Coupled ϵ-NTU Method to Design and Evaluate the Performance of Energy Exchangers With Coupled Heat and Mass Transfer Open Access

The classical $ϵ$-number of transfer units (NTU) method is widely used to design and evaluate the performance of heat and mass energy exchangers. In energy exchangers, where the heat and mass transfer are coupled, i.e., the magnitude of heat transfer impacts the magnitude of mass transfer and vice-versa, the classical $ϵ$-NTU method fails to capture the outlet fluid conditions of the energy exchanger accurately. It cannot be used for designing/evaluating the performance of energy exchangers where heat and mass transfer are coupled. The coupled $ϵ$-NTU model uses modified heat and mass capacity ratios to capture the effects of coupled heat and mass transfer. The use of the coupled $ϵ$-NTU model to design and evaluate the performance of energy exchangers is illustrated, specifically on a liquid-air-membrane energy exchanger (LAMEE), but the model can be extended to other coupled energy exchangers. The coupled $ϵ$-NTU model is validated using a numerical model of a LAMEE in counterflow and crossflow configurations. The validation is completed for over 14,500 test points representing a wide range of operating conditions. The average error in estimating sensible and moisture transfer effectiveness using the coupled $ϵ$-NTU method is less than ±1.5% for both configurations, compared to the numerical model illustrating the robustness of the coupled $ϵ$-NTU model. Of the 14,500 tested points, the error in estimating sensible or moisture transfer effectiveness is greater than 4% for less than 5% of the test points.

The $ϵ$-NTU method is widely used by engineers in two main ways: (i) to estimate the performance of heat exchangers (performance methodology) and (ii) to design heat exchangers (design methodology). For the performance methodology, the number of transfer units (NTU) of the heat exchanger is known, and the effectiveness, i.e., the outlet temperatures of the hot and the cold streams, is calculated [1]. In contrast, in the design methodology, the NTU of the exchanger is calculated to achieve a specified effectiveness [2]. In both cases, the inlet conditions of the exchanger are known. Figure 1 illustrates the two concepts of performance and design methodology for heat exchangers. The $ϵ$-NTU method has been studied extensively in classical heat exchanger books [1,3] and applies to exchangers where the heat and moisture transfer are independent. The $ϵ$-NTU method does not apply when heat and mass transfer are coupled, as in liquid-to-air membrane energy exchangers (LAMEEs) [4]. In LAMEEs, heat and water vapor are transferred between an air stream and a liquid desiccant stream. When water vapor transfers between these streams, it involves a phase change that affects the temperature of the liquid desiccant. This temperature change influences the sensible heat transfer between the streams, making the heat and moisture transfer dependent on each other. This paper presents procedures to solve the coupled $ϵ$-NTU method for both design and performance of coupled heat and mass exchangers, with particular focus on a LAMEE. The method presented in this paper can be extended to similar coupled energy exchangers.

Liquid-air-membrane energy exchangers are built with membranes separating air and liquid (solution) streams, as shown in Fig. 2. Heat and moisture are transferred between the air and the liquid stream, depending on the air's temperature and humidity ratio and the solution concentration. LAMEEs have predominant use in applications involving drying or humidifying air streams and have potential applications in liquid desiccant air-conditioning systems [4,5] as well as energy recovery from exhaust air in buildings [6]. Since moisture can pass through the membrane, the dew-point temperature at the membrane surface is reduced, delaying the formation of frost when LAMEEs are used in cold climates. When frost forms on the surface of an exchanger, it reduces the efficiency of the exchanger, and periodic energy-intensive defrost cycles are needed to maintain operation [7,8].

For uncoupled energy exchangers, the effectiveness (⁠$ϵs$ and $ϵm$⁠) is bounded between 0 and 1. In coupled energy exchangers, however, the release or absorption of phase change energy can cause unusual temperature changes. For example, the hotter stream can cool below the inlet temperature of the colder stream, or the outlet temperature of the hotter stream can be higher than its inlet temperature. Which means $ϵs$ and $ϵm$ are not bounded between 0 and 1 for coupled energy exchangers like LAMEEs [9].

Various numerical and analytical methods for estimating performance of energy exchanger have been investigated in the literature and the methods specific to LAMMEs are presented in the following subsections.

Several numerical models have been developed in the literature to solve for heat and moisture transfer rates in a LAMEE. Vali et al. [10] developed a finite difference-based numerical model for a countercrossflow LAMEE and found the deviations of heat and mass transfer estimates from experimental data were ±10%. They found that the temperature change of the solution depends on both the sensible heat transfer with the air and the phase change energy release associated with the moisture transfer. This confirms the strong intercoupling of the heat and moisture transfers and the dependence of effectiveness values on the inlet conditions of the air and solution streams. A transient model of a LAMEE was developed by Seyed-Ahmadi et al. [9,11] to estimate the effect of step changes in inlet airstream conditions on the system's transient response. Alipour Shotlou and Pourmahmoud [12] created a 3D CFD model to investigate the effects of exchanger structure on heat and mass transfer in a LAMEE, including the effects of vortices at the entry of the LAMEE. The 3D CFD model was also used to predict frost formation in the LAMEE and could successfully distinguish between regions of frost and condensation in the LAMEE.

The numerical models discussed have been extensively studied and validated in the literature; they are computationally intensive and challenging to use in cases where the yearly performance of a LAMEE in an HVAC system needs to be simulated at a subhourly interval. To avoid this problem, a few iterative analytical models were also investigated in the literature. Zhang [13] proposed an iterative analytical model to estimate the heat and moisture transfer in hollow-fiber membrane contactors with a liquid desiccant used for dehumidifying air in typical summer conditions. This model was modified by Ge et al. [14] to be applicable to a countercrossflow LAMEE. Fix et al. [15] presented a generalized ϵ-NTU framework for membrane-based humidification and dehumidification systems, where they defined the driving force for moisture transfer using vapor pressure differences instead of humidity ratios of the fluid streams. This framework is highly effective for vacuum-based membrane dehumidification but does not discuss the coupled heat and mass transfer phenomenon present in LAMEEs.

Zhang and Niu [16] proposed a model similar to the $ϵ$-NTU method to estimate $ϵs$ and $ϵm$ in a dehumidifier with sensible and latent energy exchange. The model, however, does not consider the dependence of effectiveness on inlet conditions with average errors in $ϵs$ of 7.9% and $ϵm$ of 8.5%. While the numerical and analytical models presented here are specific to LAMEEs, similar numerical and analytical models exist for other forms of coupled heat and moisture exchangers [16–21].

Although several of the numerical and iterative analytical models discussed are fairly accurate for predicting the performance of coupled heat and moisture exchangers for typical summer air-conditioning applications, they do not adequately explain how $ϵs$ and $ϵm$ depend on the inlet conditions of the air and solution streams, as noted in various studies [10,21,22]. Furthermore, they require high levels of programing to arrive at a method engineers can use, which is often not feasible during the initial design stages.

The main contrast between the classical and the coupled $ϵ$-NTU methods is the use of $Cre$ and $me*$ instead of $Cr$ and $m*$⁠. The use of $Cre$ and $me*$ connects the effectiveness of a LAMEE with the inlet conditions of the solution and the air stream, which was omitted by previous analytical models [16] and necessary for an accurate representation of the coupled nature of heat and mass transfer. Furthermore, if the heat and mass transfer are uncoupled, the values of $Cre$ and $me*$ would be equal to $Cr$ and $m*$⁠, respectively.

Kamali et al. [22] demonstrated that if the temperature profile (humidity ratio profile) of the air stream is linear along the exchanger and also if $Uh$(⁠$Um$⁠) is constant throughout the exchanger, then the coupled $ϵ$-NTU model, Eqs. (3) and (4), provides an exact solution for $ϵs$(⁠$ϵm$⁠), despite the coupling between heat and moisture exchange. They used the measured inlet and outlet conditions from an experimental test on a small-scale LAMEE [24,25] and also a hollow fiber membrane contactor [13] to calculate $me*$ and $Cre$ for each experimental data point. The calculated $Cre$ and $me*$ values were used in Eqs. (3) and (4), to obtain estimations for $ϵs$ and $ϵm$⁠. The maximum discrepancy between the model estimations and the experimental values of $ϵs$ and $ϵm$ did not exceed 16%. This illustrates the ability of the coupled $ϵ$-NTU method to estimate the effectiveness of a LAMEE.

The coupled $ϵ$-NTU method can be used readily for the design of coupled energy exchangers, i.e., to estimate the NTU_h (NTU_m) required to heat/cool, humidify/dehumidify the air stream to specified outlet conditions without requiring any computer simulations. However, the main limitation to using the coupled $ϵ$-NTU method is that it cannot be used directly to estimate the performance of a coupled exchanger, i.e., to estimate $ϵs$ (⁠$ϵm)$ from known NTU_h (NTU_m) values, since prior knowledge of outlet conditions is required to calculate $Cre$ and $me*$⁠. In fact, if the outlet conditions of the air and the solution streams are known, the effectiveness values can be calculated readily using Eqs. (1) and (2).

Therefore, the main objective of this paper is to present a model to estimate $Cre$ and $me*$ based on the inlet temperature and humidity ratio of the air and solution streams for a counterflow LAMEE, and to extend the model to a crossflow LAMEE. Once the values of $Cre$ and $me*$ for a specific operating condition are known, $ϵs$ and $ϵm$ can be estimated using Eqs. (3) and (4) for a given NTU_h (NTU_m). The paper also illustrates the accuracy of the coupled $ϵ$-NTU method used to design and evaluate the performance of a LAMEE by validating it with a numerical model of a LAMEE. The following subsection discusses the governing equations of a LAMEE and the methodology to use the coupled $ϵ$-NTU method for design and performance evaluation of a LAMEE.

In this paper, the coupled $ϵ$-NTU is illustrated on a LAMEE and the expressions for $Cre$ and $me*$ presented are specific for a LAMEE. A similar procedure for other energy exchangers, such as cooling towers [26] can be followed. The governing equations for heat and moisture transfer in a LAMEE are presented based on Ref. [22]. The governing equations are the base for the model that estimates $Cre$⁠, and $me*$⁠, which are then used to estimate $ϵs$ and $ϵm$ in the coupled $ϵ$-NTU model. The assumptions for the model are as follows:

1. The exchanger is operating under steady-state conditions.

2. Fluid properties are uniform along the height of the exchanger.

3. Heat and moisture transfer occurs only between the two streams through the porous membrane.

4. Heat and moisture transfer occurs only in the direction perpendicular to the membrane.

5. The deformation of the membrane and the maldistribution of flow as a result of deformation have a negligible impact [27].

6. The overall heat and moisture transfer coefficients are constant inside the LAMEE.

7. The mass flow and heat capacity rates of the fluids are constant inside the LAMEE.

8. The membrane is impermeable to liquid and permeable only to vapor.

It has been assumed that the mass flowrate of the dry air is equal to that of the moist air because, for typical HVAC applications, $Wair$ is typically less than 30–40 g/kg_da, and the percentage changes in $m˙air$ are nearly two orders of magnitude smaller than the percentage changes in $Wair$⁠.

The solution stream in a LAMEE is a liquid desiccant, which can absorb or release moisture to the air stream based on the operating conditions (i.e., it can dehumidify or humidify the air stream). Typically, the liquid desiccant is an aqueous salt solution. By varying the type of salt or the concentration of the salt solution, the conditions of the solution stream can be altered.

In Eq. (19),$hfg (kJ/kg)$ is the specific heat of evaporation of water. It is assumed in these equations that the latent heat of evaporation/condensation is absorbed from or released into the solution stream and not the air stream. This assumption is valid for LAMEEs operating under atmospheric conditions because the membranes are impermeable to liquids at such pressures, allowing only vapor exchange between the streams, and phase change occurs on the solution side [10].

The performance methodology is used to estimate $Cre$⁠, $me*$⁠, $ϵs,$ and $ϵm$ based on the inlet conditions of the air stream and the solution stream using the governing principles presented in Eqs. (10)–(21). The solution requires an iterative technique because of the coupling of $Cre$⁠, and $me*$⁠, with $ϵs$ and $ϵm$⁠, since the outlet conditions are not known.

In the next section, the model is validated against a wide range of inlet operating conditions representing both humidification and dehumidification of the air stream using the liquid desiccant for both counterflow and crossflow configurations, followed by a discussion of the limitations of the proposed model.

In this section, the coupled $ϵ$-NTU method is verified using a numerical model. The numerical model uses a finite difference method to solve the energy and mass balance equations for the air and the solution side of an exchanger. The coupled $ϵ$-NTU method for performance methodology is verified first, followed by validation for use in the design methodology. Furthermore, the coupled and the classical $ϵ$-NTU methods are compared with experimental data available in the literature for both counter and crossflow configurations in the  Appendix.

The coupled $ϵ$-NTU applied to a LAMEE is validated using a numerical model for three distinct data sets, shown in Table 1. The first dataset, Set A, corresponds to conditions where the air is cooled and dehumidified. The test conditions reflect cases when the air transfers heat and moisture to the liquid desiccant stream. Set B corresponds to the cases where the air is heated and humidified; the air stream gains heat and moisture from the liquid stream. The third test set, Set C, corresponds to a special case in refrigeration and heat pumps where a LAMEE might be used to simultaneously cool/heat and dry the air stream to avoid frosting in subzero conditions [7,8]. In all three test sets, NTU_h varies from 1.5 to 4.5, and the ratio of NTU_h/NTU_m varies from 0.4 to 0.8. Furthermore, the ratio of the air mass flowrate to that of the solution (⁠$m*$⁠) varies from 0.2 to 0.8.

Figure 4 illustrates $ΔT(air)$ for the three datasets, which consist of 14,690 points for both the counterflow and crossflow configurations. As seen in Fig. 4, the test points fall closely on the X = Y line, and there are no significant systematic errors for the test points. The average RMS error between $ΔT(air)$ and $ΔW(air)$ differentials estimated by the coupled $ϵ$-NTU and the numerical model in the air stream, for the two flow configurations, are presented in Table 2.

Figure 5 illustrates $ΔT(sol)$ in the solution stream. While for most of the test points, there is a very close agreement, it can be seen that the comparison of $ΔT(sol)$ starts to diverge marginally for the humidification test points at larger, negative values of $ΔT(sol)$ which is also seen in the deviation of $ΔW(air)$ for the humidification test cases presented in Fig. 6. The cause of the divergence is that the temperature and humidity profiles at these points have a more significant deviation from the linear profiles assumed as the basis for the coupled $ϵ$-NTU method. The RMS error in estimating $ΔT(sol)$ and $ΔW(sol)$ for the solution side is presented Table 3.

The average values of $ΔTnorm,air/sol$ and $ΔWnorm,air$ for 14,690 test points, are presented in Table 4 for the counterflow and crossflow configurations. Of the data points considered, only 0.1% of the tested points have $ΔTnorm,air$ and $ΔWnorm,air$ greater than 10%. The low average $ΔTnorm,air$ and $ΔWnorm,air$ indicate that the coupled $ϵ$-NTU model is accurate in estimating coupled heat and mass transfer in energy exchangers.

However, the maximum $ΔTnorm,air$ and $ΔWnorm,air$ for counterflow configuration can be as high as 28% and 63% indicating that the coupled $ϵ$-NTU method is prone to errors under certain operating conditions further described in Sec. 3.4.

To validate the coupled $ϵ$-NTU model as a tool for designing energy exchangers, the inlet operating conditions listed in Table 1 and the corresponding outlet conditions generated by the numerical model are used to calculate $Cre, me*, ϵs,$ and $ϵm$⁠. These values are then used to estimate $NTUh*$ and $NTUm*$ using Eqs. (28) and (29) of the coupled $ϵ$-NTU method. For the purposes of designing an energy exchanger, $NTUh*$ and $NTUm*$ would be used to determine the size (A) and overall heat transfer coefficients (U_h, U_m) of the exchanger, as given by Eqs. (5) and (6). To validate the methodology $NTUh*$ and $NTUm*$ are used along with the temperature, humidity and m* values from Table 1 to estimate $ϵs*$ and $ϵm*$ using the numerical model, which are compared with $ϵs$ and $ϵm$ to assess the validity of the coupled $ϵ$-NTU model. The validation procedure for design methodology is shown in Fig. 7(a).

Figure 7(b) represents the differences in sensible effectiveness and moisture effectiveness, i.e., the temperature and humidity differential values calculated using Eqs. (32) and (33) for the cases where $Cre, me*, ϵs,$ and $ϵm$ are bounded by 0 and 1 so that Eqs. (28) and (29) are continuous. The maximum error in the sensible effectiveness is 6.1%, and mass transfer effectiveness is 10.9%. The average error is ±0.85% in sensible effectiveness and ±0.6% in mass transfer effectiveness, indicating that the coupled $ϵ$-NTU is a good design tool for the cases where $Cre, me*, ϵs,$ and $ϵm$ are bounded by 0 and 1. An interesting feature of Fig. 7(b) is that the errors in mass transfer effectiveness are higher when the errors in sensible effectiveness are lower and vice versa.

The $ϵs$ and $ϵm$ values obtained using the coupled $ϵ$-NTU method with the performance methodology and the classical $ϵ$-NTU method are compared to the effectiveness values obtained using the numerical model. It can be observed from Fig. 8(a) that the sensible effectiveness values (⁠$ϵs*$⁠) obtained using the coupled $ϵ$-NTU method for the 14,690 points fall mostly within the ±5% bounds of the numerical model with the maximum deviation being 28%. However, using the classical $ϵ$-NTU method would yield differences as large as 70% for the same dataset. Figure 8(b) illustrates the moisture transfer effectiveness comparison and shows that using the classical $ϵ$-NTU method to estimate moisture transfer effectiveness would yield errors as high as 160%, which emphasizes the need for the coupled $ϵ$-NTU method. It can be observed that the coupled $ϵ$-NTU method diverges from the numerical model when the effectiveness values estimated by the numerical model are greater than 1 or less than 0, which is because Eqs. (3) and (4) would return an effectiveness value between 0 and 1 for any values of NTU_h, NTU_m, $Cre,$ and $me*$ where Eqs. (3) and (4) are mathematically valid.

Figures 9(a) and 9(b) show the temperature differential (Eq. (30)) and sensible effectiveness for different values of H*, estimated by both the coupled, and classical $ϵ$-NTU methods and the numerical method. The key point is that the classical $ϵ$-NTU method does not consider how effectiveness values depend on outlet conditions. This means the effectiveness and temperature differential remains constant, regardless of operating conditions, as shown in Fig. 9. When H* is zero, meaning there is negligible moisture transfer, the values from the classical and coupled $ϵ$-NTU methods match the numerically estimated values. In this case, both methods give the same results. However, as H* increases and moisture transfer become significant, the classical $ϵ$-NTU method becomes inaccurate. In these situations, the coupled method must be used to design or estimate the performance of a coupled energy exchanger.

The validation of the coupled $ϵ$-NTU model with the numerical model yielded minimal average errors of less than ±1.5% for over 14,500 test points. However, there are points where the maximum error in estimating sensible and moisture transfer effectiveness can be as high as ±28% or ±63%, respectively. Figure 10 illustrates the variation of $ΔTnorm,air$ and $ΔWnorm,air$ with H* across 14,690 data points. It can be observed that the magnitude of $ΔTnorm,air$ is highest at larger H* values, indicating that the error in estimating sensible effectiveness increases when latent heat transfer dominates over sensible heat transfer. Conversely, when sensible heat transfer is dominant, the magnitude of $ΔWnorm,air$ is higher. However, at higher H* values, where latent energy transfer contributes the most to the overall heat exchange, errors in estimating sensible effectiveness become less consequential, as they have a minimal impact on total energy transfer. Similarly, at low H* values, where latent heat transfer is minimal, errors in estimating moisture transfer effectiveness are also less significant, as sensible heat transfer is the primary contributor to the overall energy exchange.

Figure 11 represents the error histograms for $ΔTnorm,air$ and $ΔWnorm,air$ and it can be observed that most points have errors of less than 5%, and only 4% of the tested dataset have $ΔTnorm,air$ or $ΔWnorm,air$ greater than 5%, indicating high errors occur only in certain cases at which the temperature and humidity profiles significantly differ from the linearity assumption, which is the basis for the coupled $ϵ$-NTU method. Future work of this paper will investigate and map the regions where the coupled $ϵ$-NTU model is less accurate and the reasons for lower accuracy. The effect of energy exchanger parameters and operating conditions on the accuracy will also be discussed. Furthermore, a model will also be presented for the cases where the solution is the fluid with minimum specific heat capacity. In general, the coupled $ϵ$-NTU model is an effective method to estimate the sensible and moisture transfer effectiveness of an energy exchanger which has coupled heat and moisture transfer.

Kamali et al. [22] introduced a coupled $ϵ$-NTU method to address heat and mass transfer in coupled energy exchangers, such as liquid-to-air membrane energy exchangers. The coupled $ϵ$-NTU model is more effective than the classical $ϵ$-NTU method to (i) evaluate the performance of coupled energy exchangers and (ii) design energy exchangers. The coupled $ϵ$-NTU method is applied to the case of a LAMEE but can be applied to any energy exchanger that has coupled heat and moisture transfer with phase change. The accuracy of the coupled $ϵ$-NTU method is verified using a numerical model. The average error in estimating the sensible effectiveness and moisture transfer effectiveness of a counterflow LAMEE is ±1%, while for a crossflow LAMEE it is ±1% and ±1.5%, respectively. In the design methodology, the coupled $ϵ$-NTU method is used to estimate NTU_h and NTU_m to attain outlet conditions. The effectiveness values obtained using the NTU values estimated by the coupled $ϵ$-NTU model closely matched with the actual numerical effectiveness values with average errors being less than ±1%. However, the coupled $ϵ$-NTU method leads to significant errors when the temperature and humidity profiles deviate significantly from a linear profile. When used for performance methodology, the maximum sensible effectiveness errors are ±28% and moisture transfer effectiveness errors are ±63%.

Overall, the coupled $ϵ$-NTU model presents a method to estimate coupled heat and moisture transfer within an energy exchanger with good accuracy. The proposed analytical method will aid in the design of energy exchangers and lead the path to gaining a better understanding of the dependence of $ϵs$ and $ϵm$ on the inlet air and solution operating conditions.

Professor Robert Besant was the cosupervisor of Houman Kamali during his M.Sc. studies at the University of Saskatchewan in 2014. Professor Beasant unfortunately passed away before the publication, and we recognize his contributions to this work. This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher acknowledges the U.S. government license to provide public access under the DOE Public Access Plan¹.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

A =

area (m²)

C =

heat capacity rate (kW/K)

c_p =

specific heat capacity (kJ/(kg K))

$Cr$ =

heat capacity ratio

$Cre$ =

effective heat capacity ratio

h =

height of the exchanger (m)

$hfg$ =

heat of evaporation of water (kJ/kg)

H* =

operating factor

l =

dimensionless position along the length of the exchanger

L =

length of the exchanger (m)

$m˙$ =

mass flow rate (kg/s)

$m˙W$ =

moisture transfer rate (kg/s)

$m*$ =

mass flow ratio

$me*$ =

effective mass capacity ratio

NTU_h =

number of heat transfer units

NTU_m =

number of mass transfer units

q_l =

latent heat transfer rate (kW)

q_s =

sensible heat transfer rate (kW)

T =

temperature (K)

$Uh$ =

overall heat transfer coefficient (kW/(m² K))

$Um$ =

overall mass transfer coefficient (kg/(m² s))

$Wair$ =

humidity ratio of air (g/kg_da)

$Wsol$ =

equilibrium humidity ratio of solution (g/kg_da)

$x$ =

position along the length of the exchanger $(m)$

$Xsol$ =

concentration of salt solution

$Δ$ =

difference operator

$ϵm$ =

moisture transfer effectiveness of energy exchanger

$ϵs$ =

sensible effectiveness of energy exchangers

HX =

heat exchanger

LAMEE =

liquid-to-air membrane energy exchanger

MX =

moisture exchanger

The coupled $ϵ$-NTU model used for this study is compared using experimental data from Ref. [14]. The validation is performed for 13 points, and the test conditions, including air and solution inlet temperatures, NTU_h, NTU_m, and E_sol are presented in Table 5.

Figure 12(a) compares $ϵs$ values obtained from experimental $(ϵs_exp)$⁠, numerical $(ϵs_num)$⁠, coupled $ϵ$-NTU method (⁠$ϵs*$⁠) and classical $ϵ$-NTU method $(ϵs_classical)$⁠. It can be observed that the maximum deviation between $ϵs_exp$ and $ϵs*$ is 20%, with an average deviation of around 6%. While $ϵs_exp$ and $ϵs_classical$ had a maximum deviation of 60% and an average deviation of 19%, indicating that the coupled $ϵ$-NTU method is superior to the classical method. However, it remains susceptible to errors, as discussed in the results and discussion section. In the case of moisture transfer effectiveness, $ϵm_exp$ and $ϵm*$ have a maximum deviation of 4.5% with an average deviation of 3%. Furthermore, it must be noted that the uncertainty in calculating effectiveness ranges from 5% to 25%, depending on the operating conditions.

The coupled $ϵ$-NTU model for a crossflow LAMEE is also compared with experimental data from Ref. [29]. The air and the solution inlet temperatures are set as 30 °C and 20 °C, respectively, while the air relative humidity is 70% and the solution concentration is 39% for all the tests presented in Table 6. Among the eight test points, the average difference between sensible effectiveness estimated by the coupled $ϵ$-NTU and the experimental values is 5%, while the difference is 4.5% for moisture transfer effectiveness.