# Modeling and Simulation of Either Co-Current or Countercurrent Operated Reverse-Osmosis-Based Air Water Generator

^{*}

## Abstract

**:**

## 1. Introduction

^{3}[4]. However, there are many regions without sufficient natural drinking water resources. In regions where access to seawater or polluted water is available, desalination or wastewater treatment plants can be used to provide clean water [5]. In case seawater is available, reverse osmosis with membranes is normally used due to the high energy efficiency [6]. If no access to liquid water exists, the water must be transported to the recipients by land or air. In many regions affected by water shortages, large facilities that provide clean water in sufficient quantities do not exist, and transport by land or air seems not feasible. Another potential source is the atmosphere, where the water is stored in the form of water vapor. The earth’s atmosphere contains so much water vapor that in its liquid state it would have a volume of about 13,000 km

^{3}, which is about one seventh the volume of fresh water on the earth’s surface [4].

## 2. Materials and Methods

#### 2.1. Absorbents

#### 2.2. Conceptual Design of Reverse Osmosis Based Air Water Generator

#### 2.2.1. Co-Current Multi-Stage Reverse Osmosis

#### 2.2.2. Countercurrent Multi-Stage Reverse Osmosis

#### 2.3. Modeling

#### 2.3.1. Absorber

#### Assumptions

- The pressure p in the aqueous lithium bromide solution is constant.
- The total pressure ${p}_{tot}$ of the air is constant.
- The liquid film is flat and has no surface waves.
- The film thickness is considered constant along the height of the absorber column.
- The inlet mass flow rate of the solution and the inlet volume flow rate of the air are assumed to be constant and are calculated according to Appendices B and C of [9].
- The conditions of the air and solution are constant at a given height of the absorber.

#### Correlations

#### Calculations of the Absorber

#### Calculations of the Ventilator

#### Solution Algorithm

#### 2.3.2. Reverse Osmosis Process

#### Assumptions

- No temperature changes over the membranes, the pressure exchangers or the pumps.
- The membrane has a salt rejection of 100%, so no salt flows through the membrane.
- Concentration polarization phenomena in the membrane are not considered.
- Water mass transfer through the membrane is calculated using a membrane constant.
- The representative membrane module used has a pressure drop of 1 bar; therefore, this pressure drop is distributed linearly over the membrane.
- No leakages between the streams in the pressure exchangers.

#### Calculations of the Reverse Osmosis Membrane Modules

#### Calculations of the Pressure Exchangers and Pumps

#### Solution Algorithm

#### 2.4. Simulations

^{3}/h, and a solution mass flow of 1.5 kg/s were used as input parameters. Furthermore, different values for the partial pressure of water vapor ${p}_{w}$ and the ambient air temperature ${T}_{air}$ were used as boundary conditions for a series of simulations.

## 3. Results

#### 3.1. Co-Current Multi-Stage Reverse Osmosis

#### 3.2. Countercurrent Multi-Stage Reverse Osmosis

## 4. Discussion

^{3}, and for co-current operation with variable booster pump pressures, it is between ∼230–1480 kWh/m

^{3}. For countercurrent operation with fixed booster pump pressures, the specific energy demand is between ∼230–1260 kWh/m

^{3}, and for countercurrent operation with variable booster pump pressures, it is between ∼230–1240 kWh/m

^{3}. The simulations with variable booster pump pressures resulted in energy demands that were equal to or smaller than those with fixed booster pump pressures; however, they required more reverse osmosis membrane module stages. Under ideal conditions, the values simulated in this paper are slightly lower than those determined by Wahlgren for condensation processes, which require between 270 and 550 kWh/m

^{3}[27].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AWG | Air water generator/air water generation |

LCOW | Levelized cost of water |

PX | Pressure exchanger |

Symbols | |

a | Activity $[-]$ |

A | Area $\left[{\mathrm{m}}^{2}\right]$ |

${A}_{memb}$ | Membrane constant $[\mathrm{kg}/\left(\mathrm{s}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}\right)]$ |

c | Concentration $[\mathrm{mol}/{\mathrm{m}}^{3}]$ |

${c}_{p}$ | Specific heat capacity (at constant pressure) $[\mathrm{J}/(\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\left)\right]$ |

d | Diameter $\left[\mathrm{m}\right]$ |

D | Diffusion coefficient $[{\mathrm{m}}^{2}/\mathrm{s}]$ |

h | Specific enthalpy $[\mathrm{J}/\mathrm{kg}]$ |

$\dot{H}$ | Enthalpy flow $\left[\mathrm{W}\right]$ |

J | Mass flux $[\mathrm{kg}/\left(\mathrm{s}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\right)]$ |

${L}_{char}$ | Characteristic length $\left[\mathrm{m}\right]$ |

$\dot{m}$ | Mass flow rate $[\mathrm{kg}/\mathrm{s}]$ |

N | Number of elements $[-]$ |

p | Pressure $\left[\mathrm{Pa}\right]$ |

P | Power $\left[\mathrm{W}\right]$ |

$\dot{Q}$ | Heat flow $\left[\mathrm{W}\right]$ |

R | Universal gas constant $[\mathrm{J}/(\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\left)\right]$ |

s | Gap thickness $\left[\mathrm{m}\right]$ |

T | Temperature $\left[\mathrm{K}\right]$ |

$vs.$ | Velocity $[\mathrm{m}/\mathrm{s}]$ |

V | Volume $\left[{\mathrm{m}}^{3}\right]$ |

$\tilde{V}$ | Molar volume $[{\mathrm{m}}^{3}/\mathrm{mol}]$ |

$\dot{V}$ | Volume flow rate $[{\mathrm{m}}^{3}/\mathrm{s}]$ |

${w}_{i}$ | Mass fraction of component i $[\mathrm{kg}/\mathrm{kg}]$ |

x | Position $\left[\mathrm{m}\right]$ |

${x}_{i}$ | Mole fraction of component i $[\mathrm{mol}/\mathrm{mol}]$ |

${X}_{i}$ | Mass load of water per component i $[\mathrm{kg}/\mathrm{kg}]$ |

Indices | |

a | Air |

$abs$ | Absorption |

$avg$ | Average |

$el$ | Electric |

$elem$ | Element |

f | Feed |

g | Gas |

h | Hydraulic |

i | i-th element |

$in$ | Inlet |

j | Solvent |

$out$ | Outlet |

p | Permeate |

r | Retentate |

$sol$ | Solution |

$tot$ | Total |

$vap$ | Vapor |

w | Water |

Greek Symbols | |

$\alpha $ | Heat transfer coefficient $[\mathrm{W}/\left({\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\right)]$ |

$\beta $ | Mass transfer coefficient $[\mathrm{m}/\mathrm{s}]$ |

$\gamma $ | Activity coefficient $[-]$ |

$\delta $ | Film thickness $\left[\mathrm{m}\right]$ |

$\zeta $ | Drag coefficient $[-]$ |

$\eta $ | Efficiency $[-]$ |

$\vartheta $ | Temperature ${[}^{\xb0}\mathrm{C}]$ |

$\lambda $ | Thermal conductivity $[\mathrm{W}/(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\left)\right]$ |

$\mu $ | Chemical potential $[\mathrm{J}/\mathrm{mol}]$ |

$\nu $ | Kinematic viscosity $[{\mathrm{m}}^{2}/\mathrm{s}]$ |

$\rho $ | Density $[\mathrm{kg}/{\mathrm{m}}^{3}]$ |

$\mathrm{\Pi}$ | Osmotic pressure $\left[\mathrm{Pa}\right]$ |

Dimensionless Numbers | |

$\mathrm{Le}=\lambda /\left(D\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}{c}_{p}\right)$ | Lewis number |

$\mathrm{Nu}=\alpha \phantom{\rule{0.166667em}{0ex}}{L}_{char}/\lambda $ | Nusselt number |

$\mathrm{Pr}=\nu \phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}{c}_{p}/\lambda $ | Prandtl number |

$\mathrm{Re}=\rho \phantom{\rule{0.166667em}{0ex}}v\phantom{\rule{0.166667em}{0ex}}{L}_{char}/\eta $ | Reynolds number |

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**Figure 1.**Multi-stage reverse osmosis [9].

**Figure 2.**Process schematic for an AWG with absorption and co-current multi-stage reverse osmosis from [9].

**Figure 3.**Process schematic for an AWG with absorption and countercurrent multi-stage reverse osmosis.

**Figure 4.**Energy demand per cubic meter of water [kWh/m

^{3}] (

**a**,

**c**) and number of necessary reverse osmosis membrane modules (

**b**,

**d**) for fixed (

**a**,

**b**) and optimized (

**c**,

**d**) booster pump pressure. The following applies to all figures. The white areas on the left side represent conditions where, for the chosen absorber dimensions, not enough water can be extracted from the air because the required salt mass fraction is too high and the solution starts to crystallize. To determine whether the solution begins to crystallize, the solid liquid equilibrium of aqueous lithium bromide is used [26]. The white areas in the lower right corners represent conditions where the air is oversaturated and therefore no representative statements can be made.

**Figure 5.**Energy demand per cubic meter of water (kWh/m

^{3}) (

**a**,

**c**) and number of necessary reverse osmosis membrane modules (

**b**,

**d**) for fixed (

**a**,

**b**) and optimized (

**c**,

**d**) booster pump pressure. The following applies to all figures. The white areas on the left side represent conditions where, for the chosen absorber dimensions, not enough water can be extracted from the air because the required salt mass fraction is too high and the solution starts to crystallize. To determine whether the solution begins to crystallize, the solid liquid equilibrium of aqueous lithium bromide is used [26]. The white areas in the lower right corners represent conditions where the air is oversaturated and therefore no representative statements can be made.

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**MDPI and ACS Style**

Fill, M.; Kleingries, M.
Modeling and Simulation of Either Co-Current or Countercurrent Operated Reverse-Osmosis-Based Air Water Generator. *Membranes* **2021**, *11*, 913.
https://doi.org/10.3390/membranes11120913

**AMA Style**

Fill M, Kleingries M.
Modeling and Simulation of Either Co-Current or Countercurrent Operated Reverse-Osmosis-Based Air Water Generator. *Membranes*. 2021; 11(12):913.
https://doi.org/10.3390/membranes11120913

**Chicago/Turabian Style**

Fill, Marc, and Mirko Kleingries.
2021. "Modeling and Simulation of Either Co-Current or Countercurrent Operated Reverse-Osmosis-Based Air Water Generator" *Membranes* 11, no. 12: 913.
https://doi.org/10.3390/membranes11120913