2. Theory
2.6 Parameter estimation for breakthrough curve modelling
The required input parameters for the adsorption models applied in the present study are summarised in Figure 8.
Figure 8. Required input parameters for single-solute and mixture adsorption models.
LDF = linear driving force; IAST = ideal adsorbed solution theory; c0 = inlet concentration (mg/L);
vF = linear filter velocity (m/s); V& = volumetric flow rate (L/s), mA = adsorbent mass (g), pB = bed density (-); εB = bed porosity (-); dP = particle diameter (m); A = column cross section (m²), L = length of the column (m), rP = particle radius (m); KF = parameter of the Freundlich isotherm (mg/g)/(mg/L)n; n = parameter of the Freundlich isotherm (-); KL = parameter of the Langmuir isotherm (L/mg); kFaVR = volumetric film mass transfer coefficient (s-1); DL = liquid film diffusion coefficient (m²/s); kS*
= intraparticle mass transfer coefficient (s-1), kS*
(0) = intrinsic mass transfer coefficient (s-1); ω = empirical parameter that describes the strength of the influence of the adsorbed amount (-); DS = intraparticle solid diffusion coefficient (m²/s)
Figure 8 shows that each model needs fixed values for the adsorbate(s)/adsorber system, equilibrium isotherm parameters as well as film and intraparticle mass transfer coefficients.
In the following, the determination of the equilibrium isotherm parameters and kinetic coefficients is described in more detail.
2.6.1 Equilibrium isotherm parameters
To determine equilibrium data, the bottle-point (batch) method is usually applied. The time required to reach equilibrium is typically between some hours and some weeks and has to be investigated in preliminary kinetic studies. After the equilibrium is established, the adsorbed mass can be calculated using the material balance equation for batch adsorption processes:
) regression analysis of log q values versus log c values, whereas Langmuir parameters (KL and qm) can be gained by linear regression analysis of c/q values versus c values.
Single-solute NOM uptake at acidic pH is a two-component system due to the anion of the applied acid. The Freundlich parameters for the single-solute NOM at acidic pH can be estimated from equilibrium experiment results with constant c0 and varying mA/VL ratios. The equilibrium data are plotted in the form of c/c0 values versus mA/VL ratios. Then, the IAST can be applied in an inverse mode, which implies fitting Freundlich parameters to the measured single-solute NOM equilibrium data.
BTC modelling for multi-solute systems requires knowledge of the concentrations and isotherm parameters of all components. In the case of water treatment, NOM is not a defined mixture. Therefore, the concentrations and the isotherm parameters of the different NOM fractions cannot be directly derived from DOC measurements. Applying the LC-OCD analysis for each equilibrium experiment could be a possible, but time-consuming and expensive solution. Another method, which is used in the present work, is to apply a fictive component approach, the so-called adsorption analysis, after Sontheimer et al. (1988) and Johannsen and Worch (1994). The principle of adsorption analysis consists in defining NOM fractions (mostly 3-5) with different adsorption performances characterised by individual Freundlich isotherm parameters (KF and n). For simplification, n is normally held constant and only different KF values are used to characterise the graduation of the adsorption strength. Further, a search routine based on the IAST is used to find that concentration distribution of the NOM fractions which allows the best fitting of the experimental data of the DOC isotherm.
2.6.2 Kinetic parameters
The volumetric film mass transfer coefficient kFaVR (s-1) strongly depends on the hydrodynamic conditions within the reactor. Thus, if the kFaVR value is needed as input parameter for a BTC model, it cannot be determined using batch experiments. Consequently, kFaVR values are often estimated by empirical correlations, which relate the film mass transfer coefficient to hydrodynamic conditions, adsorption characteristics and adsorbate properties.
In the present work, each kFaVR value is obtained by empirical correlations from the dimensionless Reynolds (Re) and Schmidt (Sc) numbers after Williamson et al. (1963) or Wilson and Geankoplis (1966) (see Equations (20) and (21), respectively). First, the dimensionless Sherwood number (Sh) is calculated and subsequently the film mass transfer kF (m/s) is found from the definition of Sh.
where dP is the particle diameter (m) and ν is the kinematic viscosity (m²/s).
The liquid phase diffusion coefficient DL (m²/s) is either known from the literature or can also be obtained by the empirical correlation (Worch, 1993)
53
where T is the temperature (K), η is the dynamic viscosity (Pa·s) and M is the molecular mass of the solute (g/mol).
Additionally to kF, the volumetric mass transfer area aVR (m-1) is needed for the calculation of the kFaVR value. It can be calculated by Equation (24) for spherical adsorber materials.
P B
VR d
a =6⋅(1−ε ) (24)
The intraparticle (solid) mass transfer coefficient k*S (s-1) as well as the respective diffusion coefficient DS (m²/s) can be obtained by separate batch kinetic experiments under conditions where only the surface diffusion is rate-determining (high shaking velocity). For the analysis of the kinetic curves (c/c0 versus time), the balance equation (19) is combined with the LDF kinetic approach proposed by Glueckauf (1955a) and solved numerically.
In some cases, the linear solid-phase concentration gradient (LDF approach) is a too rough simplification to calculate the k value. Then, a better approximation can be reached, if the S*
*
k value is not considered as a constant but as a parameter that depends on the adsorbed S
amount. This dependence can be described after Worch (2012) with the following equation ) transfer resistance increases with increasing loading.
Further, to avoid time-consuming experiments, the intraparticle (solid) mass transfer coefficient can be approximately predicted by an empirical equation after Hesse and Worch (1997) with
The correlation gives acceptable results for defined micropollutants. Contrary, the calculated values for NOM containing water are often too high in comparison with parameters estimated by BTC fitting (Worch, 2012). Therefore, a specific correlation was proposed for NOM under the assumption that the mass transfer coefficients are the same for all NOM fractions (Hess, 2001), which is defined as the total concentration of all adsorbable NOM fractions expressed as DOC (mg/L).
Further, DS can be derived from the k*S value following the Glueckauf (1955a) approach mass transfer (film-phase, solid-phase or both) controls the uptake process.
0 transfer, b) for 0.5 < Bi ≤ 30 by the film and intraparticle mass transfer and c) for Bi > 30 by the intraparticle mass transfer. The influence of Bi on the BTCs is shown in Figure 9.
Figure 9. Influence of Biot numbers (Bi) on the breakthrough curves calculated by LDF model (--- ideal breakthrough curve).
It can be seen in Figure 9 that low Bi values refer to film mass transfer controlled uptake with an earlier breakthrough in comparison to the ideal BTC. High Bi numbers indicate an intraparticle mass resistance control on the adsorption process, which results in a long tailing of the BTC. Medium values point to both film and intraparticle mass transfer controlled uptake and leads to symmetrical BTC.
film and intraparticle mass