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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.