )in out

d VcFc Fc

dt=

V const, well mixed

inFcdc Fc

dt V V =

0inc =

dc Fc

dt V=

dc Fdt

c V=

Initial conditions: 0c c= 0=t

Solve:0

lnc Ft

c V=

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time const:F

V

whenF

Vt =

0

ln 1c

c=

0

0.368c

c=

concentration decreased by 63%

When the reaction with a reaction rate r occurs in the reactor:

inFcdc Fc r

dt V V

= + r = reaction rate/volume

Plug Flow Bioreactor

A

FU =

( )| |z z z

cz A zUAc UAc A zr

t+

= +

Assume constant area and express the equation in a differential form,

( )c cUrt z

= +

At steady state, the concentration changes with position, but not time. We assume that

the velocity of the liquid flow is constant. Then

cU r

z

=

If we definez

EU

= and substitute into the above equation then,

c cU rz t

= =

It is clear that the equation looks just like that for a batch reactor.

In a bioreactor for cell growth the reaction that we are interested in is basically that for

cell growth and for substrate consumption.

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We can express the growth rate dependence on substrate concentration in a Mono-kinetic

equation, then the equations become

1m

s

xsdx

dz K s U

=

+

max1 1

( )s

xsds

dz Y K s U

=

+

Let the initial conditions be z=0, x=x0, s=s0, the concentration profiles of cell and

substrate can be solved just like a batch reactor and are shown in Figure ________.

Example:

Aerobic microorganism growing in a plug flow bioreactor. The concentration of oxygen

and cells at the inlet are: 0 0.2mmoles

l= and 0 1

gx

l= . The maximum specific

growth rate is 1 hr-1 . The consumption of oxygen can be assume to follow Monod

kinetics with 0.01s mmoleK l= and the yield coefficient of biomass based on oxygen is

02

1

30 0x

gcellsY

mmole= . The cross sectional area of the reactor is 2100cmA = .

What is the maximum reactor length that can be used for the flow rate (F) ofmin

1 l and

min10 l ?

Assumptions:

(i) const volume

(ii) well mixed

(iii) oxygen is the e growth-limiting nutrient

It is clear that oxygen concentration will be used quickly depleted in a plug flow reactor

used for cell cultivation. Therefore, plug flow reactor is rarely used for large scale

operation of cell cultivation. Rather, a stirred tank or other mixing vessel type of reactors

with a continuous supply of oxygen from the gas phase by air sparging, are frequentlyused. Plug flow reactor is more often used in enzymatic reactions and in bioseparations.

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They are also used in applications which are relatively small in scale, such as in tissueengineering applications.