Kinetic analysis of the transformation of phthalate esters in a series of stoichiometric reactions in anaerobic wastes

Abstract

Phthalates such as dimethyl phthalate, dimethyl terephthalate (DMT), diethyl phthalate (DEP), di(2-ethylhexyl) phthalate and mono(2-ethylhexyl) phthalate (MEHP) are degraded to varying degrees under anaerobic conditions in waste treatment systems. Here we kinetically analyse the enzymatic hydrolyses involved and the subsequent stoichiometric reactions. The resulting model indicates that the degradation of the alcohols released and the transformation of the phthalic acid (PA) result in biphasic kinetics for the methane formation during transformation of DMT, DEP and MEHP. The ester hydrolysis and the PA transformation to methane appear to be the two rate-limiting steps. The PA-fermenting bacteria, which have biomass-specific growth rates between 0.04 and 0.085 day⁻¹, grow more slowly than the other bacteria involved. Anaerobic microorganisms that remove intermediate products during phthalic acid ester conversion appear to be important for the efficiency of the ultimate phthalate degradation and to be inhibited by elevated hydrogen partial pressures. The model was based on (and the simulations corresponded well with) data obtained from experimental waste treatment systems.

Appendix

The Models

Enzymatic transformation of phthalic acid diesters via their corresponding monoesters

The following equations, expressing the relations shown in Fig. 1, were used in the basic model:

$$\begin{aligned} & \begin{array}{*{20}l} {{\frac{{{\text{dPD}}}}{{{\text{d}}t}} = - \nu _{{{\text{m}}1}} \,E\frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}},} \hfill} \\ {{\frac{{{\text{d}}E}}{{{\text{d}}t}} = k_{1} \,B_{1} - k_{2} E,} \hfill} \\ {{\frac{{{\text{dPM}}}}{{{\text{d}}t}} = \nu _{{{\text{m}}1}} \,E\frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}} - \nu _{{{\text{m}}2}} \,E\frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}},} \hfill} \\ {{\frac{{{\text{dROH}}}}{{{\text{d}}t}} = \nu _{{{\text{m1}}}} \,E\frac{{{\text{PD}}}}{{K_{{PD}} + {\text{PD}}}} + \nu _{{{\text{m}}2\,}} E\frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}} - \rho _{{{\text{m}}1}} B_{1} \frac{{{\text{ROH}}}}{{K_{P} + {\text{ROH}}}},} \hfill} \\ {{\frac{{{\text{d}}B_{1} }}{{{\text{d}}t}} = Y_{1} \,\rho _{{{\text{m1}}}} B_{1} \frac{{{\text{ROH}}}}{{K_{P} + {\text{ROH}}}} - k_{{{\text{d1}}}} B_{1} ,} \hfill} \\ {{\frac{{{\text{dPA}}}}{{{\text{d}}t}} = \nu _{{{\text{m}}2}} \,E\frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}} - \rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}},} \hfill} \\ {{\frac{{{\text{d}}B_{2} }}{{{\text{d}}t}} = Y_{2} \,\rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}} - k_{{{\text{d}}2}} B_{2} } \hfill} \\ \end{array} \\ & \\ \end{aligned} $$

(18)

, where PD, PM, E, ROH and PA are the molar concentrations in the water phase of the diester, monoester, enzyme, alcohol (product), phthalic acid, respectively; K _PD, K _PM, K _ROH and K _PA are the half-saturation constants of the corresponding substrates; B ₁ and B ₂ are the concentrations of active alcohol-and PA-degrading microorganisms, respectively; ρ _m1 and ρ _m2 are the maximum specific rates of alcohol and PA utilization, respectively; Y ₁ and Y ₂ are the biomass yields due to alcohol and PA utilization, respectively; k _d1 and k _d2 are the corresponding biomass decay coefficients; ν _m1 and ν _m2 are the maximum specific rates for enzymatic hydrolysis of PD and PM, respectively; and k ₁ and k ₂ are the coefficients for production and decay of the hydrolytic enzyme, respectively. Molar (mole l⁻¹≡M) was chosen as the concentration unit.

Reduced model: introduction of the dimensionless variables

$$\frac{{{\text{PD}}}}{{{\text{PD}}_{0} }} = p_{1} ;\,\frac{{{\text{PM}}}}{{{\text{PD}}_{0} }} = p_{2} ;\,\frac{{{\text{PA}}}}{{{\text{PD}}_{0} }} = p_{3} ;\,\frac{E}{{E_{0} }} = e;\,\frac{{{\text{ROH}}}}{{E_{0} }} = r;\,\frac{{B_{1} }}{{B_{{10}} }} = b_{1} ;\,\frac{{B_{2} }}{{B_{{20}} }} = b_{2} ;\,\nu _{{m1}} \frac{{E_{0} }}{{{\text{PD}}_{0} }}t = \tau $$

, where PD₀, E ₀, B ₁₀, B ₂₀ are the characteristic values of PD, E, B ₁ and B ₂, the basic model (Eq. (18)) is rewritten as:

$$\begin{array}{*{20}l} {{\frac{{dp_{1} }}{{d\tau }} = - e\frac{{p_{1} }}{{m_{1} + p_{1} }},} \hfill} \\ {{\varepsilon \frac{{de}}{{d\tau }} = n_{1} b_{1} - n_{2} e,} \hfill} \\ {{\frac{{dp_{2} }}{{d\tau }} = e\frac{{p_{1} }}{{m_{1} + p_{1} }} - n_{3} e\frac{{p_{2} }}{{m_{2} + p_{2} }},} \hfill} \\ {{\varepsilon \frac{{dr}}{{d\tau }} = e\frac{{p_{1} }}{{m_{1} + p_{1} }} + n_{3} e\frac{{p_{2} }}{{m_{2} + p_{2} }} - n_{4} b_{1} \frac{r}{{m_{3} + r}},} \hfill} \\ {{\frac{{db_{1} }}{{d\tau }} = a_{1} {\left( {n_{4} b_{1} \frac{r}{{m_{3} + r}} - n_{5} b_{1} } \right)},} \hfill} \\ {{\frac{{dp_{3} }}{{d\tau }} = n_{3} e\frac{{p_{2} }}{{m_{2} + p_{2} }} - n_{6} b_{2} \frac{{p_{3} }}{{m_{4} + p_{3} }},} \hfill} \\ {{\frac{{db_{2} }}{{d\tau }} = a_{2} {\left( {n_{6} b_{2} \frac{{p_{3} }}{{m_{4} + p_{3} }} - n_{7} b_{2} } \right)}} \hfill} \\ \end{array} $$

(19)

, where the dimensionless parameters are the following:

$$\begin{array}{*{20}c} {\varepsilon = \frac{{E_0 }}{{\text{PD}_0 }};\,a_1 = \frac{{Y_1 \text{PD}_0 }}{{B_{10} }};\,a_2 \frac{{Y_2 \text{PD}_0 }}{{B_{20} }};\,\text{m}_1 = \frac{{K_{\text{PD}} }}{{\text{PD}_0 }};\,\text{m}_2 = \frac{{K_{PM} }}{{\text{PD}_0 }};\,\text{m}_3 = \frac{{K_P }}{{E_0 }};\,\text{m}_4 = \frac{{K_{PA} }}{{\text{PD}_0 }};} \\ {n_1 = \frac{{k_1 B_{10} }}{{\nu _{\text{m}1} E_0 }};\,n_2 = \frac{{k_2 }}{{\nu _{\text{m}1} }};\,n_3 = \frac{{\nu _{\text{m}2} }}{{\nu _{\text{m}1} }};\,n_4 = \frac{{\rho _{\text{m}1} B_{10} }}{{\nu _{\text{m}1} E_0 }};\,n_5 = \frac{{k_{d1} B_{10} }}{{\nu _{\text{m}1} E_0 Y_1 }};\,n_6 = \frac{{\rho _{\text{m}2} B_{20} }}{{\nu _{\text{m1}} E_0 }};\,n_7 = \frac{{k_{d2} B_{20} }}{{\nu _{\text{m}1} E_0 Y_2 }}} \\\end{array} $$

.

According to the Tikhonov theorem (Tikhonov 1952), at ɛ→0 and similar values in the corresponding parts of equations in the system (Eq. (19)), the differential equations for enzyme and alcohol concentrations can be substituted with the corresponding algebraic equations. For that, the characteristic enzyme concentration should be much smaller than the characteristic PD concentration (in molar units). The enzyme and alcohol concentrations are regarded as being quickly tuned to the PD, PM, B ₁ and B ₂ values, implying that the rate of enzyme decay very soon becomes equal to the enzyme production rate, at which point the rate of the alcohol consumption equals the alcohol production rate due to the PD and PM hydrolysis. This is equivalent to Bodenstein's empirical quasi-stationary assumption; when the derivatives of the components with high reaction ability are very close to zero (enzyme and alcohol concentrations in the model Eq. (18):

$$\frac{{dE}}{{dt}} = 0,\,\frac{{dROH}}{{dt}} = 0$$

(20)

.

Thus, system (Eq. (18)) can be reduced to the following model:

$$\begin{array}{*{20}l} {{\frac{{d{\text{PD}}}}{{dt}} = - \nu _{{{\text{m1}}}} \frac{{k_{1} B_{1} }}{{k_{2} }}\frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}} = - \rho *B_{1} \frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}},} \hfill} \\ {{\frac{{d{\text{PM}}}}{{dt}} = \frac{{k_{1} B_{1} }}{{k_{2} }}{\left( {\nu _{{{\text{m}}1}} \frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}} - \nu _{{{\text{m}}2}} \frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}}} \right)} = B_{1} {\left( {\rho *\frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}} - \rho **\frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}}} \right)},} \hfill} \\ {{\frac{{dB_{1} }}{{dt}} = Y_{1} B_{1} {\left( {\rho *\frac{{{\text{PD}}}}{{K_{{{\text{PD}}}} + {\text{PD}}}} + \rho **\frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}}} \right)} - k_{{d1}} B_{1} ,} \hfill} \\ {{\frac{{d{\text{PA}}}}{{dt}} = \rho **B_{1} \frac{{{\text{PM}}}}{{K_{{{\text{PM}}}} + {\text{PM}}}} - \rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}}} \hfill} \\ {{\frac{{dB_{2} }}{{dt}} = Y_{2} \rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}} - k_{{d2}} B_{2} } \hfill} \\ \end{array} $$

(21)

, where ρ* and ρ** are the maximum PD and PM consumption rates, respectively, i.e.

$$\rho * = \nu _{\text{m}1} \frac{{k_1 }}{{k_2 }}\,$$

and

$$\rho ** = \nu _{{{\text{m}}2}} \frac{{k_{1} }}{{k_{2} }}$$

By not considering hydrolysis at the enzymatic level, the reduced model includes the Monod functions for PD and PM consumption, as well as for biomass growth of the alcohol-degrading microorganisms. The coefficient for biomass yield (Y ₁) corresponds to the intermediate product (R–OH; alcohol).

Degradation of residual organic material in inocula

The solid waste material used as inocula in the degradation studies contained not only microorganisms, but also residual MSW. This organic matter also contributes to methane production and has to be accounted for in some of our simulations. The decay of the organic matter was expressed as follows:

$$\frac{{dW}}{{dt}} = - kW$$

(22)

where W is the residual organic matter concentration of the inoculum in the studied culture systems and k is its first-order rate constant. This expression was used in cases where the target substrates were MEHP and EH (Figs. 5 and 6), i.e. when methane formed in the control cultures was not subtracted from those of the targets.

Acetate and hydrogen formation, and acetoclastic and hydrogenotrophic methanogenesis

Methane is produced from acetate (Eq. 13) or hydrogen and carbon dioxide (Eq. 14) by acetoclastic or hydrogenotrophic methanogens, respectively. In our case, the acetate and hydrogen is generated from PA and/or from the alcohol released during phthalate ester hydrolysis (see (Eq. (18), above). The formation of methane was modelled as follows:

$$\begin{array}{*{20}l} {{\frac{{d{\text{Ac}}}}{{dt}} = a*{\left( {1 - Y_{1} } \right)}\rho _{{{\text{m}}1}} B_{1} \frac{{{\text{ROH}}}}{{K_{p} + {\text{ROH}}}} + 3{\left( {1 - Y_{2} } \right)}\rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}} + c*kW - \rho _{{{\text{m}}3}} B_{3} \frac{{{\text{Ac}}}}{{K_{{{\text{Ac}}}} + {\text{Ac}}}},} \hfill} \\ {{\frac{{dB_{3} }}{{dt}} = Y_{3} \,\rho _{{m3}} B_{3} \frac{{{\text{Ac}}}}{{K_{{{\text{Ac}}}} + {\text{Ac}}}} - k_{{d3}} B_{3} ,} \hfill} \\ {{\frac{{dH_{2} }}{{dt}} = b*{\left( {1 - Y_{1} } \right)}\rho _{{{\text{m}}1}} B_{1} \frac{{{\text{ROH}}}}{{K_{p} + {\text{ROH}}}} + 3{\left( {1 - Y_{2} } \right)}\rho _{{{\text{m}}2}} B_{2} \frac{{{\text{PA}}}}{{K_{{{\text{PA}}}} + {\text{PA}}}} + {\text{d}}*kW - \rho _{{{\text{m}}4}} B_{4} \frac{{H_{2} }}{{K_{{H_{2} }} + H_{2} }},} \hfill} \\ {{\frac{{dB_{4} }}{{dt}} = Y_{4} \rho _{{m4}} B_{4} \frac{{H_{2} }}{{K_{{H_{2} }} + H_{2} }} - k_{{d4}} B_{4} ,} \hfill} \\ {{\frac{{d{\text{CH}}_{4} }}{{dt}} = V*{\left\{ {{\left( {1 - Y_{3} } \right)}\rho _{{{\text{m}}3}} B_{3} \frac{{{\text{Ac}}}}{{K_{{{\text{Ac}}}} + {\text{Ac}}}} + {\left( {1 - Y_{4} } \right)}\rho _{{{\text{m}}4}} B_{4} \frac{{H_{2} }}{{K_{{H_{2} }} + H_{2} }}} \right\}}} \hfill} \\ \end{array} $$

(23)

, where Ac is the acetate concentration; H ₂ is the hydrogen partial pressure; K _Ac and $K_{{h_{2} }}$ are the corresponding half-saturation constants; B₃ and B₄ are the concentrations of acetoclastic and hydrogenotrophic methanogens, respectively; ρ_m3 and ρ_m4 are the maximum specific rates of acetate and hydrogen consumed by acetoclastic and hydrogenotrophic methanogens, respectively; Y₃ and Y₄ are the biomass yields of acetoclastic and hydrogenotrophic methanogens, respectively; k_d3 and k_d4 are the corresponding decay coefficients of biomass; CH₄ is the quantity of methane gas formed; a and b are the stoichiometric coefficients of acetate and hydrogen production during the corresponding alcohol transformation, respectively; c and d are the stoichiometric coefficients of acetate and hydrogen production during residual waste degradation; and V is the liquid volume.

The partial gas pressures of carbon dioxide, hydrogen and methane were computed according to Vavilin et al. (1995), i.e.

$$\frac{{dP_{i} }}{{dt}} = \frac{{{\text{RT}}}}{{V^{{{\text{gas}}}} }}{\left( { - {\text{TR}}_{i} + {\sum\limits_{i = 1}^{i = 3} {{\text{TR}}_{i} \frac{{P_{i} }}{{P_{t} }}} }} \right)}$$

(24)

, where R is the universal gas constant, T is the temperature (^oK), V^gas is the volume of the gas phase, P_t is the total gas pressure, and TR_i is the rate of mass transfer exchange between gaseous and liquid phases (TR₁, TR₂ and TR₃, were used for carbon, methane and hydrogen, respectively). It was assumed that hydrogen gas and methane were insoluble in water.

Hydrogen gas inhibition

Methane production from the degradation of organic material in the inocula may be inhibited by hydrogen if its partial pressure exceeds a certain level. Such levels may occur e.g. during degradation of the alcohol moieties. Therefore, a hydrogen gas inhibiting effect (I) was included for the first-order rate constants of residual organic material degradation (Figs. 5 and 6) and DEHP hydrolysis (Fig. 7) according to:

$$I = \frac{1}{{1 + {\left[ {\frac{{P_{{{\text{H}}2}} }}{C}} \right]}^{2} }}$$

(25)

, where P _H2 is the partial pressure of hydrogen and C is the inhibiting hydrogen partial pressure constant.

As mentioned previously, strong inhibition of the degradation of different PDs has been observed under acidogenic conditions (Ejlertsson, 1997). However, for the degradation studies modelled in the present paper, a phosphate-carbonate buffer (pH 7) was used together with methanogenic inocula. Therefore, we assumed that the pH was buffered, which could otherwise have decreased below 6 due to acetate production and PA generation.

Simplified first-order kinetics of DEHP hydrolysis in continuous-flow reactors

The following equation was considered for DEHP hydrolysis in a continuous-flow reactor:

$$\frac{{d\,{\text{DEHP}}}}{{dt}} = \frac{1}{{t_{r} }}{\left( {{\text{DEHP}}_{{\inf }} - {\text{DEHP}}_{{eff}} } \right)} - k_{m} I \times {\text{DEHP}}_{{eff}} $$

(26)

where DEHP_inf and DEHP_eff are the influent and effluent DEHP concentrations, t _r is the retention time, k _m is the first-order rate constant of DEHP hydrolysis, and I is the inhibiting function.