# The Role of the Morphological Characterization of Multilayer Hydrophobized Ceramic Membranes on the Prediction of Sweeping Gas Membrane Distillation Performances

^{*}

## Abstract

**:**

## 1. Introduction

_{min}) [2,3,4]. Polymeric membranes, such as polypropylene, polytetrafluoroethylene, or polyvinylidene difluoride, and its modifications are typically used in MD operations because they have an LEP

_{min}at room temperature higher than 2–3 bar [3,7,8,9,10].

_{min}decreases with increasing temperature. A critical wetting temperature in the range from 130 to 135 °C was found for FAS-grafted carbon-based membrane titania [20,21], indicating that maximum operating temperature values should not exceed 100–120 °C, values at which LEP

_{min}drops below 1 bar. Similar values have been obtained for alumina membrane grafted with FAS [22].

## 2. Materials and Methods

#### 2.1. Membranes and Modules

_{IN}) and outer diameter (d

_{OUT}) of the fiber, number of fibers (N

_{f}), the inner surface area (A

_{IN}), the total fiber length (L

_{tot}), and the inner shell diameter (d

_{S}), as depicted in Figure 1a.

_{eff}).

_{min}) values obtained with pure water according to the “flooding curve” method introduced by Varela-Corredor et al. [20,21] (Table 1). In agreement with the results obtained by Varela-Corredor et al. [21], in which the critical wetting temperature for that material was measured in the range of 130 to 135 °C, all the bundles used in the present work show the required hydrophobic character when employed at pressures below LEP

_{min}.

_{m}) and the mean pore diameter (dp

_{m}) calculated according to this method, abbreviated in the following as the “average-membrane-morphology” (AMM) method, are listed in Table 3 for each bundle.

#### 2.2. Experimental Set-Up and Procedures

_{min}at the corresponding liquid-side temperature to avoid membrane wetting.

_{min}value of 0.9 bar, as reported in [21].

_{NaCl}), as shown in Table 4.

_{1}), and the time at which the second sample was taken is referred to as (t

_{2}). Consequently, the salinity of NaCl (S

_{NaCl}) is measured at time (t

_{2}), and the total mass in the liquid phase (m

_{tot, liquid phase}(t

_{2})) can be evaluated, assuming total salt rejection, as shown in Equation (1), since the total mass of NaCl contained in the liquid side is known as the initial value. The experimental water flux (J

_{w}) through the bundle during the period (t

_{2}− t

_{1}) can be finally calculated according to Equation (2), with reference to the inner surface.

## 3. SGMD of NaCl-Water Solutions across Capillary Bundles: Model Equations

- Steady-state conditions;
- Total NaCl rejection: the membrane is a perfect barrier and thus only water permeates;
- Gas phase behaves as an ideal gas mixture;
- No heat loss in the module (well-insulated module);
- Parallel flow of liquid and gas streams within the module.

#### 3.1. Local Model: Heat and Mass Transfer across The Membrane

_{w}) can be described as the combination of molecular and Knudsen diffusion through a stagnant gas (air) [23,33], represented by Equation (3), in which the mass transfer coefficient of the membrane (k

_{w,m}) is defined in a direct way.

_{w,m}can be expressed by the relations reported in Equation (4):

_{Weq,j}) can be estimated from the Bosanquet equation [4,34] by using the Knudsen diffusivity of each layer j (D

_{W,Kn,j}) and the molecular diffusivity of water in air (D

_{WG}). The membrane temperature (T

_{m}) is calculated as the arithmetic mean of the temperatures at the two membrane interfaces (T

_{Lm}) and (T

_{Gm}).

_{w,m}can be expressed by the relations reported in Equation (5), in which the average values of the pore diameter (d

_{pm}) and the porosity-tortuosity ratio ((ε/τ)

_{m}) of the membrane are used.

_{S,L}and k

_{w,G}represent the mass transfer coefficient of salt in the liquid phase and the mass transfer coefficient of water in the gas phase, respectively. The mass transfer coefficients are calculated according to the relationships reported in Appendix A and Appendix B.

_{w,Lm}) can be calculated with the modified Raoult’s law (Equation (8)), which takes into account the non-ideality of salt solutions:

_{w,Lm}) represents the activity coefficient of water, which should be calculated at the conditions existing at the liquid/membrane interface.

_{net}) transferring through the membrane and the thermal boundary layer of the gas phase is represented by Equation (9a). The heat balance at the liquid/membrane interface is represented by Equation (9b), taking into account liquid evaporation:

_{L}and h

_{G}represent the convective heat transfer coefficients of the liquid side and of the gas side, respectively. λ

_{w}(T

_{Lm}) is the molar latent heat of vaporization of water that must be calculated at the temperature existing at the liquid/membrane interface.

^{−1}K

^{−1}) is very negligible, compared to the conductivity of solid titania (~7.8 Wm

^{−1}K

^{−1}) (Appendix B), heat conduction across the membrane is controlled by conductivity of the solid. Because of the very thin layer of the membrane, ${k}_{m}^{cond.}$ can be estimated to be in the range from 1700 to 2400 Wm

^{−2}K

^{−1}; this value, compared with the heat transfer coefficient values in the gas phase, which are typically in the range from 1 to 10 Wm

^{−2}K

^{−1}, allows the heat conduction across the membrane to be neglected with respect to convective transfer in the sweeping gas. Consequently, Equation (10) can be neglected, the membrane temperature (T

_{m}) can be calculated as (T

_{m}= T

_{Lm}) and Equation (9) can be simplified into Equation (11).

_{w,m}) are obtained from gas permeation measurements, independent of SGMD operations. Indeed, the same parameters can be used to simulate and/or describe any MD process, e.g., both direct contact and vacuum membrane distillation operations.

#### 3.2. Module Simulation

_{eff}) (Figure 1c) between the inlet and outlet nozzles of the shell.

_{L}), the interstitial gas velocity (v

_{0,G}), and the friction factor (f) are also defined.

_{w}) values, as defined in Equation (23), referring to the internal area of the module, so as to be comparable with the experimental data, elaborated according to the procedure represented by Equation (1).

## 4. Results and Discussion

_{w}) for the same test conditions given in Table 4 and the corresponding experimental flux values (experimental J

_{w}) is first reported. Simulations were performed with the membrane parameters obtained according to the LBL and AMM method. The results are shown in Figure 4.

_{w}) can also be expressed by the relationships reported in Equation (24):

_{lm,a}represents the logarithmic mean partial pressure of air across the membrane, which is frequently incorporated into the mass transfer coefficient [4].

_{w,Lm}x

_{w,Lm}) of Equation (24) decreases and the water flux decreases. However, the effect is not remarkable in the salinity range investigated; the effect of increasing temperature from 90 to 100 ºC can far outweigh the effect of increasing salinity from 10 to 45 g/kg.

_{min}, and, therefore, to prevent wettability of the membrane, the P

_{L}-P

_{G}difference must be decreased, which implies an increase in pressure in the gas. Since the flux decreases with increasing gas pressure, while it increases with increasing liquid temperature, it is clear that a functional optimum will exist between liquid temperature and gas pressure. A detailed analysis of the process will be needed to identify the best operating ranges.

^{2}h) with α-Si

_{3}N

_{4}membranes grafted with dimethyl-dichlorosilane [41] up to 21 kg/(m

^{2}h) with alumina membranes grafted with 1H, 1H, 2H, 2H-perfluorooctyltriethoxysilane [43].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

Latin Letters | SI Units | |

${A}_{IN}$ | Inner surface Area | [m^{2}] |

${\tilde{C}}_{p},{\widehat{C}}_{p}$ | Molar, mass, heat capacity at constant pressure | [J mol^{−1} K^{−1}], [J kg^{−1} K^{−1}] |

${d}_{IN}$ | Inner diameter of a fiber | [m] |

${d}_{OUT}$ | Outer diameter of a fiber | [m] |

${d}_{lm,m}$ | Logarithmic mean diameter of the membrane ${d}_{lm,m}=\frac{{d}_{OUT}-{d}_{IN}}{\mathrm{ln}\frac{{d}_{OUT}}{{d}_{IN}}}$ | [m] |

${d}_{lm,j}$ | Logarithmic mean diameter of the membrane layer j | [m] |

${d}_{S}$ | Shell diameter | [m] |

${d}_{p,j}$ | Pore diameter of the membrane layer j | [m] |

${d}_{pm}$ | Mean pore diameter of the membrane | [m] |

${D}_{W,Kn}$ | Knudsen diffusion coefficient of water | [m^{2} s^{−1}] |

${D}_{WG}$ | Molecular diffusion coefficient of water in gas | [m^{2} s^{−1}] |

${D}_{Weq}$ | Equivalent diffusion coefficient of water | [m^{2} s^{−1}] |

D | Molecular diffusion coefficient | [m^{2} s^{−1}] |

f | Fanning factor | [dimensionless] |

${G}_{in}$ | Inlet stream of gas | |

${G}_{out}$ | Outlet stream of gas | |

$G{z}_{H},G{z}_{M}$ | Graetz number for heat, mass transfer | [dimensionless] |

h | Convective heat transfer coefficient | [W m^{−2} K^{−1}] |

${J}_{w}$ | Mass flux of water across the membrane (defined in Equation (2)) | [kg m^{−2} s^{−1}) |

${k}_{w}$ | Mass transfer coefficient of water | [m s^{−1}] |

${k}_{S,L}$ | Mass transfer coefficient of salt in liquid | [m s^{−1}] |

${k}^{\mathit{cond}.}$ | Thermal conductivity coefficient | [W m^{−1} K^{−1}] |

${k}_{m}^{cond.}$ | Pseudo-thermal conductivity of the membrane (defined in Equation (9a)) | [W m^{−2} K^{−1}] |

${L}_{\mathit{eff}}$ | Effective length of membrane module (Figure 1c) | [m] |

${L}_{\mathit{tot}}$ | Total length of membrane module | [m] |

${L}_{\mathit{in}}$ | Inlet stream of liquid | |

${L}_{\mathit{out}}$ | Outlet stream of liquid | |

${m}_{w,\mathit{liquid}\mathit{phase}}$ | Mass of water in the liquid side | [kg] |

${m}_{\mathit{tot},\mathit{liquid}\mathit{phase}}$ | Total mass of solution in the liquid side | [kg] |

M | Molar mass | [kg mol^{−1}] |

$\stackrel{\xb7}{n}$ | Molar flow rate | [mol s^{−1}] |

${N}_{w}^{\prime}$ | Transmembrane Molar flow rate of water per unit length per fiber | [mol m^{−1} s^{−1}] |

${N}_{f}$ | Number of fibers | [dimensionless] |

Nu | Nusselt number | [dimensionless] |

P | Pressure | [Pa] |

Pr | Prandtl number | [dimensionless] |

${P}_{w}^{*}$ | Vapor pressure of water | [Pa] |

${Q}_{L}^{\prime}$ | Heat flow rate per unit length per fiber in the liquid thermal boundary layer | [W m^{−1}] |

${Q}_{\mathit{net}}^{\prime}$ | Net transmembrane heat flow rate per unit length per fiber | [W m^{−1}] |

Re | Reynolds number | [dimensionless] |

${R}_{g}$ | Universal gas constant | [J mol^{−1} K^{−1}] |

${S}_{\mathit{Nacl}}$ | Salinity of NaCl solution | [g NaCl kg^{−1} Solution] |

Sc | Schmidt number | [dimensionless] |

Sh | Sherwood number | [dimensionless] |

T | Temperature | [K] |

${v}_{\mathit{L}}$ | Liquid velocity in lumen-side (defined in Equation (15)) | [m s^{−1}] |

${V}_{0,\mathit{G}}$ | Gas interstitial velocity in shell-side (defined in Equation (21)) | [m s^{−1}] |

x | Mole fraction in liquid phase | [dimensionless] |

y | Mole fraction in gas phase | [dimensionless] |

z | Axial coordinate in membrane module | [m] |

Greek Letters | SI Units | |

${\gamma}_{w,Lm}$ | Activity coefficient of water at liquid/membrane interface | [dimensionless] |

$\delta $ | Thickness | [m] |

${\left(\epsilon /\tau \right)}_{j}$ | Porosity-tortuosity ratio of the membrane layer j | [dimensionless] |

${\left(\epsilon /\tau \right)}_{m}$ | Mean porosity-tortuosity ratio of the membrane | [dimensionless] |

${\epsilon}_{p}$ | Packing factor of the membrane module | [dimensionless] |

$\eta $ | Dynamic viscosity | [Pa s] |

$\lambda $ | Molar latent heat of vaporization | [J mol^{−1}] |

$\rho $ | Density | [kg m^{−3}] |

Superscripts and Subscripts | ||

a | Air | |

G | Gas side | |

Gb | At gas bulk | |

Gm | At gas/membrane interface | |

IN | Inlet section | |

j | Layer j (j = S for support, j = 1 for layer1, j = 2 for layer 2, j = 3 for layer 3) | |

L | Liquid side | |

Lb | At liquid bulk | |

Lm | At liquid/membrane interface | |

s | Salt | |

solid | Solid portion of the membrane | |

w | Water |

## Appendix A. Mass and Heat Transfer Correlations

**Table A1.**Correlations for heat and mass transfer in forced convection in unbaffled shell and tube configuration.

Side | Correlation | Validity Range | Reference |
---|---|---|---|

Tube | $\begin{array}{c}Nu=3.66+\frac{0.0668\text{}G{z}_{H}}{1+0.04\text{}G{z}_{H}^{2/3}}\\ Sh=3.66+\frac{0.0668\text{}G{z}_{M}}{1+0.04\text{}G{z}_{M}^{2/3}}\end{array}$ (A.1) | $\begin{array}{l}Re<2100\\ G{z}_{H},G{z}_{M}<100\end{array}$ | [44,45] |

$\begin{array}{l}Nu=0.116\left(R{e}_{}^{2/3}-125\right)P{r}_{}^{1/3}\left[1+{\left(\frac{{d}_{IN}}{{L}_{eff}}\right)}^{2/3}\right]\\ Sh=0.116\left(R{e}_{}^{2/3}-125\right)S{c}_{}^{1/3}\left[1+{\left(\frac{{d}_{IN}}{{L}_{eff}}\right)}^{2/3}\right]\end{array}$ (A.2) | $\begin{array}{l}2100<Re<{10}^{4}\\ 60<\frac{{L}_{tube}}{{d}_{IN}}<250\end{array}$ | [46,47] | |

Shell (parallel flow) | $\begin{array}{l}Nu=0.128{d}_{eq}^{0.6}R{e}_{}^{0.6}P{r}_{}^{1/3}\\ Sh=0.128{d}_{eq}^{0.6}R{e}_{}^{0.6}S{c}_{}^{1/3}\end{array}$ (A.3) ${d}_{eq}=\frac{\left({d}_{S}^{2}-{N}_{f}{d}_{OUT}^{2}\right)}{\left({d}_{S}+{N}_{f}{d}_{OUT}\right)}$(in inches) | $\begin{array}{l}80<{d}_{eq}Re<2\times {10}^{4}\\ inches\end{array}$ | [44] |

**Table A2.**Dimensionless numbers used in Table A1.

Heat Transfer | Mass Transfer |
---|---|

$Re=\frac{\rho \text{}{v}^{\ast}\text{}{l}^{\ast}}{\eta}$ | |

$Nu=\frac{h\text{}{l}^{\ast}}{{k}^{cond.}}$ | $Sh=\frac{k\text{}{l}^{\ast}}{{D}_{}}$ |

$Pr=\frac{\eta \text{}{\widehat{C}}_{p}}{{k}^{cond.}}$ | $Sc=\frac{\eta \text{}}{\rho \text{}{D}_{}}$ |

$G\text{}{z}_{\text{}H}=Re\text{}Pr\frac{{d}_{IN}}{{L}_{eff}}$ | $G\text{}{z}_{\text{}M}=Re\text{}Sc\frac{{d}_{IN}}{{L}_{eff}}$ |

${l}^{*}={d}_{IN}\text{}(for\text{}tube),\text{}{l}^{*}={d}_{OUT}\text{}(for\text{}shell),\text{}{v}^{*}={v}_{L}\text{}(for\text{}tube),\text{}{v}^{*}={v}_{0,G}\text{}(for\text{}shell)$ |

## Appendix B. Relevant Chemical-Physical Properties

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**Figure 1.**Scheme of membranes and modules: (

**a**) Capillary bundle; (

**b**) Cross-section of the multilayer membrane of a capillary; (

**c**) Bundle-housing arrangement in counter-current flow pattern; (

**d**) System coordinates for the equations used in the plug-flow (parallel flow) model.

**Figure 3.**SGMD of water-salt solutions: expected (

**a**) Composition profiles; (

**b**) Temperature profiles across a generic membrane section.

**Figure 5.**The effect of the inlet operating conditions on the modeled flux in case of countercurrent flow for the bundle B2758 at the operating conditions reported in Table 7. (

**a**) Liquid inlet temperature; (

**b**) Liquid inlet velocity; (

**c**) Liquid inlet salinity; (

**d**) Gas inlet interstitial velocity.

**Table 1.**Geometric parameters and LEP

_{min}values at the corresponding temperature of the capillary bundles.

Code | d_{IN}(mm) | d_{OUT}(mm) | N_{f}(fibers) | L_{tot}(cm) | d_{S}(cm) | L_{eff}(cm) | A_{IN}(cm ^{2}) | LEP_{min} (at T)(bar) |
---|---|---|---|---|---|---|---|---|

B2754 | 1.56 | 3.20 | 37 | 20 | 3.60 | 13 | 363 | 4.2 (25 °C) |

B2755 | 1.56 | 3.20 | 37 | 20 | 3.60 | 13 | 363 | 4 (25 °C) |

B2756 | 1.56 | 3.20 | 37 | 20 | 3.60 | 13 | 363 | 6.2 (25 °C) |

B2888 | 1.9 | 3.54 | 37 | 20 | 3.60 | 13 | 442 | 0.30–0.39 (130 °C) ^{§} |

B2758 | 1.9 | 3.20 | 22 | 20 | 2.50 | 17 | 263 | 6.9 (25 °C) ^{§} |

^{§}from [21].

**Table 2.**Morphological properties of the four layers of the capillary bundles, as estimated by the LBL method [23].

Layer 3 | Layer 2 | Layer 1 | Support | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Code | d_{p}(nm) | (ε/τ) | δ (µm) | d_{p}(nm) | (ε/τ) | δ (µm) | d_{p}(nm) | (ε/τ) | δ (µm) | d_{p}(nm) | (ε/τ) | δ (µm) |

B2754 | 548 | 0.0029 | 10 | 250 | 0.34 | 30 | 800 | 0.20 | 30 | 4500 | 0.11 | 750 |

B2755 | 534 | 0.0032 | 10 | 250 | 0.34 | 30 | 800 | 0.20 | 30 | 4500 | 0.11 | 750 |

B2756 | 435 | 0.0044 | 10 | 250 | 0.34 | 30 | 800 | 0.20 | 30 | 4500 | 0.11 | 750 |

B2888 | 328 | 0.0069 | 10 | 250 | 0.34 | 30 | 800 | 0.20 | 30 | 4500 | 0.11 | 750 |

B2758 | 68 | 0.084 | 10 | 250 | 0.34 | 30 | 800 | 0.20 | 30 | 4500 | 0.11 | 580 |

**Table 3.**Average membrane morphological properties of the capillary bundles, as estimated by the AMM method [23].

Average Values | ||
---|---|---|

Code | dp_{m} (nm) | (ε/τ) |

B2754 | 468 | 0.27 |

B2755 | 1232 | 0.053 |

B2756 | 354 | 0.44 |

B2888 | 337 | 0.38 |

B2758 | 87 | 3.414 |

**Table 4.**Operating conditions in SGMD of NaCl-water solutions (symbols refer to Figure 2 and Notation).

Liquid Inlet to Tube-Side (L_{in}) | Gas Inlet to Shell-Side (Gin) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Trial | T2 (°C) | P3 (bar) | F3 (L/h) | S_{NaCl} (g/kg) | v_{L,IN}(m/s) | ΔP (mbar) | T1 (°C) | P1 (bar) | F0 (m ^{3}_{STP}/h) | v_{0,G,IN}(m/s) | Bundle |

B | 61.5 | 4.95 | 100 | 18.79 | 0.39 | - | 43.0 | 4.10 | 5.15 | 0.56 | B2755 |

C | 88.9 | 2.55 | 100 | 18.92 | 0.39 | - | 49.0 | 2.20 | 2.91 | 0.60 | B2755 |

D | 90.9 | 2.60 | 100 | 19.68 | 0.39 | - | 61.0 | 2.25 | 2.70 | 0.57 | B2756 |

E | 89.9 | 2.45 | 100 | 18.24 | 0.39 | - | 51.5 | 1.90 | 1.87 | 0.45 | B2754 |

F | 89.6 | 2.30 | 100 | 18.31 | 0.39 | - | 55.5 | 1.90 | 1.82 | 0.44 | B2754 |

H | 64.6 | 3.34 | 100 | 19.50 | 0.45 | 170 | 41.7 | 4.05 | 1.71 | 0.58 | B2758 |

I | 89.7 | 3.98 | 100 | 19.67 | 0.45 | 250 | 56.1 | 3.95 | 0.24 | 0.63 | B2758 |

J | 64.1 | 2.90 | 100 | 19.82 | 0.45 | 200 | 43.1 | 2.70 | 1.51 | 0.58 | B2758 |

K | 89.5 | 4.84 | 100 | 20.03 | 0.45 | 212 | 60.8 | 4.86 | 4.12 | 0.90 | B2758 |

L | 40.9 | 2.30 | 100 | 18.58 | 0.45 | 310 | 39.3 | 2.13 | 2.05 | 0.98 | B2758 |

M | 72.6 | 2.98 | 100 | 18.74 | 0.45 | 310 | 52.5 | 2.88 | 2.73 | 1.01 | B2758 |

N | 50.3 | 5.13 | 104 | 18.90 | 0.46 | 325 | 44.5 | 5.00 | 4.64 | 0.96 | B2758 |

O | 87.1 | 5.08 | 105 | 19.13 | 0.47 | 308 | 64.5 | 5.10 | 4.66 | 1.00 | B2758 |

P | 110.3 | 5.33 | 103 | 19.58 | 0.46 | 329 | 69.8 | 5.23 | 4.87 | 1.03 | B2758 |

Q | 110.2 | 5.25 | 100 | 19.93 | 0.45 | 290 | 69.3 | 5.10 | 4.76 | 1.03 | B2758 |

R | 70.2 | 5.18 | 150 | 17.97 | 0.40 | 331 | 57.3 | 5.20 | 4.84 | 0.43 | B2888 |

S | 89.3 | 5.13 | 150 | 18.75 | 0.40 | 296 | 61.5 | 5.25 | 4.87 | 0.44 | B2888 |

T | 90.5 | 5.03 | 150 | 19.58 | 0.40 | 251 | 61.8 | 5.00 | 4.63 | 0.44 | B2888 |

U | 91.10 | 5.05 | 150 | 19.95 | 0.40 | 264 | 62.0 | 5.08 | 4.66 | 0.43 | B2888 |

**Table 5.**Balance equations in the liquid side (plug flow model) and auxiliary variables (see Figure 1d and Notation).

Equation | Equation | |
---|---|---|

$\frac{d{\dot{n}}_{L}}{dz}={N}_{w}^{\prime}\cdot {N}_{f}\text{\hspace{1em}}$ | Total mass balance | (12) |

$\frac{d{\dot{n}}_{s}}{dz}=0$ | NaCl mass balance | (13) |

$\frac{d{T}_{Lb}}{dz}=\frac{{Q}_{L}^{\prime}\cdot {N}_{f}}{{\dot{n}}_{L}\text{}{\tilde{C}}_{{p}_{,L}}}$ | Heat balance | (14) |

$\frac{d{v}_{L}}{dz}=\frac{4{N}_{w}^{\prime}{M}_{w}}{\pi {d}_{IN}^{2}{\rho}_{L}}$ | Liquid velocity | (15) |

$\frac{d{P}_{L}}{dz}=4f{\rho}_{L}\frac{{v}_{L}^{2}}{2{d}_{IN}}\text{\hspace{1em}}$ $at\text{}\mathrm{Re}2300\text{}f=16/\mathrm{Re},\text{}at\text{}2300\mathrm{Re}5000\text{}f=0{.079\mathrm{Re}}^{-0.25}$ | Pressure drop | (16) |

$\begin{array}{c}at\text{}z={L}_{tot}\\ {\dot{n}}_{L}={\dot{n}}_{L,IN},\text{}{\dot{n}}_{s}={x}_{s,Lb,IN}\text{}{\dot{n}}_{L,IN},{T}_{Lb}={T}_{L,IN},\text{}{P}_{L}={P}_{L,IN},{v}_{L}={v}_{L,IN}\end{array}$ | Boundary conditions |

**Table 6.**Balance equations in the gas side (plug flow model) and auxiliary variables (see Figure 1d and Notation).

Equation | Equation | |
---|---|---|

$\frac{d{\dot{n}}_{G}}{dz}={N}_{w}^{\prime}\cdot {N}_{f}$ | Total mass balance | (17) |

$\frac{d{\dot{n}}_{a}}{dz}=0$ | Air mass balance | (18) |

$\frac{d\text{\hspace{0.17em}}{T}_{Gb}}{dz}=\frac{{Q}_{net}^{\prime}\cdot {N}_{f}}{{\dot{n}}_{G}\text{\hspace{0.17em}}{\tilde{C}}_{P,G}}$ | Heat balance | (19) |

$\left(\frac{1}{{\rho}_{G}}\right)\frac{d{\rho}_{G}}{dz}=\left(\frac{-1}{{T}_{Gb}}\right)\frac{d{T}_{Gb}}{dz}+\left(\frac{1}{{P}_{G}}\right)\frac{d{P}_{G}}{dz}+\left(\frac{{M}_{w}-{M}_{a}}{{M}_{G}}\right)\frac{d{y}_{w,Gb}}{dz}$ | Ideal gas law | (20) |

${\rho}_{G}\left(\frac{d{v}_{0,G}}{dz}\right)+{v}_{0,G}\left(\frac{d{\rho}_{G}}{dz}\right)=\frac{4{N}_{w}^{\prime}{M}_{w}}{\pi \left(\frac{{d}_{S}^{2}}{{N}_{f}}-{d}_{OUT}^{2}\right)}$ ${v}_{0,G}=\frac{{\dot{n}}_{G}{M}_{G}/{\rho}_{G}}{\left(1-{\epsilon}_{p}\right)\left(\frac{\pi}{4}{d}_{S}^{2}\right)},{\epsilon}_{p}={N}_{f,tot}{\left(\frac{{d}_{OUT}}{{d}_{S}}\right)}^{2}$ | Gas phase velocity | (21) |

$\frac{d{P}_{G}}{dz}=\frac{8{N}_{f}{\eta}_{G}{v}_{0,G}}{{d}_{S}^{2}\left(\frac{1}{2}\left(\frac{\mathrm{ln}\left({\epsilon}_{p}\right)}{1-{\epsilon}_{p}}+\frac{3-{\epsilon}_{p}}{2}\right)\right)}\text{\hspace{1em}}$ | Pressure drops (equivalent annulus model [35] | (22) |

$\begin{array}{c}at\text{}z=0\\ {\dot{n}}_{G}={\dot{n}}_{G,IN},\text{\hspace{1em}}{\dot{n}}_{a}={\dot{n}}_{a,IN},\text{\hspace{1em}}{T}_{Gb}={T}_{G,IN},\text{}{P}_{G}={P}_{G,IN}\\ {v}_{0,G}={v}_{0,G,IN},\text{\hspace{1em}}{\rho}_{G}={\rho}_{G,IN},\text{\hspace{1em}}{y}_{w}{}_{,Gb}={y}_{w,Gb,IN}\end{array}$ | Boundary conditions |

**Table 7.**Operating conditions used for simulations reported in Figure 5a–d.

Figure | 5a | 5b | 5c | 5d |
---|---|---|---|---|

T_{L,IN} (°C)
| * | * | * | 100 |

P_{L,IN} (bar)
| * | 2 | 2 | 2 |

S_{NaCl,IN} (g/kg)
| 20 | * | * | 45 |

v_{L,IN} (m/s)
| 0.5 | * | 0.5 | 0.5 |

T_{G,IN} (°C)
| 45 | 45 | 45 | 45 |

P_{G,IN} (bar)
| * | 1.7 | 1.7 | 1.7 |

Relative humidity of air (%) | 0 | 0 | 0 | * |

v_{0,G,IN} (m/s)
| 1 | 1 | 1 | * |

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## Share and Cite

**MDPI and ACS Style**

Fawzy, M.K.; Varela-Corredor, F.; Boi, C.; Bandini, S.
The Role of the Morphological Characterization of Multilayer Hydrophobized Ceramic Membranes on the Prediction of Sweeping Gas Membrane Distillation Performances. *Membranes* **2022**, *12*, 939.
https://doi.org/10.3390/membranes12100939

**AMA Style**

Fawzy MK, Varela-Corredor F, Boi C, Bandini S.
The Role of the Morphological Characterization of Multilayer Hydrophobized Ceramic Membranes on the Prediction of Sweeping Gas Membrane Distillation Performances. *Membranes*. 2022; 12(10):939.
https://doi.org/10.3390/membranes12100939

**Chicago/Turabian Style**

Fawzy, Mohamed K., Felipe Varela-Corredor, Cristiana Boi, and Serena Bandini.
2022. "The Role of the Morphological Characterization of Multilayer Hydrophobized Ceramic Membranes on the Prediction of Sweeping Gas Membrane Distillation Performances" *Membranes* 12, no. 10: 939.
https://doi.org/10.3390/membranes12100939