Comparison of DCMD and AGMD Membrane Distillation Configurations for Different Polymeric Membranes

1. Introduction

Membrane distillation (MD) is an emerging thermal separation process that utilizes a hydrophobic porous membrane to selectively permit the transport of water vapor from a heated feed solution to a cooler permeate side. The driving force in MD is the vapor pressure gradient across the hydrophobic membrane surfaces, influenced by temperature differences and other operational and structural parameters [1,2,3,4,5]. Recent advancements in superhydrophobic membranes have demonstrated potential for enhancing mass flux by mitigating temperature polarization effects, which otherwise reduce the vapor pressure gradient. Such improvements have been explored through the development of novel membranes and modifications to existing commercial ones [6,7,8,9].

Compared to conventional desalination technologies, such as reverse osmosis (RO), MD offers several notable advantages. These include high rejection rates for non-volatile solutes, operation under low pressures, and compatibility with low-grade waste heat or renewable energy sources. The energy efficiency of MD distinguishes it from pressure-driven processes like nanofiltration (NF) and RO, positioning it as a viable alternative for diverse applications, including the removal of organic contaminants and heavy metals from wastewater [10,11]. Additionally, MD achieves near-total rejection of non-volatile solutes, making it particularly suitable for treating high-salinity brines, such as RO concentrate streams or integrating with multiple-effect distillation (MED) systems [12,13,14,15,16,17].

To evaluate and compare the performance of three commercial hydrophobic membranes for MD, an experimental setup was developed for the characterization of a flat-plate direct contact membrane distillation (DCMD) module. This setup allowed controlled variation of operational parameters, including feed and permeate temperatures and flow rates. DCMD is characterized by its simplicity, involving direct contact between the both the hot feed stream and the condensing liquid with the membrane, facilitating vapor transport and condensation [18,19]. However, alternative configurations, such as air-gap membrane distillation (AGMD), also hold significant promise. The AGMD configuration incorporates an air gap between the membrane and the condensing surface, acting as a thermal barrier that reduces heat losses through the membrane and enhances thermal efficiency [18,20]. While DCMD are known for offering ease of operation and higher vapor fluxes, AGMD provides superior thermal efficiency, requiring a trade-off based on specific process requirements. Thus, a second experimental setup was assembled, again with full control of the operational variables, to allow for critical comparisons of the two configurations with different commercially available polymeric membranes.

Therefore, this study provides a detailed comparative analysis of DCMD and AGMD configurations, employing three different membranes made of polyvinylidene fluoride (PVDF), polytetrafluoroethylene (PTFE), and polyethylene (PE). Experiments over a wide range of NaCl concentrations, from 7 to 70 g/L, in conjunction with theoretical modelling, were utilized to assess the effects of key operational parameters - feed concentration, feed temperature, and flow rates - on permeate flux and thermal efficiency. The present analysis aims to, while verifying the relative merits of the DCMD and AGMD configurations, inspect whether such differences are more or less marked for the three different membrane materials with their specific structural characteristics.

2. Materials and Methods

2.1. Membranes and Module Configuration

Three different commercial polymeric membranes were used in the experiments, namely, PVDF, PTFE, and PE. The membranes were characterized in terms of their structure, wettability, and thermal properties, as can be found in [5,21,22]. The detailed characteristics of each membrane, including thickness, mean pore size, and porosity, are provided in Table 1.

The membranes were tested in both the DCMD and AGMD configurations using flat-sheet modules designed and built for these setups. The DCMD module had an effective membrane area of 0.005062 m², while the AGMD module had an effective membrane area of 0.006532 m².

To design the geometry of a DCMD module with rectangular channels that ensure flow uniformity within the microchannels, CFD (computational fluid dynamics) simulations were conducted, with the platform OpenFOAM (2021). These simulations varied the shape and characteristics of the inlet and outlet plenums on both the feed and distillate sides (Naveira-Cotta et al., 2023). For the analyzed range of volumetric flow rates, the chosen best configuration with eight microchannels per side, is shown in Figure 1. Velocity profiles obtained from the OpenFOAM simulations for various volumetric flow rates (0.2, 0.8, and 1.2 L/min) are shown in Figures 2. Figure 2a, at a central plane of the channels’ length, illustrates the uniformity of the velocity profiles at the various channels. Figure 2b provides velocity profiles at three different cross-sections (near the inlet, center, and outlet), reconfirming the fairly uniform distributions at different longitudinal positions.

The DCMD module’s flow uniformity for all analyzed flow rates supports the simplification assumption in the theoretical model towards formulating the heat and mass transfer process for a single representative microchannel. The AGMD module was designed as shown in Figure 3 and incorporates a 2 mm air gap between the membrane surface and the condensation plate.

2.2. Experimental Setup

A desalination bench was designed and built at LabMEMS/UFRJ for the evaluation of the distillation unit performance and characterization of commercial and modified superhydrophobic membranes, initially with a module in the DCMD configuration. The proposed experimental system (Figure 4) consists of two hydraulic circuits: a feed circuit described by the orange lines, where the saline water flows, and a permeate circuit described by the blue lines, where the distillate water flows.

In the DCMD configuration, the membrane is the only existing barrier that separates the feed and the permeate streams. The heating of the feed stream and the cooling of the permeate stream are carried out with the aid of two compact heat exchangers connected to two ultra-thermostatic baths to control the inlet temperatures of both streams. The specification of operating parameters for this experimental setup are provided in Table 2.

The experimental setup includes temperature sensors (K-type thermocouples – chromel/alumel) at the inlet and outlet ports of the distillation module, four pressure sensors (Rucker, model RTP-420 0-0.6 bar), two microturbine volumetric flow rate meters (Badger BV1000-TRN-025-B), two probes for measuring feed and permeate electrolytic conductivity (Digimed DM32), and two fluid reservoirs, supported by precision scales (Marte AD 5002), where the instantaneous mass of the feed and permeate are measured and acquired to compute the mass flow rates.

The utilization of NaCl concentrations exceeding 35 g/L in membrane distillation experiments is crucial for accurately simulating the conditions found in saline water sources, such as industrial and mining wastewater and reverse osmosis brines. Membrane distillation is a promising desalination technology capable of treating high-salinity waters, including petroleum reservoirs production water, while achieving high solute rejection rates [23,24].

3. Theoretical Model

The modelling and simulation of the heat and mass transfer phenomena that occur in membrane distillation processes are crucial to theoretically evaluate the transport mechanisms and the interaction of parameters that influence the efficiency of the process. This task is essential to reduce costs and time in the experimental analysis for the various combinations of operational parameters, as well as to achieve a proper design and operational optimization. In this sense, in addition to the influence of membrane characteristics, one of the main factors that affects the performance of the membrane distillation process is the temperature polarization phenomenon, evaluated by the temperature polarization coefficient (TPC) as:

$$TPC={\displaystyle \frac{T_{fm}-T_{pm}}{T_f-T_p}}$$

(1)

Temperature polarization refers to the effect that the thermal boundary layer causes in reducing the temperature difference between the membrane surfaces ($T_{fm}$- temperature at the feed-membrane interface, and $T_{pm}$ - temperature at the membrane-permeate interface) with respect to the temperature difference between the fluid streams ($T_f$ - average temperature of the fluid in the feed stream, and $T_p$ - average temperature of the fluid in the permeate stream). The mass transport of the volatile component (vapor) occurs through the pores of the membrane while heat is transferred both by the membrane solid matrix and by the vapor itself [25]. Also, the ratio between the total feed concentration and the concentration at the liquid-vapor interface on the feed side is called concentration polarization coefficient, and can be determined from [26]:

$$\mathrm{C}\mathrm{P}\mathrm{C}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{f}\mathrm{m}}}{C_f}}$$

(2)

3.1. Heat Transfer

From the feed channel to the permeate channel, the heat transfer process in a DCMD configuration is modelled across the three regions (feed stream, membrane, and permeate stream) as a lumped system, as shown in Figure 5: (i) the convective heat transfer $\left({\dot{Q}}_{conf\_f}\right)$ across the thermal layer of the feed stream in contact with the membrane, given by Eq. (2) (ii) the transmembrane heat transport (${\dot{Q}}_m$) described by Eq. (3) and (iii) heat transfer by convection (${\dot{Q}}_{conv\_p}$) across the thermal boundary layer of the permeate stream in contact with the other side of the membrane [25,27,28,29].

$${\dot{\mathrm{Q}}}_{{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}_{\mathrm{f}}}={\mathrm{h}}_{\mathrm{f}}{\mathrm{A}}_{\mathrm{m}}({\mathrm{T}}_{\mathrm{f}}-{\mathrm{T}}_{\mathrm{f}\mathrm{m}})$$

(3)

$$\dot{{\mathrm{Q}}_{\mathrm{m}}}={\dot{\mathrm{Q}}}_{\mathrm{v}\mathrm{a}\mathrm{p}\_\mathrm{m}}+{\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\_\mathrm{m}}={\mathrm{A}}_{\mathrm{m}}\left[{\mathrm{N}}_{\mathrm{w}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{V}}+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{m}}}{{\mathsf{\delta }}_{\mathrm{m}}}}({\mathrm{T}}_{\mathrm{f}\mathrm{m}}-{\mathrm{T}}_{\mathrm{p}\mathrm{m}})\right]$$

(4)

$${\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\_\mathrm{p}}={\mathrm{h}}_{\mathrm{p}}{\mathrm{A}}_{\mathrm{m}}({\mathrm{T}}_{\mathrm{p}\mathrm{m}}-{\mathrm{T}}_{\mathrm{p}})$$

(5)

where $\dot{Q}$is the heat transfer rate [W], $h$ is the convective heat transfer coefficient [W/m² K], $A_m$ is the area of the membrane normal to the heat flux [m²], $N_w$ is the permeate flux [ kg/m² s], $\Delta H_V$ is the latent heat of vaporization [J/kg], $k_m$ is the effective thermal conductivity of the membrane [W/m K] and ${\delta }_m$ is the nominal thickness of the membrane [m].

The effective conductivity, $k_m$, can be estimated from the thermal conductivities of the solid and vapor phases $k_s$ and $k_g$, and the membrane porosity ε [-]. The parallel-resistance model (thermal conductivity in series) is the most used effective thermal conductivity model, followed by series-resistance (thermal conductivity in parallel) and the Maxwell I model [29,30,31,32]. In all these models, the fluid thermal conductivity in the bulk membrane, $k_{mg}$, is made equal to the water vapor thermal conductivity, which is considered a temperature-dependent property. For the interface regions, however, the liquid water thermal conductivity is used. Here, the effective conductivity, $k_m$, is described by the isostrain (parallel) model, Eq. (6), which the literature indicates as adequate for membranes with solid and gas phases approximately parallel to each other [33,34,35,36]

$${\mathrm{k}}_{\mathrm{m}}=\left(1-\mathsf{\epsilon }\right){\mathrm{k}}_{\mathrm{s}}+\mathsf{\epsilon }{\mathrm{k}}_{\mathrm{g}}$$

(6)

In the steady-state regime, the three Eqs. (3), (4) and (5) can be combined to determine the surface temperatures at the membrane interfaces ($T_{fm}$ and $T_{pm}$), once the heat transfer coefficients are computed, equating the heat transfer rates in each region:

$${\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\_\mathrm{f}}={\dot{\mathrm{Q}}}_{\mathrm{m}}={\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\_\mathrm{p}}$$

(7)

By means of the energy balance of Eq. (7), and rewriting $k_m/{\delta }_m$ as $h_m$, the membrane surface temperatures on the feed ($T_{fm}$) and permeate ($T_{pm}$) sides can be obtained as:

$${\mathrm{T}}_{\mathrm{f}\mathrm{m}}={\displaystyle \frac{\left({\mathrm{h}}_{\mathrm{m}}\left[{\mathrm{T}}_{\mathrm{p}}+\left(\frac{{\mathrm{h}}_{\mathrm{f}}}{{\mathrm{h}}_{\mathrm{p}}}\right){\mathrm{T}}_{\mathrm{f}}\right]+{\mathrm{h}}_{\mathrm{f}}{\mathrm{T}}_{\mathrm{f}}-{\mathrm{N}}_{\mathrm{w}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{V}}\right)}{{\mathrm{h}}_{\mathrm{m}}+{\mathrm{h}}_{\mathrm{f}}\left(1+\frac{{\mathrm{h}}_{\mathrm{m}}}{{\mathrm{h}}_{\mathrm{p}}}\right)}}$$

(8)

$${\mathrm{T}}_{\mathrm{p}\mathrm{m}}={\displaystyle \frac{\left({\mathrm{h}}_{\mathrm{m}}\left[{\mathrm{T}}_{\mathrm{f}}+\left(\frac{{\mathrm{h}}_{\mathrm{p}}}{{\mathrm{h}}_{\mathrm{f}}}\right){\mathrm{T}}_{\mathrm{p}}\right]+{\mathrm{h}}_{\mathrm{p}}{\mathrm{T}}_{\mathrm{p}}+{\mathrm{N}}_{\mathrm{w}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{V}}\right)}{{\mathrm{h}}_{\mathrm{m}}+{\mathrm{h}}_{\mathrm{p}}\left(1+\frac{{\mathrm{h}}_{\mathrm{m}}}{{\mathrm{h}}_{\mathrm{f}}}\right)}}$$

(9)

The present geometric and operating conditions warrant laminar flow regime on both fluid streams, so the following correlation for Nusselt number, previously employed for the DCMD configuration [33,35], is adopted:

$$\mathrm{N}\mathrm{u}\equiv {\displaystyle \frac{\mathrm{h}{\mathrm{d}}_{\mathrm{h}}}{\mathrm{k}}}=4.36+{\displaystyle \frac{0.036\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}\left(\frac{{\mathrm{d}}_{\mathrm{h}}}{\mathrm{L}}\right)}{1+0.0011{\left[\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}\left(\frac{{\mathrm{d}}_{\mathrm{h}}}{\mathrm{L}}\right)\right]}^{2/3}}}, \mathrm{R}\mathrm{e}<2100$$

(10)

where k is the fluid thermal conductivity [W/m K], $\rho$ is the density [kg/m³], µ is the dynamic viscosity [Pa.s], and $c_p$ is the specific heat [J/kgK], $\overline{u}$ is the feed or permeate mean velocity [m/s], d_h is the hydraulic diameter of the channel, Re is the Reynolds number, $Pr$ is the Prandtl number, and $L$ is the channel length.

3.2. Mass Transfer

The permeate flux through the membrane ($N_w$) is described by the Dusty-Gas model, based on the Maxwell-Stefan relationships. Neglecting surface diffusion, the Dusty-Gas model is written as [34]:

$${\displaystyle \frac{{\mathrm{J}}_{\mathrm{w}}}{{\mathrm{D}}_{\mathrm{w}}^{\mathrm{k}}}}+{\displaystyle \frac{{\mathrm{p}}_{\mathrm{a}}{\mathrm{J}}_{\mathrm{w}}-{\mathrm{p}}_{\mathrm{w}}{\mathrm{J}}_{\mathrm{a}}}{{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}}}=-{\displaystyle \frac{1}{\mathrm{R}{\mathrm{T}}_{\mathrm{m}}}}\nabla {\mathrm{p}}_{\mathrm{w}}$$

(11)

where $J_w$ and $J_a$ are the molar fluxes of water vapor and air [mol/m² s], $p_w$ and $p_a$ are the partial pressures of water vapor and air inside the pore [Pa], $D_w^k$ is the Knudsen diffusivity, $D_{wa}^o$ is the molecular diffusivity of water in the air, $T_m$ is the thermodynamic average temperature inside the pores, and R is the universal gas constant.

Therefore, the permeate flow, $N_w$, is determined from the permeate molar flow, $J_w$, from which is the result of Eq. (11) neglecting viscous transport and surface diffusion, zero airflows ($J_a\cong 0$) inside the pores and considering Dalton’s law [37,38]:

$${\mathrm{N}}_{\mathrm{w}}={\mathrm{J}}_{\mathrm{w}}{\mathrm{M}}_{\mathrm{w}}$$

(12)

$${\mathrm{J}}_{\mathrm{w}}={\mathrm{C}}_{\mathrm{h}\mathrm{p}}{\displaystyle \frac{{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}}{\mathrm{R}{\mathrm{T}}_{\mathrm{m}}{\mathsf{\delta }}_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}}}}\ln \left({\displaystyle \frac{{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}-{\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}{\mathrm{p}}_{\mathrm{f}}^{\mathrm{v}\mathrm{a}\mathrm{p}}}{{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}-{\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}{\mathrm{p}}_{\mathrm{p}}^{\mathrm{v}\mathrm{a}\mathrm{p}}}}\right)$$

(13)

where ${\delta }_{appa}$ is the membrane apparent thickness, $M_w$ is the molecular mass of water, $D_{eff}$ is the effective diffusion coefficient, and $p_f^{vap}$ and$p_p^{vap}$ are the partial vapor pressures at the membrane interface on the feed and permeate sides, respectively.

Considering that the phenomenon of capillary depression is a consequence of the hydrophobicity of the membrane, [39,40] proposed the two correction factors for the distillate mass flux, namely, $C_{hp}$ and ${\delta }_{appa}$. The curved region formed at the entrance of the membrane pores increases the total area of the liquid-vapor interface according to the hydrophobicity degree (related to the contact angle, θ), which varies for each membrane. This effect is accounted for by the factor $C_{hp}$. Additionally, due to this increase on the interfacial surface, the distance between the two liquid/vapor interfaces is reduced, thus an apparent membrane thickness (${\delta }_{appa}$) was also introduced in the mass flux relation [41]:

$${\mathrm{C}}_{\mathrm{h}\mathrm{p}}={\displaystyle \frac{2}{\left[1+\sin \left(\mathsf{\theta }\right)\right]}}$$

(14)

$${\mathsf{\delta }}_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}}=\mathsf{\delta }-({\mathsf{\delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\_\mathrm{f}\mathrm{e}\mathrm{e}\mathrm{d}}+{\mathsf{\delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\_\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}})$$

(15)

The interface thickness (${\delta }_{{int}_{feed}}$) for the feed/mebrane interface, and ${\delta }_{{int}_{perm}}$ for the membrane/distillate interface are evaluated as the lengths of the portion of the pore volume that is occupied by the liquid at the pore channel entrance, which is approximated as the volume of a spherical cap, being a function of the mean pore diameter, d_p, and the contact angle, θ [41]:

$${\mathsf{\delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{p}}}{2\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\theta }\right)}}\left[\sin \left(\mathsf{\theta }\right)-1\right]$$

(16)

The vapor partial pressure at the membrane interface, $p_i^{vap}$, is a function of the temperature, and it is estimated according to the Antoine equation [34]:

$${\mathrm{p}}_{\mathrm{i}}^{\mathrm{v}\mathrm{a}\mathrm{p}}=\mathrm{e}\mathrm{x}\mathrm{p}\left(23.1964-{\displaystyle \frac{3816.44}{{\mathrm{T}}_{\mathrm{m},\mathrm{i}}-46.13}}\right)$$

(17)

where $T_{m,i}$ is the fluid/membrane interface temperature (in Kelvin). For the feed side, the saline concentration (NaCl) must be considered, since the saline concentration will reduce the partial pressure. For non-ideal binary mixtures, the vapor partial pressure is often described as:

$${\mathrm{p}}_{\mathrm{i}}^{\mathrm{v}\mathrm{a}\mathrm{p}\mathrm{\text{'}}}={\mathrm{x}}_{\mathrm{w}}{\mathrm{a}}_{\mathrm{w}}{\mathrm{p}}_{\mathrm{i}}^{\mathrm{v}\mathrm{a}\mathrm{p}}$$

(18)

where $x_w$ is the water molar fraction, $p_i^{vap}$ is the water vapor partial pressure given by the Antoine equation, Eq. (14), and $a_w$ is the activity coeficient, related to the feed sodium chloride molar fraction ${\mathcal{X}}_{NaCl}$, as:

$${\mathrm{a}}_{\mathrm{w}}=1-\mathrm{0,5}{\mathcal{X}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}-10{\mathcal{X}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}^2$$

(19)

Finally, the effective diffusion coefficient is expressed as function of the Knudsen diffusivity $D_w^k$, and the molecular diffusivity of water in air, $D_{wa}^o$, besides the total pressure inside the pores, P:

$${\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\left({\displaystyle \frac{{\mathrm{D}}_{\mathrm{w}}^{\mathrm{k}}{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}}{{\mathrm{D}}_{\mathrm{w}\mathrm{a}}^{\mathrm{o}}+\mathrm{P}{\mathrm{D}}_{\mathrm{w}}^{\mathrm{k}}}}\right)$$

(20)

$${\mathrm{D}}_{\mathrm{w}}^{\mathrm{k}}={\displaystyle \frac{\mathsf{\epsilon }{\mathrm{d}}_{\mathrm{p}}}{3\mathsf{\tau }}}\sqrt{{\displaystyle \frac{8\mathrm{R}{\mathrm{T}}_{\mathrm{m}}}{\mathsf{\pi }{\mathrm{M}}_{\mathrm{w}}}}}{\mathrm{N}}_{\mathrm{w}}={\mathrm{J}}_{\mathrm{w}}{\mathrm{M}}_{\mathrm{w}}$$

(21)

$$D_{wa}^o=4.46\bullet {10}^{-6}{\displaystyle \frac{\epsilon }{\tau }}{T}_m^{2.334}$$

(22)

where ε is the membrane porosity, τ is the membrane tortuosity [42] and $d_p$ is the membrane pore diameter.

Several tortuosity models can be found in the literature [43,44,45,46,47]. A critical review of some tortuosity models was performed in [48]. Recently, to better understand the effects of the tortuosity on the distillate mass flux in DCMD systems, and to better evaluate their adequacy in describing this output parameter, [49], evaluated eleven tortuosity models (eight based on geometrical hypothesis and three based on fractal theory), comparing their predictions with experimental results. The geometry-based tortuosity models are supported by distribution of Euclidean distances between pores of the membrane structure. On the other hand, fractal theory-based porous media assumes that molecules diffuse along a path within a fractal system. Nevertheless, in both Euclidean and fractal models, tortuosity is always estimated from relationships in terms of membrane porosity. In the present work, for the morphology of the porous membrane used, we use the expression proposed by Iversen et al. (1997a), Jonsson (1980a) and Mackie et al. (1955):

$$\mathsf{\tau }={\displaystyle \frac{{\left(2-\mathsf{\epsilon }\right)}^2}{\mathsf{\epsilon }}}$$

(23)

For AGMD, the distillate mass flux can be calculated by compensating for the resistance of the air space, as follows [53]

$${\mathrm{J}}_{\mathrm{w}}={\displaystyle \frac{{(\mathrm{P}}_{\mathrm{w}\mathrm{a}}-{\mathrm{P}}_{\mathrm{c},\mathrm{f}})}{{\mathrm{R}}_{\mathrm{a}\mathrm{g}}}}$$

(24)

where,

$${\mathrm{R}}_{\mathrm{a}\mathrm{g}}={\left[{\displaystyle \frac{\mathsf{\epsilon }\mathrm{D}\mathrm{P}}{{\mathsf{\delta }}_{\mathrm{a}}{\mathrm{P}}_{\mathrm{a}\mathrm{g},\mathrm{l}\mathrm{o}\mathrm{g}}\mathrm{R}}}{\displaystyle \frac{{\mathrm{M}}_{\mathrm{v}}}{\left({\mathrm{T}}_{\mathrm{a}\mathrm{g},\mathrm{a}\mathrm{v}\mathrm{g}}+273.15\right)}}\right]}^{-1}$$

(25)

where P_c,f is the vapor pressure at the air gap interface of the condensing plate, R_ag is the air gap resistance, δ_a is the air gap width, P_ag,log is the log mean pressure in the air gap, and T_ag,log is the log mean temperature in the air gap. The total resistance in the AGMD model can then be calculated by:

$${\mathrm{R}}_{\mathrm{A}\mathrm{G}\mathrm{M}\mathrm{D}}={\mathrm{R}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{a}\mathrm{g}}$$

(26)

Then,

$${\mathrm{R}}_{\mathrm{A}\mathrm{G}\mathrm{M}\mathrm{D}}=\left({\displaystyle \frac{\mathsf{\tau }\mathsf{\delta }}{\mathsf{\epsilon }}}+{\mathsf{\delta }}_{\mathrm{a}\mathrm{g}}\right){\displaystyle \frac{{\mathrm{P}}_{\mathrm{a}\mathrm{g},\mathrm{l}\mathrm{o}\mathrm{g}}}{\mathrm{D}{\mathrm{P}}_{\mathrm{t}}}}{\displaystyle \frac{{\mathrm{R}}_{\mathrm{g}}{\mathrm{T}}_{\mathrm{a}\mathrm{g},\mathrm{a}\mathrm{v}\mathrm{g}}}{{\mathrm{M}}_{\mathrm{v}}}}+{\left[{\displaystyle \frac{2\mathsf{\epsilon }\mathrm{r}}{\mathsf{\tau }\mathsf{\delta }}}{\left({\displaystyle \frac{8{\mathrm{M}}_{\mathrm{v}}}{\mathsf{\pi }{\mathrm{R}}_{\mathrm{g}}{\mathrm{T}}_{\mathrm{m}}}}\right)}^{0.5}\right]}^{-1}$$

(27)

where δ, δ_ag, ε, τ and r are the thickness of the membrane, the thickness of the air gap, the porosity of the membrane, the tortuosity of the pores and the average pore size of the membrane, respectively. P_a,log is the logarithmic mean air pressure between the two sides of the membrane, D is the diffusion coefficient of water vapor through the air gap, P_t is the total air and water vapor pressure.

3.3. Implementation of the Computational Model

The heat and mass transfer characteristics of the DCMD system (given by Eqs. (1-22)) are evaluated through a computational program developed and implemented in Matlab-Simulink platform (R2019a), for the theoretical prediction of the distillate mass flux, between other variables. Since there is an interdependency between distillate mass flux and the temperatures at the membrane/fluid interfaces, i.e., $N_w=f\left(T_{fm},T_{pm}\right)$, $T_{fm}=f\left(N_w\right),$ and $T_{pm}=f\left(N_w\right)$, it is necessary to employ an iterative procedure for the simultaneous convergence of all dependent variables. The flowchart of the iterative algorithm is depicted in Figure 6. Initially, all morphological membrane parameters must be defined, physical properties, module geometry, and process operating conditions. The thermophysical properties of water (density, specific heat, thermal conductivity and viscosity) are evaluated according to the database of the EES software (Engineering Equation Solver). The physical and chemical properties for the saline solutions were evaluated according to the correlations presented in [54]. After estimating the temperatures at the interfaces of the membrane, and determining the heat and mass transfer parameters, the convergence for the distillate mass flux is verified, in each iterative step, to within a predefined relative error criterion, $tol={10}^{-6}$.

Three performance metrics were used to evaluate the improvement associated with the implementation of an air gap configuration as compared to a direct contact configuration. The parameters are the gain output ratio (GOR), thermal efficiency (TE), and specific energy consumption (SEC). The GOR, defined as the ratio of heat transferred to water vapor against the total heat input, remains equal to or less than 1 in cases when latent heat is not recovered, and is given by:

$$\mathrm{G}\mathrm{O}\mathrm{R}={\displaystyle \frac{{\dot{\mathrm{m}}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{v}}}{{\dot{\mathrm{Q}}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}}}$$

(28)

Another essential parameter for such comparisons is the specific energy consumption (SEC), which assesses the energy efficiency of the process. By measuring the amount of energy consumed to produce a specific amount of distillate, SEC is a valuable indicator for optimizing the relationship between the energy input and the output obtained in membrane distillation. Minimizing SEC not only implies energy savings but also promotes more sustainable and economical practices. This parameter can be calculated as:

$$\mathrm{S}\mathrm{E}\mathrm{C}={\displaystyle \frac{{\dot{\mathrm{Q}}}_{\mathrm{p}}{\mathsf{\rho }}_{\mathrm{w}}}{{\mathrm{J}}_{\mathrm{w}}\mathrm{A}}}$$

(29)

Thermal efficiency plays a critical role when membrane distillation is driven by sustainable heat sources such as solar energy. Maximizing thermal efficiency means optimizing the conversion of thermal energy into useful work, contributing not only to process efficiency, but also to the ongoing search for more sustainable and energy-efficient industrial solutions. The thermal efficiency of the membrane distillation process is defined as the fraction of heat transfer due to mass transfer through the membrane pores and total heat transferred through the membrane [55]. The thermal efficiency parameter is ideal for MD systems with little or no condensation energy recovery. The thermal efficiency of the MD process can be expressed by:

$$\mathrm{T}\mathrm{E}={\displaystyle \frac{\dot{{\mathrm{Q}}_{\mathrm{p}}}}{{\dot{\mathrm{Q}}}_p+{\dot{Q}}_{cond}}}={\displaystyle \frac{{\mathrm{J}}_{\mathrm{w}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{v}}\mathrm{A}}{{\mathrm{J}}_{\mathrm{w}}\mathsf{\Delta }{\mathrm{H}}_{\mathrm{v}}\mathrm{A}+{\mathrm{h}}_{\mathrm{m}}({\mathrm{T}}_{\mathrm{f}\mathrm{m}}-{\mathrm{T}}_{\mathrm{p}\mathrm{m}})}}$$

(30)

The interconnection of these indicators provides a comprehensive approach to understand the overall performance and improve efficiency and productivity in membrane distillation units.

4. Results and Discussion

This section presents the results obtained from both the bench experiments and the mathematical modelling. This study aimed to maximize the system performance and compare the effects on the main parameters for the three different membranes. The operating conditions including temperatures, salinity, and mass flow rates are as given in Table 2.

4.1. Distillate Fluxes in DCMD and AGMD Modules

Figure 7 illustrates the experimental results for distillate mass flux as a function of the different combinations of module configurations (DCMD and AGMD) and types of membranes (PVDF, PTFE and PE) used in the membrane distillation process, as indicated in the legends. It can be observed that the DCMD configuration provided significantly higher distillate fluxes compared to the AGMD configuration, especially for the PE and the PTFE membranes.

In Figure , it can be observed the influence of different NaCl concentrations (7, 35, and 70 g/L) on the performance of the membranes in both configurations (DCMD and AGMD). In the DCMD configuration with the PTFE membrane, Figure 8.c, the distillate flux values remained high and consistent, ranging from 21.87 kg/m²h (for 70 g/L) to 22.15 kg/m²h (for 7 g/L). This behavior indicates the adequacy of the PTFE membrane for high salinity conditions, attributed to its high hydrophobicity and porosity, which favor vapor transfer even under high salinity conditions. In contrast, the PE membrane in the DCMD configuration, Figure 8.a, showed a reduction in distillate flux as the NaCl concentration increased, starting from 23.82 kg/m²h (7 g/L), dropping to 21.07 kg/m²h (35 g/L), then to 20.01 kg/m²h (70 g/L). This behavior reflects a noticeable sensitivity to saline conditions, limiting its performance in environments with high NaCl concentrations.

In the AGMD configuration, the distillate fluxes were considerably lower regardless of the membrane type or salt concentration. For instance, the PVDF membrane exhibited fluxes ranging from 12.52 kg/m²h (for 7 g/L) to 11.32 kg/m² h (for 70 g/L). The lowest distillate flux overall was obtained through the PE membrane with the AGMD module, 10.27 kg/m² h for 70 g/L.

Figure 9 shows the required heat transfer rate for each of the three membranes, with both the DCMD and AGMD modules and the three NaCl concentrations (7, 35 and 70 g/L), now at the fixed feed temperature of 70°C, permeate temperature at 32.5°C, and feed and permeate flow rates of 0.5 L/min. The DCMD configuration consistently requires higher heat transfer rates for all membranes, reconfirming the direct relationship between the higher distillate flux and the higher energy consumption. The PVDF membrane presented the highest thermal energy consumption with the DCMD module, with consistent values of required thermal power over 150 W for all NaCl concentrations. This performance reflects its high efficiency in vapor transfer, as corroborated by the higher distillate flux values in Figure . On the other hand, the PTFE and PE membranes in the DCMD configuration showed a slight reduction in heat consumption as the NaCl concentration increased. For the PTFE membrane, consumption decreased from 155.47 W (7 g/L) to 145.06 W (70 g/L), while for the PE membrane, the decrease was from 124.21 W (7 g/L) to 104.74 W (70 g/L). This moderate reduction indicates a lower sensitivity in maintaining the vapor pressure gradient under high salinity conditions.

In the AGMD configuration, energy consumption was lower, but not in every case. For instance, the PE membrane in AGMD configuration required slightly more thermal energy than the same membrane in DCMD configuration at 70 g/L. The PTFE membrane led to thermal energy consumption ranging from 118.53 W (at 7 g/L) to 99.34 W (70 g/L), reflecting the greater thermal resistance of this configuration. For the PE membrane, however, the case of highest salinity (70 g/L) led to the highest energy requirements in the AGMD configuration, higher than in the DCMD module for the same salinity, as previously pointed out. Although the PVDF membrane demanded higher thermal energy values, such as 152.44 W for 7 g/L, these were still considerably lower than those observed in the DCMD configuration.

Figure 10 shows the comparative performance of the two module configurations in the distillation process, in terms of the ratio of permeate fluxes and the ratio of required heat input in the feed for AGMD and DCMD. These ratios are influenced by both the feed flow rate and the type of membrane used, as shown in Figure 10. For lower flow rates (0.2 L/min), Figure 10a, greater dispersion is observed in the values of both ratios, indicating that the differences in performance between the processes are more striking under these conditions. As the flow rate increases (0.5 L/min and 0.8 L/min), the points become more grouped, indicating greater uniformity in the behavior of the processes. PVDF membranes showed greater variation in the values of the ration analyzed, reflecting greater sensitivity to operating conditions. PTFE membranes, in turn, showed more uniform behavior in all conditions, which can be attributed to their thermal and hydrophobic properties. PE membranes, on the other hand, showed inferior performance in terms of permeate flux, especially for lower flow rates, limiting their applicability in scenarios that demand greater mass transport. The PVDF membrane results lead to larger values of the permeate fluxes ratio, thus closer values of the permeate fluxes between the two modules, with lower values of the heat input ratio, thus less power requirements for the AGMD module with respect to DCMD one. All graphs refer to a NaCl concentration of 70 g/L.

4.2. Concentration and Temperature Polarization

Figure 11 show results for the thermal polarization coefficient (TPC) and concentration polarization coefficient (CPC), for the two membrane module configurations and the three different membranes. Figure 11a shows that the thermal polarization coefficient (TPC) gradually decreases with increasing feed temperature for all membranes analyzed (PE, PVDF, and PTFE). In the DCMD configuration, the TPC values ranged from 0.179 (70 g/L) to 0.192 (7 g/L) for the PE membrane and from 0.142 to 0.220 for the PVDF membrane, while for the PTFE membrane, ranging from 0.106 to 0.229. These results indicate a progressive reduction in the ratio of the temperature difference between the membrane surfaces and the difference between the two fluid streams, as the feed temperature increases and the permeate stream is kept at the same inlet temperature, which reflects increased vapor transport efficiency under higher temperature conditions. In the AGMD configuration, TPC values remained relatively uniform. The PE membrane exhibited variations between 0.274 (70 g/L) and 0.326 (7 g/L), while the PVDF membrane varied between 0.302 and 0.347. The PTFE membrane showed values ranging from 0.334 to 0.386.

Figure 11b shows a non-monotonic behavior for the concentration polarization coefficient (CPC) in the DCMD configuration, with maximum values recorded around 75 °C in the cases shown here. This behavior can be attributed to the increased solute accumulation near the membrane surface within this temperature range, resulting from the higher distillate flux observed in the DCMD configuration. CPC values in the DCMD configuration reached peaks of 1.11 (PE), 1.14 (PVDF), and 1.19 (PTFE), while in the AGMD configuration, the values were lower and more uniform. This reflects a reduced concentration polarization, attributed to the presence of the air gap and the lower permeate flow, which limits the accumulation of solutes near the membrane surface. When comparing the TPC and CPC values between the different types of membranes at the feed temperature of 75 °C, as illustrated in Figure 11c, slight differences are observed. The PTFE membrane presented the highest TPC (0.368, AGMD) and CPC (1.19, DCMD) values, indicating larger temperature polarization (TPC) in AGMD and concentration polarization (CPC) in DCMD, in comparison with the PE (TPC-AGMD: 0.361; CPC - DCMD: 1.11) and PVDF (TPC- AGMD: 0.367; CPC-DCMD: 1.14) membranes. The AGMD configuration exhibited more uniformity in relation to both TPC and CPC, compared to the DCMD configuration.

4.3. Comparison of Energy Efficiency

The Gain Output Ratio (GOR), Specific Energy Consumption (SEC), and Thermal Efficiency (TE) were analyzed in Figs. 12, considering the effects of feed temperature and salinity. The results illustrated in Figure 8 show that the GOR varies moderately with increasing feed temperature in the DCMD process. For example, the PE membrane has slightly decreasing GOR values (0.72–0.62), while the PVDF and PTFE membranes exhibit gradual increase, reaching 0.47 and 0.52, respectively, at 85°C. This behavior reflects the increase in the vapor pressure gradient, which facilitates mass transfer through the membrane, resulting in greater permeate production. Comparing the materials, it is observed that PE outperforms PVDF and PTFE in DCMD. However, in AGMD, lower GOR values are observed, with PE and PVDF membranes increasing to 0.47 and 0.42, respectively, at 85°C. The PTFE membrane achieves the highest GOR in AGMD, reaching 0.48, benefiting from the air barrier that minimizes thermal losses. PE in DCMD demonstrates significant advantages, with GOR up to 60% higher than PTFE in DCMD. In AGMD, the performance is more balanced. Figure 8 confirms that the SEC decreases with increasing feed temperature in most situations. In DCMD, PE achieves the lowest SEC values, starting at 828.48 (65°C) and becoming fairly uniform at higher temperatures. Conversely, PVDF and PTFE membranes show higher initial SEC values, with PVDF reducing from 2252.31 to 1154.81 and PTFE varying between 1980.98 and 1180.67. These data show that AGMD in general consumes less thermal energy compared to DCMD. In addition, the intermediate thermal properties of the PVDF membrane resulted in a balanced performance between the two configurations. Figure 8 illustrates that DCMD presents a higher TE, which improves with increasing feed temperature. The PTFE membrane achieves the highest TE in DCMD, rising from 0.75 to 0.95, followed by PE with values up to 0.81 at 85°C. The PVDF exhibits a moderate increase in TE, reaching 0.76. In AGMD, PTFE also presents superior TE, rising to 0.60 at 85°C, PE to 0.58 at 85°C whereas PVDF demonstrate lower efficiencies (with 0.40 at 85°C). The results for TE demonstrate that AGMD reduces thermal losses, with improvements of up to 107% for PTFE compared to DCMD.

The NaCl rejection coefficient for the three different membranes — PE (polyethylene), PVDF (polyvinylidene fluoride) and PTFE (polytetrafluoroethylene) — under different operating conditions, considering variations in the feed temperature (65 °C to 85 °C) and in the NaCl concentrations, remained above 99.8%, indicating high efficiency of the membranes in retaining the solute, even with thermal and concentration parametric variations.

4.4. Comparison of Experimental Results and the Mathematical Models

Figure 13 compare results of the permeate fluxes obtained experimentally and predicted by the mathematical models previously described for the DCMD and AGMD membrane distillation configurations. Each graph represents a specific combination of configuration (DCMD and AGMD) and membrane type (PE, PVDF and PTFE), and shows the relation between the experimental values (on the x-axis) and the predicted values (on the y-axis) for different NaCl concentrations (7, 35 and 70 g/L) and different flow rates (0.2, 0.5, and 0.8 L/min). The figures include reference lines for 15% relative deviation, which allows to easily assess the accuracy of the mathematical model with respect to the experimental data. It can be seen that, in all graphs, most of the points are concentrated close to the line of equality (y=x), indicating that the mathematical model presents good agreement with the experimental data for both configurations and for all membranes analyzed. The great majority of points lie within the bounds of the ±15% deviation lines, indicating that the model is a reliable tool in predicting MD performance under a variety of operating conditions. Some slight discrepancies can be noted, especially for the PVDF membrane in the DCMD configuration, where some points exceed the 15% deviation bounds.

5. Conclusions

This study comparatively analyzed the DCMD and AGMD configurations in the membrane distillation process, using different membrane materials (PE, PVDF and PTFE) and evaluating their performances under a wide range of operating conditions. The results highlight that the choice of configuration and membrane type significantly influences thermal efficiency, destillate flux, energy consumption and resistance to polarization and salinity phenomena. The DCMD configuration showed superior performance in terms of distillate flux, especially when associated with the PTFE membrane, whose pertinent properties — high porosity, high hydrophobicity, and thermal resistance — favored vapor transfer, resulting in fluxes of up to 33.59 kg/m²h. However, this configuration required more thermal energy, reflected in the high energy consumption values, which reached up to 180 W. On the other hand, the AGMD configuration demonstrated lower distillate productivity, but offered advantages in terms of energy consumption and temperature variability, characteristics that make it suitable for operations that prioritize energy savings and thermal control. The effect of salinity was more evident in the PE membrane, which showed a significant reduction in distillate flux with increasing NaCl concentration, indicating less robustness for applications in highly saline environments. In contrast, the PTFE membrane maintained consistent performance even at NaCl concentrations of up to 70 g/L, confirming its suitability for severe conditions in applications. The PVDF membrane proved to be sensitive to high temperature and salinity conditions, highlighting the need for adjustments in its application to avoid performance limitations. The temperature (TPC) and concentration (CPC) polarization coefficients indicated that the DCMD configuration, although favoring high distillate fluxes, is more susceptible to polarization effects, especially in thick membranes such as the PTFE one. In contrast, AGMD presented greater uniformity in CPC values, standing out as an alternative to mitigate the effects of polarization in prolonged processes or in highly concentrated solutions. The energy efficiency indicators corroborated the observed trends. The GOR was maximized at lower temperatures, especially for the PE membrane in DCMD (up to 0.8 at 65°C), while AGMD presented more balanced values due to the lower heat losses associated with the air gap. The specific energy consumption (SEC) was consistently lower in the AGMD configuration, with emphasis on the PTFE membrane, which reached 1110.93 kWh/m³ at 85°C, confirming its efficiency in high temperature scenarios. Finally, the validation of the mathematical model revealed good agreement with the experimental data, with deviations predominantly within the 15% deviation bounds. This adherence demonstrates the reliability of the model as a predictive tool for process improvement in different configurations and membrane types, although adjustments are recommended for specific cases, such as in the behavior of the PTFE membrane in the DCMD configuration under extreme conditions.

In summary, the results of this study provide a solid basis for the selection of membranes and configurations in membrane distillation applications, considering the specific demands of energy efficiency, salt rejection rate, and permeate fluxes. The superior performance of the PTFE membrane in the DCMD configuration positions it as the most suitable choice among those here considered for applications requiring high distillate fluxes, while the AGMD, with lower energy consumption, presents itself as a viable alternative for scenarios that prioritize thermal efficiency and lower susceptibility to polarization. These findings contribute significantly to the advancement of membrane distillation technology, offering valuable guidelines for its development and implementation in a wide variety of application contexts.