Energy production from the controlled mixing of seawater and fresh water has the advantage that it can be operated continuously unlike processes based on solar and wind energy. The main process in this area and the process that motivated this work is Pressure-Retarded Osmosis (PRO). The pioneers of reverse osmosis recognised the conceptual simplicity of PRO [1] but concern about costs has always been a problem [2,3,4]. The first pilot plant seeking to develop osmotic energy was opened by the Norwegian energy company Statkraft in 2009 but the pilot plant was closed in 2014 due to the low power density (power per unit of membrane area) exhibited by the prototypes and the questionable economic feasibility of the process [5]. Today it is generally accepted that if PRO is to be commercially viable then it will be necessary to use resources with a higher salinity than seawater, for example brine from a reverse osmosis desalination plant [6, 7], but the power output will still be modest. However in a recent purely theoretical paper a thought provoking theoretical analysis by Yaroshchuk [8] suggested that under some conditions *and* with a membrane displaying ‘leakiness’, a ‘breakthrough’ mode might occur even to the extent that with the appropriate membranes reverse solute diffusion would be eliminated. Indeed there was the prediction that with these appropriate membranes there would be co-current flow of both solvent and solute against a concentration gradient.

Previous exploration of the implications of the effect of minor deviations from ideal semi-permeability using the Spiegler-Kedem (S-K) model have confined themselves to a single value of the reflection coefficient [8]. In this article a range of values are used. As noted elsewhere [9], whilst irreversible thermodynamic arguments were used to derive the solute and solvent transport equations of the S-K model, the membrane itself was treated as a black box". The novel variants of the S-K model introduced here in the context of forward osmosis are not fundamentally based but one form does avoid the incongruity (indeed thermodynamically inappropriateness) arising from the other formulations of the S-K model.

It has been suggested that the total solute flux can change sign, and that the “dramatic change in the behaviour is ultimately caused by the change in the direction of solute flow through the membrane” [8]. This specific possibility has been refuted elsewhere where it was also shown (contrary to statements in [8]) that the use of thick support layers is undesirable.

### Theory

Following the S-K equations given in [8], these can be simplified for FO systems by to omitting the pressure term because there should be no overall pressure drop across the barrier layer. The resulting equations are:

$$ {J}_v=\chi \sigma .\beta RT\frac{dc}{dx} $$

(1)

$$ {J}_s=-\omega \frac{dc}{dx}+\left(1-\sigma \right)c{J}_v $$

(2)

where *c* is the reference (virtual) solute concentration (as given in [10] and used in [8]), *ω* is the solute permeability, *σ* is the solute reflection coefficient, *J*_{s} is the solute flux, *J*_{v} is the solvent flux and *χ* is the hydraulic permeability. *β* is the van’t Hoff factor, *R* the universal gas constant and *T* the absolute temperature of the system. The implicit assumption in adopting this approach will be discussed later.

Now using Eq. 1 to define \( \frac{dc}{dx}/{J}_v \) and substituting the result into Eq.2 it is found that the concentration in the solute flux equation cannot be taken to be a variable because the other terms are all invariant. Consequentially all terms in (3) are fixed.

$$ \frac{J_s}{J_v}=\frac{-\omega }{\chi \sigma \beta RT}+\left(1-\sigma \right)c $$

(3)

In [8] the concentration *c* in Eq.2 was a variable but the final term in Eq. 3 would be better written as: (1 − *σ*)*c*_{i} where *c*_{i} is the concentration of solute at the boundary between the support layer and the barrier layer. Consequently a sounder alternative for the barrier layer would have been:

$$ \frac{B}{A\sigma \beta RT}-\left(1-\sigma \right){c}_i=\left(-\frac{J_s}{J_v}\right) $$

(4)

where *A* and *B* are the standard ‘A’ parameter and ‘B’ parameter for water flux and salt flux respectively.

The consequences that follow from the adoption of Eq. 4 are explored below. The layout of the system is illustrated in Fig. 1. For the study of a potential breakthrough mode, this orientation is the one of interest [8]. The arrows show the direction of the actual fluxes but mathematically they are taken as positive in the positive x-direction which co-insides with the water flux, *J*_{v}. Hence the set of equations for the support, the barrier layer and the draw side boundary layer are related to each other as follows:

$$ \frac{c_i\exp \left(-{Pe}_s\right)-{c}_f}{1-\exp \left(-{Pe}_s\right)}=\frac{B}{A\sigma \beta RT}-\left(1-\sigma \right){c}_i=\frac{c_d-{c}_m\exp \left({Pe}_{bl}\right)}{\exp \left({Pe}_{bl}\right)-1}=\left(-\frac{J_s}{J_v}\right) $$

(5)

where *Pe*_{s} is the Peclet number for the support layer, *Pe*_{bl} is the Peclet number for the draw-side layer and *c*_{m} is the concentration at the interface between the membrane and the draw solution. *c*_{d} is the bulk concentration on the draw side. The external concentration on the feed-side (which is at the low concentration *c*_{f}) is insignificant compared with the internal concentration polarisation within the support and a separate term for the mass transfer coefficient on the feed-side has not been included.

As components of Eq. 5 are referred to individually below they are listed out as:

$$ \frac{c_i\exp \left(-{Pe}_s\right)-{c}_f}{1-\exp \left(-{Pe}_s\right)}=\left(-\frac{J_s}{J_v}\right) $$

(5a)

$$ \frac{B}{A\sigma \beta RT}-\left(1-\sigma \right){c}_i=\left(-\frac{J_s}{J_v}\right) $$

(5b)

$$ \frac{c_d-{c}_m\exp \left({Pe}_{bl}\right)}{\exp \left({Pe}_{bl}\right)-1}=\left(-\frac{J_s}{J_v}\right) $$

(5c)

The use of Eq. (4) is consistent with the ‘leakiness’ giving rise to a convective term that does not vary across the membrane. Initially this paper was going to have an exclusive focus upon exploring (i) the set of equations given by Eq. (5) as an alternative to those in [8]; and (ii) the effect of varying the reflection coefficient. However a new three-parameter model was briefly explored due to increasing concern about the validity of applying the S-K model to FO with concentration invariant parameters. Now FO is the simplest of setting for the application of the Spiegler-Kedem model so when this model yields irrational results this indicates that some detail in the S-K model is wrong. This might be the use of concentration invariant parameters. Given an increasing concern about whether the S-K model captures the basic physics, a purely empirical equation was developed as an alternative.

Combining Eqs. (1) and (2) one obtains:

$$ {J}_s=-\omega \frac{dc}{dx}+\left(1-\sigma \right) c\chi \sigma .\beta RT\frac{dc}{dx} $$

(6)

Now the concentration in the barrier layer at its interface with the draw solution is higher than that at the interface with the support layer and so for constant *J*_{s} Eq. (6) suggests that \( \frac{dc}{dx} \) will vary with position within the membrane (unless *σ* = 1) but the invariant value of the volumetric flux *J*_{v} with position will via Eq. (1) suggest that \( \frac{dc}{dx} \) is constant. There is thus (unless the product of concentration and hydraulic permeability, *cχ* is constant) an apparent contradiction at the heart of the S-K model when it is applied to FO. This point illustrates the cautioning comments given elsewhere [9] and noted above. Furthermore the permeability of salt in polymers is not a constant and for sulfonated polymers, the salt diffusion coefficient increases markedly as the external salt concentration increases [11]. So notwithstanding the established nature of the Spielger-Kedem (S-K) model [12] it should not be seen as so superior as to be above challenge.

An additional reason for asserting that the S-K model, as particularised in [8], is invalid is that it is of the form that one obtains when two resistances are in parallel. If one considers an electrical circuit where there is a current, *I*, flowing through two parallel resistances (*R*_{1} and *R*_{2}) driven by a potential difference *V*, then

$$ I=\left(1/{R}_1+1/{R}_2\right)V $$

(7)

Comparing Eqs. 6 and 7, the parallels are obvious. However if the second channel associated with (1 − *σ*) is considered to consist of pores, the pores will be in contact with the bulk fluids. In reverse osmosis the flow through such channels is sustained by pressure but in FO there is no pressure difference between the two sides and osmosis can only be sustained by perm-selectivity. Whilst there might be a mitigating effect due to convective flow of solvent it is poorly modelled by the above equation. Caution must be taken in the use of the S-K model and it should not be over-interpreted.

The S-K model is a three-parameter model well established for reverse osmosis but rarely used for FO. Previously it has been used to describe the transport across the active layer of a FO/PRO membrane [13]. A comparison with the standard two-parameter solution-diffusion (S-D) model was made and little difference was found [12]. Having noted concerns regarding the S-K model, at least as it is applied to FO the scope of the paper has been extended to include a purely empirical equation. This equation (i) tends of the S-D model at low flux and (ii) \( \left(\frac{J_s}{J_v}\right) \) declines with increasing *J*_{v} but the ratio is never positive i.e. it tends to zero and does not transition into a regime of supposed co-current flow of solute and solvent.

$$ \left(-\frac{J_s}{J_v}\right)=\frac{B}{A\sigma \beta RT}\exp \left(-a{J}_v\right) $$

(8)

where ‘a’ would be determined from experiments. Herein the concern is to establish the influence of ‘a’ upon the flux-draw concentration relationship.