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Ozone dissolution in aqueous systems treatment of the experimental data
Experimental Thermal and Fluid Science 28 (2004) 395–405 www.elsevier.com/locate/etfs
Ozone dissolution in aqueous systems treatment of the experimental data A.K. Bin
*
Faculty of Chemical and Process Engineering, Warsaw University of Technology, ul. Wary nskiego 1, 00-645 Warszawa, Poland
Abstract The major concern of the paper was to establish how the treatment method of the experimental data collected during a semibatch ozone dissolution process in aqueous systems using gas–liquid absorbers could influence the determination of the fundamental process parameters such as ozone solubility (or the Henry’s law constant values) and mass transfer coefficients. Statistically the best were the sigmoidal Boltzmann and Weibull fits, however, they provide the product of the mass transfer coefficient ðkL aÞ and the saturation ozone concentration in the liquid phase. In order to determine the Henry’s constant an independent evaluation of kL a is necessary. Ozone decomposition term might play relatively small role in the overall mass balance of ozone in the liquid phase so the question of the reaction order of ozone decomposition in such a case may be of less significance. 2003 Elsevier Inc. All rights reserved. Keywords: Ozone dissolution; Data treatment; Gas–liquid system
1. Introduction Ozonation process in the liquid phase is most frequently accomplished by injecting the ozone gas (mixtures of air–ozone or oxygen–ozone) through a sparger into the liquid, so that the fundamental feature of that process should be categorised as absorption. Usually the studies on ozone absorption in the aqueous systems are carried out in stirred-tank reactors or in bubble columns. A part of such studies has been aimed at determination of the mass (ozone) transfer coefficients, on ozone decomposition and on ozone solubility. Determination of ozone solubility in aqueous systems is often accompanied by the simultaneous studying of ozone decomposition in the liquid phase. Typical experimental procedure (a dynamic method) involves bubbling of a gas mixture containing ozone through a porous element into a liquid sample (water or aqueous solution), maintained at the isothermal and almost constant ozone partial pressure conditions (semi-batch run). Determination of ozone solubility in water or in aqueous solu-
*
Fax: +48-22-825-14-40. E-mail address: [email protected] (A.K. Bi n).
0894-1777/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2003.03.001
tions has a long history. Probably the first paper on this topic is that due to Sch€ one [1]. A number of the compilation attempts of the published data can be found in the references (e.g. [2–10]). The dissolved ozone concentration is monitored as a function of time and the resulting points (curves) are subject to further analysis. Typical examples of such data are shown in Fig. 1, where the experimental data due to Kuo et al. [11] have been replotted. Also, as an illustration, the Boltzmann fit (cf. Appendix A) is shown for that data. It is evident from these graphs that depending upon the experimental conditions different curves CL ðtÞ are obtained. The Boltzmann fit that represents the group of the so-called sigmoidal fits, approximates the considered experimental data points very well, thus preliminary indicating that the changes of the dissolved ozone concentration with time may successfully be described by Eq. (A.5). The more rigorous discussion of the experimental data treatment follows in the next section. The major point of this approach is aimed at a critical analysis how the method of the experimental data treatment collected during the ozone dissolution batch process in an aqueous system can influence the determination of parameters such as ozone solubility (or the Henry’s law constant values), the mass transfer coefficient, kL a, and ozone decomposition terms in that system.
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A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
Nomenclature volumetric mass transfer coefficient in the liquid phase (s1 ) kd , k1 , k2 , k3 kinetic constants of ozone self-decomposition (s1 , mol1m , m3m3 , s1 ) K, K1 , K2 dimensionless kinetic constants L, L1 dimensionless parameters M, M1 dimensionless parameters m power exponent in Eq. (1) QG gas volumetric flow rate (m3 s1 ) S solubility (¼ 1=HA ) T temperature (C) t time (s) uG gas superficial velocity in the bubble column (m s1 ) yi , ^yi observed and predicted value, respectively
kL a
Greek letters u parameter defined by Eq. (C.9) k parameter in the Weibull fit
2.0
2.0
1.5
1.5
CL (x 104 mM)
CL (x 104 mM)
A, A1 , A2 fitting parameters a specific interfacial area (m1 ) B fitting parameter in the Weibull fit CG molar concentration of ozone in the gas phase (mol m3 ) CL molar concentration of ozone in the liquid phase (mol m3 ) CL equilibrium molar concentration of ozone in the liquid phase (mol m3 ) CLss steady-state molar concentration of ozone in the liquid phase (mol m3 ) d power exponent in the Weibull fit D fitting parameter in the Boltzmann fit DL molecular diffusivity in the liquid phase (m2 s1 ) ei residual (¼ yi ^yi ) HA dimensionless Henry’s law constant h, h0 depth of the liquid layer in the bubble column (m)
1.0
0.5
0.0
0
(a)
pH = 2.2
t = 25oC QG = 0.51 dm3/min CG = 48.5 mg O3/min
pH = 4.1
1000
2000
1.0
QG (dm3/min)
(b)
0.5
pH = 6.2 pH = 7.1
3000
0.5
o
25 C pH = 2.2 CG = 39.95 mg O3/dm3
1.0 1.5 2.0
0.0
4000
0
Time (s)
1000
2000
3000
4000
Time (s)
Fig. 1. Dissolved ozone concentration vs. time [11]. (O, d, ,
)
Experimental data points; curves––Boltzmann fit.
2. Treatment of the experimental data
2.1. Constant gas composition model: CG ¼ CG;inlet
Interpretation of the collected experimental data requires an assumption of the mixing behaviour in both phases [12]. Different models can be taken into account (cf. Table 1). If the analytical solutions are available, they can be used to fit to the measured concentration profiles yielding the best estimates for the parameters such as kL a, kd , and CL . Experimental data obtained in the batch mode measurements can be treated using some of the models listed in Table 1. The following examples illustrate application of the constant gas composition and complete mixing models (no. 1 and 2 in Table 1), accounting for ozone decomposition in the liquid phase.
This model is in fact based on the assumption (cf. [13]) that the changes of the ozone concentration in the gas phase during bubbling can be neglected, i.e. CG const, and whence also ozone equilibrium concentration in the liquid, CL const (under the isothermal and approximately isobaric conditions). The process of ozone dissolution (absorption) has most frequently been described by the following expression (the more rigorous method is given in Appendix C) dCL ¼ kL aðCL CL Þ kd CLm dt
ð1Þ
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
397
Table 1 Types of model applicable in the absorption data treatment (batch mode) No.
Type of model
Remarks
1. 2.
Constant gas composition Complete mixing (CSTR)
3. 4. 5. 6. 7. 8.
Plug flow (PF) Multistage mixing Axial dispersion (ADM) Multistage mixing with backward flow Two-bubble-class Gas-PF ; Liq-ADM
9.
Gas-PF ; Liq-CSTR
CG ¼ CG;inlet , liquid phase well mixed. Analytical solution available for first-order ozone decomposition Gas phase well mixed, gas composition variable. Analytical solution available for first-order ozone decomposition Plug flow in the gas phase Gas pathway treated like a series of stirred cells Plug flow in the gas and/or liquid phases superimposed with some degree of axial dispersion Gas phase is considered to consist of many perfectly mixed cells in series superimposed with a backward flow between cells Accounting for differentiated bubble size in the heterogeneous regime by assuming two bubble size classes Set of two mass balance equations containing the dispersion terms and the appropriate initial and boundary conditions. Numerical solutions necessary Analytical solution available when uG ¼ const ¼ uG;inlet and if the accumulation in the gas phase can be neglected
Ozone consumption (decomposition or other chemical reactions) is accounted for by the second term. At the initial condition t ¼ 0, CL ¼ 0 and under the previously made assumptions Eq. (1) can readily be integrated for m ¼ 1 and 2. The results of this integration are as follows: CL m ¼ 1 CL ¼ f1 exp½ð1 þ KÞðkL atÞg ð2Þ 1þK where K ¼ kd =ðkL aÞ. LðM 1Þ ðM þ 1Þ m ¼ 2 CL ¼ CL ð3Þ 2KðL þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mþ1 exp½ðkL atÞM, and where M ¼ 1 þ 4K , L ¼ M1 kd CL K¼ . kL a A somewhat different type of Eq. (1) is of interest dCL ¼ ðkL aÞðCL CL Þ þ k3 CL k2 CL2 dt
ð4Þ
which after integration yields L1 ðM1 þ K1 Þ þ ðM1 K1 Þ CL ¼ CL ð5Þ 2K2 ðL1 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3 k2 CL 1, K2 ¼ , M1 ¼ K12 þ 4K2 , and with K1 ¼ kL a kL a K1 M1 exp½M1 ðkL atÞ. L1 ¼ K1 þ M1 2.2. Complete mixing model In this model, it is assumed that the phase is well mixed while the solute concentration varies with time. The mass (ozone) balance equations for both phases are as follows: gas phase dCG uG 1 eG ¼ ðCG CG Þ ðkL aÞ ðCLi CL Þ dt eG hL eG ð6Þ
liquid phase dCL ¼ kL aðCL CL Þ kd CLm ð7Þ dt Assuming the first-order ozone decomposition (m ¼ 1) in the liquid phase the set of Eqs. (6) and (7) can analytically be solved yielding the following dimensionless ozone concentrations in the gas and liquid phases, respectively (cf. [14,15]): CGþ ¼ c3 þ
1 þ c3 r2 1 þ c3 r1 expðr1 hÞ þ expðr2 hÞ r1 r2 r2 r1
CLþ ¼ c3 þ
c3 1 ð1 þ c3 r2 Þð1 þ a1 þ r1 Þ expðr1 hÞ þ a1 a1 ðr1 r2 Þ
þ
ð1 þ c3 r1 Þð1 þ a1 þ r2 Þ expðr2 hÞ a1 ðr2 r1 Þ
ð8Þ
ð9Þ
where CGþ ¼
CG ; CG
a1 ¼ tm ðkL aÞ
CLþ ¼
CL HA ; CG
1 eG ; eG H A
c 1 ¼ 1 þ a1 þ a2 þ a3 ;
h¼
t uG t ¼ ; tm eG hL
a2 ¼ tm ðkL aÞ;
a3 ¼ tm kd ;
c 2 ¼ a2 þ a1 a3 þ a3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ a3 c1 þ c21 4c2 c3 ¼ ; ; r1 ¼ 2 c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 c21 4c2 r2 ¼ 2 Some authors (e.g. [3,11,16–18]) suggested that m ¼ 1:5 in Eq. (1). Also, the complex forms of the decomposition term have been considered (e.g. [5,17,20,22]). However, these require numerical solution, e.g. using the standard fourth-order Runge–Kutta scheme (R–K) so that further fitting procedure is then much more complicated and tedious. Recently, Qiu [14] studied ozone decomposition and solubility in aqueous solutions of pH varying from 2 to 9. Based on his own experiments and the results of many other authors, as well as on considerations on the
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A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
mechanism of ozone self-decomposition in aqueous solutions, he concluded that the first-order ozone decomposition kinetics provided statistically the best correlation of the rate data over the wide range of pH (from 2 to 13.5 at 25 C)––Eq. (10): kd ¼ 20½OH 1=2 þ 900½OH ðs1 Þ
ð10Þ
The derived expressions can be fitted to the experimental data by performing a typical regression procedure based on minimisation of the sum of squares of deviations. As a result of such a treatment the ‘‘best’’ values of the searched kinetic parameters (CL ; kL a; kd ; k2 ; k3 ) or their combinations can be estimated. If the values of any of these parameters are known from some other independent determinations this fact can (and should) also be taken into account. However, it has to be realised that possible dependency between the fitting parameters may exist and this has to be carefully considered in choosing the most adequate parameters to be fitted. After performing preliminary tests the following fitting procedures for a data set of Kuo et al. [11] have been applied: 1. Boltzmann fit (four parameters found from regression, i.e. A1 , A2 , t0 and D),
2. m ¼ 1 (Eq. (2)) (two parameters found from regression, i.e. ðkL auÞCL and ðkL au þ kd Þ), 3. m ¼ 2 (Eq. (3)), assuming a value of ðkL auÞCL from the previous procedure (two parameters found from regression, ðkL auÞ and kd ), 4. m ¼ 2 with a constant value of CL given by the authors (two parameters found from regression, i.e. ðkL auÞ and kd ), 5. Weibull fit with d ¼ 1 (two parameters found from regression, i.e. ðkL auÞCL and ðkL au þ kd Þ), 6. Complete mixing model (Eq. (9)) with constants values of CL given by the authors (two parameters found from regression, ðkL auÞ and kd ). The results of the applied fitting procedures for the experimental data of Kuo et al. [11] are listed in Table 2. A survey of the numerical values contained in Table 2 indicates that usually the statistical quality of regression is good (R2 > 0:99). However, the differences between the procedures are better reflected by the square root of the sum of squares of the residuals (Norm)––cf. Fig. 2. From this figure, one can conclude that procedure no. 1 (the Boltzmann fit) provides the best overall fit to the experimental data under consideration. The goodness of a fit can also be assessed by analysing the residuals [23].
Table 2 The results of the fitting procedures for the data sets of Kuo et al. [11]; experimental conditions are indicated in Fig. 1 Run 8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
Proc. no. 2 (m ¼ 1)
Proc. no. 1 (Boltzmann fit) ðkL auÞCL (·104 m s1 )
CLss (·104 m)
R2
ðkL auÞCL (·104 m s1 )
CLss (·104 m)
kL au þ kd (s1 )
R2
0.00314 0.00358 0.00289 0.00255 0.00238 0.00396 0.00644 0.00904
1.810 1.768 1.533 1.446 1.299 1.399 1.446 1.644
0.9991 0.9993 0.9999 0.9859 0.9986 0.9986 0.9976 0.9991
0.00389 0.00405 0.00393 0.00382 0.00254 0.00455 0.00657 0.00916
1.843 1.788 1.569 1.516 1.302 1.410 1.444 1.645
0.00211 0.00227 0.00250 0.00252 0.00195 0.00323 0.00455 0.00557
0.9959 0.9976 0.9945 0.9857 0.9988 0.9976 0.9974 0.9992
0.00220 0.00233 0.00259 0.00269 0.00192 0.00325 0.00447 0.00556
0.9969 0.9980 0.9954 0.9884 0.9989 0.9977 0.9976 0.9992
Proc. no. 3 (m ¼ 2) 8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
0.00389 0.00405 0.00393 0.00382 0.00254 0.00455 0.00657 0.00916
Proc. no. 5 (Weibull fit with d ¼ 1) 1.827 1.781 1.552 1.487 1.302 1.405 1.443 1.644
0.9962 0.9978 0.9950 0.9869 0.9988 0.9977 0.9974 0.9992
Proc. no. 4 (m ¼ 2)
8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
0.00403 0.00416 0.00405 0.00405 0.00250 0.00458 0.00646 0.00915
1.837 1.784 1.565 1.505 1.304 1.409 1.446 1.645
Proc. no. 6 (complete mixing)
ðkL auÞ (s1 )
CLss (·104 M)
R2
ðkL auÞ (s1 )
CLss (·104 M)
kd (s1 )
R2
0.00175 0.00181 0.00169 0.00165 0.00133 0.00243 0.00351 0.00511
1.833 1.779 1.554 1.492 1.291 1.403 1.440 1.643
0.9970 0.9984 0.9970 0.9904 0.9970 0.9985 0.9942 0.9986
0.00273 0.00291 0.00277 0.00265 0.00198 0.00342 0.00489 0.00693
1.843 1.788 1.569 1.515 1.301 1.410 1.444 1.644
0.000278 0.000359 0.000654 0.000723 0.000501 0.000628 0.000797 0.000334
0.9960 0.9977 0.9947 0.9859 0.9987 0.9977 0.9973 0.9992
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
399
3. Discussion Procedures 1–3 and 5 do not allow direct determination of CL or kL a but provide only the values of their product. Procedures 2 and 5 provide also the estimated values of the sum ðkL a u þ kd Þ. All the considered procedures enable one to estimate the values of the steady-state asymptote, CLss . Figs. 4–6 display the
0.014 proc. 1 proc. 2 proc. 3 proc. 4 proc. 5 proc. 6 Kuo et al. (1977)
(kLaϕ) CL* (x 104 M/s)
0.012
Fig. 2. Values of the square root of the sum of squares of the residuals (Norm) for the experimental data of Kuo et al. [11].
Some examples of the residuals distribution for the experimental data of Kuo et al. [11] are shown in Fig. 3. From these graphs it can be concluded that both approximations, i.e. the Boltzmann fit (procedure no. 1) and the Weibull fit (procedure no. 5) provide the smallest distributions of the residuals.
0.006 0.004 0.002
8_2
8_3
9_1
9_3
1.0e-5 Run 8_4
5.0e-6
ei (M)
0 -2e-6 -4e-6
m=1 m=2 m=2 g BF WF (d=1) CMM
-6e-6
(a)
-8e-6 0
500
1000
1500
2000
2500
3000
0.0 -5.0e-6 m=1 m=2 m=2 g BF WF (d=1) CMM
-1.0e-5
3500
-1.5e-5
0
500
1000
(b) 1500
2000
2500
3000
Time (s)
Time (s) 6e-6
6e-6
Run 9_1
Run 9_3
m=1 m=2 m=2 g BF WF (d=1) CMM
4e-6
4e-6 2e-6
ei (M)
2e-6 0
0 -2e-6
-2e-6
-4e-6
(c) -4e-6
9_2
0
500
1000
1500
(d) 2000
Time (s)
2500
3000 3500
9_4
Fig. 4. Values of the product ðkL auÞCL for the experimental data of Kuo et al. [11].
2e-6
ei (M)
8_4
Run No.
Run 8_1
4e-6
ei (M)
0.008
0.000 8_1
6e-6
-1e-5
0.010
-6e-6
0
500
1000
1500
2000
Time (s)
Fig. 3. Values of the residuals for the experimental data of Kuo et al. [11].
m=1 m=2 m=2 g BF WF (d=1) CMM
2500
3000
3500
400
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405 1e+1
2.0 proc. 1 proc. 2 proc. 3 proc. 4 proc. 5 proc. 6
1.9
1.7
1e+0 1e-1
kd (s-1)
CLss (x 10 mM)
1.8
[24] [19] [17] [7] [25] [26]
1.6 1.5
[27] (25oC) [28] [29] [22] [5] [21] (nitric acid) [21] (acetic acid)
[14] (25oC)
1e-2 1e-3
1.4
1e-4 1.3 1.2 8_1
t = 20oC 1e-5 8_2
8_3
8_4
9_1
9_2
9_3
9_4
1e-6
Run No.
0
2
4
6
8
10
12
pH
Fig. 5. Values of the asymptote CLss for the experimental data of Kuo et al. [11].
Fig. 7. Ozone self-decomposition first-order rate constant vs. pH at 20 C (cf. [19,26–28]). 0.008
0.002
0.000 8_1
8_2
8_3
8_4
9_1
9_2
9_3
9_4
Run No. Fig. 6. Values of the sum ðkL auÞ þ kd for the experimental data of Kuo et al. [11].
obtained values of these parameters or their combinations in a graphical form. From these graphs, one can conclude that in the case of the product ðkL auÞCL the different treatment of the experimental data will yield values differing on average by 13–21% (absolute value) from that obtained from procedure 1 (Boltzmann fit), except of that for procedure 6 (complete mixing), which produced the highest difference (on average 68%). In the case of the sum ðkL a u þ kd Þ––Fig. 6, both procedures 2 and 5 lead to practically the same values (the difference between them is on average 3%) while those found from procedure 6 are higher by about 33%. The estimated values of the steady-state asymptote, CLss , are very close to each other (on average ±1%). If separation of kL a u, CL and/or kd from their products or sums is attempted, one has to know some of them (e.g. kL a u or kd values) from independent sources or measurements. For further considerations, a survey of the published or available experimental data on ozone decomposition rate in water and diluted aqueous solutions may be helpful (Fig. 7). At low pH (pH < 5)
0.008 0.007
(1/s)
(1/s) (kLa ϕ) + kd
0.004
(kLa ϕ ) + kd
proc. 2 proc. 5 proc. 6
0.006
the values of the first-order self-decomposition rate, kd , are of the order of 104 s1 . As it can be seen in Table 2 this could give 5–20% contribution of kd in the sum ðkL a u þ kd Þ. At any rate, only at low pH-values procedures 2 and 5 would yield the possibility of estimating the values of kL a u since in this range of pH the contribution of ozone self-decomposition term in the sum ðkL a u + kd Þ is sufficiently small––most likely within the accuracy range of the experimental procedure. The results of such estimations are shown in Fig. 8 for the data of Kuo et al. [11] collected at pH ¼ 2.2 and 25 C. Procedures 2 and 5 yield almost the same values, differing from each other within the mean absolute error 1.7%, while in the case of procedure 6 they are systematically higher by 25–44% from those obtained from procedure 2. The original data of Kuo et al. [11] when treated in the way described above are lower by 40–53% when compared with the results from procedure 2.
0.006
Kuo et al. data [11] proc. 2 proc. 5 proc. 6
pH = 2.2
0.005 0.004 0.003 0.002 0.001 0.0
0.5
1.0
1.5
2.0
2.5
QG (L/min) Fig. 8. Dependence of the sum ðkL auÞ þ kd vs. QG for the experimental data of Kuo et al. [11].
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
When the values of ðkL auÞ are known or estimated, it is possible to split the product ðkL auÞCL to calculate ozone equilibrium solubility, CL , and then the Henry’s law constant (or dimensionless ozone solubility, S). The results of such calculations are presented in Table 3. From this data it is seen that both procedures 2 and 5 give the same values of HA or S, while taking the values of the product ðkL auÞCL estimated from procedure 1 (the Boltzmann fit) and using the values of ðkL auÞ based on the original data of Kuo et al. [11], the Henry’s law constant is lower by 38% than the mean value obtained from the former procedures. Kuo et al. [11] based their calculations on a value of S ¼ 0:21 (or HA ¼ 4:76). It is then evident that depending on the fitting procedure and the assumed values of other parameters, e.g. the mass transfer coefficient, ðkL a uÞ, different values of the Henry’s law constant are obtained. The effect of pH on the estimated values of HA is not very clear though according to the data of some authors [4,7,9,14,18,20, 21,30,33,34] this effect could be considered. An apparent influence of pH on ozone solubility (and hence on the value of the Henry’s law constant) in aqueous systems may be attributed to the limited accuracy of the decomposition term evaluation and the assumed procedure of the experimental data treatment. On the other hand, since the experiments carried out at different pHvalues require introduction of acids and/or buffers, this means that in such cases we operate with diluted solutions and hence we can expect effects of these additions on ozone solubility as compared to that in the ‘‘pure’’ solvent (water). The previous determinations of ozone solubility findings were based on different assumptions regarding the ozone decomposition term as well as on the data treatment procedure. For instance, in evaluating ozone solubility in aqueous systems Roth and Sullivan [4] or Andreozzi et al. [30] assumed the first-order reaction of
401
ozone self-decomposition with respect to dissolved ozone concentration, while Kuo et al. [11], Sotelo et al. [20] and Miyahara et al. [18] postulated that the power exponent in Eq. (1) should be taken as 1.5 or 2, depending on the accompanying dissolved salts (or buffers). An independent determination of the mass transfer coefficient in the applied absorber is also recommended by most of these authors. The experimental data collected during the batch runs are treated typically by considering the steady-state asymptote [4,7,18,21,29,30] dCL m ¼ 0 ) kL aðCL CLss Þ ¼ kCLss ð11Þ dt With the independently known values of kL a, k, m, and the measured (or extrapolated) CLss , it is then possible to calculate the equilibrium ozone concentration in the liquid phase, CL , and hence the Henry’s law constant. Sotelo et al. [20] assessed first ozone self-decomposition reaction order and then verified these findings using a plot directly resulting from Eq. (1): dCL þ kCLm ¼ kL aðCL CL Þ ð12Þ dt i.e. as dCL =dt þ kCLm ¼ f ðCL Þ. A linear plot is expected yielding kL a and CL . This method requires an earlier and independent quantitative information on the ozone selfdecomposition term. Caprio et al. [6] performed a set of experiments on ozone solubility in water by using a procedure which avoided any equilibrium disturbance on the analytical samples. In this procedure the batch system was led into a steady state identified with the phase (absorption) equilibrium with respect to the gas inlet concentration. A simple ozone mass balance allowed calculation of the ozone Henry’s constant based on the known volumes of both phases in the absorption vessel. Negligible ozone decomposition in the liquid phase was assumed. The authors verified the accuracy of their method by
Table 3 Estimated values of the dimensionless Henry’s constant, HA , and ozone solubility, S, for the data Kuo et al. [11], runs 8 and 9 (T ¼ 25 C) Run
pH
Fitting procedure 2 (m ¼ 1)
8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
2.2 4.1 6.2 7.1 2.2 2.2 2.2 2.2
Mean value at pH ¼ 2.2 Std. dev. a
B).
1 (BF)a
5 (d ¼ 1)
HA
S
HA
S
HA
S
5.48 5.65 6.44 6.67 6.39 5.90 5.76 5.06
0.182 0.177 0.155 0.150 0.156 0.169 0.174 0.198
5.50 5.66 6.46 6.71 6.38 5.91 5.76 5.06
0.182 0.177 0.155 0.149 0.157 0.169 0.174 0.198
3.81 3.34 4.14 4.69 3.89 3.82 3.36 2.96
0.262 0.300 0.241 0.213 0.257 0.261 0.298 0.338
5.72 0.50
0.164 0.014
5.72 0.49
0.164 0.015
3.57 0.40
0.283 0.035
In this calculations the product ðkL auÞCL has been separated using ðkL auÞ values resulting from the original data of Kuo et al. [11] (cf. Appendix
402
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
changing the volume of water filling the absorber, which should provide a straight line of the overall ozone content and the water volume. Their data can be regarded as being the most accurate among those reported in the references and are accepted as an International Ozone Association (IOA) standard [10]. Anselmi et al. [31] studied ozone mass transfer in a stirred vessel. They concluded that the best fit to their experimental points was obtained when the secondorder decomposition of dissolved ozone was assumed. However, in the later phase of the data treatment they neglected the decomposition term since according to their estimates its contribution in the overall ozone mass balance was below 2%. Further conclusions can be gained by comparing the rates of ozone absorption expressed by the derivative dCL =dt calculated using different methods of data treatment against the measured ozone concentrations in the liquid phase. Fig. 9 shows examples for four data sets of Kuo et al. [11]. One can notice that practically a linear dependence between the derivative dCL =dt and CL does exist, while the deviations from this trend occur at the beginning of the absorption experiment (low values of CL and hence its possible larger experimental errors). It is interesting to note that there are relatively small differences between the values of this derivative calcu-
lated using different methods of the experimental data fitting. This would indicate that the ozone decomposition term might play a relatively small role in the overall mass balance of ozone in the liquid phase so that the question of the order of ozone decomposition will be of less significance. In this context the problem of a model that would adequately describe the studied process will emerge when considering model selection. Apart from the previously presented discussion it should be remarked that that problem has been considered by a number of authors [35–38]. From these papers it can be concluded that the range and precision of the independent variables affect the results and credibility of the regression equations. For example, it could mean that including an additional term in the regression equation might not necessarily improve the expected accuracy because of harmful effects of collinearity. Hence, often simple models can be recommended provided that there are no further arguments (theoretical or experimental evidences) that would support more complex models. It should be pointed out that the recommended fits are equally successful when applied to treat the experimental data of many other authors [14,18,20–22,24,25, 29,32].
5e-7
5e-7
Run 8_2
num. dif. m=1 m=2 m=2 (g) WF (d=1) BF CMM
4e-7 3e-7
4e-7
dCL /dt (M/s)
Run 8_1
2e-7 1e-7
3e-7 2e-7 1e-7
(b)
(a) 0 0.00000
0.00005
num. diff. m=1 m=2 m=2(g) WF (d=1) BF CMM
0.00010
0.00015
0 0.00000
0.00020
0.00005
0.00010
CL (M)
0.00015
3e-7
7e-7 Run 9_3
2e-7
num. diff. m=1 m=2 m=2 (g) WF (d=1) BF CMM
6e-7
num. diff. m=1 m=2 m=2 (g) WF (d=1) BF CMM
5e-7
dCL /dt (M/s)
Run 9_1
dCL /dt (M/s)
0.00020
CL (M)
1e-7
4e-7 3e-7 2e-7 1e-7
(c) 0 0.00000
(d)
0.00005
0.00010
CL (M)
0.00015
0 0.00000
0.00005
0.00010
CL (M)
Fig. 9. Rate of ozone absorption vs. measured ozone concentration in the liquid phase [11].
0.00015
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
4. Conclusions A number of different fitting procedures have been applied to analyse the experimental data on ozone transfer to water collected by Kuo et al. [11] in a bubble column. The major concern of this treatment was to establish how the method of the data treatment could influence the determination of ozone solubility (or the Henry’s law constant values) in water and/or aqueous systems. It was demonstrated that the fitting procedure affects the values of the Henry’s constant. From the statistical point of view the best fits to the experimental data, considered in the form of CL ðtÞ, were achieved using the Boltzmann and the Weibull fits. These two fits, however, provide statistically best values of the product CL and kL a so that in order to determine the Henry’s law constant an independent evaluation of the mass transfer coefficient, kL a, is necessary. Thus, the accuracy of determination of this process parameter will inevitably influence the accuracy of the estimated Henry’s law constant in the ozone–water system. It was also found that the ozone decomposition term might play a relatively small role in the overall mass balance of ozone in the liquid phase so the question of the order of ozone decomposition in such a case may be of less significance. This conclusion must be treated with caution since depending on the experimental conditions ozone decomposition can be much more rapid (e.g. at pH > 9). Among the experimental procedures applied, the one by Caprio et al. [6] seems to yield the most accurate values of ozone solubility in water. The suggested IOA standard of ozone solubility in water is apparently based on the experimental data of these authors.
Appendix A. Sigmoidal fits To approximate the experimental data obtained in batch absorption experiments two types of sigmoidal fits seem to be of particular interest: the Boltzmann and Weibull fits. Boltzmann fit The Boltzmann fit is based on the following expression CðtÞ ¼
A1 A2 0 þ A2 1 þ exp tt D
ðA:1Þ
It can be seen from Eq. (A.1) that the Boltzmann fit should provide four fitting parameters: A1 , A2 , t0 and D. The first two of them have the same unit as CðtÞ, while t0 and D are expressed in time units. As t ! 1, CðtÞ ! A2 so that A2 is an asymptote of the expression (A.1). At t¼0 Cð0Þ ¼
A1 A2 þ A2 6¼ 0 1 þ exp tD0
ðA:2Þ
403
which demonstrates that expression (A.1) would formally not fulfil the initial condition: at t ¼ 0, Cð0Þ ¼ 0. However, this discrepancy is usually small and can be neglected for further considerations. An interesting feature resulting from the Boltzmann fit can be derived by differentiating CðtÞ with respect to time and then manipulating it to obtain the following expression: exp½ðtt0 Þ=D ðA A Þ 1 2 D dC ¼ ðA:3Þ
tt0 2 dt 1 þ exp D
Since A1 A2 0 ¼ C A2 and 1 þ exp tt D t t A A C A1 0 1 2 1¼ ; exp ¼ D C A2 C A2 substituting these terms into Eq. (A.2) and further rearrangement finally yields dC 1 ¼ ðA1 CÞðA2 CÞ dt DðA1 A2 Þ 1 ½A1 A2 þ ðA1 þ A2 ÞC C 2 ¼ DðA2 A1 Þ
ðA:4Þ
The shape of Eq. (A.4) reveals that if the experimental data on ozone dissolution with its simultaneous decomposition can be well fitted with the Boltzmann fit, in fact they should obey the following overall kinetic form dC ¼ k1 ðC CÞ þ k3 C k2 C 2 ðA:5Þ dt where k1 ¼ kL a (the mass transfer coefficient in the liquid phase), C is the equilibrium ozone concentration dissolved in the liquid phase, and the term k3 C k2 C 2 accounts for ozone decomposition. A comparison of Eq. (A.4) with Eq. (A.5) readily gives A1 A2 ; DðA2 A1 Þ 1 k2 ¼ DðA2 A1 Þ
k1 C ¼
k3 k1 ¼
A 1 þ A2 ; DðA2 A1 Þ
Weibull fit The Weibull fit is described by the following expression d
CðtÞ ¼ A ðA BÞ expððktÞ Þ
ðA:6Þ
Again, similarly as for the Boltzmann fit, this expression does not fulfil the initial condition that at t ¼ 0, Cð0Þ ¼ 0. In fact, Cð0Þ ¼ B 6¼ 0. As t ! 1, CðtÞ ! A so that A is an asymptote of the expression (A.6). For the special case when d ¼ 1, Eq. (A.6) will almost completely correspond to Eq. (2). This can be demonstrated by considering the derivative of Eq. (A.6) and comparing the result with Eq. (1) taking m ¼ 1. In that case k1 CL ¼ kA and k1 þ k ¼ k.
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A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
Appendix B. Mass transfer coefficients Kuo et al. [11] measured the mass transfer coefficients in an oxygen–water system (Table 4). This data can be recalculated for the ozone–water system assuming the penetration model of mass transfer and taking the values of diffusivities of both species as 2.44 · 109 and 1.90 · 109 m2 s1 at 25 C [39], respectively. 0:5 DL;ozone ðkL aÞozone ¼ ðkL aÞoxygen DL;oxygen ¼ 0:882ðkL aÞoxygen
ðB:1Þ
Appendix C. Ozone absorption in a bubble column
ðC:1Þ
• gas phase (with uG const) uG
dCG þ ðkL aÞðCL CL Þ ¼ 0 dh
ðC:2Þ
The first term of the RHS in Eq. (C.1) contains an integral mean driving force in the liquid phase calculated for a gas bubble as it rises across the liquid layer. At the interface the Henry’s law is obeyed: CL ¼
CG HA
ðC:3Þ
Let us assume a quasi-stationary process of ozone dissolution from a gas bubble which would mean that CL const during the rising time of a bubble (typically this is a matter of a few seconds). Integration of Eq. (C.2) yields ðkL aÞh CG ¼ ðCG;in CL HA Þ exp þ CL HA uG H A ðC:4Þ
Table 4 Values of the mass transfer coefficients Kuo et al. [11] QG , dm3 min1
kL a (oxygen), s1
kL a (ozone), s1
ua
kL au, s1
0.5 1.0 1.5 2.0
0.0017 0.0027 0.0038 0.0043
0.00132 0.00210 0.00276 0.00335
0.8899 0.9108 0.9213 0.9280
0.00117 0.00191 0.00254 0.00311
a
Calculated from Eq. (C.9)––cf. Appendix C.
Integral mean of CL is defined as Z h0 1 CL ¼ CL ðhÞ dh h0 0 which after using Eq. (C.5) gives CG;in uG HA CL ¼ CL HA kL a ðkL aÞh 1 exp þ CL u G HA
ðC:5Þ
ðC:6Þ
ðC:7Þ
This expression can then be used in Eq. (C.1):
Ozone mass balance in both phases (assuming negligible ozone accumulation in the gas phase) can be expressed as follows: • liquid phase Z h0 dCL 1 ¼ ðkL aÞ ðCL CL Þ dh kCLm h0 0 dt
After substitution Eq. (C.3) becomes CG;in ðkL aÞh CL exp þ CL CL ¼ uG HA HA
dCL ¼ ðkL aÞuðCL;in CL Þ kCLm dt where uG HA ðkL aÞh0 u¼ 1 exp kL ah0 uG H A
ðC:8Þ
ðC:9Þ
Eq. (C.8) differs from Eq. (1) by containing a factor u ( 6 1) and that the equilibrium ozone concentration CL;in (¼ CG;in =HA ) refers to ozone concentration in the inlet gas. Under the typical experimental conditions u ¼ 0:6–0:9.
References [1] E. Sch€ one, Ber. Chem. Ges. 6 (1873) 1224–1229. [2] A.E. Rawson, Studies in ozonisation. Part 2. Solubility of ozone in water, Water Water Eng. 57 (1953) 102–111. [3] V.I. Matrozov, S.A. Kashtanov, A.M. Stepanov, B.A. Tregubov, Ozone solubility in water, Zh. Prikl. Khim. 48 (8) (1975) 1838– 1841 (in Russian). [4] J.A. Roth, D.E. Sullivan, Solubility of ozone in water, Ind. Eng. Chem. Fundam. 20 (1981) 137–140. [5] J.A. Roth, D.E. Sullivan, Kinetics of ozone decomposition in water, Ozone Sci. Eng. 5 (1983) 37–49. [6] V. Caprio, A. Insola, P.G. Lignola, G. Volpicelli, A new attempt for the evaluation of the absorption constant of ozone in water, Chem. Eng. Sci. 37 (1982) 122–123. [7] L.F. Kosack-Channing, G.R. Helz, Solubility of ozone in aqueous solution of 0–0.6 M ionic strength at 5–30 C, Environ. Sci. Technol. 17 (1983) 145–149. [8] Yu.A. Aleksandrov, B.I. Tarunin, M.L. Perepletchikov, Solubility of ozone in liquids, Zh. Prikl. Khim. 57 (10) (1983) 2386–2397 (in Russian). [9] A. Ouederni, J.C. Mora, R.S. Bes, Ozone absorption in water: mass transfer and solubility, Ozone Sci. Eng. 9 (1987) 1–12. [10] W.J. Masschelein, Unit Processes in Drinking Water Treatment, Marcel Dekker, Inc., New York, 1992, Chapter 3. [11] C.H. Kuo, K.Y. Li, C.P. Wen, J.L. Weeks, Absorption and decomposition of ozone in aqueous solutions, AIChE Symp. Ser. 73 (166) (1977) 230–241. [12] A.K. Bi n, M. Roustan, Mass transfer in ozone reactors, in: Proceedings of the International Specialized Symposium IOA 2000, Fundamental and Engineering Concepts for Ozone Reactor Design, 1–3 March, 2000, Toulouse, pp. 89–131.
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405 [13] A.K. Bi n, Ozone dissolution and decomposition in aqueous systems, in: Proceedings of the 13th World Ozone Congress, 27– 31.10.1997, Kyoto, vol. 1, B1-3-2, 1997, pp. 369–374. [14] Y. Qiu, Kinetic and mass transfer studies of the reactions between dichlorophenols and ozone in liquid–liquid and gas–liquid systems, Ph.D. Dissertation, Mississippi State University, 1999. [15] Y. Qiu, C.H. Kuo, M.E. Zappi, Performance and simulation of ozone absorption and reactions in a stirred-tank reactor, Environ. Sci. Technol. 35 (1) (2001) 209–215. [16] M.L Kilpatrick, C.C. Herrick, M. Kilpatrick, The decomposition of ozone in aqueous solutions, J. Am. Chem. Soc. 78 (1956) 1784– 1789. [17] J.L. Sotelo, F.J. Beltran, F.J. Benitez, J. Beltran-Heredia, Ozone decomposition in water: kinetic study, Ind. Eng. Chem. Res. 26 (1987) 39–43. [18] T. Miyahara, M. Hirokawa, M. Ueda, K. Yoshida, Solubility of ozone into water in a bubble column, Kagaku Kogaku Ronbun. 20 (1994) 497–503. [19] D.E. Sullivan, J.A. Roth, Kinetics of ozone self-decomposition in aqueous solution, water––1979, AIChE Symp. Ser. 7 (197) (1980) 142–149. [20] J.L. Sotelo, F.J. Beltran, F.J. Benitez, J. Beltran-Heredia, Henry’s law constant for the ozone water system, Water Res. 10 (1989) 1239–1246. [21] F. De Smedt, Fundamental study of the behaviour of ozone in water: application in the cleaning of semiconductors, Ph.D. Dissertation, Katholieke Universiteit Leuven, Belgium, 2000. [22] Y.-Ch. Hsu, T.-Y. Chen, J.-H. Chen, Ch.-W. Lay, Ozone transfer into water in a gas-inducing reactor, Ind. Eng. Chem. Res. 41 (1) (2002) 120–127. [23] J. Wisniak, A. Polishuk, Analysis of residuals––a useful tool for phase equilibrium data analysis, Fluid Phase Equilib. 164 (1999) 61–82. [24] M.D. Gurol, P.C. Singer, Kinetics of ozone decomposition: a dynamic approach, Environ. Sci. Technol. 16 (7) (1982) 377–383. [25] R. Munter, Theory and practice of natural and wastewater ozonation, D.Sc. Thesis, Kiev, Ukraine, 1990.
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[26] T. Morioka, N. Motoyama, H. Hoshikawa, M. Okada, T. Moniwa, J. Jpn. Waterworks Assoc. 60 (7) (1991) 7–17. [27] K. Watanabe, T. Kinugasa, K. Higaki, Ozone absorption in bubble column, in: Memoir of Niihama National College of Technology, vol. 27, 1991, pp. 48–52. [28] Y. Zheng, Kinetic and mass transfer studies of the reactions between cresols and ozone in liquid–liquid and gas–liquid system, Ph.D. Dissertation, Mississippi State University, 1993. [29] J.J. Wu, S.J. Masten, Mass transfer of ozone in semibatch stirred reactor, J. Environ. Eng. 127 (12) (2001) 1089–1099. [30] R. Andreozzi, V. Caprio, I. Ermellino, A. Insola, V. Tufano, Ozone solubility in phosphate-buffered aqueous solutions: effect of temperature, tert-butyl alcohol, and pH, Ind. Eng. Chem. Res. 35 (4) (1996) 1467–1471. [31] G. Anselmi, P.G. Lignola, C. Raitano, G. Volpicelli, Ozone mass transfer in stirred vessels, Ozone Sci. Eng. 6 (1984) 17–28. [32] A. Kercß, A.M. Saatcßi, Computerized continuous monitoring and analysis of ozone in solution, Ozone Sci. Eng. 18 (1996) 469–476. [33] F.J. Beltran, J.F. Garcia-Araya, J.M. Encinar, Henry and mass transfer coefficients in the ozonation of wastewaters, Ozone Sci. Eng. 19 (1997) 281–296. [34] E. Rischbieter, H. Stein, A. Schumpe, Ozone solubilities in water and aqueous salt solutions, J. Chem. Eng. Data 45 (2000) 338– 340. [35] N. Brauner, M. Shacham, Role of range and precision of the independent variable in regression of data, AIChE J. 44 (1998) 603–611. [36] Sh.L. Quinn, D.W. Bacon, Th.J. Harris, Assessing the precision of model predictions and other functions of model parameters, Can. J. Chem. Eng. 77 (1999) 723–737. [37] M. Schacham, N. Brauner, Considering precision of experimental data in construction of optimal regression models, Chem. Eng. Process. 38 (1999) 477–486. [38] S. Singh, On establishing the credibility of a model for a system, Trans. Inst. Chem. Eng. 76 (A) (1998) 657–668. [39] P.N. Johnson, R.A. Davis, Diffusivity of ozone in water, J. Chem. Eng. Data 41 (1996) 1485–1487.
Ozone dissolution in aqueous systems treatment of the experimental data A.K. Bin
*
Faculty of Chemical and Process Engineering, Warsaw University of Technology, ul. Wary nskiego 1, 00-645 Warszawa, Poland
Abstract The major concern of the paper was to establish how the treatment method of the experimental data collected during a semibatch ozone dissolution process in aqueous systems using gas–liquid absorbers could influence the determination of the fundamental process parameters such as ozone solubility (or the Henry’s law constant values) and mass transfer coefficients. Statistically the best were the sigmoidal Boltzmann and Weibull fits, however, they provide the product of the mass transfer coefficient ðkL aÞ and the saturation ozone concentration in the liquid phase. In order to determine the Henry’s constant an independent evaluation of kL a is necessary. Ozone decomposition term might play relatively small role in the overall mass balance of ozone in the liquid phase so the question of the reaction order of ozone decomposition in such a case may be of less significance. 2003 Elsevier Inc. All rights reserved. Keywords: Ozone dissolution; Data treatment; Gas–liquid system
1. Introduction Ozonation process in the liquid phase is most frequently accomplished by injecting the ozone gas (mixtures of air–ozone or oxygen–ozone) through a sparger into the liquid, so that the fundamental feature of that process should be categorised as absorption. Usually the studies on ozone absorption in the aqueous systems are carried out in stirred-tank reactors or in bubble columns. A part of such studies has been aimed at determination of the mass (ozone) transfer coefficients, on ozone decomposition and on ozone solubility. Determination of ozone solubility in aqueous systems is often accompanied by the simultaneous studying of ozone decomposition in the liquid phase. Typical experimental procedure (a dynamic method) involves bubbling of a gas mixture containing ozone through a porous element into a liquid sample (water or aqueous solution), maintained at the isothermal and almost constant ozone partial pressure conditions (semi-batch run). Determination of ozone solubility in water or in aqueous solu-
*
Fax: +48-22-825-14-40. E-mail address: [email protected] (A.K. Bi n).
0894-1777/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2003.03.001
tions has a long history. Probably the first paper on this topic is that due to Sch€ one [1]. A number of the compilation attempts of the published data can be found in the references (e.g. [2–10]). The dissolved ozone concentration is monitored as a function of time and the resulting points (curves) are subject to further analysis. Typical examples of such data are shown in Fig. 1, where the experimental data due to Kuo et al. [11] have been replotted. Also, as an illustration, the Boltzmann fit (cf. Appendix A) is shown for that data. It is evident from these graphs that depending upon the experimental conditions different curves CL ðtÞ are obtained. The Boltzmann fit that represents the group of the so-called sigmoidal fits, approximates the considered experimental data points very well, thus preliminary indicating that the changes of the dissolved ozone concentration with time may successfully be described by Eq. (A.5). The more rigorous discussion of the experimental data treatment follows in the next section. The major point of this approach is aimed at a critical analysis how the method of the experimental data treatment collected during the ozone dissolution batch process in an aqueous system can influence the determination of parameters such as ozone solubility (or the Henry’s law constant values), the mass transfer coefficient, kL a, and ozone decomposition terms in that system.
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A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
Nomenclature volumetric mass transfer coefficient in the liquid phase (s1 ) kd , k1 , k2 , k3 kinetic constants of ozone self-decomposition (s1 , mol1m , m3m3 , s1 ) K, K1 , K2 dimensionless kinetic constants L, L1 dimensionless parameters M, M1 dimensionless parameters m power exponent in Eq. (1) QG gas volumetric flow rate (m3 s1 ) S solubility (¼ 1=HA ) T temperature (C) t time (s) uG gas superficial velocity in the bubble column (m s1 ) yi , ^yi observed and predicted value, respectively
kL a
Greek letters u parameter defined by Eq. (C.9) k parameter in the Weibull fit
2.0
2.0
1.5
1.5
CL (x 104 mM)
CL (x 104 mM)
A, A1 , A2 fitting parameters a specific interfacial area (m1 ) B fitting parameter in the Weibull fit CG molar concentration of ozone in the gas phase (mol m3 ) CL molar concentration of ozone in the liquid phase (mol m3 ) CL equilibrium molar concentration of ozone in the liquid phase (mol m3 ) CLss steady-state molar concentration of ozone in the liquid phase (mol m3 ) d power exponent in the Weibull fit D fitting parameter in the Boltzmann fit DL molecular diffusivity in the liquid phase (m2 s1 ) ei residual (¼ yi ^yi ) HA dimensionless Henry’s law constant h, h0 depth of the liquid layer in the bubble column (m)
1.0
0.5
0.0
0
(a)
pH = 2.2
t = 25oC QG = 0.51 dm3/min CG = 48.5 mg O3/min
pH = 4.1
1000
2000
1.0
QG (dm3/min)
(b)
0.5
pH = 6.2 pH = 7.1
3000
0.5
o
25 C pH = 2.2 CG = 39.95 mg O3/dm3
1.0 1.5 2.0
0.0
4000
0
Time (s)
1000
2000
3000
4000
Time (s)
Fig. 1. Dissolved ozone concentration vs. time [11]. (O, d, ,
)
Experimental data points; curves––Boltzmann fit.
2. Treatment of the experimental data
2.1. Constant gas composition model: CG ¼ CG;inlet
Interpretation of the collected experimental data requires an assumption of the mixing behaviour in both phases [12]. Different models can be taken into account (cf. Table 1). If the analytical solutions are available, they can be used to fit to the measured concentration profiles yielding the best estimates for the parameters such as kL a, kd , and CL . Experimental data obtained in the batch mode measurements can be treated using some of the models listed in Table 1. The following examples illustrate application of the constant gas composition and complete mixing models (no. 1 and 2 in Table 1), accounting for ozone decomposition in the liquid phase.
This model is in fact based on the assumption (cf. [13]) that the changes of the ozone concentration in the gas phase during bubbling can be neglected, i.e. CG const, and whence also ozone equilibrium concentration in the liquid, CL const (under the isothermal and approximately isobaric conditions). The process of ozone dissolution (absorption) has most frequently been described by the following expression (the more rigorous method is given in Appendix C) dCL ¼ kL aðCL CL Þ kd CLm dt
ð1Þ
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
397
Table 1 Types of model applicable in the absorption data treatment (batch mode) No.
Type of model
Remarks
1. 2.
Constant gas composition Complete mixing (CSTR)
3. 4. 5. 6. 7. 8.
Plug flow (PF) Multistage mixing Axial dispersion (ADM) Multistage mixing with backward flow Two-bubble-class Gas-PF ; Liq-ADM
9.
Gas-PF ; Liq-CSTR
CG ¼ CG;inlet , liquid phase well mixed. Analytical solution available for first-order ozone decomposition Gas phase well mixed, gas composition variable. Analytical solution available for first-order ozone decomposition Plug flow in the gas phase Gas pathway treated like a series of stirred cells Plug flow in the gas and/or liquid phases superimposed with some degree of axial dispersion Gas phase is considered to consist of many perfectly mixed cells in series superimposed with a backward flow between cells Accounting for differentiated bubble size in the heterogeneous regime by assuming two bubble size classes Set of two mass balance equations containing the dispersion terms and the appropriate initial and boundary conditions. Numerical solutions necessary Analytical solution available when uG ¼ const ¼ uG;inlet and if the accumulation in the gas phase can be neglected
Ozone consumption (decomposition or other chemical reactions) is accounted for by the second term. At the initial condition t ¼ 0, CL ¼ 0 and under the previously made assumptions Eq. (1) can readily be integrated for m ¼ 1 and 2. The results of this integration are as follows: CL m ¼ 1 CL ¼ f1 exp½ð1 þ KÞðkL atÞg ð2Þ 1þK where K ¼ kd =ðkL aÞ. LðM 1Þ ðM þ 1Þ m ¼ 2 CL ¼ CL ð3Þ 2KðL þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mþ1 exp½ðkL atÞM, and where M ¼ 1 þ 4K , L ¼ M1 kd CL K¼ . kL a A somewhat different type of Eq. (1) is of interest dCL ¼ ðkL aÞðCL CL Þ þ k3 CL k2 CL2 dt
ð4Þ
which after integration yields L1 ðM1 þ K1 Þ þ ðM1 K1 Þ CL ¼ CL ð5Þ 2K2 ðL1 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3 k2 CL 1, K2 ¼ , M1 ¼ K12 þ 4K2 , and with K1 ¼ kL a kL a K1 M1 exp½M1 ðkL atÞ. L1 ¼ K1 þ M1 2.2. Complete mixing model In this model, it is assumed that the phase is well mixed while the solute concentration varies with time. The mass (ozone) balance equations for both phases are as follows: gas phase dCG uG 1 eG ¼ ðCG CG Þ ðkL aÞ ðCLi CL Þ dt eG hL eG ð6Þ
liquid phase dCL ¼ kL aðCL CL Þ kd CLm ð7Þ dt Assuming the first-order ozone decomposition (m ¼ 1) in the liquid phase the set of Eqs. (6) and (7) can analytically be solved yielding the following dimensionless ozone concentrations in the gas and liquid phases, respectively (cf. [14,15]): CGþ ¼ c3 þ
1 þ c3 r2 1 þ c3 r1 expðr1 hÞ þ expðr2 hÞ r1 r2 r2 r1
CLþ ¼ c3 þ
c3 1 ð1 þ c3 r2 Þð1 þ a1 þ r1 Þ expðr1 hÞ þ a1 a1 ðr1 r2 Þ
þ
ð1 þ c3 r1 Þð1 þ a1 þ r2 Þ expðr2 hÞ a1 ðr2 r1 Þ
ð8Þ
ð9Þ
where CGþ ¼
CG ; CG
a1 ¼ tm ðkL aÞ
CLþ ¼
CL HA ; CG
1 eG ; eG H A
c 1 ¼ 1 þ a1 þ a2 þ a3 ;
h¼
t uG t ¼ ; tm eG hL
a2 ¼ tm ðkL aÞ;
a3 ¼ tm kd ;
c 2 ¼ a2 þ a1 a3 þ a3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ a3 c1 þ c21 4c2 c3 ¼ ; ; r1 ¼ 2 c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 c21 4c2 r2 ¼ 2 Some authors (e.g. [3,11,16–18]) suggested that m ¼ 1:5 in Eq. (1). Also, the complex forms of the decomposition term have been considered (e.g. [5,17,20,22]). However, these require numerical solution, e.g. using the standard fourth-order Runge–Kutta scheme (R–K) so that further fitting procedure is then much more complicated and tedious. Recently, Qiu [14] studied ozone decomposition and solubility in aqueous solutions of pH varying from 2 to 9. Based on his own experiments and the results of many other authors, as well as on considerations on the
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A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
mechanism of ozone self-decomposition in aqueous solutions, he concluded that the first-order ozone decomposition kinetics provided statistically the best correlation of the rate data over the wide range of pH (from 2 to 13.5 at 25 C)––Eq. (10): kd ¼ 20½OH 1=2 þ 900½OH ðs1 Þ
ð10Þ
The derived expressions can be fitted to the experimental data by performing a typical regression procedure based on minimisation of the sum of squares of deviations. As a result of such a treatment the ‘‘best’’ values of the searched kinetic parameters (CL ; kL a; kd ; k2 ; k3 ) or their combinations can be estimated. If the values of any of these parameters are known from some other independent determinations this fact can (and should) also be taken into account. However, it has to be realised that possible dependency between the fitting parameters may exist and this has to be carefully considered in choosing the most adequate parameters to be fitted. After performing preliminary tests the following fitting procedures for a data set of Kuo et al. [11] have been applied: 1. Boltzmann fit (four parameters found from regression, i.e. A1 , A2 , t0 and D),
2. m ¼ 1 (Eq. (2)) (two parameters found from regression, i.e. ðkL auÞCL and ðkL au þ kd Þ), 3. m ¼ 2 (Eq. (3)), assuming a value of ðkL auÞCL from the previous procedure (two parameters found from regression, ðkL auÞ and kd ), 4. m ¼ 2 with a constant value of CL given by the authors (two parameters found from regression, i.e. ðkL auÞ and kd ), 5. Weibull fit with d ¼ 1 (two parameters found from regression, i.e. ðkL auÞCL and ðkL au þ kd Þ), 6. Complete mixing model (Eq. (9)) with constants values of CL given by the authors (two parameters found from regression, ðkL auÞ and kd ). The results of the applied fitting procedures for the experimental data of Kuo et al. [11] are listed in Table 2. A survey of the numerical values contained in Table 2 indicates that usually the statistical quality of regression is good (R2 > 0:99). However, the differences between the procedures are better reflected by the square root of the sum of squares of the residuals (Norm)––cf. Fig. 2. From this figure, one can conclude that procedure no. 1 (the Boltzmann fit) provides the best overall fit to the experimental data under consideration. The goodness of a fit can also be assessed by analysing the residuals [23].
Table 2 The results of the fitting procedures for the data sets of Kuo et al. [11]; experimental conditions are indicated in Fig. 1 Run 8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
Proc. no. 2 (m ¼ 1)
Proc. no. 1 (Boltzmann fit) ðkL auÞCL (·104 m s1 )
CLss (·104 m)
R2
ðkL auÞCL (·104 m s1 )
CLss (·104 m)
kL au þ kd (s1 )
R2
0.00314 0.00358 0.00289 0.00255 0.00238 0.00396 0.00644 0.00904
1.810 1.768 1.533 1.446 1.299 1.399 1.446 1.644
0.9991 0.9993 0.9999 0.9859 0.9986 0.9986 0.9976 0.9991
0.00389 0.00405 0.00393 0.00382 0.00254 0.00455 0.00657 0.00916
1.843 1.788 1.569 1.516 1.302 1.410 1.444 1.645
0.00211 0.00227 0.00250 0.00252 0.00195 0.00323 0.00455 0.00557
0.9959 0.9976 0.9945 0.9857 0.9988 0.9976 0.9974 0.9992
0.00220 0.00233 0.00259 0.00269 0.00192 0.00325 0.00447 0.00556
0.9969 0.9980 0.9954 0.9884 0.9989 0.9977 0.9976 0.9992
Proc. no. 3 (m ¼ 2) 8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
0.00389 0.00405 0.00393 0.00382 0.00254 0.00455 0.00657 0.00916
Proc. no. 5 (Weibull fit with d ¼ 1) 1.827 1.781 1.552 1.487 1.302 1.405 1.443 1.644
0.9962 0.9978 0.9950 0.9869 0.9988 0.9977 0.9974 0.9992
Proc. no. 4 (m ¼ 2)
8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
0.00403 0.00416 0.00405 0.00405 0.00250 0.00458 0.00646 0.00915
1.837 1.784 1.565 1.505 1.304 1.409 1.446 1.645
Proc. no. 6 (complete mixing)
ðkL auÞ (s1 )
CLss (·104 M)
R2
ðkL auÞ (s1 )
CLss (·104 M)
kd (s1 )
R2
0.00175 0.00181 0.00169 0.00165 0.00133 0.00243 0.00351 0.00511
1.833 1.779 1.554 1.492 1.291 1.403 1.440 1.643
0.9970 0.9984 0.9970 0.9904 0.9970 0.9985 0.9942 0.9986
0.00273 0.00291 0.00277 0.00265 0.00198 0.00342 0.00489 0.00693
1.843 1.788 1.569 1.515 1.301 1.410 1.444 1.644
0.000278 0.000359 0.000654 0.000723 0.000501 0.000628 0.000797 0.000334
0.9960 0.9977 0.9947 0.9859 0.9987 0.9977 0.9973 0.9992
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
399
3. Discussion Procedures 1–3 and 5 do not allow direct determination of CL or kL a but provide only the values of their product. Procedures 2 and 5 provide also the estimated values of the sum ðkL a u þ kd Þ. All the considered procedures enable one to estimate the values of the steady-state asymptote, CLss . Figs. 4–6 display the
0.014 proc. 1 proc. 2 proc. 3 proc. 4 proc. 5 proc. 6 Kuo et al. (1977)
(kLaϕ) CL* (x 104 M/s)
0.012
Fig. 2. Values of the square root of the sum of squares of the residuals (Norm) for the experimental data of Kuo et al. [11].
Some examples of the residuals distribution for the experimental data of Kuo et al. [11] are shown in Fig. 3. From these graphs it can be concluded that both approximations, i.e. the Boltzmann fit (procedure no. 1) and the Weibull fit (procedure no. 5) provide the smallest distributions of the residuals.
0.006 0.004 0.002
8_2
8_3
9_1
9_3
1.0e-5 Run 8_4
5.0e-6
ei (M)
0 -2e-6 -4e-6
m=1 m=2 m=2 g BF WF (d=1) CMM
-6e-6
(a)
-8e-6 0
500
1000
1500
2000
2500
3000
0.0 -5.0e-6 m=1 m=2 m=2 g BF WF (d=1) CMM
-1.0e-5
3500
-1.5e-5
0
500
1000
(b) 1500
2000
2500
3000
Time (s)
Time (s) 6e-6
6e-6
Run 9_1
Run 9_3
m=1 m=2 m=2 g BF WF (d=1) CMM
4e-6
4e-6 2e-6
ei (M)
2e-6 0
0 -2e-6
-2e-6
-4e-6
(c) -4e-6
9_2
0
500
1000
1500
(d) 2000
Time (s)
2500
3000 3500
9_4
Fig. 4. Values of the product ðkL auÞCL for the experimental data of Kuo et al. [11].
2e-6
ei (M)
8_4
Run No.
Run 8_1
4e-6
ei (M)
0.008
0.000 8_1
6e-6
-1e-5
0.010
-6e-6
0
500
1000
1500
2000
Time (s)
Fig. 3. Values of the residuals for the experimental data of Kuo et al. [11].
m=1 m=2 m=2 g BF WF (d=1) CMM
2500
3000
3500
400
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405 1e+1
2.0 proc. 1 proc. 2 proc. 3 proc. 4 proc. 5 proc. 6
1.9
1.7
1e+0 1e-1
kd (s-1)
CLss (x 10 mM)
1.8
[24] [19] [17] [7] [25] [26]
1.6 1.5
[27] (25oC) [28] [29] [22] [5] [21] (nitric acid) [21] (acetic acid)
[14] (25oC)
1e-2 1e-3
1.4
1e-4 1.3 1.2 8_1
t = 20oC 1e-5 8_2
8_3
8_4
9_1
9_2
9_3
9_4
1e-6
Run No.
0
2
4
6
8
10
12
pH
Fig. 5. Values of the asymptote CLss for the experimental data of Kuo et al. [11].
Fig. 7. Ozone self-decomposition first-order rate constant vs. pH at 20 C (cf. [19,26–28]). 0.008
0.002
0.000 8_1
8_2
8_3
8_4
9_1
9_2
9_3
9_4
Run No. Fig. 6. Values of the sum ðkL auÞ þ kd for the experimental data of Kuo et al. [11].
obtained values of these parameters or their combinations in a graphical form. From these graphs, one can conclude that in the case of the product ðkL auÞCL the different treatment of the experimental data will yield values differing on average by 13–21% (absolute value) from that obtained from procedure 1 (Boltzmann fit), except of that for procedure 6 (complete mixing), which produced the highest difference (on average 68%). In the case of the sum ðkL a u þ kd Þ––Fig. 6, both procedures 2 and 5 lead to practically the same values (the difference between them is on average 3%) while those found from procedure 6 are higher by about 33%. The estimated values of the steady-state asymptote, CLss , are very close to each other (on average ±1%). If separation of kL a u, CL and/or kd from their products or sums is attempted, one has to know some of them (e.g. kL a u or kd values) from independent sources or measurements. For further considerations, a survey of the published or available experimental data on ozone decomposition rate in water and diluted aqueous solutions may be helpful (Fig. 7). At low pH (pH < 5)
0.008 0.007
(1/s)
(1/s) (kLa ϕ) + kd
0.004
(kLa ϕ ) + kd
proc. 2 proc. 5 proc. 6
0.006
the values of the first-order self-decomposition rate, kd , are of the order of 104 s1 . As it can be seen in Table 2 this could give 5–20% contribution of kd in the sum ðkL a u þ kd Þ. At any rate, only at low pH-values procedures 2 and 5 would yield the possibility of estimating the values of kL a u since in this range of pH the contribution of ozone self-decomposition term in the sum ðkL a u + kd Þ is sufficiently small––most likely within the accuracy range of the experimental procedure. The results of such estimations are shown in Fig. 8 for the data of Kuo et al. [11] collected at pH ¼ 2.2 and 25 C. Procedures 2 and 5 yield almost the same values, differing from each other within the mean absolute error 1.7%, while in the case of procedure 6 they are systematically higher by 25–44% from those obtained from procedure 2. The original data of Kuo et al. [11] when treated in the way described above are lower by 40–53% when compared with the results from procedure 2.
0.006
Kuo et al. data [11] proc. 2 proc. 5 proc. 6
pH = 2.2
0.005 0.004 0.003 0.002 0.001 0.0
0.5
1.0
1.5
2.0
2.5
QG (L/min) Fig. 8. Dependence of the sum ðkL auÞ þ kd vs. QG for the experimental data of Kuo et al. [11].
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
When the values of ðkL auÞ are known or estimated, it is possible to split the product ðkL auÞCL to calculate ozone equilibrium solubility, CL , and then the Henry’s law constant (or dimensionless ozone solubility, S). The results of such calculations are presented in Table 3. From this data it is seen that both procedures 2 and 5 give the same values of HA or S, while taking the values of the product ðkL auÞCL estimated from procedure 1 (the Boltzmann fit) and using the values of ðkL auÞ based on the original data of Kuo et al. [11], the Henry’s law constant is lower by 38% than the mean value obtained from the former procedures. Kuo et al. [11] based their calculations on a value of S ¼ 0:21 (or HA ¼ 4:76). It is then evident that depending on the fitting procedure and the assumed values of other parameters, e.g. the mass transfer coefficient, ðkL a uÞ, different values of the Henry’s law constant are obtained. The effect of pH on the estimated values of HA is not very clear though according to the data of some authors [4,7,9,14,18,20, 21,30,33,34] this effect could be considered. An apparent influence of pH on ozone solubility (and hence on the value of the Henry’s law constant) in aqueous systems may be attributed to the limited accuracy of the decomposition term evaluation and the assumed procedure of the experimental data treatment. On the other hand, since the experiments carried out at different pHvalues require introduction of acids and/or buffers, this means that in such cases we operate with diluted solutions and hence we can expect effects of these additions on ozone solubility as compared to that in the ‘‘pure’’ solvent (water). The previous determinations of ozone solubility findings were based on different assumptions regarding the ozone decomposition term as well as on the data treatment procedure. For instance, in evaluating ozone solubility in aqueous systems Roth and Sullivan [4] or Andreozzi et al. [30] assumed the first-order reaction of
401
ozone self-decomposition with respect to dissolved ozone concentration, while Kuo et al. [11], Sotelo et al. [20] and Miyahara et al. [18] postulated that the power exponent in Eq. (1) should be taken as 1.5 or 2, depending on the accompanying dissolved salts (or buffers). An independent determination of the mass transfer coefficient in the applied absorber is also recommended by most of these authors. The experimental data collected during the batch runs are treated typically by considering the steady-state asymptote [4,7,18,21,29,30] dCL m ¼ 0 ) kL aðCL CLss Þ ¼ kCLss ð11Þ dt With the independently known values of kL a, k, m, and the measured (or extrapolated) CLss , it is then possible to calculate the equilibrium ozone concentration in the liquid phase, CL , and hence the Henry’s law constant. Sotelo et al. [20] assessed first ozone self-decomposition reaction order and then verified these findings using a plot directly resulting from Eq. (1): dCL þ kCLm ¼ kL aðCL CL Þ ð12Þ dt i.e. as dCL =dt þ kCLm ¼ f ðCL Þ. A linear plot is expected yielding kL a and CL . This method requires an earlier and independent quantitative information on the ozone selfdecomposition term. Caprio et al. [6] performed a set of experiments on ozone solubility in water by using a procedure which avoided any equilibrium disturbance on the analytical samples. In this procedure the batch system was led into a steady state identified with the phase (absorption) equilibrium with respect to the gas inlet concentration. A simple ozone mass balance allowed calculation of the ozone Henry’s constant based on the known volumes of both phases in the absorption vessel. Negligible ozone decomposition in the liquid phase was assumed. The authors verified the accuracy of their method by
Table 3 Estimated values of the dimensionless Henry’s constant, HA , and ozone solubility, S, for the data Kuo et al. [11], runs 8 and 9 (T ¼ 25 C) Run
pH
Fitting procedure 2 (m ¼ 1)
8/1 8/2 8/3 8/4 9/1 9/2 9/3 9/4
2.2 4.1 6.2 7.1 2.2 2.2 2.2 2.2
Mean value at pH ¼ 2.2 Std. dev. a
B).
1 (BF)a
5 (d ¼ 1)
HA
S
HA
S
HA
S
5.48 5.65 6.44 6.67 6.39 5.90 5.76 5.06
0.182 0.177 0.155 0.150 0.156 0.169 0.174 0.198
5.50 5.66 6.46 6.71 6.38 5.91 5.76 5.06
0.182 0.177 0.155 0.149 0.157 0.169 0.174 0.198
3.81 3.34 4.14 4.69 3.89 3.82 3.36 2.96
0.262 0.300 0.241 0.213 0.257 0.261 0.298 0.338
5.72 0.50
0.164 0.014
5.72 0.49
0.164 0.015
3.57 0.40
0.283 0.035
In this calculations the product ðkL auÞCL has been separated using ðkL auÞ values resulting from the original data of Kuo et al. [11] (cf. Appendix
402
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
changing the volume of water filling the absorber, which should provide a straight line of the overall ozone content and the water volume. Their data can be regarded as being the most accurate among those reported in the references and are accepted as an International Ozone Association (IOA) standard [10]. Anselmi et al. [31] studied ozone mass transfer in a stirred vessel. They concluded that the best fit to their experimental points was obtained when the secondorder decomposition of dissolved ozone was assumed. However, in the later phase of the data treatment they neglected the decomposition term since according to their estimates its contribution in the overall ozone mass balance was below 2%. Further conclusions can be gained by comparing the rates of ozone absorption expressed by the derivative dCL =dt calculated using different methods of data treatment against the measured ozone concentrations in the liquid phase. Fig. 9 shows examples for four data sets of Kuo et al. [11]. One can notice that practically a linear dependence between the derivative dCL =dt and CL does exist, while the deviations from this trend occur at the beginning of the absorption experiment (low values of CL and hence its possible larger experimental errors). It is interesting to note that there are relatively small differences between the values of this derivative calcu-
lated using different methods of the experimental data fitting. This would indicate that the ozone decomposition term might play a relatively small role in the overall mass balance of ozone in the liquid phase so that the question of the order of ozone decomposition will be of less significance. In this context the problem of a model that would adequately describe the studied process will emerge when considering model selection. Apart from the previously presented discussion it should be remarked that that problem has been considered by a number of authors [35–38]. From these papers it can be concluded that the range and precision of the independent variables affect the results and credibility of the regression equations. For example, it could mean that including an additional term in the regression equation might not necessarily improve the expected accuracy because of harmful effects of collinearity. Hence, often simple models can be recommended provided that there are no further arguments (theoretical or experimental evidences) that would support more complex models. It should be pointed out that the recommended fits are equally successful when applied to treat the experimental data of many other authors [14,18,20–22,24,25, 29,32].
5e-7
5e-7
Run 8_2
num. dif. m=1 m=2 m=2 (g) WF (d=1) BF CMM
4e-7 3e-7
4e-7
dCL /dt (M/s)
Run 8_1
2e-7 1e-7
3e-7 2e-7 1e-7
(b)
(a) 0 0.00000
0.00005
num. diff. m=1 m=2 m=2(g) WF (d=1) BF CMM
0.00010
0.00015
0 0.00000
0.00020
0.00005
0.00010
CL (M)
0.00015
3e-7
7e-7 Run 9_3
2e-7
num. diff. m=1 m=2 m=2 (g) WF (d=1) BF CMM
6e-7
num. diff. m=1 m=2 m=2 (g) WF (d=1) BF CMM
5e-7
dCL /dt (M/s)
Run 9_1
dCL /dt (M/s)
0.00020
CL (M)
1e-7
4e-7 3e-7 2e-7 1e-7
(c) 0 0.00000
(d)
0.00005
0.00010
CL (M)
0.00015
0 0.00000
0.00005
0.00010
CL (M)
Fig. 9. Rate of ozone absorption vs. measured ozone concentration in the liquid phase [11].
0.00015
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
4. Conclusions A number of different fitting procedures have been applied to analyse the experimental data on ozone transfer to water collected by Kuo et al. [11] in a bubble column. The major concern of this treatment was to establish how the method of the data treatment could influence the determination of ozone solubility (or the Henry’s law constant values) in water and/or aqueous systems. It was demonstrated that the fitting procedure affects the values of the Henry’s constant. From the statistical point of view the best fits to the experimental data, considered in the form of CL ðtÞ, were achieved using the Boltzmann and the Weibull fits. These two fits, however, provide statistically best values of the product CL and kL a so that in order to determine the Henry’s law constant an independent evaluation of the mass transfer coefficient, kL a, is necessary. Thus, the accuracy of determination of this process parameter will inevitably influence the accuracy of the estimated Henry’s law constant in the ozone–water system. It was also found that the ozone decomposition term might play a relatively small role in the overall mass balance of ozone in the liquid phase so the question of the order of ozone decomposition in such a case may be of less significance. This conclusion must be treated with caution since depending on the experimental conditions ozone decomposition can be much more rapid (e.g. at pH > 9). Among the experimental procedures applied, the one by Caprio et al. [6] seems to yield the most accurate values of ozone solubility in water. The suggested IOA standard of ozone solubility in water is apparently based on the experimental data of these authors.
Appendix A. Sigmoidal fits To approximate the experimental data obtained in batch absorption experiments two types of sigmoidal fits seem to be of particular interest: the Boltzmann and Weibull fits. Boltzmann fit The Boltzmann fit is based on the following expression CðtÞ ¼
A1 A2 0 þ A2 1 þ exp tt D
ðA:1Þ
It can be seen from Eq. (A.1) that the Boltzmann fit should provide four fitting parameters: A1 , A2 , t0 and D. The first two of them have the same unit as CðtÞ, while t0 and D are expressed in time units. As t ! 1, CðtÞ ! A2 so that A2 is an asymptote of the expression (A.1). At t¼0 Cð0Þ ¼
A1 A2 þ A2 6¼ 0 1 þ exp tD0
ðA:2Þ
403
which demonstrates that expression (A.1) would formally not fulfil the initial condition: at t ¼ 0, Cð0Þ ¼ 0. However, this discrepancy is usually small and can be neglected for further considerations. An interesting feature resulting from the Boltzmann fit can be derived by differentiating CðtÞ with respect to time and then manipulating it to obtain the following expression: exp½ðtt0 Þ=D ðA A Þ 1 2 D dC ¼ ðA:3Þ
tt0 2 dt 1 þ exp D
Since A1 A2 0 ¼ C A2 and 1 þ exp tt D t t A A C A1 0 1 2 1¼ ; exp ¼ D C A2 C A2 substituting these terms into Eq. (A.2) and further rearrangement finally yields dC 1 ¼ ðA1 CÞðA2 CÞ dt DðA1 A2 Þ 1 ½A1 A2 þ ðA1 þ A2 ÞC C 2 ¼ DðA2 A1 Þ
ðA:4Þ
The shape of Eq. (A.4) reveals that if the experimental data on ozone dissolution with its simultaneous decomposition can be well fitted with the Boltzmann fit, in fact they should obey the following overall kinetic form dC ¼ k1 ðC CÞ þ k3 C k2 C 2 ðA:5Þ dt where k1 ¼ kL a (the mass transfer coefficient in the liquid phase), C is the equilibrium ozone concentration dissolved in the liquid phase, and the term k3 C k2 C 2 accounts for ozone decomposition. A comparison of Eq. (A.4) with Eq. (A.5) readily gives A1 A2 ; DðA2 A1 Þ 1 k2 ¼ DðA2 A1 Þ
k1 C ¼
k3 k1 ¼
A 1 þ A2 ; DðA2 A1 Þ
Weibull fit The Weibull fit is described by the following expression d
CðtÞ ¼ A ðA BÞ expððktÞ Þ
ðA:6Þ
Again, similarly as for the Boltzmann fit, this expression does not fulfil the initial condition that at t ¼ 0, Cð0Þ ¼ 0. In fact, Cð0Þ ¼ B 6¼ 0. As t ! 1, CðtÞ ! A so that A is an asymptote of the expression (A.6). For the special case when d ¼ 1, Eq. (A.6) will almost completely correspond to Eq. (2). This can be demonstrated by considering the derivative of Eq. (A.6) and comparing the result with Eq. (1) taking m ¼ 1. In that case k1 CL ¼ kA and k1 þ k ¼ k.
404
A.K. Bin / Experimental Thermal and Fluid Science 28 (2004) 395–405
Appendix B. Mass transfer coefficients Kuo et al. [11] measured the mass transfer coefficients in an oxygen–water system (Table 4). This data can be recalculated for the ozone–water system assuming the penetration model of mass transfer and taking the values of diffusivities of both species as 2.44 · 109 and 1.90 · 109 m2 s1 at 25 C [39], respectively. 0:5 DL;ozone ðkL aÞozone ¼ ðkL aÞoxygen DL;oxygen ¼ 0:882ðkL aÞoxygen
ðB:1Þ
Appendix C. Ozone absorption in a bubble column
ðC:1Þ
• gas phase (with uG const) uG
dCG þ ðkL aÞðCL CL Þ ¼ 0 dh
ðC:2Þ
The first term of the RHS in Eq. (C.1) contains an integral mean driving force in the liquid phase calculated for a gas bubble as it rises across the liquid layer. At the interface the Henry’s law is obeyed: CL ¼
CG HA
ðC:3Þ
Let us assume a quasi-stationary process of ozone dissolution from a gas bubble which would mean that CL const during the rising time of a bubble (typically this is a matter of a few seconds). Integration of Eq. (C.2) yields ðkL aÞh CG ¼ ðCG;in CL HA Þ exp þ CL HA uG H A ðC:4Þ
Table 4 Values of the mass transfer coefficients Kuo et al. [11] QG , dm3 min1
kL a (oxygen), s1
kL a (ozone), s1
ua
kL au, s1
0.5 1.0 1.5 2.0
0.0017 0.0027 0.0038 0.0043
0.00132 0.00210 0.00276 0.00335
0.8899 0.9108 0.9213 0.9280
0.00117 0.00191 0.00254 0.00311
a
Calculated from Eq. (C.9)––cf. Appendix C.
Integral mean of CL is defined as Z h0 1 CL ¼ CL ðhÞ dh h0 0 which after using Eq. (C.5) gives CG;in uG HA CL ¼ CL HA kL a ðkL aÞh 1 exp þ CL u G HA
ðC:5Þ
ðC:6Þ
ðC:7Þ
This expression can then be used in Eq. (C.1):
Ozone mass balance in both phases (assuming negligible ozone accumulation in the gas phase) can be expressed as follows: • liquid phase Z h0 dCL 1 ¼ ðkL aÞ ðCL CL Þ dh kCLm h0 0 dt
After substitution Eq. (C.3) becomes CG;in ðkL aÞh CL exp þ CL CL ¼ uG HA HA
dCL ¼ ðkL aÞuðCL;in CL Þ kCLm dt where uG HA ðkL aÞh0 u¼ 1 exp kL ah0 uG H A
ðC:8Þ
ðC:9Þ
Eq. (C.8) differs from Eq. (1) by containing a factor u ( 6 1) and that the equilibrium ozone concentration CL;in (¼ CG;in =HA ) refers to ozone concentration in the inlet gas. Under the typical experimental conditions u ¼ 0:6–0:9.
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