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GENERAL RESEARCH Measurement of Overall Volumetric Mass Transfer Coefficients for Carbon Dioxide in a Well-Mixed Reactor Using a pH Probe Gordon A. Hill* Department of Chemical Engineering, 57 Campus DriVe, UniVersity of Saskatchewan, Saskatoon SK S7N 5A9, Canada
Carbon dioxide can serve as a source of carbon for photosynthetic cell cultures, but it must reside in the aqueous phase prior to uptake. Carbon dioxide is a gas at room temperature and pressure, and its solubility in water is very low. It resides in the aqueous phase as four different species, carbon dioxide, CO2; carbonic acid, H2CO3; bicarbonate ion, HCO3-; and carbonate ion, CO3), whose equilibrium concentrations are pH dependent. With these factors taken into account, a model has been developed to allow the pH profile of an aqueous phase, undergoing a sudden application of bubbles containing carbon dioxide, to be used as a measure of the transient dissolved carbon dioxide concentration in a reactor. Experimentally measured carbon dioxide profiles were used to calculate the overall volumetric mass transfer coefficients for carbon dioxide into a well-mixed reactor as a function of temperature, stirring speed, and aeration rate. Finally, an empirical correlation is provided to predict the overall volumetric mass transfer coefficient of carbon dioxide as a function of these three operating variables. Introduction Carbon dioxide is a ubiquitous chemical that is produced in vast quantities by oxidation reactions of substrates containing organic carbon. Typical oxidation reactions include the combustion of fuels such as coal in power plants and the metabolism of foods such as glucose in living cells. Carbon dioxide is an important component of the greenhouse gas effect at the Earth’s surface, helping to maintain surface temperature at conditions that support life.1 However, the steady increase in carbon dioxide concentration in the atmosphere is now believed to be a significant contributing factor to global warming.2 Much of the released carbon dioxide is captured in oceans or other water bodies where it can be converted back to organic carbon by living cells such as photosynthetic algae. Several attempts have been made in recent years to develop photobioreactors using carbon dioxide as the source of carbon for cell cultures.3 An initial but important design feature of any photobioreactor is the transfer of carbon dioxide from the gas phase to the aqueous phase, since living cells consume only the dissolved molecules of carbon dioxide. Although this is an important feature of the reactor design, no direct study has heretofore been made of the rate of carbon dioxide transfer from air bubbles into the aqueous phase. Instead, researchers commonly assume that the mass transfer rate of carbon dioxide can be estimated using values for oxygen mass transfer rates that have been extensively measured.4-7 Brune and Novak8 developed a model to use dynamic pH shifts during a batch culture to calculate the growth conditions of algal cells. Using this technique, they later reported that, compared to other algae, Chlorella sp. achieved the highest maximum specific growth rate on dissolved carbon dioxide * E-mail:
[email protected].
measured at 0.070 h-1 at a temperature of 27 °C and light intensity of 600 ft-candles.9 Livansky10 determined the desorption rate of carbon dioxide from a stirred solution without bubbling using pH profiles. Dahod11 allowed dissolved carbon dioxide to diffuse through silicone tubing into a nitrogen gas stream to measure steady-state dissolved carbon dioxide concentrations, but that method would be too slow to determine transient concentration changes. In this work, the shift in pH values caused by the constant bubbling of carbon dioxide into a well-mixed reactor is modeled. Experimental dynamic pH data are then used to calculate the overall volumetric mass transfer coefficients of carbon dioxide into the aqueous phase in a wellmixed reactor as functions of temperature, mixing speed, and aeration rate. Experimental Section Carbon dioxide and air were set to stable flowrates using high precision rotameters (Cole Parmer catalog no. C-03217) and calibrated mass flowmeters (Cole Parmer model 32712) placed in series in each line. The two streams were then mixed to generate gas flows containing 10% carbon dioxide by volume and sparged into a well-mixed, baffled reactor (New Brunswick Bioflo III). The reactor was operated with 2.45 L of reverse osmosis (RO) or salty water. The depth of the aqueous phases was 17.6 cm. A six-blade Rushton-type impeller (diameter 6.45 cm, width 1.3 cm) was used to mix the aqueous phase and was located 7.0 cm above the bottom of the reactor. The gas mixture was sparged through four equally spaced, 1.0 mm diameter orifices located on a 5.0 cm diameter sparging loop, positioned three cm below the rotating impeller. Three variables affecting the overall mass transfer coefficient were investigated, including temperature (15-40 °C), impeller speed (150-600 rpm), and total gas flowrate (0.2-2.0 L/min). Using reverse osmosis water, 20 runs were performed at strategic combinations of the design
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variables according to the Central Composite Rotatable Design strategy (CCRD).12 One extra run was performed with salty water at the midpoint of the design variables. For each run, the reactor was initially filled with fresh water and was brought to the correct temperature and stirring rate using automatic control that is part of the Bioflo III system. The gas flow was not connected to the reactor for several minutes prior to beginning a run so that a representative gas sample could be obtained to provide an accurate carbon dioxide gas-phase concentration. A Hach model sension3 pH meter was connected to a computer with HachLink software to collect pH data at 10 s intervals. A Hach model 51910 pH probe was immersed 5 cm below the top surface of the aqueous phase to measure the instantaneous pH of the well-mixed aqueous phase. The pH probe responded instantaneously to sudden pH changes when moving between calibration standards. The pH probe was calibrated prior to each run. At time zero, the gas flow was connected to the reactor and pH data was automatically collected until steady-state pH values were achieved, which could take up to 20 min depending on operating conditions. For one run, 71.75 g of NaCl (2.85%) was mixed into the reverse osmosis water prior to beginning the flow of gas. Carbon dioxide gas samples were collected just before and after each run and were analyzed on a Hewlett-Packard model 5890 gas chromatograph using a Carbosieve S-II packed column and thermal conductivity detector (TCD). Operating conditions were inlet at 150 °C, detector at 190 °C, and oven temperature starting at 100 °C for 2 min, then ramped to 150 °C at 30 °C/ min and held at 150 °C for 2 min. Maximum relative measurement error was (2% of the values measured. Theory Upon dissolution in water, carbon dioxide undergoes three chemical reactions involving four chemical species (carbon dioxide, carbonic acid, bicarbonate ion, and carbonate ion) according to K0
H2O + CO2,aq 798 H2CO3 K1
H2CO3 798 H+ + HCO3K2
HCO3- 798 H+ + CO3)
(1) (2) (3)
In the above reactions, the equilibrium constants (K0, K1, and K2) are known functions of temperature and salt concentrations that span those found in seawater.13 The value of K0 is such that the concentration of dissolved CO2 far exceeds that of H2CO3 at all conditions, but as custom dictates, in the species and charge balances that follow, the dissolved CO2 and H2CO3 neutral species are considered as one component and called carbonic acid, H2CO3. The rates of reactions 2 and 3 are much faster than that of reaction 1, which would, therefore, be the rate-limiting reaction step.14 In this work, however, reaction 1 was calculated to be 100 times higher than the maximum mass transfer rates, and therefore, reaction equilibrium conditions were assumed to exist in the aqueous phase for all chemical species. Given temperature and salt concentration and using the equations of Mook and de Vries,13 the equilibrium constants will be known values in the following three equations:
[H2CO3] )
[H+][HCO3-] K1
(4)
[H+][CO3)] K2
(5)
[HCO3-] ) [H+] )
1 × 10-14 [OH-]
(6)
The solubility of CO2 in water is well-represented by Henry’s law according to
PCO2 ) H‚[H2CO3]
(7)
where PCO2 is the partial pressure of carbon dioxide in the air above the aqueous phase (atm), H is Henry’s coefficient (atm‚ L/mol, which is also a known function of temperature and salt concentration), and [H2CO3] is the dissolved concentration of carbonic acid in the aqueous phase (mol/L). With knowledge of Henry’s coefficient and equilibrium constants, for a case where air containing a known amount of CO2 (i.e., PCO2 is known) is being bubbled through water, eq 4-7 represent a set of four equations with five unknown species. A unique solution is made possible by the fact that there can be no net charge in the water so that
[H+] ) [OH-] + [HCO3-] + 2[CO3)]
(8)
Solutions of these five nonlinear equations provide the equilibrium concentrations of all the carbonate species and the pH of water for any value of PCO2. These equations were solved on Excel using formulas for Henry’s coefficient and equilibrium constants given by Mook and deVries,13 which were verified with data from the International Critical Tables15 and Butler.16 The pH of pure and salty (3%) water and dissolved CO2 species concentrations at 25 °C and PCO2 ranging from atmospheric (0.000 378 atm) up to 0.2 atm is shown in Figure 1. The carbonate ion concentrations are negligible for all partial pressures of CO2. However, it is noteworthy that the pH is lower in salty water than in pure water and there is significantly more total dissolved carbon dioxide and much more is in the form of bicarbonate ion. To model the dynamic flow of carbon dioxide from the gas phase into the aqueous mixture, it is necessary to incorporate the reduced carbonic acid concentration due to reactions 2 and 3. These reactions are instantaneous, with time constants measured in µs.16 From equilibrium analyses, it is clear that the carbonate ion concentration remains negligible when CO2 is bubbled into water or salty water, and since the solution becomes acidic, the hydroxide ion concentration can also be ignored. Under these conditions, eq 8 shows that the hydronium and bicarbonate ion concentrations will be the same. The mass transfer of carbon dioxide into the aqueous phase of a reactor can then be modeled by
d[H2CO3] ) KLa([H2CO3]* - [H2CO3]) × dt
(
1-
K1
)
K1 + (K1[H2CO3])0.5
(9)
[H2CO3]* represents the saturation concentration of carbonic acid in the liquid phase (mol/L), and KLa is the overall volumetric mass transfer coefficient (h-1). The last term in
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Figure 3. Fitting of eq 9 (line) to determine KLa for carbon dioxide dissolving into the well-mixed reactor (symbols ) experimental data, case of Figure 2). Figure 1. Equilibrium concentrations and pH in water (solid lines) and salty water (3%, dashed lines) when gas containing carbon dioxide at PCO2 is bubbled into a well-mixed reactor.
model that predicted this relationship was developed in this work (shown by the solid line in Figure 2) and is given by
[H2CO3] ) [H2CO3]min +
([H2CO3]max - [H2CO3]min) 1 + exp
Figure 2. Buildup of dissolved carbonic acid in the well-mixed reactor demonstrating a time delay of 100 s (symbols ) experimental data, line ) eq 10, Q ) 1.1 L/min, T ) 27.5 °C, RPM ) 375).
brackets on the right-hand side accounts for the instantaneous shift in carbonic acid to bicarbonate ion, which reduces the apparent rate of accumulation of carbonic acid in the aqueous phase. Equation 9 was solved for carbonic acid concentrations using the fourth-order Runge-Kutta numerical method and Excel software. But since there is a simple, inverse relationship between carbonic acid and hydronium ion concentrations given by eq 4, solving eq 9 also provides a prediction of the pH dynamics when carbon dioxide is bubbled through the aqueous phase in the reactor. Results and Discussion pH data generated in each run suffered from significant time delay required for the gas containing carbon dioxide to establish a steady-state condition in the reactor. This included replacement of the air in the piping and sparger, in the bubbles in the aqueous phase of the reactor, and in the headspace (1.2 L) above the well-mixed water phase. This delay phenomenon is shown in Figure 2 for a typical set of experimental pH data. This type of time delay, followed by sharp increase in concentration and finally followed by a plateau to steady-state concentration, is seen often in the biochemical engineering literature, typically for batch growth when cells must rearrange their metabolic pathways to consume new substrates. An interesting empirical
(
)
- (t - tC) Wn
(10)
where tC represents an estimate of the time to the midpoint of the rise in concentration (see Figure 2), W represents an estimate of the span of the rising curve in time units (see Figure 2), and n is a fitting parameter. In the case of these experiments, the maximum value of carbonic acid was the saturation concentration (a priori known from eq 7) and the minimum carbonic acid concentration was zero. Plotting the data in the form of Figure 2 permitted determination of the data point that represented the initiation of fully developed mass transfer into the reactor aqueous phase, and then this point was used as time zero for fitting eq 9 to determine KLa. Figure 3 demonstrates this analysis for the case of Figure 2, where the air flowrate was 1.10 L/min, the mixing speed was 375 rpm, and the temperature was 27.5 °C. For this run, the measured CO2 concentration in the air was 9.48% (v/v), giving a saturated carbonic acid concentration of 0.002 81 M. The best-fit value of KLa for this data set was found to be 41.4 h-1. Using the best-fit values of KLa for all 20 runs performed at the CCRD locations, an empirical predicting equation was developed for KLa (h-1) that included all terms significant at the 90% confidence level,
KLa ) 33.9 + 7.96T* + 15.7Q* + 18.8RPM* + 6.46Q*2 + 8.25T*Q* (11) where the asterisk terms are the dimensionless, CCRD coded locations of temperature (T in °C), gas flowrate (Q in L/min), and mixing speed (in rpm), given by
T*)
T - 27.5 7.432
(12)
Q*)
Q - 1.1 0.5351
(13)
RPM - 375 133.8
(14)
RPM* )
Equation 11 indicates that the measured KLa values increased as all three variables were increased and there was a significant
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Figure 4. Parity plot showing scatter of best fit KLa data to prediction of eq 11.
interaction effect between temperature and air flowrate. KLa values varied between 20 and 120 h-1. The accuracy of the empirical model is demonstrated in Figure 4, where the maximum scatter of the data was found to be (22 h-1 at all measured values of KLa, demonstrated by the dashed lines in Figure 4. No other directly measured experimental values of KLa for carbon dioxide into an aerated well-mixed reactor have been published for comparison purposes, but many investigators have reported on mass transfer rates for oxygen into well-mixed reactors. The classic work of Robinson and Wilke17 involved studying oxygen transfer at 30 °C using a similar vessel in both water and saltwater. They reported that, in pure water, the overall volumetric oxygen mass transfer coefficient ranged from 40 to 200 h-1 over similar mixing and aeration rates used in this study, and KLa increased with both the mixing and aeration rates as observed here for carbon dioxide. It is not too surprising that the overall volumetric mass transfer coefficients would be slightly higher for oxygen, since the diffusion coefficient of oxygen in water is 20% higher that that of carbon dioxide.18 Temperature increases the diffusion coefficient of molecules in liquids,19 which causes the overall mass transfer coefficient to increase with temperature, as found in this study. The interaction of temperature and air flowrate also enhanced the overall mass transfer coefficient, which may have been due to bubble formation effects at the orifices. Robinson and Wilke17 also reported that the addition of ionic salts increased the overall volumetric mass transfer coefficient compared to those measured in pure water. In this work, using a solution of 2.85% NaCl, the value of KLa decreased from a mean value of 34 h-1 to a value of 28 h-1 at 375 rpm, 27.5 °C, and an aeration rate of 1.1 L/min. Although this decrease disagrees with the trend observed by Robinson and Wilke17 for oxygen mass transfer, the lower value is well inside the maximum scatter observed in the pure water experiments, suggesting that the addition of salt resulted in no change in the KLa value for carbon dioxide mass transfer into water. From visual observations, it was expected that the measured KLa values would increase since, because of lower coalescence, bubble size was noticeably smaller when salt was present in the water phase. However, the ionic charge effect at the bubble surface may have reduced the ability of carbon dioxide molecules to diffuse away from the increased bubble surface area, resulting in no net change in the overall volumetric mass transfer coefficient. Figure 5 shows the effects of temperature, aeration rate, and mixing speed on the overall mass transfer coefficient as
Figure 5. Effect of operating variables on KLa for carbon dioxide dissolving into water in a well-mixed reactor (lines ) eq 11, T range ) 15-40 °C, Q range ) 0.2-2.0 L/min, RPM range ) 150-600).
predicted by eq 11 over the ranges of conditions studied in this investigation. In each case, when one variable was changed, the other two variables were held constant at the midpoint values of this study. Within the ranges of variables studied, it is clear that the mixing speed had the greatest effect on the mass transfer coefficient followed by aeration rate and finally temperature. Within the accuracy of the measurements, the mixing speed and temperature caused the value of KLa to increase linearly, while aeration rate resulted in a nonlinear rise in KLa values. Livansky reported a linear increase in the mass transfer coefficient with temperature for desorption of dissolved carbon dioxide from a nonaerated water surface.10 Given the similarities in diffusion coefficients of oxygen and carbon dioxide in water, this study confirms that the use of diffusion coefficient ratios to predict the overall mass transfer coefficient from gas bubbles into water in a well-mixed reactor is a good technique; however, the effect of ionic charge on mass transfer may be different for each chemical species. Conclusions pH measurements have been used to directly measure the overall mass transfer coefficient of carbon dioxide dissolving from bubbles into a well-mixed reactor. With RO water present in the reactor, the value of KLa ranged from 20 to 120 h-1 over temperature ranges from 15 to 40 °C, aeration rates from 0.2 to 2.0 L/min, and stirring speeds from 150 to 600 rpm. KLa values were slightly below those found for oxygen in well-mixed reactors, indicating that the prediction of carbon dioxide transfer by using the diffusion coefficient correction factor is quite reasonable. Salt addition to the water phase did not improve the mass transfer rate of carbon dioxide. An empirical correlation is presented to show the increasing rates of carbon dioxide transfer as a function of all three operating variables: temperature, aeration rate, and mixing speed. Acknowledgment This study was made possible by financial support provided from the Natural Sciences and Engineering Research Council of Canada (NSERC). Nomenclature H ) Henry’s coefficient (atm‚L/mol) K ) equilibrium constants in reactions 1-3 KLa ) overall mass transfer coefficient (h-1)
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n ) exponent in equation 10 PCO2 ) partial pressure of carbon dioxide (atm) Q ) aeration rate (L/min) t ) time (s) tC ) time to midpoint of rising curve in equation 10 (s) W ) span of rising curve in equation 10 (s) Literature Cited (1) Appenzeller, T. The case of the missing carbon. Nat. Geographic 2004, 205 (2), 88. (2) Weart, S. R. The discoVery of global warming; Harvard University Press: Cambridge, MA, 2003. (3) Borowitzka, M. A. Commercial production of microalgae: Ponds, tanks, tubes and fermenters. J. Biotechnol. 1999, 70, 313. (4) Talbot, P.; Gortares, M. P.; Lencki, R. W.; de la Noue, J. Absorption of CO2 in algal mass culture systems: A different characterization approach. Biotechnol. Bioeng. 1991, 37, 834. (5) Royce, P. N. C.; Thornhill, N. F. Estimation of dissolved carbon dioxide concentrations in aerobic fermentations. AIChE J. 1991, 37, 1680. (6) Molina Grima, E.; Sanchez Perez, J. A.; Garcia Camacho, F.; Robles Medina, A. Gas-liquid transfer of atmospheric CO2 in microalgal cultures. J. Chem. Technol. Biotechnol. 1993, 56, 329. (7) Babcock, R. W.; Malda, J.; Radway, J. C. Hydrodynamics and mass transfer in a tubular airlift photobioreactor. J. Appl. Phycol. 2002, 14, 169. (8) Brune, D. E.; Novak, J. T. The use of carbonate equilibrium chemistry in quantifying algal carbon uptake kinetics. Eur. J. Appl. Microbiol. Biotechnol. 1981, 13, 71. (9) Novak, J. T.; Brune, D. E. Inorganic carbon limited growth kinetics of some freshwater algae. Water Res. 1985, 19, 215.
(10) Livansky, K. Losses of CO2 in outdoor mass algal cultures: determination of the mass transfer coefficient KL by means of measured pH course in NaHCO3 solution. ArchiV. Hydrobiol. 1990, 85, 87. (11) Dahod, S. K. Dissolved carbon dioxide measurement and its correlation with operating parameters in fermentation processes. Biotechnol. Prog. 1993, 9, 655. (12) Diamond, W. J. Practical experimental designs for engineers and scientists, 2nd ed.; Van Nostrand Reinhold: New York, 1989; pp 268275. (13) Mook, W. G.; deVries, J. J. EnVironmental isotopes in the hydrological cycle: Principles and applications. Volume 1: Introduction, theory, methods, reView; IAEA Publication: Vienna, Austria, 2005; pp 143166. (14) Johnson, K. S. Carbon dioxide hydration and dehydration kinetics in seawater. Limnol. Oceanogr. 1982, 27, 849. (15) National Research Council. International Critical Tables; McGrawHill: New York, 1929; Vol. III. (16) Butler, J. N. Ionic equilibrium, a mathematical approach; AddisonWesley: Reading, MA, 1964; p 207. (17) Robinson, C. W.; Wilke, C. R. Oxygen absorption in stirred tanks: A correlation for ionic strength effects. Biotechnol. Bioeng. 1973, 15, 755. (18) Perry, R. H., Green D. W., Eds. Perry’s Chemical Engineers’ Handbook, 7th ed.; McGraw-Hill: New York, 1997; pp 2-330-2-331. (19) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264.
ReceiVed for reView February 26, 2006 ReVised manuscript receiVed May 13, 2006 Accepted June 8, 2006 IE060242T
