Abstract

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.

1 Introduction

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.

Fig. 1
Difference between processes followed for performance and design methodology of heat exchangers
Fig. 1
Difference between processes followed for performance and design methodology of heat exchangers
Close modal

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].

Fig. 2
Schematic of a counterflow LAMEE. A membrane separates the air and solution channels, allowing heat and moisture to pass through.
Fig. 2
Schematic of a counterflow LAMEE. A membrane separates the air and solution channels, allowing heat and moisture to pass through.
Close modal

1.1 Performance Parameters of an Energy Exchanger.

The parameters used to quantify the performance of an energy exchanger like LAMEEs are sensible effectiveness (ϵs) and moisture transfer effectiveness (ϵm) [1]. The ϵs is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate, as shown in Eq. (1). Similarly, ϵm is the ratio of the actual moisture transfer rate to the maximum possible moisture transfer rate, as shown in Eq. (2)
(1)
(2)

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.

1.2 Numerical Methods to Estimate Heat and Moisture Transfer in Liquid-Air-Membrane Energy Exchangers.

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.

1.3 Analytical Methods to Estimate Heat and Moisture Transfer in Liquid-Air-Membrane Energy Exchangers.

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 [1621].

1.4 Coupled ϵ-Number of Transfer Units Method.

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.

Kamali et al. [22,23] filled this gap by introducing the coupled ϵ-NTU model for counterflow LAMEEs. The coupled ϵ-NTU model essentially extends the concepts of the classical ϵ-NTU model to be applicable to estimate the coupled heat and moisture transfer in an energy exchanger. The coupled ϵ-NTU model, given in Eqs. (3) and (4), expresses ϵs and ϵm in terms of an effective heat capacity rate ratio Cre, and effective mass flowrate ratio me* as follows:
(3)
(4)
The number of transfer units NTUh and the number of moisture transfer units NTUm are presented in the following equations:
(5)
(6)
where Uh(W/(m2·K)) and Um(kg/(s·m2)) are the overall heat and moisture transfer coefficients, respectively. A(m2) is the total heat and moisture transfer area of the exchanger. Cmin and m˙min correspond to the heat capacity and mass flowrate of the air stream in this study. Cmin is defined as shown in Eq. (7). It is possible that the liquid desiccant can be the fluid with minimum specific heat capacity, but this case has not been investigated in this study
(7)
The dimensionless parameters Cre and me* are defined as shown in the following equations:
(8)
(9)

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.

1.5 Focus of This Study.

The coupled ϵ-NTU method can be used readily for the design of coupled energy exchangers, i.e., to estimate the NTUh (NTUm) 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 NTUh (NTUm) 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 NTUh (NTUm). 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.

2 Methodology

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.

The differential heat transfer rate dqs(W) and differential moisture transfer rate dm˙W(kg/s) of a differential element in a LAMEE can be calculated as
(10)
(11)
where l is the dimensionless position along the length of the exchanger (see Fig. 2)
(12)
Assuming no mass loss to the environment, the principle of conservation of mass can be used to express the ideal mass balance equation for LAMEEs, i.e., the change in mass flowrate of the moist air because of moisture transfer would be equal to the change in mass flowrate of the solution
(13)
In terms of the change in the humidity ratio of the air stream, the mass balance results in
(14)

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/kgda, 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.

The concentration of the salt solution, Xsol, is defined as the ratio between the mass flowrate of the dry salt, m˙salt(kg/s), in the solution stream and the total mass flowrate of the solution stream, m˙sol(kg/s)
(15)
In terms of the change in the concentration of the solution stream, the mass balance results in the following equation:
(16)
Assuming there is no energy exchange with the environment, the differential change in the air stream's temperature can be calculated as shown in Eq. (17). The change in temperature of the solution side is a result of heat exchange with the air stream and the release/absorption of the phase change energy associated with the moisture exchange between the streams. The differential change in the temperature of the solution stream can be calculated as shown in Eq. (18), where dqL is the differential latent energy exchange between the air and solution streams is calculated from Eq. (19)
(17)
(18)
(19)

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].

Therefore, the ideal energy balance equation in LAMEEs can be expressed as shown in Eqs. (20a) and (20b). Cr is the ratio of minimum to maximum specific heat ratios of fluid streams. The second term on the right-hand side of Eq. (20a) is the latent heat term and leads to coupling between heat and mass transfer
(20a)
(20b)
(21)

2.1 Coupled ϵ-Number of Transfer Units Method Performance Methodology.

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.

The energy balance expression, presented in Eq. (20a), can be rewritten as shown in the following equation:
(22)
Using the definitions of ϵs and ϵm with Eq. (22), one can arrive at the following equation:
(23)
Simonson and Besant [9] proposed an operating factor H*, defined in Eq. (24), which represents the potential for latent heat transfer (phase change energy because of moisture transfer) to that of sensible heat transfer. Substituting the expression for H* into Eq. (23), the expression for Cre is obtained as shown in Eq. (25)
(24)
(25)
To estimate Cre using Eq. (25), the values of ϵs and ϵm must be known, which are also coupled to Cre and me*. Furthermore, it is not straightforward to develop an expression similar to Eq. (25) for me* given the complex relation of the solution stream's equilibrium humidity ratio (Wsol) to its temperature. The liquid solution (desiccant) used in this study is LiCl, and the concentration and temperature of the solution are used to obtain Wsol using correlations presented in Ref. [28]. An iterative methodology is employed to solve for Cre,me*,ϵs, and ϵm as shown in Fig. 3. The initial guesses for ϵsandϵm are carried out by using Eqs. (3) and (4) and setting the values of Cre and me* equal to Cr and m*, respectively. The iterative procedure presented in Fig. 3 is carried out until the change in the values of Cre,me*,ϵs, and ϵm between the current and the previous iteration, is less than 10e−6. Equations (22)(25) can also be applied to a crossflow energy exchanger, where the expression to estimate ϵs and ϵm are different from Eqs. (3) and (4) and are calculated using the following equations:
(26)
(27)
Fig. 3
Procedure for solving the coupled ϵ-NTU performance methodology to estimate Cre, me*, ϵs, and ϵm
Fig. 3
Procedure for solving the coupled ϵ-NTU performance methodology to estimate Cre, me*, ϵs, and ϵm
Close modal

2.2 Coupled ϵ-Number of Transfer Units Method Design Methodology.

In the design methodology, NTU is calculated to develop an exchanger that achieves a specified effectiveness for a given sent of operating conditions. Using the coupled ϵ-NTU method is straightforward to design an energy exchanger, as Cre,me* can be readily calculated using known input and the required desired output conditions. Equations (28) and (29) are used to estimate the NTUh and NTUm for a counterflow energy exchanger. However, it must be noted that the coupled ϵ-NTU method cannot be used when the fraction inside the natural logarithm is negative as it returns an imaginary value. Therefore, for this study the values of Cre,me*,ϵs, and ϵm are bounded between 0 and 1 while using the coupled ϵ-NTU method for designing a LAMEE. A similar procedure can be followed for the design of a crossflow energy exchanger, however the equations for NTUh and NTUm are different, and are not as straightforward to solve
(28)
(29)

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.

3 Validation of the Coupled ϵ-Number of Transfer Units Method With a Numerical 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.

3.1 Validation of the Coupled ϵ-Number of Transfer Units Performance Methodology.

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, NTUh varies from 1.5 to 4.5, and the ratio of NTUh/NTUm 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.

Table 1

Validation data sets used to estimate the accuracy of the proposed coupled ϵ-NTU model

DatasetTair (°C)Tsol (°C)Wair (g/kgda)Xsol (msalt/msolution)
Dehumidification (Set A)30–4015–2512–2425–35
Humidification (Set B)20–3035–457–1425–35
Winter (Set C)0–(−10)−10–(−20)1–325–35
DatasetTair (°C)Tsol (°C)Wair (g/kgda)Xsol (msalt/msolution)
Dehumidification (Set A)30–4015–2512–2425–35
Humidification (Set B)20–3035–457–1425–35
Winter (Set C)0–(−10)−10–(−20)1–325–35
For inlet operating conditions presented in Table 1, the outlet temperature and humidity ratios of the air and solution streams can be estimated by the coupled ϵ-NTU and the numerical methods. These estimates can be compared using the temperature and humidity ratio differential defined in Eqs. (30) and (31). The results of these comparisons are plotted in Figs. 46 
(30)
(31)
Fig. 4
Comparison between ΔT(air) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Fig. 4
Comparison between ΔT(air) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Close modal
Fig. 5
Comparison between the ΔT(sol) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Fig. 5
Comparison between the ΔT(sol) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Close modal
Fig. 6
Comparison between the ΔW(air) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Fig. 6
Comparison between the ΔW(air) as estimated by the coupled ϵ-NTU model and the numerical model for: (a) counterflow and (b) crossflow LAMEE
Close modal

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.

Table 2

Average RMS errors for ΔT(air) and ΔW(air) for counterflow and crossflow configurations

DifferentialCounterflowCrossflow
Temperature0.13 °C0.17 °C
Humidity0.1 g/kgda0.1 g/kgda
DifferentialCounterflowCrossflow
Temperature0.13 °C0.17 °C
Humidity0.1 g/kgda0.1 g/kgda

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.

Table 3

Average RMS errors for ΔT(sol) and ΔW(sol) for counterflow and crossflow configurations

DifferentialCounterflowCrossflow
Temperature0.05 °C0.05 °C
Humidity0.05 g/kgda0.05 g/kgda
DifferentialCounterflowCrossflow
Temperature0.05 °C0.05 °C
Humidity0.05 g/kgda0.05 g/kgda
The RMS errors given in Tables 2 and 3 do not give a complete picture of the accuracy of the coupled ϵ-NTU model. To further investigate the errors associated with the coupled ϵ-NTU model, the normalized temperature and humidity ratio differentials defined in Eqs. (32) and (33) are used. The normalized temperature and humidity ratio differentials applied on the air side represent the magnitude of the difference in sensible and moisture effectiveness values estimated by the coupled ϵ-NTU (ϵ*) and the numerical model given air is the fluid with minimum heat capacity and mass flowrate
(32)
(33)

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.

Table 4

Average normalized errors for counter and crossflow configurations

ConfigurationΔTnorm,airΔWnorm,airΔTnorm,sol
Counterflow1%1%0.4%
Crossflow1.5%1%0.5%
ConfigurationΔTnorm,airΔWnorm,airΔTnorm,sol
Counterflow1%1%0.4%
Crossflow1.5%1%0.5%

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.

3.2 Validation of the Coupled ϵ-Number of Transfer Units Design Methodology.

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 (Uh, Um) 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).

Fig. 7
(a) Validation process for design methodology, (b) difference in the sensible and moisture transfer effectiveness estimated by the coupled ϵ-NTU when used for designing an energy exchanger compared to a numerical model
Fig. 7
(a) Validation process for design methodology, (b) difference in the sensible and moisture transfer effectiveness estimated by the coupled ϵ-NTU when used for designing an energy exchanger compared to a numerical model
Close modal

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.

3.3 Comparison of Coupled ϵ-Number of Transfer Units With Classical ϵ-Number of Transfer Units Methods.

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 NTUh, NTUm, Cre, and me* where Eqs. (3) and (4) are mathematically valid.

Fig. 8
Comparison of: (a) sensible and (b) moisture transfer effectiveness values estimated using coupled ϵ-NTU and classical ϵ-NTU with the numerical model. The dotted lines represent ±5% bounds.
Fig. 8
Comparison of: (a) sensible and (b) moisture transfer effectiveness values estimated using coupled ϵ-NTU and classical ϵ-NTU with the numerical model. The dotted lines represent ±5% bounds.
Close modal

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.

Fig. 9
Variation of: (a) the temperature differential of the air stream (Eq. (30)) and (b) sensible effectiveness with H* as estimated by the classical, coupled ϵ-NTU and numerical methods. Tsol = 22 °C, Tair = 33 °C, NTUh = 3, NTUm = 1.2, and m* = 0.5.
Fig. 9
Variation of: (a) the temperature differential of the air stream (Eq. (30)) and (b) sensible effectiveness with H* as estimated by the classical, coupled ϵ-NTU and numerical methods. Tsol = 22 °C, Tair = 33 °C, NTUh = 3, NTUm = 1.2, and m* = 0.5.
Close modal

3.4 Errors Associated With Coupled ϵ-Number of Transfer Units Method.

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.

Fig. 10
ΔTnorm,air and ΔWnorm,air errors associated with the coupled ϵ-NTU method used to estimate the performance of a LAMEE for input conditions from Table 1
Fig. 10
ΔTnorm,air and ΔWnorm,air errors associated with the coupled ϵ-NTU method used to estimate the performance of a LAMEE for input conditions from Table 1
Close modal

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.

Fig. 11
Histograms illustrating the frequency of  ΔTnorm,air and ΔWnorm,air for the validation conducted for 14,690 data points
Fig. 11
Histograms illustrating the frequency of  ΔTnorm,air and ΔWnorm,air for the validation conducted for 14,690 data points
Close modal

4 Conclusion

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 NTUh and NTUm 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.

Acknowledgment

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.

Data Availability Statement

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

Nomenclature

A =

area (m2)

C =

heat capacity rate (kW/K)

cp =

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

NTUh =

number of heat transfer units

NTUm =

number of mass transfer units

ql =

latent heat transfer rate (kW)

qs =

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/kgda)

Wsol =

equilibrium humidity ratio of solution (g/kgda)

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

Appendix: Comparison of Coupled ϵ-Number of Transfer Units With Experimental Data

A.1 Counterflow.

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, NTUh, NTUm, and Esol are presented in Table 5.

Table 5

Operating parameters for validation tests for counterflow LAMEE. Experimental data obtained from Ref. [14].

Testmair (kg/h)Tair_in (°C)Wair_in (kg/kgda)msol (kg/h)Tsol_in (°C)Esol (%)NTUhNTUm
T15.2930.10.01438.312432.05.02.6
T23.6630.00.01435.642432.06.03.4
T32.6230.00.01424.082432.07.04.2
T45.4125.00.01428.432432.05.02.7
T55.2235.20.01428.152432.05.02.6
T65.3130.00.02028.352432.05.02.7
T75.3430.10.00828.192432.05.04.0
T85.3230.00.01454.172432.15.03.2
T95.3230.00.014212.452432.05.02.8
T105.3329.90.01388.212832.15.02.8
T115.3330.00.01388.203232.05.03.7
T125.3230.00.01428.642437.05.02.6
T135.3130.00.01397.872427.65.02.6
Testmair (kg/h)Tair_in (°C)Wair_in (kg/kgda)msol (kg/h)Tsol_in (°C)Esol (%)NTUhNTUm
T15.2930.10.01438.312432.05.02.6
T23.6630.00.01435.642432.06.03.4
T32.6230.00.01424.082432.07.04.2
T45.4125.00.01428.432432.05.02.7
T55.2235.20.01428.152432.05.02.6
T65.3130.00.02028.352432.05.02.7
T75.3430.10.00828.192432.05.04.0
T85.3230.00.01454.172432.15.03.2
T95.3230.00.014212.452432.05.02.8
T105.3329.90.01388.212832.15.02.8
T115.3330.00.01388.203232.05.03.7
T125.3230.00.01428.642437.05.02.6
T135.3130.00.01397.872427.65.02.6

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.

Fig. 12
Comparison of (a) sensible and (b) moisture transfer effectiveness obtained from experiments, numerical model, coupled and classical ϵ-NTU methods—experimental data obtained from Ref. [14]
Fig. 12
Comparison of (a) sensible and (b) moisture transfer effectiveness obtained from experiments, numerical model, coupled and classical ϵ-NTU methods—experimental data obtained from Ref. [14]
Close modal
A.1.1 Comparison With Experimental Data Crossflow.

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.

Table 6

Operating parameters for validation tests for crossflow LAMEE. Experimental data obtained from Ref. [29].

NTUhm*ϵs_expϵs*ϵs_numϵs_classicalϵm_expϵm*ϵm_numϵm_classical
2267.169.871.082.750.055.753.149.5
2370.876.176.083.550.856.453.851.9
2473.579.078.684.251.156.754.153.2
4275.378.781.195.7574.278.876.369.5
4381.087.987.096.6575.679.877.673.5
4484.991.489.991.4775.080.378.275.5
8285.783.290.499.591.694.692.785.2
8389.493.694.499.791.895.494.089.6
8491.896.996.199.892.895.794.591.7
NTUhm*ϵs_expϵs*ϵs_numϵs_classicalϵm_expϵm*ϵm_numϵm_classical
2267.169.871.082.750.055.753.149.5
2370.876.176.083.550.856.453.851.9
2473.579.078.684.251.156.754.153.2
4275.378.781.195.7574.278.876.369.5
4381.087.987.096.6575.679.877.673.5
4484.991.489.991.4775.080.378.275.5
8285.783.290.499.591.694.692.785.2
8389.493.694.499.791.895.494.089.6
8491.896.996.199.892.895.794.591.7

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