Thin film theory and boundary layer theory, an approach for analysing heat and mass transfer in spacer-filled Direct Contact Membrane Distillation

Heat and mass transfer in spacer-filled direct contact membrane distillation were experimentally investigated and explained in more detail based on the combination of the thin film theory and boundary layer theory. Experimental results were used to calculate mass transfer coefficient through the boundary layer instead of existing Sherwood number correlations or modeling results. Consequently, the change of direct contact membrane distillation (DCMD) performance in terms of mass fluxes, temperature polarization coefficient (TPC), and concentration polarization coefficient (CPC) could be analyzed more clearly through the fluctuation of mass transfer coefficient through the boundary layer, and boundary layer thickness. Moreover, the results also revealed that the effect of membrane pore size on the mass transfer coefficient through the boundary layer, boundary layer thickness, and internal heat transfer coefficient was insignificant. In contrast, membrane pore size considerably influenced the membrane permeability coefficient, TPC, and CPC.


Indian journal of Engineering
and Matsuura, 2011; Ashoor et al., 2016).The temperature difference at the membrane surface causes the transmembrane vapour pressure gradient allowing the water molecules to transport to permeate side from feed side through membrane pores in DCMD (Khayet and Matsuura, 2011; Drioli et al., 2015).Although all DCMD experiments are conducted under identical conditions, the variance in permeate production in DCMD can be attributed to two factors.The initial factor is the varying design and configurations of the DCMD module and the membrane morphology, which results in inaccuracies in mass transfer calculations (Khayet and Matsuura, 2011;Schofield et al., 1990).
Then, the internal heat transfer coefficient is defined (Tewodros et al., 2022) Experimental methods cannot define this parameter (Khayet, 2011;El-Bourawi et al., 2006;Camacho et al., 2013).To solve this problem, the analogy between heat and mass transfer (in Table 1) Lokare and Vidic, (2019), Qtaishat et al., (2008), Schofield et al., (1990), Bahmanyar et al., (2012), Ve et al., (2021a), or the simulated results Chang et al., (2017), Katsandri, (2017a), Katsandri, (2017b), Saeed et al., (2015), Taamneh and Bataineh, (2017) are applied to define Sherwood number.Then, the mass transfer coefficient of particles through the boundary layer is defined.However, these methodologies had their constraints.The mass transfer characteristics don't appear to be sufficiently explained under various operating conditions due to the reliance on the Sherwood number or simulated work for the mass transfer coefficient.As a result, it is necessary to explore another method that is independent of the Sherwood number correlation to determine the mass transfer coefficient through the boundary layer.This study aims to clarify the fluctuation mass fluxes, TPC, and CPC through the implication of thin film theory and boundary layer theory under the impact of different operating conditions, as shown in (Figure 1).Furthermore, the effect of membrane pore size on the DCMD performance was explained more clearly based on this approach regarding the analysis of the mass transfer coefficient of particles, mass transfer coefficient through membrane pore, and boundary layer thickness.
Figure 1 The entire framework of this study

Heat transfer
Direct contact membrane distillation is a complex process involving both heat and mass transfer.Initially, the convective heat is transferred across the liquid boundary layer to the membrane surface at the feed side.Then, heat in the form of conduction and vapour latent heat passes through membrane pores.Finally, the convective heat is removed from the membrane surface across the liquid boundary layer on the permeate side.
The heat transfer rate across the boundary layer of both sides of the DCMD module: The heat transfer rate through the membrane can be expressed (Khayet and Matsuura, 2011): The thermal conductivity of membrane (k m ) is calculated according to the Maxwell (Type II) model, as mentioned in previous studies (García-Payo and Izquierdo-Gil, 2004; Ve et al., 2021b) At the steady state: The feed and permeate membrane surface temperature can be calculated from Equations ( 1), ( 2), (3), (5): The heat transfer coefficients hi, and the overall heat transfer coefficient (H) are determined by Eq. ( 8) and Eq. ( 9): In this investigation, the spacer altered the flow direction; therefore, the Nusselt number in the feed and permeate side (Nui) can be evaluated: The spacer characteristics are showed in (Table 2).Based on those measurements, the hydraulic diameter (dh), and spacer porosity (ɛ) were calculated (Ve et al., 2019).
Where the partial pressures of water vapour being a function of surface temperatures and solution salinity as explained by Antoine and Sharqawy's equations (Khayet and Matsuura, 2011;Schofield et al., 1987;Nayar et al., 2016).The membrane permeability coefficient (Cm) was determined by (Ding et al., 2003;Ve et al., 2024).All mentioned parameters in Eq. ( 12) are explained in the Nomenclature section.
For saline water on the feed side, predicting mass flux requires consideration of the concentration polarization effect, a crucial factor adversely affecting permeate flux.In practical scenarios, the solution concentration at the membrane surface is higher than that at bulk (Aas et al., 2017).To examine the impact of the concentration polarization effect on mass flux, thin film theory can be employed Curcio and Drioli, (2005) Eq. ( 13) and Eq. ( 14) describe the dependence of permeate flux (Jw) on the bulk solute concentration (Sf), solute concentration on the membrane surface (Sm, f), the mass transfer coefficient through the boundary layer (ks), and feed density (ρf). And ks (m.s-1) can be defined by Eq.( 15): Where (δ) (m) is the boundary layer thickness, and D (m2.s-1) is the diffusion coefficient of the solute, which is determined by (Boudinar et al., 1992).

Temperature polarization
TPC is considered a limiting factor for DCMD efficiency, and TPC can be expressed using Eq. ( 18).The decrease of the membrane surface temperature on the feed side and the increase of the membrane surface temperature on the permeate side results in reduced partial vapor pressure difference.As a result, the reduction in the driving force resulted in a negative influence on freshwater production in the DCMD system.
Based on the TPC value, the DCMD system can be considered to have a poor or good design.If the TPC value is lower than 0.2, the DCMD process is controlled by heat transfer resistance, and this leads to a poorly designed DCMD module.If the TPC value is larger than 0.6, the DCMD process is controlled by mass transfer resistance within the membrane, and the DCMD system has a good fluid dynamic property.Usually, TPC values from 0.4 to 0.8 satisfy the DCMD module (Khayet and Matsuura, 2011; Lawson and Lloyd, 1997).

Experimental setup
In this investigation, a DCMD setup with an effective membrane of 150 x 150 mm in length, and width was established.The channel height of DCMD was 4 mm.To support the membrane, plastic spacers shown in Table 2 were placed on both sides of DCMD.Two commercial PTFE membranes with nominal pore size of 0.22 µm (PTFE022), and 1 µm (PTFE1) were applied.Both membranes had polypropylene layer support and a 75% void fraction.A solution of NaCl with 20000 ppm and 40000 ppm concentration was prepared.
To measure the solution concentration in the feed and permeate side, a C-100 device was used.
On the feed side, hot saline water was heated to the appropriate temperature by BE-25L-T of 2500 W and pumped into the feed channel.On the permeate side, the CW-5000 chiller was used to cool down fresh water to a fixed temperature, which was pumped counter-currently.Four PT100 (± 0.50C) were used to measure the temperatures at inlets and outlets of feed and permeate channels.
Two flowmeter sensors (±10% of the reading) were used to measure the volume flow on both sides.All the data was collected by a data operating conditions are shown in (Table 3).In each case, the experiments were repeated three times to ensure the accuracy of the measured results.

Effect of feed inlet temperature on heat and mass transfer
This section examines the impact of inlet temperature at the feed side on heat and mass transfer rates.The operating conditions are the Case I of (Table 3).As can be seen from Figure 3  In addition, Figure 4 and Figure 5 showed that the improvement of permeate flux was also proved through the evaluation of the internal heat transfer coefficient on the feed side, the total heat transfer coefficient, the thickness of the boundary layer, and the mass transfer coefficient through the boundary layer.The rise of inlet temperature at the feed side led to the increase of both the heat transfer coefficient, and this caused the decrease in thermal boundary layer thickness (Bouchrit et al., 2015).While the hf went up unremarkably at nearly 4% for both membranes, there was a remarkable increase in the total heat transfer coefficient because of the rise in the heat of vaporization (Eq.( 9)) Suleman et al., (2021), with nearly 18.5% and 19.8% for PTFE022 and PTFE1, respectively in the investigated feed inlet temperature.
There is no doubt that the boundary layer thickness dropped by nearly 8% when the inlet temperature at the feed side rose from 400C to 500C for both membranes, as shown in (Figure 5).As a result, ks increased by nearly 7.2% for the PTFE022 and PTFE1 membranes.Figure 4, and Figure 5 revealed that the pore size of the membrane didn't affect the internal heat transfer coefficient, the boundary layer thickness, as well as the mass transfer coefficient through the boundary layer.However, the pore size of the membrane influenced the membrane permeability coefficient, and the total heat transfer coefficient significantly.At 500C of feed inlet temperature, the membrane permeability coefficient and the total heat transfer coefficient for PTFE1 were more significant than that for PTFE022, with nearly 20% and 10.1%, respectively.From Figure 7, the CPC rose insignificantly with nearly 4.4%, and 5.5% for PTFE022 and PTFE1, respectively, when the inlet temperature at the feed side went up from 400C to 500C.The improvement of mass fluxes due to the temperature caused the rise of CPC, as shown in Eq. ( 14).Additionally, both shear stress and viscosity of the feed solution reduced as the feed inlet temperature rose could result in a higher fouling potential on the feed membrane interface, hence the non-volatile substance accumulated more and more.As a result, the CPC was higher as the feed inlet temperature rose (Bouchrit et al., 2015).As can be seen from Figure 7, the PTFE1 membrane had larger CPC values than that for the PTFE022 membrane.This result is due to not only the higher mass flux for PTFE1 but also the higher rate of evaporation of water molecules through the larger pore size of PTFE1.Consequently, the accumulation of non-volatile substances on the feed PTFE1 membrane surface was more than that on the feed PTFE022 membrane surface.

Effect of volume flow rate on heat and mass transfer
The experimental conditions were the Case II of (Table 3).Through the fluctuation of the heat transfer coefficient, mass transfer coefficient through the boundary layer, and boundary layer thickness, the impact of volume flow rate and the membrane pore size on freshwater production could be explained.From Figure 8, the enhanced permeate fluxes were insignificant for both membranes.The rise in volume flow rate led to the increase of the Reynolds number, then the heat transfer coefficient rose, as shown in Eq. ( 8) and Eq.The dominant factor for this result is the permeability coefficient.From Figure 10, there was an insignificant difference between the boundary layer thickness and the mass transfer coefficient through the boundary layer.However, there was a considerable difference in permeability coefficient for PTFE022 and PTFE1 membranes.The mass transfer coefficient through the PTFE1 membrane pore was much larger, nearly 20% than that for the PTFE022 membrane in all investigated volume flow rates.Figure 11 and Figure 12 described that there was an increase in TPC and a decrease in CPC when the volume flow rate went up in both membranes.The boundary layer thickness decreased due to the increase of the volume flow rate, so the temperature difference between the bulk feed side and the feed membrane interface was lower.Consequently, the transmembrane temperature difference was more prominent, and the TPC was higher (Ve et al., 2021a).Furthermore, there were two main factors regarding the reduction of CPC related to the increase in the volume flow rate.Firstly, as mentioned in Eq. ( 14), the enhancement of permeate production in the investigated range of volume flow rate led to the CPC increase.Secondly, a decrease in the shear stress caused a lower concentration difference between the bulk and membrane surface Ve et al., (2021a); therefore, the concentration polarization effect is reduced.Also, according to Bouchrit et al., (2015), the latter factor was considered dominant for decreasing CPC in this case.For membranes with larger pore sizes, more water molecules evaporated from the feed side to the permeate side.Thus, the temperature difference between at bulk solution and at membrane surface was more considerable, and the non-volatile matters accumulated more and more on the membrane surface.Consequently, the temperature polarization effect and concentration polarization effect were more severe.Therefore, in comparison to the PTFE022 membrane, the PTFE1 membrane obtained lower TPC values and higher CPC values, as shown in (Figure 11, 12).

Effect of feed concentration on heat and mass transfer rates
This section explored the impact of feed concentration on heat and mass transfer rates.The experimental conditions were Case III of (Table 3).From Figure 13, there was a slight decrease of nearly 1.5% in mass flux for both membranes when the feed concentration rose from 20,000 ppm to 40,000 ppm.As mentioned in previous studies Bouchrit et al., (2015), Alklaibi and Lior, (2006), the feed concentration has no or little influence on mass fluxes if the feed concentration was in the range of (17,750 ppm -50,000 ppm).
Further, according to Bouchrit et al., (2015), the mass flux dropped considerably by 56% when the feed concentration rose to 177,500 ppm.The water activity was reduced because of the rise of the mole fraction of the feed solution, which caused the drop in the difference of partial vapor pressure between two sides of DCMD channels, as shown in Eq. ( 19) (Khayet and Matsuura, 2011; concentration, which led to a slight change in the partial vapor pressure difference.As shown in Figure 15, the TPC showed a minimal increase in feed concentration.This slight increase in TPC can be attributed to two main factors.Firstly, as illustrated in Figure 16, hf decreased slightly, with nearly 1.3% for both membranes.This decrease is a result of the marginal increase in both the kinematic and dynamic viscosity of saline water.Thus, the Reynolds number reduced slightly when the concentration of feed solution was in the range of 20,000 ppm -40,000 ppm (Alkhudhiri et al., 2012).Secondly, the expansion of the boundary layer thickness and lower mass fluxes led to a decreased heat flux on the feed side.This resulted in a lower temperature gradient in the boundary liquid layer (T_f-T_(m,f)), which was the dominant factor contributing to the increase of TPC as the feed concentration rose (Martıńez-Dıéz and Vázquez-González, 1999; Suleman et al., 2021;Termpiyakul et al., 2005).

Figure 15
The effect of feed concentration on the TPC Figure 16 The impact of feed concentration on convective heat transfer coefficient As indicated in Eq. ( 13) and Eq. ( 14), the decline in mass flux resulting from the rise in feed concentration caused a decrease in CPC, meaning that the concentration difference between the bulk solution and the membrane surface was reduced (Bouchrit et al., 2015).
However, there was only a 0.2% and 0.4% reduction in CPC for PTFE022 and PTFE1, respectively.Additionally, PTFE1 exhibited a lower TPC (Figure 15) and a higher CPC (Figure 17) compared to PTFE022, attributed to the greater evaporation of water molecules through a larger membrane pore size.

Figure 17
The effect of feed concentration on the CPC in spacer-filled DCMD configuration

CONCLUSIONS
The thin film theory and boundary layer theory were used to investigate heat and mass transfer characteristics in spacer-filled DCMD configurations under the impact of different experimental conditions and various pore sizes of hydrophobic PTFE membranes.The relationship between the mass transfer coefficient through the boundary layer, the boundary layer thickness, and DCMD performance was explained in more detail based on this approach.The mass transfer coefficient through the boundary layer, boundary layer thickness, internal heat transfer coefficient, TPC, and CPC underwent substantial changes under the impact of operating conditions.
The feed inlet temperature was the most influential factor on permeate fluxes compared to volume flow rate and feed concentration.
At 500C feed inlet temperature, the mass fluxes rose more than 93% for PTFE022 and PTFE1 membranes compared to the recorded values at 400C.The substantial improvement of mass fluxes can be attributed to the significant change in the total heat transfer coefficient.In contrast, the rise of mass fluxes under the effect of volume flow rate can be attributed to the substantial change in the internal heat transfer coefficient instead of the fluctuation of the total heat transfer coefficient.However, the pore size of membranes exerted a pronounced impact solely on the membrane permeability coefficient, TPC, and CPC.Notably, there was negligible effect of the membrane pore size on the mass transfer coefficient of particles, boundary layer thickness, and the internal heat transfer coefficient.
distillation (MD) has been becoming more popular in saline water treatment technology due to the technological breakthrough of membrane production and the implementation of renewable energy (Khayet, 2013; Khayet and Matsuura, 2011).Direct contact membrane distillation (DCMD) has been more attractive than other technologies due to its simple configuration (Khayet . If the saline solution was on the feed side of DCMD, mass transfer phenomena should consider the concentration polarization effect.The mass flux tends to decrease owing to this effect (Khayet, 2011; Plamenov, 2017; El-Bourawi et al., 2006; Qtaishat et al., 2008; Chen et al., 2009).To enhance the accuracy of concentration polarization coefficient (CPC) estimation, it is essential to theoretically determine the solution concentration at the feed membrane surface (Bouchrit et al., 2015; Andrjesdóttir et al., 2013; Khayet, 2011; El-Bourawi et al., 2006).

3 ⁄
Laminar (Li et al., 2002) Sh = 0.065 × Re 0.875 ×Sc 0.25 Laminar (Schock and Miquel, 1987) Sh = 0.023 × Re 0.875 ×Sc 0.25 Turbulent (Alkhudhiri et al., 2012) Sh = 2 × Re 0.483 ×Sc 1 Laminar (Alkhudhiri et al., 2012) ℎ = 0.0588. 0.55 . 0.33 Laminar (Ve et al., 2021a) Ve et al., (2021a) has proposed an approach that combined thin film theory and boundary layer theory to find out the mass transfer correlation through boundary layer in spacer-filled DCMD.The study investigated the impact of feed inlet temperature, velocity, and feed concentration on mass transfer with various spacers.The relationship between operating conditions and boundary layer thickness, mass transfer coefficient, and shear stress are clearly explained based on this approach.However, this research didn't reveal the impact of the operating conditions on the fluctuation in permeate flux, temperature polarization coefficient (TPC), and CPC.Furthermore, this research didn't investigate the effect of different commercial membranes on the change in DCMD performance.
, the experimental mass flux increased exponentially due to the increase in feed inlet temperature.The exponential relationship between partial vapor pressure and temperature was the main reason (Khayet and Matsuura, 2011;Schofield et al., 1987).Mass fluxes at 400C were 8.3 kg/m2-h, and 9.5 kg/m2-h for PTFE022 and PTFE1, respectively.At 500C, there was a significant rise of more than 93% in mass fluxes for both membranes.To conclude, the feed inlet temperature affected significantly the mass flux enhancement in DCMD(Bouchrit et  al., 2015; Alkhudhiri et al., 2012; Alklaibi and Lior, 2006; Singh and Sirkar, 2014).

Figure 3
Figure 3 Experimental mass fluxes for experimental conditions in Case I

Figure 4 Figure 5
Figure 4 The impact of feed inlet temperature on (a) convective heat transfer coefficient, and (b) total heat transfer coefficient in spacerfilled DCMD channels

Figure 6
Figure 6 TPC vs. feed inlet temperature in spacer-filled DCMD configuration

Figure 7
Figure 7 CPC vs. feed inlet temperature in spacer-filled DCMD configuration

Figure 8
Figure 8 The influence of volume flow rate on mass flux

Figure 9 Figure 10
Figure 9 The impact of volume flow rate on (a) convective heat transfer coefficient, and (b) total heat transfer coefficient in spacer-filled DCMD channels

Figure 11 Figure 12
Figure 11 The effect of volume flow rate on the TPC in spacer-filled DCMD configuration

Figure 13 Figure 14
Figure 13The impact of feed concentration on mass flux