Breakthrough curves Breakthrough curves Breakthrough curves Breakthrough curves
General approach
Fixed beds are generally used in water treatment. Water is applied directly to one end and forced through the packing adsorbent by gravity or pressure.
The pollutants present in the water are removed by transfer onto the adsorbent. The region of the bed where the adsorption takes place is called the mass transfer zone, adsorption zone or adsorption wave. As a function of time, for a constant inlet flow, the saturated zone moves through the contactor and approaches the end of the bed. Then, the effluent concentration equals the influent concentration and no more removal occurs. This phenomenon is termed breakthrough. An illustration is given in Figure 5.
time
time C/C0
0
0
tb
adsorption wave
saturated zone adsorption zone
C0 C0 C0 C0 C0
z
1
Figure 5. Schematic breakthrough curve and column saturation.
118 BIOREMEDIATION OF AQUATIC & TERRESTRIAL ECOSYSTEMS
Utilization of breakthrough curves
Some information can be extracted from the breakthrough curve. The breakthrough time is determined by reporting the ratio Cb/C0 = 0.05 or 0.1, i.e., when the pollutant outlet concentration is between 5 to 10 % of the outlet concentration. This percentage is a function of the desired water quality.
The total amount of solute removed (Qmax removed) from the feed stream upon complete saturation is given by the area above the effluent curve (C vs.
t, Fig. 5), that is:
0 0
0∞ dt 1 q
=
∫
= ε + − εmax removed 0
Q Q ((C - C) ) SHC ( ) SH (13)
q0: adsorption capacity in equilibrium with C0 (mg g-1) The solute removed at t = tb is given approximately by:
tb
tb removed= 0
Q Q(C - C) (14)
An example is given in Figure 6. Lanthanum is removed by Pseudomonas aeruginosa trapped in a gel (Texier et al. 1999, 2000, 2002). From equations 13 and 14, the data presented in Table 5 are determined and can be used to design processes.
Table 5. Design parameters obtained from breakthrough curves.
C0 (mmol L-1) tb (min) Qmax (mmol g-1) Qtb (mg g-1)
2 84 208 23
4 50 247 29
6 39 342 36
Modeling the breakthrough curves
Many models, either more or less sophisticated, are available in the literature (Ruthven 1984, Tien 1994). In this paragraph, we give three classic models useful for describing the breakthrough curves or some important operating and design data. For all the models, the assumptions are the following:
- the system is in a steady state, i.e. the flow and inlet concentrations are constant,
- there is no chemical or biological reaction, only a mass transfer occurs,
- the temperature is constant.
Bohart Adams model
This model is based on two kinetic equations of transfer from the fluid phase and accumulation in the inner porous volume of the material. A simple equation is obtained giving the breakthrough time (tb) as a function of the operating conditions:
0 1
= − −
0 0
0 0 0 b
N Z U Ln C
C U kN C
tb (15)
or
0
0 0 0
N
(Z - Z ) C U
tb= (16)
where
tb : breakthrough time (h)
k : adsorption kinetic constant (Lg-1min-1) C0 : inlet concentration (mg L-1)
U0 : velocity in the empty bed (m h-1)
Figure 6. Breakthrough curves from a fixed bed biosorption experiment;
lanthanum removal onto Pseudomonas aeruginosa. U0 = 0.76 m h-1 Z = 300 mm -500 < dp < 1,000 mm (Adapted from Texier et al. 1999, 2000, 2002).
120 BIOREMEDIATION OF AQUATIC & TERRESTRIAL ECOSYSTEMS
N0 : adsorption capacity (mg L-1) Z : filter length (m)
Z0 : adsorption zone (m)
The two parameters (N0 and Z0 (or k)) are experimentally determined. In order to illustrate the utilization of this approach, the results are presented in Figure 7 and Table 6 (Texier et al. 2002). These lab experiments were performed with Pseudomonas aeruginosa trapped in a polyacrylamide gel adsorbing lanthanide ions at different operating conditions. From this example, some conclusions can be proposed:
- the biosorption capacities decrease with the water velocity in the column. The mass transfer zone (Z0) is found to be < 2 mm for U0 = 0.76 m h-1 and 144 mm U0 = 2.29 m h-1,
- the size has no real influence (125 < dp < 1,000 mm),
- better capacities are obtained at higher initial concentrations, - the adsorption capacities are proportional to the bed depth,
although the influence of this parameter is weak.
These results are in agreement with Volesky and Prasetyo (1994) who showed that this sorption model was useful for the determination of the key design parameters.
Figure 7. Breakthrough curves of lanthanum adsorbed onto Pseudomonas aeruginosa trapped in a polyacrylamide gel (C0 = 2 mol L-1, U0=0.76 m h-1, 500 < dp
< 1000 mm, pH = 5).
Mass transfer model
The relations used for this model are:
— a mass balance between the aqueous phase and the solid phase,
— a mass transfer equation assuming a linear driving force approximation,
— the Freundlich equation (equation 6).
An equation describing the breakthrough curves is found:
1
0 1
1
n n−
= + -rt C( ) C
t Ae (17)
where
n : Freundlich equation parameter C(t) : concentration at time t (mg L-1) C0 : initial concentration (mg L-1)
A, r : equation parameters determined experimentally
This approach has been successfully applied to pilot unit adsorption in a large number of studies (Clark 1987).
Table 6. Estimation of the characteristic biosorbent process parameters for lanthanum adsorbed onto Pseudomonas aeruginosa trapped in a polyacrylamide gel (adapted from Texier et al. 2002).
U0 C0 Z dp tp Qmax Qtp N0 K
(mh-1) (mmol L-1) (mm) (mm) (min) (mmol g–1) (mg g–1) (mg g–1) (Lg–1min–1)
0.23 2 250 500-1000 228 205 23 23 0.2
0.54 2 250 500-1000 81 199 22 23 0.3
0.76 2 250 500-1000 60 197 22 19 0.7
0.76 2 300 500-1000 84 208 23 21 0.8
0.76 2 400 500-1000 96 217 19 19 0.5
0.99 2 250 500-1000 52 171 22 21 1.2
1.38 2 250 500-1000 31 152 16 15 1.6
2.29 2 250 500-1000 12 126 7 15 1.9
0.76 2 300 250-500 102 222 30 23 0.4
0.76 2 300 125-250 90 206 27 23 0.2
0.76 4 300 500-1000 50 247 29 25 0.6
0.76 6 300 500-1000 39 342 36 33 0.4
122 BIOREMEDIATION OF AQUATIC & TERRESTRIAL ECOSYSTEMS Homogeneous Surface Diffusion Model (HSDM) and Equilibrium Column Model (ECM) equations
Crittenden and co-workers (1976, 1978, 1980) developed a model based on the surface diffusion of adsorbate. Numerous applications have been performed (Montgomery 1985). In a fixed bed, the following assumptions are made:
— there is no radial dispersion; the concentration gradients exist only in the axial direction,
— plug flow exists within the bed,
— surface diffusion (kinetics limiting the mass transfer) is much greater than pore diffusion thus the contribution of pore diffusion is neglected. The adsorbent has a homogeneous surface and the diffusion flux is described by Fick's law s
dx
=
J D dC .
— a linear driving force relation describes the external mass transfer from the liquid to the external surface of the solid,
— the Freundlich equation gives the adsorption equilibria between the solid and liquid phases.
An exhaustive development of this model has been presented in previous publications (Montgomery 1985). Table 7 summarizes the different equations required to describe the mechanisms.
The set of equations cannot be directly solved analytically. Solutions may be obtained using orthogonal collocation techniques. The partial differential equations are reduced to differential equations that are integrated. Computer software and calculus methodologies are described in some adsorption books and journals (Tien 1994, Basmadjian 1997, Thomas and Crittenden 1998).
Kratochvil and Volesky (2000) proposed a heavy metal ion mixture model. The assumptions are a constant feed composition, isotherm operations, uniform packing materials, homogeneity of the bed and no precipitation in the bed. The equations integrate the description of ion exchange reactions, the molar balance for sorbing species, the axial diffusion and a mass transfer equation. They applied this model to a mixture of copper and cadmium onto a packed bed of Sargassum algal biosorbent in the calcium form. An example of a classical breakthrough curves is presented in Figure 8.
Table 7. Homogeneous Surface Diffusion Model (HSDM) equations.
Purpose Equation
Solid phase mass balance 2 2
D s Freundlich isotherm equation q = KC1/n
where
kf : external mass transfer coefficient (s-1) Ds : surface diffusion coefficient (m2 s-1) R : particle radius (m)
j : sphericity (dimensionless) r : radial length of spherical shell (m) z : axial direction (m)
ra : adsorbent density (kg m-3)
A neural network
A new approach for the modeling of breakthrough curves is to use a statistical tool: neural networks. These are an association of several neurons (Fig. 9) connected together to make a network. This kind of approach has been applied to the adsorption of organics onto activated carbon fibers (Faur-Brasquet and Le Cloirec 2001, 2003) and lanthanide ion removal onto immobilized Pseudomonas aeruginosa (Texier et al. 2002). In this study, several architectures of neural network were tested, as shown in Figure 10, in order to model the breakthrough curves.
124 BIOREMEDIATION OF AQUATIC & TERRESTRIAL ECOSYSTEMS
Figure 9. Presentation of a specific neuron.
It appears that the prediction ability is satisfactory for the first part of the curve (C/C0 < 0.25) when the metal ion begins to be released from the column. The choice of the input parameters and the neuron network architecture is important for the prediction of experimental data.
Considering that the most interesting part of the breakthrough curve to Figure 8. Breakthrough curves for multicomponent biosorption onto a biosorbent (adapted from Kratochvil and Volesky 2000).
Input Parameters
Connection weight Mathematical Parameters
Output Parameters
0 0.5 1 1.5
0 0.5 1 1.5
Cu2+
Cd2+
C0Qvt/qmax C/C0
Figure 10. Neural network architectures used for modeling the breakthrough curves of lanthanide ions in the biosorption column system.
Figure 11. Agreement between experimental and predicted C/C0 values with a neural network (C0, Z, Re, and t) applied to the lanthanum breakthrough curve (C0 = 2 - 6 mM, U0 = 0.76 - 2.29 m h-1, Z = 250 - 420 mm).
126 BIOREMEDIATION OF AQUATIC & TERRESTRIAL ECOSYSTEMS evaluate column performance is the first one that corresponds to the metal ion release, a comparison between the experimental and the calculated data (Fig. 11) partly illustrates the feasibility of using neural networks for biosorption. However, continued investigations are required to extend the prediction ability of such a numerical approach.