1 Introduction

Purified terephthalic acid (PTA) is a feedstock of the polyester industry. Global production of PTA was more than 50 million tons in 2014 (Sun, 2014). As a result, a huge amount of wastewater is discharged following the hydrogenation step of the PTA process. PTA wastewater contains mostly p-toluic (PT) acid, which is generated by the catalytic hydrogenation of the key impurity, 4-carboxybenzaldehyde (Li et al., 2014). Evidence indicates that aromatic acids have serious side effects on environmental protection and human health, thus, the direct discharge of PTA wastewater into natural waterways is strictly forbidden (Macarie et al., 1992; Daramola et al., 2011). At present, PTA wastewater is treated mostly by anaerobic biological methods to degrade the aromatic acid (Kleerebezem et al., 2005) before discharge. The bio-treatment has a low degradation efficiency for aromatic acids (Thiruvenkatachari et al., 2007), and the wastewater treatment is costly and troublesome (Liu and Cheng, 2014). About 1.5–2.0 t of wastewater, which originally comes from deionized water, has to be discharged to produce 1 t of PTA. Therefore, a more resource-saving and cost effective method is urgently required for PTA wastewater treatment.

In recent years a novel extraction method using p-xylene (PX) as the extractant for recovering PT acid and water from PTA wastewater has been proposed (Nakao et al., 2006; Wang et al., 2014). PT acid originates as an intermediate of PX oxidation to PTA, and is therefore easy to extract using PX. The extractant rich in PT acid after extraction is fed back directly to the oxidation step of the PTA process without separation of solute and solvent. In this way PT acid, together with PX, converts into PTA product. This method is cost effective and most of the PT acid in wastewater can be recovered. Nevertheless, the recommended apparatus, i.e., an extraction column (Wang et al., 2014) and mixer-settler (Nakao et al., 2006), will inevitably cause the emulsification of solvents and entrainment by water, which leads to secondary pollution in water requiring further purification (Zhu et al., 2015). This situation is more serious for PTA wastewater treatment, because the initial concentration of PT acid in water is quite low (500–900 g/m3), so a very small amount of emulsion entrainment is intolerable.

Non-dispersive solvent extraction (NDSE) is therefore the best choice for the extraction of PTA wastewater. The structure of an NDSE module is similar to that of a tube-shell heat exchanger, which contains thousands of hollow fibers. Water and solvent flow separately through the tube and shell side, and contact each other at the porous walls of the hollow fibers. NDSE is a continuous phase extraction without dispersion and subsequent emulsification. The water raffinate treated by NDSE contains little PT acid but no PX emulsion. It is expected that the aqueous solution with a PT acid content of less than 100 g/m3 can be recycled in the hydrogenation step of the PTA process as fresh solvent solution (Li et al., 2010), so most water raffinate from NDSE can be re-used. Thus, the simplest liquid-liquid extraction configuration is sufficient for purifying the PTA wastewater, and the PT acid does not need to be recovered from the extractant after NDSE. With the advantages of no emulsification and high extraction efficiency (Pabby and Sastre, 2013), NDSE has been used in many applications, including the chemical separation of pesticides (Đorđević et al., 2014), bisphenol A (Gupta et al., 2014), toluene (Barati et al., 2014), fermentation products (Shirazian and Ashrafizadeh, 2011), zinc chloride and hydrochloric acid (Lum et al., 2014), 1-butanol and acetone (Moreno et al., 2014), cephalexin (Hao et al., 2014), Co (Vernekar et al., 2013), Lu (Kumrić et al., 2012), U (Dixit et al., 2013), Zn (Samaniego et al., 2007), and Cu (Ren et al., 2008). However, there have been no published studies of NDSE applied to PTA wastewater.

In this study, an attempt was made to develop a novel NDSE method for simultaneously recovering PT acid from wastewater and recycling the water resource, while avoiding solvent emulsification and entrainment. Firstly, experiments were carried out to evaluate the feasibility of NDSE for PTA wastewater treatment, and to investigate the impacts of operating conditions. Then, we carried out modeling and simulation of extraction of PT acid by PX in NDSE. The rate determining step was identified. The effects of the operating and membrane parameters on membrane module performance were investigated by the simulation. Finally, optimal parameters are suggested for industrial NDSE design.

2 Experiments

A lab-scale membrane module was assembled for NDSE experiments (Table 1). The housing of the membrane module was stainless steel sealed with PX resistant epoxy at each end. The polypropylene hollow fiber used was a commercial product from the Hangzhou H-Filtration Membrane Technology & Engineering Co., Ltd., China. The length of the fiber was designed to be as long as possible to ensure that the experimental NDSE performance was close to that of an industrial module (usually 1.0 m). However, large numbers of fibers were not needed in the module because the mass transport behavior is identical for each fiber. A similar design, using only a single hollow fiber, was used by Kumrić et al. (2012). The essential structural and physical parameters of the fiber were provided by the supplier (Table 1).

Table 1 Specifications of the hollow fiber and membrane module
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The setups of the NDSE experiments are shown in Fig. 1. The hollow fiber membrane was immersed in PX for 24 h to ensure that the porous film of hollow fibers was fully saturated by PX. The aqueous solution was pumped through the lumen side by using a digital single-piston solvent pump (Lab-Allance Series I, USA), while the extractant flowed countercurrently through the shell side. The flow rates were calibrated by measuring the outlet flow volumes over a certain time interval with an overall uncertainty of ±1%. A slight over pressure (about 5 kPa) between the lumen and shell sides at any axial position was maintained by adjusting the value on the lumen outlet pipeline. The temperature was controlled by immersing the membrane module in a water bath at the assigned temperature.

Fig. 1
figure 1

Schematic diagram of the NDSE experimental setup

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The following operating parameters were investigated: extraction time te (or contact time, represented by the aqueous residence time in the lumen side for a single pass), aqueous/organic volumetric flow ratio (a/o), initial PT acid concentration (Cf), and temperature (T). The extraction time in experiments ranged from 20 to 70 s, the a/o from 3.0 to 9.0, the initial PT acid concentration from 300 to 900 g/m3, and the temperature from 308.2 to 328.2 K. Pre-experiments indicated that an extraction time of 70 s was sufficient to reduce the PT acid concentration to the re-use standard. The above ranges cover available conditions of the existing PTA process while taking into account the appropriate NDSE operating conditions to realize the re-use of water raffinate.

The lumen and shell side outlet streams were sampled after the NDSE experiments reached a steady state (after running for about 20 min), and the PT acid concentration of the solutions was analyzed by means of a high performance liquid chromatography (HPLC). Details of the analytical method can be found in Kong et al. (2013). Three parallel samples were prepared and analyzed for each experimental run to evaluate the relative deviation, which was shown to be less than 1%. The data presented in Section 4.1 are the mean values obtained in the duplicate experiments. The mass conservation of PT acid between the outlet streams and the aqueous inlet stream was also measured and was found to have a relative deviation of less than 1%.

3 Theory

3.1 Model formulation

A membrane module usually contains a huge number of hollow fibers (about 10 000 to 70 000). However, the modeling of the membrane module can be equivalently transformed into the case of a single fiber surrounded by a concentric cylinder solvent flow layer (Fig. 2, see “Happel’s free surface model” below). The aqueous phase flows in the lumen side and the organic phase in the shell side. Both are assumed to be fully developed countercurrent laminar flows. The aqueous phase enters at z=0. The hydrophobic membrane is considered non-wetted with aqueous solution but filled with PX in the micro-pores. An aqueous/organic interface therefore forms at the mouth of membrane pores adjacent to the lumen side. PT acid in the lumen aqueous phase diffuses firstly to the interface, where it is extracted by the organic phase, then permeates through the membrane, spreads to the shell side, and finally is brought out by the organic phase.

Fig. 2
figure 2

Transport scheme of PT acid in a countercurrent flow pattern with Happel’s free surface model

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The model developed to simulate the mass transport in NDSE contains solutions of the continuity equations in all the sub-regions, i.e., the lumen side, the membrane, and the shell side, with appropriate velocity distributions. The differential form of mass balance (also called the convection and diffusion equation) can be written as (Bringas et al., 2009)

$${V_i}{{\partial {C_i}} \over {\partial z}} = {D_i}{1 \over r}{\partial \over {\partial r}}\left({r{{\partial {C_i}} \over {\partial r}}} \right),$$
(1)

where V, C, and D with subscript i refer to the velocity distribution, the concentration of PT acid, and the diffusivity and physical property in the ith sub-region, respectively; z and r are the axes of cylindrical coordinates. The term on the left-hand side illustrates the axial convective transport mass flux of PT acid, while that on the right-hand side represents the radial diffusive transport mass flux. Note that the axial diffusion is supposed to be negligible.

The velocity distribution in the lumen side follows Newtonian laminar flow (Eq. (3) in Table 2). In the porous membrane, where the velocity is zero, the mass transport is by diffusion only. Hence, the concentration profile can be expressed analytically by

$${C_{\rm{m}}} = \left({{{\left. {{C_{\rm{m}}}} \right\vert }_{r = {r_2}}} - {{\left. {{C_{\rm{m}}}} \right\vert }_{r = {r_1}}}} \right)\ln (r{{/}}{r_1})/\ln ({r_2}{{/}}{r_1}) + {\left. {{C_{\rm{m}}}} \right\vert _{r = {r_1}}},$$
(2)

where r1 and r2 are the fiber inner and outer radii, and the subscript m denotes the quantity in the membrane. In the shell side, Happel’s free surface model (Happel, 1959) is adopted to describe the shell side velocity Vo (Eqs. (4a) and (4b) in Table 2). This model simplifies the shell side into a concentric cylinder flow layer surrounding the fiber core. The volumetric rate of the shell side is assumed to be evenly distributed around each fiber with a slip condition at the outer interface of the shell side.

Table 2 Mathematical expressions in relation to Eq. (1)
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The boundary conditions of the lumen and shell sides are expressed by Eqs. (5a)–(6c) in Table 2. Note that the concentrations on both sides at the interface r=r1 satisfy the phase equilibrium relationship Cm=f(Ca), which will be discussed in Section 3.2.1.

3.2 Model parameters

3.2.1 Estimation of phase equilibrium parameters and diffusivities

The phase equilibrium relationship of PT acid in PX and water phases has been described by the non-random two liquid (NRTL) activity coefficient model (Kong et al., 2013). The model, however, has a complicated expression. For simplicity in calculation, a quadratic form of the phase equilibrium relationship has been re-correlated, which also shows high fitting precision (R2>0.997, Fig. 3).

$$\left[ {{C_{\rm{o}}}} \right] = a\left[ {{C_{\rm{a}}}} \right] + b{\left[ {{C_{\rm{a}}}} \right]^2},$$
(7)

where [Ca] and [Co] represent equilibrium concentrations of PT acid in water and PX, respectively. Values of a and b are listed in Table 3. A comparison between experimental and calculated results is shown in Fig. 3.

Fig. 3
figure 3

Quadratic form of phase equilibrium relationship between water and PX for PT acid. Points are experimental data from Kong et al. ( 2013 ), and lines are calcu-lation results from Eq. (7)

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Table 3 Equilibrium parameters in Eq. (7) and diffusivities of PT acid in water and PX phases
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For convenience, in the following computation on temperature dependence, the values of a and b can be further correlated with temperature (R2>0.995) by

$$a = - 51{.}05 + 0{.}2767T - 3{.}601 \times {10^{ - 4}}{T^2},$$
(8a)
$$b = 177{.}6 - 0{.}7254T + 1{.}820 \times {10^{ - 6}}{T^3},$$
(8b)

where T is the value of thermodynamic temperature.

Diffusivities of PT acid in water (Da) and PX (Do) are calculated using the Wilke-Chang equation (Wilke and Chang, 1955). The results are listed in Table 3. The diffusivity of PT acid in membrane pores is calculated by multiplying Do by the membrane porosity (ε) and dividing by membrane tortuosity (τ). Moreover, the temperature dependence of diffusivities can be correlated (R2>0.999) by

$${D_{\rm{a}}} = - 7{{{.}769}} \times {{1}}{{{0}}^{ - {{9}}}}{{ + 2}}{{{.}850}} \times {{1}}{{{0}}^{ - 11}}T{{,}}$$
(9a)
$${D_{\rm{o}}} = - {{9}}{{{.}115}} \times {{1}}{{{0}}^{ - {{9}}}}{{ + 3}}{{{.}862}} \times {{1}}{{{0}}^{ - 11}}T{{.}}$$
(9b)

3.2.2 Estimation of pressure drops and breakthrough pressure

To avoid penetration of PX through the membrane layer into the lumen side, the aqueous solution needs to maintain a higher pressure than the shell side at any axial position. The pressure profiles along the lumen and shell sides should therefore be calculated to estimate the maximum pressure difference. Following Biełuszka et al. (2014), the pressure drops in the lumen side Δpa and the shell side Δpo can be calculated by

$$\Delta {p_{\rm{a}}} = - {{8{{\bar V}_{\rm{a}}}{\mu _{\rm{a}}}z} \over {r_1^2}},$$
(10a)
$$\Delta {p_{\rm{o}}} = - {{8\left({r_3^2 - r_2^2} \right){{\bar V}_{\rm{o}}}{\mu _{\rm{o}}}z} \over {4r_2^2r_3^2 - r_2^4 - 3r_3^4 + 4r_3^4\ln ({r_3}{{/}}{r_2})}},$$
(10b)

where μ is the viscosity of the liquid.

The pressure difference between the water and PX phases, however, must be larger than zero but smaller than the breakthrough pressure Pcr. For the countercurrent flow of aqueous and organic phases, the sum of the pressure differences should satisfy the following restriction:

$$\Delta {p_{\rm{a}}} + \Delta {p_{\rm{o}}} < {P_{{\rm{cr}}}}{.}$$
(11)

Overpressure will lead to the aqueous phase leaking through the membrane into the shell side. Pcr can be estimated using the Young-Laplace equation (Drioli and Giorno, 2010):

$${P_{{\rm{cr}}}} = - {{4\Theta \gamma \cos {\theta _{\rm{c}}}} \over {{d_{\rm{p}}}}},$$
(12)

where Θ, γ, θc, and dp are the geometric factor (equal to 1 for cylindrical pores), the interfacial tension, the liquid-solid contact angle, and the maximum diameter of pores, respectively.

3.3 Numerical solution of model equations

The model equations are hard to solve analytically due to the non-linear equilibrium relationship on the boundary conditions Eq. (7). So the models in the lumen and shell sub-regions were solved numerically using COMSOL software (version 4.3a) in a 2D axi-symmetric coordinate system. The sub-regions should be scaled up due to the large difference between the r- and z-directions. The lumen and shell sub-regions were meshed automatically with mesh generated by COMSOL, having quite a fine mesh near all the boundaries/interfaces by adopting the boundary distribution. Note that the membrane sub-region was not meshed. The concentration profile of the membrane was calculated from Eq. (2) after the computation of the other two sub-regions. Finite element analysis is combined with error control using the numerical solver of SPOOLES with a computational time of about 20 min and a relative tolerance of less than 10−6.

To demonstrate the accuracy of numerical programming, a case of the classical Graetz problem was solved and compared with the analytical solution. For the lumen side with wall concentration equal to zero, the analytical solution takes the form of a Fourier series (Graetz, 1885):

$$C(r,z) = \sum\limits_{n = 1}^\infty {{A_n}{Y_n}(r)} \exp \left({ - \lambda _n^2z} \right),$$
(13)

where A n , Y n , and λ n are the associated constant, the eigenfunction, and the eigenvalue, respectively. Fig. 4 shows the numerical and analytical solutions of dimensionless concentration profiles (Cf=1). The relative errors between the two solutions are less than 1×10−4, which means that the numerical algorithm is sufficiently accurate.

Fig. 4
figure 4

Comparison of numerical and analytical solutions for the Graetz problem. Points are from the analytical solution, and lines are numerical results

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A grid independence test was conducted to check the mass conservation of PT acid in the sub-regions. The number of cells was varied from 80 000 to 120 000. After 100 000 cells, a further increase in cell number led to a less than 0.1% change in the mass conservation. Considering both computational time and accuracy, all the computations were conducted with 100 000 cells.

To evaluate NDSE performance, three indexes were defined as follows, and used as measures of the aqueous outlet PT acid concentration (Cout), efficiency of the membrane module (Em), and efficiency of the solvent (Es):

$${C_{{\rm{out}}}} = {{\int\nolimits_0^{{r_1}} {r{V_{\rm{a}}}{{\left. {{C_{\rm{a}}}} \right\vert }_{z = 1}}{\rm{d}}r} } \over {\int\nolimits_0^{{r_1}} {r{V_{\rm{a}}}{\rm{d}}r} }},$$
(14a)
$${E_{\rm{m}}} = ({C_{\rm{f}}} - {C_{{\rm{out}}}}){u_{\rm{a}}}/{V_{\rm{m}}},$$
(14b)
$${E_{\rm{s}}} = {C_{{\rm{out}},{\rm{o}}}}/{C{\ast}} = (a{{/}}o)({C_{\rm{f}}} - {C_{{\rm{out}}}})/(a{C_{\rm{f}}} + bC_{\rm{f}}^2),$$
(14c)

where ua is the overall volumetric flow rate of the aqueous solution in a module, Vm is the volume of the module, and Cf, Cout,o, and C* are the initial feed PT acid concentration, the organic outlet PT acid concentration, and equilibrium concentration, respectively. Cout measures the degree of wastewater purification, which should be lower than 100 g/m3 for re-use in the PTA process. Em denotes the capacity of the module. A higher value of Em means the membrane module runs more efficiently. Es represents the efficiency of solvent usage. A higher value of Es means less solvent is needed in extraction. The maximum of 1.0 denotes the limit of the extraction at aqueous/organic equilibrium.

4 Results and discussion

4.1 Experimental results

Figs. 5–7 show the experimental results for the aqueous outlet PT acid concentration (Cout) with various operating parameters. Fig. 5 illustrates the effects of the extraction time (te) and initial PT acid concentration (Cf). Cout decreases dramatically from 1.30 to 0.585 mol/m3 as te increases from 20 to 70 s (Cf=3.67 mol/m3), indicating that te is a very sensitive parameter. This is expected for a hydrophobic membrane wetted by an extractant with a large partition coefficient (Shen et al., 2009). A longer extraction time indicates a longer axial conventional time compared with the radial diffusion time, so a higher proportion of PT acid will diffuse to the organic phase leading to better extraction performance. When te is longer than 60 s, Cout can be reduced to lower than 0.734 mol/m3 (100 g/m3) for water re-use in all cases. Thus, 20–70 s is a suitable range for te from a practical point of view. A remarkable rise in the feed concentration Cf (300–900 g/m3) slightly increases the outlet concentration Cout (72.3–89.2 g/m3, te=70 s), which means that the higher is the impurity concentration in the wastewater, the more effective is the NDSE method. Younas et al. (2008) suggested that the higher extraction rate caused by a higher initial concentration is more obvious with a short extraction time, and after a sufficiently long time the influence of the initial concentration on the aqueous outlet concentration can be negligible. This can be further explained by Fig. 3 which shows that a higher Cf will lead to a higher partition coefficient (m=[Co]/[Ca]) and overall mass transport coefficient (Section 4.2), and so will facilitate the mass transport, but will tend to be unremarkable in terms of the decrease in the aqueous PT acid concentration.

Fig. 5
figure 5

Effects of extraction time (t e ) and initial PT acid concentration (C f ) on the aqueous outlet PT acid concen-tration (C out ) with a/o=9.0 and T=318 K (lines are simulation results)

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Fig. 6
figure 6

Effects of volumetric ratio ( a/o ) and initial PT acid concentration ( C f ) on the aqueous outlet PT acid concen-tration ( C out ) with t e =50 s and T =318 K (lines are simulation results)

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Fig. 7
figure 7

Effects of temperature ( T ) and extraction time ( t e ) on the aqueous outlet PT acid concentration ( C out ) with a/o =9.0 and C f =3.67 mol/m 3 (500 g/m 3 ) (lines are simulation results)

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Fig. 6 shows the variation in Cout with various aqueous/organic volumetric flow ratios (a/o) and Cf values. Cout exhibits a notably near-linear increase from 49.0 to 96.7 g/m3 (Cf=3.67 mol/m3) with a/o in all the experiments. The influence of a/o is because of the less extractant being available for a higher ratio of feed to PX, as already noted in previous studies (Agarwal et al., 2013; 2014). A smaller a/o will therefore improve the performance of NDSE but increase the cost of solvent use. In addition, Cout is not as sensitive to a/o as it is to the extraction time.

Fig. 7 indicates the effects of temperature on Cout with different extraction time. For a certain te, Cout seems to have a minimum value in the temperature range investigated. Cout changes from 99.0 to 96.5 g/m3 then to 104 g/m3 with T varying from 308 to 313 K then to 328 K, respectively (te=50 s). Temperature rise has two opposing effects on the extraction: accelerating the diffusion of PT acid or the rate of extraction, and decreasing the equilibrium partition coefficient m or extraction capacity of the PX phase. Therefore, there should be an optimal temperature for a certain te. This phenomenon is further investigated over a wider range by computational simulation in Section 4.4.

The extraction process under experimental conditions was simulated by Eqs. (1)–(9) to validate the model and parameters. A comparison between computed and experimental results is shown in Figs. 5–7. The relative errors between them are less than 6%, which demonstrates that the model parameters were all properly estimated. The positive deviation between the experimental and computational Cout may come from the non-ideality of velocity distributions in the entrance of the hollow fiber. In the following sections, the simulation results are used to investigate further the impacts of a wide range of operating conditions, so as to select proper extracting conditions.

4.2 Concentration profiles in all sub-regions

The concentration profiles were calculated under typical industrial operating conditions: ua= 0.204 m3/h, te=50 s, Cf=6.61 mol/m3 (900 g/m3), a/o= 9.0, T=318 K, r1=200 µm, r2=250 µm, r3=400 µm, rm=60 mm (module inner radius), n=22 522, L=1.0 m, ϕ=0.391, ε=0.45, and τ=2.2. The aqueous outlet concentration of PT acid in this case is 0.704 mol/m3 (95.9 g/m3). Fig. 8 depicts the PT acid concentration profiles in three parts representing, from left to right, the lumen side, the membrane, and the shell side, respectively. The PT acid concentration in aqueous solution notably decreases along the radial direction from the center to the membrane wall of the hollow fiber. At the aqueous-membrane interface the concentration shows an abrupt jump from the aqueous phase to the PX phase, as a result of a much higher partition ratio (m≫1) of PT acid in the PX phase than that in the aqueous phase. The PT acid concentration in the PX phase, Co, maintains a high platform on the membrane and shell side, but decreases slightly along the radial direction on the shell side. In the axial direction, the concentrations of Ca and Co decrease dramatically as a result of countercurrent contact of aqueous and organic phases. The impurity PT acid transfers from the lumen aqueous phase to the membrane organic phase, then penetrates the membrane, and is swept out from the module by PX.

Fig. 8
figure 8

PT acid concentration profiles (mol/m 3 ) in all the sub-regions under typical industrial operating conditions. The three parts, from left to right, represent the lumen side, the membrane, and the shell side, respectively. At the aqueous inlet boundary z =0 m, while at the organic inlet boundary z =1 m

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To quantitatively evaluate the resistances in the sub-regions, the individual mass transfer coefficients (k i ) in all the sub-regions are introduced as follows (McCabe et al., 1993):

$$\left\{ {\begin{array}{*{20}c} {{k_{\rm{a}}} \approx {D_{\rm{a}}}/{r_1},\quad \quad \quad }\\ {{k_{\rm{m}}} \approx {D_{\rm{m}}}/({r_2} - {r_1}),}\\ {{k_{\rm{o}}} \approx {D_{\rm{o}}}/({r_3} - {r_2}){.}} \end{array}} \right.$$
(15)

The contributions of ki to the aqueous solution based on the overall mass transfer coefficient (Ka) can be calculated by the well-known resistance-in-series model (Gameiro et al., 2008):

$${1 \over {{K_{\rm{a}}}}} = {1 \over {{k_{\rm{a}}}}} + {{{r_1}} \over {{k_{\rm{m}}}m{r_{{\rm{lm}}}}}} + {{{r_1}} \over {{k_{\rm{o}}}m{r_2}}},$$
(16)

where rlm is the logarithmic mean radius of membrane. m represents the slope of a linear-relationship partition coefficient equation as

$$m = {\left. {{{{\rm{d}}{C_{\rm{m}}}} \over {{\rm{d}}{C_{\rm{a}}}}}} \right\vert _{r = {r_1}}} = a + 2b{\left. {{C_{\rm{a}}}} \right\vert _{r = {r_1}}}{.}$$
(17)

In the case shown in Fig. 8, the values of 1/ka, r1/(kmmrlm), and r1/(komr2) on the right hand side of Eq.(16) at z=0.5 m are 1.55×105, 5.37×103, and 2.97×103, respectively. The order of magnitude indicates that the aqueous mass transfer resistance predominates, i.e., Ka≈ka, which means the rate determining step of the overall mass transfer process is in the lumen side. This situation is similar to that described by Kiani et al. (1984) as a “high value of m”. Fig. 3 shows that m is in the range of 4–12 which means most PT acid initially in the aqueous phase tends to concentrate in the organic phase when contacting PX. As a result, at the contacting surface of the boundary between the lumen side and membrane, PT acid in the aqueous phase will decrease dramatically while the concentration gradient will become steep. In the lumen side, there is purely axial laminar flow without any radial or tangential velocity. The only radial mass transfer mechanism is molecular diffusion. Table 3 shows that the PT acid diffusivity Da is of the order of 10−9 m2/s, so low diffusivity restricts radial mass transfer in the lumen side, which thereby becomes the rate determining step of the overall mass transfer process. Note that in Fig. 6, a/o still shows a relatively notable effect on Cout with constant te, which means that flow in the shell side is also important in affecting the aqueous/solvent ratio.

4.3 Influence of membrane parameters

The effects of the fiber inner radius and porosity-to-tortuosity ratio (ε/τ) on NDSE performance need to be investigated for selection of membrane modules. Enlarging r1 will decrease the pressure drop in the lumen side and hence the pumping costs (Wickramasinghe et al., 1991), as well as enhance the strength of the fiber, while ε/τ varies among different commercial membrane products. Usually, when designing a membrane module, the packing density of hollow fibers (defined as the volume fraction of hollow fibers occupied in a module) is a constant value as high as possible (about 0.40 for a module with high flow rate, provided by the membrane supplier), and the wastewater treatment load of a module should be set unchanged. Keeping packing density and load constant, increasing r1 means prolonging the extraction time, reducing the number of hollow fibers and the aqueous-organic contact area. Simulations were run under the typical conditions ua=0.204 m3/h, a/o=9.0, Cf=6.61 mol/m3 (900 g/m3), T=318 K, Δr2=r2−r1=50 µm, rm=60 mm, L=1.0 m, and ϕ=0.391. Note that usually the length of fiber for commercial products is 1.0 m, so L was set as 1.0 m in all simulations. To maintain a constant membrane thickness (50 µm), the free surface radii r3 and r1 were varied simultaneously. The ε/τ ranged from 0.067 to 0.45, corresponding to the typical values reported by Gabelman and Hwang (1999).

Simulation results (Fig. 9) show the variation in Cout with various values of r1 and ε/τ. A smaller r1 results in a lower Cout, and higher efficiencies of Em and Es calculated from Eqs. (14b) and (14c). A decrease in r1 leads to two opposite effects: a shorter radial distance and larger contact area for PT acid diffusion, and a shorter extraction time when keeping packing density and flow rate unchanged (e.g., r1=150 µm, te=43.9 s; r1=300 µm, te=57.3 s). The former becomes predominant (Fig. 9). This result seems to contradict the findings of Younas et al. (2008), who observed the opposite effect of r1 for NDSE of aroma compounds, i.e., a larger inner radius increased the mass transfer rate. The discrepancy originates from differences in the conditions in the two studies. In the study of Younas et al. (2008) the number of fibers was kept constant while the packing density was allowed to vary, but in this study, the fiber packing density was kept constant while the number of fibers varied. At a fixed overall flow rate and fiber number, an increase in fiber radius increases extraction time and packing density, both of which undoubtedly promote mass transfer in extraction, as in the case of Younas et al. (2008).

Fig. 9
figure 9

Effects of the fiber inner radius ( r 1 ) and porosity-to-tortuosity ratio ( ɛ/τ ) on the aqueous outlet PT acid concentration ( C out ), keeping the fiber packing density, membrane thickness, and overall volumetric flow rate unchanged

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Fig. 9 also shows that Cout decreases with a rise in ε/τ. For a large ε/τ of 0.20–0.45, the influence becomes unremarkable. A similar trend is found for Em and Es. A large ε/τ leads to high diffusivity in the membrane. However, the membrane resistance is not a dominant factor for mass transfer, so the effect is limited. Considering the balance between NDSE performance and fiber lifetime, an r1 of 200–250 µm is suggested, while ε/τ should be larger than 0.20.

4.4 Influence of operating parameters

The ratio of aqueous and organic phase volumetric flow rates a/o and temperature are the most important operating parameters, in addition to extraction time, which should be carefully designed in view of the required effluent PT acid content, industrial restrictions, and the cost of NDSE. The maximum PX amount available in a conventional PTA plant is 1/3 of the wastewater, which means a/o should be larger than 3. The temperature of wastewater in practice is in the range of 30–80 °C. A lower temperature extends the lifetime of the hollow fiber, but increases cooling costs.

Fig. 10 illustrates the effects of the te and a/o on the aqueous outlet concentration. All parameters except te, ua, and a/o were fixed at the typical values used in Section 4.2. As expected, Cout decreases with te and increases with a/o. Prolonging te will facilitate the radial transport of PT acid from aqueous solution to the membrane, while decreasing a/o would enhance the axial convection in the shell side, both of which reduce Cout. te is more sensitive over short (20–60 s) than over long periods. In view of the Cout requirement, treating efficiency and cost, te should be set around 60 s in the design and an a/o should be selected that is as large as possible near the contour of Cout 0.734 mol/m3 (100 g/m3).

Fig. 10
figure 10

Effects of the extraction time ( t e ) in the lumen side and volumetric ratio ( a/o ) on the aqueous outlet PT acid concentration ( C out )

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Fig. 11 shows the variation in the module efficiency Em with te and a/o. Em dramatically decreases with te and slightly declines with a/o, which is easy to understand from Eq. (14b) as Em is directly proportional to (Cf−Cout) and inversely proportional to the aqueous residence time (extraction time te). Changes in te and a/o in the range only slightly alter the concentration difference (Cf−Cout) which cannot catch up with 1/te. A short time means high efficiency and outlet PT acid content. So there are contradictory requirements between water cleanness and treatment efficiency that require a compromise in the design of NDSE.

Fig. 11
figure 11

Effects of the extraction time ( t e ) in the lumen side and volumetric ratio ( a/o ) on the efficiency of the membrane module ( E m )

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Fig. 12 depicts the variation in solvent efficiency Es with te and a/o. Es is between 0.07 and 0.44 in the considered range, far smaller than 1.0, which means the extractant PX was in the unsaturated state in the purification of the wastewater. a/o shows a remarkable effect on Es while te seems to have less effect, which is clear from Eq. (14c). Less solvent usage means a lower cost and a higher impurity content. A compromise choice for the water/solvent ratio seems appropriate at an a/o of about 9.0.

Fig. 12
figure 12

Effects of the extraction time ( t e ) in the lumen side and volumetric ratio ( a/o ) on solvent efficiency ( E s )

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The pressure difference between the lumen and shell sides and the breakthrough pressure were calculated to prevent breakthrough of both PX and aqueous phases in the te ranges of interest. The results show the maximum pressure difference is 6.62 kPa while te=20 s and a/o=3.0, which is far beyond the breakthrough pressure of 500 kPa. Moreover, the pressure drop along the membrane module is not appreciable, which can reduce the operating costs of the process (Fadaei et al., 2011).

The effect of temperature on Cout is shown in Fig. 13. In the simulation, other parameters used were the typical values described in Section 4.2. According to Eqs. (14b) and (14c), Em and Es will simultaneously reach maximum values when Cout is at the minimum point for a certain te. A close inspection of Fig. 13 shows that the optimal temperature moves towards a higher value as te decreases (from 306 to 328 K with te reducing from 110 to 20 s). Compared with Fig. 7, the existence of the Cout minimum and shift tendency is shown more clearly in Fig. 13, especially for the case of a short extraction time. As pointed out in Section 4.1, a temperature rise will accelerate PT acid diffusion but decrease the solvent extraction capacity. The Cout minimum is a balance between these two opposing effects. This phenomenon become more evident in the situation of a short extraction time where mass transfer proceeds faster, and the kinetic and thermodynamic factors dominate at low and high temperature intervals, respectively. For the te and a/o suggested above, an appropriate T was chosen as 318 K.

Fig. 13
figure 13

Effects of the operating temperature ( T ) and the extraction time ( t e ) on the aqueous outlet PT acid concen-tration ( C out )

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5 Conclusions

NDSE with PX as extractant to recover both PT acid and water from PTA wastewater was shown to be feasible and effective by means of experiment and model simulation. The effects of membrane and operating parameters te, a/o, T, Cf, r1, and ε/τ on the PT acid content in the raffinate Cout and the membrane, and on solvent efficiencies Em and Es, were experimentally and numerically investigated. The rate determining step of mass transport was found to be within the aqueous solution. The extraction time te and fiber radius r1 are, therefore, the two parameters to which Cout is most sensitive. Shrinkage of the radius r1 can remarkably accelerate PT acid transport, decrease Cout, and enhance the extraction efficiency Em at the cost of increasing the pressure drop and shortening the membrane lifetime. Prolonging te will reduce PT acid content in raffinate at the cost of reducing membrane efficiency. A larger a/o ratio results in less extractant usage but increasing PT acid content, and a/o shows near-linear influences on Cout and solvent efficiency Es, but has little effect on Em. Temperature has opposing effects on mass transport: a higher T increases PT acid diffusivities, but reduces the partition coefficient in the organic phase. Therefore, there exists an optimal T value for NDSE performance. The optimal temperature moves towards a higher value as te decreases. A higher feed PT acid concentration Cf will enhance the extraction efficiency and slightly increase Cout. In view of the availability in industrial PTA processes and the necessary compromise between water cleanness and extraction cost, the membrane and operating parameters appropriate for industrial design are suggested as follows: te=50–60 s, a/o=9.0, T=318 K, r1=200–250 µm, and ε/τ=0.20–0.45. Under these conditions, PTA wastewater can be purified and re-used, together with PT acid recovery, with acceptable efficiencies of the membrane module and solvent.