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Submerged-helical module design for pressure retarded osmosis:A conceptual studyusing computational fluid dynamics

Aschmoneit, Fynn Jerome; Hélix-Nielsen, Claus

Published in:Journal of Membrane Science

Link to article, DOI:10.1016/j.memsci.2020.118704

Publication date:2021

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Aschmoneit, F. J., & Hélix-Nielsen, C. (2021). Submerged-helical module design for pressure retardedosmosis:A conceptual study using computational fluid dynamics. Journal of Membrane Science, 620, [118704].https://doi.org/10.1016/j.memsci.2020.118704

Journal of Membrane Science 620 (2021) 118704

Available online 10 October 20200376-7388/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Submerged-helical module design for pressure retarded osmosis:A conceptual study using computational fluid dynamics

F.J. Aschmoneit , C. Helix-Nielsen *

Department of Environmental Engineering, Technical University of Denmark, Bygningstorvet, 2800, Kgs. Lyngby, Denmark

A R T I C L E I N F O

Keywords: Osmotic power Virtual prototyping Geometry optimization Computational fluid dynamics Renewable energy

A B S T R A C T

Recent research efforts in taking pressure retarded osmosis closer to commercialization have concentrated on the development of pressure retarded osmosis-specific membranes. The module design, on the other hand, has not attracted much attention, although the need for pressure retarded osmosis-specific module designs was identi-fied, when traditional desalination module designs were proven unfeasible for pressure retarded osmosis ap-plications. This work introduces a novel, pressure retarded osmosis-specific module design. The submerged- helical module is a low packing density, spacer-free design, which implicates a significantly lower pressure drop along the draw stream with a power density, similar to more densely packed module types. In this article a theoretical model for the performance of the submerged-helical module design is developed.

The model shows that the submerged-helical module may yield a significant net energy generation, under specific geometry configurations and operation conditions. Its results indicate that the design is advantageous to spiral-wound modules and plate-and-frame modules for pressure retarded osmosis, in terms of operation costs.

1. Introduction

Pressure retarded osmosis (PRO) is a process converting the Gibbs free energy in water streams with different salinities into hydraulic pressure: A semi-permeable membrane separates the low concentrated feed stream from the higher concentrated, and pressureized draw stream. The osmotic gradient across the membrane drives the perme-ation of water into the draw stream. This excess volume in the draw stream is led to a hydro-turbine converting the kinetic energy in the pressureized flow to electric energy [1].

The PRO process poses the foundation for alternative technologies for the generation of emission-free, green energy, with a small envi-ronmental impact. One desired application is the power generation from mixing fresh- and seawater in estuarine environments, which has a global power generation potential of 1600–2000 TWh/a [2,3]. As opposed to power generation with wind- or solar energy, exploitation of saline gradients from sea- and freshwaters allows a permanent, disruption-free, osmotic power generation. In 2009, the Norwegian power utility Statkraft put the first pilot PRO power plant in operation. Specially designed PRO membranes were utilized in traditional desali-nation membrane module designs. The project was discontinued, due to insufficient and uneconomical power generation. It was concluded that

a seawater PRO technology would need a power density of at least 5 W/m2, in order to be competitive with other forms of green energy [4]. The power density can be significantly improved by using draw solu-tions with a higher salinity than seawater. Examples for sources for these hypersaline draw solutions are reverse osmosis desalination concen-trates, hypersaline lakes, such as the dead sea, salt domes, geothermal water of industrial brines [5,6]. Although the feasibility of the PRO technology has been shown for hypersaline draw solutions, their commercialization is a rare exception, reducing hypersaline PRO to niche applications [5].

The reasons for the PRO inefficiency are known and can be divided in membrane-related performance limits and application inefficiencies. The membrane performance is a trade-off between water permeability, salt rejection and support material, which all affect the water perme-ation rate, while the support material also defines the mechanical resistance against the hydraulic pressure difference. Internal concen-tration polarization (ICP) was identified to severely reduce the mem-brane flux in osmotic pressure driven processes [7,8]. Research activities have therefore concentrated on the improvement of specially designed PRO membranes with a high water permeation and a thin porous sup-port material [2,9]. The application inefficiencies, on the other hand, are a trade-off between the external concentration polarization (ECP, the

* Corresponding author. Danmarks Tekniske Universitet Denmark. E-mail address: clhe@env.dtu.dk (C. Helix-Nielsen).

Contents lists available at ScienceDirect

Journal of Membrane Science

journal homepage: http://www.elsevier.com/locate/memsci

https://doi.org/10.1016/j.memsci.2020.118704 Received 30 April 2020; Received in revised form 30 August 2020; Accepted 31 August 2020

Journal of Membrane Science 620 (2021) 118704

2

draw stream dilution in the membrane vicinity), membrane fouling, and the operation costs. ECP and membrane fouling were identified to severely affect the water flux [7,10]. All these performance aspects are controlled through the applied membrane-embedding module type and the operation of it. It was found that traditional FO or RO module types, such as the spiral-wound module (SWM) and the plate-and-frame module (PL-FR) are not ideal for PRO, and that there is a need for PRO-specific module types or membrane topologies [11].

One concept for alternative membrane topologies of hollow fiber membranes enhances the water flux as a result of altered fiber designs.

In [12] it is shown how the helical twisting of a membrane micro-channels significantly increases the mixing characteristics inside the channels. For the designed gas-liquid contactor the authors report a gas uptake enhancement of up to 80%, compared to straight microchannels, concluding that the helically twisted micorchannels are an advanta-geous geometry design.

In [13], Culfaz et al. investigate how hollow fibers with a non-round, ‘microstructured’ cross section affect the fouling propensity in aerated bioreactors under various aeration conditions. It is shown that micro-structured hollow fibers reveal a potentially higher polarization resis-tance. Based on the same principle, Luelf et al. present a detailed fabrication method for various complex, microstructured topologies, including multi-channel, helically twisted hollow fibers. Associated flow simulations and magnetic resonance imaging analyses reveal secondary flow patterns in the cross flow. They conclude that the presented hollow fiber topologies presumably improve the mass transfer through reduced polarization effects [14]. Other concepts are concerned with the modi-fication of spacer geometries in draw channels. While spacer optimiza-tions are mostly based on net-shaped topologies, Fritzmann et al. investigate how flow-aligned, twisted spacer bands affect the water flux and fouling propensity in submerged, aerated filtration systems. They conclude that twisted spacers significantly increase the membrane flux, due to the locally deflected flow and resulting higher shear forces on the membranes [15].

The above referenced studies demonstrate that the twisting of fiber- structured membranes yield greater efficiencies, due to better mixing. In the presented study, it is investigated to what extend the geometry design feature of twisting membranes can be translated to flat sheet membranes.

The presented Submerged-Helical module (SHM) design is based on the idea to utilize flat sheet membranes themselves to mix the draw

solution and reduce the ECP and membrane fouling, which may be regarded as a combination of the two above mentioned concepts for flat sheet membranes: Similar to SWM, the membranes are designed as double layer sheets, enclosing tricot spacers, which act as feed channels. These sheets are referred to as membrane envelope sheets (MES), see Fig. 1, left. A cylindrical pressure chamber (PRC) holds several MES, which are twisted around their longitudinal axes, and which feed channels are led through the top- and bottom cylinder walls, see Fig. 1, right. All MES are arranged on a hexagonal array with a spacing of dm. The PRC has inlets and outlets on its top- and bottom sides and acts as the draw channel. As the draw solution is pushed through the PRC, it flows alongside the MES, while being deflected by the helical shape of the MES. This deflection yields a higher shear stress on the membrane surface, which reduces ECP. Contrary to other PRO module types is the low packing density, which implies great volume flow rates and low dilution of the draw stream. In order to limit this effect, a part of the draw stream is recirculated.

The objective of this work is to assess the feasibility of the SHM design for PRO power generation and therewith lay the foundation for subsequent design of experiments. The study is based on a computa-tional fluid dynamics (CFD) model which is executed on a set of repre-sentative operation- and geometry parameters. Analogous to Ref. [16] the generated simulation results are processed in terms of scaling law relations, revealing more insights of the underlying physics, and allowing to relate back to a wide range of operation- and geometry parameters.

This article first introduces the governing equations for fluid- and mass transport for the inwards of the module geometry and the mem-brane, respectively. Thereafter, the CFD model of the SHM design is developed and executed on a range of representative operation- and geometry parameters. A set of scaling laws is derived based on the re-sults from the CFD simulations, which allow for the performance pre-diction of the SHM design for various geometry and operation conditions. Finally, it is assessed in how far the SHM design compares with SWM and PL-FR modules for PRO with seawater and brines as draw solutions.

2. Theory

In order to model the microscopic water and solute transport within the membrane and the macroscopic water and solute transport through

Fig. 1. (a) Sketch of a membrane envelope sheet (MES): Two membranes in AL-DS configuration enclose a tricot spacer, which acts as the feed channel. The MES are submerged in the pressure chamber (PRC), which feed inlets and outlets are traversed through the PRC walls.

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

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the embedding module, it is common to isolate the membrane water and solute transport, model these separately in the ICP porous medium model, and embed this in the macroscopic fluid dynamics model. Therefore, instead of modeling membrane transport and module flow simulta-neously, this approach alternatingly models each part of these depen-dent processes until convergence. If the mass transfer characteristics of a module geometry are known, the ECP does not need to be resolved through the fluid dynamics model, but can be calculated through the ECP film theory model. This section outlines the fluid dynamics model, the ICP porous medium model and the ECP film theory model.

2.1. Fluid dynamics model

The fluid dynamics model defines the evolution of the flow variables velocity, u, dynamic pressure, p′ , and solute concentration, c, through the conservation laws for flow momentum (the Navier-Stokes equations) and mass (the continuity equation), and for the solute concentration (the convection-diffusion equation), respectively.

The model itself is defined through a set of assumptions, which set its governing equations. In the present case, the following assumptions are made on the water solution and the flow conditions:

Λ1 The fluid is incompressible and Newtonian. Λ2 The fluid density ρ is independent of the solute concentration. Λ3 The fluid viscosity μ is independent of the solute concentration. Λ4 External forces, such as gravity, are insignificant. Λ5 The solute diffusion coefficient D is independent of the solute

concentration. Assumptions Λ2 and Λ3 are limited to reasonably low solute con-

centrations and their applicability must be assessed with respect to the draw solute of interest. In the case of NaCl solutions with a mass fraction of 3% (relating to an osmotic pressure of ∼ 26 bar), the induced errors in the density and the viscosity are < 2% and ∼ 5%, respectively. Assumption Λ5 is limited to NaCl, which diffusion coefficient for water does not show significant changes with its concentration [17]. Through the above assumptions, the governing equations can be derived in the following:

The conservation of total mass is governed by the continuity equa-tion:

∂tρ+ ∂j(ρuj

)= 0 (1)

Under assumptions Λ1 and Λ2, the continuity equation simply states that the velocity field must be divergence-free:

∂juj = 0 (2)

The Navier-Stokes equation for the i-th velocity component is given by:

∂t(ρui)+ uj∂j(ρui)= − ∂ip′

+ ∂j[μ(∂jui + ∂iuj

)+ δi,jξ∂kuk

](3)

Under assumptions Λ1 − Λ4, and through employing equation (2), the Navier-Stokes equations are simplified significantly:

∂tui + uj∂jui = −1ρ∂ip

′

+ ν∂jjui (4)

The evolution of the solute concentration is governed by the convection-diffusion equation (5), which is reduced to equation (6) under assumptions Λ1 and Λ5.

∂tc= ∂j(D∂jc

)− ∂j

(c uj

)(5)

∂tc=D∂jjc − uj∂jc (6)

Assumptions Λ1 − Λ5 and the resultant equations (2), (4) and (6) define the fluid dynamics model.

2.2. ICP porous medium model

The water and solute transport through TFC membranes are gov-erned by the osmotic pressures in the membrane vicinity on both sides, the hydraulic pressure difference across the membrane and the mem-brane characteristics water permeability A, reverse solute coefficient B, and its structural parameter S. In order to simplify the water and solute transport within the membrane, and therewith make the model suitable for combining it with the fluid dynamics model, the flow and the solute concentration are modeled as flow through a porous medium, which yields a local analytical, transcendental function. Within the fluid dy-namics model, the whole TFC membrane can then be treated as an infi-nitely thin sheet, which permeates water and absorbs solute, based on the solute concentration in the membrane vicinity, and the above mentioned membrane characteristics.

Solute concentration polarization within and around the membrane affect the membrane water flux, see Fig. 2. The concentrative ECP, cs,f −

cf , and the dilutive ECP, cd − ca,d, are boundary layer effects, which are resolved in the fluid dynamics model. The ICP porous medium model de-termines the water flux jw and the reverse solute flux js, based on the adjacent concentrations cs,f and ca,d, and thereby models the ICP and the unknown ca,f . The model defines the local water flux jw and the local membrane draw osmotic pressure πa,d, which are implemented in the fluid dynamics model as boundary conditions.

This model is based on the following assumptions: λ1 The osmotic pressure is proportional to the solute concentration,

i.e. the van’t Hoff equation is valid: π = ξc, where ξ is the proportion-ality factor.

λ2 The dynamic pressure is negligible compared to the hydraulic pressure difference.

λ3 There’s no hydraulic pressure on the feed side. λ4 The feed stream is freshwater and feed side concentrative ECP is

therefore negligible: cs,f = cf = 0[18]. Assumption λ1 is valid for NaCl concentration of up to 1.5 M, cor-

responding to an osmotic pressure of ∼ 72 bar[19]. The dynamic pres-sure is of magnitude O (100) Pa, while the applied hydraulic pressure is at least of O (105) Pa, justifying assumption λ2. Similarly, the driving pressure in the feed channel is insignificant compared to the hydraulic pressure difference (assumption λ3).

The water flux across the active layer can be decomposed in an os-motic part and an hydraulic pressure part:

Fig. 2. Cross section of a TFC membrane is AL-DS configuration: The light gray area represents the porous support of width δS, the thin dark area is the active separation layer. The black curve illustrates a possible concentration profile within the membrane and its vicinity. The indices f ,dstand for feed and draw, and s, astand for support- and active layer. The water and solute fluxes are indicated by the arrows.

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

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jw = jπw − jp

w

= AΔπ − AΔp= A

(πa,d − πa,f

)− A

(pd − pf

)(7)

In PRO, the hydraulic flux component jpw is positive, but smaller than the osmotic membrane flux jπ

w, and therefore counteracting the osmotic membrane flux.

The reverse solute flux is defined as:

js = − BΔc= − B

(ca,d − ca,f

) (8)

In flat sheet membranes, the water flux and the reverse salt flux are constant throughout the porous support layer. The solute transport equation can therefore be written as:

js = c(x)jw − εD∂xc(x), (9)

where ε is the support porosity and D the solute diffusion coefficient. This inhomogeneous first order ordinary differential equation has a solution for the concentration profile of the form:

c(x)= αexp(

jwxεD

)

+ β (10)

Substituting the profile ansatz (10) in the transport equation (9) and evaluating it at the support feed side at x = 0 with c(0) = cs,f , yields the solution concentration profile within the support:

c(x)=(

cs,f −js

jw

)

exp(

jw

εDx)

+js

jw(11)

Evaluating the profile equation (11) at the support draw side, facing the active layer at x = τδS, with the support thickness δS, its tortuosity τ, and c(τδs) = ca,f , yields:

ca,f =

(

cs,f −js

jw

)

exp(

jwSD

)

+js

jw(12)

By substituting equation (8) in equation (12), and employing the van’t Hoff equation (assumption λ1), equation (12) can be rearranged [20]:

jw =A

⎧⎪⎪⎨

⎪⎪⎩

πa,d − πs,f exp(

jwSD

)

1 + Bjw

[

exp(

jwSD

)

− 1] − Δp

⎫⎪⎪⎬

⎪⎪⎭

(13)

Through assumptions λ2 and λ3, the local pressure difference is approximated through the constant hydraulic pressure on the draw side, pd. Assumptions λ4 and λ5 remove the dependence on the feed-side osmotic pressure. The resultant simplified water flux equation only de-pends on the membrane characteristics (A,B, S) and the draw-side os-motic- and hydraulic pressure, πa,d, pd:

jw =A

⎧⎪⎪⎨

⎪⎪⎩

πa,d

1 + Bjw

[

exp(

jwSD

)

− 1] − pd

⎫⎪⎪⎬

⎪⎪⎭

(14)

In the vicinity of the membrane, the solute transport is governed by the water flux jw and the diffusion against the concentration gradient:

js

(x → x+a,d

)= c(x)jw − D∂xc(x)

≈ c(x)jw − Dc(x + γ) − c(x)

γ

(15)

The discretization of the concentration derivative allows to solve for the membrane draw concentration, which is transformed to

ca,d =c(xa,d + γ

)+ js

γD

1 + jwγD

(16)

The reverse solute flux follows from equations (7) and (8), and assumption λ1 as:

js(xa,d

)= −

Bξ

(jw

A+Δp

)

(17)

Substituting equation (17) in (16), and employing λ2 and λ3, yields the osmotic pressure on the membrane draw-side:

πa,d =

π(xa,d + γ

)− γB

D

(jwA + pd

)

1 + jwγD

(18)

2.3. ECP film theory model

The film theory model assumes that the ECP is confined to a fully developed boundary layer of thickness δBL. This allows to define the draw concentration at the membrane surface and at a distance δBL from the membrane as:

c(δS) = ca,dc(δS + δBL) = cd

(19)

The solute flux Js and the water flux Jw are assumed constant in this boundary layer. The transport equation can therefore be written as:

Js = cJw − D∂xc (20)

This equation is solved analogous to the solute transport in the support layer in the ICP porous medium model, using the boundary con-ditions in equation (19), yielding an expression for the solute concen-tration on the membrane surface:

ca,d =

(

cd −Js

Jw

)

exp(

−JwδBL

D

)

+Js

Jw(21)

If the hydraulic pressure difference equals half the osmotic pressure difference, pd = πd/2, the flux ratio Js/Jw is defined through the mem-brane characteristics:

Js

Jw= −

2BξA

(22)

Implementing equation (21) into the water flux equation of the ICP porous medium model, equation (14), and substituting equation (22) and the mass transfer coefficient kd = D/δBL yields the water flux equation with modeled ECP, for pd = πd/2:

Jw =A

⎧⎪⎪⎨

⎪⎪⎩

(

πd +2BA

)

exp(

− Jwkd

)

− 2BA

1 + BJw

[

exp(

JwSD

)

− 1] −

πd

2

⎫⎪⎪⎬

⎪⎪⎭

(23)

The mass transfer coefficient kd is a function of the cross flow and the module geometry.

3. SHM design characterization

The SHM design is characterized in terms of the geometry’s inherent mass transfer characteristics and its pressure drop, in dependence of the cross flow and the draw osmotic pressure. Through knowledge about how the mass transfer coefficient changes under various geometry and operation conditions, equation (23) can be used to evaluate the SHM water flux under these conditions. The non-dimensional representation of the mass transfer is given by the Sherwood number Sh = kd dh/D, which scales for flows in ducts as:

Sh=φ0

(dh

l

)φ1

Reφ2 , (24)

in which the coefficients φi are characterizing the geometry and flow dependence in terms of the hydraulic diameter dh and the Reynolds

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

5

number Re. Similarly, the pressure drop is characterized in terms of the friction

number f = δp dh/(2ρ U2 l). The dimension-less friction number scales with the Reynolds number, and is therefore a function of the flow magnitude, the viscosity and density:

f = γ0Reγ1 (25)

Several series of CFD simulations were conducted on various ge-ometry and operation conditions, providing data against which the scaling laws were fitted, which yield the SHM characteristics in terms of φi and γi.

The CFD analysis was conducted with OpenFOAM v1806 [21] and the simulations were executed on DTU’s high performance computers. A sample case of the CFD simulations can be downloaded from GitHub [22].

The geometry of the CFD model is a reduced SHM design, taking advantage of the symmetry planes of the hexagonal array of individual MES. The MES surface is defined through the membrane width wm, its length lm, and the twisting rate ε ([ε] = revolutions/m). The MES are parametrized through:

ψ =(rcos(2πεz), rsin(2πεz), z) r∈[−

wm

2,wm

2

], z ∈ [0, lm] (26)

The reduced computational domain is one half of one hexagonal column, in which the presence of neighboring MES is modeled through the boundary conditions, see Fig. 3, left. The boundary conditions are illustrated in Fig. 3: The outlet (I) is at the domain top, opposite of the inlet (which is not seen in this perspective). The membrane (II) is modeled as an infinitely thin sheet. It extends from the inlet to the outlet as parametrized in equation (26). Periodic boundaries (III) account for the symmetry along the extended surface of the membrane surface, and symmetry planes (IV) mimic the effect of neighboring MES. At the inlet a uniform velocity U and a uniform osmotic pressure πd are prescribed. The membrane is defined as a flow inlet, with a non-uniform profile according to equation (14). Similarly, the osmotic pressure at the membrane is non-uniform and modeled through equation (18). The membrane parameters are A = 5.3 l/(m2 h bar), B = 2.0 l/(m2 h) and S = 600 μm, which are taken from Ref. [23], and which are in the range of modern commercial PRO membranes [8]. The hydraulic pressure pd

in equations (14) and (18) is uniform throughout the computational

domain and therefore decoupled of dynamic pressure variations p′ in the computational domain, see assumption λ2.

The computational domain was designed in SALOME [24] and sub-sequently used to project the computational mesh onto the domain surfaces through OpenFOAM’s blockMesh tool. For the considered operating conditions, Reynolds numbers of up to Re = O (105) were assumed, and therefore, turbulence effects must be expected. The tur-bulence energy dissipation was modeled through the k − ω SST model, which was initialized with k = 10− 6 m2/s2 and ω = 0.8 1/s.

In order to estimate the discretization error, a grid convergence index (GCI) analysis was conducted. It was identified that the solute concentration at the membrane surface is most sensitive with respect to the mesh resolution. Various mesh resolutions were tested and the relative change in the concentration field at the membrane surface was recorded. The GCI was conducted with the geometry parameters ε =

1 m− 1, dm = 0.2 m, wm = 0.15 m and lm = 1 m, and the operating con-ditions were set to U = 0.1 m/s and πd = 30 bar. The relative error with respect to a continuous solution (i.e. the GCI value) was estimated to 1.2 % for a discretization of 675 cells/m parallel to the membrane sur-face and 1350 cells/m in the membrane normal direction.

The CFD simulations were run on geometry configurations with varying twisting rates ε, membrane widths wm and module lengths lm, with a fixed membrane distance dm = 20 cm, see Table 1. The operation conditions varied in terms of the cross flow velocity, U ∈ [0.01,0.05] m/s, an osmotic pressure of πd ∈ {15,30}bar, and a draw hydraulic pressure that always equals half the osmotic pressure, pd = πd/2. All simulations were based on the previously described membrane characteristics.

Along the membrane surface, the local mass transfer coefficient was

Fig. 3. (a) Computational domain geometry sketch and boundary condition definition: (I) outlet; (II) membrane; (III) periodic boundaries; (IV) symmetry. (b) Local mass transfer on membrane surface and flow streamlines through SHM.

Table 1 Geometry and operation parameters for CFD simulations.

id ε[m− 1] wm/dm lm[m] πd[bar] dh[m]

c1 0 0.75 1 15 0.46 c2 0.5 0.25 2 30 1.38 c3 1 0.5 1 30 0.68 c4 1 0.75 1 15 0.45 c5 1 0.75 1 30 0.45

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

6

measured as the ratio of the concentration in the membrane normal direction and the concentration difference between the bulk region and the membrane surface:

kd

D=

∂nccd − ca,d

(27)

Fig. 3(b) shows the intensity of the local mass transfer according to equation (27).

The simulation results for the mass transfer are plotted in Fig. 4, in terms of the Sherwood number, against the Reynolds number Re =

Udh/ν. The data show a strong dependence on the Reynolds number Re, and on the twisting rate ε (see c1, c2, c3). The Sherwood number also correlates positively with the membrane width, although the effect is much smaller compared to the Reynolds number and the twisting rate (see c3, c4). The applied osmotic pressure does not affect the mass transfer (see c4, c5).

The pressure drop along the SHM is measured as δp. It is expressed in terms of the friction number through:

f =δp dh

2ρ U2 lm, (28)

The friction number decreases with greater Reynolds number, see Fig. 5. The twisting rate greatly affects the friction number (see c1, c2, c3), while the membrane width has no significant effect (see c3, c4), and the osmotic pressure has no effect (see c4, c5).

In order to incorporate the twisting rate ε in the SHM’s hydraulic diameter, the hydraulic diameter is defined through the void volume of one hexagonal column, Vhex = (

3

√/2)d2

mlm, and the surface area of one MES, Am:

dh =4Vhex

Am(29)

The membrane surface area is given by the integration of the local surface normal vector:

Am = 2∫ lm

0

∫ wm/2

− wm/2‖∂rψ × ∂zψ‖2drdz

= 2lm

∫ wm/2

− wm/2

(2πεr)2+ 1

√

dr

= lm

{

wm

(πεwm)2+ 1

√

+1πε ln

[πεwm +

(πεwm)2+ 1

√ ]}

,

(30)

where the factor 2 is due to the membrane double layer. The most significant geometry parameters for the SHM geometry are

the twisting rate ε and the membrane width wm. Fig. 3(b) indicates how

the mass transfer is greater along the outside edges of the twisted MES, which is affected by ε and wm. Both parameters influence the membrane surface area (see equation (30)), and therefore also the hydraulic diameter dh. But this dependency does not cover the extend to which these parameters affect the Sherwood number and the friction number. The power laws (24) and (25) are therefore modified to allow the scaling analysis with respect to these fundamental parameters. Through intro-ducing the normalized twisting rate ε′

= ε/(1 m− 1), the power laws (24) and (25) are extended to:

Sh=(φ0 +φ1ε′

)

(dh

lm

)φ2

Reφ3 (31)

f =(γ0 + γ1ε′

)Reγ2 (32)

Fitting the modified mass transfer power law (31) against the mass transfer data in Fig. 4, yields the SHM mass transfer characteristics as: φ0 = 2.39, φ1 = 3.41, φ2 = 0.18 and φ3 = 0.58. Similarly, the pressure drop power law (32) is fitted against the data in Fig. 5, which yields the SHM pressure drop characteristics as: γ0 = 1.38, γ1 = 1.66, and γ2 = −

0.43. The resultant best-fit curves are displayed in Figs. 4 and 5 as solid lines. It is seen that the derived power laws cover the parameter varia-tions very well. Only in the case of flat MES (c1), the cross flow effect on the Sherwood number and friction number is slightly less pronounced. The c1 data sets were therefore omitted in the power law fitting. Nevertheless, the c1 data sets illustrate qualitatively how flat MES relate to twisted MES.

4. Feasibility analysis

The scaling analysis allows for a more broad performance analysis of the SHM design, without the need for running CFD simulations on every configuration. The water flux in the SHM system is calculated through equation (23), in which the mass transfer coefficient kd is obtained from the power law (31). Similarly, the pressure drop is obtained from the friction factor power law (32) and equation (28). Fig. 6 shows the membrane power density Ψ = Jwpd against the cross flow velocity, for different design and operation parameters. The four configurations are all based on the membrane parameters, which are previously introduced for the CFD simulations. The configurations indicate how the twisting rate ε, the membrane width wm, and the draw osmotic pressure πd affect the power density. The system length was therefore fixed to 1 m and the sheet distance dm was set to 20 cm. Comparing c1 and c2 shows that an increased membrane width yields a higher power density. Similarly, by comparing c1 and c3, a greater power density due to an increased twisting rate is indicated. These two findings show how effectively the

Fig. 4. Mass transfer characteristics of SHM design: The CFD data series refer to Table 1. The data is fitted against the power law in equation (31).

Fig. 5. Pressure drop characteristics of SHM design: The CFD data series refer to Table 1. Note that the data series c4 and c5 are overlapping and appear as disks in the plot. The data is fitted against the power law in equation (32).

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

7

helical MES reduce ECP and therefore yield higher power densities. The configuration c4 is based on a higher osmotic pressure, which yields a higher power density than its equivalent c1. Configuration c4 is included in order to visualize the relative extend to which c2 and c3 improve the power density.

The above qualitative power density analysis solely concentrates on the power generation, disregarding the work done to drive the system, which is integrated in the following.

One design aspect of the SHM is its very low packing density, which implies that great volumes of draw solution have to be pre-treated, pressureized and driven through the system. Consequently, the power required for pressurizing and driving the draw stream is greater than the power output for all but very low cross flow velocities. Another two consequences of the low packing density are the low dilution of the draw stream as it passes through the system, and the low pressure drop. To counter-act the high energy demand of pressurizing the vast draw so-lution volume, a recirculation of the draw stream is implemented in the system.

Fig. 7 shows the schematics of an SHM-PRO system, including the bypass (B), which recirculates a fraction β of the inlet draw flow rate Qd. The SHM with draw stream recirculation has several effects on the system performance:

• The draw stream is diluted and the SHM is operated with an effective osmotic pressure, πeff

d < πd, which reduces the output power Pout =

δQ pd. • Only a fraction 1 − β of the draw stream must be pressureized

through the high pressure pump (Php), which reduces the operation power, compensating for possibly low efficiencies of the pressure exchanger (PX), ηpx.

• Due to the low pressure drop through the SHM, the work done by the booster pump, PB = Qdδp, is significantly smaller compared to dense module types.

In order to evaluate the effective osmotic pressure for the draw recirculation, an osmotic pressure balance equation is solved. The os-motic pressure flux behind the bypass is:

Qdπeffd = βQdπout

d + (1 − β)Qdπd, (33)

which is reduced to:

πeffd = βπout

d + (1 − β)πd, (34)

The osmotic pressure behind the PRO module, πoutd , depends on the

added volume flow δQ through the PRO process:

Fig. 6. Power density against cross flow velocity indicating how the membrane width and twisting rate affect the power density compared to an increase in os-motic pressure.

Fig. 7. SHM-PRO system schematics of draw stream with hydro-turbine (TRB), PRO module (SHM), booster pump (PB), bypass (B), high pressure pump (PHP) and pressure exchanger (PX). The state variables indicate the local flow rate, hydraulic pressure and osmotic pressure.

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

8

πoutd =

11 + δQ/Qd

πeffd (35)

The PRO volume flow δQ is a function of the water flux, which itself is a function of the effective osmotic pressure:

δQ=AmJw(πeff

d

)(36)

Therefore, the effective osmotic pressure must be evaluated numer-ically, by iterating through equations (34)–(36), using an initial effective osmotic pressure equal to the inflow osmotic pressure πd.

The SHM system performance is defined through the difference be-tween the power output and the power input, which both are a function of the recirculation fraction β. The following analysis presents, how the recirculation affects the SHM system performance, and how it relates to PRO systems utilizing SWM or PL-FR modules. For the sake of compa-rability all configurations are tested with a fixed total membrane area of 100 m2and a same inflow rate entering the system Qd

′

. This implies for the SHM configurations that β governs the recirculated draw stream flow rate Qd

′

/(1 − β) and not the inflow of the pure solute stream. Therefore, the state variables in Fig. 7 are substituted with Qd→Qd

′

/(1 − β) to relate to this comparative study. Through fixing the flow rate entering the system, the work done by the high pressure pump is similar for all configurations. The analysis is therefore independent of draw stream pre-treatment and pressure-exchanger inefficiencies, and concentrates solely on the system performance in terms of power output and work done by the booster pump:

P= δQ pd −Qd

′

1 − βδp (37)

The comparative analysis was conducted with osmotic pressures π ∈

{27,54}bar, relating to PRO applications with seawater and brines, respectively.

The SHM configuration in this analysis was using a reasonably small membrane spacing of dm = 15 cm with slim membranes of wm = 13 cm, allowing for a 1 cm glue line on either side of the MES. The twisting rate was set to ε = 5 m− 1 and the total module length was lm = 10 m. The SHM system contained 26 MES in order to reach the target membrane area of 100 m2.

The SWM configuration was adopted from a 8040 RO module with a 28 mil spacer. However, the SWM was virtually elongated to 2.44 m, in order to yield the target membrane area of 100 m2. Its mass transfer and pressure drop are defined through the power laws Sh =

1.9Sc1/3(Re dh/lm)1/2[25] and f = 22.9Re− 0.6. The PL-FR configuration assumed membrane sheets of 1.25 m width

and 4 m length, with a draw channel width of 1.5 mm, which is not spacer-filled. The module contained 10 sheets in order to reach the 100 m2membrane area. The mass transfer and the pressure drop were defined through Sh = 1.85(Sc Re dh/lm)1/3and f = 24/ Re, respectively [25].

All configurations were run with the same inflow rate Qd′

∈ [0,5]l/ s. This inflow range was chosen, because it relates to standard operation conditions of a single SWM with cross flow velocities in the range [0, 31]cm/s. The configurations all used the same membrane parameters: A =

5.3 l/(m2 h bar), B = 2.0 l/(m2 h) and S = 600 μm. The water flux and pressure drop were calculated through equations (23) and (28), in which the configurations’ respective Sherwood number and friction number scaling laws were employed. The hydraulic pressure on the draw side was always set to half the (effective-) osmotic pressure.

The SHM configurations were tested for recirculation fractions β ∈

{0.98, 0.97, 0.9, 0}. Their resultant cross flow velocities and effective osmotic pressures, with respect to the maximum inflow Qd,max

′ , are summarized in Table 2.

It can be seen that, under this operation mode, the recirculation fraction has a significant influence on the cross flow velocity (U

(1 − β)− 1). As a result of the recirculation-dependent cross flow velocity, the effective draw stream osmotic pressure is diluted significantly to at most 92% of the inflow osmotic pressure.

Figs. 8 and 9 show the absolute configuration performance according to equation (37) against the system inflow rate Qd

′ , for the inflow os-motic pressures πd = 27 bar and πd = 54 bar, respectively. All configu-rations exhibit concave performance curves, which is the result of the counter-acting sublinear increase of water flux, and the quadratic in-crease of pressure drop, with respect to the cross flow velocity. The performance is therefore always positive for small inflow rates and be-comes negative for larger inflow rates. In the case of seawater - fresh-water PRO (Fig. 8), the SHM performance is significantly influenced by the recirculation fraction β. The small variation of β in SHM1 and SHM2 has no significant influence on the effective osmotic pressure, but due to the substantial difference in cross flow velocity, the pressure drop in SHM1 is greater, which yields a lower peak performance in SHM1. Similarly, SHM3 exhibits a slightly higher peak performance than SHM2, for comparable osmotic pressure, but significantly lower cross flow ve-locity. In the case of no recirculation (SHM4), the system is operated at the inflow osmotic pressure πd and the cross flow velocity is only 1 cm/s, for Qd

′

= 5 l/s. At this cross flow, the pressure drop is insig-nificant. However, the SHM design feature of helical MES is not effective at such small cross flow velocities, and the performance is low compared to the peak performance of configurations with recirculation. The two more densely packed PL − FR and SWM modules show a lower peak performance, compared to the SHM configurations with recirculation (SHM1SHM2SHM3). For the SWM, the pressure drop dominates the performance, which is related to its embedded spacer.

In the brine - freshwater PRO application, with an inflow osmotic pressure of 54 bar, all configurations perform analogically to the seawater - freshwater PRO. Because the inflow rate Qd

′ is similar in both applications, the driving power is the same. But since the greater os-motic pressure yields greater membrane flux, the total performance is greater, compared to seawater PRO.

The SHM configurations in both applications indicate, that for a

Table 2 SHM configurations with different recirculation fractions and resulting cross flow and relative dilution with respect to maximum inflow Qd,max

′

.

config β Umax [cm /s] πeffd,min/27 bar πeff

d,min/54 bar

SHM1 0.98 49.9 0.93 0.92 SHM2 0.97 33.2 0.93 0.92 SHM3 0.9 10.0 0.94 0.93 SHM4 0.0 1.0 1.0 1.0

Fig. 8. Power gain in seawater-PRO against system inflow for SHM systems with various recirculation rates and SWM and PL-FR configurations.

F.J. Aschmoneit and C. Helix-Nielsen

Journal of Membrane Science 620 (2021) 118704

9

given SHM geometry, there exists an ideal operation mode in terms of β and Qd

′

, that maximizes the power generation. Note that the energy consumption does not include draw pre-treatment and pressurization. It is similar in all considered configurations and is directly proportional to the inflow rate Qd

′ .

5. Conclusions & outlook

A novel module design for PRO has been introduced and its perfor-mance analyzed. The SHM design concept is based on low packing density, spacer-free draw solute mixing through helical membrane sheets, and the recirculation of the draw stream. The performance of the SHM geometry and its operation has been quantified through a set of CFD simulations, which results have been the basis for the subsequent scaling-law analysis. Through the scaling-law analysis, simple power laws for the mass transfer and pressure drop have been derived, covering a wide parameter space. A feasibility analysis has been conducted in order to find favorable operation conditions and geometry parameters. Is was shown that the SHM design can reach the 5 W/m2 commercial threshold for a draw osmotic pressure of 30 bar and utilizing a commercially available membrane. In a comparative study of a 100 m2

PRO system with seawater and brine showed that the SHM power gain is very sensitive with respect to the recirculation rate. The study also suggests that the SHM design may yield a greater power gain than more densely packed SWM and PL-FR module types.

This article indicates that the SHM design may potentially be a very well performing module type for PRO. However, the presented study is based on a series of assumptions and their consequent models, which could not be experimentally validated. This article should therefore be regarded as a preliminary concept feasibility study.

Various challenges will have to be met, in order to develop this concept into a technology: A lab-scale set up must be built through which the fundamental design feature of twisted membranes is vali-dated, in terms of water flux increase and pressure drop decrease. Furthermore, it will be investigated how the membrane twisting affects the fouling propensity and the twisting-induced strain. Following this, the design will be optimized with respect to actual application scenarios, defining the mechanical construction, including the optimal number of MES and positioning of draw inlets and outlets. Finally, the effect of implementing an aeration system to the SHM design for limiting ECP and fouling should be considered.

CRediT authorship contribution statement

F.J. Aschmoneit: Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. C.

Helix-Nielsen: Formal analysis, Investigation, Resources, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the Innovation Fund Denmark, Innova-tionsfonden, under the MEMENTO project with grant number 4106- 00021B.

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Fig. 9. Power gain in brine-PRO against system inflow for SHM systems with various recirculation rates and SWM and PL-FR configurations.

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F.J. Aschmoneit and C. Helix-Nielsen