Introduction

The use of heating and air conditioning systems in buildings is on the rise [1], this has led to an increase in the buildings sector’s share of energy consumption and CO2 emissions [2], which has led to studying different methods to improve the use of alternative sources of energy [3, 4]. Therefore, the use of environmentally friendly and energy-efficient alternatives to the vapor compression air conditioning systems is becoming increasingly more important. A promising alternative is an air conditioning system that consists of an evaporative cooling system and a liquid desiccant-based dehumidifier. Liquid desiccant dehumidifiers are either adiabatic or internally cooled [5]. It has been shown that the moisture removal rate of an internally-cooled liquid desiccant dehumidifier can be higher by up to 47% [6] and 54% [7] relative to an adiabatic liquid desiccant dehumidifier.

Internally-cooled liquid desiccant dehumidifiers include 3-fluid [8, 9] and 4-fluid designs [10,11,12]. The three fluids in a 3-fluid design are supply air, liquid desiccant, and a coolant (chilled water or refrigerant). The dehumidifier is a compact heat and mass exchanger that consists of a stack of channels. Supply air and liquid desiccant flow in the supply channels, while chilled water flows in the exhaust channels. The 4-fluid design is a liquid desiccant dehumidifier with internal evaporative cooling, the four fluids are supply air, liquid desiccant, exhaust air, and water. Similar to the 3-fluid design, the dehumidifier consists of supply and exhaust channels. Supply air and liquid desiccant flow in the supply channels, where the dehumidification takes place, while exhaust air and water flow in the exhaust channels, where the evaporative cooling takes place.

Heat and mass exchangers are typically classified according to flow arrangement and it is well known that their performance is dependent on the flow arrangement. A number of studies [13,14,15,16,17] have been conducted to compare the performance of different flow arrangements of 3-fluid internally-cooled liquid desiccant dehumidifiers. Liu et al. [13] analyzed the performance of 6 flow arrangements of 3-fluid internally-cooled liquid desiccant dehumidifiers using a two-dimensional (2D) heat and mass transfer model. The three fluids were supply air, liquid desiccant, and cooling water. It was concluded that the two flow arrangements where the supply air and liquid desiccant are in counter-flow have the best performance, the cooling water was in cross-flow and in counter-flow with the supply air in these two flow arrangements. Liu et al. [14] compared the performance of 3-fluid internally-cooled liquid desiccant dehumidifiers with 10 different flow arrangements. The three fluids were supply air, liquid desiccant, and cooling water. The comparison was performed using a heat and mass transfer model of the dehumidifier. The study concluded that, for best performance, the supply air should be in counter-flow with the cooling water and in cross-flow with the liquid desiccant. Li and Yao [15] analyzed the performance of 10 different flow arrangements of 3-fluid internally-cooled liquid desiccant dehumidifiers. The three fluids were supply air, liquid desiccant, and cooling water. The comparison was performed using a heat and mass transfer model of the dehumidifier. It was shown that the best dehumidification performance was obtained when the supply air was in counter-flow with the liquid desiccant and cooling water. In another study, using the same model, Li and Yao [16] compared the performance of 10 different flow arrangements of 3-fluid internally-cooled liquid desiccant dehumidifiers in terms of cooling capacity. It was concluded that the highest cooling capacity is obtained when the supply air flows in cross-flow with the liquid desiccant and in counter-flow with the cooling water. Using a 2D heat and mass transfer model, Guan et al. [17] compared the performance of 3 flow arrangements of 3-fluid internally-cooled liquid desiccant systems. The three fluids were supply air, liquid desiccant, and refrigerant. In the first flow arrangement, the air and liquid desiccant flow in parallel-flow. In the second flow arrangement, the air and liquid desiccant flow in cross-flow, and in the third flow arrangement, the air and liquid desiccant flow in counter-flow. Considering the combined performance of the dehumidifier and the regenerator, it was concluded that the counter-flow arrangement showed the best performance.

Although there are few studies that compared the performance of different flow arrangements of 3-fluid internally-cooled liquid desiccant dehumidifiers, to the best knowledge of the authors, no study has compared the different flow arrangements of 4-fluid internally-cooled liquid desiccant dehumidifiers. Therefore, a comparison of the thermal performance of 10 flow arrangements of 4-fluid internally-cooled liquid desiccant dehumidifiers is performed in this study. The 10 flow arrangements of a 4-fluid internally-cooled liquid desiccant dehumidifier considered in this study are shown in Fig. 1. In all flow arrangements, the liquid desiccant and water flow from the top to bottom. The flow arrangements are named according to the flow directions of the supply air and liquid desiccant. For example, in the flow arrangements parallel 1–3, both the supply air and liquid desiccant enter the dehumidifier at the same end and move in the same direction, while the flow direction of the exhaust air, with respect to the supply air, changes from parallel-flow, parallel 1, to cross-flow, parallel 2, and to counter-flow, parallel 3.

Fig. 1
figure 1

Schematic showing the 10 different flow arrangements of 4-fluid internally-cooled liquid desiccant dehumidifiers

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This study has two objectives, the first and main objective is to compare 10 flow arrangements of 4-fluid internally-cooled liquid desiccant dehumidifiers in terms of the supply air humidity ratio change and supply air enthalpy change. The second objective is to evaluate the effect of nine input parameters, namely supply air temperature and humidity ratio, exhaust air temperature and humidity ratio, liquid desiccant temperature, concentration, and flow rate, supply air mass flow rate, and exhaust to supply air mass flow rate ratio, on the performance of the 10 flow arrangements of 4-fluid liquid desiccant dehumidifiers. The comparison is conducted using a 2D model of the heat and mass transfer in 4-fluid liquid desiccant dehumidifiers. The model is validated by comparing its predictions to experimental measurements from literature.

The remainder of this paper is structured as follows. ‘Experimental’ gives a brief description of the experimental setup, including the specifications of the prototype and the test conditions. ‘Numerical model’ presents the heat and mass transfer model, including the governing equations, boundary conditions, and transfer coefficients. In ‘Results and discussion’, the results are presented and discussed. Finally, the conclusions of the study are presented in ‘Conclusions’.

Methods

Experimental

The experiments conducted by Woods and Kozubal [11] were used to validate the numerical heat and mass transfer model. A brief description of the dehumidifier used in conducting the experiments is as follows. The core of the dehumidifier was a heat exchanger that consisted of a stack of 36 pairs of supply and exhaust channels. Schematic of a channel pair is shown in Fig. 2. As shown in the figure, the supply air and liquid desiccant flow in cross-flow, with the supply air flowing left to right (horizontal) and the liquid desiccant (lithium chloride) flowing from top to bottom (vertical direction). While the exhaust air and water flow in parallel-flow, both flow vertically from top to bottom. The specifications of the dehumidifier are listed in Table 1. To ensure the uniform distribution of liquid desiccant and water, the walls of the supply and exhaust channels were covered with wicking material. In the supply channels, membranes were attached on the wicking material. The dehumidifier was tested by varying the inlet conditions of the supply air, exhaust air, and liquid desiccant. The test conditions are listed in Table 2 [11].

Fig. 2
figure 2

Schematic of a pair of supply and exhaust channels

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Table 1 Specifications of the dehumidifier
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Table 2 Experimental test conditions [11]
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Numerical model

Figure 3 shows a schematic of a pair of supply and exhaust channels having the flow arrangement of cross 1, as shown in Fig. 1. This flow arrangement is selected for presenting the heat and mass transfer model since it has the same flow arrangement as the experimental setup. The supply air flows in the positive x direction, in cross-flow with respect to the exhaust air and liquid desiccant, which are flowing in the negative y direction. It should be noted that according to the flow arrangement of the dehumidifier, the governing equations and the corresponding boundary conditions will slightly change. In order to simplify the governing equations, the following assumptions are made [18, 19]:

  1. 1.

    The system is insulated; therefore, no heat transfer occurs between the device and the surroundings.

  2. 2.

    The thickness of the channels is small relative to the channel length and width; therefore, the variations of temperature and humidity ratio normal to the flow are neglected, reducing the problem to be two-dimensional.

  3. 3.

    The flow is incompressible, laminar, fully developed, and steady.

  4. 4.

    The mass flow rates are constant.

  5. 5.

    The heat and mass transfer coefficients are constant.

  6. 6.

    Heat and mass transfer analogy holds.

Fig. 3
figure 3

Schematic of the dehumidifier used to develop the heat and mass transfer model

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Heat and mass transfer governing equations

By applying the stated assumptions, the differential equations that govern the heat and mass transfer in the internally-cooled liquid desiccant dehumidifier are given as follows. In these equations, subscripts s, e, LD, and wf refer to the supply channel, exhaust channel, liquid desiccant, and water film, respectively.

Energy balance of the supply air is given by

$${\dot{m}}_{\mathrm{s}}{C}_{\mathrm{p},\mathrm{s}}\frac{\partial {T}_{\mathrm{s}}}{\partial x}=UW({T}_{\mathrm{LD}}-{T}_{\mathrm{s}}),$$
(1)

where U is the overall heat transfer coefficient, determined by

$$U={\left(\frac{1}{{H}_{\mathrm{s}}}+\frac{{t}_{\mathrm{membrane}}}{{k}_{\mathrm{membrane}}}\right)}^{-1},$$
(2)

where H is the convective heat transfer coefficient, tmembrane is the thickness of the membrane, and kmembrane is the thermal conductivity of the membrane.

Energy balance of the exhaust air is given by

$$-{\dot{m}}_{\mathrm{e}}{C}_{\mathrm{p},\mathrm{e}}\frac{\partial {T}_{\mathrm{e}}}{\partial y}={H}_{\mathrm{e}}L({T}_{\mathrm{wf}}-{T}_{\mathrm{e}}).$$
(3)

The specific heats of the supply and exhaust air are determined by the following equation [20]:

$${C}_{\mathrm{p}}={C}_{\mathrm{p},\mathrm{air}}+\omega {C}_{\mathrm{p},\mathrm{vapor}},$$
(4)

where ω is the humidity ratio, Cp,air (1007 J kg−1 K−1) is the specific heat of air, and Cp,vapor (1870 J kg−1 K−1) is the specific heat of water vapor at room temperature.

Moisture balance of the supply air is given by

$${\dot{m}}_{\mathrm{s}}\frac{\partial {\omega }_{\mathrm{s}}}{\partial x}={U}_{\mathrm{m},\mathrm{s}}W({\omega }_{\mathrm{LD}}-{\omega }_{\mathrm{s}}),$$
(5)

where Um,s is the overall mass transfer coefficient and ωLD is humidity ratio of air in equilibrium with the liquid desiccant film. The overall mass transfer coefficient is calculated using the following equation:

$${U}_{\mathrm{m},\mathrm{s}}={\left(\frac{1}{{H}_{\mathrm{m},\mathrm{s}}{\rho }_{\mathrm{air}}}+\frac{{t}_{\mathrm{membrane}}}{{\rho }_{\mathrm{air}}{D}_{\mathrm{membrane}}}\right)}^{-1}$$
(6)

ωLD is determined by [20]

$${\omega }_{\mathrm{LD}}=0.622\frac{{\mathrm{P}}_{\mathrm{LD}}}{{P}_{\mathrm{atm}}-{P}_{\mathrm{LD}}}.$$
(7)

The properties of the liquid desiccant solution, including the saturated vapor pressure above the liquid desiccant film, PLD, density, ρLD, thermal conductivity, kLD, specific heat, Cp,LD, and enthalpy of dilution, hdilution, are determined by correlations from Conde [21].

Moisture balance of the exhaust air is given by

$$-{\dot{m}}_{\mathrm{e}}\frac{\partial {\omega }_{\mathrm{e}}}{\partial y}={H}_{\mathrm{m},\mathrm{e}}{\rho }_{\mathrm{air}}L({\omega }_{\mathrm{wf}}-{\omega }_{\mathrm{e}}).$$
(8)

The saturated air humidity ratio, ωwf, is determined by Eq. (7), substituting PLD with Pg, the pressure of the saturated vapor at the exhaust air temperature, which is calculated as follows [22, 23]:

$${P}_{\mathrm{g}}=(611.21){\mathrm{e}}^{\left(\left(18.678-\frac{{T}_{\mathrm{e}}}{234.5}\right)\left(\frac{{T}_{\mathrm{e}}}{257.14+{T}_{\mathrm{e}}}\right)\right)}.$$
(9)

Energy balance of the liquid desiccant film is given by

$${-\dot{m}}_{\mathrm{LD}}{C}_{p,\mathrm{LD}}\frac{\partial {T}_{\mathrm{LD}}}{\partial y}+UL\left({T}_{\mathrm{LD}}-{T}_{\mathrm{s}}\right)+{U}_{\mathrm{ms}}L\left({\omega }_{\mathrm{LD}}-{\omega }_{\mathrm{s}}\right)\left({{h}_{\mathrm{dilution}}+h}_{\mathrm{fg}}\right)+{U}_{\mathrm{plate}}L\left({T}_{\mathrm{LD}}-{T}_{\mathrm{wf}}\right)=0,$$
(10)

where hdilution is the enthalpy of dilution of the liquid desiccant and hfg is the enthalpy of evaporation of water, which is determined by [24]

$${h}_{\mathrm{fg}}={10}^{3}(2501-2.369 {T}_{\mathrm{wf}}).$$
(11)

The overall heat transfer coefficient of the plastic plate, Uplate, is calculated as follows:

$${U}_{\mathrm{plate}}={\left(\frac{{t}_{\mathrm{wick}}}{{k}_{\mathrm{LD}}}+\frac{{t}_{\mathrm{plate}}}{{k}_{\mathrm{plate}}}+\frac{{t}_{\mathrm{wick}}}{{k}_{\mathrm{wf}}}\right)}^{-1},$$
(12)

where kLD, kplate, and kwf are the thermal conductivity of the liquid desiccant, plastic plate, and water, respectively.

Energy balance of the water film is given by

$${H}_{\mathrm{e}}L\left({T}_{\mathrm{wf}}-{\mathrm{T}}_{\mathrm{e}}\right)+{h}_{\mathrm{me}}{\rho }_{\mathrm{air}}L\left({\omega }_{\mathrm{wf}}-{\omega }_{\mathrm{e}}\right){h}_{\mathrm{fg}}-{U}_{\mathrm{plate}}L\left({T}_{\mathrm{LD}}-{T}_{\mathrm{wf}}\right)=0.$$
(13)

Equations (1), (3), (5), (8), (10), and (13) govern the heat and mass transfer in the internally-cooled liquid desiccant dehumidifier under the stated assumptions.

Boundary conditions

The governing equations are completed by the following boundary conditions. The inlet conditions of the supply air are specified, Eqs. (14) and (15). At y = W, the top boundary of the supply channel, the liquid desiccant temperature is specified, Eq. (16). The exhaust air temperature and humidity ratio are specified at the inlet of the exhaust channel, at y = W, Eqs. (17) and (18).

$${T}_{\mathrm{s}}(0, y)={T}_{\mathrm{s},\mathrm{ in}}$$
(14)
$${\omega }_{\mathrm{s}}(0, y)={\omega }_{\mathrm{s},\mathrm{ in}}$$
(15)
$${T}_{\mathrm{e}}(x, W)={T}_{\mathrm{e},\mathrm{ in}}$$
(16)
$${\omega }_{\mathrm{e}}(x, W)={\omega }_{\mathrm{e},\mathrm{ in}}$$
(17)
$${T}_{\mathrm{LD}}(x, W)={T}_{\mathrm{LD},\mathrm{ in}}$$
(18)

Heat and mass transfer coefficients

The convective heat transfer coefficients in the supply and exhaust channels are estimated as follows:

$$H=\frac{\mathrm{Nu}{k}_{\mathrm{air}}}{{d}_{\mathrm{h}}}.$$
(19)

The Nusselt number, Nu, for parallel plates is assumed to be equal to the average of the constant wall temperature boundary condition (7.541) and constant heat flux (8.235) boundary condition values [25], since the boundary condition at the channel walls is neither constant temperature nor constant heat flux.

The mass transfer coefficient in the supply and exhaust channels are calculated as follows:

$${h}_{\mathrm{m}}=\frac{\mathrm{Sh}{D}_{\mathrm{va}}}{{d}_{\mathrm{h}}},$$
(20)

where Sh is the Sherwood number, and Dva is the mass diffusivity of water vapor in air. Given that heat and mass transfer analogy holds, the Sherwood number is considered equal to the Nusselt number.

Computational grid

The heat and mass transfer equations were discretized and solved using the finite element method. The computational domain, shown in Fig. 4, was discretized using a structured grid with quadrilateral elements. The length and width of the domain were divided into 20 and 40 elements, respectively. The elements along the length and width were distributed using a symmetric arithmetic sequence with an element ratio of 6. The total number of elements in the computational grid is 800. To ensure the selected grid provides grid-independent results, two additional grids with 3200 and 12,800 elements were created. Using the additional two grids, the obtained outlet conditions of the supply air changed by less than 0.1%. Therefore, it was verified that the selected grid provided grid-independent results.

Fig. 4
figure 4

Computational domain grid

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Results and discussion

Numerical model validation

To validate the numerical model, the predicted supply air outlet humidity ratio of the experimental tests conditions listed in Table 2 are compared to the measured values [11]. Figure 5 compares the predictions of the numerical model with measured values of the supply air outlet humidity ratio. The data labels in the figure represent the test numbers from Table 2. As shown in the figure, the model’s predictions match the measured values within 6.7%. The average difference between the predicted and measured values is about 3.7%. At 16 out of the 20 tests, the model predicted higher outlet humidity ratios compared to the measured values.

Fig. 5
figure 5

Numerical model predictions vs experimental measurements of the supply air outlet humidity ratio

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Performance comparison—reference case

Using the test conditions 8 from Table 2 as input conditions, the performance of the 10 flow arrangements of the 4-fluid liquid desiccant dehumidifier is compared in terms of the supply air humidity ratio change, Δωs (ωs,in − ωs,out), and the supply air enthalpy change, Δhs (hs,in − hs,out). Figure 6a, b shows the supply air humidity ratio change and enthalpy change of the 10 flow arrangements in percentage, in which the flow arrangement counter 3 is taken as the reference (100%) since it has the lowest change in supply air humidity ratio and enthalpy.

Fig. 6
figure 6

Comparison of all flow types in terms of a the change in supply air humidity ratio and b the change in supply air enthalpy. The lowest performing flow arrangement, counter 3, is taken as the reference case (100%)

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As shown in Fig. 6a, flow arrangement counter 1 has the highest change in supply air humidity ratio, 104.3%, and enthalpy, 110.5%. The dehumidifiers with the flow arrangements, parallel 1, cross 2, and counter 3, have the lowest performances, in these flow arrangements the supply air and exhaust air are in parallel-flow. While the dehumidifiers with the flow arrangements, parallel 3, cross 4, and counter 1, have the highest performance, in these flow arrangements the supply air and exhaust air are in counter-flow. Therefore, it can be concluded that the flow direction of the supply and exhaust air with respect to each other has the most effect on the performance.

Effect of input conditions

In this section, the effect of nine input conditions on the supply air humidity ratio change and supply air enthalpy change are evaluated. In each case the input conditions that are not varied are kept constant and equal to the test conditions 8 from Table 2.

Effect of supply air inlet temperature

Figure 7 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the supply air inlet temperature, Ts,in, for all 10 flow arrangements. As shown in the figure, increasing the supply air inlet temperature from 24 to 46 °C reduces the change in supply air humidity ratio by about 1.5 g kg−1 (20%), while it increases the change in supply air enthalpy by about 11 kJ kg−1 (90%). The respective decrease and increase in the supply air humidity ratio change and enthalpy change can be explained as follows. As the supply air inlet temperature increases, the temperature of the liquid desiccant solution along the dehumidifier increases as well. This increase in temperature leads to an increase in the vapor pressure, and consequently the humidity ratio, of the air in equilibrium with the liquid desiccant, which decreases the dehumidification potential. The difference in performance between the flow arrangements is relatively small; however, as shown in Fig. 7b, d, which are zoomed-in views of the plots in Fig. 7a, c, it can be noted that the difference in performance between the flow arrangements increases at higher supply air inlet temperatures.

Fig. 7
figure 7

The effect of supply air inlet temperature, Ts,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of supply air inlet humidity ratio

Figure 8 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the supply air inlet humidity ratio, ωs,in, for all 10 flow arrangements. The removed moisture from the supply air increases by about 7 g kg−1 (140%) and the enthalpy change of the supply air increases by about 14 kJ kg−1 (100%), by increasing the supply air inlet humidity ratio from 15 to 29 g kg−1. Increasing the supply air inlet humidity ratio improves the moisture transfer between the supply air and desiccant solution. The difference in performance between the flow arrangements increases with increasing supply air inlet humidity ratio.

Fig. 8
figure 8

The effect of supply air inlet humidity ratio, ωs,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of exhaust air inlet temperature

Figure 9 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the exhaust air inlet temperature, Te,in, for all 10 flow arrangements. Increasing the exhaust air inlet temperature from 24 to 46 °C reduces the change in supply air humidity ratio and enthalpy by about 0.7 g kg−1 (9.5%) and 3.5 kJ kg−1 (18%), respectively. Increasing the exhaust air inlet temperature while its humidity ratio is kept constant increases its wet bulb temperature, which is the lowest attainable temperature in the exhaust channel. Therefore, by increasing the exhaust air temperature, the evaporative cooling potential in the exhaust channel reduces, leading to higher temperatures of the liquid desiccant and consequently lowering the obtained change in the supply air humidity ratio and enthalpy.

Fig. 9
figure 9

The effect of exhaust air inlet temperature, Te,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of exhaust air inlet humidity ratio

Figure 10 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the exhaust air inlet temperature, ωe,in, for all 10 flow arrangements. Increasing the exhaust air inlet humidity ratio from 15 to 29 g kg−1 reduces the change in supply air humidity ratio and enthalpy by about 1 g kg−1 (14.5%) and 5.3 kJ kg−1 (27%), respectively. Increasing the exhaust air inlet humidity ratio while its temperature is kept constant increases its wet bulb temperature. Therefore, the cooling potential of the exhaust channel is reduced, which leads to higher temperatures of the liquid desiccant and therefore lowering the obtained change in the supply air humidity ratio and enthalpy.

Fig. 10
figure 10

The effect of exhaust air inlet humidity ratio, ωe,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of liquid desiccant inlet temperature

Figure 11 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the liquid desiccant inlet temperature, TLD,in, for all 10 flow arrangements. Increasing the liquid desiccant inlet temperature from 24 to 38 °C reduces the change in supply air humidity ratio and enthalpy by about 0.2 g kg−1 (2.9%) and 1.2 kJ kg−1 (6.2%), respectively. This shows that the effect of the liquid desiccant inlet temperature on the performance of the dehumidifier is very small. Since the liquid desiccant is cooled by the evaporation of the water in the exhaust channel, its inlet temperature has a negligible effect on the change of the supply air humidity ratio and enthalpy. This will be shown clearly in ‘Contours of temperature and humidity ratio in the supply channel’.

Fig. 11
figure 11

The effect of liquid desiccant inlet temperature, TLD,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of liquid desiccant concentration

Figure 12 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the liquid desiccant concentration, CLD,in, for all 10 flow arrangements. Increasing the liquid desiccant concentration from 0.32 to 0.46 increases the change in supply air humidity ratio and enthalpy by about 4.8 g kg−1 (108%) and 9.3 kJ kg−1 (70.5%), respectively. When the liquid desiccant temperature is fixed, an increase in its concentration leads to a decrease in the vapor pressure of the air that is in equilibrium with it in the supply channel. The lower the vapor pressure, the higher the mass transfer potential. Therefore, increasing the liquid descant concentration leads to an increase in the supply air humidity ratio change.

Fig. 12
figure 12

The effect of liquid desiccant concentration, CLD,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of liquid desiccant flow rate

Figure 13 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the liquid desiccant flow rate, vLD, for all 10 flow arrangements. The plots of Δωs and Δhs are almost horizontal, that is, the flow rate of the liquid desiccant has very small effect on the performance of the dehumidifiers, which is in agreement with the experimental results [11]. Increasing the liquid desiccant flow rate from 6 to 20 mL min−1 increases the change in supply air humidity ratio and enthalpy by about 0.04 g kg−1 (0.6%) and 0.2 kJ kg−1 (1%), respectively. Given the use of membranes in the supply channels, the mass flow rate of the liquid desiccant relative to the supply air and exhaust air mass flow rates is very small, their ratio ranges from 3 to 25%. The small thermal mass of the liquid desiccant leads to a large change in its temperature along the dehumidifier, this is shown in ‘Contours of temperature and humidity ratio in the supply channel’, minimizing its effect on the outlet conditions of the dehumidifier.

Fig. 13
figure 13

The effect of liquid desiccant flow rate, vLD,in, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of supply air mass flow rate

Figure 14 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the supply air mass flow rate, \({\dot{m}}_{\mathrm{s}}\), for all 10 flow arrangements. Increasing the supply air mass flow rate from 2 to 8 g s−1 reduces the change in supply air humidity ratio and enthalpy by about 5.3 g kg−1 (54%) and 15.2 kJ kg−1 (56.4%), respectively. As the supply air mass flow rate increases, the air flows faster in the supply channel, reducing the contact time between the air and the liquid desiccant, which decreases the absorption of moisture from the supply air, leading to lower supply air humidity ratio and enthalpy change.

Fig. 14
figure 14

The effect supply air mass flow rate, \({\dot{m}}_{\mathrm{s}}\), on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Effect of exhaust to supply air mass flow rate ratio

Figure 15 shows the supply air humidity ratio change, Δωs, and the supply air enthalpy change, Δhs, as functions of the exhaust to supply air mass flow rate ratio, r, for all 10 flow arrangements. Increasing the exhaust to supply air mass flow rate ratio from 0 to 1, increases the change in supply air humidity ratio and enthalpy by about 4.2 g kg−1 (134%) and 19.4 kJ kg−1 (1830%), respectively. Increasing r from 0–0.2, leads to a sharp increase in the performance of the dehumidifiers. While this increase in performance almost flattens with further increase in r. The reason for this behavior can be explained as follows. The exhaust channel is the heat sink of the system. The higher the amount of air in the exhaust channel, the more cooling is provided to the supply channel, resulting in a decrease in the supply air and liquid desiccant temperatures. As pointed out before, the lower the temperature of the liquid desiccant along the dehumidifier, the higher the mass transfer potential, which lead to an increase in the supply air humidity ratio and enthalpy change.

Fig. 15
figure 15

The effect exhaust to supply air mass flow rate ratio, r, on a, b the humidity ratio change of the supply air and c, d the enthalpy change of the supply air for the 10 different flow arrangements of the 4-fluid dehumidifier

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Contours of temperature and humidity ratio in the supply channel

Using the inlet conditions of test 8 specified in Table 2, the contours of temperature and humidity ratio of the airstream in the supply channel for all 10 flow arrangements obtained using the two-dimensional model are shown in Figs. 16 and 17, respectively. The supply air inlet temperature and humidity ratio are 35 °C and 18.6 g kg−1, respectively. The length and width of the channels are 0.4 m. The contours demonstrate the importance of using a two-dimensional model. Referring to Fig. 1, where the 10 flow arrangements are illustrated, it can be seen that there is two-dimensional variation in the temperature and humidity ratio contours in the flow arrangements when at least one of the fluid streams is perpendicular to the other streams.

Fig. 16
figure 16

Contours of temperature of the airstream in the supply channel for all 10 flow arrangements

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Fig. 17
figure 17

Contours of humidity ratio of the airstream in the supply channel for all 10 flow arrangements

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Temperature and humidity ratio distribution in the counter 1 and 3 flow arrangements

Given that the dehumidifier’s flow arrangements counter 1 and 3 have the highest and lowest performance, respectively, the distribution of the fluids temperature and humidity ratio along the dehumidifiers are analyzed. The temperature distribution of the supply air, exhaust air, liquid desiccant, and water film along the dehumidifiers with flow arrangements counter 1 and 3 are shown in Fig. 18a and c. The test 8 conditions from Table 2 are used as the inlet conditions. The arrows in the figure indicate the direction of the supply and exhaust air flow in the channels.

Fig. 18
figure 18

The variation of a temperature and b humidity ratio of the 4 fluids in the counter-flow 1 dehumidifier. The variation of c temperature and d humidity ratio of the 4 fluids in the counter-flow 3 dehumidifier

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The supply air enters the dehumidifiers at 35 °C and exits at 34.6 °C from the counter 1 flow arrangement while it exits at 35.6 °C from the counter 3 flow arrangement. The highest temperature difference between the supply and exhaust air streams in the counter 1 flow arrangement reaches about 4 °C at about 60% of the width. While the highest temperature difference between the supply and exhaust air streams in the counter 3 flow arrangement reaches about 2.2 °C at about 40% of the width.

It can be seen that the liquid desiccant temperature variation along the width for both flow arrangements follows the trend of the water film since it is losing heat to it. Here, the negligible effect of the liquid desiccant inlet temperature and flow rate can be seen. The liquid desiccant’s temperature changes sharply after passing about 5% (from 100 to 95%) of the width.

Figure 18b and d shows the humidity ratio distribution of the supply air, ωs, exhaust air, ωe, air in equilibrium with the liquid desiccant film, ωLD, and air in equilibrium with the water film, ωwf, along the dehumidifiers width with flow arrangements counter 1 and 3. The supply air enters the dehumidifiers at a humidity ratio of 18.6 g kg−1 and exits at 11.67 g kg−1 from the counter 1 flow arrangement while it exits at 11.96 g kg−1 from the counter 3 flow arrangement. In both flow arrangements, the exhaust air humidity ratio increases along the direction of the flow and exits at a saturation state.

Conclusions

In this study, 10 different flow arrangements of 4-fluid liquid desiccant dehumidifiers were compared in terms of the supply air humidity ratio change and supply air enthalpy change. The comparison was performed using a 2D numerical model of the heat and mass transfer in the dehumidifier, which was validated against experimental measurements from literature. The main conclusions of this study are as follows:

  • The optimum performance (maximum supply air humidity ratio and enthalpy decrease) was obtained by the dehumidifier’s flow arrangement counter 1, in which the supply air is in counter-flow with the liquid desiccant and the exhaust air.

  • The lowest performance (minimum supply air humidity ratio and enthalpy decrease) was obtained by the dehumidifier’s flow arrangement counter 3, in which the supply air is in counter-flow with the liquid desiccant and in parallel-flow with the exhaust air.

  • The maximum difference between the highest and lowest performing dehumidifier’s flow arrangements in terms of the supply air humidity ratio decrease was 4.3%, while it was 10.5% in terms of the supply air enthalpy decrease.

  • The flow direction of the supply and exhaust air with respect to each other has more effect on the performance than the flow direction of the supply and liquid desiccant with respect to each other.

  • The liquid desiccant inlet temperature and flow rate have negligible effect on the performance of the dehumidifier in all 10 flow arrangements. While increasing the liquid desiccant concentration improves the performance of the dehumidifier.