6.2. Concentration polarization fouling

Concentration polarization (CP) phenomenon entails the formation of a boundary double layer along the membrane surface, with salt concentration considerably higher than that of the starting injected solution as revealed in Figure 6 [42–44]. Cb is the salt concentration within the boundary layer; Cs is the salt concentration at the inner membrane surface, and Cp is the lower salt content of the freshwater on the pass through side.

As indicated in Figure 6, the flow comes to pass in the boundary layers of the feed/concentrate spacers; two different types are encountered: a convective flow of fresh feed solution from the bulk and diffusion flow of repelled drain salts, coming back into the feed flow. In that concern, the semipermeable ROM is designed to give higher rate of convective flow than the diffusion flow, as the salts and particulate solids discarded tend to pile up with highest salt contents on the inner surface of the ROM. The concentration of solid particulates in the boundary layer

Figure 6. Boundary layers in a membrane-feed spacer. RO, reverse osmosis.

6.1. External and internal fouling

rejected salts; it can be viewed as [40, 41]:

• Construction of mineral deposits (scale)

typically pH < 3 and alkali at pH > 12.

6.2. Concentration polarization fouling

lower salt content of the freshwater on the pass through side.

irreversible changes.

1. External or "surface" fouling (EF)

2. Internal fouling (IF)

12 Wastewater and Water Quality

inorganic matters

paths:

The classification of the fouling phenomenon depends on the location of the accumulated

EF involves accumulation of rejected salts on the surface of the membranes by three distinct

• Construction of cake of rejected solids, particulates, colloids, and other organic and/or

• Biofilm construction, i.e., growth of colonies of microorganisms on the surface of the

Typically, the three mechanisms can occur in any combination at any given time. However,

IF is a regular loss of membrane productivity due to changes in its chemical structure either by physical compaction or by chemical degradation. Physical deterioration of the membrane may result from long-term application of feed stream at pressures higher than that designed for the ROMs; they are designed to handle 83 bars for sea water reverse osmosis membranes and/or by their continued setup at source water temperatures above 45C, the limit of safe membrane operation. Chemical deterioration results from continuous exposure to strong oxidants, e.g., chlorines, bromines, ozones, permanganates, peroxides chemicals, and very strong acids,

The difference between EF and IF is somewhat clear; EF could be completely reversed by chemical cleaning, while IF causes permanent damage of the micropores, resulting in an

Concentration polarization (CP) phenomenon entails the formation of a boundary double layer along the membrane surface, with salt concentration considerably higher than that of the starting injected solution as revealed in Figure 6 [42–44]. Cb is the salt concentration within the boundary layer; Cs is the salt concentration at the inner membrane surface, and Cp is the

As indicated in Figure 6, the flow comes to pass in the boundary layers of the feed/concentrate spacers; two different types are encountered: a convective flow of fresh feed solution from the bulk and diffusion flow of repelled drain salts, coming back into the feed flow. In that concern, the semipermeable ROM is designed to give higher rate of convective flow than the diffusion flow, as the salts and particulate solids discarded tend to pile up with highest salt contents on the inner surface of the ROM. The concentration of solid particulates in the boundary layer

membranes, rapidly attachable by excretion of extracellular materials

external membrane fouling of ROMs is most frequently caused by biofouling.

leads to critical negative significances on the ROM function. They include increased osmotic pressure, increased salt extract, creation of hydraulic opposition of water stream, and Induction of scale and fouling on the ROM.

Concentration polarization cannot be evaded; it can only be reduced before taking any corrective measures; concentration polarization should be quantified. This quantification occurs in three separate consecutive paths. They can be emphasized as balancing the chemical and mass balance equations across the boundary layer, balancing the transport equations over the ROM and determination of solute transport equations within the pores of the ROM. System performance can be predicted by simultaneous solution of all these three equations. Based on the type of concentration polarization, there are two classes of models: an osmotic pressurecontrolled model and a gel layer-controlling model.

## 6.3. Osmotic pressure controlled model [OPCM]

In this situation, solute particles form a viscous boundary layer concluded on the surfaces of ROMs [45–47]. Solute concentration increases from the bulk to membrane surface concentration across the mass transfer barrier layer. In this case, the width of the mass transfer boundary layer is constant. At any cross section of the boundary layer for the concentration gradient, dc dy, at the steady state, the solute mass steadiness leads to

$$
\left[ \left( \upsilon\_w \ c - \upsilon\_w \ c\_p \right) + D \ \frac{dc}{dy} \right] = 0 \tag{2}
$$

where vw is the permeate flux in m3 /m<sup>2</sup> .s; c and cp are the bulk and permeate concentrations in kg/m<sup>3</sup> ; and D is the solute diffusivity in m<sup>2</sup> /s.

Integrating the above equation across the thickness of the mass transfer boundary layer, the governing equation of the flux is obtained as

$$w\_w = \left(\frac{D}{\delta}\right) \ln\left(\frac{c\_m - c\_p}{c \bullet - c\_p}\right) = k \ln\left(\frac{c\_m - c\_p}{c \bullet - c\_p}\right) \tag{3}$$

Δπ <sup>¼</sup> <sup>π</sup><sup>m</sup> � <sup>π</sup><sup>P</sup> <sup>¼</sup> <sup>a</sup>1½ �þ cm � cP <sup>a</sup><sup>2</sup> cm<sup>2</sup> � cP

model or the osmotic pressure-controlling model.

6.4. Solution diffusion model for RO/NF

could be rewritten as

and vo

where

<sup>α</sup> <sup>¼</sup> <sup>a</sup> Δp

results:

where <sup>β</sup> <sup>¼</sup> <sup>B</sup>

vo w <sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>3</sup> cm<sup>3</sup> � cP

where the constant coefficient is known as the difference between a1and a<sup>n</sup> and real retention could be defined by cm and cP indicated as the coefficients across the ROM phases, respectively, namely, the upstream and downstream phases. Therefore, Eq. (7) can be written in terms of the single parameter cm using Eq. (2), to reduce the system variables to cm and vw, instead of the existing three parameters. The new variables can be attained by solving Eqs. (4) and (6) using an iterative algorithm like the Newton-Raphson equations. This model is known as classic-film

The real retention is a partition coefficient, or really the solute flux across the membrane considered using the solution diffusion model described earlier; linear relationship is considered between π and c in the case of salt solution, π ¼ ac [42, 43, 48, 49]. In practice, Eq. (6) and the film theory equation, Eq. (4), are only considered. Therefore, the osmotic pressure model

The above equation can be equated with the film theory equation and the following equation

<sup>w</sup> <sup>½</sup><sup>1</sup> � <sup>α</sup>ð Þ cm � cP � ¼ <sup>k</sup> ln cm � cP

where B is a constant. Combining Eqs. (8) and (10), the following equation is obtained:

<sup>w</sup> <sup>½</sup><sup>1</sup> � <sup>α</sup>ð Þ cm � cP � ¼ <sup>B</sup> cm � cP

<sup>1</sup> � <sup>α</sup>cm <sup>þ</sup> <sup>α</sup>cP <sup>¼</sup> <sup>β</sup> cm � cP

c ∘ � cP

cP

cP

vw cP ¼ B cð Þ <sup>m</sup> � cP (11)

vw <sup>¼</sup> <sup>v</sup><sup>o</sup>

<sup>w</sup> ¼ LP ΔP are the pure water flux.

vo

From the solution diffusion model, the solute flux is written as

vo

The above equation can be simplified as.

<sup>3</sup> <sup>þ</sup> …… <sup>þ</sup> an½ � cmn � cPn (8)

Recent Drifts in pH-Sensitive Reverse Osmosis http://dx.doi.org/10.5772/intechopen.75897 15

<sup>w</sup> ½ � 1 � α ð Þ cm � cP (9)

(10)

(12)

(13)

This equation is well known as the film theory equation. In the above equation, k is the mass transfer coefficient in m/s, δ is the mass transfer boundary layer thickness in m, and cm, cp, and c are solute concentrations at the membrane-feed solution interface, in the permeate and in the bulk, generally expressed in kg/m3 , respectively. The mass transfer coefficient is estimated from the following equations depending on the channel geometry and flow regimes. In the rectangular channel, the mass transfer coefficient is estimated using the following Sherwood number relations. For laminar flow (Leveque's equation):

$$Sh = \frac{k \, d\_\epsilon}{D} = 1.85 \left( ReSc \, \frac{d\_\epsilon}{L} \right)^{\frac{1}{5}} \tag{4}$$

where, Sh,Re, and Sc are the numbers related to Sherwood, Reynolds, and Schmidt, respectively, de is the equivalent diameter in m, and L is the length of the membrane in m. For turbulent flow, Leveque's equation gives rise to (Dittus-Boelter equation):

$$Sh = 0.023 \text{(Re)}^{0.8} \text{(Sc)}^{0.33} \tag{5}$$

In the case of flow through the tube with diameter d in m, the mass transfer coefficient is estimated for laminar flow (Leveque's equation) (Gekas and Hallstrom 1987):

$$Sh = \frac{k \, d}{D} = 1.62 \left( ReSc \, \frac{d\_\epsilon}{L} \right)^{\frac{1}{2}} \tag{6}$$

In addition, for the turbulent flow, it is calculated from Eq. (5). Now, the transport equation in the flow channel, Eq. (4), must be coupled with the transport law through the porous membrane. It is expressed as Darcy's law:

$$
\sigma\_w = \mathcal{L}\_P \left(\Delta P - \Delta \pi\right) \tag{7}
$$

where Δπ is the osmotic pressure difference between the membrane sides that effectively are related to the quantity of matter, especially the concentration and inversely proportional to the molecular weight of solute; it is a linearly proportional to concentration in the case of a typical salt or lower molecular weight solutes. However, it deviates from linearity in the case of polymers, proteins, and higher molecular weight solutes. In Eq. (2) there are three unknowns, namely, vw, cp, and the finally predicted cm. The comprehensive correlation between the osmotic pressure and concentration, π ¼ ac, could be pragmatic equation for osmotic pressure difference at the ROM surface as

Recent Drifts in pH-Sensitive Reverse Osmosis http://dx.doi.org/10.5772/intechopen.75897 15

$$
\Delta \pi = \pi\_m - \pi\_P = a\_1[\mathfrak{c}\_m - \mathfrak{c}\_P] + a\_2\left[\mathfrak{c}\_m\right.\\
\left. \left. \left. \mathfrak{c}\_P \right. \right] + a\_3\left[\mathfrak{c}\_m\right. \left. \left. \left. \mathfrak{c}\_P \right] \right. \right] + \dots \\
\dots \\
+ a\_n[\mathfrak{c}\_m\mathfrak{n} - \mathfrak{c}\_P\mathfrak{n}] \qquad \text{(8)}
$$

where the constant coefficient is known as the difference between a1and a<sup>n</sup> and real retention could be defined by cm and cP indicated as the coefficients across the ROM phases, respectively, namely, the upstream and downstream phases. Therefore, Eq. (7) can be written in terms of the single parameter cm using Eq. (2), to reduce the system variables to cm and vw, instead of the existing three parameters. The new variables can be attained by solving Eqs. (4) and (6) using an iterative algorithm like the Newton-Raphson equations. This model is known as classic-film model or the osmotic pressure-controlling model.

#### 6.4. Solution diffusion model for RO/NF

The real retention is a partition coefficient, or really the solute flux across the membrane considered using the solution diffusion model described earlier; linear relationship is considered between π and c in the case of salt solution, π ¼ ac [42, 43, 48, 49]. In practice, Eq. (6) and the film theory equation, Eq. (4), are only considered. Therefore, the osmotic pressure model could be rewritten as

$$
\sigma\_w = \upsilon\_w^\circ \left[1 - \mathfrak{a} \left(\mathfrak{c}\_m - \mathfrak{c}\_P\right)\right] \tag{9}
$$

where

Integrating the above equation across the thickness of the mass transfer boundary layer, the

This equation is well known as the film theory equation. In the above equation, k is the mass transfer coefficient in m/s, δ is the mass transfer boundary layer thickness in m, and cm, cp, and c are solute concentrations at the membrane-feed solution interface, in the permeate and in the

from the following equations depending on the channel geometry and flow regimes. In the rectangular channel, the mass transfer coefficient is estimated using the following Sherwood

<sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>85</sup> ReSc de

where, Sh,Re, and Sc are the numbers related to Sherwood, Reynolds, and Schmidt, respectively, de is the equivalent diameter in m, and L is the length of the membrane in m. For

In the case of flow through the tube with diameter d in m, the mass transfer coefficient is

<sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>62</sup> ReSc de

In addition, for the turbulent flow, it is calculated from Eq. (5). Now, the transport equation in the flow channel, Eq. (4), must be coupled with the transport law through the porous mem-

where Δπ is the osmotic pressure difference between the membrane sides that effectively are related to the quantity of matter, especially the concentration and inversely proportional to the molecular weight of solute; it is a linearly proportional to concentration in the case of a typical salt or lower molecular weight solutes. However, it deviates from linearity in the case of polymers, proteins, and higher molecular weight solutes. In Eq. (2) there are three unknowns, namely, vw, cp, and the finally predicted cm. The comprehensive correlation between the osmotic pressure and concentration, π ¼ ac, could be pragmatic equation for osmotic pressure

Sh <sup>¼</sup> <sup>0</sup>:023 Re ð Þ<sup>0</sup>:<sup>8</sup>

<sup>¼</sup> <sup>k</sup> ln cm � cp c ∘ � cp 

L <sup>1</sup>

L <sup>1</sup>

3

vw ¼ L<sup>P</sup> ð Þ ΔP � Δπ (7)

3

, respectively. The mass transfer coefficient is estimated

ð Þ Sc <sup>0</sup>:<sup>33</sup> (5)

(3)

(4)

(6)

ln cm � cp c ∘ � cp 

governing equation of the flux is obtained as

14 Wastewater and Water Quality

bulk, generally expressed in kg/m3

brane. It is expressed as Darcy's law:

difference at the ROM surface as

vw <sup>¼</sup> <sup>D</sup> δ 

number relations. For laminar flow (Leveque's equation):

Sh <sup>¼</sup> k de

turbulent flow, Leveque's equation gives rise to (Dittus-Boelter equation):

estimated for laminar flow (Leveque's equation) (Gekas and Hallstrom 1987):

Sh <sup>¼</sup> k d

<sup>α</sup> <sup>¼</sup> <sup>a</sup> Δp and vo <sup>w</sup> ¼ LP ΔP are the pure water flux.

The above equation can be equated with the film theory equation and the following equation results:

$$v\_w^o \left[1 - \alpha (c\_m - c\_P)\right] = k \ln \left[\frac{c\_m - c\_P}{c\_\bullet - c\_P}\right] \tag{10}$$

From the solution diffusion model, the solute flux is written as

$$
\sigma\_w \ c\_P = B(\mathfrak{c}\_m - \mathfrak{c}\_P) \tag{11}
$$

where B is a constant. Combining Eqs. (8) and (10), the following equation is obtained:

$$
\sigma\_w^\rho \left[ 1 - \alpha (c\_m - c\_P) \right] = B \left[ \frac{c\_m - c\_P}{c\_P} \right] \tag{12}
$$

The above equation can be simplified as.

$$1 - \alpha c\_m + \alpha c\_P = \beta \left[\frac{c\_m - c\_P}{c\_P}\right] \tag{13}$$

where <sup>β</sup> <sup>¼</sup> <sup>B</sup> vo w From the above equation, the membrane surface concentration is obtained as

$$\mathcal{L}\_m = \mathcal{c}\_P \left[ 1 + \left( \frac{1}{\beta + \alpha c\_P} \right) \right] \tag{14}$$

cav <sup>¼</sup> cm � cP ln cm cP � �

> vw k

h i <sup>þ</sup> ð Þ <sup>1</sup> � <sup>σ</sup> ð Þ <sup>c</sup> <sup>∘</sup> � cP <sup>e</sup>

From the mentioned equations between (16) and (21) in cm, vw, and cP, we can iteratively obtain

The MSDM cons are described by the hypotheses that the mass transfer boundary layer is fully developed, whereas the corresponding entrance length required is substantial. Furthermore, physical properties such as diffusivity and viscosity considerably do not vary with concentration, while mass transfer coefficients are calculated from heat-mass transfer analogies applica-

On the other hand, the film theory-based osmotic pressure model presents a simple and quick method for quantifying system performance. In order to overcome these cons, the twodimensional mass transfer boundary layer equation can be solved, and/or detailed pore flow models can be incorporated. Many studies are available including these intricacies of the model.

In this approximation, the gels of concentrated solutes are deposited over the ROM surface with certain thickness in a uniformly fixed distribution of the solutes, and an outer mass transfer boundary film is formed [52, 55]. In that, the film theory, in which the solute concentration extends from feed concentration and gel concentration undergoing drastic variation in viscosity, diffusivity, and density, can be applied to obtain the equation of permeate flux as

vw <sup>¼</sup> <sup>k</sup> ln cg

The pH-sensitive characteristics of PAm-ZTS membrane were achieved upon static adsorption modes of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ using both Langmuir and Freundlich Iso-

The utilization of Langmuir and Freundlich isotherms to depict the complexation process of binding metal ions in the polymer has previously been investigated using the washing and

7. pH-responsive characteristics of PAm-ZTS membrane

therms, as well as the reverse osmosis dynamic mode.

7.1. Langmuir and Freundlich isotherms

co

2 4

ln 1 <sup>þ</sup> <sup>c</sup> <sup>∘</sup> �cP cP � � <sup>e</sup>

� �

vw k

3

Recent Drifts in pH-Sensitive Reverse Osmosis http://dx.doi.org/10.5772/intechopen.75897

� � (22)

5 (21)

17

vw k

By combining Eqs. (4) and (19), we get

a system prediction.

ble for impervious conduits.

6.7. Gel layer-controlling model (GLCM)

cP <sup>¼</sup> <sup>β</sup> vw

c ∘ � cp � � e

Putting cm from the above equation into Eq. (9), we get

$$\frac{\beta \upsilon\_w \circ}{\alpha c\_P + \beta} - k \ln \left[ \frac{c\_P}{(\alpha c\_P + \beta)(c \circ - c\_P)} \right] = 0 \tag{15}$$

Once more a trial-and-error formula for cP is tried using a standard iterative technique.

#### 6.5. Kedem-Katchalsky model [KKM]

KKM is considered as another alternate to osmotic pressure one, in which the imperfect retention of the solutes by the RO/NF/UF membranes is incorporated by a reflection coefficient in the equation of the final output flux [50, 51]:

$$
\sigma\_w = \mathcal{L}\_P(\Delta P - \sigma \Delta \pi) \tag{16}
$$

where σ is the reflection coefficient. Using π in the above equation gives rise to the following flux equality:

$$
\sigma\_w = \mathcal{L}\_P \left[ \Delta P - a \sigma (\mathcal{c}\_m - \mathcal{c}\_P) \right] \tag{17}
$$

Turning back to the film theory, the concentration on the ROM surface could be rewritten by

$$\mathfrak{c}\_{m} = \mathfrak{c}\_{P} + (\mathfrak{c}^{\rho} - \mathfrak{c}\_{P})\mathfrak{e}^{\frac{\mathfrak{c}\_{\overline{\nu}}}{k}} \tag{18}$$

Combining Eqs. (15) and (17), the following equation is obtained:

$$\upsilon\_{w} = \mathcal{L}\_{P} \left[ \Delta P - a \sigma \left( (c^{\rho} - c\_{P}) e \frac{\upsilon\_{w}}{k} \right) \right] \tag{19}$$

By means of Eq. (19), cP could be conveyed in terms of cmby using Eq. (10), followed by solving Eq. (18).

#### 6.6. Modified solution diffusion model [MSDM]

The solute transports across the RO/NF/UF membranes are given by adopting both the convective transport and the diffusive transport of the solutes across the voids of the membranes and writing the corresponding flux equation as [52–54]

$$
\upsilon\_w \ c\_P = B(\mathfrak{c}\_m - \mathfrak{c}\_P) + (1 - \sigma)\upsilon\_w \ c\_{av} \tag{20}
$$

where

$$c\_{av} = \frac{c\_m - c\_P}{\ln\left(\frac{c\_m}{c\_P}\right)}$$

By combining Eqs. (4) and (19), we get

From the above equation, the membrane surface concentration is obtained as

Putting cm from the above equation into Eq. (9), we get

6.5. Kedem-Katchalsky model [KKM]

flux equality:

16 Wastewater and Water Quality

Eq. (18).

where

in the equation of the final output flux [50, 51]:

βvw ∘

cm ¼ cP 1 þ

<sup>α</sup>cP <sup>þ</sup> <sup>β</sup> � <sup>k</sup> ln cP

Once more a trial-and-error formula for cP is tried using a standard iterative technique.

KKM is considered as another alternate to osmotic pressure one, in which the imperfect retention of the solutes by the RO/NF/UF membranes is incorporated by a reflection coefficient

where σ is the reflection coefficient. Using π in the above equation gives rise to the following

Turning back to the film theory, the concentration on the ROM surface could be rewritten by

<sup>o</sup> ð Þ � cP <sup>e</sup>

h i � �

By means of Eq. (19), cP could be conveyed in terms of cmby using Eq. (10), followed by solving

The solute transports across the RO/NF/UF membranes are given by adopting both the convective transport and the diffusive transport of the solutes across the voids of the membranes

<sup>o</sup> ð Þ � cP <sup>e</sup>

vw

vw k

vw cP ¼ B cð Þþ <sup>m</sup> � cP ð Þ 1 � σ vw cav (20)

cm ¼ cP þ c

vw ¼ L<sup>P</sup> ΔP � aσ c

Combining Eqs. (15) and (17), the following equation is obtained:

6.6. Modified solution diffusion model [MSDM]

and writing the corresponding flux equation as [52–54]

1 β þ αcP � � � �

<sup>α</sup>cP <sup>þ</sup> <sup>β</sup> � �ð Þ <sup>c</sup> <sup>∘</sup> � cP " # (14)

(19)

¼ 0 (15)

vw ¼ LPð Þ ΔP � σΔπ (16)

vw ¼ L<sup>P</sup> ½ � ΔP � aσð Þ cm � cP (17)

<sup>k</sup> (18)

$$c\_P = \frac{\beta}{\sigma\_w} \left[ \left( c \bullet - c\_p \right) e^{\frac{\nu\_w}{k}} \right] + \left[ \frac{\left( 1 - \sigma \right) \left( c \bullet - c\_P \right) e^{\frac{\nu\_w}{k}}}{\ln \left( 1 + \left( \frac{c \bullet - c\_P}{c^p} \right) e^{\frac{\nu\_w}{k}} \right)} \right] \tag{21}$$

From the mentioned equations between (16) and (21) in cm, vw, and cP, we can iteratively obtain a system prediction.

The MSDM cons are described by the hypotheses that the mass transfer boundary layer is fully developed, whereas the corresponding entrance length required is substantial. Furthermore, physical properties such as diffusivity and viscosity considerably do not vary with concentration, while mass transfer coefficients are calculated from heat-mass transfer analogies applicable for impervious conduits.

On the other hand, the film theory-based osmotic pressure model presents a simple and quick method for quantifying system performance. In order to overcome these cons, the twodimensional mass transfer boundary layer equation can be solved, and/or detailed pore flow models can be incorporated. Many studies are available including these intricacies of the model.

#### 6.7. Gel layer-controlling model (GLCM)

In this approximation, the gels of concentrated solutes are deposited over the ROM surface with certain thickness in a uniformly fixed distribution of the solutes, and an outer mass transfer boundary film is formed [52, 55]. In that, the film theory, in which the solute concentration extends from feed concentration and gel concentration undergoing drastic variation in viscosity, diffusivity, and density, can be applied to obtain the equation of permeate flux as

$$w\_w = k \ln \left(\frac{c\_\S}{c^\diamond}\right) \tag{22}$$

### 7. pH-responsive characteristics of PAm-ZTS membrane

The pH-sensitive characteristics of PAm-ZTS membrane were achieved upon static adsorption modes of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ using both Langmuir and Freundlich Isotherms, as well as the reverse osmosis dynamic mode.

#### 7.1. Langmuir and Freundlich isotherms

The utilization of Langmuir and Freundlich isotherms to depict the complexation process of binding metal ions in the polymer has previously been investigated using the washing and enrichment methods of the PEUF process [22, 24]. However, the different metal ions were subjected directly to reverse osmosis in the absence of any binding polymers. In this case, the assumption that the concentration of metal ions in the permeates, Cpi, symbolizes the concentration of metal that is free in the solution, Yi, is prepared.

The Langmuir isotherm equation is given by [20, 56, 57]

$$Q = \frac{Q\_{\text{max}} \, Y\_i}{K\_{\text{L}} + Y\_i} \tag{23}$$

Q ¼ KF Yi

is the metal free in solution (mg/l), and n is a constant. Freundlich equation gives a linear form

Figure 7 displays the linear regression fits of the Langmuir isotherm to the data obtained for particular metal ions in solution with PAm at pH 3 upon PAm-ZTS surface. The Langmuir isotherm fitted the test data very well (R2 values >0.98). Figure 8 exhibits the fits of the experimental data to the Freundlich isotherm at the same pH. Although this model fits the data intelligently well, the fit was not as good as the Langmuir model. This issue discloses that the Langmuir isotherm offers a better description of the binding of metal ions to PAm-ZTS than the Freundlich isotherm [63–66]. However, for all cases, the Qmax asset value was found in

These rates can be applicable when considering the retention of the metal ions during the RO

Figure 8. Freundlich isotherm model fits to the experimental data for binding of single metal ions at pH 3.

where Q is the amount of metal ion, KF is the Freundlich equilibrium constant (mg1�<sup>n</sup> g�<sup>1</sup>

[58–62]:

process.

the following order [20]:

Ce4+ > Pr3+ > Sm3+ > Gd3+ > Dy3+ > Ho3+

<sup>n</sup> (25)

Recent Drifts in pH-Sensitive Reverse Osmosis http://dx.doi.org/10.5772/intechopen.75897

lnQ ¼ nln Yi þ ln KF (26)

l n ), Yi 19

where: Q is the amount of metal ion, whereas Qmax is the maximum capacity of polymer (mg metal/g membrane).

Yi is the metal free in solution (mg/l).

KL is the Langmuir equilibrium constant (mg/l).

Langmuir equation gives a linear form:

$$\frac{1}{Q} = \frac{K\_{\rm L}}{Q\_{\rm max}} + \frac{1}{Y\_i} + \frac{1}{Q\_{\rm max}} \tag{24}$$

The Freundlich isotherm equation is given by

Figure 7. Langmuir isotherm model fits to the experimental data for binding of single metal ions at pH 3.

$$Q = \mathbb{K}\_F \text{ Y}\_i^n \tag{25}$$

where Q is the amount of metal ion, KF is the Freundlich equilibrium constant (mg1�<sup>n</sup> g�<sup>1</sup> l n ), Yi is the metal free in solution (mg/l), and n is a constant. Freundlich equation gives a linear form [58–62]:

$$
\ln Q = n \ln \, Y\_i + \ln K\_F \tag{26}
$$

Figure 7 displays the linear regression fits of the Langmuir isotherm to the data obtained for particular metal ions in solution with PAm at pH 3 upon PAm-ZTS surface. The Langmuir isotherm fitted the test data very well (R2 values >0.98). Figure 8 exhibits the fits of the experimental data to the Freundlich isotherm at the same pH. Although this model fits the data intelligently well, the fit was not as good as the Langmuir model. This issue discloses that the Langmuir isotherm offers a better description of the binding of metal ions to PAm-ZTS than the Freundlich isotherm [63–66]. However, for all cases, the Qmax asset value was found in the following order [20]:

$$\text{Ce}^{4+} > \text{Pr}^{3+} > \text{Sm}^{3+} > \text{Gd}^{3+} > \text{Dy}^{3+} > \text{Ho}^{3+} $$

enrichment methods of the PEUF process [22, 24]. However, the different metal ions were subjected directly to reverse osmosis in the absence of any binding polymers. In this case, the assumption that the concentration of metal ions in the permeates, Cpi, symbolizes the concen-

> <sup>Q</sup> <sup>¼</sup> QmaxYi K<sup>L</sup> þ Yi

where: Q is the amount of metal ion, whereas Qmax is the maximum capacity of polymer (mg

1 <sup>Q</sup> <sup>¼</sup> <sup>K</sup><sup>L</sup> Qmax þ 1 Yi þ 1 Qmax

Figure 7. Langmuir isotherm model fits to the experimental data for binding of single metal ions at pH 3.

(23)

(24)

tration of metal that is free in the solution, Yi, is prepared. The Langmuir isotherm equation is given by [20, 56, 57]

metal/g membrane).

18 Wastewater and Water Quality

Yi is the metal free in solution (mg/l).

Langmuir equation gives a linear form:

KL is the Langmuir equilibrium constant (mg/l).

The Freundlich isotherm equation is given by

These rates can be applicable when considering the retention of the metal ions during the RO process.

Figure 8. Freundlich isotherm model fits to the experimental data for binding of single metal ions at pH 3.

#### 7.2. Rejection of metal ions

Solute rejections of PAm-ZTS membrane under environmental pH values of 3 and 8 are performed to further evaluate the pH-responsive gating function of membrane. The feed solution is prepared by dissolving Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ in pH buffer with different concentrations mg/l, and the buffers of pH 3 or pH 8 is freshly prepared by adding HCl or NaOH in DI water. The experimental conditions of filtration tests are the same usually used for hydraulic permeability measurements at 0.1 MPa. All the Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ solutions are used as feed solution only within 48 h after preparation. In the filtration test, membranes are conditioned with buffers of pH 3 and pH 8 firstly [67, 68].

Then the permeability of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ solutions is monitored until the stabilization of membrane is reached and the filtrate is collected. Concentrations of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ ions in the filtrate solution are measured with Buck Scientific 210 VGP Atomic Absorption Spectrophotometer. The function of examination of the permeate samples for the relevant metal allows the calculation of the observed retention value (Ri) of each metal ion using [9, 42]

$$R\_i = \left(1 - \frac{\mathbb{C}\_{Pi}}{\mathbb{C}\_{\text{ft}}}\right) \times 100\tag{27}$$

Figures 9 and 10 show the rejection coefficient values of single metal ions for different feed metal concentrations using RO Mode at pH ~3 and ~8, respectively. Generally, a high rejection asset value of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ was observed at low concentrations. Increasing the metal ion content in the feed solution results in a marked decrease in the metal ion rejection, which may be attributed to the closed gates of the membrane. As pH-responsive membrane, PAm-ZTS showed differential rejections according to the pH conditioned. At pH 3, Gd3+ and Ce4+ were subjected to highest rejection, while Sm3+ and Ho3+ were rejected with the weakest rates; their rejection coefficients have lowered to less than ten percent at higher concentrations. On the other hand, Pr3+ showed the greatest rejection, while Sm3+ indicated the quietest rejection. At despicable concentrations, most of the ions are highly rejected at disgusting concentrations that reach about eighty percent. These asset values drop to about thirty to forty percent at higher concentrations. The variation of the rejection as a function of pH in pH-responsive membranes may be explained by the variation of PAm-ZTS conformation because of pHdependent dissociation of amide hydroxyl. In addition, protonation of amide groups

Recent Drifts in pH-Sensitive Reverse Osmosis http://dx.doi.org/10.5772/intechopen.75897 21

To verify the reversibility and durability of pH-responsive open and closed gating function of the membrane pores, the fluxes of membranes are tested with alternate change of buffer pH between 8 and 3, repeatedly. To characterize the pH-responsive performance, a

Figure 10. Rejection values of single metal ions at pH 8 for different feed metal concentrations using RO Mode.

under acidic conditions could be observed [68, 69].

special coefficient, called pH-responsive coefficient K, is defined as

where Cpi is the concentration of metal ion, i in the pass through, and Cfi is the concentration of metal ion, i in the primary feed solution.

Figure 9. Rejection values of single metal ions at pH 3 for different feed metal concentrations using RO Mode.

Figures 9 and 10 show the rejection coefficient values of single metal ions for different feed metal concentrations using RO Mode at pH ~3 and ~8, respectively. Generally, a high rejection asset value of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ was observed at low concentrations. Increasing the metal ion content in the feed solution results in a marked decrease in the metal ion rejection, which may be attributed to the closed gates of the membrane. As pH-responsive membrane, PAm-ZTS showed differential rejections according to the pH conditioned. At pH 3, Gd3+ and Ce4+ were subjected to highest rejection, while Sm3+ and Ho3+ were rejected with the weakest rates; their rejection coefficients have lowered to less than ten percent at higher concentrations. On the other hand, Pr3+ showed the greatest rejection, while Sm3+ indicated the quietest rejection. At despicable concentrations, most of the ions are highly rejected at disgusting concentrations that reach about eighty percent. These asset values drop to about thirty to forty percent at higher concentrations. The variation of the rejection as a function of pH in pH-responsive membranes may be explained by the variation of PAm-ZTS conformation because of pHdependent dissociation of amide hydroxyl. In addition, protonation of amide groups under acidic conditions could be observed [68, 69].

7.2. Rejection of metal ions

20 Wastewater and Water Quality

each metal ion using [9, 42]

metal ion, i in the primary feed solution.

Solute rejections of PAm-ZTS membrane under environmental pH values of 3 and 8 are performed to further evaluate the pH-responsive gating function of membrane. The feed solution is prepared by dissolving Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ in pH buffer with different concentrations mg/l, and the buffers of pH 3 or pH 8 is freshly prepared by adding HCl or NaOH in DI water. The experimental conditions of filtration tests are the same usually used for hydraulic permeability measurements at 0.1 MPa. All the Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ solutions are used as feed solution only within 48 h after preparation. In the filtration

Then the permeability of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ solutions is monitored until the stabilization of membrane is reached and the filtrate is collected. Concentrations of Ce4+, Pr3+, Sm3+, Gd3+, Dy3+, and Ho3+ ions in the filtrate solution are measured with Buck Scientific 210 VGP Atomic Absorption Spectrophotometer. The function of examination of the permeate samples for the relevant metal allows the calculation of the observed retention value (Ri) of

Ri <sup>¼</sup> <sup>1</sup> � CPi

Figure 9. Rejection values of single metal ions at pH 3 for different feed metal concentrations using RO Mode.

Cfi 

where Cpi is the concentration of metal ion, i in the pass through, and Cfi is the concentration of

� 100 (27)

test, membranes are conditioned with buffers of pH 3 and pH 8 firstly [67, 68].

To verify the reversibility and durability of pH-responsive open and closed gating function of the membrane pores, the fluxes of membranes are tested with alternate change of buffer pH between 8 and 3, repeatedly. To characterize the pH-responsive performance, a special coefficient, called pH-responsive coefficient K, is defined as

Figure 10. Rejection values of single metal ions at pH 8 for different feed metal concentrations using RO Mode.

$$K = \frac{\text{FLUX}\_{pH\_3}}{\text{FLUX}pH\_8} \tag{28}$$

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where the numerator and denominator represent the transmembrane fluxes at pH 3 and pH 8, respectively. The membrane showed fast response as a function of pH for its potential applications; the fluxes at pH 8 are around 375 l/(m2 h), while, with changing the feed to pH 3 buffer, the fluxes across the membrane decreased quickly to around 123 l/(m<sup>2</sup> h) within the first recording period, about 40 s. Therefore, the pH-responsive coefficient was about 0.328, indicating a good response of the membrane at the mentioned pHs. This value is mainly a fraction, which contradicts to others found in literature, as other membranes showed a reversed behavior at the same tested pHs [67].
