Case 1:18-cv-00924-CFC Document 399-6 Filed 10/07/19 Page 1 of 50 PageID #: 30704
`
`548
`
`~hese results are eminently rcaso b
`eigenvalues is equal lo zero
`
`Re(J.1)>0
`.
`.
`
`s~stem dynamic characteristic: ;;"c:,' case arises and further analysis i~e;::: par~ of th: system
`_n_a le m view of Eq. (9.33). If the Jar
`eigenvalues to check the inequ rt ~r example, Ref. 10). Fortunately we
`
`0
`
`BIOCHEMICAL ENGINEERING FUNDA
`MENTALS
`
`·
`n the orher hand
`. .
`, c, Is unstable if. any eigenvalue h
`as pos111ve real part;
`anyj
`
`(9.39)
`
`ed
`to etermme local
`.d n J Jes istcd above. First, suppose rhnt 'ih ;c n~t co~pute all the
`has been expanded to pr
`e etermrnnnt m Eq. (9.34)
`ov, e an mth-order algebraic equation.
`J."' + B ;, .. - ,
`J.+B
`I• +· · · + B
`I
`., • O
`Now we
`m - 1
`(9.40)
`can app y the Hurwitz criterio
`'
`.
`.
`parts if and only if the following condition~s ::•:e;sscrts that all roots of (9.40) have negative real
`
`B, > 0
`
`de{~' BJ] B, > O
`
`r BJ
`
`det ~ B, B, > 0
`B, BJ
`
`··1
`
`(9.41}
`
`det
`
`B, B3 B,
`0
`I B, B,
`0
`0 B, B3
`0
`0 > O
`0
`I
`B2
`·---.......... -------
`•.•...
`..•
`.•. 8,,.
`
`1mll~ gr?wth and examine the dyn~::ev:~~ situation where a single substrate
`
`.
`
`. As an example, we shall inves .
`
`.
`
`on of the Monod chemostat model
`pphcatron of the general unsteady-st t
`a~d substrate and use of Monod'
`a e ?'lass balance (9.27) to both bioph
`.
`yields
`s expression (7.10) for the spec,·fi
`h ase
`c growt
`rate
`
`and
`
`dx d =: D(xo - x) + µm .. sx
`s+ K s
`
`t
`
`ds
`dt == D(s0 -
`
`1
`s) _ _ µmuSX
`Yx,s x + Ks
`
`(9.42a)
`
`(9.42b)
`
`For t~e. case of sterile feed (x == 0
`non-tnv1al one given earlier in Eqs. (~. l ~)ere dare two possible steady states, the
`an (7.15) and the " washout" solution
`
`t C. F._ Walter, " Kinetic and Biolo ical a
`
`.
`
`.
`
`::: t Grrysola (eds.), Biochemical Re:ulator;d::ote~1cal_ Control Mechanisms," p. 335 in E K
`
`'
`
`cw ork, 1972.
`
`. un
`c unisms rn Eucurymic Cells John w ·1
`1 ey & Sons,
`•
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`549
`
`x a 0, s a s0 • We can determine which of these steady states will be observed in a
`continuous culture by determining their stability. Local stability can be studied
`using the linearized form of Eqs. (9.42). The results of such a local-stability study
`of the Monod chemostat are summarized below.
`
`D
`>
`
`µ,. .. so
`-
`K,+s0
`
`Nontrivial steady-state
`[Eqs. (7.14, 7.15)]
`
`Unstable
`
`Stable
`
`Washout steady-state
`
`Stable
`
`Unstable
`
`Another prediction of this analysis is that concentrations cannot approach
`their steady-state values in a damped oscillatory fashion. As such oscillatory
`phenomena have been observed experimentally, this substrate-and-cell model is
`insufficent to predict all dynamic features of some reactors.
`Other weaknesses in the dynamic model of the Monod chemostat are
`known. It predicts instantaneous response of the specific growth rate to a change
`in substrate concentration: experimentally, a lag is present (see Prob. 10.6).
`Moreover, growth-rate hysteresis and variations in the yield factor have been
`established. Steady oscillations have been found in several experimental studies
`(Fig. 9.6). Consequently, while the Monod chemostat model is quite successful
`for steady-state purposes in many cases, it has numerous drawbacks as a dy(cid:173)
`namic representation.
`By introducing additional variables into the model, i.e., by giving it more
`"structure," some of the phenomena unexplained by the Monod model can be
`accounted for. The need for structured models in such cases rests on conceptual
`
`30
`
`e ..;20
`::a.
`-::,
`<, ..,
`v >
`2 10
`l:
`
`x-x x-
`
`-X X-
`
`X x-
`
`- X
`
`x-
`
`- x
`
`0.3
`
`g
`0.2 ~
`l ;;
`0. 1 8°
`
`~
`
`o ,__..__-'---'---L--JL--L--..L....--'---'--_Jo
`0
`3
`4
`5
`6
`7
`8
`10
`9
`:?
`Tim•. h
`
`Figure 9.6 These sustained osc1ll11tions or pyruvate concentration ( •) were observed in continuous
`culture of E. coli. No tice that the cell concentration (crosses) remains approximately constant. ( Re·
`printed from B. Siky la, "Continuous Culliva1ion of Microorganisms," Suom. Kemislil., vol. 38, p. /80,
`/965.)
`
`APPX 0284
`
`

`

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`
`550
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`551
`
`can be quite important. It has been suggested that this model with its unusual
`characteristics may help explain the operating difficulties which are common in
`anaerobic digestion processes. Some aspects of substrate-inhibition effects in
`CSTRs will be explored further in Chap. 14.
`Mixed culture systems involving multiple cellular species can exhibit
`complicated dynamic behavior. We shall investigate these types of bioreaction
`systems in detail in Chap. 13. There, we shall also introduce additional general
`mathematical methods and results useful for analyzing and describing reactor
`dynamics.
`
`9.3 REACTORS WITH NONIDEAL MIXING
`
`Now we depart from the ideal cases of completely mixed tanks or plug-flow
`tubular reactors, situations which can be approximated under small-scale labora(cid:173)
`tory conditions, and consider more realistic conditions encountered in larger
`scale process reactors. We shall be concerned in this section with methods to
`characterize mixing and flow patterns in reaction vessels, with application of this
`knowledge for reactor design, and with examination of some of the interactions
`which arise between biological or biocatalyzed reactions and the mixing and flow
`patterns in the vessel. First, we consider mixing times in agitated tanks to intro(cid:173)
`duce important time scales, to show the existence of large-scale circulation
`patterns in reactor vessels, and to get some feeling for orders of magnitudes of
`the circulation times encountered in different bioreactor situations.
`
`9.3.1 Mixing Times in Agitated Tanks
`
`The mixing time denotes the time required for the tank composition to achieve a
`specified level of homogeneity following addition of a tracer pulse at a single
`point in the vessel. The tracer might be a salt solution, an acid or base, or a
`heated or cooled pulse of fluid. The circulation characteristics of the vessel and
`mixing time can be measured by continuously monitoring the tracer concentra(cid:173)
`tion at one or several points in the vessel. As shown schematically in Fig. 9.7,
`different types of reactor internals and agitators give rise to different circulation
`and mixing time characteristics. In the sketched responses in Fig. 9.7, periodic
`patterns in the tracer concentration are evident, indicating a characteristic
`number of bulk circulations of Huid required before achieving composition
`homogeneity. The circulation time is also important because it indicates approxi(cid:173)
`mately the characteristic time interval during which a cell or biocatalyst sus(cid:173)
`pended in the agitated fluid will circulate through different regions of the reactor,
`possibly encountering different reaction conditions along the way. Then, as men(cid:173)
`tioned before, one must consider whether or not the Huctuations encountered are
`of sufficient magnitude and on an appropriate time scale to influence local kinetic
`behavior significantly. We shall return in the conclusion of this section on mixing
`to examination of some experimental studies of mixing effects on biocatalyst
`performance.
`
`.
`porn~s mentioned earlier .
`rn Chap. 7 and in th.
`transient situation th
`time scale of the e'n . e balanced-growth approx ~s c~apter's introduction I
`•
`b. I
`v1ronment l h
`1mat1on d
`• n a
`
`BIOCl-!EMrc
`AL ENGINEERING rUNDAMENTA
`LS
`
`:i~·~ro~:'. 7::-~:il~!;, bt~:~:ct~o:"::':e;~::·:::!!;.:t•h:0:;;:•::.:! ::;
`
`e synthesis). In such
`ics model should be
`Ponents (or pseudo
`in 5';'/ ;:\e alrea~y ex:~:~n~;:s; S~c. 7.4) of the c;;i;h:~:~ to include more
`.
`... Applied to
`wo component st
`:iveral experimental featu;e:ntrnuous. c_ulture dynam;~ct~~7d model of Williams
`. e analogy between the M not anticipated by the M Is model reproduces
`on~d growth-rate equation onod model. Extendin
`res, Je~reson and Smith
`termed~ate species whic~l lJ include in their dynamic (7.10) and enzyme kinet~
`is an analog of the
`~amk_rishna, Fredrickso
`chemostat model an in(cid:173)
`tn their dynamic model.~~ and Tsuchi~a [12] conside;:zy~~~s~bstrate complex.
`structure lo the nutrient ph a sense, this approach can b n '~ rb1tor of cell growth
`S h A completely different as':·
`.
`e viewed as adding more
`c roeder [13], who conside v1ewp_omt has been taken
`r the influence of floccula . by Lee, Jackman, and
`process. We have alread
`
`_individ u::°c:1ft i~h~:~;~. gro"'.th
`;~:!:::~~;~:gJ~gates :al~:;e;;~~- ~:~a~e
`1
`.
`. o IC processes With·
`exam l
`I erent from tho
`1crob1al
`.
`p e, would have to d·m
`Ill such floes could
`se in individual di
`b .
`rop~ase is viewed as hav~ use into the floe lo reach c:per~e~ c7lls; nutrients, for
`lls ~n Its mterior. Thus th
`ent kinetics but a/so w'th lllg two components (fl
`the two different mor~hote. possibility of interch~~~=n~ _md_iv_iduals) with differ~
`~veraU yield-factor fluctu o~1cal forms of behavior Tho llld1v1dual cells between
`~:~o~'e Monod model- :;;o;~r growth-r~te hys;eres~s,re:~~tinr model exhibits
`e compatible with exp
`.
`s ower responses
`Y s model.
`et anoth
`errmental findi
`er conceptual atta k .
`and B
`ngs than
`ungay [14] Th
`c
`rs apparent ·
`· . ey propose that b
`Processes h .
`tn the model of y
`_the cell is ~/:!u~~
`~ e:utri~nt into th~c:~~~ ~:::~~tances in the m:~:;~a!:u:~•
`~: th~ former quantity wh:;~ ';~tan: t~ the external nu~~:~~ concentration wiihin
`h se on this view point exh .b. ect Y influences the cell's
`concentration, and it
`as often been observed exp '. Its lags in response toe ~rowth rate. The model
`.
`In closing our re .
`erimentally.
`nvrronmental changes
`tlally important phen~:: of chemostat dynamics we
`, as
`"':here_ excessive nutrient ~~~-~?t embodied in the ~:~u~d note another poten(cid:173)
`g1ven rn Eq. (7.32)
`I Its growth, the specific o model. In situations
`growth-rate expres~ion
`
`1:
`
`.
`
`.
`
`µ z: ~
`s
`K.. + s + ;;./K.
`should be used A
`· h
`.
`"
`chemostat
`ca ti
`·
`·
`this specific
`n y differently fro
`Wit
`can be complex, :~:::~~~:~~~;o~ditions. ;y::::~~h::: l~ere can ~t:;;;
`m the classical Monod h growth rate can behav
`steady states for
`
`ec s not considered in
`
`I
`r or such a system
`a ocal stability analysis
`
`APPX 0285
`
`

`

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`
`552
`
`lnjec1ionl
`
`(a)
`
`lnjec1ion==;l,
`
`(b)
`
`,r-EJeclrode
`
`®
`
`®
`Many baffles
`small impeue'r
`
`BIOCHEMICAL ENGINEERING FUNDAMENTALS
`
`Diffusion ro forced
`vortex con lroJs
`
`t
`IM -----:,,J~
`
`~Time
`
`Number of circ. 4
`
`lnjcclion-=;-1,
`
`(c)
`
`tEJec1rode
`
`®
`
`~Time
`
`Number of c:irc.( ~Z)
`
`®
`A few baffles,
`large impeller
`
`lnjeclion1
`
`(d)
`
`f,Elccrrode
`
`r--""'-'
`®
`
`IM~
`
`~Time
`
`Number of circ. 7
`
`Propeller wirh
`draf1 tube
`
`l
`
`IM
`
`~
`Figure 9,7 Di/Tercnr impellers a
`~Time
`.
`.
`response paucrns observed foll:. internals-'~ agilaled ranks and lhc
`.
`from the tracer concentration in rh'"i pulse rnJection of tracer. The I t corrcspondrng characteristic
`e rnal, completely mixed stare ( A•r. a teN,r are ploued as deviations
`'.lier ogata [JS/J.
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`553
`
`Mixing time correlations developed for Newtonian fluids and mycelial cul(cid:173)
`tures and experimental data on mixing times in microbial polysaccharide solu(cid:173)
`tions may be found in the review of Charles (18]. Mixing times of 29 to 104 s
`were measured in fermentation tanks of size 2.5 to 160 m 3• In some cases mixing
`times of several minutes have been reported. Studies of a 25 m3 deep jet aeration
`system revealed a mixing time of 80 s in water. Charles reported mixing times of
`around 6 min in I % xanthan solution at 300 rpm with no air flow, decreasing to
`around one minute at 500 rpm and with 0.25 % air flow. On the other hand,
`mixing times in the range of 2 to 3 s have been mentioned in small reactors.
`These figures give a sense of the order of magnitude range which might be ex(cid:173)
`pected at different reactor scales in different types of bioreactor fluids. Here our
`concern is with large-scale fluid circulation and possible composition and tem(cid:173)
`perature nonuniformities. Finer scale considerations having to do with turbu(cid:173)
`lence and its interaction with mass transfer and cells is provided in Chap. 8.
`Clearly consideration of a single circulation time in, say, an agitated tank is a
`conceptual approximation. If we monitor the sojourn of various parcels of fluid
`from the impeller region, different paths through the vessel will be followed by
`different fluid parcels, giving rise to correspondingly different circulation times.
`Bryant [I 9] has described how the circulation time distribution f,(t) can be experi(cid:173)
`mentally determined by use of a small, neutrally bouyant radio transmitter and a
`monitoring antenna placed in the vessel. By definition, fc(t) dt is the fraction of
`circulations which have circulation time between t ~nd t + dt. Bryant indicates
`that the circulation time distribution for agitated tQpks can usually be well ap(cid:173)
`proximated by the functional form of a log-normal distribution
`re ) = _ I _
`[ -
`r-;::_ exp
`u1...; 2,r
`The two parameters in this representation, the log-mean circulation time r, and
`log-mean circulation time standard deviation u1, are related to the mean circula(cid:173)
`tion time f and standard deviation u off, by
`
`(In t -
`2 2
`<11
`
`t1)
`
`2
`
`]
`
`(9.43)
`
`Jc l
`
`(9.44a)
`
`(9.44b)
`Experimental measurements give f and u which can then be used in Eq. (9.44) to
`determine parameter values for f, in Eq. (9.43). Later in this chapter we shall
`apply the circulation time distribution concept to calculate effects of fluid circula(cid:173)
`tion on overall bioreactor performance.
`
`9.3.2 Residence Time Distributions
`
`Let us now try to imagine what happens to a small parcel of fluid after it has
`entered a continuous-flow bioreactor. Because of mixing in the vessel, this fluid
`will be broken into smaller parts, which separate and disperse throughout the
`vessel. Thus, some fraction of this lluid element will rapidly find its way to the
`
`APPX 0286
`
`

`

`Case 1:18-cv-00924-CFC Document 399-6 Filed 10/07/19 Page 4 of 50 PageID #: 30707
`
`BIOCHEMICAL ENGINEERING FUNDAMENTALS
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`555
`
`Dc1,•.ror
`
`-c,
`\ I "' tr;i,·cr
`
`\.'Ollll."lllra(1011 (rcspnnSl',
`c,,
`
`hell lra<cr
`Ll>l1Lcn1r.11ion ls1111111lus)
`
`- - l:.nJucrH
`r - - - sln:~m
`
`Whal is needed in many cases is not the ff function but the residence-time(cid:173)
`distribution (RTD) function t 0(t), which is defined by
`S(t) dt = fraction of fluid in exit stream which has been in
`vessel for lime between t and t + dt
`
`(9.47)
`
`Thus, for example, the fraction of the exit stream which has resided in the vessel
`for times smaller than t is
`
`I S(x) dx
`
`It follows from definition (9.47) that
`t CZI C(x)dx = 1
`
`(9.48)
`
`A simple thought experiment will now serve to clarify the relationship be(cid:173)
`tween the 6 and :F functions. Returning to the stimulus-response experiment of
`Fig. 9.8, let us imagine the vessel contents to consist of two different types of
`fluid. Fluid I contains tracer at concentration c•, and fluid II is devoid of tracer.
`Consequently, all elements of fluid I must have entered the vessel al some time
`greater than zero. Then, any fluid I in the effluent at time t has been in the system
`for a time less than t. On the other hand, fluid II had to be in the vessel at t = 0
`since only fluid I has entered since then. All fluid II elements in the effluent at
`time r consequently have resiqence times greater than t. Assuming that we know
`the C function, we can write the exit tracer concentration c(t) as the sum of the
`fluid I and fluid II contributions:
`
`r~
`C(x) dx + O· J, S(x) dx
`
`c(t) = c•·
`
`0
`
`fl
`§ (t) = L t'(x) d.x
`
`Combining Eqs. (9.49) and (9.46) produces the desired relationship
`
`(9.49)
`
`(9.50)
`
`"~ r----------
`
`c,, - - - - - -
`
`Trmc -
`Figure 9.8 Schematic diagram sh
`.
`.
`Tmrc -
`step tracer input.
`owing expcrrmenral measurement of lh
`e response (.9- curve) lo a
`
`h'J
`effluent stream
`' w I e other portions of.
`.
`
`.
`
`e e uent stream is a mixture of t1u·d
`rea_ctor for different Jen ths
`.
`I
`
`:~mes ~efore entering the exit pipe. Vie~~;~\;and~r ab~ut the vessel for varying
`~ erent y, this scenario indicates that
`:•:;:~ :;;: :~:~:,~' ::::~~:it ~;:~;=~~~~~~:h:r 1~:':~:~~• a~~~~;~
`e ~ments which have resided .
`
`. w
`d"
`or etermimng the residence 1·
`1me 1stnbu-
`
`t1on are reviewed next
`w
`·
`e shall consider first
`, b.
`an ar llrary vessel with one feed
`and it will be ass
`!~i~ into the feedu;:: :~ro~h:m:;;e;~i~h~t t thc~e is no ba:kn~i~::i::~n!:!~:j
`i_xrng c~aracteristics of the vessel w
`m o t e vessel. In order to probe th
`~smg an inert tracer: at some datu' ~ cond~ct a stimulus-response exper'
`e
`at concentration c• into the 'e d 1· m time designated l = 0 we introd
`iment
`w
`·
`11 e me and m ·
`·
`.
`'
`uce tracer
`amtam this tracer feed for t > 0 Th
`c ~onuor the system response (in th.
`expetm;n1, as w~IJ a!~~: !ii~;;~;: 1
`specific stimulus Fi
`is case the exit tracer co
`.
`.
`en
`~!t~ati~ally the gener:~e~t;~~~~) ~f :~::
`n er these conditions, the rcspons eta ;x1t ~oncentration response c(t)
`h
`w ere
`e o a umt-step tracer ·
`·
`input c0 (t) = H(t)
`
`H(t) ..., {0
`
`I
`
`l < 0
`
`t ,?:?: 0
`.
`, .
`.
`.
`is obtained by dividing the c(t) functi
`tracer feed concentration c• used in o;h o~tamed_ in the above experiment by the
`response of the mixing vessel, is called lh: ;~enm_ent. T.he result, the unit-step
`,unction (Fig. 9 8)·
`·
`·
`c(t)
`.?(t) c: - = res
`•
`c•
`ponse to umt•step input of tracer
`
`(9.45)
`
`(9.46)
`
`o;:
`
`which can be differentiated with respect to t to provide the alternative form
`dJT(r) = &'(t)
`dt
`
`(9.51)
`
`We note first from Eq. (9.51) that &'(t) can be obtained by differentiating an
`experimentally determined ff curve. Also, the theory of linear systems states that
`
`1 Standard terminology rrom statistics would indicate that S is a density foncuon with /F the
`corresponding distribution. The language above is so firmly embedded in the reaction engineering
`literature, however, that it would cause confusion to alter it here,
`
`APPX 0287
`
`

`

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`
`556
`
`BIOCHEMICAL ENGINEERING FUNDAMENTALS
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`•
`
`557
`•
`
`2
`
`•
`
`•
`
`is identical to the nominal holdm~ time
`.
`.
`which says that the mean residence t1mte
`ply for a single phase in a multiphase
`I t" nship does no ap
`t I st or adsor-
`•
`I with heterogeneous ca a y
`of the vessel. This re a ,o
`.
`.
`'b
`.
`to a single phase m a vesse
`d in terms of the d1stn u-
`:::t~h~o:econd moment n!2 ish?'1hos:s ~~:na~:~~~y:f the squares of deviations
`2 _ m - m 1, w 1c
`•
`tion's vanance u -
`and reactor analysis include_ the _m~erna
`from the mean residence time:
`. .
`1
`Related functions useful m mix~?) dt denotes the fraction of flmd w1t/11n the
`age distribution function _ /(t), where I ~ r a time between t and t + dt. A mass
`I Whl.ch has been m the vesse o
`vesse
`b
`·
`balance can be applied to O tam
`/(t).= r- 1[1 - !F(t)).
`h th t A(t) dt is the probability that a
`The intensity function A(t), d~fine: sue to/for time t leaves in th~ next ~h~rt
`
`(9.60)
`
`fluid element which has been m ; t. re:~agnosing deviations from ideal m1xmg
`time interval dt, is especial~y te ~-o: introduced earlier by [16]
`
`regimes. A(t) is related to t e unc '
`S(t)
`t\(t) = I _ !F(t) = -
`
`dln[l-
`dt
`
`§( )]
`t
`
`(9.61)
`
`the time derivative of the unit-step response is the unit-impulse response, which
`reveals that &'(t) can be interpreted as the reponse of the vessel to the input of a
`unit tracer impulse at time zero. While an impulse is a mathematical idealization,
`we can approximate it experimentally by introducing a given amount of tracer
`into the vessel in a short pulse of high concentration.
`In addition to the experimental methods just described for determining the
`RTO function, we can sometimes evaluate it if a mathematical or conceptual
`model of the mixing process is available. Considering the ideal CSTR as an
`example, the unsteady-state mass balance on (nonreactive) tracer is
`de
`F
`-
`"" -- (c - c)
`dt
`VR
`O
`To determine the result of an :P" experiment for this system, we take
`c(O) = 0
`t ~ 0
`c0 ( t) :c c*
`The solution to Eq. (9.52) under conditions (9.53) and (9.54) reveals
`
`(9.53)
`(9.54)
`
`(9.52)
`
`.f7(t) =: c(t) ::::; 1 - e - (FIYR)I
`c*
`(9.55)
`Applying formula (9.51} to the result in Eq. (9.55) reveals that the RTO for a
`CSTR is
`
`F
`8(t) = - e - Ft/YR
`VR
`
`(9.56)
`
`The physical perspective of the PFR introduced above readily reveals its
`RTO. If a tracer pulse is introduced in the feed, it flows through the vessel
`without mixing with adjacent fluid and emerges after a time L/u. Thus, the tracer
`pulse in the exit has exactly the same form as the pulse fed into the PFR, except
`that it is shifted in time by one vessel holding time. Deviation from such behavior
`is evidence of breakdown in the plug-flow assumption.
`Often when dealing with distribution functions such as 8(t}, it is helpful to
`consider the moments of the distribution. The kth moment of 8(t) is defined by
`
`9 9 shows how A(t) behaves for
`.
`For an ideal CSTR, A{t) is a con~tant. F1~~;: 1~ general, whenever A(t) has a
`several types of nonidealities in st1rre!ev%ixi~g vessel has stagnant regions or
`.
`and subsequent decrease,
`it is now well established
`ma~1mu~ fast and slow flows from inlet to e~uent.
`reg1~1~:ough we cannot discuss ~II the de~a:!: :r~ixing (further discussion of
`that the RTD does not charactenze ~II asp ·11 be found in the references). The
`this point from a variety of perspectives wt
`
`, ......
`~ ',
`! ' ,e ---------
`' ..... ~-
`----...........
`/
`\
`I
`r~~..,../
`~-------
`I
`lo
`I
`I
`
`C
`
`----------
`
`--
`
`- -
`- - - - - - - - - -~ --
`'
`
`I,
`
`O
`
`/
`
`.,,,
`
`•
`Dimensionless time 6 = ti t
`
`in stirred tank reactors
`. .
`1 pes of imperf eel muung
`d
`ti 1
`between inlet an ou c
`· dicative of different Y
`D
`·
`.
`Figure 9.9 lntens1ty funcu;::s m
`inlet and outlet (normal case); 8, e ay betwet:n inlet and outlet
`I
`(Ref. 16): A, Short delay
`between inlet and outlet; D, Bypass
`tween
`ffi .
`t' ring· C Bypass
`due lo insu icic~t s tr
`• . • ufficient stirring.
`and stagnant regions due to ms
`
`(9.57)
`·,·
`Since a unit amount of tracer is introduced into a vessel to observe its 8 curve,
`and since all tracer eventually must leave the vessel, we know that
`
`k = 0, I, 2, ...
`
`~
`-
`<
`
`1
`
`(9.58)
`The first moment 1111 is the mean of the RTO, or the mean residence time f. Under
`the conditions of zero back diffusion stated at the start of this section, it can be
`proved [2] for a single phase fluid in an arbitrary vessel that
`
`VR
`f = m, = -
`F
`
`(9.59)
`
`APPX 0288
`
`

`

`Case 1:18-cv-00924-CFC Document 399-6 Filed 10/07/19 Page 6 of 50 PageID #: 30709
`
`ANALYSIS OF BIOLOGICAL REACTORS
`
`559
`.
`ial inhomogeneities in reach~•
`•
`De;ooN <ND
`cale reactors in which substant
`does not define how fluid
`.
`dowd~/nn!a~~;~sr The RTD for the vessel a~ a :t~~~erent reaction conditions.
`c~nthi~ t~he vessei circulates throughd domt a::sprovide an estimate under condi-
`w1
`9 62) can be use on y
`.
`Consequently, Eq . . < ble nonuniformities exist in the b1oreda~tor~ome biochemical
`t" ons where apprec1a
`fl cs of cells foun m
`H
`
`rated so that app icatton
`
`ber that livmg cells con-
`
`t Certainly the individual cells olr_ ~ of Eq. (9.62) seems ap~ealing. ow-
`=~•0•" ,:,~e~;~~~le pit[a11 exists he":;hw;h7.;~•:::;:;apt and respond ID •:e!:
`tain_' :;'t!:::ca~;:!,:~~::,;,~~:~~a:~es wh;~\:.":~i:~:~;"~f !,i;/:~f ep:a,:i'
`
`bined and interactive influences oh
`envtr
`during the batch in a dtrec_t y
`·s to behave hke
`the com
`. b h
`f these phases c ange
`Comp::i•t~~:i:~~ c::seq:ently, if a cell ?r flo~;~h• :.;:~~:~ i: is necessary in
`coup I
`small bateh reactor observed m a
`d" ng fluid) also remain segregated
`the ""7~hat the cell's environment (th_e surroun ~ition do not play a critical role
`be loosened and use of Eq.
`stem. II changes in medmm :omp
`genera
`:; ::: :;:hsl'.iological '.eactions, this rj:~f:::h°f:stances shall appear later m
`be better ratmnahzed. Examp
`)
`(9.62 can
`. er a single half-order
`. .
`. fluence of micromtxmg, constd
`h
`this chapter.
`Returning now to t e m
`irreversible reaction.
`
`S - - -+ p
`
`k .,
`2
`r = s
`
`(9.63)
`
`e ated but has the same
`sel which is completely se~r g
`roximation to
`. d
`.
`occurring m a Rst1;e i:i;ht view the half-order reaction asf an b:~~ate concentra(cid:173)
`h.
`"th
`RTD as a CST .
`e
`ther narrow range o su
`.
`the Michaelis-Men ten form over. a r;9 63) in a batch reactor and usmg t ts wt
`tions. By computing s(t) for reaction
`.
`
`Eq. (9.56) in Eq. (9.62) we find
`
`k[
`
`( - 2so)]}
`
`kf
`
`s = s,{ I - '• [ I - exp
`lace in an ideal CSTR, which has
`k
`.
`h, d if the same reaction ta es_ p
`1·
`·1 the effluent substrate
`h maximum mtxedness tmt ,
`On the other an , . .
`by definition microm1xmg at t e
`
`(9.64)
`
`.
`
`4so ]}
`I + (kf)2
`
`I +
`
`(9.65)
`
`s ,. s0{1 - (k
`
`f)2 [ -
`
`2·
`
`concentration is
`
`tio!s in vessels wi_th adrbi!rary i::~d:i:~e i!1
`
`~itre":nce betwee~ th:::t;;::;:::~
`
`d"
`So
`mixedness con t-
`.
`conversion under maxm~um
`.
`.
`. es is described m Ref. 20.
`A eneral method for calculatm~
`: fortunate that rea~tor pe~:i~:~~:~~
`' F
`example, for an trreverst
`.
`. .
`From a practical es1gn v
`0
`0
`:~~;':!"um
`often not l?o s7nsiti~;R m;~:
`order reaehon m _a
`d ' ximum mixedness ,s less than I pe "d an adequate
`complete segreg~t•?n an w:ato be inexact, Eq. (9.62) may prov1 e
`in cases where tt is kno
`
`558
`
`RTD indicates how long various "pieces" of effluent fluid have been in the
`reactor, but it does not tell us when fluid elements of different ages are intermixed
`in the vessel.
`
`BIOCHEMICAL ENGINEERING FUNDAMENTALS
`
`This point can perhaps be clarified by considering two limiting cases. In the
`first instance, suppose that fluid elements of all ages are constantly being mixed
`together. In other words, the incoming feed material immediately comes into
`intimate contact with other fluid elements of all ages. Such a situation, usually
`termed a state of maximum mixedness, prevails in the ideal CSTR. At the other
`extreme, fluid elements of different ages do not intermix at all while in the vessel
`and come together only when they are withdrawn in the effluent stream. In this
`case, which is called complete segregation, reaction proceeds independently in
`each fluid element: the reaction processes in one segregated clump of fluid are
`unaffected by the reaction conditions and rates prevailing in nearby fluid ele(cid:173)
`ments. Between these two limiting situations falls a continuum of small-scale
`mixing, or micromixing.
`
`The RTO of a reactor is completely independent of its micromixing charac(cid:173)
`teristics. Often, this is not a limitation because micromixing has a small effect on
`reactor performance. On the other hand, micromixing can influence reactor per(cid:173)
`formance significantly in special situations. It has been suggested [16) that
`the sensitivity of reactor performance to micromixing can be usefully assessed by
`calculating reactor performance under the special cases of maximum mixedness
`and complete segregation. If substantial difference is obtained, the reactor is
`sensitive to micromixing and will be difficult to scale up. In such situations,
`predictability of scale-up will be enhanced by using a PFTR or something ap(cid:173)
`proximating it (see next section) because PFTR performance is micromixing in(cid:173)
`sensitive regardless of the reaction network or kinetics.
`ln a reactor with complete segregation, each independent fluid element be(cid:173)
`haves like a small batch reactor. The effluent fluid is a blend of the products of
`these batch reactors, which have stayed in the system for different lengths of time.
`Restating this in mathematical terms, let c,it) be the concentration of component
`i in a batch reactor after an elapsed time t, where the initial reaction mixture in
`the batch system has the same composition as the flow-reactor feed steam. So
`long as significant heating or volume change is not caused by the reaction(s), it
`makes no difference whether one or many different reactions are occurring. A
`fraction lf(t) dt of the reactor effluent contains fluid elements with residence times
`near t and hence concentrations near c,it). Summing over all these fractions
`gives the exit concentration c1 for the completely segregated reactor:
`
`c, == l"' c1b(t)8(t) dt
`
`(9.62)
`
`In deriving Eq. (9.62), we assumed that reaction conditions (T, pH, dissolved
`oxygen, etc.) were effectively uniform in the "small-batch reactor" and in the
`mixing vessel characterized by the RTD. Here "effectively uniform" means that
`any differences in conditions which do exist have negligible or acceptably small
`effects on the bioreaction processes of interest. This assumption may well break
`
`APPX 0289
`
`

`

`Case 1:18-cv-00924-CFC Document 399-6 Filed 10/07/19 Page 7 of 50 PageID #: 30710
`
`560
`
`BIOCHEMICAL ENGINEERING FUNDAMENTALS
`
`DESIGN AND ANALYSIS OF BIOLOGICAL REACTORS
`
`561
`
`approximation of reaction performance. [By series expansion of Eqs. (9.64) and
`(9.65), develop an expression for the relative difference, JI - s(9.64)/s(9.65)1.J
`For all single reactions with order less than unity, maximum substrate
`conversion is achieved at maximum mixedness. Complete segregation provides
`greatest conversions for reaction orders greater than one. As may be justified
`using the superposition principle for linear systems, the degree of micromixing
`does not influence first-order reactions. Thus, for one or more reactions with
`first-order kinetics, Eq. (9.62) may be used independent of micromixing state.
`Another exploitation of RTD data is the evaluation of parameters in various
`nonideal-flow reactor models. A variety of such models is considered in the next
`section.
`
`9.3.3 Models for Nonideal Reactors
`
`Obviously the RTD contains useful in(prmation about flow and mixing within
`fun~f!ons can be used is to assess the extent of
`the vessel. One way the 8 and §
`deviation from an idealized reactor. For example, we calculated 8 and§ above
`for an ideal CSTR. By comparing these curves with those for a real vessel, we can
`get an idea of how well the actual system approximates complete mixing.
`Different kinds of nonidealities often have distinctive manifestations in the
`observed response functions. We can gain some appreciation of them by con(cid:173)
`structing models representing various sorts of deviations from the idealized mix(cid:173)
`ing system. Examples are shown in Fig. 9.IO. Case (a) is the ideal CSTR, case (b)
`
`Co mbined model sd1e,n;,tic
`
`~,.._ r±, 1-·
`~
`
`F
`
`:r
`
`L
`I '
`
`G
`
`I - - - - - -:..::--------
`
`Fr11'
`
`I - (J
`
`J!f.
`V
`
`Ft/ V
`
`Fr/ I'
`
`, ..
`
`Fr1V
`(<J)
`
`Ft/I' ")
`Figure 9.10 s; and ~ functions for (a) an ideal CSTR, (b) a CSTR with bypassing, and (c) a CSTR
`with a dead zone,
`
`Ft/ V
`(h)
`
`F, so
`
`F, Xi, S 1
`
`Figure 9.11 Model _for an incompletely mixed CSTR
`with II stagnant region.
`
`and there is a dead volume (1 - a)V~ in
`ars immediately in the ~ function,
`involves bypassing of the feed_ stream,
`case (c). When there is byp~ssmg, tracera ap:rethe If curve than in the ideal case.
`while a dead region results m faster de~. ~ model for a nonideal continuous-flow
`Another useful and ~ommonly a~~ ~~eat CSTRs (Fig. 9.11 ). Here the reactor
`stirred reactor uses tw~ mte~connect
`aller completely mixed regions. The feed
`contents have been divided mto two s?1 1 'whose volume is a fraction Cl of the
`and effluent streams pass throug~ ref on h • nges material with stagnant region 2
`total reactor volume. In turn, region excMa
`d growth kinetics with constant
`F, If we assume
`ono
`• •
`·
`th·
`at volumetric flow rate
`·
`d
`·be steady-state cond1t1ons m
`is
`yield factor, the following mass balances escn
`system:
`
`X1 ,.. Yx:s(so - S1)
`
`region I substrate
`
`X2 - X1 == Yx/S(s 1 -
`
`s 2)
`
`region 2 substrate
`
`(9.66a)
`
`(9.66b)
`
`X2 + ayµm .. K, + S1
`S1
`
`X == (1 + yD)X1
`I
`
`region I cells
`
`(9.66c)
`
`region 2 cells
`
`(9.66d)
`
`where
`
`_ ,!'R D == !._ == nominal dilution rate
`Y - F'
`VR
`
`(9.67)
`
`t for this model is obtained by setting so == s i == S2