6232
Ind. Eng. Chem. Res. 2003, 42, 6232-6235
Kinetics of Air Absorption by Water in Sparged Agitated Pressure Vessels Apostolos G. Vlyssides,* Elli Maria P. Barampouti, and Sofia T. Mai Laboratory of Organic Chemical Technology, School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zographou 157 00, Greece
In this work, the transfer rate of air to water in a sparged agitated pressure vessel was studied. The mass-transfer coefficient of air was proved to depend on the mixing energy per liquid volume (Pg/V), the air flow rate per liquid volume (Qa/V) as well as the vessel’s pressure and temperature. The derived equation, which correlates these parameters with the mass-transfer coefficient, can be used for the design of a pressure vessel included in a dissolved air flotation process. Introduction Sparged agitated pressure vessels are often used for various gas-liquid reactions. A process in which the operation of such systems is of vital importance is dissolved air flotation (DAF). The aim for any aeration process should be to attain the required rate of oxygen transfer at a minimum cost. A number of correlations have been proposed for the rate of air absorption in sparged agitated pressure vessels.1,2 Most of those correlations are applicable just to atmospheric pressure and to long retention times. Nevertheless, there are processes such as DAF that include pressure vessels that need to operate at pressures greater than atmospheric in order to achieve large air mass transfer in limited volume vessels and retention time. Thus, in such cases, the up-to-date available correlations cannot be of any use. In this work, an effort was made in order to correlate the design and operational parameters of an agitated pressure vessel, a part included in a DAF process of vital importance, with the mass-transfer coefficient. Theoretical Approach Absorption Kinetics. The basic equation that describes the mass-transfer or air uptake rate in aeration practice3 takes the form
dC ) kLR(C∞ - C) dt
(1)
The integration of eq 1 using the limits 0-C and 0-t results in eq 2.
∫0CC∞dC- C ) ∫0tkLa dt
(2)
When eq 2 is solved, eq 3 is obtained.
(
)
C∞ ) kLat ln C∞ - C
(3)
C ) C∞(1 - 10-KLat)
where KLa ) 0.434kLa (5)
The solution of eq 5 is not easy because in the laboratory only the variables C in relation to t can be measured, while kLa and C∞ have to be evaluated. The method first proposed by Thomas4 can be adjusted in this case, and the following equations can be used.
F1 ) 1 - 10-x
(
F2 ) 2.3x 1 +
(6)
2.3x -3 6
)
(7)
where F1 and F2 are two independent mathematical functions of parameter x. However, eq 6 can be transformed in series as
1 1 1 F1 ) 2.3x 1 - (2.3x) + (2.3x)2 - (2.3x)3 + ... 2 6 24 (8)
[
]
and eq 7 can be transformed in series as
1 1 1 F2 ) 2.3x 1 - (2.3x) + (2.3x)2 - (2.3x)3 + ... 2 6 24 (9)
[
]
The first three terms of the above two series are similar, and the small differences between the rest of the terms will affect F1 and F2 very little. Hence, F1 ∼ F2. Subsequently, because eq 5 has the form of F1, it can be transformed through eq 7 into eq 10 by replacement of KLat into parameter x.
[
C ) 2.3KLat 1 +
-3 2.3 KLat C∞ 6
]
(10)
Then eq 10 can be rearranged into following expression:
When eq 3 is transformed, the resulting expression is
C ) C∞(1 - e-kLat)
and when the logarithms are changed, the expression is
(4)
* To whom correspondence should be addressed. Fax: ++30 210 772 3269. E-mail:
[email protected].
(2.3KLa)2/3 t 1/3 1 + t ) C (2.3KLaC∞)1/3 6C∞1/3
()
(11)
Equation 11 is linear with axes (t/C)1/3 and t of the form (t/C)1/3 ) A + Bt where
10.1021/ie030363i CCC: $25.00 © 2003 American Chemical Society Published on Web 10/28/2003
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6233
A ) (2.3KLaC∞)-1/3
(12)
The turnover τ is calculated by the equation
τ ) QL/V
and
B ) (2.3KLa)2/3/6C∞1/3
(13)
The intersection of the straight line with the vertical axes determines parameter A, and parameter B is determined by the line’s slope. When eqs 12 and 13 are solved, KLa and C∞ are obtained as shown.
KLa ) 2.61
B B or kLa ) 4.8387 A A
C∞ ) 1/6A2B
(14)
Assuming that any bubble remains in the liquid phase up to a turnover time, then the residence time of bubbles is
τR ) 1/τ
kLa ) f(P,T,Pg/V,Vs)
(16)
kLa ) f(P,T,Pg/V,N,Vs)
(17)
kLa ) f(P,T,QL/V,Vs)
(18)
(23)
We define a value for D32 and calculate the geometric characteristics of the bubble (volume Vb and surface area Ai) and the fractional air holdup according to the following equation:7
(15)
Thus, using the preceding equations, KLa and C∞ can be estimated using experimental data of time variation of C. To accomplish the main goal of this work, these parameters must be correlated with the design and operational parameters of a sparged agitated pressure vessel. Design parameters that are considered are the liquid’s retention time, the geometrical characteristics, and the power of the impeller. As far as the operational parameters are concerned, the air flow rate, the vessel’s pressure, and the impeller’s revolutions are the most representative ones. Subsequently, an effort in this direction was made. kLa Relations. Values of kLa, the mass-transfer coefficient, have been expressed by most investigators as a power function of one, two, or all of the independent variables shown in following equations:1
(22)
(
() ( )
D32
)
µ µa
0.25
P0 V
)
0.4 1/0.65
1.3σ0.6
(24)
The surface aeration is negligible when7
( )( ) Nd2F µ
0.7
Nd Vs
0.3
e 25 000
(25)
Assuming spherical bubbles and negligible surface aeration, the specific interfacial area, R, is calculated by the following equation:7
R ) 6/D32
(26)
When surface aeration is significant, then the specific interfacial area R must be modified as follows:7
R′ ) R × 10-4
[( ) ( ) Nd2F µ
0.7
Nd Vs
0.3
]
- 25000
(27)
The total bubble area is calculated by the equation
A ) VR
(28)
The number of bubbles per turnover is According to the equation described by Michel and Miller5 and modified by Uhl et al.,1 the gassed turbine power Pg is calculated as follows:
Pg ) 1.27 × 10-7N2.92d5.84Qa-0.41
P0 ) (K3µ/gi)N3d5
Vt ) NbVb
for NRe,d < 300
(20b)
where K2 and K3 are constants depending on the impeller shape and
Qg ) Vtτ
The impeller discharge rate QL is calculated by the following equation:6
QL ) K1Nd3
(21)
where K1 is a constant depending on the impeller shape.
(31)
If the new value of Qg is different from Qa, then a new value of D32 is defined and the calculations for eqs 2431 are repeated until Qg ) Qa. The terminal bubble-rise velocity Vr is calculated by the following equation:
Vr )
NRe,d ) d2NF/µ
(30)
Now the air flow rate could be recalculated from the following equation:
for NRe,d > 10 000 (20a)
and
(29)
The total bubble volume per turnover is
(19)
The ungassed turbine power P0 is calculated according to following equations:
P0 ) (K2F/gi)N3d5
Nb ) A/Ai
D322g4/5(F - Fa)4/5 µ3/5 × 10F1/5
(32)
The superficial gas velocity Vs is calculated by the following equation:
[
Vs ) Vr
Rσ0.6 P0 0.4 0.2 215 γ V
( )
]
(33)
6234 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
Figure 1. Experimental apparatus: A, agitator; AC, air compressor; CV, control valve; ER, electronic regulator; ETD, electric thermal device; EV, electric valve; FICR, flow indicator controller recorder; FIR, flow indicator recorder; LC, level controller; M, motor; MR, mechanical regulator; NRV, nonreturn valve; PC, pressure controller; PR, pressure regulator; PRV, pressure relief valve; PV, pressure vessel; Qae, air effluent; Qai, air influent; Qwe, water effluent; Qwi, water influent; TICR, temperature indicator controller.
Figure 3. Influence of Vs on kLa for the entire range of the parameters examined.
desirable pressure. The liquid temperature is controlled by TICR, the impeller’s revolutions by a combination of MR and ER, and last the air flow rate by CV1 and FICR1. A PC circuit, which includes a CV3 and a PR, achieves the constant pressure during each experiment. Water may be removed from the vessel with air. Then, LC and thus EV and PR2 are activated so that freshwater is added to the system. In the experiments conducted, the temperature was constant at 20 °C and the pressure was varied from 1 to 8 atm, rotation from 260 to 980 rpm, and Qa from 1 to 5 m3/h. In each case, a data record of the attained air concentration in the liquid C in the equation to time t was collected. The air concentration was measured as the difference in the indications of FIR1 and FIR2, which denotes air absorption by the liquid phase. Apart from KLa and C∞, which were calculated as mentioned above using the experimental data, Pg and Vs are estimated using eqs 19 and 33, respectively. Results
Figure 2. Pressure vessel.
Methods and Materials Apparatus. To estimate KLa and C∞, various experiments were conducted using the apparatus presented in Figure 1. In detail, the general characteristics of the pressure vessel are shown in Figure 2. It includes an impeller whose revolutions can be controlled. Underneath the impeller, there is an air dispersion system. A compressor feeds air, and the flow rate Qa is set at the desirable value by a control valve CV1. Temperature control is achieved by an electrical resistance system. Furthermore, in this apparatus, pressure can be controlled by air removal. Experimental Procedure First of all, the pressure vessel is filled with water, is sealed, and is fed with air without stirring until the
Having completed the experimental series, a data record of six parameters (P, N, Pg, Vs, KLa, C∞) was collected. For each of the tests performed, plots of ln KLa versus ln Vs were made. From Figure 3, it is obvious that there is a strong linear correlation of ln KLa and ln Vs for constant Qa and varying pressure P and revolutions per minute N, while parallel lines are formed for different air flow rates Qa. By regression analysis, we concluded that the best representation of the data is achieved by the following equation:
kLa ) 74.35Vs-0.22(Qa/V)1.0011
(34)
Furthermore, plots of ln KLa versus ln(Pg/V) were constructed for the whole range of the experimental data. Figure 4 illustrates only a part of them and indicates that there is a strong linear correlation of ln KLa and ln(Pg/V) for constant Qa and P. By regression analysis of these parallel lines, the following equation was derived:
kLa ) 946.24(Pg/V)0.3315(Qa/V)0.696P-0.0984
(35)
Another important variable for the design of a sparged agitated pressure vessel is the equilibrium solubility of
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6235
Nomenclature
Figure 4. Influence of Pg/V on kLa for Qa ) 2 m3/h (P ) 3, 5, and 7 atm) and Qa ) 3 m3/h (P ) 4, 6, and 8 atm).
A ) total bubble area, cm2 Ai ) bubble surface area, cm2 C ) air concentration at time t, mg/L C∞ ) equilibrium solubility of air in water at saturation for system conditions, mg/L d ) impeller diameter, ft D ) vessel diameter, m D32 ) Sauter mean bubble diameter (diameter such that the surface/volume ratio of the entire bubble population is represented), m F1, F2 ) mathematical functions of variable x g ) gravitational constant, m/s2 gi ) conversion factor, 32.2 HL ) liquid-side height, m K1 ) constant depending on the impeller shape K2, K3) constants depending on the impeller shape kLa ) mass-transfer coefficient, h-1 N ) impeller speed, s-1 [refs 20 and 21], min-1 NRe,d ) Reynolds number Nb ) number of bubbles per turnover P ) pressure, atm Pg ) gassed turbine power, hp P0 ) ungassed turbine power, hp Qa ) air flow rate, ft3/min [ref 19], m3/h QL ) impeller discharge rate, ft3/s T ) temperature, °C V ) liquid volume, ft3 Vb ) bubble volume, cm3 Vr ) terminal bubble-rise velocity, m/s Vs ) superficial gas velocity, m/s Vt ) total bubble volume per turnover, cm3 x ) variable of F1 and F2 mathematical functions Greek Letters
Figure 5. Theoretical (Henry’s law) and experimental influence of P on C∞.
air in water at saturation for system conditions C∞, which was calculated from the experimental data according to the methodology described. It was observed, as expected from the application of Henry’s law in the gaseous mixture of air, that C∞ depended solely on the system’s pressure (Figure 5).
R ) specific interfacial area, cm-1 γ ) liquid specific gravity ) fractional air holdup µ ) water viscosity, kg/m/s µa ) air viscosity, kg/m/s F ) water density, lb/ft3 [ref 20], kg/m3 Fa ) air density, kg/m3 σ ) surface tension, dyn/cm τ ) turnover, s-1 τR ) resident time of bubbles in the liquid-phase volume, s
Conclusions When eq 19 is substituted into eq 35, the following equation is derived, according to which the masstransfer coefficient is directly related to the system’s design parameters:
kLa ) 4.8968N0.96798d1.93596Qa0.56P-0.0984V-1.0275 (36) Given the pressure vessel’s characteristics (V and d), when the air flow rate Qa, the impeller’s revolutions N, and the vessel’s pressure P are changed, according to eq 36, the desirable mass-transfer coefficient kLa can be achieved. Such an equation can be proven to be a powerful tool in the design of a sparged agitated pressure vessel and in the control of its operation as well. As far as control is concerned, it is obvious through eq 36 that by changing the impeller’s revolutions, the mass-transfer coefficient kLa can always be at a desirable level, so that feeding fluctuations cannot influence the DAF’s efficiency.
Literature Cited (1) Uhl, W. V.; Winter, L. R.; Heimark, L. E. Mass transfer in large secondary treatment aerators; AIChE Symposium Series 167; AIChE: New York, 1976; Vol. 73. (2) Yagi, H.; Yoshida, F. Gas Absorption by Newtonian and nonNewtonian fluids in sparged agitated vessels. Ind. Eng. Chem. Process Des. Dev. 1975, 14 (4), 488. (3) WPCF Manual of Practice No. 5. Aeration in Wastewater Treatment; Water Pollution Control Federation: Washington DC, 1970, (4) Thomas, A. H. Water Sewage Works 1950, 97, 123-124. (5) Michel, B. J.; Miller, S. A. AIChE J. 1962, 8 (2), 262. (6) Ludwig, E. W. Design for Chemical and Petrochemical Plants; Gulf Publishing Co.: London, 1977. (7) Perry, H. R.; Chilton, H. C. Chemical Engineer’s Handbook, 5th ed.; McGraw-Hill: New York, 1976.
Received for review April 28, 2003 Revised manuscript received September 19, 2003 Accepted September 25, 2003 IE030363I
