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Chapter 11 Non-ideal Flow and Behavior of RealBioreactors

So far we have restricted our discussion to two ideal flow patterns, well-mixed and plug flow. Although real bioreactors never fully follow these flow patterns, a large number of designs and operations approximate these ideals with negligible error. However, some cases involving in biological reaction systems provide the flow patterns which are obviously different from these two ideal conditions. For example, fermenters in cascade which are widely used for ethanol fermentation process; when bioreactors are scaled up, some stagnant zones may be developed, especially for high- viscous non-Newtonian biofluids.

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Furthermore, bypassing or short-circuiting flows also inevitably exist inside continuously operated bioreactors. The theory of nonidealflow in bioreacter engineering mainly focuses on describing all of these phenomena mathematically.

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11.1 Residence Time Distribution Function

In an ideal plug-flow or batch reactor, all the elements of material leaving the reactors have been inside them for exactly the same amount of time. The time reaction elements has spent in a reactor is called residence time.

The ideal plug flow and batch reactors are the only two classes of reactors in which all the reaction elements present the same residence time. In all of the other reactors the elements in the feed spend different times inside the reactors before they finally leave; that is, there is a distribution of residence times for the reaction elements within the reactors. For example, consider the CSTR;

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the feed introduced into it at any given time becomes completelymixed with the material already inside the reactor. In other words, some of the elements entering the CSTR leave it immediately, because product stream is being continuously withdrawn from the reactor; at the same time, others maybe remain in the reactor forever because all the material is never removed from the reactor at one time.

Residence time distribution (RTD) function (E function) is proposed to describe residence time presented by different reaction

elements inside a reactor, therefore:

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=∫∝

Edt

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Figure 11.1 Residence Time Distribution (RTD) Function

ERTD, or E curve

Total area

Fraction of exit streamolder than t 1

tt1

= 1

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The fraction of exit stream of age between t and t + dt is Edt, and the fraction younger than age t1 is:

whereas the fraction of material older than t1 is:

The direct reason responsible for RTD is mixing within a reactor, thus, RTD can be used to describe all kinds of non-ideal flow presented within real reactors mathematically.

∫1

0

tEdt

∫∫ −=∝ 1

1 01

t

tEdtEdt

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Experimental Methods

Since we plan to characterize the extent of non-ideal flow by means of RTD, we should like to know how to evaluate E for any flow. For this we resort to the experimental method named stimulus-response technique.

In our problem the stimulus is usually a tracer input into the feed, whereas the response is a time record of the tracer leaving the vessel. Any material that can be detected and which does not disturb the flow pattern in the vessel can be used as tracer, and any type of input signal such as a random signal, a periodic signal, a step signal, or a pulse signal, etc. may be used.

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TimeTr

acer

con

cent

ratio

n

Trac

er c

once

ntra

tion

TimeTr

acer

con

cent

ratio

n

TimeTr

acer

con

cent

ratio

nTime

Tracer input

Stimulus

Tracer output

ResponseVessel

(a) Random input signal

InputResponse Input

Response

(b) Cyclic input signal

(c) Step input signal (d) Pulse input signal

Input

Response

ResponseInput

Figure 11.2 Commonly used timulus-response techniques

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The F Curve:

With no tracer initially present anywhere, impose a step input of tracer of concentration C0 on the fluid stream entering the vessel. Then a time record of tracer concentration in the exit stream from the vessel, measured as C/C0, is called the F curve.

The C Curve:

With no tracer initially present anywhere, impose an idealized instantaneous pulse of tracer on the feed stream. Such an input is often called a delta function or impulse. The normalized response is then called the C curve.

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Therefore,

where

∫∝

=0

1)( dttC

∫∝

==0

)( and )()( dttcQQtctC

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Figure 11.3 Typical response signal, called

F

t

the F curve, in response to step input

00

1Step input signal

F curve

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Figure 11.4 Typical downstream signal, called the C curve,

C

Area = 1

tt

in response to an upstreamδ-function input signal

00

Ideal pulse input

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Let us relate F and C with E for a closed vessel, the closed vessel being defined as one in which fluid enters and leaves solely by plug flow, thus with a flat velocity profile. Varying velocities, back diffusion, swirls, and eddies are not permitted at the entrance and exit. Real vessels often reasonably satisfy this assumption.

To relate E with C for steady-state flow we note that the RTD for any batch of entering fluid must be the same as for any leaving batch. If this were not so, material of different ages would accumulated within the vessel, thus violating the steady-state assumption.

Now imagine the following experiment. At time t = 0 introduce a pulse of tracer into the stream. The C curve for the tracer

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then records when these molecules leave, in other words, their distribution of ages. Since the C curve represents the RTD for that particular batch of entering fluid, it must also be the RTD for any other batch, in particular, any batch in the exit stream. So we have:

C = E

To relate E with F, imagine a steady flow. At time t = 0 a step input signal is switched on, and then, record the rising concentration of tracer in the exit stream, the F curve, representing tracer and only tracer in the exit stream younger than age t at any time t > 0. Thus, we have:

∫=tEdtF

0

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and

the mean residence time, or holding time, or space-time:

We expect that the mean of the E curve is given by :

EdtdF

=

DFVt 1or ==τ

t

))((0∫ ∫∝ ∝

=t

EdtFdtV

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and

Now, we summarize as follows:

EtttEdtdtEdt

FVt ==== ∫∫ ∫

∝∝ ∝

00][

dtdFCE ==

∫ ∫==t t

CdtEdtF0 0

EC ttt ==

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The Mean:

or

the Variance σ2:

∫∫∝

∝

=

0

0

Cdt

tCdtt

∑ ∆∑ ∆

=ii

iii

tCtCtt

2

0

02

0

02

2)(

tCdt

Cdtt

Cdt

Cdttt−=

−=σ

∫∫

∫∫

∝

∝

∝

∝

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or

Based on the normalized condition of E curve:

and

222

2 )(t

tCtCt

tCtCtt

ii

iii

ii

iii −∑ ∆∑ ∆

=∑ ∆

∑ ∆−≅σ

∑∫ ∆≅=∝

tEttEdtt ii0

20 0

222 )( tEdttEdttt −=−=σ ∫ ∫∝ ∝

22 ttEt ii −∑ ∆≅

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The Dirac Delta Function:

δ(t − t0) = ∞ at t = t0

δ(t − t0) = 0 elsewhere

and

a < t0 < b

1)( 0 =−δ∫∝

∝−dttt

)()()( 00 tfdttfttb

a=−δ∫

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11.2 The RTD in Ideal Reactors11.2.1 RTDs in Batch and Plug-flow Reactors

The RTDs in plug-flow reactors and ideal batch reactors are the simplest to consider. All the elements leaving such reactors have spent precisely the same amount of time within the reactors. TheRTD in such a case is the Dirac delta function, that is:

This means before and after τ no any materials is out of bioreactors, and just at the time τ all of the materials is harvested.

)()( τ−δ= ttE

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11.2.2 Single CSTR RTDMass balance for tracer gives:

Integrating with the initial condition c = c0 at t = 0 gives:

and

τ−

=t

ectc 0)(

τ====

τ−

∝τ

−

τ−

∝

∫∫

t

t

t

e

dtec

ec

dttc

tctCtE

0 0

0

0)(

)()()(

dtdcVFc =−0

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Figure 11.5 E curve for PFR

t0

0

E(t)

τ

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Figure 11.6 E curve for a CSTR

t0

E(t)

τ1

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11.3 Models for Non-Ideal Flow

Many types of models can be used to characterize non-ideal flow presented within real bioreactors. One-Parameter Models such as the tanks-in-series model and the dispersion model are widely used in bioreaction engineering. The dispersion model is mainly used to design medium continuous sterilization processes discussed previously or some enzyme catalytic reactions carried out within tubular reactors. We also introduce Two-Parameter Model of CSTR with Bypassing and Dead Space at the end of this section, by which some large scale stirred fermenters can be described properly.

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11.3.1 The Tanks-in-Series Model

The RTD will be analyzed from a tracer pulse injected into the first reactor and gives:

)()( 3 tCtE =

Figure 11.7 Tanks in series

Pulse

Feed, F

1 2 3

Product stream, C3 (t)

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Mass balance on the tracer within the first tank gives:

Mass balance on the tracer within the second reactor gives:

Therefore,

1)(1τ

−=

t

etC

)]()([)(21

2 tCtCFdt

tdCVi −=

i

t

iietC

dttdC τ

−

τ=

τ+

1)()( 22

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With the initial condition C2(t) = 0 at t = 0, we have:

similarly,

thus,

i

t

iettC τ−

τ=)(2

i

t

i

ettC τ−

τ= 3

2

3 2)(

i

t

i

ettE τ−

τ= 3

2

2)(

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Generalizing this equation to a series of n CSTRs gives its RTD:

We introduce:

and

i

t

ni

ne

nttE τ

−−

τ−=

)!1()(

1

inttτ

=τ

=θ

θ−−

−θ

=θ nn

ennnE

)!1()()(

1

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Figure 11.8 Tanks-in-series response to a pulse input signal

E(θ)

θ

n=2

n=4

n=10

10

n=1 (CSTR)1.0

n=∝ (PFR)

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∫ θθ−θ=τσ

=σ ∝θ 0

22

22 )()1( dE

∫ θθ+∫ θθθ−∫ θθθ= ∝∝∝000

2 )()(2)( dEdEdE

1)(02 −∫ θθθ= ∝ dE

∫ −θ−θ

θ= ∝ θ−−

0

12 1

)!1()( de

nnn n

n

1)!1( 0

1 −∫ θθ−

= ∝ θ−+ den

n nnn

nnn

nn

n

n 11)!1()!1( 2 =−

+−

= +

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So,

Also, we can estimate the model parameter n based on the properties of E(θ):

The general mass balance on any component A in the ith tank gives:

2

2

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στ

=σ

=θ

n

)1(1

max )!1()1()( −−

−

−−

=θ nn

ennnE

5 for 2% error )1(2

><−π

≅ nnn

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For biomass:

Rearranging:

0 , ,1 , =+−− iAiiAiA rVFCFC

01 =µ+−− iiii xVFxFx

1−µ−= i

ii x

VFFx

1−µ−= ix

DD

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Mass balance within the (i-1)th, (i-2)th, …, tanks gives:

11.3.2 CSTR with Biomass Recycle

Cell concentration in a single CSTR can be increased by recycling biomass from the product stream back to the bioreactor. By this way, much higher rate of substrate conversion and product formation can be achieved. With cell recycling, the critical dilution rate for washout is also increased allowing greater operating flexibility.

1132

1

))......()((x

DDDDx

i

i

i−

−

µ−µ−µ−=

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F, Si

VxSP

Fr = αF, xr = βx

Recycle stream

Product stream

Feed streamFermenter

Cell separator

Figure 11.9 CSTR with biomass recycling

F, S, xf, P

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We define:

and

Mass balance for biomass gives:

and

FFr==α

stream feed of rateflow c volumetritheslurry biomass recycling of rateflow c volumetrithe

xxr==β

fermenter theinside biomass of ionconcentrat theslurry recycling withincontained biomass of ionconcentrat the

0)( =+−µ+ xFFxVxF rrr

)1(1 −βα−µ

=D

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P = Dx

cD D

x

D cr

with cell recycling

Bio

mas

s con

cent

ratio

n

and

pro

duct

ivity

xP = Dx

withoutcell recycling

Figure 11.10 CSTR with cell recycling