# Study of Pressure Retarded Osmosis Process in Hollow Fiber Membrane: Cylindrical Model for Description of Energy Production

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## Abstract

**:**

_{o}= 10,000 μm. At the end of this manuscript, the calculated mass transfer rates were compared to those measured. It was stated that the curvature effect in using a capillary membrane must not be left out of consideration when applying hollow fiber membrane modules due to their relatively low lumen radius. The presented model provides more precise prediction of the performance in the case of hollow fiber membranes.

## 1. Introduction

_{w}

_{,ro}r

_{o}= J

_{w}r, i.e., at r = r J

_{w}= J

_{w,r}

_{o}r

_{o}/r. The mass transfer rate was defined as J = −DdC/dr. Thus, the solute transfer rate was defined for every transport layer. Accordingly, the authors obtained a model with five unknown parameters, C

_{m}, C

_{s}, C

_{sp}, J

_{w}and J

_{s}, as well as five implicit equations with five known transport parameters, which were solved by “guessing a water flux and iterated until the anticipated water flux” corresponds to the equations given for the individual transport layers [21]. The question might be whether there is only one solution of a problem with five variables and when there is more, than which one is the real one. Cheng et al. [22] studied the forward osmosis-pressure retarded osmosis hybrid process using hollow fiber membrane modules. They described the mass transport using a cylindrical coordinate, as it was applied by Siversten et al. [21]. They have taken into account that the water flux continuously varies due to the membrane radius. In order to get the water flux by easy integration of the transfer rate equations, “a concept of linear water flux and linear reverse salt flux has to be defined”, namely ξ

_{w}= J

_{w}2π(r), and ξ

_{s}= J

_{s}2π(r) [9,22]. Then the known water flux expressions defined for flat-sheet were modified with ξ

_{w}(and ξ

_{s}). These modified expressions were used to calculate the water and solute fluxes, taking into account the cylindrical effect. After predicting the water flux, the structural parameter was determined without considering the effect of the external mass transfer resistances [22].

## 2. Materials and Methods

## 3. Theory

_{max}(r)] is as [24,25]:

_{o}):

_{o}is the lumen radius of the cylindrical membrane, m]). The obtained differential equation, with constant parameters, for the ith sub-layer (i = 1, …, N) is as:

_{i}and S

_{i}represent the constants of the general solution. Suitable internal and outer boundary conditions should determine their values. The internal boundary conditions, written for every single sub-layer will be as follows:

_{i}= 1 + iΔR; ΔR = δ

_{d}/(Nr

_{o}) for the liquid boundary layer on the draw side; ΔR = δ

_{s}/(Nr

_{o}) for the membrane support layer:

_{m}= 1 + NΔR)

^{o}and C

_{m}(see Figure 1), respectively) and the membrane porous support layer (its inlet and outlet concentrations are C

_{s}and C

_{f}, respectively). The effect of the fluid boundary layer of the feed side is generally not higher than 5% [24], thus its effect on the overall mass transfer resistance is neglected here for the sake of simplicity. Applying the internal (Equations (7) and (9)) and the external (Equations (10) and (11)) boundary conditions, one can get three N × N dimensional determinants (see e.g., [24]). These determinants are used to determine the T

_{i}and S

_{i}(i = 1 − N) values, applying the known determinant’s laws [23]. The solution of this special problem (not shown here in detail; several examples are shown in Nagy’s book [24] on it: see e.g., pp. 144–150, 177–183 or 279–282, where the same calculation methodology is used) is, e.g., for the boundary layer, as follows (δ

_{d}is the thickness of the boundary layer):

_{1}is as:

_{1}and S

_{1}(note the S

_{1}value can also be predicted by Equation (10) in the knowledge of the value of T

_{1}) can be calculated by applying Equations (14)–(18). Then, the inlet mass transfer rate of both the transport resistance can be calculated, on the basis of Equation (9), as shown below.

#### 3.1. Expressing the Mass Transfer Rates

_{1}and S

_{1}, the individual mass transfer rates and the overall mass transfer rate can easily be defined. Then the curvature effect on the overall inlet mass transfer rates and thus its effect on the membrane performance can be discussed. Next, the mass transfer rate expressions will be given and discussed.

#### The Inlet Mass Transfer Rates of the Single Mass Transport Layers

^{o}is the bulk concentration; C

_{m}is the concentration on the selective layer):

#### 3.2. The Overall Inlet Cylindrical Mass Transfer Rate

_{ov}. Equations (12)–(23) should calculate the boundary and the membrane support layer, taking into account that all parameters in these expressions, having subscript, are different in the two transport layers due to the fact that the change in the R

_{i}and ${\overline{R}}_{i}$ values is different. Accordingly, the three inlet mass transfer rates, i.e., those for the liquid boundary layer, of the active membrane layer and for the porous support layer, will be, respectively, as:

_{s}value in Equation (24) is multiplied by a factor of r

_{o}/(r

_{o}− δ

_{s}).

_{s}means the concentration of the surface of the selective and support layer i):

_{i}= 1 + iΔR; ΔR = δ

_{s}/(Nr

_{o}); ${\overline{R}}_{i}={R}_{i-1}+\Delta R/2$; ${\overline{R}}_{1}\simeq 1+\Delta R/2$; the thickness of the active layer can be negligible; C

_{f}is the concentration of the feed side):

_{sp}denotes the concentration on the surface of the support and feed boundary layer; when the resistance of the feed boundary layer is zero then C

_{sp}≅ C

_{f}):

_{m}will be as:

_{s}will be as, applying Equations (28) and (31):

_{w}values (Equation (31)).

## 4. Results and Discussion

#### 4.1. Effect of the Inlet Solute Concentration and the Draw Side Mass Transfer Coefficient as a Function of Radius

_{w}values, using the trial-and-error method, to the measured ones. The osmotic pressure is predicted in this study by OLI software (OLI Stream Analyzer 2.0 software [27], which gives the real osmotic pressure values as a function of the solute concentration [28].

#### 4.1.1. The Effect of Draw Side Salt Concentration

_{o}values on the water flux is seen. Its values are strongly increasing with the stronger curvature of the membrane, i.e., with the decrease in the membrane radius. For instance, the values of the water flux are 9.9 × 10

^{−6}m/s, at r

_{o}= 10

^{4}μm (or at flat-sheet membrane) and 16 × 10

^{−6}m/s, at r

_{o}= 100 μm, and C

_{d}= 4 M. This means 1.6-fold of increase on the “density” of performance. This change is rather remarkable and non-negligible, e.g., in the performance prediction of the scale-up of the industrial apparatus. On the other hand, the deviation in the water flux in the function of the membrane radius gradually decreases with the lowering of the radius. At C

_{d}= 0.6 M, this difference falls between 2.74 × 10

^{−6}m/s and 3.23 × 10

^{−6}m/s, which is not more than 18%. However, it should be noted that the change in the water flux as a function of the lumen radius also depends on membrane transport parameters, such as water and salt permeability, or structural parameter and the value of the thickness of the membrane support layer, δ

_{d}. These effects will be shown later. The water flux results obtained by the flat-sheet membrane, applying Equation (36), give excellent agreement with those obtained by the presented approached model at r

_{o}values of 10,000 μm. This is 1 cm, which corresponds to the radius of the tube membranes. Accordingly, the cylindrical effect might already be neglected at this scale.

_{d}= 3 M in the radius range investigated, its values are 11.8 W/m

^{2}and 8.05 W/m

^{2}, in the radius range of 100 and 10,000 μm. In studying the effect of the parameters of membrane properties, it can be seen that the effect of the radius on the produced energy density can strongly depend on them, as will be shown later in this paper.

#### 4.1.2. Effect of the Solute Transfer Coefficient, k_{d}, on the Power Density

_{d}values (note k

_{f}→ ∞). As can be seen, the draw side mass transfer coefficient values also have remarkable effect on the power density, which depends rather strongly on the lumen radius. The mass transfer coefficient is mainly determined by the hydrodynamic conditions of the fluid phase. In case of a moderately stirred draw phase or at the moderate fluid flow rate of about 25–35 cm/s, the mass transfer coefficient is measured to be 1.5–1.9 × 10

^{−5}m/s [5]. As can be seen, the value of power density is essentially lower at k

_{d}= 1.9 × 10

^{−5}m/s than, e.g., without mass transfer resistance in the draw fluid phase. On the other hand, the curvature of the curves in Figure 5 is similar to a function of the radius, at different values of k

_{d}. Deviations of the curves at, e.g., k

_{d}= 2.5 × 10

^{−5}m/s and 5 × 10

^{−5}m/s, are about 9% and 8%, at membrane radius of 100 μm and 10,000 μm, respectively. This means that the effect of the k

_{d}values hardly depends on the membrane lumen radius.

#### 4.2. Effect of the Membrane Properties, A, B and S

_{s}, the tortuosity, τ, and the porosity, ε, of the membrane support layer, namely S = δ

_{s}τ/ε. Models involve the S values and, accordingly, the thickness of the support layer should be measured separately. Knowledge of its value is important during our evaluation methodologies, considering the effect of the lumen radius.

#### 4.2.1. The Effect of the Membrane Water Permeability

^{−7}m/sbar, listed in Table 1 is a rather low value. The measured data are similar or 2–3 fold higher. Figure 6 illustrates the effect of the A value in rather a wide range, and it is varied between 1 × 10

^{−7}and 100 × 10

^{−7}m/sbar.

_{o}values. The increasing tendency of the water fluxes is similar to each other, at the different values of r

_{o}. The lifting gradient of the increase in curves starts to slow at higher permeability values, due to the rapid solution of the draw solution at higher values of A.

^{−7}and 1 × 10

^{−7}m/sbar, the increase in J

_{w}values is 344% (J

_{w}= 20.4 × 10

^{−6}m/s and 8.92 × 10

^{−6}m/s) and 14%, respectively, related to the J

_{w}data for r

_{o}= 100 μm and those for flat-sheet membrane, respectively. This behavior proves that the cylindrical membrane’s water flux can be remarkably affected by the cylindrical space, depending on the lumen radius. This change is illustrated more clearly by Figure 7, which gives the normalized water flux (J/J

_{∞}) vs. lumen radius at different water permeability values. As can be seen, the main change occurs below r

_{o}< 500 μm. This is in harmony with data obtained at different values of draw side solute concentration (Figure 2). The simple cause of it is that the space variation gradually increases with the decrease in the radius. It reaches its highest value at the lowest radius, here at r

_{o}= 100 μm. Capillary membrane modules with lesser radius (lesser than 100 μm) cannot probably be produced.

_{o}= 100 μm and the flat-sheet membrane at ΔP = 10 bar. Here it can also clearly be seen that the increasing tendency of the power density is rather prominent, especially in a water permeability range of (3–30) × 10

^{−7}m/sbar. The water flux and, accordingly, the power density gradually increases with the decrease in the lumen radius. Thus, it is obvious that the lowest possible radius can serve the maximum value of the membrane performance, related to the membrane surface. Perhaps it is worth mentioning that the increase in the power density reaches close to 200% at a hydraulic pressure difference of ΔP = 10 bar, and at A = 50 × 10

^{−7}m/sbar, considering radius values of r

_{o}= 100 μm and 10,000 μm or for the flat-sheet membrane (values of power density obtained were 8.53 W/m

^{2}and 4.4 W/m

^{2}, respectively). Note that the total surface of a capillary module with a lower radius is obviously lower than a capillary with a larger radius. Here, the performance values are compared to the surface of a capillary module. It is thought that the water flux values and the power density are characteristic parameters for the judgment of the membrane performance.

#### 4.2.2. Effect of the Salt Permeability, B

^{−7}and 100 × 10

^{−7}m/s. Taking into account the expression of Yip and Elimelech [30], namely that B = ξA

^{3}(ξ is a constant), the value of B can vary in a wide range depending on A values in a polymeric membrane. According to this figure, the water flux decreases relatively remarkably when B > 4 × 10

^{−7}m/s, but one should keep in mind the values of other parameters as they are given in Table 1.

#### 4.2.3. Effect of the Thickness of the Porous Support Layer, δ_{s}

_{s}τ/ε. It is generally accepted that the value of τ/ε can vary between 2 and 7. In this study, it would be important to know the real thickness of the porous support layer. Due to the special investigation of this process in the literature, this value is not given in the published papers. According to our measurements, it varies between about 100–300 μm. In this study, the S value was chosen to be 500 μm, while the thickness of the membrane porous layer is 100 μm, in most of our simulation data. Here, in this subsection, δ

_{s}was changed between 100 μm and 500 μm, while the value of S was kept at 500 μm. Accordingly, the value of τ/ε varied between 1 and 5.

_{s}on the water flux is perhaps moderate (at r

_{o}= 100 μm, the increase in water flux is not more than 9%). As it has been experienced, at r

_{o}= 10,000 μm (or obtained by a flat-sheet membrane), the effect of the support layer curvature does not practically affect the value of the water flow rate (broken line). This means that, decisively, the lumen radius, and, accordingly, the inlet mass transfer rate, affects the overall solute transfer rate and the water flux.

#### 4.3. Comparison of the Measured and Predicted Data

#### 4.3.1. Validation of the Cylindrical Model by Measured Data

_{d}= 0.5 M.

_{o}= 940 μm and δ

_{s}= 150 μm, while Equation (37) was applied for the flat-sheet operation mode. The measured values (■) and their evaluation by the flat-sheet model (Equation (37)) in case of k

_{d}= k

_{f}→∞ are plotted by a broken line. The predicted data fit well with the measured one. On the other hand, the application of the developed cylindrical model gives much higher values than the measured ones at k

_{d}= k

_{f}→ ∞ (dotted line), which is more significant than that in Figure 3, in the lumen radius, of r

_{o}= 940 μm. Might it be the consequence of the neglect of the mass transfer resistance of the draw side? The question arises whether the draw side boundary layer

^{’}s external mass transfer resistance causes this large deviation. For illustration, the effect of k

_{d}, the water flux as a function of draw concentration, is plotted in this figure, under different values of external mass transfer coefficients. The effect of the draw concentration on the water flux is obtained by the cylindrical model (by Equation (31)) at r

_{o}= 940 μm and δ

_{s}= 150 μm. Water flux, predicted by the model given by Equation (31) with three different values of k

_{d}(k

_{d}= 1.0, 1.5, 4.0 and →∞; k

_{f}was assumed to be infinite in every case), can be strongly affected by the values of k

_{d}, its decrease gradually lowers the deviation of the predicted data from those of the measured ones.

_{d}value can be neglected at Re = 1500 (Re = v2r

_{o}/D) as it was done by the authors in Ref. [18]. Accordingly, the volume flow rate is 0. 5 cm/s (5 × 10

^{−3}m/s), which is a rather low value. Bui et al. [5] measured the values of the external mass transfer coefficients, k

_{d}, and k

_{f}and they have obtained that k

_{d}= 1.74–1.84 × 10

^{−5}m/s and 2.0–2.1 × 10

^{−5}m/s for 0–1.5 M NaCl with cross-flow velocities between 0.21–0.31 m/s. Manickam and McCutcheon [10] reported similar results, namely k

_{d}= 2.0 × 10

^{−5}m/s at 0.26 m/s cross-flow velocity. In this case, however, the volume velocity in the capillary is much lower. However, the convective velocity of 0.5 cm/s cannot provide remarkable dispersion; thus, the streaming can remain laminar in that the transfer resistance can strongly alter the transfer rate of the solute and water (if Re < 2100–2300 then the flow in the capillary is laminar). Applying the Graetz–Léveque expression for the determination of the Sherwood number, i.e., Sh = (ReScd/L)

^{0.33}[24], the mass transfer coefficient of the draw side will be as k

_{d}≅ 1.2 × 10

^{−5}m/s. The water flux for this mass transfer coefficient is also plotted in Figure 12 (dot-dot-broken line). In our opinion, these predicted results are close to the measured one, proving the applicability of the cylindrical model presented. According to these results, it is needed to compare the results of the presented model with other measured data.

#### 4.3.2. Model Validation by Measured Data of Wan and Chung

_{d}= k

_{f}→ ∞ is plotted by a dashed line (C

_{d}= 1 M NaCl; C

_{f}= 0). The values of A (A = 9.7 × 10

^{−7}m/cbar = 3.5 Lmh) and B (B = 0.78 m/s − 0.97 × 10

^{−7}m/s = 0.28 Lmh − 0.35 Lmh as a function of the pressure difference) were measured by the authors. The value of the structural parameter was obtained by fitting the model data to the measured ones. Thus, it was obtained that S = 480 µm. As it can be seen, the measured and the predicted data have an excellent agreement. The effect of the hydraulic pressure difference on the water flux, predicted by using the cylindrical model (by Equation (31)) at r

_{o}= 288 μm and δ

_{s}= 225 μm is plotted by a solid line.

## 5. Conclusions

_{s}membrane thickness) and operating conditions (C

_{d}, k

_{d}). It was stated that the inlet solute concentration strongly influences the water flux, depending on the lumen radius. In the lumen radius range of 150–1000 μm, this change can reach 200–300%, depending on the draw concentration. Similarly, the water permeability also strongly affects the water flux, depending on the lumen radius. The change in the power density can reach 4–6 fold or more with the increase in the water permeability and/or with a decrease in the lumen radius. According to Figure 13, the power density obtained by using a cylindrical membrane is 66% more than that by using a model for a flat-sheet membrane applying the same values of the transport parameters. The major statements in this paper, according to the results are as follows:

- the mass transfer rate (solute and water) through the cylindrical membrane essentially differs from that of the flat-sheet membrane;
- the deviation strongly depends on the membrane radius between r
_{o}= 150–1000 µm, depending on the values of the transport parameters (A, B, S, k_{d}); - with the decrease in the membrane radius, the cylindrical effect increases and the increase can reach 4–6 fold;
- the developed cylindrical mathematical model serves as a simple iterative calculation method for the prediction of the cylindrical membrane performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | water permeability, m/sbar |

B | solute permeability, m/s |

C | solute concentration, kmol/m^{3} |

D | solute diffusion coefficient, m^{2}/s |

J | water flux, m/s |

N | number of sublayers |

P | pressure, bar |

r | membrane radius, m |

r_{o} | lumen radius, m |

R | dimensionless radius, m, μm |

${\overline{\mathit{R}}}_{i}$ | average radius inside the sublayers |

v | convective (volumetric) flow, m/s, m^{3}/m^{2} s |

Greek | |

δ | thickness of transport layers, m, μm |

ξ | parameter, |

π | osmotic pressure, bar |

Subscript | |

d | draw solution |

f | feed |

m | draw side active layer |

o | surface at r = r_{o} |

ov | overall |

s | support layer or solute |

w | water |

Upper script | |

o | bulk phase |

## Appendix A

_{i+}

_{1}and S

_{i}

_{31}for cases when i > 1, taking into account Equations (6) and (9), one can give for the ith internal interface the following two boundary conditions [24]:

_{i}and S

_{i}are known from the calculation in the previous step, thus introducing values of κ

_{i}and ψ

_{i}as they are given by Equations (A3) and (A4):

_{i+}

_{1}and S

_{i+}

_{1}will be after some manipulations of the above expressions, as:

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**Figure 1.**Schematic illustration of the three mass transport resistances (draw side fluid boundary layer, membrane dense active and porous support layers) of the cylindrical membrane taken into account in the prediction of the overall solute and water transport rates in the prediction of the mass transport properties of the capillary membrane. It contains the important notations (C

^{o}, C

_{m}, C

_{s}, C

_{f}, δ

_{d}, δ

_{s}, r

_{o}, r

_{m}), as well.

**Figure 2.**Water flux as a function of the inlet draw salt concentration applying the parameters listed in Table 1, with exception of the C

_{d}values. (S = 5 × 10

^{−4}m; δ

_{s}= 100 μm, thus τ/ε = 5; k

_{d}= 1.9 × 10

^{−5}m/s, thus δ

_{d}= 80 μm; k

_{f}→ ∞).

**Figure 3.**Normalized water flux as a function of the membrane’s lumen radius, which are related to the values obtained for flat-sheet membrane, J

_{∞}(parameters as they are listed in Table 1).

**Figure 4.**The power density, at ΔP = 10 bar, as a function of the lumen radius, at different values of the inlet, draw phase concentration (other parameter values as they are listed in Table 1).

**Figure 5.**Power density (at ΔP = 10 bar) as a function of the lumen radius of the capillary module, at different values of the draw side mass transfer coefficient, k

_{d}, applying seawater–river water pair (C

_{d}= 0.6 M; C

_{f}= 0.015 M; S = 500 μm, and δs = 100 μm; other parameters as listed in Table 1; the lines represent the calculated data, ■ points are calculated data for flat-sheet membrane).

**Figure 6.**Water flux as a function of the water permeability, at different values of the membrane lumen radius, varied between 100 μm and 10,000 μm. Other parameter values are as they are listed in Table 1.

**Figure 7.**Normalized water flux (J

_{o}is the water flux obtained for flat-sheet membrane, by Equation (36)) as a function of the lumen radius. Other parameters’ values are as written in Table 1).

**Figure 8.**Power density as a function of water permeability, at different values of the lumen radius, varied between 100 μm and 10,000 μm. (parameter values as they are given in Table 1).

**Figure 9.**Water flux as a function of the solute permeability at r

_{o}= 100 μm and by flat-sheet membrane (broken line), without hydraulic pressure difference (for other parameter values see Table 1).

**Figure 10.**Water flux as a function of the thickness of the membrane support layer, at different values of r

_{o}, with the value of S = 500 μm (ΔP = 0) (other parameters are listed in Table 1).

**Figure 11.**The measured [18] and the predicted water flux as a function of the draw concentration; the osmotic pressure was predicted by both the van’t Hoff equation and the OLI software [27] to show the difference between their applications; Equation (37) was used in both cases. (S = 600 μm; A = 6.9 × 10

^{−7}m/sbar; B = 0.56 × 10

^{−7}m/s, D = 1.61 × 10

^{−9}m

^{2}/s; δ

_{s}= 150 μm; k

_{d}= k

_{f}→∞; ΔP = 0; value of δ

_{s}was determined on the basis of a SEM image; Figure 2c in Ref. [18]; 1 × 10

^{−6}m/s = 3.6 L/m

^{2}h).

**Figure 12.**Water fluxes as a function of the draw concentration applying the cylindrical model developed (solid lines), at different values of draw side mass transfer coefficient. For comparison, the effect of the concentration, using the flat-sheet model (broken lines), is also given with zero mass transfer resistance, and measured points (■) are also plotted, as given in Figure 11. (S = 1370 μm; r

_{o}= 940 μm; its value was determined by means of SEM image in Figure 2c in Ref. [18]; A = 2.6 × 10

^{−7}m/sbar; B = 0.56 × 10

^{−7}m/s, D = 1.61 × 10

^{−9}m

^{2}/s; δ

_{s}= 150 μm; k

_{f}→ ∞; ΔP = 0; k

_{d}→ ∞ for flat-sheet; k

_{d}with variable values are given in the figure, by the cylindrical model).

**Figure 13.**Water fluxes as a function of the pressure difference (∆P) for comparison of the measured (■) [33] data to those obtained by theoretical evaluations, applying the flat-sheet (dashed line) as well as cylindrical model presented (solid line) (polyethersulfone–polyethylene hollow fiber membrane; fiber id.: 575 µm; fiber od.: 1025 µm; membrane thickness, δ

_{s}= 225 µm; fiber length: 15 cm; number of fibers: 3; packing density: 2.5%).

Parameters | Values |
---|---|

Feed solute concentration | 0.015 M |

Solute concentration in the draw solution | 0.6 M, or varies |

Membrane transport parameters | A = 1.9 × 10^{−7} m/sbar or variesB = 5.02 × 10 ^{−7} m/s or variesS = 5 × 10 ^{−4} m (δ_{s} = 100 μm, τ/ε = 5. or varies) |

External mass transfer coefficients in draw side, | 1.9 × 10^{−5} m/s, thus δ_{d} = 80 μm |

External mass transfer coefficient in feed side | → ∞ |

Diffusion coefficient at high salinity solution | 1.5 × 10^{−9} m^{2}/s |

Diffusion coefficient at low salinity solution | 1.5 × 10^{−9} m^{2}/s |

Hydraulic pressure difference | 0, 10 bar or others |

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**MDPI and ACS Style**

Nagy, E.; Ibrar, I.; Braytee, A.; Iván, B.
Study of Pressure Retarded Osmosis Process in Hollow Fiber Membrane: Cylindrical Model for Description of Energy Production. *Energies* **2022**, *15*, 3558.
https://doi.org/10.3390/en15103558

**AMA Style**

Nagy E, Ibrar I, Braytee A, Iván B.
Study of Pressure Retarded Osmosis Process in Hollow Fiber Membrane: Cylindrical Model for Description of Energy Production. *Energies*. 2022; 15(10):3558.
https://doi.org/10.3390/en15103558

**Chicago/Turabian Style**

Nagy, Endre, Ibrar Ibrar, Ali Braytee, and Béla Iván.
2022. "Study of Pressure Retarded Osmosis Process in Hollow Fiber Membrane: Cylindrical Model for Description of Energy Production" *Energies* 15, no. 10: 3558.
https://doi.org/10.3390/en15103558