Biochemical Engineering

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1 Biocheical Engineering Jaes M. ee Departent of Cheical Engineering Washington State University Pullan, WA Chapter 9 Agitation and Aeration ntroduction Basic Mass-Transfer Concepts Correlation for Mass-Transfer Coefficient Measureent of nterfacial Area Correlations for a and D Gas Hold-Up Power Consuption Deterination of Oxygen-Absorption Rate Correlation for k a Scale-Up Shear-Sensitive Mixing Noenclature Probles References ast Update: August 10, 001

2 9 ii Agitation and Aeration 001 by Jaes M. ee, Departent of Cheical Engineering, Washington State University, Pullan, WA This book was originally published by Prentice-Hall nc. in 199. You can download this file and use it for your personal study of the subject. This book cannot be altered and coercially distributed in any for without the written perission of the author. f you want to get a printed version of this text, please contact Jaes ee. All rights reserved. No part of this book ay be reproduced, in any for or by any eans, without perission in writing fro the author.

3 Chapter 9 Agitation and Aeration 9.1. ntroduction One of the ost iportant factors to consider in designing a ferenter is the provision for adequate ixing of its contents. The ain objectives of ixing in ferentation are to disperse the air bubbles, to suspend the icroorganiss (or anial and plant tissues), and to enhance heat and ass transfer in the ediu. Since ost nutrients are highly soluble in water, very little ixing is required during ferentation just to ix the ediu as icroorganiss consue nutrients. However, dissolved oxygen in the ediu is an exception because its solubility in a ferentation ediu is very low, while its deand for the growth of aerobic icroorganiss is high. For exaple, when the oxygen is provided fro air, the typical axiu concentration of oxygen in aqueous solution is on the order of 6 to 8 g/. Oxygen requireent of cells is, although it can vary widely depending on icroorganiss, on the order of 1 g/ h. Even though a ferentation ediu is fully saturated with oxygen, the dissolved oxygen will be consued in less than one inute by organiss if not provided continuously. Adequate oxygen supply to cells is often critical in aerobic ferentation. Even teporary depletion of oxygen can daage cells irreversibly. Therefore, gaseous oxygen ust be supplied continuously to eet the requireents for high oxygen needs of icroorganiss, and the oxygen transfer can be a ajor liiting step for cell growth and etabolis. Mixing provided by a laboratory shaker apparatus is adequate to cultivate icroorganiss in flasks or test tubes. Rotary or reciprocating action of a shaker is effective to provide gentle ixing and surface aeration. For bench-, pilot-, and production-scale ferenters, the ixing is usually provided by echanical agitation with or without aeration. The ost widely used arrangeent is the radial-flow ipeller with six flat blades ounted on a disk (Figure 9.1), which is called flat-blade disk turbine or Rushton turbine. Radial-flow ipellers (paddles and turbines) produce flow radially fro the turbine blades toward the side of the vessel, where the flow splits into two

4 9- Agitation and Aeration directions: one part goes upward along the side, back to the center along the liquid surface, and down to the ipeller region along the agitating shaft; and the other goes downward along the side and botto, then back to the ipeller region. On the other hand, the axial flow ipellers (propellers and pitched blade paddles) generate flow downward to the tank botto, then up the side and back down the center to the ipeller region. Therefore, the flat-blade disk turbine has the advantage of liiting the short-circuiting of gas along the drive shaft by forcing the gas, introduced fro below, along the path into the discharge jet. Mass-Transfer Path: The path of gaseous substrate fro a gas bubble to an organelle in a icroorganis can be divided into several steps (Figure 9.) as follows: 1. Transfer fro bulk gas in a bubble to a relatively unixed gas layer. Diffusion through the relatively unixed gas layer 3. Diffusion through the relatively unixed liquid layer surrounding the bubble 4. Transfer fro the relatively unixed liquid layer to the bulk liquid 5. Transfer fro the bulk liquid to the relatively unixed liquid layer surrounding a icroorganis 6. Diffusion through the relatively unixed liquid layer 7. Diffusion fro the surface of a icroorganis to an organelle in which oxygen is consued Steps 3 and 5, the diffusion through the relatively unixed liquid layers of the bubble and the icroorganis, are the slowest aong those outlined previously and, as a result, control the overall ass-transfer rate. Agitation and aeration enhance the rate of ass transfer in these steps and increase the interfacial area of both gas and liquid. n this chapter, we study various correlations for gas-liquid ass transfer, interfacial area, bubble size, gas hold-up, agitation power consuption, and voluetric ass-transfer coefficient, which are vital tools for the design and operation of ferenter systes. Criteria for the scale-up and shear sensitive ixing are also presented. First of all, let's review basic ass-transfer concepts iportant in understanding gas-liquid ass transfer in a ferentation syste.

5 9.. Basic Mass-Transfer Concepts A Agitation and Aeration Molecular Diffusion in iquids When the concentration of a coponent varies fro one point to another, the coponent has a tendency to flow in the direction that will reduce the local differences in concentration. Molar flux of a coponent A relative to the average olal velocity of all constituent J A is proportional to the concentration gradient dc A /dz as dca JA= DAB (9.1) dz which is Fick's first law written for the z-direction. The D AB in Eq. (9.1) is the diffusivity of coponent A through B, which is a easure of its diffusive obility. Molar flux relative to stationary coordinate N A is equal to CA dca N = A ( NA NB) DAB C + dz (9.) where C is total concentration of coponents A and B and N B is the olar flux of B relative to stationary coordinate. The first ter of the right hand side of Eq. (9.) is the flux due to bulk flow, and the second ter is due to the diffusion. For dilute solution of A, N J (9.3) Diffusivity: The kinetic theory of liquids is uch less advanced than that of gases. Therefore, the correlation for diffusivities in liquids is not as reliable as that for gases. Aong several correlations reported, the Wilke-Chang correlation (Wilke and Chang, 1955) is the ost widely used for dilute solutions of nonelectrolytes, D! AB ( ξm ) = µ A B 0.6 VbA T (9.4) When the solvent is water, Skelland (1974) recoends the use of the correlation developed by Other and Thakar (1953). D! AB = µ V ba 13 (9.5)

6 9-4 Agitation and Aeration The preceding two correlations are not diensionally consistent; therefore, the equations are for use with the units of each ter as S unit as follows: D! AB diffusivity of A in B, in a very dilute solution, /s M olecular weight of coponent B, kg/kol B T teperature, K µ solution viscosity, kg/ s V ba solute olecular volue at noral boiling point, 3 /kol /kol for oxygen [See Perry and Chilton (p.3-33, 1973) for extensive table] ξ association factor for the solvent:.6 for water, 1.9 for ethanol, 1.5 for ethanol, 1.0 for unassociated solvents, such as benzene and ethyl ether. Exaple 9.1 Estiate the diffusivity for oxygen in water at 5 C. Copare the predictions fro the Wilke-Chang and Other-Thakar correlations with the experiental value of /s (Perry and Chilton, p. 3-5, 1973). Convert the experiental value to that corresponding to a teperature of 40 C. Solution: Oxygen is designated as coponent A, and water, coponent B. The olecular volue of oxygen V ba is /kol. The association factor for water ξ is.6. The viscosity of water at 5 C is kg/ s (CRC Handbook of Cheistry and Physics, p. F-38, 1983). n Eq. (9.4) n Eq. (9.5)! D AB! D AB [.6(18)] 98 9 = =.5 10 /s ( ) ( 0.056) = =.7 10 /s ( ) ( 0.056) f we define the error between these predictions and the experiental value as

7 !! ( D ) AB ( D ) predicted AB! ( D ) Agitation and Aeration 9-5 = experiental % error 100 AB experiental The resulting errors are 9.6 percent and 9. percent for Eqs. (9.4) and (9.5), respectively. Since the estiated possible error for the experiental value is ± 0 percent (Perry and Chilton, p. 3-5, 1973), the estiated values fro both equations are satisfactory.! Eq. (9.4) suggests that the quantity DABµ T is constant for a given liquid syste. Though this is an approxiation, we ay use it here to estiate the diffusivity at 40 C. Since the viscosity of water at 40 C is kg/ s fro the handbook, 4!! D AB at 40 C =.5 10 = / s f we use Eq. (9.5), D µ! AB 1.1 is constant, !! 9 D AB at 40 C =.5 10 = / s Mass-Transfer Coefficient The ass flux, the rate of ass transfer q G per unit area, is proportional to a concentration difference. f a solute transfers fro the gas to the liquid phase, its ass flux fro the gas phase to the interface N G is qg NG = = kg( CG CG ) (9.6) i A where C G and C Gi is the gas-side concentration at the bulk and the interface, respectively, as shown in Figure 9.3. k G is the individual ass-transfer coefficient for the gas phase and A is the interfacial area. Siilarly, the liquid-side phase ass flux N is q N = = k( C C ) i (9.7) A where k is the individual ass-transfer coefficient for the liquid phase.

8 9-6 Agitation and Aeration Gas iquid C G k /k G C G C i C Gi C Gi C C C i Figure 9.3 Concentration profile near a gas-liquid interface and an equilibriu curve. Since the aount of solute transferred fro the gas phase to the interface ust equal that fro the interface to the liquid phase, N = N (9.8) Substitution of Eq. (9.6) and Eq. (9.7) into Eq. (9.8) gives CG CG k i = C C k G i G (9.9) which is equal to the slope of the curve connecting the (C, C G ) and (C i, C Gi ), as shown in Figure 9.3. t is hard to deterine the ass-transfer coefficient according to Eq. (9.6) or Eq. (9.7) because we cannot easure the interfacial concentrations, C i or C Gi. Therefore, it is convenient to define the overall ass-transfer coefficient as follows: * * ( C C ) ( C C ) N = N = K = K (9.10) G G G G * where C G is the gas-side concentration which would be in equilibriu with the existing liquid phase concentration. Siilarly, C * is the liquid-side concentration which would be in equilibriu with the existing gas-phase concentration. These can be easily read fro the equilibriu curve as shown in Figure 9.4. The newly defined K G and K are overall ass-transfer coefficients for the gas and liquid sides, respectively.

9 Agitation and Aeration 9-7 C G C Gi C G * C C i C * Figure 9.4 The equilibriu curve explaining the eaning of * C G and Exaple 9. Derive the relationship between the overall ass-transfer coefficient for liquid phase K and the individual ass-transfer coefficients, k and k G. How can this relationship be siplified for sparingly soluble gases? Solution: According to Eqs. (9.7) and (9.10), * ( ) * C k ( C C ) = K C C (9.11) Therefore, by rearranging Eq. (9.11) * 1 1 C C = K k C C Since i i * ( C C) + ( C C ) 1 i = k C C * ( C C ) 1 1 = + k k C C i i i i (9.1) k ( C C ) = k ( C C ) (9.13) G G G By substituting Eq. (9.13) to Eq. (9.1), we obtain i * C C 1 1 = + = + K k k C C k k M G G G G i i (9.14)

10 9-8 Agitation and Aeration which is the relationship between K, k, and k G. M is the slope of the line connecting (C, C G ) and ( C, C G ) as shown in Figure 9.4. * For sparingly soluble gases, the slope of the equilibriu curve is very steep; therefore, M is uch greater than 1 and fro Eq. (9.14) K k (9.15) Siilarly, for the gas-phase ass-transfer coefficient, K G k (9.16) Mechanis of Mass Transfer Several different echaniss have been proposed to provide a basis for a theory of interphase ass transfer. The three best known are the two-fil theory, the penetration theory, and the surface renewal theory. The two-fil theory supposes that the entire resistance to transfer is contained in two fictitious fils on either side of the interface, in which transfer occurs by olecular diffusion. This odel leads to the conclusion that the ass-transfer coefficient k is proportional to the diffusivity D AB and inversely proportional to the fil thickness z f as k D G AB = (9.17) z f Penetration theory (Higbie, 1935)assues that turbulent eddies travel fro the bulk of the phase to the interface where they reain for a constant exposure tie t e. The solute is assued to penetrate into a given eddy during its stay at the interface by a process of unsteady-state olecular diffusion. This odel predicts that the ass-transfer coefficient is directly proportional to the square root of olecular diffusivity k D AB = πt e 1/ (9.18) Surface renewal theory (Danckwerts, 1951) proposes that there is an infinite range of ages for eleents of the surface and the surface age distribution function φ(t) can be expressed as st φ() t = se (9.19)

11 Agitation and Aeration 9-9 where s is the fractional rate of surface renewal. This theory predicts that again the ass-transfer coefficient is proportional to the square root of the olecular diffusivity k ( sd ) 1/ = (9.0) All these theories require knowledge of one unknown paraeter, the effective fil thickness z f, the exposure tie t e, or the fractional rate of surface renewal s. ittle is known about these properties, so as theories, all three are incoplete. However, these theories help us to visualize the echanis of ass transfer at the interface and also to know the exponential dependency of olecular diffusivity on the ass-transfer coefficient Correlation for Mass-Transfer Coefficient Mass-transfer coefficient is a function of physical properties and vessel geoetry. Because of the coplexity of hydrodynaics in ultiphase ixing, it is difficult, if not ipossible, to derive a useful correlation based on a purely theoretical basis. t is coon to obtain an epirical correlation for the ass-transfer coefficient by fitting experiental data. The correlations are usually expressed by diensionless groups since they are diensionally consistent and also useful for scale-up processes. The diensionless group iportant for correlations can be derived by using Buckingha Pi theory as shown in the following exaple. Exaple 9.3 The ass transfer coefficient k of oxygen transfer in ferenters is a function of Sauter ean diaeter D 3, diffusivity D AB, and density ρ c viscosity µ c of continuous phase (liquid phase). Sauter-ean diaeter D 3 can be calculated fro easured drop-size distribution fro the following relationship, D 3 = n i= 1 n i= 1 AB nd i nd i 3 i i (9.1) Deterine appropriate diensionless paraeters that can relate the ass transfer coefficient by applying the Buckingha-Pi theore.

12 9-10 Agitation and Aeration Solution: The first step of Buckingha-Pi theore is to count the total nuber of paraeters. n this case, there are five paraeters: k, D 3, D AB, ρ c, and µ c, all of which can be expressed with three principle units: ass M, length, and tie T. Therefore, Nuber of paraeter: n = 5 Nuber of principle diension: r = 3 n developing the diensionless groups, every diensionless group will contain r =3 of repeating paraeters and the total nuber of diensioness group will be n as Nuber of repeating paraeter: = r =3 Nuber of diensionless group: n = 5 3 = Therefore, we need to choose three repeating paraeters. Since we want to develop a correlation for k, we want it to be only one diensionless group and therefore, cannot be a repeating paraeter. You can choose any three out of D 3, D AB, ρ, and µ, although you ay end up different set of diensionless groups depending on how you select the. f we choose D 3, D AB, and ρ as repeating paraeters, two diensionless groups can be constructed as Π = D D ρ k (9.) a b c 1 3 AB c Π = D D ρ µ (9.3) a' b' c' 3 AB c c Now, the exponents of the above equations can be deterined so that both groups can be diensionless by substituting each paraeter with their diensions as Π = (9.4) a b 3 c 1 ( /T) (M/ ) (/T) Collecting all exponents for M, T, and unit, M: c = 0 T: b 1 = 0 b = 1 : a + b 3c = 0 a = 1 Therefore, the first diensionless group is

13 Π Agitation and Aeration 9-11 kd = D D ρ k = (9.5) AB c DAB which is known as Sherwood nuber N Sh. Siilarly, Π ρ µ µ c = D3DAB c c = (9.6) DABρc which is known as Schidt nuber, N Sc. Earlier studies in ass transfer between the gas-liquid phase reported the voluetric ass-transfer coefficient k a. Since k a is the cobination of two experiental paraeters, ass-transfer coefficient and interfacial area, it is difficult to identify which paraeter is responsible for the change of k a when we change the operating condition of a ferenter. Calderbank and Moo-Young (1961) separated k a by easuring interfacial area and correlated ass-transfer coefficients in gas-liquid dispersions in ixing vessels, and sieve and sintered plate colun, as follows: 1. For sall bubbles less than.5 in diaeter, or k 1/3 /3 ρµ g Sc ρ c c = 0.31N (9.7) 1/3 1/3 Sh 0.31 Sc Gr N = N N (9.8) where N Gr is known as Grashof nuber and defined as 3 3 D ρ g ρ Grashof nuber, NGr = (9.9) c µ The ore general fors which can be applied for both sall rigid sphere bubble and suspended solid particle are c or k 3 1/3 /3 ρµ cg Sc ρ c DAB = N D (9.30) 1/3 1/3 Sh Sc Gr N = + N N (9.31)

14 9-1 Agitation and Aeration Eqs. (9.30) and (9.31) were confired by Calderbank and Jones (1961), for ass transfer to and fro dispersions of low-density solid particles in agitated liquids which were designed to siulate ass transfer to icroorganiss in ferenters.. For bubbles larger than.5 in diaeter, or k 1/3 1/ ρµ g Sc ρ c c = 0.4N (9.3) 1/ 1/3 Sh 0.4 Sc Gr N = N N (9.33) Based on the three theories reviewed in the previous section, the exponential dependency of olecular diffusivity on the ass-transfer coefficient is expected to be soe value between 0.5 and 1. t is interesting to note that the exponential dependency of olecular diffusivity in the preceding correlations is /3 or 1/, which is within the range predicted by the theories. Exaple 9.4 Estiate the ass-transfer coefficient for the oxygen dissolution in water 5 C in a ixing vessel equipped with flat-blade disk turbine and sparger by using Calderbank and Moo-Young's correlations. Solution: The diffusivity of the oxygen in water 5 C is /s (Exaple 9.1). The viscosity and density of water at 5 C is kg/ s (CRC Handbook of Cheistry and Physics, p. F-38, 1983) and kg/3 (Perry and Chilton, p. 3-71, 1973), respectively. The density of air can be calculated fro the ideal gas law, 5 PM (9) ρair = = = 1.186kg/ 3 RT (98) Therefore the Schidt nuber, N Sc 4 µ = = = ρd (.5 10 ) Substituting in Eq. (9.7) for sall bubbles, AB 3

15 k = 0.31(357.) = / /s Substituting in Eq. (9.3) for large bubbles, k = 0.4(357.) = 1/ /s Agitation and Aeration /3 ( )( )(9.81) ( ) 4 1/3 ( )( )(9.81) ( ) Therefore, for the air-water syste, Eqs. (9.7) and (9.3) predict that the ass-transfer coefficients for sall and large bubbles are and /s, respectively, which are independent of power consuption and gas-flow rate. Eq. (9.3) predicts that the the ass-transfer coefficient for the oxygen dissolution in water 5 C in a ixing vessel is /s, regardless of the power consuption and gas-flow rate as illustrated in the previous exaple proble. opes De Figueiredo and Calderbank (1978) reported later that the value of k varies fro to /s, depending on the power dissipation by ipeller per unit volue (P /v ) as 0.33 P k v (9.34) This dependence of k on P /v was also reported by Prasher and Wills (1973) based on the absorption of CO into water in an agitated vessel, as follows: P = 1/ k 0.59DAB vµ c 0.5 (9.35) which is diensionally consistent. However, this equation is liited in its use because the correlation is based on only one gas-liquid syste. Akita and Yoshida (1974) evaluated the liquid-phase ass-transfer coefficient based on the oxygen absorption into several liquids of different physical properties using bubble coluns without echanical agitation. Their correlation for k is

16 9-14 Agitation and Aeration k D gd gd 1/ 3 1/4 3/8 3 ν c 3 3 ρc = 0.5 D AB D AB ν σ c (9.36) where ν c and σ are the kineatic viscosity of the continuous phase and interfacial tension, respectively. Eq. (9.36) is applicable for colun diaeters of up to 0.6, superficial gas velocities up to 5 /s, and gas hold-ups to 30 percent Measureent of nterfacial Area To calculate the gas absorption rate q for Eq. (9.7), we need to know the gas-liquid interfacial area, which can be easured eploying several techniques such as photography, light transission, and laser optics. The interfacial area per unit volue can be calculated fro the Sauter-ean diaeter D 3 and the volue fraction of gas-phase H, as follows: 6H a = (9.37) D The Sauter-ean diaeter, a surface-volue ean, can be calculated by easuring drop sizes directly fro photographs of a dispersion according to Eq. (9.1). Photographic easureent of drop sizes is the ost straightforward ethod aong any techniques because it does not require calibration. However, taking a clear picture ay be difficult, and reading the picture is tedious and tie consuing. Pictures can be taken through the base or the sidewall of a ixing vessel. To eliinate the distortion due to the curved surface of a vessel wall, the vessel can be iersed in a rectangular tank, or a water pocket can be installed on the wall (Skelland and ee, 1981). One liitation of this approach is that the easureent of drop size is liited to the regions near the wall, which ay not represent the overall dispersion in a ferenter. Another ethod is to take pictures by iersing the extension tube with a objective lense in the tank as described by Hong and ee (1983). Drop size distribution can be indirectly easured by using the lighttransission technique. When a bea of light is passed through a gas-liquid dispersion, light is scattered by the gas bubbles. t was found that a plot of the extinction ratio (reciprocal of light transittance, 1/T) against interfacial area per unit volue of dispersion a, gave a straight line, as follows (Vereulen et al., 1955; Calderbank, 1958): 3

17 1 Agitation and Aeration a T = + (9.38) n theory, 1 is unity and is a constant independent of drop-size distribution as long as all the bubbles are approxiately spherical. The light transission technique is ost frequently used for the deterination of average bubble size in gas-liquid dispersion. t has the advantages of quick easureent and on-line operation. The probes are usually ade of irror-treated glass rods (Vereulen et al., 1955), internally blackened tubes with irrors (Calderbank, 1958), or fiber optic light guide (Hong and ee, 1983) Correlations for a and D Gas Sparging with No Mechanical Agitation eaving the vicinity of a sparger, the bubbles ay break up or coalesce with others until an equilibriu size distribution is reached. A stable size is achieved when turbulent fluctuations and surface tension forces are in balance (Calderbank, 1959). Akita and Yoshida (1974) deterined the bubble-size distribution in bubble coluns using a photographic technique. The gas was sparged through perforated plates and single orifices, while various liquids were used. The following correlation was proposed for the Sauter-ean diaeter: gd V C s ν gd c C D 3 gd 6 Cρ = c D C σ and for the interfacial area ad C 1 gdcρ = 3 σ gdc ν 0.1 H 1.13 (9.39) (9.40) where D C is bubble colun diaeter and V s is superfical gas velocity, which is gas flow rate Q divided by the tank cross-sectional area. Eqs. (9.39) and (9.40) are based on data in coluns of up to 0.3 in diaeter and up to superficial gas velocities of about 0.07 /s Gas Sparging with Mechanical Agitation Calderbank (1958) correlated the interfacial areas for the gas-liquid dispersion agitated by a flat-blade disk turbine as follows:

18 9-16 Agitation and Aeration 1. For V s < 0.0 /s, for a ( P v) ρ V s = 1.44 c 0.6 σ V t 1/ (9.41) where ND N Re < 0,000 i V s N Re i is the ipeller Reynolds nuber defined as N DNρ = (9.4) µ Re i c 0.7 The interfacial area for ( ) 0.3 N / 0,000 Re ND i V s > can be calculated fro the interfacial area a 0 obtained fro Eq. (9.41) by using the following relationship. log.3a 0.7 ND ( 5 = ) N i a V 10 Re 0 c s 0.3 (9.43). For V s > 0.0 /s, Miller (1974) odified Eq. (9.41) by replacing the aerated power input by echanical agitation P with the effective power input P e, and the terinal velocity V t with the su of the superficial gas velocity and the terinal velocity V s + V t. The effective power input P e cobines both gas sparging and echanical agitation energy contributions. The odified equation is a ( P v) ρ V s = 1.44 e c 0.6 σ Vt V + s 1/ (9.44) Calderbank (1958) also correlated the Sauter-ean diaeter for the gas-liquid dispersion agitated by a flat-blade disk turbine ipeller as follows: 1. For dispersion of air in pure water, 0.6 σ D3 = H ( P v) ρ (9.45) c. n electrolyte solutions (NaCl, Na SO 4, and Na 3 PO 4 ),

19 Agitation and Aeration 9-17 D = σ d H µ ( P v) ρ µ c c 3. n alcoholic solution (aliphatic alcohols), 0.5 (9.46) D = σ d H µ ( P v) ρ µ c c 0.5 (9.47) where all constants are diensionless except in Eq. (9.45). Again the preceding three equations can be odified for high gas flow rate (V s > 0.0 /s) by replacing P by P e, as suggested by Miller (1974) Gas Hold-Up Gas hold-up is one of the ost iportant paraeters characterizing the hydrodynaics in a ferenter. Gas hold-up depends ainly on the superficial gas velocity and the power consuption, and often is very sensitive to the physical properties of the liquid. Gas hold-up can be deterined easily by easuring level of the aerated liquid during operation Z F and that of clear liquid Z. Thus, the average fractional gas hold-up H is given as H Z Z Z F = (9.48) F Gas Sparging with No Mechanical Agitation: n a two-phase syste where the continuous phase reains in place, the hold-up is related to superficial gas velocity V s and bubble rise velocity V t (Sridhar and Potter, 1980): Vs H = (9.49) V + V Akita and Yoshida (1973) correlated the gas hold-up for the absorption of oxygen in various aqueous solutions in bubble coluns, as follows: s 1/1 H 0.0 gdcρ V c s = 4 (1 H ) σ gd C t 1/8 3 gdc ν c (9.50) Gas Sparging with Mechanical Agitation: Calderbank (1958) correlated gas hold-up for the gas-liquid dispersion agitated by a flat-blade disk turbine ipeller as

20 9-18 Agitation and Aeration r Side View Top View gauge Figure 9.5 A torque table to easure the power consuption for echanical agitation. H 1/ 0.4 1/ 0. VH s 4 +( ) ( ) V c s = P v ρ V 0.6 t V t σ (9.51) where has a unit () and V t = 0.65 /s when the bubble size is in the range of 5 diaeter. The preceding equation can be obtained by cobining Eqs. (9.41) and Eq. (9.45) by eans of Eq. (9.37). For high superficial gas velocities (V s > 0.0 /s), replace P and V t of Eq. (9.51) with effective power input P e and V t + V s, respectively (Miller, 1974) Power Consuption The power consuption for echanical agitation can be easured using a torque table as shown in Figure 9.5. The torque table is constructed by placing a thrust bearing between a base and a circular plate, and the force required to prevent rotation of the turntable during agitation F is easured. The power consuption P can be calculated by the following forula P= π rnf (9.5) where N is the agitation speed, and r is the distance fro the axis to the point of the force easureent. Power consuption by agitation is a function of physical properties, operating condition, and vessel and ipeller geoetry. Diensional analysis provides the following relationship:

21 Agitation and Aeration 9-19 P ND N D D H D = ρnd T W f ρ,,,,, g D µ D D (9.53) The diensionless group in the left-hand side of Eq. (9.53) is known as power nuber N P, which is the ratio of drag force on ipeller to inertial force. The first ter of the right-hand side of Eq. (9.53) is the ipeller Reynolds nuber N Re i, which is the ratio of inertial force to viscous force, and the second ter is the Froude nuber N Fr, which takes into account gravity forces. The gravity force affects the power consuption due to the foration of the vortex in an agitating vessel. The vortex foration can be prevented by installing baffles. For fully baffled geoetrically siilar systes, the effect of the Froude nuber on the power consuption is negligible and all the length ratios in Eq. (9.53) are constant. Therefore, Eq. (9.53) is siplified to P ( ) N = α N β (9.54) Figure 9.6 shows Power nuber-reynolds nuber correlation in an agitator with four baffles (Rushton et al., 1950) for three different types of ipellers. The power nuber decreases with an increase of the Reynolds nuber and reaches a constant value when the Reynolds nuber is larger than 10,000. At this point, the power nuber is independent of the Reynolds nuber. For the noral operating condition of gas-liquid contact, the Reynolds nuber is usually larger than 10,000. For exaple, for a 3-inch ipeller with an agitation speed of 150 rp, the ipeller Reynolds nuber is 16,5 when the liquid is water. Therefore, Eq. (9.54) is siplified to N = constant for N > 10,000 (9.55) p The power required by an ipeller in a gas sparged syste P is usually less than the power required by the ipeller operating at the sae speed in a gas-free liquids P o. The P for the flat-blade disk turbine can be calculated fro P o (Nagata, 1975), as follows: D 1.96 P DT D D N DN Q 10 = 3 P o D T ν g ND log 19 Re Re (9.56)

22 9-0 Agitation and Aeration 100 N P 10 Flat six-blade turbine 1 Propeller E+00 1.E+01 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 NRe i Figure 9.6 Power nuber Reynolds nuber correlation in an agitator with four baffles (each 0.1D T ). (Rushton, et al., 1950) Exaple 9.5 A cylindrical tank (1. diaeter) is filled with water to an operating level equal to the tank diaeter. The tank is equipped with four equally spaced baffles whose width is one tenth of the tank diaeter. The tank is agitated with a 0.36 diaeter flat six-blade disk turbine. The ipeller rotational speed is.8 rps. The air enters through an open-ended tube situated below the ipeller and its voluetric flow rate is /s at 1.08 at and 5 C. Calculate the following properties and copare the calculated values with those experiental data reported by Chandrasekharan and Calderbank (1981): P = 697 W; H = 0.0; k a = s 1. a. Power requireent b. Gas hold-up c. Sauter-ean diaeter d. nterfacial area e. Voluetric ass-transfer coefficient Solution: a. Power requireent: The viscosity and density of water at 5 C is kg/ s (CRC Handbook of Cheistry and Physics, p.

23 Agitation and Aeration 9-1 F-38, 1983) and kg/ 3 (Perry and Chilton, p. 3-71, 1973), respectively. Therefore, the Reynolds nuber is ρnd (997.08)(.8)(0.36) NRe = = = 406,357 4 µ which is uch larger than 10,000, above which the power nuber is constant at 6. Thus, P o = ρ = = N D 6(997.08)(.8) (0.36) 794 W The power required in the gas-sparged syste is fro Eq. (9.56) P (.8) (0.36) = 7 3 P o log (.8)0.36 Therefore, P = 687 W b. Gas hold-up: The interfacial tension for the air-water interface is kg/s (CRC Handbook of Cheistry and Physics, p. F-33, 1983). The volue of the the dispersion is π 3 v = 1. (1.) = The superficial gas velocity is 4Q 4( ) Vs = = = /s πdt π1. Substituting these values into Eq. (9.51) gives H H = / / 4 (687 /1.43) The solution of the preceding equation for H gives H = 0.03 c. Sauter-ean diaeter: n Eq. (9.45)

24 9- Agitation and Aeration D = (687/1.43) = = d. nterfacial area a: n Eq. (9.37) 6H 6 (0.03) a = = = 37.7 D e. Voluetric ass-transfer coefficient: Since the average size of bubbles is 4, we should use Eq. (9.3). Then, fro Exaple 9.4 Therefore, ka k = /s 1 = (37.7) = 0.017s 4 1 The preceding estiated values copare well with those experiental values. The percent errors as defined in Exaple 9.1 are 1.4 percent for the power consuption, 15 percent for the gas hold-up, and 1.7 percent for the voluetric ass-transfer coefficient Deterination of Oxygen-Absorption Rate To estiate the design paraeters for oxygen uptake in a ferenter, you can use the correlations presented in the previous sections, which can be applicable to a wide range of gas-liquid systes in addition to the air-water syste. However, the calculation procedure is lengthy and the predicted value fro those correlations can vary widely. Soeties, you ay be unable to find suitable correlations which will be applicable to your type and size of ferenters. n such cases, you can easure the oxygen-transfer rate yourself or use correlations based on those experients. The oxygen absorption rate per unit volue q a /v can be estiated by qa * * Ka( C C) ka( C C) v = = (9.57) Since the oxygen is sparingly soluble gas, the overall ass-transfer coefficient K is equal to the individual ass-transfer coefficient k. Our objective in ferenter design is to axiize the oxygen transfer rate with the iniu power consuption necessary to agitate the fluid, and also

25 Agitation and Aeration 9-3 Table 9.1 Solubility of Oxygen in Water at 1 at. a Teperature Solubility C ol O / g O / a Data fro nternational Critical Tables, Vol., New York: McGraw-Hill Book Co., 198, p. 71. iniu air flow rate. To axiize the oxygen absorption rate, we have to axiize k, a, C * C. However, the concentration difference is quite liited for us to control because the value of C * is liited by its very low axiu solubility. Therefore, the ain paraeters of interest in design are the ass-transfer coefficient and the interfacial area. Table 9.1 lists the solubility of oxygen at 1 at in water at various teperatures. The value is the axiu concentration of oxygen in water when it is in equilibriu with pure oxygen. This solubility decreases with the addition of acid or salt as shown in Table 9.. Norally, we use air to supply the oxygen deand of ferenters. The axiu concentration of oxygen in water which is in equilibriu with air C * at atospheric pressure is about one fifth of the solubility listed, according to the Henry's law, C * p = O (9.58) H O ( T) where po is the partial pressure of oxygen and H ( ) O T is Henry's law constant of oxygen at a teperature, T. The value of Henry's law constant can * be obtained fro the solubilities listed in Table 9.. For exaple, at 5 C, C

26 9-4 Agitation and Aeration Table 9. Solubility of Oxygen in Solution of Salt or Acid at 5 C. a Conc. Solubility, ol O / ol/ HCl H SO 4 NaCl a Data fro F. Todt, Electrocheishe Sauer-stoffessungen, Berlin: W. de Guy & Co., is 1.6 ol/ and p O is 1 at because it is pure oxygen. By substituting! these values into Eq. (9.58), we obtain H O (5 C) is at /ol. Therefore, the equilibriu concentration of oxygen for the air-water contact at 5 C will be * 0.09 at C = = 0.64 ol O / = 8.43 g/ at /ol deally, oxygen-transfer rates should be easured in a ferenter which contains the nutrient broth and icroorganiss during the actual ferentation process. However, it is difficult to carry out such a task due to the coplicated nature of the ediu and the ever changing rheology during cell growth. A coon strategy is to use a synthetic syste which approxiates ferentation conditions Sodiu Sulfite Oxidation Method The sodiu sulfite oxidation ethod (Cooper et al., 1944) is based on the oxidation of sodiu sulfite to sodiu sulfate in the presence of catalyst (Cu ++ or Co ++ ) as Cu or Co Na SO3+ O Na SO4 (9.59) This reaction has following characteristics to be qualified for the easureent of the oxygen-transfer rate:

27 Agitation and Aeration The rate of this reaction is independent of the concentration of sodiu sulfite within the range of 0.04 to 1 N.. The rate of reaction is uch faster than the oxygen transfer rate; therefore, the rate of oxidation is controlled by the rate of ass transfer alone. To easure the oxygen-transfer rate in a ferenter, fill the ferenter with a 1 N sodiu sulfite solution containing at least M Cu ++ ion. Turn on the air and start a tier when the first bubbles of air eerge fro the sparger. Allow the oxidation to continue for 4 to 0 inutes, after which, stop the air strea, agitator, and tier at the sae instant, and take a saple. Mix each saple with an excess of freshly pipetted standard iodine reagent. Titrate with standard sodiu thiosulfate solution (Na S O 3 ) to a starch indicator end point. Once the oxygen uptake is easured, the k a ay be calculated by using Eq. (9.57) where C is zero and C * is the oxygen equilibriu concentration. The sodiu sulfite oxidation technique has its liitation in the fact that the solution cannot approxiate the physical and cheical properties of a ferentation broth. An additional proble is that this technique requires high ionic concentrations (1 to ol/), the presence of which can affect the interfacial area and, in a lesser degree, the ass-transfer coefficient (Van't Riet, 1979). However, this technique is helpful in coparing the perforance of ferenters and studying the effect of scale-up and operating conditions. Exaple 9.6 To easure k a, a ferenter was filled with 10 of 0.5 M sodiu sulfite solution containing M Cu ++ ion and the air sparger was turned on. After exactly 10 inutes, the air flow was stopped and a 10 saple was taken and titrated. The concentration of the sodiu sulfite in the saple was found to be 0.1 ol/. The experient was carried out at 5 C and 1 at. Calculate the oxygen uptake and k a. Solution: The aount of sodiu sulfite reacted for 10 inutes is = 0.9ol/ According to the stoichioetric relation, Eq. (9.59) the aount of oxygen required to react 0.9 ol/ is

28 9-6 Agitation and Aeration Therefore, the oxygen uptake is = 0.145ol/ 3g O ol = 600s 3 (0.145ole O ) g/s The solubility of oxygen in equilibriu with air can be estiated by Eq. (9.58) as C p O = = = * 4 HO ( T) (793at /ol)(1ol/3g) (1at)(0.09ol O ol air) g/ Therefore, the value of k a is, according to the Eq. (9.57), ka = = = C ( g/ 0) 3 qa / v g/s * 3 C s Dynaic Gassing-out Technique This technique (Van't Riet, 1979) onitors the change of the oxygen concentration while an oxygen-rich liquid is deoxygenated by passing nitrogen through it. Polarographic electrode is usually used to easure the concentration. The ass balance in a vessel gives dc() t * = ka [ C C( t)] (9.60) dt ntegration of the preceding equation between t 1 and t results in * C C( t1) ln * C C( t) ka = (9.61) t t 1 fro which k a can be calculated based on the easured values of C (t 1 ) and C (t ). 1

29 Direct Measureent Agitation and Aeration 9-7 n this technique, we directly easure the oxygen content of the gas strea entering and leaving the ferenter by using gaseous oxygen analyzer. The oxygen uptake can be calculated as q = ( Q C Q C ) (9.6) a in O,in out O,out where Q is the gas flow rate. Once the oxygen uptake is easured, the k a can be calculated by using Eq. (9.57), where C is the oxygen concentration of the liquid in a ferenter * and C is the concentration of the oxygen which would be in equilibriu with the gas strea. The oxygen concentration of the liquid in a ferenter can be easured by an on-line oxygen sensor. f the size of the ferenter is rather * sall (less than 50 ), the variation of the C C in the ferenter is fairly sall. However, if the size of a ferenter is very large, the variation can be * significant. n this case, the log-ean value of C C of the inlet and outlet of the gas strea can be used as * ( C C ) M = ln Dynaic Technique * * ( C C ) ( C C) in * * ( C C ) ( C C ) out in out (9.63) By using the dynaic technique (Taguchi and Huphrey, 1966), we can estiate the k a value for the oxygen transfer during an actual ferentation process with real culture ediu and icroorganiss. This technique is based on the oxygen aterial balance in an aerated batch ferenter while icroorganiss are actively growing as dc dt * ( ) O X = kac C r C (9.64) where r O is cell respiration rate [g O /g cell h]. While the dissolved oxygen level of the ferenter is steady, if you suddenly turn off the air supply, the oxygen concentration will be decreased (Figure 9.7) with the following rate dc dt O X = r C (9.65)

30 9-8 Agitation and Aeration air off C dc dt Figure 9.7 Dynaic technique for the deterination of k a. t since k a in Eq. (9.64) is equal to zero. Therefore, by easuring the slope of the C vs. t curve, we can estiate ro C X. f you turn on the airflow again, the dissolved oxygen concentration will be increased according to Eq. (9.64), which can be rearranged to result in a linear relationship as C = C + r C * 1 dc O X ka dt (9.66) The plot of C versus dc/ dt + ro C X will result in a straight line which has the slope of 1/(k a) and the y-intercept of C * Correlation for k a Bubble Colun Akita and Yoshida (1973) correlated the voluetric ass-transfer coefficient k a for the absorption of oxygen in various aqueous solutions in bubble coluns, as follows: σ = 0.6 AB ν c T ρ c ka D D g H (9.67) which applies to coluns with less effective spargers. n bubble coluns, for 0 < V s < 0.15 /s and 100 < P g /v < 1100 W/ 3, Botton et al. (1980) correlated the k a as ka Pg / v = (9.68)

31 Agitation and Aeration 9-9 where P g is the gas power input, which can be calculated fro p v g V s = (9.69) Mechanically Agitated Vessel For aerated ixing vessels in an aqueous solution, the ass-transfer coefficient is proportional to the power consuption (opes De Figueiredo and Calderbank, 1978) as 0.33 P k v (9.70) The interfacial area for the aerated ixing vessel is a function of agitation conditions. Therefore, according to Eq. (9.41), 0.4 P 0.5 a Vs v Therefore, by cobining the above two equations, k a will be P ka v 0.77 V 0.5 s (9.71) (9.7) Nuerous studies for the correlations of k a have been reported and their results have the general for as b P b3 ka = b 1 Vs v (9.73) where b 1, b 1, and b 1 vary considerably depending on the geoetry of the syste, the range of variables covered, and the experiental ethod used. The values of b and b 3 are generally between 0 to 1 and 0.43 to 0.95, respectively, as tabulated by Sidean et al. (1966). Van't Riet (1979) reviewed the data obtained by various investigators and correlated the as follows: 1. For coalescing air-water dispersion, P ka 0.06 = v 0.4 V 0.5 s (9.74)

32 9-30 Agitation and Aeration 1. For noncoalescing air-electrolyte solution dispersions, P ka 0.00 = v 0.7 V 0. s (9.75) both of which are applicable for the volue up to.6 3 ; for a wide variety of agitator types, sizes, and D /D T ratios; and 500 < P /v < 10,000 W/ 3. These correlations are accurate within approxiately 0 percent to 40 percent. Exaple 9.7 Estiate the voluetric ass-transfer coefficient k a for the gas-liquid contactor described in Exaple 9.4 by using the correlation for k a in this section. Solution: Fro Exaple 9.4, the reactor volue v is , the superficial gas velocity V s is /s, and power consuption P is 687 W. By substituting these values into Eq. (9.74), ka = = 0.018s Scale-Up Siilitude For the optiu design of a production-scale ferentation syste (prototype), we ust translate the data on a sall scale (odel) to the large scale. The fundaental requireent for scale-up is that the odel and prototype should be siilar to each other. Two kinds of conditions ust be satisfied to insure siilarity between odel and prototype. They are: 1. Geoetric siilarity of the physical boundaries: The odel and the prototype ust be the sae shape, and all linear diensions of the odel ust be related to the corresponding diensions of the prototype by a constant scale factor.

33 Agitation and Aeration Dynaic siilarity of the flow fields: The ratio of flow velocities of corresponding fluid particles is the sae in odel and prototype as well as the ratio of all forces acting on corresponding fluid particles. When dynaic siilarity of two flow fields with geoetrically siilar boundaries is achieved, the flow fields exhibit geoetrically siilar flow patterns. The first requireent is obvious and easy to accoplish, but the second is difficult to understand and also to accoplish and needs explanation. For exaple, if forces that ay act on a fluid eleent in a ferenter during agitation are the viscosity force F V, drag force on ipeller F D, and gravity force F G, each can be expressed with characteristic quantities associated with the agitating syste. According to Newton's equation of viscosity, viscosity force is F V du = µ A dy (9.76) where du/dy is velocity gradient and A is the area on which the viscosity force acts. For the agitating syste, the fluid dynaics involved are too coplex to calculate a wide range of velocity gradients present. However, it can be assued that the average velocity gradient is proportional to agitation speed N and the area A is to D, which results. F V µ ND (9.77) The drag force F D can be characterized in an agitating syste as F P o D (9.78) DN Since gravity force F G is equal to ass ties gravity constant g, F G 3 ρd g (9.79) The suation of all forces is equal to the inertial force F as, 4 F = F V + F F F D N D + G = ρ (9.80) Then dynaic siilarity between a odel () and a prototype (p) is achieved if ( FV) ( FD) ( FG) ( F) = = = (9.81) ( F ) ( F ) ( F ) ( F ) V p D p G p p

34 9-3 Agitation and Aeration or in diensionless fors: F F = F F V D G p p p V F F = F F D F F = F F The ratio of inertial force to viscosity force is G (9.8) 4 ρ ρ which is the Reynolds nuber. Siilarly, F D N D N N Rei F = µ ND = µ = (9.83) V ρ ρ F D N N D 1 = = = (9.84) F P / D N P N V o o P 4 ρ 3 G ρd g g F D N D N = = = NFr (9.85) F Dynaic siilarity is achieved when the values of the nondiensional paraeters are the sae at geoetrically siilar locations. ( NRe ) ( Re ) i p = N i ( NP) p = ( NP) (9.86) ( N ) = ( N ) Fr p Fr Therefore, using diensionless paraeters for the correlation of data has advantages not only for the consistency of units, but also for the scale-up purposes. However, it is difficult, if not ipossible, to satisfy the dynaic siilarity when ore than one diensionless group is involved in a syste, which creates the needs of scale-up criteria. The following exaple addresses this proble.

35 Exaple 9.8 Agitation and Aeration 9-33 The power consuption by an agitator in an unbaffled vessel can be expressed as P ρnd N D = f, 3 5 µ g ρnd Can you deterine the power consuption and ipeller speed of a 1,000-gallon ferenter based on the findings of the optiu condition fro a geoetrically siilar one-gallon vessel? f you cannot, can you scale up by using a different fluid syste? Solution: Since V p /V = 1,000, the scale ratio is, ( D) p 1/3 =1,000 =10 ( D ) (9.87) To achieve dynaic siilarity, the three diensionless nubers for the prototype and the odel ust be equal, as follows: P P = ρnd o o ρnd p (9.88) ρnd ρnd = µ µ p ND ND = g g p (9.89) (9.90) f you use the sae fluid for the odel and the prototype, ρ p = ρ and µ p = µ. Canceling out the sae physical properties and substituting Eq. (9.87) to Eq. (9.88) yields N 5 p ( Po) p = 10 ( Po ) N 3 (9.91) The equality of the Reynolds nuber requires N = 0.01 N (9.9) p

36 9-34 Agitation and Aeration On the other hand, the equality of the Froude nuber requires 1 Np = N (9.93) 10 which is conflicting with the previous requireent for the equality of the Reynolds nuber. Therefore, it is ipossible to satisfy the requireent of the dynaic siilarity unless you use different fluid systes. f ρp ρ and µ p µ, to satisfy Eqs. (9.89) and (9. 90), the following relationship ust hold. µ 1 µ = ρ 31.6 ρ (9.94) Therefore, if the kineatic viscosity of the prototype is siilar to that of water, the kineatic viscosity of the fluid, which needs to be eployed for the odel, should be 1/31.6 of the kineatic viscosity of water. t is ipossible to find the fluid whose kineatic viscosity is that sall. As a conclusion, if all three diensionless groups are iportant, it is ipossible to satisfy the dynaic siilarity. The previous exaple proble illustrates the difficulties involved in the scale-up of the findings of sall-scale results. Therefore, we need to reduce the nuber of diensionless paraeters involved to as few as possible, and we also need to deterine which is the ost iportant paraeter, so that we ay set this paraeter constant. However, even though only one diensionless paraeter ay be involved, we ay need to define the scale-up criteria. As an exaple, for a fully baffled vessel when p N Rei >10,000, the power nuber is constant according to Eq. (9.55). For a geoetrically siilar vessel, the dynaic siilarity will be satisfied by P P = ρnd o o ρnd p (9.95) f the fluid eployed for the prototype and the odel reains the sae, the power consuption in the prototype is