AGITATION AND AERATION Although in many aerobic cultures, gas sparging provides the method for both mixing and aeration - it is important that these two aspects of fermenter design be considered separately. The following diagram shows the major resistances for transfer of oxygen from a gas bubble to the cell. The eight mass transfer steps are: i. Transfer from the interior of the gas bubble to the gas-liquid interface ii. Movement across the gas-liquid interface iii. Diffusion through the relatively stagnant liquid film surrounding the bubble iv. Transport through the bulk liquid v. Diffusion through the relatively stagnant liquid film surrounding the cells vi. Movement across the liquid-cell interface vii. If the cells are a floc, clump or solid particle Diffusion through the solid to the individual cell viii. Transport through the cell wall, membrane and cytoplasm to the site of reaction
These resistances occur in series and the largest of them will be rate controlling. Thus the entire mass transfer pathway may be modelled using a single mass transfer correlation. We can eliminate some of these resistances from consideration as rate controlling. Since gas-phase mass diffusivities are typically much higher than liquid-phase diffusivities, the resistance of the gas film within the bubble can neglected relative to the liquid film surrounding the bubble. Similarly, the gasliquid interfacial resistance to transport is small. Provided the liquid is well-mixed, transport through the liquid phase is generally rapid and may be neglected. Intracellular oxygen transfer resistance is negligible because of the small distances involved.three mass transfer resistances and the reaction rate remain. These mass transfer resistances are the two liquid film resistances and the intraparticle resistance. Depending on the size of the microbial particle (for example, either a single cell of 1-2 microns or a microbial pellet of several millimetres in diameter), any one of these resistances may be controlling. In the case of bacteria or yeast cells, their very small size (and hence large interfacial area) relative to that of a gas bubble will result in a liquid film surrounding the gas bubble being the rate determining step in oxygen transport. On the other hand, large microbial pellets or fungal hyphae may be of a size comparable to that of a bubble (approximately 4-5 mm) and resistance in the liquid film surrounding the solid may dominate. The intraparticle resistance results from diffusion and reaction of oxygen within the pellet or fungal matte and depends on the effective diffusivity of oxygen and relative rates of reaction. The rheology of the liquid phase also has a strong influence on the rate of mass transfer. Within a liquid of low viscosity, very little agitation is required to maintain a well-mixed liquid phase and mass transfer through the liquid bulk is unhindered. With a more viscous broth, mixing is less efficient and the presence of a yield stress may result in regions of stagnant fluid near the walls and internal of the reactor. This can result in a substantial resistance to transport through the bulk liquid phase. The liquid phase rheology also influences the velocity of the gas bubble or microbial solid phase relative to that of the fluid. Thus the liquid film resistances will be increased. Thus, two extremes may be considered - one in which diffusion through the liquid film controls and the other where bulk mixing patterns influence the rate of transport. Mixing is often represented by the dimensionless group the Power Number (N P ) Where : N P = P / (n 3 D 5 ρ ) P = External Power from Agitator (J sec -1 ) n = Impeller Rotation Speed ( sec -1 ) D = Impeller Diameter ( m ) ρ = Density of Fluid ( kg m -3 )
The Power Number has a fundamental relationship with the Reynolds Number (N Re ). Where: N Re = D 2 N ρ / µ µ = Viscosity ( N sec m -2 ) This is shown in the following figure: The Power Number allows for the calculation of the power required for and degree of mixing characterized by the Reynolds Number. There are a few important points to note here: 1) The above figure relates to a fermenter of fixed geometry with specific agitators placed at defined places within the fermenter. 2) Each reactor type must be individually characterized and may only be effectively scaled up if the dimension relationships are maintained during scale-up
3) There are some correlations for mass transfer as a function of overall variables (described later)
Three flow regions may be identified: Laminar Region: The laminar region corresponds to N Re < 10 for many impellers. For stirrers with very small wall clearance, such as an anchor and helical ribbon mixer, laminar flow persists until N Re = 100 or greater In this region: or N P I / N Re P = k 1 ( D 3 N 2 µ ) Where k is a proportionality constant.
The value of k for various impellers are: Impeller Type k 1 N P (at N Re = 10 5 ) Ruston Turbine 70 5-6 Paddle 35 2 Marine Impeller 40 0.35 Anchor 420 0.35 Helical Ribbon 1000 0.35 The Power requirement for laminar flow is independent of the density of the fluid but directly proportional to the fluid viscosity. Turbulent Region The Power Number is independent of the Reynolds Number in turbulent flow. Therefore: and N P = Constant P = N P ρ N 3 D 5 The N P for turbines is much higher than for most other impellers, indicating that impellers transmit more power to the fluid than other designs. The Power required for turbulent flow is independent of the viscosity of the fluid but proportional to the fluid density. The turbulent region is fully developed at N Re > 10 3-10 4 for most small impellers in baffled vessels. Without baffles, turbulence is not fully developed until N Re > 10 5. Even at this Reynolds Number, the value of N P (non-baffled) is reduced to between half to one tenth of the N P (baffled).
Transition Region: Between laminar and turbulent flow lies the transition region. Both fluid density and fluid viscosity affect Power requirements in this region. There is normally a gradual transition from laminar to fully-developed turbulent flow pattern and the Reynolds Number range for the transition region depends on the system geometry. Hence, depending on the operating regime, the density and viscosity of the fluid may exert considerable influence on the system behaviour. Impeller Types There are many different impeller types and each will have a different effect with respect to two important concepts in mixing: 1. Mixing and Circulation Patterns within the Bioreactor (Radial and Axial Impellers) 2. Shear Stress Imparted by the baffle on the bioreactor contents
Radial Impeller Axial Impeller
In addition, each will have a viscosity range over which they are efficient.
The rheological properties of microbial broths are primarily determined by the cell concentration and cell morphology. High cell concentrations tend to produce viscous broths. Filamentous morphology often leads to non-newtonian behaviour. Metabolic products such as polysaccharides, extracellular proteins and solid substrates may also lead to non-newtonian behaviour.
Pseudoplastic and dilatant fluids follow the Ostwald-de Waele or Power Law: where: τ = K γ n τ = Shear Stress K = Consistency Index γ = Shear Rate n = Flow Behaviour Index When n < 1, the fluid exhibits pseudoplastic behaviour. When n > 1, the fluid is dilatant and when n = 1, the fluid exhibits Newtonian behaviour. For Power law fluids, the apparent viscosity is expressed as: µ = τ / γ = K γ n-1 Apparent viscosity decreases (shear thinning) with increasing shear rate for pseudoplastic fluids and increases with shear rate (shear thickening)for dilatant fluids. Some fluids do not produce motion until some finite shear stress has been exceeded and flow initiated. Once they yield stress has been exceeded, Bingham plastics behave like Newtonian Fluids; a constant ratio K P exists between the shear stress and change in shear rate: τ = τ o + K P γ Another common plastic behaviour is described by the Casson Equation: τ 0.5 = τ o 0.5 + K P γ 0.5 Once the yield stress has been exceeded, the behaviour of the Casson fluid is pseudoplastic. Hence, rheological behaviour may change considerably during the course of a fermentation.
Broths containing low concentrations of approximately spherical particles are usually Newtonian and their viscosity varies with cell concentration. Yeast and bacterial cultures exhibit this behavior and this behavior may be represented by: where: µ S = µ L (1 + f (J,φ)) µ S Viscosity of the suspension µ L = Viscosity of the suspending liquid J = Geometric ratio for the particles φ = Volume fraction of the particles At low volume fraction, yeast suspensions may be represented by Einstein s Equation: µ S = µ L ( 1 + 2.5 φ ) At higher concentrations (up to 14% volume fractions), the vand Equation represents the data more closely: µ S = µ L ( 1 + 2.5 φ + 7.52φ 2 )
As the cell concentration increases, the apparent viscosity increases in a nonlinear manner. Various correlations have been proposed, typically of the form: where: η x M η = apparent viscosity = rate of stress / rate of strain (this relationship is linear for Newtonian fluids) X = Biomass Concentration M = Exponent = 1.0-2.65 Filamentous Molds = 0.3-3.0 for Pellets Returning to Oxygen Mass Transfer: Here, the mass transfer coefficient k L a (min -1 ) is introduced which describes the combined mass transfer rate ( k L ) and the interfacial area through which the mass transfer occurs ( a ). Since there is no accumulation of oxygen at steady state at any place within the fermenter (including the solid (cell) / liquid interface) : where: do 2 / dt = k L a (C SAT - C) - QO 2. X C SAT = Saturated Oxygen Concentration in the Liquid phase ( mol / L ) C = Actual Oxygen Concentration in the Liquid Phase ( mol / L ) QO 2 = Specific oxygen Uptake Rate ( mol / g / h ) X = Biomass Concentration ( g / L ) k L a = Oxygen mass Transfer Coefficient ( h -1 ) For practical design purposes, methods for estimating k L a are necessary. Three methods are commonly used: 1) Steady-State Method 2) Non Steady-State Method 3) Sulfite Method
Steady-State Method The first method assumes steady state and no holdup of gases in the fermenter. Hence: k L a ( C SAT - C ) = QO 2. X The specific oxygen uptake rate is estimated using gas phase measurements of the concentration of oxygen and carbon dioxide in the exit steam and using the known volumetric flow rate entering the fermenter, the known composition of air entering the fermenter and a nitrogen balance to estimate the volumetric flow exiting the fermenter. The cell concentration, the actual dissolved oxygen concentration and the fermenter volume are known or can be measured. Fully aerated sterile media ia used to estimate C SAT. Unsteady State Method This method involves turning off the air supply and then turning it back on again after he dissolved oxygen level has appreciably fallen.
In the unsteady-state equation: dc / dt = k L a ( C SAT - C ) - QO 2. X The specific rate of oxygen uptake and the cell concentration are assumed to remain constant over the relatively short period of the test. Hence: dc / dt = k L a. C SAT k L a. C - constant = constant - k L a. C constant = constant - k L a. C Hence, a plot of dc / dt v- C gives a straight line with a slope of -k L a For example: Time (h) Dissolved Oxygen Concentration ( mg / L ) 0 0.52 0.5 0.70 1.0 0.93 2.0 1.23 3.0 1.55 4.0 1.80 5.0 2.00 gives k L a = 0.171 min -1 Sulfide Method This is a chemical method based on the reaction: NaSO 3 + 0.5O 2 = Na 2 SO 4 Cu 2+ / Co 2+ This reaction is very fast and it assumed that the rate limiting step is the transfer of oxygen from the gas phase to the liquid phase and that the dissolved oxygen concentration is zero. The inherent assumption is that the sulfite solution has the same rheological properties as fermentation broth is clearly not accurate. Thus, this method should
be considered to give the maximum capacity for oxygen transfer for a given aeration system. Oxygen solubility is a function of: 1. Temperature 2. Salts 3. Organics
An empirical correlation exists for correcting values of oxygen solubility in water for the effects of cations, anions and sugars:
where:
Hence, this is a major variable from fermentation to fermentation and within a particular fermentation. There are correlations available for these effects. The experimental estimation of k L a is supplemented by a large number of correlations. The simplest of these are correlations for stirred fermenters containing non-coalescing, non-viscous media: k L a = 2 x 10-3 ( P / V ) 0.7. u G 0.2 P = Power ( W ) V = Fluid Volume ( m 3 ) u G = Superficial Gas Velocity ( m /s ) Another common correlation is the relationship between gassed power and ungassed power. This attempts to correlate the equivalent power that must be expended in a gassed system to obtain the equivalent amount of mixing as in a ungassed system under known conditions (eg impeller diameter and rotational speed). These correlations are highly specific to particular systems. Other correlations exist for a range of practically important operational factors: 1. Bubble Coalescence and Breakup 2. Gas Hold Up 3. Effect of Surfactants and Viscous Liquids 4. Liquid Circulation Patterns On the issue of Liquid Circulation Patterns, there are increasingly sophisticated methods available for analysis of these: 1. Flow Visualisation Using Dyes and Optical Fibres 2. Computer Analysis Using Non-Ideal Reactor Analysis 3. Velocity Gradients Measurements 4. Computational Fluid Dynamics Relevant to the Liquid Circulation patterns is the Kolmogaroff Scale of Mixing. Turbulant flow occurs when the fluid now longer travels along streamlines but moves across them as cross currents. The kinetic energy of turbulent flow is directed into regions of rotational flow called eddies. Masses of eddies of various size exist during turbulent flow. Large eddies are continuously formed by the stirrer and these are broken down into smaller eddies which produce even smaller eddies, Eddies possess kinetic energy which is continuously transferred to eddies of smaller size. This process is referred to as dispersion. When the eddies become very small, they can no longer sustain rotational motion and their kinetic energy is converted to and dissipated as heat. Most stirrer energy is lost as heat very little from collision with the fermenter walls.
The degree of homogeneity possible as a result of dispersion is limited by the size of the smallest eddies which may be formed in a particular fluid. This size is given as the the Kolmogaroff Scale of Mixing or Kolmogaroff Scale of Turbulence ( λ ). where: λ = ( ν 3 / ε ) 0.25 λ = Characteristic Dimension of the Smallest Eddy ( m ) ν = Kinematic Viscosity of Fluid ( = µ / ρ ) ( m 2 s -1 ) µ = Viscosity ( kg m -1 s -1 ) ρ = Density ( kg m -3 ) ε = Rate of Energy Dissipation / Mass of Fluid ( W.s kg -1 ) Hence, the greater the power input, the smaller the eddy. λ is also dependent on viscosity and smaller eddies are produced in less viscous liquids. For low viscosity liquids, λ is normally in the range of 30-100µm. This is the smallest scale of mixing that can be achieved by dispersion. To achieve a smaller scale, we must rely on diffusion. Although slower, within eddies of 30-100 µm, this is achieved in I sec.