1. Introduction
Membrane distillation (MD) is a thermally driven process where the driving force is a vapor pressure difference created by a temperature difference across a porous hydrophobic membrane. MD can use low-grade heat from different sources, such as the sun, geothermal wells, and industrial processes, to produce ultra-fresh water [
1,
2,
3]. MD is also an interesting candidate to achieve zero liquid discharge and crystallization from different solutions due to its ability to treat highly concentrated solutions, such as brine from desalination facilities [
4,
5,
6,
7]. The use of MD for simultaneous recovery of freshwater and minerals from different sources of impaired water makes it relevant to achieving sustainability and a circular economy [
6,
8].
MD can be operated in several configurations, including air gap, vacuum, sweep gas, and direct contact. Direct contact membrane distillation (DCMD) is the simplest configuration of the process in terms of the equipment and modules involved [
9,
10,
11]. In DCMD, the membrane is in direct contact with the feed solution on one side and with the permeate on the other side. The driving force (i.e., vapor pressure difference) is induced by keeping the feed solution at a higher temperature than the permeate stream, creating a positive heat transfer through the membrane. Water and volatile compounds from the liquid feed evaporate, travel through the membrane pores, and are condensed at the membrane surface on the permeate side.
In DCMD, heat is transferred from the feed side to the permeate side due to the transport of water vapor through the membrane pores and conduction through the membrane [
12,
13]. As a result, the temperature at the membrane surface differs from its value in the bulk of the solution. The mass (vapor) flux across the membrane is directly linked with the difference in vapor pressures at the membrane surface on the feed and permeate sides, where the mass transfer coefficient of the membrane appears as a constant. The temperature difference between the bulk solution and membrane surface is known as temperature polarization, which decreases the effective driving force across the membrane and results in a reduction in transmembrane vapor flux [
13]. The mass transfer coefficient of the membrane is a function of membrane properties including pore size, overall porosity, thickness, and the
τ which, depending upon the membrane pore size and mean free path of the water vapor, can be calculated according to different models [
3]. Determination of the vapor pressure at the surface requires knowledge of the membrane surface temperatures, which are linked with the bulk temperatures through heat transfer coefficients [
14]. The thermal conductivity of the membrane (
) affects the heat conducted across the membrane and therefore directly influences the total heat transport across the membrane and hence the temperature at the membrane surface [
15]. Thus, the determination of flux is associated with the calculation of surface temperatures and the mass transfer coefficient of the membrane.
Understanding heat and mass transport in MD is important to design, improve, and optimize the process and module design [
3,
16,
17,
18]. Numerous semi-empirical correlations have been proposed to calculate the Nusselt number (
Nu) for heat transfer coefficient in MD channels (see
Section 2.3) [
13,
19]. The ultimate selection of the correlation for the
Nu for a given fluid is a function of the applied hydrodynamics and the system configuration (e.g., flat sheet, hollow fiber). Likewise,
Nu, several correlations have been proposed to describe the
, including the parallel resistance model, the series resistance model, and the Maxwell I model, as described in
Section 2.3. In all these correlations, effective membrane thermal conductivity models account for the membrane porosity, the thermal conductivity of stagnant air within the pore, and the thermal conductivity of the membrane material. High
decreases the temperature gradient across the membrane, which results in lower vapor flux [
20]. For
τ, which is inversely linked with the vapor flux, three different approaches have been adopted [
21,
22]: (i) use it as an adjustable parameter in the model; (ii) use a constant value (usually between 1 and 2, but occasionally greater than 2) for the
τ; and (iii) use theoretical approaches to link the membrane porosity with the
τ.
Despite their fundamental importance, heat and mass transport in MD are poorly understood [
13,
23,
24]. The current state-of-the-art modeling of MD approaches has two major limitations regarding the use of various correlations for the
Nu,
, and
τ of the membrane. Firstly, they compare the validity of various correlations proposed for any of the three parameters (
Nu,
τ, and
) for a fixed combination of the other two parameters. In other words, the state-of-the-art approaches do not test the validity of various combinations of correlations for the
Nu,
, and
τ of the membrane. For instance, in some studies for the determination of a suitable correlation for the
Nu, it was assumed that the
could be represented by the parallel model [
13,
25]. Phattaranawik et al. considered the suitability of
correlations in their DCMD model but neglected the correlations for
τ [
26]. Kim et al. studied the effect of using eleven different correlations for the
τ on the flux and concluded that the use of an inappropriate correlation can incorporate a significant error in the predicted flux [
27]. However, the study was carried out by assuming that heat transport within the membrane and in feed and permeate channels can be described by using a fixed combination of correlations for
and
Nu. This approach is clearly very specific to the membrane and operating setup applied in each of the studies, and its validity for a broad set of membrane and module characteristics cannot be guaranteed. The second important limitation of the current state-of-the-art is that the validation of different correlations for
Nu,
, and
τ has been tested by comparing the theoretical and experimental values of vapor flux only [
11,
15,
27,
28,
29]. The potential of these models to predict the outlet temperatures of feed and permeate, which are crucial to calculating thermal and cooling energy demand, respectively, is broadly neglected in the current literature.
The overall objective of the current study is to analyze the capability of various combinations of state-of-the-art correlations for Nu, , and τ to predict the experimental flux and outlet temperatures for hollow fiber membrane modules. The ultimate objective is to find the best-suited combination of Nu, , and τ to predict the experimental data (flux and outlet temperatures).
4. Observed Tendencies
All different combinations of adjustable parameters and the corresponding
R2 values are shown in
Table A1 in the
Appendix A. The results have been reported in decreasing order with respect to
Rtot2. From
Table A1, it is evident that
Rtot2 values vary significantly for different combinations, and even though it is not possible to clearly evaluate the prediction accuracy for every correlation, some tendencies can be observed.
It is noteworthy that among the best-performing models, the most commonly used correlation for the determination of is . This indicates that can describe the heat transfer across the membrane more accurately than the remaining correlations. Contrarily, most of the models with poor overall prediction performance use correlation. Models based on especially show poor prediction of the outlet temperatures, suggesting that and are not weighted appropriately in the series resistance model () and that the focus of this correlation may be flux prediction.
A similar tendency is observed when examining tortuosity correlations, where
, when paired with the appropriate correlations for
Nu and
, best predicts the experimental outputs. The poorest prediction of the correlations for
τ follows the order
,
, and
. Generally, a tendency is observed where a lower value of
τ corresponds to a lower prediction ability, except for
and
. This suggests that the optimal value of
τ is around 2.0 for a membrane with high porosity, which is also reported in a study by Khayet et al. [
3]. The value of
τ has a direct impact on the permeability coefficient,
B, which is used to determine the transmembrane flux. Higher values of
τ entail lower
B and thereby also lower predicted flux. This observation leads to the conclusion that most of the correlations used for
τ tend to overestimate the flux.
When observing the Nu correlations, no clear tendency was observed. This is evident from the fact that and are present in both the models with the highest and lowest prediction accuracy. This might also suggest that the choice of the Nu correlation is of less significance than those of τ and . Furthermore, these observed tendencies for the correlations of τ, Nu, and emphasize the importance of making the correct choice of model for theoretical DCMD modeling.