Cell Kinetics and Fermenter Design Ref.: Ref.: Chap 6, Biochemical Engineering, James M. Lee Prepared by: Engr.. Ray Limuel Engr Limue l C. Laureano La ureano
Introduction Understanding Understanding the the growth kinetics of microbial, animal, or plant cells is important for the design and operation of fermentation systems employing them. Cell kinetics deals with te rate of cell growth and how it is affected by various chemical and physical physical conditions.
Introduction Understanding Understanding the the growth kinetics of microbial, animal, or plant cells is important for the design and operation of fermentation systems employing them. Cell kinetics deals with te rate of cell growth and how it is affected by various chemical and physical physical conditions.
Introduction Cell kinetics is the result of numerous complicated complicated networks networks of biochemical and chemical reactions reactions and an d transport phenomena, which involves multiple phases and multicomponent systems. During the course of growth, the heterogeneous mixture of young and old cells is continuously changing and adapting itself in the media environment environment which is also continuously changing in physical and chemical conditions. As a result, accurate accurate
Introduction mathematical modeling of growth kinetics is impossible to achieve. Even with such a realistic model, this approach is usually useless because the model may contain many parameters which are impossible to determine.
Introduction Assumptions are made to be able to arrive at simple models which are useful for fermenter design and performance predictions. The simplest model is the unstructured, distributed model which is based on the following assumptions: 1. Cells can be represented by a single component, such as cell mass, cell number,
Introduction or concentration of protein, DNA, or RNA. This is true for balanced growth, since douling of cell mass for balanced growth is accompanied by doubling of all other measurable properties of the cell population. 2. The population of cellular mass is distributed uniformly throughout the culture. The cell suspension can be regarded as homogeneous solution. The heterogeneous nature of cells can
Introduction be ignored. The cell concentration can be expressed as dry weight per unit volume. The medium is formulated so that only one component may be limiting the reaction rate. All other components are present at sufficiently high concentrations, so that minor changes do not significantly affect the reaction rate.
Introduction Fermenters are also controlled so that environmental parameters such as pH, temperature, and dissolved oxygen concentration are maintained at a constant level.
Various Models for Cell Kinetics Cell Components Population
Unstructured
Structured
Distributed
Cells are represented by a single component, which is uniformly distributed throughout the culture.
Multiple cell components, uniformly distributed throughout the culture interact with each others.
Segregated
Cells are represented by a single component, but they form a heterogeneous mixture.
Cells are composed of multiple components and form a heterogeneous mixture.
Definitions CX
= cell concentration, dry cell weight per unit volume
CN
= cell number density, number of cells per unit volume
= cell density, wet cell weight per unit volume of cell mass
dCX/dt = change of dry cell concentration with time rX
= growth rate of cells on a dry weight basis
dCN/dt = change of cell number density with time rN
= growth rate of cells on a number basis
= division rate of cells on a number basis = dlog2CN/dt
Definitions dCX/dt and rX: dCX/dt – change of cell concentration in fermenter, which may include the effects of the input and the output flow rates, cell recycling, and other operating conditions of a fermenters rX – actual growth rate of the cells *rX = dCX/dt in BATCH operation dCN/dt and dCX and dt:
Growth rate based on the number of cells and that based on cell weight are not necessarily the same because the average size of the cells may vary considerably from one phase to another. When the mass of an individual cell increases without division, the growth rate based on cell weight increases, while that based on the number of cells stays the same.
Definitions
During exponential growth period, which is the phase that we are most interested in from an engineer’s point of view, the growth rate based on the cell number and that based on cell weight can be assumed to be proportional to each other.
dCX/dt, dCN/dt, dlog2CN/dt () Sometimes, the growth rate can be confused with the division rate, which is defined as the ratio of cell division per unit time. N C N C N 0 2
Definitions Average division rate:
N t
1 t
log2 C N log2 C N 0
Division rate at time t:
d log2 C N dt
Definitions The growth rate defined as the change in cell number with time is the slope of the C N vs t curve, while the division rate is the slope of the log 2CN vs t curve. As explained later, the division rate is constant during the exponential growth period, while the growth rate is not.
Growth Cycle for Batch Cultivation If you inoculate unicelullar microorganisms into a fresh sterilized medium and measure the cell number density with respect to time and plot it, you may find that there are six phases of growth and death. 1. Lag phase: A period of time when the change of cell number is zero. 2. Accelerated growth phase: The cell number starts to increase and the division rate increases to reach a maximum.
Growth Cycle for Batch Cultivation 3. Exponential growth phase: The cell number increases exponentially as the cells start to divide. The growth rate is increasing during this phase, but the division rate which is proportional to dlnCN0/dt , is constant at its maximum value. 4. Decelerated growth phase: After the growth rate reaches a maximum, it is followed by the deceleration of both growth rate and division rate.
Growth Cycle for Batch Cultivation 5. Stationary phase: The cell population will reach a maximum value and will not increase any further. 6. Death phase: After nutrients available for the cells are depleted, cells will start to die and the number of viable cells will decrease.
Growth Cycle for Batch Cultivation
Lag Phase The lag phase (or initial stationary, or latent) is an initial period of cultivation during which the change of cell number is zero is negligible or zero. Even though the cell number does not increase, the cells may grow in size during this period. Length of this period is affected by: (a) Type and age of MOs (b) Size of inoculum (c) Culture conditions
Lag Phase If
MOs are inoculated from a medium with a low nutrient concentration to a medium with a high concentration – longer lag phase period because the cells must produce the enzymes necessary for the metabolization of the available nutrients If the MOs are moved from high to low nutrient concentration – lag phase period is short
Lag Phase Size of inoculum – if a small
amount of cells are inoculated into a large volume, longer lag phase period *For a large-scale operation of the cell culture, it is our objective to make this lag phase as short as possible. At the end of the lag phase, when growth begins, the division rate increases gradually and reaches a maximum value in the exponential growth phase. This is called the accelerated growth phase.
Exponential Growth Phase In unicellular organisms, the progressive doubling of cell number results a continually increasing rate of growth in the population. A bacterial culture undergoing balanced growth mimics a first-order autocatalytic chemical reaction (Carberry, 1976; Levenspiel, 1972). Therefore, the rate of the cell population increase at any particular time is proportional to the number density (CN) of bacteria present at that time:
Exponential Growth Phase r N
dC N dt
C N
Where: µ = specific growth rate (h –1)
1 dC N C N dt
d ln C N dt
Exponential Growth Phase
d ln C N dt
d log2 C N ln 2 ln 2 dt
If µ is constant with time during the exponential growth period, then C N
C N 0
dC N C N
t
dt t 0
upon integration: C N C N 0 exp t t 0
Exponential Growth Phase Doubling time, td, is given by t d
ln 2
1
Classification of microorganisms in terms of growth-rate on temperature Temperature, C Group
Minimum
Optimum
Maximum
Thermophiles Mesophiles Psychrophiles Obligate Facultative R. Y. Stanier, M. Doudoroff, and E. A. Adelberg , “The Microbial World,” 3d ed., p. 316, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1970.
Factors Affecting the Specific Growth Rate Substrate Concentration: Monod equation – empirical expression based on the form of equation normally associated with enzyme kinetics or gas adsorption:
maxC S K S C S
KS is equal to the concentration of nutrient when the specific growth rate is half of its maximum value (µmax).
Factors Affecting the Specific Growth Rate Several other models have been proposed to improve the Monod equation:
maxC S 2
K I 1 C S K I 2C S
max 1 e C S / K S
max
1 K S C S
max n C S
Factors Affecting the Specific Growth Rate If several substances can limit the growth of a microorganism, the following model is employed: max
C 1
C 2
K 1 C 1 K 2 C 2
...
A term ke is used when the limiting nutrient is the energy source for the culture
maxC S K S C S
k e
Factors Affecting the Specific Growth Rate Product Concentration: As cells grow they produce metabolic by-products which can accumulate in the medium. The growth of microorganisms is usually inhibited by the products, whose effect can be added to the Monod equation as follows: C S K P max K S C S K P C P N
C S C P 1 max K S C S C Pm
Factors Affecting the Specific Growth Rate Which CPm is the maximum product concentration above which cells cannot grow due to product inhibition
Other conditions: The specific growth rate of MOs is also affected by medium pH, temperature, and oxygen supply. The optimum pH and temperature differ from one MO to another.
Stationary Phase and Death Phase The growth of microbial populations is normally limited either by the exhaustion of available nutrients or by the accumulation of toxic products of metabolism. As a consequence, the rate of growth declines and growth eventually stops. At this point a culture is said to be in the stationary phase. The stationary phase is usually followed by a death phase in which the organism in the population die. Death occurs either because of the depletion of the cellular reserves of energy, or the accumulation of toxic products. Like growth, death is an exponential function. In some cases, the organisms not only die but also disintegrate, a process called lysis.
Batch or Plug-Flow Fermenter An ideal stirred fermenter is assumed to be well mixed so that the contents are uniform in compositions at all times. Another ideal fermenter is the plug-flow fermenter, the analysis of which is analogous to the ideal batch fermenter.
Batch or Plug-Flow Fermenter In a tubular-flow fermenter , nutrients and microorganisms enter one end of a cylindrical tube and the cells grow while they pass through. Since the long tube and lack of stirring device prevents complete mixing of the fluid, the properties of the flowing stream will vary in both longitudinal and radial direction. However, the variation in the radial direction is small compared to that in the
Batch or Plug-Flow Fermenter longitudinal. The ideal tubular-flow fermenter without radial variations is called a plug-flow fermenter (PFF). In reality, the PFF fermenter is hard to be found. However, the packed-bed fermenter and multi-staged fermenter can be approximated as PFF.
Batch or Plug-Flow Fermenter Even though the steady-state PFF is operated in a continuous mode, the cell concentration of an ideal batch fermenter after time t will be the same as that of a steady-state PFF at the longitudinal location where the residence time is equal to t.
Batch or Plug-Flow Fermenter
Batch or Plug-Flow Fermenter If liquid medium is inoculated with a seed culture, the cell will start to grow exponentially after the lag phase. dC X dt
r X C X
Integrating the above eqn gives C X
C X 0
dC X r X
C X
C X 0
dC X
C X
t
dt t t 0 t 0
Batch or Plug-Flow Fermenter According to the prior eqn, the batch growth time t – t0 is the area under the 1/rX versus CX curve between CX0 and CX as shown below.
Batch or Plug-Flow Fermenter At this time just note that the curve is U shaped, which is characteristic of autocatalytic reactions:
SX XX The rate for an autocatalytic reaction is slow at the start because the concentration of X which acts as a biological catalyst is low. It increases as cells multiply and reaches a maximum rate. As the substrate is depleted and the toxic products accumulate, the rate decreases to a low value.
Batch or Plug-Flow Fermenter If Monod kinetics adequately represents the growth rate during the exponential period: C X
C X 0
K S C S dC X maxC S C X
t
dt t 0
The growth yield (YX/S) is defined as C X C X C X 0 Y X / S C S C S 0 C S
Batch or Plug-Flow Fermenter Cell concentration change with respect to time: K S Y X / S C X K S Y X / S C S 0 t t 0 max 1 ln ln C X 0 C S 0Y X / S C X 0 C X 0 C S 0Y X / S C S
MM eqn VS Monod eqn dC P dt
r maxC S
dC P
K M C S
dt
maxC S C X K S C S
Batch or Plug-Flow Fermenter Monod kinetic parameters, µmax and KS, cannot be estimated with a series of batch runs as easily as the MM parameters for an enzyme reaction. In the case of cell cultivation, the initial rate of reaction in a batch run is always zero due to the presence of a lag phase, during which Monod kinetics does not apply. It should be noted that even though the Monod equation has the same form as the MM equation, the rate eqn is different.
Ideal Continuous Stirred-tank Fermenter Chemostat Cellular growth is usually limited by one essential nutrient, and other nutrients are in excess. For a chemostat at steady state, the nutrient, product, and cell concentrations are constant. For this reason, the name chemostat refers to constant chemical environment (for cell culture).
Ideal Continuous Stirred-tank Fermenter
Ideal Continuous Stirred-tank Fermenter A turbidostat is a CSTF in which the cell concentration in the culture vessel is maintained constant by monitoring the optical density of the culture and controlling the feed flow rate.
Ideal Continuous Stirred-tank Fermenter
Ideal Continuous Stirred-tank Fermenter Microbial population can be maintained in a state of exponential growth over a long period of time by using a system of continuous culture. Continuous culture can be operated as chemostat or turbidostat . Chemostat – flow rate is set at a particular value and the rate of growth of the culture adjusts to this flow rate Turbidostat – turbidity is set at a constant level by adjusting the flow rate
Ideal Continuous Stirred-tank Fermenter
Ideal Continuous Stirred-tank Fermenter Material Balance Input – Ouput + Generation = Accumulation FC Xi FC X Vr X V
dC X dt
For steady-state condition CSTF: dCX/dt = 0 m
V F
C X C Xi r X
Ideal Continuous Stirred-tank Fermenter The shorter the residence time in reaching a certain cell concentration, the more effective the fermenter. If the input stream is sterile (CXi = 0), and the cells in a CSTF are growing exponentially (rX = µCX) m
1
1
D
Ideal Continuous Stirred-tank Fermenter The specific growth rate of a microorganism can be controlled by changing the medium flow rate and CS can be calculated with a known residence time and the Monod kinetic parameters as: C S
K S
m max 1
Note: eqn above is only applicable when mµmax > 1. If mµmax < 1, the growth rate of the cells is less than the rate of cells leaving the outlet stream. Consequently, all of the cells in the fermenter will be washed out.
Ideal Continuous Stirred-tank Fermenter If the growth yield is constant, then C X Y X / S C Si C S
Correlation of CX: For CP:
K S C X Y X / S C Si m max 1 K S C P C Pi Y P / S C Si m max 1
Evaluation of Monod Kinetic Parameters If a certain microorganism follows Monod kinetics, the plot of 1/µ versus 1/CS yields the values of µmax and KS by reading the intercept and the slope of the straight line. However, 1/µ approaches infinity as the substrate concentration decreases. This gives undue weight to measurements made at low substrate concentrations and insufficient weight to measurements at high substrate concentrations.
Evaluation of Monod Kinetic Parameters Linear Equations in the determination of Monod kinetic parameters: 1
C S
1
max K S
max
K S
1
max C S 1
max
max K S
C S
C S
Example 6.2 A chemostat study was performed with yeast. The medium flowrate was varied and the steady-state concentration of cells and glucose in the fermenter were measured and recorded. The inlet concentration of glucose was set at 100 g/L. The volume of the fermenter contents was 500 mL. The inlet stream was sterile.
Example Example 6.2 Flow rate F, mL/h
Cell Concentration CX, g/L
Substrate Concentration
31 50 71 91 200
5.97 5.94 5.88 5.76 0
0.5 1.0 2.0 4.0 100
CS, g/L
a. Find Find the the rat rate e equa equatio tion n for for cel celll gro growth. wth. b. What What should should be the range ange of the flow flow rat ate e to prevent prevent washout of the cells?
Example Example 6.2 Given: F CSi = 100 g/L
V = 500 mL *inlet stream is sterile
CS CX
Example Example 6.2 Given: Data: Flow rate F, mL/h
Cell Concentration CX, g/L
Substrate Concentration
31 50 71 91 200
5.97 5.94 5.88 5.76 0
0.5 1.0 2.0 4.0 100
CS, g/L
Example 6.2 Required: a. rate eqn for cell growth b. flow rate to prevent washout of cells
Example 6.2 Solution: a. Making use of the linear form of Monod eqn to solve for the kinetic parameters µ max and KS: 1
Note:
1
m
1
max
K S
1
max C S
V F
*Exclude point when F = 200 mL/h because we need to consider only the exponential growth phase.
Example 6.2 Solution: Upon linear regression: µmax = 0.2514 h –1 KS = 1.5256 g/L Therefore:
r X C X
r X
maxC S K S C S
0.2514C S 1.5256 C S
C X
C X
g L h
Example Example 6.2 Solution: b. To prevent washout of cells, the cell concentra concentration tion should be maintained so that it will be greater than zero. K S 0 C X Y X / S C Si m m ax 1
Solving for m gives: m
V F
K S C Si C Si m ax
Example Example 6.2 Solving for F F F
VC Siμ max KS CSi 0.5L 100 10 0 g/L g/L 0.2514h 1
1.5256g/L g/L 100 10 0 g/L g/L
F 0.1238 L/h
Example 11-1/p. 561] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
A strain of mold was grown in a batch culture on glucose and the following data were obtained as shown in the table. Do the following: (a) Calculate Calculate the maximum maximum net specific specific growth growth rate rate.. (b) Calculate Calculate the apparent apparent grow growth th yield. (c) What maximum maximum cell cell concen concentr tration ation could one expect if 150 g of glucose were used with the same size inoculum? (d) How many many generations generations of cells are there there in the culture for for part (c)?
Example 11-1/p. 561] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
Example 11-1/p. 561] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu (a) To obtain the maximum net specific growth rate, we calculate the rates using the finite difference scheme as illustrated in the following table:
Example 11-1/p. 561] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
From the table, we find that the maximum net specific growth rate, μ max 0.0752/h (b)YX/S
ΔCX
ΔCS
20.9 1.0 50.0 0.3
0.4004
(c) CX,max CX0 YX/SCS0 1.0 0.4004150 61.0604 g cells/ (d) n
l nCX CX0 l n2
l n61.0 1.0 l n2
5.93 6
Problem 11.1/p. 601] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
A simple, batch fermentation of an aerobic bacterium growing on methanol gave the results shown on the table. Calculate: (a) Maximum growth rate, max (b) Specific growth rate of the cells, YX/S (c) Mass doubling time, td (d) Saturation constant, KS (e) Specific growth rate, net at t = 10 h
Problem 11.1/p. 601] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
Problem 11.2/p. 601] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu The data in Table P11.2 were obtained for Pyrodictium occultum at 98C. Run 1 was carried out in the absence of yeast extract and run 2 with yeast extract. Both runs initially contained Na 2S. The vol% of the growth product H2S collected above the broth was reported as a function of time as shown in Table P11.2. (a) What is the lag time with and without the yeast extract? (b) What is the difference in the maximum specific growth rates, max, of the bacteria with and without the yeast extract? (c) How long is the stationary phase? (d) During which phase does the majority production of H2S occur?
Problem 11.2/p. 601] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
Problem 11.4/p. 604] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
The data in Table P11.4 were obtained for the effect of temperature on the fermentative production of lactic acid by a strain of Lactobacillus delbruekii . From these data, calculate the value of the activation energy for this process.
Problem 11.4/p. 604] Bioprocess Engineering: Kinetics, Biosystems, Sustainability, and Reactor Design by Shijie Liu
Supplementary Problems 1. Escherichia coli grows with a doubling time of 0.5 h in the exponential growth phase. (a) What is the value of the specific growth rate? (b) How much time would be required to grow the cell culture from 0.1 kg dry cell/m3 to 10 kg dry cell/m 3? 2. E. coli grows from 0.10 kg dry cell/m 3 to 0.50 kg dry cell/m3 in 1 h. (a) Assuming the exponential growth during this period, evaluate the specific growth rate. (b) Evaluate the doubling time during the exponential growth phase. (c) How much time would be required to grow from 0.10 kg dry cell/m 3 to 1.0 kg dry cell/m3? You may assume the exponential growth during this period.
Productivity of CSTF Normally, the productivity of the fermenter is expressed as the amount of product produced per unit time and volume. If the inlet stream is sterile (CXi = 0), the productivity of the cell is equal to CX/m.
Productivity of CSTF Cell Concentration and Residence Time for Maximum Productivity The cell productivity at steady-state CSTF C X
m
r X
maxC S K S C S
C X
The productivity is maximum when
dr X dC X
0
Productivity of CSTF Optimum cell concentration for maximum productivity C X ,opt Y X / S C Si
1
Where:
K S C Si K S
Productivity of CSTF Optimum substrate concentration: C S ,opt
C Si
1
Optimum residence time: m,opt
max 1
Comparison of Batch and CSTF Since 1/rX versus CX curve is U shaped, the following assumptions are applied: 1. The most productive fermenter system is a CSTF operated at the cell concentration as which value 1/rX is minimum because it requires the smallest residence time. 2. If the final cell concentration to be reached is in the stationary phase, the batch fermenter is a better choice than the CSTF because the residence time required for the batch is smaller than that for CSTF.
Comparison of Batch and CSTF The residence time required for a batch or steady-state PFF to reach a certain level of cell concentration is C X
b t 0
C X 0
dC X r X
Multiple Fermenters Connected in Series Choosing the optimum fermenter system for maximum productivity depends on the shape of the 1/rX versus CX curve and the process requirement, such as the final conversion. In the 1/rX versus CX curve, if the final cell concentration is less than CX,opt, one fermenter is better than two fermenters connected in series, because two CSTFs connected in series require more residence time than one CSTF does.
Multiple Fermenters Connected in Series If the final cell concentration is much larger than CX,opt, the best combination of two fermenters for a minimum total residence time is a CSTF operated at CX,opt followed by a PFF. A CSTF operated at CX,opt followed by another CSTF connected in series is also better than one CSTF.
Multiple Fermenters Connected in Series
Multiple Fermenters Connected in Series
CSTF and PFF in Series
CSTF and PFF in Series Working Equations For the CSTF C S 1
K S
m max 1
K S C X 1 Y X / S C Si m1 max 1 K S C P 1 C Pi Y P / S C Si m1 max 1
CSTF and PFF in Series Working Equations For the PFF C X 2
P 2
C X 1
Y X / S
dC X r X
C X 2
K S C S dC X
C X 1
maxC S C X
C X 2 C X 1 C S 1 C S 2
P 2 max
K S Y X / S
C X 1 C S 1Y X / S
C K S Y X / S C 1 ln X 2 ln S 1 C X 1 C X 1 C S 1Y X / S C S 2
Multiple CSTFs in Series
Multiple CSTFs in Series Working Equations Material balance on the nth steady-state CSTF
F C Xn1C Xn V n r Xn 0 r Xn Y X / S
maxC SnC Xn K S C Sn C Xn C Xn1 C Sn1 C Sn
Dilution rates: D1
F V 1
r X 1 C X 1
D2
F V 2
r X 2 C X 2
Example 6.3 Suppose you have a microorganism that obeys Monod equation: dCX dt
μ max CSCX
KS CS
where µmax = 0.7 h –1 and KS = 5 g/L. The cell yield (YX/S) is 0.65. You want to cultivate this microorganism in either one fermenter or two in series. The flow rate and the substrate concentration of the inlet stream should be 500 L/h and 85 g/L, respectively. The substrate concentration of the outlet stream must be 5 g/L.
Example 6.3 a. If you use one CSTF, what should be the size of the fermenter? What is the cell concentration of the outlet stream? b. If you use two CSTFs in series, what sizes of the two fermenters will be most productive? What are the concentration of cells and substrate in the outlet stream of the first fermenter? c. What is the best combination of fermenter types of volumes if you use two fermenters in series?
Example 6.3 Given: (a) For a single CSTF F = 500 L/h CSi = 85 g/L
µmax = 0.7 h –1 KS = 5 g/L YX/S = 0.65
CS = 5 g/L
Example 6.3 Required
Solving for V
(a) V
V
(b)CX
μ
F μ max C S
KS CS
Solution For a single CSTF D μ
F
FKS C S μ max CS
500L/h5 5 g/L V 0.7h1 5 g/L
F V
V 1 428.5714L 1 429L
Example 6.3 The outlet cell concentration is
CX YX/S CSi CS
CX 0.6585 5 g/L CX 52 g/L
Example 6.3 Given (b)
Example 6.3 Required (a) V1 and V2 (b)CX1 and CS1
Example 6.3 (a) For two CSTFs in series, the first fermenter must be operated at CX,opt and CS,opt. α
KS CSi KS
5 85
5
CX1 CX,opt YX/S CSi CS1 CS,opt τm1
τ m,opt
CSi α 1
α α 1
4.2426
0.6585
85 4.2426 1
α μ max α 1
4.2426 4.2426 1
44.7113g/L
16.2133g/L 4.2426
0.74.2426 1
1.8691h
Example 6.3 V1 τm1F 1.8691h500L/h 934.55L 935L
For the second fermenter: Input – Output + Generation = Accumulation
FCX1 CX2 V2rX 0
Example 6.3 Rearranging the OMB for V 2 gives
FCX2 CX1 KS CS2 50052 455 5 V2 0.7552 μ max CS2CX2 V2 192.3077L 193L
The total volume of the two CSTFs is V = V1 + V2 = 935 L + 193 L = 1 128 L The total volume of the two CSTFs in series is 20% smaller than a single CSTF.
Example 6.3 Solution for Requirement (c) The best combination is a CSTF operated at the maximum rate followed by a PFF. τ P2
1 μ max
KS YX/S CX2 KS YX/S CS1 1 ln ln CX1 CS1YX/S CX1 CX1 CS1YX/S CS2
1
52 50.65 50.65 16 1 ln ln 0.32h τ P2 0.7 45 160.65 45 45 160.65 5 V2 τP2F 0.32500 160L
Example 6.3 V = V1 + V2 = 950 L + 160 L V = 1 110 L
The total volume employing a CSTF followed by a PFF is 22% smaller than a single CSTF. The difference in volume of a CSTF-CSTF series versus CSTF-PFF series is negligible.
Problem 6.6 The growth rate of E. coli in synthetic medium can be expressed by Monod kinetics as rX
0.935CSC X 0.71 CS
[g/L h]
where CS is the concentration of limiting substrate, glucose. You are going to cultivate E. coli in a steady-state CSTF (working volume: 10 L) with flow rate of 7 L/h. The initial substrate concentration is 10 g/L and the cell yield constant (YX/S) is 0.60. The feed stream is sterile.
Problem 6.6 a. What will be the doubling time and the division rate of the cells in the CSTF? =0.7950 h; =1.2595 h –1
b. What will be the cell and substrate concentrations of the outlet stream? C =2.1148 g/L; C =4.7311 g/L S
X
c. If you connect one more 10-L CSTF to the first one, what will be the cell and substrate concentration in the second fermenter? C S2= d. If you increase the flow rate e. from 7 to 10 L/h for these two fermenters connected in series, what will happen and why? Make a recommendation to avoid the problem if
Problem 6.9 Suppose you have an organism that obeys the Monod equation: dCX μ max CSCX dt KS CS where µmax = 0.5 h –1 and KS = 2 g/L. The organism is being cultivated in a steady-state CSTF, where F = 100 L/h, CSi = 50 g/L, and YX/S = 0.5. a. What size of vessel will give the maximum total rate of cell production? b. What are the substrate and cell concentrations of the optimum fermenter in part (a)?
Problem 6.9 c. If the existing flow from the first fermenter in part (a) is fed to a second fermenter (CSTF), what should be the size of the second fermenter to reduce the substrate concentration to 1 g/L? d. If the existing flow from the first fermenter in part (a) is fed to a second fermenter whose size is the same as the first, what will be the cell and substrate concentrations leaving the second fermenter?
CSTF with Cell Recycling The cellular productivity in a CSTF increases with an increase in the dilution rate and reaches a maximum value. If the dilution rate is increased beyond the maximum point, the productivity will be decreased abruptly and the cells will start to be washed out because the rate of cell generation is less than that of the cell loss from the outlet stream. Therefore, the productivity of the fermenter is limited due to the loss of cells with the outlet stream. One way to improve the reactor productivity is to recycle the cell by separating the
CSTF with Cell Recycling cells from the product stream using a cross-flow filter unit.
CSTF with Cell Recycling The high cell concentration maintained using cell recycling will increase the cellular productivity since the growth rate is proportional to he cell concentration. However, there must be a limit in the increase of the cellular productivity with increased cell concentration because in a high cell concentration environment, the nutrient-transfer rate will be decreased due to overcrowding and aggregation of cells. The maintenance of the extremely high cell concentration is also not practical because the filter unit will fail more frequently at the higher cell concentrations.
CSTF with Cell Recycling If all cells are recycled back into the fermenter, the cell concentration will increase continuously with time and a steady-state will never be reached. Therefore, to operate a CSTF will recycling in a steady-state mode, we need to have a bleeding stream. The material balance for cells in the fermenter with a cell recycling unit is FC Xi BC X Vr X V
dC X dt
CSTF with Cell Recycling It should be noted that actual flow rates of the streams going in and out of the filter unit do not matter as far as overall material balance is concerned. For a steady-state CSTF with cell recycling and a sterile feed, D
m
where , the bleeding ratio, is defined as
B F
CSTF with Cell Recycling Now, D instead of D is equal to the specific growth rate. When = 1, cells are not recycled, therefore, D = . If the growth rate can be expressed by Monod kinetics, CS is given by C S
K S m max
which is valid when mmax > . The cell concentration in the fermenter can be calculated from the value of CS as Y X S C Si C S C X
Problem 6.13 A strain of yeast is being cultivated in a 30-L CSTF with a cell recycling system (cell settler) as shown in the following figure. The cell settler was designed so hat the cell concentration of its outlet stream is 30 percent of that of its inlet stream, whereas the substrate concentrations of the two streams are the same. The growth rate of the cells can be represented by the Monod kinetics with the parameters: KS = 0.05 g/L, max = 0.3 h –1, and YX/S = 0.025. Calculate the steady-state substrate and cell concentrations in the fermenter. The inlet
Problem 6.13 substrate concentration is 100 g/L and the flow rate is 20 L/h. The feed stream is sterile.
For 7/14/2015 (ST_SY14-15) Solve: 1. Problem 6.3 2. Problem 6.7 3. Problem 6.10 4. Problem 6.12
