A Review of the Mathematical Models

A review of the mathematical models for predicting the heat and masstransfer process in the liquid desiccant dehumidier

Yimo Luo, Hongxing Yang n, Lin Lu, Ronghui QiRenewable Energy Research Group (RERG), Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, China

a r t i c l e i n f o

Article history:Received 22 August 2013Received in revised form12 November 2013Accepted 20 December 2013Available online 21 January 2014

Keywords:Liquid desiccant dehumidierHeat and mass transferMathematical model

a b s t r a c t

The paper aims to overview various mathematical models for modeling the simultaneous heat and masstransfer process in the liquid desiccant dehumidier. Firstly, the dehumidication principle is introducedbriey. Then the models are interpreted in terms of two classes of dehumidiers. For the adiabaticdehumidier, the models are mainly classied into three types: nite difference model, effectiveness NTU(NTU) model, and simplied models. For the internally cooled dehumidier, there are also three kindsof models: models without considering liquid lm thickness, models considering uniform liquid lmthickness, and models considering variable liquid lm thickness. This review is meaningful forcomprehending the development process and research status of the models and choosing suitablemodels for prediction. In addition, some suggestions are proposed for the model improvement.

& 2013 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5872. Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

2.1. The mechanism of heat and mass transfer in liquid desiccant dehumidier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5882.2. Structure of the dehumidier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

3. Models for adiabatic dehumidier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5893.1. Finite difference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5893.2. Effectiveness NTU (NTU) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.3. The simplied models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5923.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

4. Models for internally cooled dehumidier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5944.1. Models without considering liquid lm thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5944.2. Models considering uniform liquid lm thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5954.3. Models considering variable liquid lm thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5964.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

1. Introduction

With the acceleration of urbanization and improvement ofpeople0 living standard, a larger proportion of building energyconsumption will be needed to keep a comfortable indoor envir-onment [1]. But it is well-known that the traditional air condition-ing system is notorious as a result of heavy dependence on electricpower, limited ability of humidity control, and occurrence of wet

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1364-0321/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.rser.2013.12.009

n Corresponding author at: Ofce ZN 816, Department of Building ServicesEngineering, The Hong Kong Polytechnic University, Hong Kong, China. Tel.:852 2766 5863.

E-mail address: [email protected] (H. Yang).

Renewable and Sustainable Energy Reviews 31 (2014) 587599

surface for breeding mildew and bacteria and so on [2]. Thus, toreduce the energy consumption in buildings and improve theindoor air quality, the liquid desiccant assisted air conditioningsystem has drawn more and more attention [37].

The major component of interest regarding heat and masstransfer of such a system is the dehumidier. Compared with theexperimental research, the simulation method is more time andcost saving. Also, some parameters in the dehumidier interior canbe observed by simulation while it is impossible to be achieved byexperiment. Most importantly, the veried simulation models areeffective tools to assess and optimize similar dehumidiers. There-fore, a large amount of studies have been done to establishreasonable mathematical models for evaluating the liquid desic-cant dehumidiers. However, there is short of detailed and specicsummary of the models until now. Thus, it is meaningful to classifyand assess the models, which will provide useful suggestion forfuture research.

In the paper, the function principle of the liquid desiccantdehumidier is introduced rstly. Based on whether there is heatremoval, the dehumidier is divided into two types: adiabatic andinternally cooled dehumidier. Correspondingly, the models aresummarized in two respects in terms of the different structures.For each model, the assumptions, governing equations, boundary

conditions and other relevant information are provided. Theapplied conditions, development process, and research status ofthe simulation models are also presented. In addition, somesuggestions are put forward for the model improvement.

2. Problem description

2.1. The mechanism of heat and mass transfer in liquid desiccantdehumidier

It is well know that in the dehumidier, complicated heat andmass transfer occurs. The driving force for heat transfer is thetemperature difference between the air and desiccant solution,and for mass transfer is the water vapor pressure differencebetween the air and the surface of the desiccant solution. Themost classic and typical mass transfer theories include lm theory,penetration theory and surface renewal theory. The theory usedmost for the dehumidier is the lm theory. It is Nernset [8] whoproposed the lm theory rst in 1904. He assumed that the wholeresistance of mass transfer in a given phase lied in a thin andstagnant region of that phase at the interface. This region is calledlm. Based on it, Whiteman [9] developed the two-lm theory.

Nomenclature

A specic surface area per unit volume [m1]CP specic heat [J kg1 K1]Csat saturation specic heat [J kg1 K1]dh hydraulic diameter [m]Dm mass diffusion coefcient [m2 s1]Dt thermal diffusivity [m2 s1]g gravity [m s2]G specic mass ow rate [kg m2 s1]G0 mass ow rate [kg s1]h specic enthalpy [kJ/kg]he enthalpy of humid air in equilibrium with liquid

desiccant [kJ kg]he,eff effective saturation enthalpy [kJ kg], Eq. (18)H height of the dehumidier [m]k heat conduction coefcient [W m1 K1]L length of the dehumidier [m]Le Lewis numberm water condensation rate [g s1] or per unit cross-

sectional area [g m2 s1], Eq. (23)m* capacity ratio similar to the one used in sensible heat

exchangersM molecular weight [g mole1]NTU number of transfer unitsNu Nusselt number (dimensionless)P pressure [Pa]Pa partial vapor pressure in air [Pa]Ps partial vapor pressure over the solution [Pa]Pt total pressure [Pa]P dimensionless vapor pressure difference ratioQ heat transferred from solution to water [kW m1]T temperature [K]T dimensionless temperature difference ratiou velocity [m s1]V volume [m3]W humidity ratio [kg H2O kg1 dry air] or the width the

dehumidier [m]

We humidity ratio of humid air in equilibrium with liquiddesiccant [kmol H2O (kmol air)1]

We,eff effective humidity ratio [kmol H2O (kmol air)1],Eq. (19)

X desiccant concentration [kg desiccant kg1solution]Xw concentration of water in solution

[kg water kg1solution]Xv concentration of water vapor in air [kgH2O kg1]

Greek letters

C heat transfer coefcient [W m2 K1]0C heat transfer coefcient corrected for simultaneous

mass and heat transfer [W m2 K1]D mass transfer coefcient [kg m2 S1]a0D molar mass transfer coefcient [kmol m

2 S1] dimensionless temperatures (TTr)/h, Eq. (33) density [kg m3] dynamic viscosity [kg m1s1] kinetic viscosity [m2 s1] air side effectivenessHE heat exchanger effectiveness latent heat of vaporization [kJ kg1] lm thickness [m] change of or difference between parameters

Subscripts

a airc criticalf cooling uid, like water, air, refrigeranti inletint interfaceo outletp primary airr secondary (return air)s desiccant solutionv water vapor

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599588

The specic transport mechanism is shown in Fig. 1. Where, PB isthe partial pressure of component B in the gas phase and XB is themole fraction of component B in the liquid phase.

The two lm theory is very easy to understand and apply, but ithas several drawbacks. Firstly, it is not reasonable to predict thatthe rate of mass transfer is directly proportional to the moleculardiffusivity. Secondly, it is difcult to decide the thickness of thetwo laminar sub layers by experiment. Finally, the convective masstransfer in the thin lms is neglected, so the theory is only suitablefor the steady mass transfer process.

In the dehumidication process, some quantity of heat is givenout as well, including the phase change heat and dilution heat. Inall of the dehumidier models, the dilution heat is neglected as itis much smaller compared with the phase change heat of watervapor [10].

2.2. Structure of the dehumidier

As for the structure of the dehumidier, according to whetherthere is heat output, the dehumidiers can be classied intoadiabatic and internally cooled dehumidier [11]. The diagramsof two dehumidier structures are shown in Fig. 2.

In an adiabatic dehumidier, the air and liquid desiccantcontact directly with each other. In the early stage, the researchwas concentrated on the structure of spray tower as a result of itssimple construction and large specic surface area [12]. However,in the spray tower, the desiccant solution is generally broken intosmall droplets, so the problem is sometimes serious of mistgeneration and carryover of liquid droplets in the air stream. Then,the packed tower was used widely because it is more compact andcan provide a higher residence time, lower liquid pressure loss and

lessen the carryover problem. In 1980, Factor and Grossmanveried the possibility of employing the packed tower as dehu-midier by theoretical analysis and experiment [13]. As for thepadding materials, the random packing like ceramic [14], plasticand polypropylene pall ring are popular rst [15,16]. Then somestructured packing materials are employed to optimize the owand reduce the resistance in the dehumidier, such as the stainlesssteel corrugated orice plate [17], celdek [18,19] and so on.

In the adiabatic dehumidier, the temperature rise of thedesiccant, resulted from the latent heat, worsens its dehumidica-tion performance, thus its dehumidication efciency is relativelylower. A solution is to increase the desiccant ow rates to achievegood dehumidication levels. However, the high desiccant owrates and the followed higher ow rates of the regenerateddesiccant solution reduce the coefcient of performance of theliquid desiccant cycle [20,21]. In addition, the desiccant particlesare much easier to be entrained by the air and therefore pollutethe indoor environment.

To solve the above problems, the internally cooled dehumidierwas developed. In an internally cooled dehumidier, besides thecontact between air and desiccant, some cold source which canprovide cool uid like air or water is added to take away the latentheat produced in the process of dehumidication, which can beregarded as an isothermal process in general. The internally cooleddehumidier has been popular since the 1990s [22,23]. As thelatent heat is removed from the dehumidier, it reduces thetemperature rise of the solution and air, resulting in efciencyimprovement [24]. Meanwhile, it allows lower desiccant ow ratesin the internally cooled dehumidier so as to reduce the pollutionproblem. But the internally cooled dehumidier has more com-plicated structure than the adiabatic one. For example, to increasethe contact area, a n structure is widely used in the internallycooled dehumidier or other heat and mass transfer devices[2529].

3. Models for adiabatic dehumidier

There are mainly three types of mathematical models, includ-ing the nite difference model, effectiveness NTU (NTU) modeland the simplied solutions.

3.1. Finite difference model

In 1980, Factor et al. [13] promoted a theoretical model topredict the performance of a countercurrent packed column airliquid contractor, based on the model for adiabatic gas absorptionput forward by Treybal in 1969. In the model, the whole dehumi-dier is divided into n parts, as shown in Fig. 3.

To simplify the complexity of the heat and mass transferprocess, several assumptions were made: (1) the ow of air anddesiccant were assumed as the slug ow, (2) the process wasadiabatic, (3) the properties of the gas and liquid were assumed

Fig. 2. The structure diagram of two dehumidiers.

Gas phase Liquidphase

Interface

GasFilm

LiquidFilmPB

XB

PB,intXB,int

Mass Transfer Direction

Fig. 1. Schematic diagram of two-lm theory.

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599 589

constant across the differential element, which meant the gradi-ents only exist at the z direction, (4) both of the heat and moisturetransfer areas were equal to the specic surface area of thepacking, (5) it was negligible of the non-uniformity of the airand solution ows, (6) in the ow direction, no heat and moisture

transfer occurred, (7) the resistance to heat transfer in the liquidphase was negligible, and (8) the interface temperature was equalto the bulk liquid temperature. Based on the above assumptions,the main governing equations were stated.

According to the mass balance in the control volume,

dGs GadW 1According to the interface mass and sensible heat transfer rates,

the air humidity change was,

dWdz

0DMvAGa

In1Ps=Pt1Pa=Pt

2

According to the interface sensible heat transfer from the air tosolution side and the energy balance on the gas side, the airtemperature change was,

dTadz

0C;aATaTsGaCp;a

3

0C;aAGaCp;vdW=dz

1expGaCp;vdW=dz=C;aA4

where C,a and 0C;a are the heat transfer coefcient (sensible) ofthe gas side and that coefcient corrected for simultaneous massand heat transfer by applying the Ackermann correction, which isone method to take into account the effect of mass transfer on thetemperature prole with an Ackermann correction factor.

Finally, the boundary conditions were: z 0 TsTs,i, GsGs,i,XXi; zH, TaTa,i, GaGa,i, WWi.

Since the above differential equations cannot be solved analy-tically, the most basic solution is a numerical integration along theheight of the dehumidier. To begin the calculation, one end of thedehumidier must be chosen as the start point. For the counter-current ow pattern, it needs to presume the outlet conditions forone of the uids. Solving the above equations from the bottom tothe top of the dehumidier, with the boundary conditions, a set ofcalculated inlet solution parameters are obtained. By comparingthe calculation results with the real values, the supposed existing

Input the inlet parameters of the

air and solution

Suppose the outlet parameters of the solution and initialize the

whole flow field

With the governing equations, calculate the relevant parameters

from j=1 to j=n

Obtain the outlet parameters of the solution

Compare the calculation values of the outlet solution with the real values

The supposed values of the outlet parameters are correct

Calculate the outlet parameters of the air

Input the physical parameters of the air and solution

Meet the accuracy requirements

Do not meet the accuracy requirements

Iteration

Fig. 4. Calculation ow chart of countercurrent pattern.

Fig. 3. Heat and moisture exchange model in the countercurrent adiabaticdehumidier.

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599590

solution variables are adjusted. The calculation will last until thenal results are very close to the real values. And the generalcalculation owchart for the nite difference model of the counterow pattern can be summarized in Fig. 4.

In the later study, Gandhidasan et al. [30] utilized the similarmodel to study various parameters on the packing height ofthe packed tower. Then, Lazzarin et al. [31] gave more specicexplanation of the calculation method in Appendix A of the litera-ture. Oberg and Goswami [32] applied a nite difference modelsimilar to Factor and Grossman0s to verify the experimental results.Taking account the insufciently wetted packing and the differentfactor when transfer the k-type mass transfer coefcients to the F-type one, Fumo and Goswami [33] improved Oberg and Goswami0smathematical model by modifying the transfer surface. In addition, acorrection factor CF was introduced to modify the correlation of thewetting surface. By comparing the results of simulation and experi-ment, it was found that the calculation results of the adapted modelagreed well with experimental results.

All of the above models introduced a coefcient 0C;a to describethe simultaneous heat and mass transfer with the Ackermanncorrection. Khan and Ball [34] promoted another solution to dealwith the simultaneous transfer process. Both heat and masstransfer processes were assumed to be gas controlled, so theinterface temperature was the temperature of the bulk liquidand the heat transfer rate across the air lm from the bulk air tothe interface was equal to that entering the liquid side,

GaCp;adTa C;aATsTadz 5

Similarly, the mass transfer across the interface was equal tothe change in humidity ratio,

GadW D;aAWeWdz 6Then, the humid air specic enthalpy change could be written

as,

dha Cp;adTadW U Cp;vTaTr 7By substituting Eqs. (6) and (7) to (5), the air enthalpy change

along the ow direction was obtained. Here it is rewritten in asimpler way appeared in the later literature,

haz

NTU ULeH

heha1Le

1

U WeW

8

In the above equations, Le and NTU were dened as

Le CDCp;a

9

NTU DAVG0a

10

In this way, the coupled heat and mass transfer were consid-ered together. In the following research, this handling method ismore popular than the Ackermann correction.

The nite difference model has been widely used for thecountercurrent dehumidier [3538]. For cross ow conguration,Liu et al. [39] proposed a model for the heat and mass transferprocess in a cross ow adiabatic liquid desiccant dehumidier/regenerator. The physical and mathematical models are describedin Figs. 5 and 6, respectively.

The governing equations of energy, water content, and solutemass balances in a differential element were,

G0aH

Uhaz

1LUG0shs

x 0 11

G0aH

UWz

1LUG0sx

0 12

dG0s UX 0 13The energy and mass transfer in the interface of the air and

desiccant solution were expressed in Eq. (8) and the followingEq. (14),

Wz

NTUL

U WeW 14

Like some other papers, Le was supposed to be one in themodel. However, the value of NTU was correlated based on thecorresponding experimental data in the paper.

Niu [40] also established a two dimensional mathematicalmodel for the cross-ow adiabatic dehumidier, and the masstransfer coefcient was gained from the experimental data. Woodsand Kozubal [41] applied the nite difference model to study theperformance of a desiccant-enhanced evaporative air conditionerand the simulation results showed good agreement with theexperimental ones.

3.2. Effectiveness NTU (NTU) model

Stevens et al. [42] reported an effective model for liquid-desiccant heat and mass exchanger, which was developed from acomputationally simple effectiveness model for cooling towers[43]. Except for the assumptions of the nite difference model, twoadditional assumptions were included. One was the assumption ofthe linear relationship of saturation enthalpy and temperature, theother one was the neglect of the water loss term for the solutionenergy balance. In addition, an effective heat and mass transferprocess was assumed.

y

z

0

L

Hx

w

Air

Desiccant

Fig. 5. Schematic of the cross ow dehumidier/regenerator [39].

L z0

H

(M,1)

(M,2)

(M,3)

(M,j)

(1,1) (2,1) (3,1) (i,1)

(1,2) (2,2) (3,2) (i,2)

(1,3) (2,3) (3,3) (i,3)

(1,j) (2,j) (3,j) (i,j)

(1,N) (2,N)(3,N) (I,N) (M,N)

x

Fig. 6. A two dimensional schematic of the cross ow dehumidier/regenerator[39].

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599 591

The main equations and calculation process of NTU modelwere summarized as follows,

(1) Calculated the Number of Transfer Units (NTU) by Eq. (10).(2) In terms of the similarity with the heat exchanger, the

effectiveness of the countercurrent ow dehumidier couldbe expressed as

1eNTU1mn

1mneNTU1mn 15

where mn was a capacitance ratio, dened analogous to thecapacitance ratio used in sensible heat exchangers, and it wasgiven in the following equations,

mn G0aCsat

G0s;iCp;s16

where Csat was the saturation specic heat, andCsat dhe=dTs.

(3) With NTU and , the air outlet enthalpy could be obtained withthe following equation,

ha;o ha;iheha;i 17

(4) Used an energy balance to calculate the solution outletenthalpy.

(5) Then an effective saturation enthalpy was found by

he;ef f ha;iha;oha;i1eNTU 18

(6) Using the enthalpy and saturated conditions, the effectivehumidity ratio Ye;ef f could be obtained.

(7) Then, by the following equation, the air outlet humidity ratiocould be calculated,

Wo We;ef f WiWe;ef f eNTU 19

(8) With the mass balance and known inlet and outlet parameters,all of the outlet parameters were acquired.

In the later study, Sadasivam and Balakrishnan [44] pointed outthat the denition of NTU based on the gas mass velocity inStevens0s model was not appropriate when the minimum owcapacity was the liquid [45]. Thus, the gas mass velocity G0a inEq. (10) was changed to the minimal mass velocity of gas andliquid.

As for the NTU model, there are much fewer literatures thanthe nite difference model. In the following study, Ren [46]developed the analytical expressions for the NTU model withperturbation technique. The model accounted for the nonlinea-rities of air humidity ratio and enthalpy in equilibrium withsolutions, the water loss of evaporation and the variation of thesolution specic heat capacity.

3.3. The simplied models

It can be found that both the nite difference model and NTUmodel require numerical and iterative computations. Thus, both ofthem are not suitable for hourly performance evaluation.

Khan and Ball [34] developed a simplied algebraic model.With the nite difference model, about 1700 groups of data wereanalyzed. The following functions were deduced,

Wo n0n1Win2Ts;in3T2s;i 20

Wo m0m1Wim2Ta;om3T2a;o 21

The above two equations can be employed to predict the exitair temperature and humidity ratio easily. However, as they weretted based on the data with certain operating solution concen-tration and solution to air mass ratio, they might not be suitablefor other conditions.

Liu et al. [47] tted some empirical correlations to estimatethe performance of a cross-ow or counter-ow liquid desiccantdehumidier. The essence of the methodology is obtaining theempirical expression of enthalpy and moisture effectiveness byexperiment.

Gandhidasan [48] reported a very simple analytical solution topredicate the rate of moisture removal for the dehumidicationprocess. Referred to previous work [49], the author promotedthe dimensionless moisture and temperature difference ratios.By combining the aforementioned two denitions, the energybalance equation could be expressed as follows,

Cp;aTTa;iTs;iMvMa

U

PtPPa;iPs;i

GsGa

Cp;sTs;oTs;i 22

According to the literature [50], the relationship between therate of moisture removal m and the partial pressure of water vaporcould be deduced as

Ps;i Pa;imPtGaP

MaMv

23

In addition, the desiccant outlet temperature was easily calcu-lated, as shown in the equation,

Ts;o Ts;iHETc;i1HE

24

Finally, substituting Eqs. (23) and (24) to (22), the moistureremoval rate m was given as,

m 1

G0sCp;sHE1HE

Ts;iTc;iG0aCp;aTTa;iTs;i

25

The method was rather simple yet it involved lots of assump-tions and limitations. Except some common assumptions, it alsorequired that the desiccant inlet temperature was different fromthe air inlet temperature, and the desiccant temperature leavingthe dehumidier was equal to that of the desiccant entering theheat exchanger.

Chen et al. [51] constructed an analytical model on the basis ofthe nite difference model for countercurrent and concurrent owpattern. The physical model is similar to that of Fig. 3. Firstly, amathematical model was built following the model promoted byKhan and Ball [34].

Then, two parameters were introduced for derivation conve-nience,

Ka LeUCp;a UTaUW 26

Ke LeUCp;a UTsUWe 27By combining the mass, energy conservation equations, mass

and energy transfer equations at the interface, the change of Kealong the ow direction was,

dKedz

mnNTUH

ha;i1mn

KeKe;oLe1Cp;a UTaKe

28

By integrating (28) from 0 to H or z, Ke at the outlet and alongthe z-axis were acquired. Based on the above results, the distribu-tion of air enthalpy, air humidity, and temperature in a counter-current adiabatic dehumidier were obtained. Then, thedistribution of the solution parameters can be calculated. In thepaper, the analytic solution of the concurrent ow heat and masstransfer process were also given out.

Ren et al. [52] derived a new analytical solution from one-dimensional differential model. By introducing some

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599592

dimensionless and dimensional groups, the conventional equa-tions of one-dimensional model were transferred to two coupledordinary differential equations, whose general solution are asfollows,

WM C1e1NTUz C2e2NTUz 29

K1C1e1NTUz K2C2e2NTUz 30

The model is just suitable for the case where the solution owrate and concentration change slightly as it assumed that thevariation of the equilibrium humidity ratio of solution dependedonly on the change of the solution temperature.

Babakhani and Soleymani developed analytical models for thecounter ow adiabatic regenerator [53] and dehumidier [54]. Theanalytical solution was deduced on the basis of the differentialequations, including the mass balance, air humidity and tempera-ture change equations listed in (1)(3), and liquid desiccanttemperature and concentration change derived from the massand energy conservation equations. To achieve the simplication,two main assumptions were applied, including the assumptions ofthe dilute gas phase and constant equilibrium humidity ratio onthe interface. Then the above equations could be solved and theintegrated analytical solution were,

W WintWiWintexpMNTUz 31

Ta C1C2expNTUz

M2MexpMNTUz 32

Ts 1RcLeC2expNTUz

M

M2MexpMNTUzTa

33

InX RmWiWintexpMNTUzC3 34

Gs GaWiWintexpMNTUzC4 35

With the above solutions, the proles of the outlet solution and airparameters were available.

Based on the above model, Babakhani [55] also developedanother analytical model which was well suited to the highdesiccant ow rate conditions.

Liu et al. [56] developed an analytical solution for a similarcross-ow packed bed liquid desiccant air dehumidier, whosenumerical model had been reported in literature [39]. In thepresent work, it was regarded that the desiccant mass ow rateand concentration kept constant in the whole process. Thus theEqs. (12)(14) were got rid of, and only Eqs. (8) and (11) were leftfor calculation, which were rewritten as

mn Uhaz

HLUhex

0 36

haz

NTUL

U hahe 37

It was found the above control equations had high similaritywith those of the cross-ow heat exchanger. Thus the methods inliterature [57,58] were referred to. As the solution expressionswere a litter complicated, here they will not be presented.

Recently, Wang et al. [59] developed a simplied yet accuratemodel for real-time performance optimization. LevenbergMar-quardt method was used to identify the parameters. The proposedmodel is suggested to be employed in real-time performancemonitoring, control and optimization. Park and Jeong [60] alsodeveloped a simplied second-order equation model as a functionof operation parameters to study their impact on the dehumidi-cation effectiveness.

3.4. Summary

To sum up, there are three models to simulate the adiabaticdehumidiers. The general comparison of them is presented inTable 1. As a matter of convenience, the information of somerepresentative models is summarized in detail in Table 2.

Table 1Comparison of the mathematical models for adiabatic dehumidier.

Classication Assumption Iteration Accuracy Applied situation

Finite difference model Least Extensive Best Component design and operation optimizationNTU model More Less Better Component design and operation optimizationSimplied models Most No Worst Annual assessment

Table 2Detail information of the mathematical models for adiabatic dehumidier.

Classication Model Flow pattern Dimensionality Treatment of coupled, heat and mass transfer

Finite difference model Factor et al. [13] Counter One-dimensional Ackermann correctionOberg et al. [32] Counter One-dimensional Introduction of enthalpyFumo et al. [33] Counter One-dimensional Ackermann correctionKhan et al. [34] Counter One-dimensional Ackermann correctionLiu et al. [39] Cross Two-dimensional Introduction of enthalpy

NTU model Stevens et al. [42] Counter One-dimensional Introduction of enthalpy

Classication Model Flow pattern Simplied method

Simplied models Khan et al. [34] Counter Correlations based on simulation resultsLiu et al. [47] Counter or cross Correlations based on experimental resultsGandhidasan [48] Counter Introduction of dimensionless parametersChen et al. [51] Counter or concurrent Introduction of parameters similar to air enthalpyRen et al. [52] Counter Introduction of dimensionless parametersBabakhani et al. [54] Counter Transfer the differential equations to nonlinear equationsLiu et al. [56] Cross Similar method of cross-ow heat exchanger

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599 593

The nite difference model is used most frequently for itsaccuracy. However, it involves complicated iterative process, so itis only suitable for the component design and operation optimiza-tion. The common assumptions have been stated in Factor0s model.However, some other simplications or improvements are made tosatisfy the real conditions. It can also be observed that counterow conguration is the most commonly used ow pattern, andit can be described by the one-dimensional model. For thedehumidier with cross ow conguration, the problem is gen-erally solved by the two-dimensional model.

For the NTU model, two additional assumptions are includedas mentioned above. Thus, the model is less accurate than thenite difference model. However, compared with the nite differ-ence model, the calculations are dramatically less extensive. Thus,the NTU model has great potential for saving time. But with thedevelopment of the computer technology, the calculated amountof the nite difference model can be accepted due to its accuracy.Therefore, much fewer researches have been done on the NTUmodel in the later studies.

From Table 2, it is concluded that different dimensionlessparameters have been introduced in the deviation processes ofthe simplied models. Meanwhile, the additive assumptions areneeded for simplication. Therefore, they are only applicable forcertain operation conditions, and different models can be chosento be suited to the real situation. The biggest advantage of thesemodels is that the iteration is avoided. As a result of their highefciency, they are often used to predict the annual energyconsumption of an air-conditioning system.

4. Models for internally cooled dehumidier

In the internally cooled dehumidier, some cooling uid isintroduced to remove the heat produced by the vapor condensa-tion, as shown in Fig. 7 [61]. Most of the models for the internallycooled dehumidier are developed upon the nite differencemethods used for the adiabatic dehumidier. The difference isthe additional consider of the heat transfer brought by the coolingmedia. In addition, because of the relatively low solution ow ratein the internally cooled dehumidier, it is easier for the solution toform a thin lm on the surface of the padding wall or plate. Briey,there are also three types of models for the internally cooleddehumidier.

4.1. Models without considering liquid lm thickness

The rst type ignores the liquid lm thickness. Khan andMartinez [62] developed a mathematical model to predict theperformance of a liquid desiccant absorber integrating indirectevaporative cooling to achieve an almost isothermal operation.The processed air and the solution owed in countercurrentdirection while the solution and water owed in parallel direction.

Primary air stream

Secondary air stream

Primary air stream

Secondary air stream

Water spray Liquid desiccant spray

P.H.E

Fig. 8. Schematic diagram of the cross-ow type plate heat exchanger [61].

Primary air

Centreline

Desiccantsolution

Thinplate

Q dx

D

Secondary air

Centreline

Thinplate

Water

Qdy

D

Fig. 9. Schematic diagram of the control volumes considered: (a) primary air-solution; (b) secondary airwater [63].

Fig. 7. Heat and moisture exchange model in the internally cooled dehumidier[61].

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599594

Saman and Alizadeh [63] also established a similar model for across-ow type plate heat exchanger (PHE) serving as internallycooled dehumidier. The schematic diagrams of the PHE arepresented in Figs. 8 and 9. It is obviously that it has the sameprinciple with that in literature [62]. The only difference lies in theconguration difference. Here, Saman0s model will be explainedin detail as it is more representative. In the paper, the PHE wasdivided into a certain number of control volumes in two orthogo-nal directions. Several assumptions were set, including no heattransfer with the environment, negligible temperature gradientbetween the solution lm and water lm, and fully cover ofsolution and water on the plate.

Thus, the change of the enthalpy and humidity ratio of theprimary air were,

dha;pdx

NTUs ULesH

he;sha;p1Les

1

UWe;sWp

38

dWpdx

NTUsH

UWe;sWp 39

Also, the mass and energy conservation equations in thecontrol volume were given as

G0a;p UdWpdx

dG0s

dx 0 40

G0a;p Udha;pdx

dG0shsdx

Q 0 41

Similarly, the change of the enthalpy and humidity ratio of thesecondary air, and mass and energy conservation equations wereexpressed as follows

dha;rdy

NTUf ULefH

he;f ha;r1Lef

1

U We;f Wr

42

dWrdx

NTUfH

U We;f Wr 43

G0a;r UdWdy

dG0f

dy 0 44

G0a;r Udha;rdy

dG0f hf dy

Q 0 45

On the basis of the above governing equations, the discretizationequations were derived for each control volume. The calculationmethod was similar to the ow chart of the nite difference modelpresented in Fig. 4, while it was more complicated as a result ofanother iteration started by the advance assumption of the coolingwater temperature. In the paper, it is noted that the lm thickness wasmentioned, but the simulation did not take it into consideration.

Liu et al. [21] compared the performances of internally cooleddehumidiers with various ow patterns. A representative heatand mass transfer model was selected for detail description, asshown in Fig. 10. Referred to literature [34], the heat and humiditychanges of the air were almost the same with the Eqs. (6) and (8).

The energy conservation equation was expressed as

G0a Uhax

G0shsx

Cp;f Gf0 LHTfy

46

As the mass conservation equation has been explained before,here it will not be presented for limited space. And the heattransfer between the desiccant solution and the cooling water was,

Tfy

NTUfL

U TsTf 47

Combined the above equations with the inlet conditions, thedistribution of the parameters of the air, desiccant solution, andthe cooling water could be obtained. The model was applied for

analyzing the role of ow patterns in depth, which was seldomreported in previous literatures. In the later study, Zhang et al. [64]employed the above model to predict the performance of aninternally-cooled dehumidier.

Yin et al. [25] built a uniform mathematical model for aninternally cooled/heated dehumidier/regenerator which wasmade up of a plate heat exchanger. It is important to point outthat the author took the non-wetted area into considerationby introducing the wetness coefcient. Meanwhile, in a controlvolume, the transfer process in the channel width was consideredto be symmetrical. In addition, to improve the accuracy of themodel, the author also applied the correlation of the mass transfercoefcient tted out by experiment [24].

Based on the one-dimensional differential equations for theheat and mass transfer processes with parallel or counterowcongurations, Ren et al. [61] developed an analytical model forinternally cooled or heated liquid desiccant-air contact units. Toincrease the accuracy, the model took the effects of solution lmheat and mass transfer resistances, the variations of solution massow rate, non-unity values of Lewis factor and incomplete surfacewetting conditions into consideration.

Recently, Qi et al. [65,66] developed a simplied model topredict the performance of the internally cooled/heated liquiddesiccant dehumidication system. Compared to previous study,all of the outlet parameters could be obtained by the correlationsquickly and accurately. Thus, it is very useful for researching thedynamic operation performance of the internally cooled/heatedliquid desiccant dehumidication system.

Khan and Sulsona [28] developed a two-dimensional and steady-state model for a vapour compression/liquid desiccant hybrid coolingand dehumidication absorber made up of the tube-n exchanger. Tosimplify the model, several assumptions were made reasonably. Themost critical one was to consider the air and refrigerant ow incountercurrent due to the large size in the air ow direction. In thisway, the complicated problem was simplied to a two-dimensionalone. The governing equations are very similar to those of literature[21] except for the mass conservation equation. There were threeiterations: one for the real refrigerant exit enthalpy, one for the correctlocal coil surface temperature and the nal one for the actual solutiontemperature.

4.2. Models considering uniform liquid lm thickness

In the second model, the lm is considered as a uniformlydistributed desiccant lm.

Fig. 10. Schematic diagram of the control volume for a counter-ow internallycooled dehumidier [21].

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599 595

Park et al. [26] developed a three dimensional numerical modelfor simulating the coupled heat and mass transfer in a cross-owinternally cooled/heated dehumidier/regenerator. The schematicdiagram was presented in Fig. 11. Some assumptions were usedbefore listing out the governing equations: the ow was consid-ered as laminar and steady; the physical properties for bothsolution and air were constant; species thermo-diffusion anddiffusion-thermo effects were negligible and thermodynamicequilibrium existed at the solutionair interface. Because of therelatively small absorption, the TEG solution lm thickness wassimplied as constant and the lm mean velocity was alsounchanged. Also, the velocity gradient in the liquid solution lmat the interface was regarded as zero and the ow of the liquidsolution and air was supposed to be fully developed at the startplace, as presented in Fig. 11.

With the assumptions, the governing equations for the liquidsolution ow were,

0 s2usy21

sg 48

usTsx

Dt;s2Tsy21

49

usXwx

Dm;s2Xwy21

50

For the air ow, the governing equations were as follows,

0 dPdz

a2uay22

51

uaTaz

Dt;a2Tay22

52

uaXvz

Dm;a2Xvy22

53

At the interface, the mass and energy balances also existed,

sDm;sXwy1

aDm;aXvy2

54

ksTsy1

kaTay2

aDm;aXvy2

55

The above equations can be discretized along the three differ-ent directions. Being combined with the boundary and interfacialconditions, the equations could be solved.

Ali et al. [6769] had used the same mathematical model tostudy the effect of the ow conguration, the inclination angle, theReynolds numbers, various inlet parameters, cu-ultrane particlesvolume fraction, and thermal dispersion on the performance of thedehumidier/regenerator.

Mesquita et al. [70] compared three different numerical modelsfor parallel plate internally cooled liquid desiccant dehumidier.The second one introduced a constant lm thickness. It wasassumed that the wall was isothermal so that the water owcould be neglected. On the basis of some other assumptions, thedehumidication process could be described by a two dimensionalmodel, as shown in Fig. 12. Being different from the rst modelwhich does not consider the lm thickness, this model took themomentum equations into consideration. Firstly, the velocityproles of the air and liquid desiccant were obtained with themomentum equations, presented as follows,

3G0ss

sg

1=356

us 3G0s

2s

2y2

y2

3

57

ua usdPdx

1a

12y22 W

2y

58

dPdx

3aus

W=22 G

0a

2aW=23

" #59

The energy and species equations of the liquid phase, gasphase, and the energy and species balances equations at theinterface are almost the same with those presented in literature[26]. Then with the velocity values and boundary conditions, allthe above equations could be solved numerically and simulta-neously with the software package Microsoft EXCEL.

Recently, Dai and Zhang [71] employed the uniform liquid lmthickness to evaluate the performance of a cross ow liquiddesiccant air dehumidier packed with honeycomb papers. Theobjective of the paper is to analyze the Nusselt and Sherwoodnumbers in the channels with honeycomb papers as the packingmaterials.

4.3. Models considering variable liquid lm thickness

The nal model introduces a variable lm thickness. Except theconstant thickness model, Mesquita et al. [70] also established

Fig. 12. Schematic diagram of control volume for a two-dimensional model [70].

L

W/2

Y1

Y2

Z (Air flow direction)

X(TEG solution flow direction)

H

Fin

surf

ace

Sym

met

ric p

late

g

Fig. 11. Schematic diagram of control volume for a three-dimensional model [26].

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599596

another variable thickness model for internally cooled liquid-desiccant dehumidiers. In the model, the lm thickness inEq. (56) varied, thus for every step of calculation, the lmthickness was recalculated, so was the liquid velocity prole inEq. (57). However, to reduce the computational time, the change inthe airow velocity prole was neglected as a result of the smalllm thickness changes. Compared the results of variable thicknessmodel with the constant thickness model, it was found that thetwo models converged to the same results at higher desiccant owrates. However, the constant thickness model underestimated thedehumidication for low desiccant ow rates.

Hueffed et al. [72] presented a simplied model for a parallel-plate dehumidier, with both adiabatic and isothermal absorption.The model used a control volume approach and accounted forthe lm thickness variation by imposing its effect on the heatand mass transfer coefcients. The specic equations of the heattransfer coefcient in terms of the lm thickness were listed as

C Nukadh

60

dh 2W2 61Then the mass transfer coefcient was obtained from the

ChiltonCoulburn analogy as

D CCp;a

DtDm;a

2=362

In each control volume, various parameters, including the lmthickness , hydraulic diameter dh, heat transfer coefcient C , andmass transfer coefcient D were calculated on the basis of theinlet conditions. In this way, the impact of the lm thicknessvariation on the heat and mass transfer process were introducedinto the model.

Recently, Peng and Pan [73] investigated the transient perfor-mance of the liquid desiccant dehumidier with a one-dimensional non-equilibrium heat and mass transfer model.Unlike the previous study, the local volume average equations ofheat and mass transfer were developed in the work, which weresolved by TriDiagonal-Matrix Algorithm (TDMA).

In the later year, Diaz [74] also developed a transient twodimensional model for a parallel-ow liquid-desiccant dehumidi-er. The geometrical model was almost the same with that ofMesquita et al. [70]. The difference of the governing equations liesin that the present model took the time item into consideration.Some non-dimensional parameters were used to simplify thecalculation process. With the model, the variations of some critical

variables over time were illustrated and the effects of oscillatorybehavior were analyzed in depth.

4.4. Summary

The detail information of the mathematical models for theinternally cooled dehumidier is listed out in Table 3.

The models for internally cooled dehumidier without con-sidering the lm thickness are very similar to those used in theadiabatic dehumidier. However, the former ones are morecomplicated due to the involvement of the cooling uid. Thesemodels ignore the effect of the velocity eld. Thus, the results ofthese models have certain discrepancy with the reality, as velocity,mass and energy have strong coupling relationship.

In the models considering uniform lm thickness, the lmthickness and velocity are usually calculated at the beginningand then keep constant through the whole calculation. Thesimulation results justied that the constant thickness modelsunder-predicted the dehumidication, especially for low desiccantmass ow rate.

For the models considering variable lm thickness, all of thevelocity, mass and energy equations are solved together. In theprocess of each iteration, the lm thickness and velocity change, sothe inuence of the ow on the heat and mass transfer can bedemonstrated. Thus, the model is most accurate.

5. Conclusion

Various mathematical models have been proposed to assess theperformance of the liquid desiccant dehumidiers. For the adia-batic dehumidier, there are mainly three kinds of models: nitedifference model, NTU model, and simplied models. The nitedifference model is used most widely for its accuracy. However, itinvolves complicated iterative process which increases the com-puter time, so it is only suitable for the design of the componentand optimization of the operating conditions. Like the nitedifference model, the NTU model also requires iteration, but itis more effective and less accurate correspondingly. The simpliedmodels require more assumptions, so they are only suitable incertain operation conditions. But due to their high efciency, theyare often used to predict the annual performance of the system.

For the internally cooled dehumidier, there are also threekinds of models: models without considering liquid lm thickness,models considering uniform liquid lm thickness, and modelsconsidering variable liquid lm thickness. The rst model ignoresthe effect of the velocity eld totally. Thus, the calculation results

Table 3Detail information of the mathematical models for internally cooled dehumidier.

Classication Model Flow pattern (air/desiccant) Flow pattern (desiccant/cooling uid) Cooling uid

Regardless of lm thickness Khan et al. [62] Counter Counter Water and airSaman et al. [63] Counter Cross Water and airLiu et al. [21] Six different congurations WaterYin et al. [25] Cocurrent Cocurrent WaterKhan et al. [28] Cross Cross AmmoniaRen et al. [61] Four possible ow arrangements

Uniform lm thickness Park et al. [26] Cross Cross R22Ali et al. [6769] Cocurrent/counter/cross Mesquita et al. [70] Counter Counter WaterDai et al. [71] Cross

Variable lm thickness Mesquita et al. [70] Counter Counter WaterHueffed et al. [72] Cross Peng et al. [73] Counter Diaz [74] Cocurrent

Y. Luo et al. / Renewable and Sustainable Energy Reviews 31 (2014) 587599 597

have certain discrepancy with the reality. The second one considerthe effect of the velocity eld, but the assumption of the steadystate and fully developed eld cannot describe the real condition,especially for low desiccant mass ow rate. The nal model is mostaccurate as it comprehensively regards the inuence of thechanged velocity eld, but the calculation becomes relativelycomplicated.

Though a large amount of researches have been conducted onmodeling and analyzing the liquid desiccant dehumidier, furtherefforts are still needed:

(1) The vast majority of the previous models assume the heat andmass transfer process to be steady state and more transientmodels to study the dynamic performance are needed.

(2) More three-dimensional models close to the reality areneeded.

(3) Most of the models focus on the outlet parameters and moreresearch are required to study the heat and mass transferprocess in the dehumidier interior.

(4) It needs to take into consideration the effect of variablephysical properties which are taken as constant in almost allexisting models.

Acknowledgments

The work described in this paper was supported by the HongKong PhD Fellowship Scheme.

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A review of the mathematical models for predicting the heat and mass transfer process in the liquid desiccant dehumidifierIntroductionProblem descriptionThe mechanism of heat and mass transfer in liquid desiccant dehumidifierStructure of the dehumidifier

Models for adiabatic dehumidifierFinite difference modelEffectiveness NTU (NTU) modelThe simplified modelsSummary

Models for internally cooled dehumidifierModels without considering liquid film thicknessModels considering uniform liquid film thicknessModels considering variable liquid film thicknessSummary

ConclusionAcknowledgmentsReferences