# Osmosis-assisted reverse osmosis, simulated to achieve high solute concentrations, low energy consumption

Here, a model is presented, to describe the flow of water through the membrane, at any point of the membrane module. The flow model can then be used to predict a flow profile along a module’s flow path, from which module performance is simulated.

### Water flow model for RO

The osmotic pressure on each side of the membrane, (pi) is estimated from the van’t Hoff equation^{52}using (R) as the ideal gas constant, (T) as absolute temperature, (VS) as the molar concentration, and (I) as the number of ions associated with each mole of dissolved solid equivalent, as shown in the equation. (1). For instance, (I) is equal to one for glucose, two for sodium acetate and three for sodium sulphate.

The difference in osmotic pressure, across a skin membrane, (Delta pi) can then be estimated, as shown in Eq. (2). Here the index (m) represents the skin surfaces of the membrane, on both sides, and the indices (h) and (I) represent the upper and lower concentration sides of the membrane.

$$Delta pi =iRTleft({C}_{m,h}-{C}_{m,l}right)$$

(2)

water flow, (J_{w}) can be estimated, as shown in the equation. (3), using (Delta P) as the hydraulic pressure difference, across the membrane, and (A M}) as the water permeability coefficient of the membrane^{15,39,60}.

$$J_{w} = A_{M} left( {Delta P – Delta pi } right)$$

(3)

The diffusive flux of the solute, away from the membrane, on the retentate side ((J_{h})) is described by a liquid film mass transfer model, as shown in Equation. (4), where (VS) is the molar concentration in mol/m^{3}and (k) is the mass transfer coefficient in m/s^{26}. Also, the indices (m), (b) and (h) refer to the interface of the membrane, the liquid mass and the retentate side of the membrane, respectively.

$$J_{h} = k_{h} left( {C_{m,h} – C_{b,h} } right)$$

(4)

(J_{w}) is the volumetric flux of water in units of m/s, forced from the retentate side to the permeate side of the membrane, while (J_{h}) is the molar diffusive flux of the solute, in units of mol/m^{2}/s, in the opposite direction to (J_{w}).

In steady state, there is no accumulation in the liquid film. On the retentate side, the rate at which ions are prevented from passing is equal to the rate at which they diffuse out of the liquid film, as shown in Eq. (5)^{35}.

$$J_{h} = J_{w} C_{b,h}$$

(5)

Equations 2, 3, 4 and 5 can be used to derive the equation. (6), to predict reverse osmosis water flow. This equation excludes (Delta pi),(C_{m,h}) and (J_{h}). Instead, it’s a function of just two variables (C_{b,h}) and (Delta P)that are tangible and easy to measure.

$$J_{w} = A_{M} k_{h} left( {frac{{Delta P – iRTC_{b,h} }}{{k_{h} + A_{M} iRTC_{b, h} }}} right)$$

(6)

Here the flux is approximated to occur in a single length dimension, perpendicular to the membrane surface. This is a widely adopted assumption for modeling mass transfer through thin films, and it is referred to as film theory in the chemical engineering literature.^{9}.

### Water flow model for OARO

The solute concentration profile, from the retentate side to the permeate side, is shown in Figure 1, for osmosis-assisted reverse osmosis (OARO). Here, unlike RO, there is a large concentration gradient at the support.

Since this concentration gradient cannot be accurately accounted for, by Eq. (4), Park et al.^{44} recommended the Internal Concentration Polarization (ICP) model, as shown in the equation. (seven). Here, (C_{b,l}) is the overall permeate concentration and (B) is the salt permeability. (K) is a constant described by Eq. (8), where (delta_{s}) is the thickness, (tau) is the tortuosity, and (upvarepsilon) is the porosity of the porous support layer, and (D) is the diffusion coefficient of the solute, in water. The rest of the parameters, in Eq. (7) are as previously defined.

$$frac{{Delta P – J_{w} /A_{M} }}{iRT} = frac{{C_{b,h} exp^{{left( {J_{w} /k_{ h} } right)}} – C_{b,l} exp^{{left( { – J_{w} K} right)}} }}{{1 + Bleft( {exp^{{ left( { – J_{w} K} right)}} – 1} right)/J_{w} }}$$

(seven)

$$K = frac{{tau delta_{s} }}{{Dupvarepsilon }}$$

(8)

### Empirical calculations of the mass transfer coefficient

The simulation of water flow in the RO and OARO cases requires a mass transfer coefficient on the retentate side, (k_{h}). This constant can be determined, by fitting the model of Eq. (6) at a range of measured flow values and their corresponding hydraulic pressure and bulk solution concentrations, in a reverse osmosis unit.

(k_{h}) can also be calculated using the empirical film-model correlation, demonstrated by Strathmann^{55}, as described below. First, the Reynolds number (N_{Re}), is calculated for liquid flow in membrane channels, according to Eq. (9), using (rho) as the density of the liquid, (mu) and liquid dynamic viscosity, (v) than the superficial velocity and (d_{H}) than the size of the flow channels.

$$N_{Re} = frac{{rho d_{H} v}}{mu }$$

(9)

The Schmidt number, (N_{Sc}) is described in Eq. (10), using (D) as the diffusion coefficient of the aqueous ion. The diffusion coefficient of acetate (1.089 × 10^{–9} m^{2}/s) is used in all calculations Buffle et al.^{14}. It is slightly lower than that of sodium cations, making it the rate-limiting diffusion coefficient.

$$N_{Sc} = frac{mu }{rho D}$$

(ten)

When calculating the Reynolds number and Schmidt number, the Sherwood number can be calculated for all Reynolds numbers less than 2100, according to the equation. (11), using (L) like the length of the flow channel, which is the length of the module, in this case.

$$N_{Sh} = 1.62N_{Re}^{0.33} N_{Sc}^{0.33} left( {d_{H} /L} right)^{0.33}$ $

(11)

The mass transfer coefficient of the liquid film can be calculated from the equation. (12).

$$k = DN_{Sh} /d_{H}$$

(12)

On the retentate side, the resistance to mass transfer is entirely attributed to the liquid film ((k_{h} = k)).

### Axial profiles for flux, concentrations and flow rates

The concentration changes along the length of the membrane module, due to the flow of water into or out of the flow channels. The flux changes due to changes in concentration. Additionally, there is a small pressure drop across the flow path. Park et al.^{44} listed the following equations to provide profiles along flow paths, for concentration, flow rate and pressure.

The pressure drop can be modeled according to the equation. (13), where (k_{money}) is the coefficient of friction.

$$frac{dP}{{dz}} = frac{{ – k_{fric} mu v}}{{d_{H}^{2} }}$$

(13)

On the retentate side, the flow rate changes, according to Eq. (14), where (z) is the distance in the fluid flow path and (w) is calculated by dividing the active area of the membrane module by its length.

$$frac{{dF_{h} }}{dz} = – wJ_{w}$$

(14)

On the retentate side, the concentration changes according to Eq. (15), where (J_{s}) is the diffusive flux of salt from the side of higher concentration to the side of lower concentration of the active layer, as described by Eq. (16).

$$frac{{dC_{b,h} }}{dz} = frac{{wleft( {C_{b,h} J_{w} – J_{s} } right)}}{{ F_{h} }}$$

(15)

$$J_{s} = Bleft( {frac{{C_{b,h} exp^{{left( {J_{w} /k_{h} } right)}} – C_{b, l} exp^{{left( { – J_{w} K} right)}} }}{{1 + Bleft( {exp^{{left( { – J_{w} K} right )}} – 1} right)/J_{w} }}} right)$$

(16)

Equations 17 and 18 illustrate the rates of flow and concentration changes on the permeate side of the membrane.

$$frac{{dF_{l} }}{dz} = – wJ_{w}$$

(17)

$$frac{{dC_{b,l} }}{dz} = frac{{wleft( {C_{b,l} J_{w} – J_{s} } right)}}{{ F_{l} }}$$

(18)

If a co-current system is to be simulated, instead of a counter-current system, (F_{l}) must adopt a negative sign in both Eqs. (17) and (18). Indeed, contrary to the fluid velocity, the flow rate is a scaler and cannot adopt negative values.

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