Simplified dynamic pressure method for kLa measurement in aerated bioreactors

Biochemical Engineering Journal 49 (2010) 165–172

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Simplified dynamic pressure method for kL a measurement in aerated bioreactors F. Scargiali ∗ , A. Busciglio, F. Grisafi, A. Brucato Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Universitá degli Studi di Palermo, Viale delle Scienze, Ed. 6 - 90128 Palermo, Italy

a r t i c l e

i n f o

Article history: Received 17 July 2009 Received in revised form 15 December 2009 Accepted 16 December 2009

Keywords: Dynamic modelling Gas–liquid mass transfer Multiphase bioreactors Oxygen transfer kL a Stirred vessels

a b s t r a c t A simplified version of the dynamic pressure method for measuring mass-transfer coefficients in gas–liquid systems is proposed. With this method oxygen concentration in the liquid phase is monitored after a sudden change of total pressure is applied to the system. With respect to the original technique introduced by Linek [14] the simplified version here proposed greatly simplifies the data treatment, yet resulting in good accuracy for most practical purposes. In practice, with the help of a simple mathematical model, it is found that the dynamic oxygen concentration response, when plotted as residual driving force versus time in a semi-log diagram, should be expected to finally settle on a straight line. From the slope of this last kL a can be immediately computed. Experimental data obtained on a lab-size stirred tank reactor confirm all model predictions, including the feature that the adoption of large pressure changes may lead to better accuracy. Mass transfer coefficient data obtained by means of the simplified dynamic pressure method (SDPM) here proposed are compared with the relevant data obtained on the same system by the most accurate physical technique (pure oxygen absorption in a pre-evacuated liquid), as well as with the literature data, resulting in SDPM full validation. All kL a data obtained are finally organized by a conventional power law correlation. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

where

The design, scale-up and optimization of industrial gas–liquid reactors and bioreactors require, among others, precise knowledge of inter-phase mass transfer performances, hence reliable information on the volumetric mass transfer coefficient, kL a [1]. The dynamic oxygen-electrode method has been widely used for kL a measurements [2–7]. For a perfectly mixed liquid phase, Eq. (1) is obtained:

  ∗  dCL kLi sbi CLi − CL = dt Nb

VL

(1)

i=1 ∗ where Nb is the total number of bubbles, sbi is bubble surface and CLi is the equilibrium oxygen concentration for each bubble. If kL , and CL∗ are identical for all bubbles at any time throughout the process, Eq. (1) can be integrated to give:

ln

CL∗ − CL

CL∗ − C0

= −kL a (t − t0 )

∗ Corresponding author. Tel.: +39 09123663714 E-mail address: [email protected] (F. Scargiali). 1369-703X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.bej.2009.12.008

(2)

Nb 

a=

sbi

i=1

VL

(3)

is the interfacial area per unit volume of liquid. Hence, if the above hypotheses are reasonably abode by, plots of ln(CL∗ − CL ) versus t will result into straight lines with a slope equal to −kL a. Certain difficulties are encountered when using this method. As a matter of fact, Eq. (2) requires that the liquid and gas phases are perfectly mixed. In all other cases precise and detailed information on the gas and liquid conditions is required in order to properly assess the mass transfer coefficient. The influence of the liquid phase flow behavior on the accuracy of measurements is recognized to be small [8]. On the contrary, the flow behavior of the gas phase may significantly affect results [3,9–11]. It is worth noting that the only case in which there is no need to resort to models for the behavior of the gas phase, is when a pure gas is absorbed in a completely degassed liquid [4,12]. In this case in fact the uniformity of bubble gas concentration is insured and the kL a values obtained using Eq. (2) are fully reliable. An interesting alternative to using a pure gas in a completely pre-evacuated liquid is given by the dynamic pressure method (DPM) [13–16]. In this last method, the effect of the inert gas is minimized

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Nomenclature a C0 CL Cp ∗ CLi

CL∗ CTot,L D DT F HN2 HO2 kL , kL,O2 kL,N2 kLi Mg N Nb Np P/VL Pg /VL R S sbi SolN2 SolO2 T ut VL

vs DPM OApE OAS SDPM

gas–liquid interfacial area per unit volume of dispersion [m−1 ] initial oxygen concentration in the liquid phase [kmol m−3 ] oxygen concentration in the liquid phase [kmol m−3 ] oxygen concentration measured by the DO probe [kmol m−3 ] equilibrium oxygen concentration for each bubble [kmol m−3 ] overall equilibrium oxygen concentration [kmol m−3 ] total liquid concentration ≈ 55.5 kmol m−3 [kmol m−3 ] gas diffusivity [m2 s−1 ] tank diameter molar gas flow-rate [kmol] Henry’s constant for nitrogen [Pa] Henry’s constant for oxygen [Pa] oxygen mass transfer coefficient [m s−1 ] nitrogen mass transfer coefficient [m s−1 ] oxygen mass transfer coefficient for a single bubble [m s−1 ] overall mole number in the gas phase [kmol] rotational speed [rpm] number of bubbles power number specific power input [W m−3 ] specific gassed power input [W m−3 ] gas universal constant [Pa m3 kmol−1 K−1 ] tank transversal section [m2 ] single bubble surface [m2 ] (1 − y)(p CTot,L /HN2 ) [kmol m−3 ] y(p CTot,L /HO2 ) [kmol m−3 ] system temperature [K] average bubble terminal rise velocity [m s−1 ] liquid volume [m3 ] superficial gas velocity [m s−1 ] dynamic pressure method pure oxygen absorption in pre-evacuated water pure oxygen absorption in air saturated water simplified dynamic pressure method

Greek letters  dimendionless time p dimendionless probe time lag  system pressure [Pa] p probe time lag [s]

and, reportedly, air absorption may be used with some confidence. The technique was found to be suitable for kL a assessment in culture media and fermentation broths with viscosities up to 279 mPa s [17]. The data treatment involved in DPM is however quite complex, as it accounts for many transient details [14]. In the present work a simplified version of the DPM, characterized by a very easy data treatment, is proposed. The procedure features are analyzed and discussed with the help of a theoretical transient model. The simplified procedure is then validated by comparison with experimental data obtained by means of the fully reliable oxygen absorption in a completely degassed liquid.

Fig. 1. Trends of the measured oxygen concentration in the liquid (Cp ) versus nondimensional time , for various dimensionless probe response times p .

2. kL a measurements Before entering into details on transient modelling, there is another source of concern that has to be addressed, namely the time lag of the probe employed for oxygen concentration measurement. 2.1. Influence of electrode response time In order to consider the influence of probe time lag on measured liquid oxygen dynamics, model computations can be carried out while varying the time lag p involved in the dissolved oxygen (DO) probe response equation: dCp CL − Cp = p dt

(4)

where Cp is the measured concentration, while CL is the actual liquid concentration. By substituting Eq. (2) in Eq. (4), under the hypothesis that C0 = 0 and t0 = 0, Eq. (5) is obtained [18]: CL∗ − Cp CL∗

=

e− − p e−/p 1 − p

(5)

where  = kL at

(6)

p = kL ap

(7)

In Fig. 1, Eq. (5) has been plotted on a semi-log diagram versus non-dimensional time . It can be seen there that in all cases the resulting curves eventually converge to the same −1 slope that characterizes zero response time probes. As a consequence if the same curves are plotted versus time (instead of dimensionless time), all final slopes will coincide with −kL a, just as in the case of no probe delay. This occurs however at increasingly larger times the larger the p . Therefore smaller and smaller residual driving forces need to be accurately assessed to read the final slope, and this is eventually limited by probe resolution and accuracy.

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167

ily measured with no need to worry about probe dynamics effects. Clearly for different probe resolution/noise characteristics, all the above reasoning has to be reworked with the help of Figs. 1 and 2 and Eqs. (5) and (8). 2.2. Pressure transient modelling

Fig. 2. Local slope of curves in Fig. 1 versus non-dimensional time , for various dimensionless probe response times p .

In order to better quantify the above considerations, one can differentiate the logarithm of both sides of Eq. (5), to get: d ln d()



CL∗ − Cp CL∗



= −1 +

(1 − p ) e(/p −) − p

In the DPM, the driving force for oxygen absorption is obtained by means of small operating pressure step-changes in the vessel. For instance, one may apply some vacuum to the system and, after steady-state conditions under reduced pressure are attained, atmospheric pressure may be suddenly restored by admitting atmospheric air to the reactor head-space. One may be tempted to regard air as if it was a single component gas, and describe the subsequent system transient as if it involved only the absorption of this pseudo-component, until saturation. Measured oxygen transient concentrations might be regarded as mimicking the single component concentration evolution. This however does not exactly coincide with what actually occurs in gas–liquid dispersions, as the simultaneous presence of at least two different components (oxygen and nitrogen, each one characterized by its own solubility and diffusivity) must be accounted for. The fundamental equations involved are the mass balances in both liquid and gas phases, that in the following simplified model will be assumed to be perfectly mixed: dCL,O2 dt dCL,N2

(8)

The second term at Eq. (8) right hand side gives in practice the deviation of local slope from the final −1 value (hence from −kL a in case of dimensional time abscissas). Eq. (8) is plotted in Fig. 2, for several values of dimensionless probe time lag p . On the same figure, an horizontal dotted line marking a 3% deviation from zero-lag probes is also reported for convenience sake. Figs. 1 and 2 quantify what should be expected from probe readings when the underlying actual liquid concentration differences follow decreasing exponential patterns. In order to exemplify how Figs. 1 and 2 may be employed in practice, let us first assume that probe resolution and noise allow 1% differences from final saturation concentration to be accurately assessed. As ln(0.01)=−4.6 the latter assumption implies that meaningful readings may be made almost all the way down the ordinate axis of Fig. 1. The question to answer is: which is the largest kL a value that can be experimentally assessed by simply reading the final slope in the relevant semi-log plots? By looking at Fig. 1 one may guess that the maximum p value practically affordable is about 0.6, as with this p curve slope seems to settle early enough on its final value for a sufficient span of accurate readings to be taken. This is confirmed by Fig. 2, where it can be seen that after =3.9 curve slope is within 3% of the final value, a difference that can be considerd negligible with respect to both graphical and experimental uncertainties. Hence the time span from =3.9 to about 5.5 (corresponding to a (CL∗ − Cp )/CL∗ span from 0.05 down to 0.01) can be employed to assess the final slope with reasonable accuracy. Notably in Fig. 2 one may see that a p of 0.8 would require useful readings to be made only after =7, and this would require in turn an unrealistic probe resolution (accurate readings of ln[(CL∗ − Cp )/CL∗ ] much smaller than 5 would be required, as it can be seen in Fig. 1). With a maximum p of 0.6 and a 6 s time-lag probe the maximum value of kL a that can be easily assessed is kL a max = 0.6/6 = 0.1 s−1 , hence a quite high value that covers almost all practical applications. With commonly available 3 s time-lag probes even higher kL a values (up to 0.2 s−1 ) can be eas-

dt

= NO2

(9)

= NN2

(10)

dMg,O2 dt dMg,N2 dt

= FO2 ,in − FO2 ,out − VL NO2

(11)

= FN2 ,in − FN2 ,out − VL NN2

(12)

where the gas overall volume, oxygen volume fraction and gas holdup can be expressed as:



Vg = y=

=



Mg,O2 + Mg,N2 RT

(13)

p Mg,O2

(14)

Mg,O2 + Mg,N2 Vg VL + Vg

(15)

The gas rate leaving the dispersion may be modelled as being proportional to gas hold-up () and average bubble rise velocity (ut ) Fg,out = Sut 

p RT

(16)

The oxygen and nitrogen interfacial flow rates per unit dispersion volume are computed as:



∗ − CL,O2 NO2 = kL,O2 a CL,O



2

∗ − CL,N2 NN2 = kL,N2 a CL,N 2





(17) (18)

According to the penetration theory, the transport coefficients of oxygen and nitrogen are assumed to be linked by the following expression: kL,N2 kL,O2



=

DN2 DO2

(19)

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Fig. 3. Simulated liquid phase concentration dynamics for kL a = 0.05 s−1 , Qg = 1 vvm, 0 = 0.5 atm, final = 1 atm. Solid line: actual liquid concentration; dashed line: measured (gauge) liquid concentration (probe time-lag = 3 s).

Finally, by neglecting gas-side mass transfer resistances, interfacial gas concentrations are expressed on the basis of gas solubilities: ∗ = ySolO2 CL,O

(20)

∗ CL,N 2

(21)

2

= (1 − y)SolN2

This simple set of equations was numerically solved for simulating the liquid and gas concentration transients following a sudden pressure change in the system. To this end, a value of kL,O2 a (hereafter referred to as kL a) was specified. This clearly also acted as the reference value with which the final result of simulated data analysis was to be compared. Oxygen and nitrogen diffusivities at T = 20 ◦ C were set at 2.5 × 10−9 and 1.9 × 10−9 m2 /s while the relevant solubilities were set at 4.02 × 109 and 7.93 × 109 Pa respectively [19]. As the pressure variation compresses the bubbles, it gives rise to a sudden decrease of interfacial area and volumetric hold-up, so that also these quantities undergo a transient that was also simulated. Despite these disturbances, the oxygen dynamics gives rise to fairly straight lines, when its residual driving force is plotted versus time in a semi-log diagram, as it can be appreciated by looking at the solid line in Fig. 3. On the same figure, the simulated measured oxygen concentrations are reported also (dotted line), in the case of a probe time-lag of 3 s. As it can be seen the probe dynamics effect, apart from the initial transient, is only that of slightly displacing the straight line response, without affecting its final slope (that can therefore be accurately assessed), as expected on the basis of Section 2.1 results. Given that a straight line is eventually obtained as in the case of OApE procedure, one may inspect whether the lines common final slope in Fig. 3 does coincide with −kL a. This is in fact known, being an input parameter of the model. The two quantities are compared in Fig. 4, where it can be seen that for small enough kL a values a negligible difference, smaller than other error sources such as experimental uncertainties, exists between the two. This implies that for small enough kL a values (e.g. kL a < 0.03 s−1 ) the dynamic technique may be employed just as if a pure gas were absorbed in a previously evacuated liquid. Notably the observed slopes are practically independent of pressure increase extent, a feature with important practical implications as larger pressure changes result into larger residual driving forces, so making for easier and more accurate measurements. Observation of Fig. 3 shows that for larger values of kL a the final slope modulus becomes smaller than kL a, with discrepancies of the order of 10 % for kL a = 0.1 s−1 and of 15 % for kL a = 0.15 s−1 . In an

Fig. 4. Slope of ln(CL∗ − CL )/(CL∗ − CL,0 ) versus actual kL a values, at various 0 (Qg = 1 vvm).

attempt to understand why under such conditions straight lines are still obtained (though with slopes not exactly coinciding with −kL a), the further assumption that gas phase mass variations with time are negligible may be added to the previous model hypotheses. Under this additional hypothesis (which should become closer and closer to reality after the initial transient) Fout ≈ Fin and the liquid phase oxygen balance can be manipulated to give: dCL,O2 yin SolO2 − C02

=

kL a dt 1 + kL a(VL SolO2 /Fin )

(22)

which implies that the O2 driving force line in Fig. 3 should actually be expected to result in a straight line with a slope given by: Slope = −

kL a 1 + kL a(VL SolO2 /Fin )

(23)

hence a slope different from kL a. This last should therefore be simply related to the experimental slope by: kL a = −

Slope 1 + Slope(VL SolO2 /Fin )

(24)

Eq. (24) was tested against many model simulations concerning widely varying kL a values. Results showed that the correction provided by Eq. (24) tends to over-correct the experimental slopes, probably because, under the above additional hypothesis, no role is left in the model equations to nitrogen solubility and diffusivity. Several attempts were unsuccessfully carried out to analytically derive a better correction for the constant slopes observed in the model-predicted oxygen concentration dynamics. In the end, by suitably modifying Eq. (24) on the basis of observed discrepancies, the following more effective correction was obtained:



kL a = −Slope

1 + Slope(VL SolN2 /Fin )

(DO2 /DN2 )

1 + Slope(VL SolO2 /Fin )

(25)

With this correction, that accounts for both the different solubilities and diffusivities, the errors incurred in the evaluation of kL a on the basis of the observed oxygen dynamics slopes are much smaller, as it can be appreciated in Fig. 5. As it can be seen, by adopting Eq. (25) over the entire kL a range here analyzed (that practically encompasses all common needs) the error is well acceptable, being always much smaller than 2%.

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Fig. 5. % difference between slope values corrected according to Eq. (25) and actual kL a for different values of kL a and starting pressures 0 . final = 1 atm, Qg = 1 vvm.

It is worth noting that the correction provided by Eq. (25) is more effective the larger the pressure difference between initial and final conditions. This adds to the measurement accuracy advantages enjoyed when larger pressure differences are adopted. Summarizing, a simplified version (SDPM) of the DPM may be proposed, in which oxygen dynamics data analysis is simply done by (i) plotting the residual driving force on a semi-log diagram, (ii) assessing the final slope value and (iii) employing Eq. (25) to obtain the relevant kL a value. The simplified version of the DPM here proposed is somewhat less accurate than the original technique introduced by Linek et al. [14], but has the advantage of being much more straightforward by substantially getting away with the computational burden of the original method, at the expense of inaccuracies smaller than experimental uncertainties. 3. Experimental The experimental apparatus employed is depicted in Fig. 6. It involved a PMMA made “standard” stirred reactor (DT = 190 mm) with a total height of 300 mm, provided with a stainless steel cover equipped with all inlet and outlet connections and a mechanical seal for shaft entrance. A 65 mm dia. “Rushton turbine” was mounted on the 17 mm dia. shaft, leaving a clearance of DT /3 from vessel bottom. The vessel was equipped with 4 vertical baffles 3 mm ◦ thick and 19 mm wide, deployed along vessel walls at 90 from each other. The vessel was filled with deionized water up to an height of 190 mm (H = DT ). The shaft was driven by a 1200 W DC motor (Mavilor MSS-12), equipped with tacho and speed control unit (Infranor SMVEN 1510) so that rotational speed was maintained constant, within 0.1%. An inexpensive Venturi vacuum pump fed by compressed air (Vaccon HVP 100) was used to evacuate the vessel down to 0.1 atm. Oxygen concentration in the liquid phase was measured by an electrode sensor (WTW CellOx 325) and control unit (WTW Oxi 340i). In order to minimize the interferences caused by gas bubbles adhering to the electrode surface, liquid was continuously withdrawn from the vessel by means of a peristaltic pump and returned to it after passing over the oxygen electrode. In such a way only few

Fig. 6. Experimental set-up.

bubbles were entrained in the external loop and the majority of these did not interact with the electrode thanks to gravity and centrifugal settling. This external loop added a (negligible) almost-pure delay of about 1 s to the measurements made. The output of the oxygen measurement unit was recorded by a data acquisition system and processed to yield the relevant value of the mass transfer coefficient kL a. The oxygen probe first order lag was experimentally found to be about 3 s [20]. As the largest kL a values here measured were of the order of 0.06 s−1 , this brings to a maximum dimensionless time parameter p of 0.18, hence to a dimensionless time  of about 1.0 for curve final slope to attain a constant value, as can be appreciated in Fig. 2. This was always small enough for many accurate readings to be taken before probe resolution impaired residual driving force readings. In all runs temperature inside the reactor was between 20 and 21 ◦ C. This was obtained by adjusting the initial temperature and exploiting the circumstance that the temperature increase during each single run was always less than 0.2 ◦ C. A static frictionless turntable and a precision balance were employed for measuring the mechanical power dissipated by the impeller at various agitation speeds and gas flow rates. Details of this apparatus are given elsewhere [21] Four different agitation speeds (500, 700, 900 and 1100 rpm) and three different gas flow rates (1/2, 1 and 2 vvm respectively) were tested. 3.1. kL a measurement techniques adopted When the SDPM was adopted three widely different starting pressures were tested (0 = 0.1, 0.5 and 0.8 atm) while final was 1 atm in all cases. Pressure inside the reactor was brought down to the desired starting pressure while feeding air to the vessel sparger at the desired flow rate. Pressure and agitation were maintained

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Fig. 8. Effect of initial pressure 0 on the oxygen concentration dynamics. N = 900 rpm, Qg = 1 vvm.

Fig. 7. Typical oxygen concentration dynamics at N = 1100 rpm, Qg = 1 vvm.

for a few minutes (typically 5 min) to lower the dissolved oxygen and nitrogen concentrations in the liquid phase close to the equilibrium values with air at 0 pressure. Atmospheric pressure was then suddenly restored by admitting air to the reactor head space through a ball valve inserted in the vessel cover and directly connected to ambient air, while continuously stirring and feeding air to the vessel sparger. Data acquisition was started shortly after the instantaneous pressure step change. A transient followed, in which the difference between the equilibrium concentration to air (oxygen and nitrogen) at 1 atm and the time dependent oxygen and nitrogen concentration in the liquid phase decayed and eventually vanished. For comparison purposes, kL a measurements were also performed via the well-established OApE procedure (pure oxygen absorption in pre-evacuated water) and the more common OAS procedure (pure oxygen absorption in air saturated water) [12]. Comparison between the three procedures was made at a fixed gas flow rate of 1 vvm. For comparison purposes with the literature experimental data, gas flow-rates of 1/2 vvm and 2 vvm were also conveniently investigated using the (now validated) SDPM. 4. Results and discussion A typical experimental oxygen concentration dynamics obtained by the SDPM (0 = 0.5 atm, N = 1100 rpm, Qg = 1.0 vvm) is shown in Fig. 7(a), where the dimensionless driving force for mass transfer is plotted versus time in a semi-log diagram. As it

can be seen, the model prediction that curve slope should eventually settle over a straight line is fully confirmed. The final slope observed is −0.0569 s−1 and the relevant kL a value, computed on the basis of Eq. (25), is 0.0601 s−1 . When the nitrogen-free procedure (OApE) was applied under the same operating conditions, the results reported in Fig. 7(b) were obtained. As it can be seen, once again (after a short transient) data points do align on a straight line with a final slope of 0.0586 s−1 ), which is to be regarded in practice as the real kL a value for the given conditions (though with some random experimental error overimposed). The fact that it compares fairly well with the relevant SDPM result may be regarded as a further validation of the SDPM here proposed. It may be worth noting that when the commonly employed pure oxygen absorption in air-presaturated water (procedure OAS) was carried out (under the same gassing and agitation speed conditions) the results shown in Fig. 7(c) were obtained. As it can be seen a straight line was obtained once again, but this time the final slope absolute value is 0.0362 s−1 , a value very different from the real kL a value. This result confirms the detrimental role played by nitrogen when OAS is adopted [4,12] and highlights the need for more reliable kL a measuring techniques, such as the OApE or the DPM, whether in the simplified version here proposed (SDPM) or not (DPM). As concerns the model prediction that the  extent should not significantly affect the results, in Fig. 8 results obtained in three runs conducted with three widely differing pressure changes (from 25% to as much as 900%, on initial pressure basis) are reported. As it can be seen in all cases the same final constant slope is observed irrespectively of the pressure change extent, fully confirming model predictions. This finding does not support the previous indications that pressure change should be small for DPM accuracy. This has interesting practical implications as the adoption of larger pressure changes simplifies the assessment of oxygen concentration differences with respect to final equilibrium conditions. SDPM experimental results obtained under 1 vvm gassing rate and various agitation speeds are reported in Fig. 9versus the relevant real values obtained with the OApE technique. As it can be seen, measurements obtained with SDPM do practically coincide with those obtained with the OApE technique, hence further vali-

F. Scargiali et al. / Biochemical Engineering Journal 49 (2010) 165–172

Fig. 9. Comparison between kL a values assessed via SDPM and OAS versus relevant real (OApE) data.

dating the SDPM. In all cases experiments were repeated typically 5 times for each agitation speed. The standard deviations observed (2–3% for OApE and SDPM) are reported as error bars in Fig. 9. On the same figure the relevant OAS measurements (standard deviation equal to 4–5% in this case) are also reported for comparison purposes. As it can be seen in all cases the resulting apparent kL a values significantly underestimate real kL a values. This is consistent with previous findings and is to be entirely attributed to measurement interference by dissolved nitrogen, as thoroughly discussed by Linek et al. [4] and Scargiali et al. [12]. As a matter of fact procedure OAS is practically identical to procedure OApE, the only difference between the two being merely the initial presence of nitrogen in the liquid phase. Clearly the OApE procedure, being virtually free from nitrogen disturbances and interpretation difficulties, is the one that leads to exact results. The SDPM here proposed is only slightly less accurate than the OApE procedure because of the presence of inert gas. Nevertheless, the inert gas effects are minimized and air can be employed instead of pure oxygen, resulting in an easier and more viable technique, that retain also the OApE data treatment simplicity.

Fig. 10. Measured mass transfer coefficients versus Linek et al. [4] correlation (Eq. (26)).

differences smaller than 10% are observed between the two data sets. 4.1.1. kL a dependence on gas flow rate and specific power input The data obtained in this work were finally correlated using a power law similar to Linek’s correlation. Correlation parameters were assessed via bivariate non-linear regression applied to all experimental data obtained, resulting in the following correlation with an R2 of 0.996: kL a = 0.0037v0.350 s

P 0.585 g

V

The now validated SDPM was used to carry out kL a measurements also at lower (1/2 vvm) and higher (2 vvm) gas flow rates. Present results may be compared with reliable literature data, obtained in the absence of initially dissolved nitrogen, such as those reported by Linek et al. [4]. This is done in Fig. 10 where the present experimental data are plotted versus relevant values predicted by Linek’s correlation:

P 0.593 g

V

(26)

As it can be seen, the present data are slightly smaller than Linek’s correlation (hence Linek’s experimental data), with increasing discrepancies the larger the gas flow rate. This may depend on minor geometrical differences between the two systems as well as on the purity of the deionized water employed in the two experimentations. As a matter of fact, mass transfer results were found to be affected by the tiniest water contaminations. In any case

(27)

In Fig. 11 the average values obtained for the three gas flow rates are reported versus Eq. (27) predictions. As in the case of Fig. 9 error bars quantifying the standard deviation observed in repeated experiments are reported also. Not surprisingly, the experimental kL a values obtained at all gas velocities compare very well with

4.1. Comparison with the literature data

kL a = 0.00495v0.4 s

171

Fig. 11. Measured mass transfer coefficients versus Eq. (27).

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Eq. (27). This does not however implies that this last should be regarded as a better correlation than Eq. (26), as clearly Linek’s data would not fit as well with Eq. (27). Instead, the (rather small) discrepancies between the two correlations should be looked at as an indication of the uncertainties (stemming from minor changes in physical features or fluids purities) affecting any correlation in the field.

[5] [6]

[7] [8]

5. Conclusions [9]

A simplified version of the DPM for measuring mass-transfer coefficients in gas–liquid systems was proposed. With respect to the original technique formerly introduced by Linek et al. [14], the simplified version here proposed skips all mathematical complexities, yet resulting in sufficient accuracy for practical purposes. Experimental validation showed that SDPM is a viable procedure that gives rise to results practically coinciding with the real kL a values obtained in the absence of nitrogen. Results also confirmed that true gas–liquid mass transfer coefficients, as obtained by procedures OApE and/or SDPM, are significantly larger than those (less reliable) measured in the same hydrodynamic conditions but absorbing pure oxygen in an airsaturated liquid phase. This highlights once again the importance of employing correct methods for kL a measurement, able to produce physically consistent data, as only such data with clear physical interpretation can lead to successful design and scale-up of apparatuses. References [1] F. Garcia-Ochoa, E. Gomez, Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview, Biotechnol. Adv. 27 (2009) 153–176. [2] K. Van’t Riet, Reviewing of measuring method and results in nonviscous gas–liquid mass transfer in stirred vessels, Ind. Eng. Chem. Proc. Des. Dev. 18 (1979) 357–364. [3] C. Chapman, L. Gibilaro, A. Nienow, A dynamic response technique for the estimation of gas–liquid mass transfer coefficients in a stirred vessel, Chem. Eng. Sci. 37 (6) (1982) 891–896. [4] V. Linek, P. Vacek, P. Benes, A critical review and experimental verification of the correct use of the dynamic method for the determination of oxygen tranfer

[10]

[11]

[12]

[13]

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[16]

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