lecture 6
TRANSCRIPT
Chemical Reaction Engineering
Lecturer : 郭修伯
Lecture 6
This course focuses on “Nonelementary reaction kinetics”.
Elementary vs. nonelementary
• Elementary: the reaction orders and stoichiometric coefficients are identical.
• Nonelementary reaction kinetics: no direct correspondence between reaction order and stoichiometry.
• Four topics will be introduced in this course:– Pseudo-steady-state hypothesis (PSSH)
– Polymerization
– Enzymatic reactions
– Bioreactors
Nonelementary reaction
• Gas-phase decomposition of azomethane (AZO)
262223)( NHCNCH
when AZO at pressure greater than 1 atm: AZON Cr 2
From experimental observation:
First - order reaction
when AZO at pressure below 50 mmHg: 2
2 AZON Cr Second - order reaction
Why? What happened?
Theory of Lindemann (1922)
• Collision or interaction between molecules forms an activated molecule, [(CH3)2N2]*
262223)( NHCNCH
*223223
1223223 ])[()()()( NCHNCHNCHNCH k
The activation can occur when translational kinetic enery is transferred into energystored in internal degrees of freedom, particularly vibrational degrees of freedom.
The concentration of the active intermediate is very difficult to measure,because AZO* is highly reactive and very short-lived (~ 10-9 s).
Translational kinetic energyCollision, photo, ...etc. Energy stored in internal
degrees of freedom
Active intermediate
• The energy must be absorbed into the chemical bonds where high-amplitude oscillations will lead to bond ruptures, molecular rearrangement, and decomposition.
• The sources of the energy:– photochemical effects of similar phenomena
– molecular collision or interaction.
• Types of active intermediates– free radicals (one or more unpaired electrons, e.g., H•)– ionic intermediates (e.g., carbonium ion)– enzyme-substrate complexes– etc.
*223223
1223223 ])[()()()( NCHNCHNCHNCH k
21* AZOAZOCkr
2232232
223*
223 )()()(])[( NCHNCHNCHNCH k
Deactivated through collision with another molecule
** 2 AZOAZOAZOCCkr
2623*
223 ])[( NHCNCH k ** 3 AZOAZOCkr
*2 3 AZON Ckr
We wanted to know why the orders of reation of rN2 are different at low and high pressures.
Difficult to measure
Actually, these are series reactions (multiple reactions):
q
iijj rr
1
**** 322
1
3
1AZOAZOAZOAZO
iiAZOAZO
CkCCkCkrr
What’s next? CAZO
*
Measurable concentration
relate to
Pseudo-Steady-State Hypothesis (PSSH)
• The active intermediate molecule has a very short lifetime low concentration
the rate of formation = the rate of disappearance
The net rate of formation of the acrive intermadiate is zero:
0* ermediateintactiver
0*** 322
1 AZOAZOAZOAZOAZO
CkCCkCkr
AZO
AZOAZO Ckk
CkC
23
21
*
*2 3 AZON Ckr
AZO
AZON Ckk
Ckkr
23
231
2
AZO
AZON Ckk
Ckkr
23
231
2
At low concentration: 32 kCk AZO 2
12 AZON Ckr
First - order reaction
Second - order reaction
At high concentration: 32 kCk AZO AZOAZON kCCk
kkr
2
312
The reaction is apparent first-order at high azomethane concentrations andapparent second-order at low azomethane concentration
Reaction mechanism
AZO
AZOAZO Ckk
CkC
23
21
*
The active intermediate, AZO*, collides with azomethane, AZO
The active intermediate, AZO*, decomposes spontaneously
The active intermediate, AZO*, is formed from AZO
The Stern-Volmer equation
High-intensity ultrasonic wave applied to water Light
wave compression microsize bubbles
Temperature rising
Generation of intermediatesChemical reaction in the bubbles
The intensity of the light given off, I, is proportional to the rate of reaction ofan activated water molecule formed in the microbubble.
hvOHOH k 23*
2 *2
*2
3)(OHOH
Ckrnsityinte 放出的光
When either carbon disulfide or carbon tetrachloride is added to the water,the intensity of sonoluminescence increases an order-of-magnitude.
聲納冷光
hvCSCS k 24*
2 *2
*2
4)(CSCS
Ckrnsityinte
However, when an aliphatic alcohol, X, is added to the solution, the intensityof sonoluminescence decreases with increasing concentration of alcohol.
Relative intensity I0/I
Alcohol concentration Cx
Sugget a mechanism consistent with experimental observation
XBCAI
I0
XBCAI
I0
)(
1
0 XBAI
I
The active intermediate collides with alcohol
productondeactivatirmediateinteX
Set CX = (X)
X is called a “scavenger” to deactivate the active intermediate清除者
)( 2CSnsityinte Active intermediate was probably formed from CS2
MCSCSM *22
M is a third body (CS2, H2O, etc.)
Purposed mechanisms:
Activation: MCSCSM k *2
12
Deactivation: MCSCSM k 22*
2
Deactivation: XCSCSX k 23*
2
Luminescence: hvCSCS k 24*
2*
24 CSCkI
Using PSSH on CS2*: 0* ermediateintactiver
0)())(())(())(( *24
*23
*2221*
2 CSkCSXkMCSkMCSkr
CS
*2
4 CSCkI
432
214
)()(
))((
kXkMk
MCSkkI
No alcohol (X=0)
42
2140 )(
))((
kMk
MCSkkI
)(1)()(
142
30 XkXkMk
k
I
I
This equation and similar equations involving scavengers are called Stern-Volmer equations.
Chain reaction
• A chain reaction consists of the following sequence:– Initiation
• formation of an active intermediate
– Propagation or chain transfer• interaction of an active intermediate with the reactant or
product to produce another active intermediate
– Termination• deactivation of the active intermediate
PSSH applied to thermal cracking of ethane
The thermal decomposition of ethane to ethylene, methane, butane andhydrogen is believed to proceed in the following sequence:
Initiation: 362 262 CHHC HCk ][ 6211 6262HCkr HCHC
Propagation: 5242
623 HCCHHCCH k ]][[ 62322 62HCCHkr HC
HHCHC k42
352
][ 5233 42 HCkr HC
2524
62 HHCHCH k ]][[ 6244 62HCHkr HC
Termination: 1045
522 HCHC k 25255 ][
5252 HCkr HCHC
(a) Use the PSSH to derive a rate law for the rate of formation of ethylene(b) Compare the PSSH solution in Part (a) to that obtained by solving the complete setof ODE mole balance
The rate of formation of ethylene ][ 52342 HCkr HC
The net rates of reaction of active intermediates CH3•, C2H5•, H• are (PSSH):
052624262
5252525252
5432
5432
HCHCHCHC
HCHCHCHCHC
rrrr
rrrrr
0262623 21 HCHCCH rrr
06242 43 HCHCH rrr
][ 6211 6262HCkr HCHC
]][[ 62322 62HCCHkr HC
][ 5233 42 HCkr HC
]][[ 6244 62HCHkr HC
25255 ][
5252 HCkr HCHC
substitute
2
13
2][
k
kCH
0][]][[ 25256232 HCkHCCHk
Purpose: replace [C2H5•]
21
625
152 ][
2][
HCk
kHC
21
625
13523 ][
2][
42
HCk
kkHCkr HC
The rate of disappearance of ethane ]][[]][[][ 624623262162HCHkHCCHkHCkr HC
The net rates of reaction of active intermediates CH3•, C2H5•, H• are (PSSH):
052624262
5252525252
5432
5432
HCHCHCHC
HCHCHCHCHC
rrrr
rrrrr
0262623 21 HCHCCH rrr
06242 43 HCHCH rrr
][ 6211 6262HCkr HCHC
]][[ 62322 62HCCHkr HC
][ 5233 42 HCkr HC
]][[ 6244 62HCHkr HC
25255 ][
5252 HCkr HCHC
substitute
21
62
21
5
1
4
3 ][2
][
HC
k
k
k
kH 2
1
625
136211 ][
2])[2(
62
HCk
kkHCkkr HC
Purpose: replace [CH3•] and [H•]
2
13
2][
k
kCH
For a constant-volume batch reactor :
21
625
13 ][
242
HCk
kkr HC
21
625
136211 ][
2])[2(
62
HCk
kkHCkkr HC
21
5
1311 ]
2)2(
6262
62
HCHCHC C
k
kkCkk
dt
dC21
5
13 62
422
HCHC C
k
kk
dt
dC
For given initial concentration of C2H6 and temperature,these two equation can be solved simultaneously
We obtain the concentration - time relationship using PSSH
Other methods can also be used…...
1. Mole balances:
C2H6 11 r
dt
dC
CH3•
CH4
C2H5•
C2H4
2. Rate laws for each species:
614212111 CCkCCkCkr
(Batch)
31
62 2CHHC k
5242
623 HCCHHCCH k
HHCHC k42
352
2524
62 HHCHCH k
1045
522 HCHC k
H2
33 r
dt
dC
44 r
dt
dC
55 r
dt
dC
77 r
dt
dC
22 r
dt
dC 122112 2 CCkCkr
2123 CCkr
245614432124 CkCCkCkCCkr
435 Ckr
614436 CCkCkr H• 66 r
dt
dC
C4H10 88 r
dt
dC
6147 CCkr
2458 2
1Ckr
All these O.D.Es can besolved simultaneously
The comparisons of the results obtained from the two methods are shown on page. 351.The two results are identical, indicatingthe validity of the PSSH underthese conditions
Reaction pathways
The second method is more frequently recently used due to the increase in computing power.
The key is to identify which intermediate reactions are important in theoverall sequence in predicting the end products.
The study of reaction pathways
Reaction pathways - smog formation
Nitrogen and oxygen react to form nitric oxide in automobile engines.The NO from automobile exhaust is oxidized to NO2 in the presence of peroxide radicals:
2NOORNOORO Nitrogen dioxide is then decomposed photochemically to give nascent oxygen:
ONOhvNO 2which reacts to form ozone:
32 OOO The ozone then involves in a whole series of reactions with hydro-carbons in the atmosphereto form aldehydes, various free radicals, and other intermediates, which react further toproduce undesirable products in air pollution:
radicalsfreealdehydesolefinO 3
OCHORRCHOCHRRCHO 3
hv OCHR
e.g.HCHOCHCHOCHCHCHCHCHO hv 22233
2
severe eye irritants
OROOR 2 P.354
Finding the reaction mechanism ...
Use both Figure 7-2 and Table 7-2 on page 354.
We will use the same idea to study “Polymerization”
A polymer is a molecule made up of repeating structural (monomer) units.
Polymerization is the process in which monomer units are linked together by chemicalreaction to form long chains. The polymer chains can be linear, branched, or cross-linked.
Polymerization reactions
Step reactions(Condensation reaction)
Chain reactions(Addition reaction)
Require bifunctional or polyfunctional monomers Require an initiator
Copolymers: polymers made up of two or more repeating units
Block
Alternating
Random
Graft
Statistical
SQSQSQ
SSSQQQ
QSSQSQ
QQQQQQ
SSSSfollow certain laws
Step polymerization reactions
• It requires that there is at least a reactive functional group on each end of the monomer
• Example:– has an amine group at one end and a carboxyl group at the other
– common functional groups are -OH, -COOH, -COCl,-NH2
• The molecular weight usually builds up slowly.• It is not meaningful to use conversion of monomer as a measure.
– The reaction will still proceed even though all the monomers has been consumed.
• We measure the progress by the parameter p which is the fraction of functional groups.
COOHCHNH 522 )( amino-caproic acid
Degree of polymerization
0
0
M
MMp
= fraction of functional groups that have been reacted
M = concentration of functional groups
pM
MX n
1
10The number of average degree of polymerization isthe average number of structural units per chain:
The number average molecular weight, is the average molecular weight of a structureunit, times the average number of structral unit per chain, plus the molecular weightof the end group, :
nM
sM nX
egM
egsnn MMXM
Interests to step polymerization
• Conversion of the functional groups
• Degree of polymerization
• Number average molecular weight
• Distribution of chain lengths, n, (i.e. molecular weight, Mn)
Determining the concentration of polymers for step polymerization
Determine the concentration and mole fraction of polymers of chain length j in terms ofinitial concentration of ARB, M0, the concentration of unreacted functional groups M,the propagation constant k and time t.
Set BRAP 1 BRAP 22BRAP jj ...
Reaction
212 PP
321 PPP
431 PPP
422 PPP
21
11
211 2
,2 1
21kP
rrkPr P
PP
21222 2321
PkPrrr PPP
31333 2431
PkPrrr PPP
22
44
224 2
,2 2
42kP
rrkPr P
PP
Rate laws
...
Two ways :A-R-BB-R-A
The net rate of reaction of P1, P2, and P3 for the first 4 reactions are:
31212
11 2221
PkPPkPkPrr P 2
2212
12 222
kPPkPkPrr P
3231213 2223
PkPPkPPkPrr P
Continue...
1
11 21
jjP PkPrr
1jjP Total concentration of functional groups = M
MkPrr P 11 21
Similarly, for j 2
1
1
2j
ijijij MkPPPkr
For a batch reactor,
The mole balance on P1: MkPdt
dP1
1 2 ktM
MM
0
0
1 2
001 1
1
ktM
MP
1
1
2j
ijijij MkPPPkr
MkPkPdt
dPr 2
21
22 2
ktM
MM
0
0
1
ktM
PkMktM
kMdt
dP
020
0
20
2
1
12
1
1
B.C.t = 0, P2=0
ktM
ktM
ktMMP
0
0
2
002 11
1
1
0
0
2
00 11
1
j
j ktM
ktM
ktMMP
generalizing
0
0
M
MMp
120 1 j
j ppMP
Chain polymerization reactions
• Chain polymerization requires an initiator (I) and proceeds by adding one repeating unit a time :
...43
32
21
1
RRM
RRM
RRM
RMI
The molecular weight in a chain usually builds up rapidly once a chain is initiated.
Free-radical polymerization
• The basic steps in free-radical polymerization:– Initiation
• formation of an active intermediate
– Propagation or chain transfer• interaction of an active intermediate with the reactant or
product to produce another active intermediate
– Termination• deactivation of the active intermediate
Initiation• An initiation step is needed to start the polymer chain growth.• It can be achieved by adding a small amount of a chemical
that decomposes easily to form free radicals:
• Initiators can be monofunctional and form the same free radicals, or they can be multifunctional and form different radicals.
• For monofunctional initiators, the reaction sequence between monomer and initiator:
II k 202
1RMI ki
• The propagation sequence between a free radical R1 with a monomer unit is:
• In general:
• The specific reaction rate kp is assumed to be identical for the addition of each monomer to the growing chain.
• The specific reaction rate ki is often taken to be equal to kp.
32
21
RMR
RMRkp
kp
1 jkp
j RMR
Propagation
• The transfer of a radical from a growing polymer chain can occur in the following ways:– Transfer to a monomer
• A live polymer Rj transfers its free radical to the monomer to from the radical R1 and a dead polymer Pj
– Transfer to another species
– Transfer of the radical to the solvent
1RPMR jkm
j
1RPCR jkc
j
1RPSR jks
j
Chain transfer
Chain transter (cont…)
• The species involved in the various chain transfer reactions are all assumed to have the same reactivity as Ri.
• The choice of solvent in which to carry out the polymerization is important.– For example, ks is 10000 times greater in CCl4 than in
benzene.
• The specific reaction rates in chain transfer are all assumed to be independent of the chain length.
Termination
• Termination to form dead polymer occurs primarily by two mechanisms:– Addition (coupling) of two growing polymers:
– Termination by disproportionation:
kjka
kj PRR
kjkd
kj PPRR
Free-radical polymerization reaction
• For example: the polymerization of styrene at 80 C initiated by 2,2-azobisisobutyronitrile:
ismpa kkkkk ~
Addition (termination)
Propagation
Chain transfer (monomer)
Chain transfer (solvent)
Initiation
Initial concentration: 0.01 M for initiator, 3 M for the monomer, and 7 M for the solvent
Rate laws for chain polymerization
• Initiation:– Only a certain fraction f will be successful in initiating
polymer chains. The rate law for the formation of the initiator free radical:
• f is the fraction of initiator free radical successful in initiating chaining and has a typical value in the range 0.2 to 0.7.
– The rate law for the formation of R1 in the initiation step:
)(2 20 IfkrIf
))((1 IMkrr iiR
Rate laws at the initiation step
Using the PSSH for the initiator free radical, I,
)(2 20 IfkrIf ))((1 IMkrr iiR
0))(()(2 20 IMkIfkr ii
)(
)(2)( 20
Mk
IfkI
i
)(2 20 Ifkri
Identical!
Rate laws for R1
• In general:
• The total loss of R1 radicals is found by adding the loss of R1 radicals in each reaction. The rate of disappearance of R1 by termination addition is given by:
• The net rate of disappearance of the free radical R1:
11 jka
j PRR
111
131212
11 ......
jjat
jaaaat
RRkr
RRkRRkRRkRkr
2221
11
111j
jsj
jcj
jmj
jdj
japi RSkRCkRMkRRkRRkMRkrr
initiation + propagation + termination (addition) + termination (disproportionation) + chain transfer
Rate laws for Rj (j 2)
• In general, the net rate of disappearance of live polymer chains with j monomer units (species j, j2)
jsjcjmi
ijdajjpj SRkCRkMRkRRkkRRMkr
11 )()(
We will then use PSSH to solve polymerization problems.
Let represent th total concentration of the radical Rj.
1
*
jjRR
1
2*)(j
tij Rkrr
jsjcjmi
ijdajjpj SRkCRkMRkRRkkRRMkr
11 )()(
2221
11
111j
jsj
jcj
jmj
jdj
japi RSkRCkRMkRRkRRkMRkrr
(j2)
Termination term
PSSH
1
0j
jr
tt
i
k
Ifk
k
rR
)(2 20*
Total free-radical concentration
The net rate of monomer consumption, -rM: )()()( mpiM rrrr
consumption by initiator
consumption by propagation
consumption by monomer chain transferLong Chain approximation (LCA)
ip rr
tpp
jjppM k
IfkMkMRkRMkrr
)(2 20*
1
The rate of disappearance of monomer
The net rate of formation of dead polymer Pj (by addition) is:
1
12
1 jk
kkjkaP RRkr
j
The rate of formation of all dead polymers
t
aa
jPp k
IfkkRkrr
j
)()(
2
1 202*
1
Enzymatically catalyzed reactions
• An enzyme, E, is a protein or proteinlike substance with catalytic properties.
• A substrate, S, is the substance that chemically transformed at an accelerated rate because of the action of the enzyme on it.
• One enzyme can catalyze only one reaction. Unwanted products are easily controlled.
• Enzymes are produced only by living organisms (bacteria, for example).
• Enzymes usually work under mild conditions (pH 4~9, 75 ~ 160 F).
Enzymes
• Most enzymes are named in terms of the reactions they catalyze (***ase), for example:
• urease: the enzyme that catalyzes the decomposition of urea
• tyrosinase: the enzyme that attacks tyrosine
• Three major types of enzyme reactions:– soluble enzyme - insoluble substrate (e.g. laundry detergents)
– insoluble enzyme - soluble substrate (similar to packed catalytic bed rxn)
– soluble enzyme - soluble substrate (e.g. many biological rxns)
our interest
Enzymatically catalyzed rxn example
The proposed mechanisms of the catalytic action of urease which causes urea to decompose into ammonia and carbon dioxide : (Levine and LaCourse, 1967)
1.The enzyme urease reacts with the substrate urea to form an enzyme-substrate complex, E•S:
*22
122 ][ ureaseCONHNHureaseCONHNH k
2.This complex can decompose back to urea and urease:
ureaseCONHNHureaseCONHNH k 222*
22 ][
3.Or, it can react with water to give ammonia, carbon dioxide, and urease:
ureaseCONHOHureaseCONHNH k 233
2*
22 2][
S E E•S
W P
The rate of disappearance of the substrate is: )())(( 21 SEkSEkrS
SEES k 1
SESE k 2
EPWSE k 3
The net rate of formation of the E•S complex is: ))(()())(( 321 SEWkSEkSEkr SE
The enzyme is not consumed by the reactions: )()()( SEEEt PSSH 0 SEr
)()(
))(()(
321
1
WkkSk
SEkSE t
)()(
))()((
321
31
WkkSk
SEWkkr t
S
)()(
))()((
321
31
WkkSk
SEWkkr t
S
Since the reaction of urea and urease is carried outin aqueous solution (water): (W) ~ constant
k’3
m
tS KS
SEkr
)(
))((3
1
23
k
kkKm
This is the form of the “Michaelis-Menten Equation”and Km is call the Michaelis constant
Vmax
(S)
-rS
Vmax
At low substrate concentration: (S) << Km
mS K
SVr
)(max
At high substrate concentration: (S) >> Km
maxVrS
In a special case, when 2maxV
rS 2/max| Vrm s
SK
Vmax/2
Km
Km is equal to the substrate concentration at whichthe rate of reaction is equal to one-half the maximum rate.
Vmax and Km characterise the enzymatic reactions described by Michaelis-Menten kinetics.Vmax is dependent on total enzyme concentration.Km is independent of total enzyme concentration.
)(
)(
)(
3max
max
t
mS
EkV
KS
SVr
Michaelis-Menten equation
Michaelis-Menten kinetics exampleSEES k 1 SESE k 2 EPWSE k 3
Curea (kmol/m3) 0.2 0.02 0.01 0.005 0.002
-rurea (kmol/m3s) 1.08 0.55 0.38 0.2 0.09
mS KS
SVr
)(
)(max
)(
1
)(
)(1
maxmaxmax SV
K
VSV
KS
rmm
S
1/-rs
(1/S) (i.e. 1/Curea)
Slope = 0.02 =maxV
Km
Intercept = 0.75 =max
1
V
urea
ureaS C
Cr
0266.0
33.1
Batch enzymatically catalyzed rxn Artificial kidney design
murea
ureaureaurea KC
CV
dt
dCr
maxA batch reactor in liquid phase:
max
00
max
lnV
CC
C
C
V
Kt ureaurea
urea
uream
)1(0 XCC
tK
XC
K
V
Xt m
urea
m
0max
1
1ln
1
Xt 1
1ln
1
t
X
Intercept = mK
Vmax
Slope =m
urea
K
C 0
Inhibition of enzyme reactions
• The rate of enzyme-catalyzed reactions is affected by pH and inhibitors.
• Three most common types of reversible inhibition:– Competitive
• Substrate and inhibitor are usually similar molecules that compete for the same site on the enzyme.
– Uncompetitive• The inhibitor deactivetes the enzyme-substrate complex, usually by attaching
itself to both the substrate and enzyme molecules of the complex.
– Noncompetitive• Enzymes containing at least two different types of sites. The inhibitor attaches
to only one type of site and the substrate only to the other.
Bioreactors
• Microorganisms and mammalian cells are used to produce a variety of products, such as insulin(胰島素 ), most antibiotics(抗生素 ), and polymers.
• Advantages:– mild reaction conditions– high yields– can catalyze successive steps in a reaction for organisms
contain several enzymes– stereospecific (立體的 ) catalyst (single desired isomer can be formed)
In general, the growth of an aerobic organism follows:
][][][][
...][][][][][
22),,( cellsmoreproductOHCO
sourcephosphatesourceoxygensourcenitrogensourcecarboncellsetcetemperaturpHconditionsmediaculture
ductProcellsMoreSubstrate cell
The rate of this reaction is proportional to the cell concentration andthe reaction is autocatalytic.
• (1) Lag phase– little increase in cell concentration– synthesizing transport proteins for moving the
substrate into the cell– synthesizing enzymes for utilizing the new substrate– beginning the work for replicating the cell’s genetic
material
time
Log cell concentration
1 432
Four phases are included in cell growth:
• (2) Exponential growth phase– the cell’s growth rate is proportional to the cell
concentration– the cells are able to use the nutrients most efficiently
• (3) Stationary phase– the cell reach a minimum biological space where the
lack of one or more nutrients limits cell growth.– many important fermentation products, including
most antibiotics, are produced in the stationary phase
• (4) Dead phase– result of either the toxic by-product and/or the
depletion of nutrient supply
Rate laws of bioreactors
ductProcellsMoreSubstrateCells conditions
The most commonly used expression is the Monod equation for exponential growth:
cg Cr
Cell growth rate Cell concentration
Specific growth rate = ss
s
CK
C
max
Maximum specific growth reaction rateMonod constant
Substrate concentration
ss
scg CK
CCr
max account for inhibition
ss
sc
n
p
pg CK
CC
C
Cr
max
*1
(one model here)
where Cp* = product concentration at which all metabolism ceases and n = experimental constant
Other cell growth rates:
Monod equation
cs
g Ck
Cr
exp1maxTessier equation
)1(max
s
cg
kC
CrMoser equation
The cell death rate is:
cttdd CCkkr )(
Concentration of a substrate toxic to the cell
Specific natural death rate constant
Specific toxic death rate constant
Stoichiometry for cell growth
• Very complex : vary with microorganism / nutrient system; vary with environmental conditions; even more complex when more than one nutrient contributes to cell growth.
• May be simplified as:
ductProcellsMoreSubstrateCells conditions
PYCYS spsccells
//
where Yc/s is the yield coefficient :cellsnewproducetoconsumedsubstrateofmass
formedcellsnewofmassY sc /
where Yp/s is the product coefficient :productformtoconsumedsubstrateofmass
formedproductofmassY sp /
Substrate consumptionProduce new cells
Maintain a cell’s daily activities
timecellsofmass
ceaintenanmforconsumedsubstrateofmassm
typical value = 0.05 h-1
The yield coefficient, Y’c/s , account for substrate consumption for maintenance:
consumedsubstrateofmass
formedcellsnewofmassY sc /
Therefore, the rate of substrate consumption for maintenance: Csm mCr
Product formation
During the growth phase
During the stationary phase
gcpp rYr /
)(/ sspp rYr
The net rate of substrate consumption:
cppsgcss mCrYrYr //
Consumption rate by cell growths
Consumption rate to form product
Consumption rate for maintenance
cppsgcss mCrYrYr //
If the product is formed during the growth phase cgcss mCrYr / gcpp rYr /
If the product is formed during the stationary phase cppsnsn mCrYr /
snsn
csnpp CK
CCkr
where Csn is the concentration of the secondary nutrient
Mass balance on cellsFor a CSTR, a mass balance on the microorganism gives :
Rate of accumulation of cells
VrrvCCvdt
dCV dgcc
c )(00
Rate of cells entering
Rate of cells leaving
Rate of net generation of live cells
Mass balance on substratesFor a CSTR, a mass balance on the substrate gives :
Rate of accumulation of substrate
VrvCCvdt
dCV sss
s 00
Rate of substrate entering
Rate of substrate leaving
Rate of net generation of substrates
Mass balance on cellsFor a batch system, a mass balance on the microorganism gives :
Rate of accumulation of cells
Vrrdt
dCV dg
c )(
Rate of net generation of live cells
Mass balance on substratesFor a batch system, a mass balance on the substrate gives :
Rate of accumulation of substrate
VmCVrYVrdt
dCV cgcss
s )(/
Rate of substrate used for cell growth
Rate of substrate used for maintenance
In the growth phase
VmCVrYdt
dCV cpps
s )(/In the stationary phase
VrYVrdt
dCV sspp
p )(/ Rate of product formation
Example: bacteria growth in a batch reactor
A fermentation process is carried out in a batch reactor. Plot the concentrations of cells,substrate, and product and growth rates as functions of time. The initial concentration is1.0 g/dm3 and the substrate concentration is 250 g/dm3.
Values of the parameters:
)/()(03.0
/7.1
33.0
52.0
/93
3
1max
3*
hcellsgsubstrategm
dmgK
h
n
dmgC
s
p
1
/
/
/
01.0
/6.5
/45.0
/08.0
hk
ggY
ggY
ggY
d
cp
sp
sc
Mass balances
Cells: Vrrdt
dCV dg
c )(
Substrate: VrVrYdt
dCV smgcs
s )(/
Product VrYdt
dCV gcp
p )(/
Rate laws
ss
sc
p
pg CK
CC
C
Cr
52.0
*max 1
cdd Ckr
csm mCr
Stoichiometry
gcpp rYr /
cdss
sc
p
pc CkCK
CC
C
C
dt
dC
52.0
*max 1
css
sc
p
pcs
s mCCK
CC
C
CY
dt
dC
52.0
*max/ 1
gcpp rY
dt
dC/
Chemostats
• Chemostats are essentially CSTRs that contain microorganisms.
• One of the most important features of the chemostat is that is allows the operator to control the cell growth rate.– By adjusting the volumetric feed rate
Skip this part. Refer to Profs. Chen & Liu