A mathematical model to predict the performance of desiccant coated evaporators and condensers

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A Mathematical model to Predict the performance OF Desiccant Coated Evaporators and condensers L.J. Hua , T.S. Ge , R.Z. Wang PII: DOI: Reference:

S0140-7007(19)30418-9 https://doi.org/10.1016/j.ijrefrig.2019.10.001 JIJR 4537

To appear in:

International Journal of Refrigeration

Received date: Revised date: Accepted date:

4 April 2019 15 August 2019 1 October 2019

Please cite this article as: L.J. Hua , T.S. Ge , R.Z. Wang , A Mathematical model to Predict the performance OF Desiccant Coated Evaporators and condensers, International Journal of Refrigeration (2019), doi: https://doi.org/10.1016/j.ijrefrig.2019.10.001

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A MATHEMATICAL MODEL TO PREDICT THE PERFORMANCE OF DESICCANT COATED EVAPORATORS AND CONDENSERS L.J. Hua, T.S. Ge, R.Z. Wang* Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, China ABSTRACT: Desiccant coated heat exchangers (DCHEs), utilizing an inner cooling source to remove sorption heat, are promising alternatives for evaporators and condensers (DCEs/DCCs) in vapor compression (VC) heat pumps. A mathematical model is necessary to facilitate the design, analysis and performance prediction of the component and the relevant systems. In this study, a three-dimensional model of DCEs/DCCs is proposed, accounting for the two-phase phenomena, the periodical switchover, the solid-side resistance, the fluid transport in multiple directions and the coupled heat and mass transfer. Study reveals that, high latent load (inlet humidity) reduces the sensible heat handling capacity of the DCEs, while the dehumidification capacity is almost independent of its sensible heat load. DCEs possess a satisfying effect of dehumidification above the dew point, thus it is unnecessary to employ low evaporation temperature. Meanwhile, the adsorption-desorption temperature difference of 30oC seems to be the optimal value for the commonly adopted evaporation temperatures (10 oC-20oC). For a specific coating thickness, there is a critical switchover period where the best performance of dehumidification is obtained. Switchover period shorter than the critical time should be avoided, and elongating the switchover cycle helps balance the ability of the system to handle the sensible and latent heat loads at the same time. KEYWORDS: evaporator; condenser; adsorption; heat and mass transfer; R410a; simulation;

1. Introduction In the field of air handling, water harvesting and energy storage [1], desiccant coated heat exchangers (DCHEs) are now arousing wide attention, for they inherit the merits of both solid desiccants and sensible heat exchangers. With the attempts of various scientists in recent years, the feasibility and superiority of DCHEs have been verified in many systems. To be specific, DCHEs could substitute the traditional devices used for latent heat handling (for example, liquid desiccant dehumidification devices and rotary wheels) in temperature and humidity independent control (THIC) air conditioners [2, 3] and fresh air-handling units [4]. Meanwhile, when incorporated in heat pumps [5-7] (named after solid desiccant heat pump, SDHP), DCHEs partially decouple the temperature and humidity control [6, 7] and simultaneously handle the sensible and latent heat of air above the dew point. In other areas regarding sorption and phase change, such as adsorption chillers and water harvesting, DCHEs also have potential application. Based on the former literature [1-7], liquids with high heat capacity are suitable as the cooling/heating media in DCHEs, while the two-phase refrigerants are better alternatives in the closed systems like SDHP. Noteworthy, the systems incorporating DCHEs normally feature periodical switchover to realize

continuous operation, and some portion of the cooling power is offset by the heat stored in DCHEs after suddenly switched from regeneration to dehumidification [1]. Two-phase media, due to their infinite thermal capacity during the evaporation/condensation, are reported to accelerate the process and lessen the offset energy [7]. The numerical studies of DCHEs are expected to facilitate the mechanism analysis, structure improvement, material screening, parameter optimization and performance prediction of the component and the relevant systems. Yet up to now, such studies are not extensive, especially for the systems adopting two-phase media. Main challenges include the multi-dimensional transport of the moisture, air and refrigerant, the transient two-phase phenomena during/after the switchover and the coupled heat and mass transfer. On the other hand, the mathematical models of desiccant wheels and conventional heat exchangers (HEs) respectively depict the similar (but somewhat simplified) mass and heat transfer process as in DCHEs, and thus are reviewed briefly in the following. The numerical models based on desiccant wheels [8, 9] can be classified into gas-side resistance (GSR) model, pseudo-gas-side (PGS) model, gas and solid-side (GSS) model and parabolic concentration profile (PCP) model [8]. In this paper, the GSS model with high precision is adopted. Yet, modifications are necessary, since DCHEs differ from rotary wheels in air duct structure. For example, radial solid-side resistance dominates in DCHEs rather than the transversal one. It is also vital to integrate the internal cooling or heating media into the model. Following discussion may indicate, the mass transfer in DCHEs shows distinct features due to the existence of heat transfer fluids. The theoretical and numerical research regarding conventional heat exchangers are abundant, including governing equation derivation [10-12], component study [12, 13], system simulation [11, 14] and correlation of the medium property [15-17]. To be specific, the above literature discusses extensively about the transient response of the HEs adopting two-phase refrigerant when encountering changing boundary conditions, since the dynamic simulation of two-phase behavior is a main focus and obstacle in this field. Without considering desiccant coating and moisture transportation, the models of HEs are not valid for DCHE systems, while the methods used in the refrigerant analysis are transplantable. Studies concentrating on DCHE simulations such as [18-21] exist in literature [22] but are dependent on experimental data or make several, simplifying assumptions at the expense of model accuracy. Yuan [18] and Jagirdar [19] extend the empirical correlations of fin efficiency in traditional HEs to the heat analysis of DCHEs. The one-dimensional GSR model is widely used in literature to identify mass transfer performance [18, 23, 24]. While a two-dimensional GSS one, coming from Tomohiro [20], is believed to be more accurate. Li proposes a more simplified method by assuming constant mass and heat resistance [21]. The above models, however, either depending on the experiments to deduce transport coefficient [21, 23, 24] or oversimplifying via neglecting influential factors (such as the temperature variation of the cooling/heating flows [19], the spreading heat resistance of the fins [20, 21, 23, 24] or the mass transfer resistance normal to the coating surface [18, 23, 24]), are not with high compatibility. Furthermore, the heat transfer media in all above literature are single-phase liquids (water or brine). As mentioned before, cooling and heating DCHEs alternatively via single-phase flow may lead to prolonged time of pre-cooling [1, 7] and obvious sensible heat losses [7]. However, the time lag would be significantly reduced when two-phase refrigerant is applied. In this paper, a detailed three-dimensional mathematical model is firstly proposed to predict the performance of the DCHEs driven by the two-phase refrigerant, which is with reasonable assumptions, independent of experiment and validated as accurate and robust.

Abbreviation DCHE

desiccant coated heat exchanger

PGR

pseudo-gas-side

GSR

gas-side resistance

PSP

parabolic concentration profile

GSS

gas and solid-side

SDHP

solid desiccant heat pump

HE

heat exchanger

VC

vapor compression



conductivity (W m-1 k-1)

Nomenclature (usually used with subscript) A

section area (m2)



dynamic viscosity (Pa s)

c

specific heat (J kg-1 K-1)



density (kg m-3)

cp

specific heat (isobaric) (J kg-1 K-1)



surface tension (N m-1)

d

copper tube diameter (m)

e

specific internal energy (J kg-1)

Subscripts

h

convective heat transfer coefficient (W m-2 k-1)

0

initial value/reference value

-1

i

specific enthalpy (J kg )

a

air flowing freely outside the desiccant surface

Nu

Nusselt number

d

porous desiccant coating (including pores)

p

pressure (Pa)

da

air confined in the desiccant pores

Pr

Prandtl number

dg

water vapor confined in the desiccant pores

Re

Reynolds number

dl

liquid water adsorbed by the desiccant

Sh

Sherwood number

f

Fin

Sc

Schmidt number

HX

DCHE (including the tubes, fins and desiccant)

t

Thickness (m)

i

Inside

T

temperature (oC)

in

Inlet

u

velocity (ms-1)

o

Outside

u

n component of velocity (m s-1)

mix

the free air mixture including water vapor

u

x component of velocity (m s-1)

r

Refrigerant

v

y component of velocity (m s-1)

tp

the two-phase refrigerant

w

z component of velocity (m s-1)

rg

saturated gas of the two-phase refrigerant

W

moisture content of the material (g g-1)

rl

saturated liquid of the two-phase refrigerant

x

quality

t

Tube

Y

air humidity ratio (g kg-1)

v

water vapor (inside air & desiccant pores)

w

water vapor flowing along the free air

Nomenclature (with specific definition) D0

Da

DG

DS

diffusion coefficient of water in air at the standard condition (m2 s-1) diffusion coefficient of water in air (m2 s-1) mass diffusivity of water vapor confined in desiccant pores (m2 s-1) surface diffusivity of the liquid water adsorbed by the desiccant (m2 s-1)

fr

friction due to the viscosity in refrigerant flow (N m-3)

fL

friction due to the viscosity in liquid refrigerant flow (N m-3)

g

gravitational acceleration (m s-2)

ilg 0

latent heat of vaporization of water at 0oC (J kg-1)

Ky

convective mass transfer rate (kg m-2 s-1)

pr 0

critical pressure of the refrigerant (Pa)

qh

scattered heat source (J m-3 s-1)

qm

scattered mass source (kg m-3 s-1)

qst

specific sorption heat of the desiccant

Wd*

Wd*   Wd  n, t  dn td

 da

Porosity of the desiccant

s

tortuosity of desiccant

td

(J kg-1)

moisture content (whose gradient along the desiccant thickness is omitted) (g g-1)

0

Geometric Nomenclature

Aa

Aa  Aa ,t  Aa , f Aa , f  2 x z    do  2td 

Aa , f

Aa ,t A fr

Ao Ar

2

4

area of the heat/mass transfer interface between the air and the desiccant coated on fins of one repeating unit (m2) Aa ,t    do  2td   Pf  t f  2td 

area of the heat/mass transfer interface between the air and the desiccant coated on tubes of one repeating unit (m2) Afr  Ly Lz frontal area (m2)





Ao  N f Pf 1   t f  2td  Pf  Ntz  1 C   z  do  ，C  min 2 



 z 2 

2



  x 2  do ,  z  do  

minimum free flow area (m2) Ar   di Pf area of the heat transfer interface between the refrigerant and the copper tubes of the DCHE (m 2)

Ca

2 Ca    do  2td   Pf  t f  2td   2 x z    do  2td  4 perimeter of air flow passage (m)  

dc

d c  d o  2  td  t f

Dh

Dh  4  ,   Aa Va ,   Ao Afr hydraulic diameter (m)



external diameter of the desiccant coated copper tube collar

la,i  Va Aa,i , i  t , f l a ,i coefficient to transform the surface heat/mass source into the scattered heat/mass source in the air-side (m)

laHX ,i  VHX Aa,i , i  t , f laHX ,i coefficient to transform the surface heat source (air) into the scattered heat source in the DCHE-side (m)

lad

lad  Vd Aa coefficient to transform the surface mass source into the scattered mass source in the desiccant-side (m)

lr

lr  Vr Ar coefficient to transform the surface heat source into the scattered heat source in the refrigerant-side (m) lrHX  VHX Ar coefficient to transform the surface heat source (refrigerant) into the scattered heat source in the

lrHX DCHE-side (m) Pf

fin pitch (m)

ro

copper tube outer radius (m)

r2

r2   x z  equivalent outer radius of the fin sector (m)

Va

Va   x z Pf volume of one repeating unit (assumed to be fulfilled by the air flow) (m 3)

Vd , f

Vd ,t

2 Vd , f  2  x z    do  2td  4 td volume of coated desiccant in one repeating unit (m3)   3 Vd ,t    do  2td   Pf  t f  td volume of coated desiccant in one repeating unit (m )

Val ,t

2 Val , f  t f  x z    do  2td   2 Val ,t   t f  do  2td  4  do 2 

VHX

VHX  Vd  V f  Vt volume of the DCHE (including the desiccant layer) in one repeating unit (m3)

Vr

3 Vr   di 2 Pf 4 volume of the internal tube one repeating unit (assumed to be fulfilled by the refrigerant) (m )

Vcu

3 Vcu    do 2  di 2  Pf 4 volume of the copper tube in one repeating unit (m )

 d ,i

 d ,i  Vd ,i VHX , i  t , f volume ratio of coated desiccant corresponding to the DCHE

 al ,i

 al ,i  Val ,i VHX , i  t , f volume ratio of coated desiccant corresponding to the DCHE

 cu

 cu  Vcu VHX volume ratio of coated desiccant corresponding to the DCHE

Val , f

2.

4 volume of the aluminum fin in one repeating unit (m3)  4 volume of the aluminum fin in one repeating unit (m3) 

Mathematical Model

(a)

(b)

(c)

(d)

Fig. 1 Detailed configuration of the example DCHE and its corresponding dimension (a)

the schematic diagram of the example DCHE (b) the repeating unit of the DCHE (c) the left view of the repeating unit (d) the front view of the repeating unit

Fig. 1 demonstrates the configuration of a sample desiccant coated heat exchanger and the fluids (air & refrigerant) flowing through it. The structure confined in the cuboid (length: L x; width: Ly; height: Lz), with the repetitive unit shown in Fig. 1 (b), is our object to conduct heat and mass transfer analysis. Based on it, some geometric parameters (for example, Aa) can be defined as listed in the geometric nomenclature above. To derive the governing equations in air side, a Cartesian coordinate is defined as shown in Fig. 1(a), with its origin coinciding with one of cuboid vertexes. While the properties of the refrigerant and DCHE are calculated along the refrigerant mainstream direction, denoted as y2-axis. This axis firstly points inward the tube at the inlet of the evaporator/condenser, and then reverses its orientation once encountering a bend tube. Besides, the moisture content in the desiccant varies throughout the desiccant coating thickness, hence another n-axis (Fig. 1(c) and 1(d)) is also imperative.

2.1 Air-side Equations (a)

(b)

(c)

Fig. 2 Schematic of heat and mass transfer (a) desiccant-side (b) air-side (c) heat exchanger

Fig. 2 tries to interpret the heat and mass transfer process of DCHEs, with Fig. 2(b) focusing on the air-side conditions. Taking the adsorption phase as an example, the unsaturated desiccant and the cold refrigerant of DCHEs altogether form a gas-solid interface of low humidity and temperature exposed to air, which then induces moisture and temperature gradient in the air-side. Thereupon, air discharges moisture and heat to the evaporator, and both of the transfer processes are accounted for by convection. The regeneration, with the same mechanism as that of the adsorption, takes place when air regains water vapor and thermal energy from the condenser. The detailed steps to obtain the air-side equations are shown in Appendix A. In short, Eq. (1), originated from mass conservation, is derived via dimension reduction and assumptions of constant density (dry air), negligible lateral (perpendicular to transversal direction) velocity and negligible relative velocity between dry air and moisture inside. RHS represents the equivalent mass source induced by convective mass transfer along the boundary, with all the complexity lumped into the mass transfer coefficient Ky (explained at length in Appendix A).   aYa    a uaYa  1 

t

x1



l

i t , f



K y Yda ,i

n  td

 Ya



(1)

a ,i

Energy conservation is then applied to the air-side (Appendix A) , with the assumptions of constant properties (density of air mixture, specific heat capacity of both dry air and water vapor), negligible pressure variation, gravity, viscosity (friction) and heat conduction. In order to get the simplified, one dimensional formulation (Eq. (2)), temperature and velocity gradient in the yz-plane are also negated. Instead, an internal heat source (  i t , f

  a  c pa +c pvYa  Ta  t

1 ha Ti  Ta  la ,i

) is introduced (explained at length in Appendix A).

  a ua  c pa +c pvYa  Ta  1      K y c pv Yda ,i x1 i t , f   la ,i



n  td



 Ya Ts ,i +

1 la ,i

 Ta , adsorption ha Ti  Ta , Ts.i   Ti , desorption 

(2) The initial and boundary conditions in this case are quite plain (Eq. (3)) .

Ya  x1 , y1 , z1 , t  0   Ya 0 ( x1 , y1 , z1 )  Ya  x =0, y1 , z1 , t   Ya ,in  y1 , z1 , t   Ta ( x1 , y1 , z1 , t  0)  Ta 0 ( x1 , y1 , z1 ) T  x =0, y , z , t   T  y , z , t  1 1 a , in 1 1  a

(3)

The convective heat and mass transfer coefficient can be calculated via Eq. (4) [19] and (5), which indicates that the Sherwood number and the Nusselt number are interconnected by Chilton-Colburn analogy [20]. c pa  aU max

c

P 3 Pf c1 c2  f   , j  0.086 Re N   dc tx  Pra 2 3  d c   Dh   Pf 0.41  0.042 N tx c1  0.361   0.158ln  N tx    , c2 ln Re dc   d c    Re 0.058 N tx c3  0.083  , c4  5.735  1.21ln  dc ln Re dc  N tx ha  j

c

4   Pf       z 

0.93

, N tx  2 0.076  x Dh 

1.42

 1.224 

ln Re dc

  

(4)

ua Afr  aU max d c , U max = a Ao a c pa Pra = a Re dc =

1.81

A p T  K y Ta   a  a Da Sha , Da  2.302 105 0  a  Va patm  T0  Sha Nua a hV  , Sca  , Nua  a a 1 1 3 3  D A a a a a Re Sc Re Pr a

a

a

(5)

a

2.2 Desiccant-side Equations Sub-graph in the left of Fig. 2 clarifies the mass transfer in the desiccant, taking the adsorption as an example. To be specific, desiccant normally has interconnected pores exposed to air and substrate atoms strongly affinitive to dissociative water molecules. Hence, the water molecules inside air can easily penetrate into the pores of unsaturated material due to molecular diffusion, then partially trapped into the potential field of adsorption [25] of the substrate, becoming adsorbed water. Also note that the water molecules, both the vapor ones and the adsorbed ones, are in most cases distributed unevenly inside the desiccant, which yield surface diffusion and bulk diffusion (including molecular diffusion and Knudsen diffusion) respectively. Namely, while molecular diffusion happens at the interface, localized adsorption, surface diffusion, molecular diffusion and Knudsen diffusion coexist inside the material. In generation phase, the direction of the concentration gradient and thus diffusion are inverted, adsorbed water is then desorbed from the substrate atoms. Based on the law of mass conservation and the assumption of even coating thickness, negligible lateral (perpendicular to n-axis) diffusion, negligible vapor-phase mass and constant physical properties (porosity and density of the desiccant), governing equation of mass transfer inside desiccant (Eq. (6)), along with the initial and boundary conditions (Eq. (7)), can be derived as followed (explained at length in Appendix A).

   d Wd  t



  d Wd      DS 0 n  n 

 Wd (n, t  0)  Wd 0 (n)   Wd (n  0, t )  0   n     d Wd  (n  td , t )  K y Yda  DS n 



(6)

(7) n  td

 Ya



Surface diffusivity Ds (Eq. (8) [8]), mass convective coefficient Ky (Eq. (5)) and correlations between Yda and Wd are imperative to enclose the above equations, provided that humidity ratio of air flowing freely outside the desiccant surface (Ya) is a known quantity.





DS Wd* , THX  =DS qst Wd* , THX  , THX : DS =

  da q D exp  0.974 106  st s 0 T  HX

 6  , D0  0.8 10 

(8)

It is assumed in our research, as in many others, that adsorbed particles are in local equilibrium with the vapor particles[25]. described by sorption isotherms. In this study, the sorption isotherm of a typical composite adsorbent (Si Sol + 30%LiCl + complete lyolysis + 5-year stable operation on component) is obtained by Micromeritics gas adsorption analyzer (ASAP2020) as shown below. Notably, our material is deemed insusceptible to temperature change [7] and with insignificant hysteresis.

Fig. 3 The sorption isotherm of the desiccant (obtained at 15oC)

2.3 DCHE-side Equations Fig. 2(c) demonstrates the heat transfer of the DCHE. The DCHE is considered as a homogenous mixture of copper, aluminum, desiccant and adsorbed water (water vapor inside the adsorbent pores is negligible). Thermal energy variation of the component is attributed to four contributors, namely, the sensible heat accompanying the moisture transfer, the heat generation/reduction due to the sorption heat effect, the air convection and the refrigerant convection. These four factors are also indicated in Eq. (10), which is deduced from energy conservation (Appendix A) with the assumption of constant heat capacity and negligible temperature variation throughout the material thickness. Eq. (11) then summarizes the initial and boundary conditions.

          j  j ,i c j   d  d ,i c pvWd*  Ti  Tt  i t , f  j  cu , al , d        cu cu  t y2  y2   1  1 Ta , adsorption 1 =  K y Ya  Yda ,i n td  qst Wd* , Ti  +c pvTs ,i   ha Ta  Ti    hr Tr  Tt  , Ts ,i   laHX ,i i t , f   laHX ,i Ti , desorption  lrHX





(9)

 T ( y , t  0)  T ( y , t  0)  T ( y ) f 2 HX 0 2  t 2  Tt ( y2  0, t )  0   y2  Tt ( y2  N tz N tx Ly , t )  0   y2

(10)

Where the fin efficiency can be calculated as Eq. (11) [19].

 m

qsorption  K1  mro  I1  mr2   I1  mro  K1  mr2   ro 2ha k f t f  Tt  Ta   qsorption  r2 2  ro 2   T f  Ta ha  K 0  mro  I1  mr2   I 0  mro  K1  mr2    Tt  Ta ha  r2 2  ro 2  Tt  Ta  2ha , qsorption  qst k y Ya  Yda , f kf tf

 (11)

Since the heat effect of sorption reaction is normally remarkable, the temperature distribution of the device is largely influenced by its mass transfer process. From another perspective, other than temperature gradient, concentration gradient can also act as driven force for heat transfer in DCHEs. This explains why the overall heat transfer performance of DCHEs is enhanced, although the desiccant coating increases the heat transfer resistance of the device.

2.4 Refrigerant-side Equations 2.4.1 Governing Equations According to the literature [10-14, 26], the following continuity, momentum and energy conservation Eq. (12)-(14) can be used to demonstrate the flow pattern of the single-phase refrigerant. In superheating and subcooling zones, the refrigerant (R410a) is treated as a homogenous mixture at thermal equilibrium. One-dimensional flow is assumed and the convective phenomena caused by the physical property variation in the transversal section can be incorporated into the coefficient hr. Meanwhile, the effects imposed by the viscosity, gravity, heat conduction and spatial variation of pressure [11] are insignificant.  r   r vr   0 t y2 2   r vr    r vr  p +   r  fr t y2 y2

  r ir  t



  r vr ir  y2



pr 1 + hr THX  Tr  t lr

(12)

(13)

(14)

The simplifying assumptions of the single-phase flow keep valid for the two-phase zone, except the homogeneous one. That is, relative ‘slip’ can be observed between the vapor and liquid velocities [11].

Thus, new definitions, such as the void fraction, average density [13, 14], average velocity or average enthalpy [26], and correlations, which relates the average variables to the corresponding vapor and liquid properties, are imperative to derive the two-phase governing equations. Necessary definitions are enumerated in the following (Eq. (15)).

 Arg  r Arg  Arl    rg Arg vrg  r  rg vrg x = r   rg Arg vrg   rl Arl vrl  r  rg vrg + 1   r   rl vrl    r  r  rg + 1   r   rl  v  r  rg vrg + 1   r   rl vrl =  r  rg vrg + 1   r   rl vrl  r  r  rg + 1   r   rl r    r  rg vrg irg + 1   r   rl vrl irl =xr irg + 1  xr  irl ir  r  rg vrg + 1   r   rl vrl 

(15)

Based on the definition Eq. (15), the mass and energy conservation laws for the evaporation or condensation zone can be derived, with the same form as those of the single-phase zone [26] (Eq. (12) and (14)). However, their physical meanings are different. The viscosity force in momentum conservation Eq. (13) is denoted by the term f r , which can be estimated by Fanning correlation [14]. When it comes to the two-phase flow, the momentum theorem yields Eq. (16). The Eq. (17), however, is a more popular version found in the literatures [14, 17]. This model, taking account of the pressure drop

f rTP

due to viscosity inside the tube, possesses the

advantages of high precision and wide flexibility.

  rg r vrg  rl 1   r  vrl    rg r vrg 2  rl 1   r  vrl 2  p +   r  f rTP t y2 y2

(16)

 1  xr 2   r vr  TP x 2     2  r     fr  pr   r vr   y2   1      t     rl r rg r    

(17)

Initial and boundary conditions are listed below.

  r ( x, y, t  0)   r 0 ( x, y ),  r ( x, y  0, t )   r ,in  vr ( x, y, t  0)  vr 0 ( x, y ), vr ( x, y  0, t )  vr ,in  ir ( x, y, t  0)  ir 0 ( x, y ), ir ( x, y  0, t )  ir ,in  p ( x, y, t  0)  p ( x, y ), p ( x, y  0, t )  p r0 r r ,in  r

(18)

2.4.2 Auxiliary Conditions In single-phase conditions, the specific formulas of the following four functions (listed in Eq. (19)-(21)) are required to enclose the Eq. (12)-(14) containing seven unknown parameters ( r , pr , Tr , ir , vr , hr , fr ). Among them, Eq. (19) is proposed as implicit expressions [27, 28] by Ding. Fanning equation (20) is adopted for the pressure drop. The convective heat transfer coefficient, in the meanwhile, is computed

by the Eq. (21) (Dittus-Boelter). Tr =Tr  pr , ir  , r =r  pr , Tr 

f r   r , vr , pr , Tr  =f r  r , vr , r  pr , Tr    fL 

(19) 2 fL 2  vr r  di  r

(20)

vd 0.079 , Rer  r r i r Rer 0.25

hr   r , vr , pr , Tr  =hr   r , vr , r  pr , Tr  , r  pr , Tr  , c pr  pr , Tr   =Nur r / di n Nur  0.023Re0.8 r Prr , Re r 

r c pr 0.4, THX  Tr r vr di , Prr  ,n   r r 0.3, THX  Tr

(21)

Where r  pr , Tr  , cpr  pr , Tr  , r  pr , Tr  is derived via linear interpolation with the source data obtained from REFPROP 7.1. To ensure the closure of the governing equations ((12), (14) and (17)) of the evaporation or condensation zone (containing nine unknown parameters ( r , pr , xr ,r , Tr , ir , vr , hr , fr )), expressions of the six functions listed in (22-27) are vital. Ding’s work [27] clarifies the relationship of the foregone enthalpy and pressure with the two-phase thermodynamic properties. Definition (15) yields (23) directly. The equations used to calculate the void fraction, evaporation heat transfer coefficient, condensation heat transfer coefficient and viscosity force are respectively put forward by Steiner [17] (Eq. (24)), Chien [15] (Eq. (25)), Jung [16, 29] (Eq. (26)) and Gronnerud [17] (Eq. (27)). xr  pr , ir  , Tr  pr , ir 

(22)

r  r , Tr   r  r , rg Tr  , rl Tr   = r rg + 1   r  rl

(23)

 r  xr , Tr ,  r , vr  = r  xr ,  rg Tr  ,  rl Tr  ,  r ,  r Tr  , vr    x x  1  xr  r  1  0.12 1  xr    r    rg   rl   rg 

 1.18 1  xr   g r   rl   rg     2  r vr  rl 0.5 

hrE  pr , xr , Tr ,  r , vr  =hrE  pr , xr ,  rg Tr  ,  rl Tr  ,  r , vr , rl Tr  , c prl Tr  , rl Tr  ,  r Tr  , Tr  hrE  Fr hrl  S r hpool  0.042   1  xr    rg  Fr  1.061exp   , Co       xr    rl   Co  rl c prl  1  xr  r vr di hrl  0.023Re rl 0.8 Prrl 0.4 rl , Re rl  , Prrl  di rl rl 0.8

S r  0.238

Co 0.238 1 ,Cf  di C f 1.11

hpool  55 pR 0.12   lg  pR  

0.5

r

g   rl   rg  0.55

M r 0.5h 0.67 , pR 

pr , h  hrTP THX  Tr  pr 0

(25)

0.25

 (24)    

hrC  pr , xr , Tr ,  r , vr  =hrC  xr , rg Tr  , rl Tr  , r , irg Tr  , irl Tr  , vr , rl Tr  , c prl Tr  , rl Tr  , Tr   2  h  22.4hrl 1    X tt 

0.81

C r

hrl  0.023Re rl 0.8 Prrl 0.4  1  xr    rg  X tt       xr    rl  0.9

Bo 

Bo0.33

rl c prl rl 1  xr  r vr di , Re rl  , Prrl  di rl rl 0.5

  rl   rg

  

(26)

0.1

h ，h  hrC THX  Tr   r vr  irg  irl  f rTP  xr , Tr ,  r , vr   f rTP  xr ,  rg Tr  ,  rl Tr  ,  r , vr , rg Tr  , rl Tr    dp  f rTP   gd    dz  L vd 2 fL 0.079 2  dp   vr r  , f L  0.25 , Re L  r r i    rl Re L  dz  L d i  rl       rl       dp   dp     rg   gd  1      1 ,    f Fr  xr  4  xr1.8  xr10 f Fr 0.5   0.25  dz  Fr       dz  Fr rl        rg  2   2 1   FrL 0.3  0.0055  ln  vr r  , FrL  1  f Fr   , FrL   FrL  gdi  rl 2  1, Fr  1  L

(27)

Where rg Tr  , rl Tr  , irg Tr  , irl Tr  can be find in Ding’s work [27] and specific formulas for rg Tr  , rl Tr  , c prl Tr  , rl Tr  ,  r Tr 

from REFPROP 7.1.

are obtained via linear interpolation with the source data obtained

3. 3.1

Simulation Method Discretization Method

(b)

(a)

(c) Fig. 4 Schematic of the CVs and nodes of spatial discretization

(a) Simplified configuration of DCHE (an array of CVs) and a top view showing the numbering strategy of nodes (b) Detailed schematic of a CVY with dimension and nodes (c) Detailed schematic of a CVN with nodes

This section aims to obtain the numerical solution of the above-mentioned equation set, namely to calculate the space distribution of the parameters of concern at set intervals (defined as time step Δt). Before that, spatial discretization of the domain is imperative. The configuration of the sample DCHE can be simplified as an array (Ntx* ny* Ntz) of infinitesimals shown in Fig. 4(a) and those infinitesimals are the appropriate control volumes (CVs) of differential equation discretization. The detailed illustration of the control volume (Fig. 4(b)) indicates the size (Δx1*Δy2*δz) of it. Notably, the grids need to be refined along n-axis to implement the calculation of desiccant conditions. That is, Eq. (6)-(8) should be discretized based on the CVs shown in Fig. 4(d) (denoted as CVN), much finer than the ones (Fig. 4(b), denoted as CVY) for other equations. Other than CVs, the nodes which represent CV properties are also defined and then numbered according to their calculation sequence. Since the differential equations are derived based on different coordinates, there are several sets of nodes in this system, marked by dots with different color in Fig. 4. Specifically, nodes applied in the calculation of Eq. (12)-(27) are numbered along the refrigerant mainstream direction (Fig. 4(a)), marked in blue and with index of i Y. While those of air-side and desiccant-side equations are marked in red and green (Fig. 4(c)) respectively. Based on the above-mentioned CVs and nodes, the mass diffusion Eq. (6) can be discretized by the Crank-Nicolson scheme. Meanwhile, an explicit formulation is adopted for the one-dimensional unsteady heat conduction Eq. (9). Hyperbolic equations [30] like (12)-(14)&(17) call for the second-order upwind scheme. (detailed in Appendix B) 3.2 Calculation Algorithm The numerical solution of Eq. (1)-(27) can be deduced based on the discrete computational domain in section 3.1 and the corresponding discrete equations listed in Appendix B, provided that necessary

geometric, material physical and operating parameters are available. Fig. 5(a) illustrates the general calculation algorithm. In short, parameters of concern are calculated one CVY after another along y2-axis at every time step. Notably, the parameters at the nodes in one CVY (Fig. 4(b)&(c)) expect iX-1 (regarded as inputs) have to be solved all at once, including the refrigerant properties at iY, DCHE temperature at iY, air conditions at iX and desiccant conditions at iN=1,2,…,IN. The detailed calculation algorithm is then demonstrated in Fig. 5(b), which explains the iteration strategy used to obtain values at nodes iX, iY and iN=1,2,…,IN via the coupled discrete Eq. (B.1)-(B.7). Fig. 5(c) further interprets the algorithm to solve discrete Eq. (B.5), (B.6) and (B.7), with three layers of iteration for two-phase refrigerant and two layers for superheated/subcooled one. All of the processes are coded in C++ and implemented in the interface of CodeBlocks.

(a)

(b)

(c)

Fig. 5 Calculation algorithm based on the discrete computational domain and discrete equations (a) Overall algorithm (b) Detailed algorithm for one CVY (c) Detailed algorithm for refrigerant module in one CVY

3.3 Optimization of time and space step In the above sections, the exact length of the space step and the time step, such as Δy2, Δn and Δt, is not designated, and random values may lead to divergent scheme. Based on the von Neumann method and criterion of positive coefficient, following restricted conditions (28)-(31) should be fulfilled to guarantee the stability of the numerical calculation. They can be simplified into inequalities listed in (32) by assigning empirical values for parameters like K y, ha and DS. Notably, since the coating thickness td is a relatively small value, the space step of CVN Δn and the corresponding time step Δt* confined by the td and Eq. (31) should also be very small. Therefore, desiccant conditions should be calculated based on not only the finer space grid Δn, but also finer time grid Δt* (indicated in Fig. 5(b)). 2  a ua  a   x1 t

k+1 K y  Ta i  1  X  2  la

(28)

k+1   k+1 2 a ua c pa +c pv Ya i 1   a c pa +c pv Ya iX  1  h X    2  a x1 t la

2 t t t  y2 2

   c    c W * k    k d d ,t pv d i  iY  j j ,t j Y  j  cu , al , d 



   c    c W * k  d d , f pv d i   j j, f j Y  j  al , d 



t * 1  2 *k N n DS Wd , THX kN



(29)



(30)

(31)

 1 t t *  a  25;  4.5 e 3;  2.5e7 x1 2t y2 2 n2

(32)

Furthermore, calculation precision and speed can be regulated via tuning step values. In Table 1, different combinations of the space step and the time step, satisfying Eq. (31), are listed and applied to a practical case. To be specific, the above algorithm is used to calculate a 4-minite operation process, in which a DCHE coated by 1.2kg silica gel (thickness 0.2mm) is switched between an evaporator (OA: 35℃, 21g/kg, 1.0m/s; inlet refrigerant: 450kg/m2s, 1.26Mpa, 0.05(quality)) and a condenser (RA: 25℃, 10g/kg, 1.0m/s; inlet refrigerant: 450kg/m2s, 2.73Mpa, 60℃) every one minute. By comparing the consumed time and the obtained value, this section attempts to find the optimized time and space step.

Table 1 Condition

Time(s)

x1 (m)

y2 (m)

n (m)

t (s)

t * (s)

1

359.105

2.25e-2

2e-2

4e-5

1

0.02

2

221.540

2.25e-2

4e-2

4e-5

1

0.02

3

142.060

2.25e-2

8e-2

4e-5

1

0.02

4

97.428

2.25e-2

16e-2

4e-5

1

0.02

5

240.690

2.25e-2

8e-2

4e-5

0.3

0.02

6

162.916

2.25e-2

8e-2

4e-5

0.5

0.02

7

150.718

2.25e-2

8e-2

4e-5

2

0.02

8

168.434

2.25e-2

8e-2

4e-5

4

0.02

9

594.504

2.25e-2

2e-2

1e-5

1

0.0025

10

190.696

2.25e-2

2e-2

2e-5

1

0.01

11

138.691

2.25e-2

2e-2

6.67e-5

1

0.02

(a)

(c)

(b)

(d)

Fig. 6 Comparison of curves obtained by different space or time step (a) SA temperature and humidity curves obtained via different value of Δy2 (b) via different value of Δt (c) via different value of Δn and Δt* (d) Local moisture content of the desiccant along the n-axis in one specific CVY (iY=1) obtained via different value of Δn and Δt*

Fig. 6 indicates that, the variation of the space step Δy2 and Δn (Δt*) within the reasonable limit imposes little effect on the SA temperature and humidity. The simulation results, however, are sensible to time step Δt. Based on the criterion of less than 1% error when compared to the results of further refined grid, Δt=1s can be regarded as an optimized value (Fig. 6(b)). In addition, Fig. 6(d) claims that the local moisture content of the desiccant is, to a small degree, overestimated/underestimated during the adsorption/desorption providing a large Δn (Δt*). After all, apart from Δt, values of the other space or time step seem insignificant, which also validates the robustness of our program indirectly. Condition 3 is applied in further calculation due to its acceptable speed and mesh-independent solution. 4.

Validation

4.1 Experimental setup

An experimental setup is established to test the performance of the SDHP system as shown in Fig. Error! Reference source not found.7. It consists of a closed refrigerant cycle and an open air one. The refrigerant cycle is built based on a traditional air-conditioning device bought at market, which contains an inverter compressor with the rated power of 736W and is designed for refrigerant R410a. Some necessary modification is made by replacing the original HEs with two DCHEs. Additionally, a needle valve is adopted to substitute the capillary tube. To modify the air route flexibly during test process, in accord with the frequent switchover of the DCHEs, several air ducts, 8 motor dampers and 2 centrifugal blowers are adopted in the re-construction of the open air cycle. Fig. 7 illustrates its control strategy in summer. Firstly, the motor dampers labeled by 1, 2, 3 and 4 are turned on, while the four-way valve and the other dampers were off. After a specific period of time, the four-way valve is electrified and thus switched, simultaneously, the 8 motor dampers change their state (the dampers labeled by 5, 6, 7 and 8 are powered and the other four motor dampers are shut off), keeping the outdoor air constantly flowing through the evaporator and the returned air through the condenser.

Fig. 7 Schematic diagram of the experimental table

During the experiments, the air temperature/humidity and the refrigerant pressure/temperature are logged in real-time along their passages. At the same time, the electricity consumption of the compressor is also monitored. The precision of the sensors for air-side temperature Ta, air-side relative humidity RHa, air flow velocity ua, refrigerant temperature Tr, refrigerant pressure pr and compressor power consumption Pcom is respectively ±0.2oC, ±1.7%RH, ±0.015m/s, ±0.15oC, ±0.5% and ±0.5%. The indirectly measurable parameters of concern include air humidity Ya, adsorbed/desorbed moisture amount per cycle D and cooling/heating capacity per cycle Q , as defined in the Eq. (33-35). 5800.2206  1.3914993  0.0048640239T  4.1764768 e 5T 2 1.4452093 e 8T 3  6.5459673ln T  T , Ta  unit : K   pvs  e  RH  pvs Ya  622 101325  RH  pvs 

(33)

N

D

ma t

Y k 1

k a , out

 Yak,in (34)

N

  qst  1.85Ta  Ya , Ta (unit :o C ) ia  1.01Ta  1000  N  iak,out  iak,in   k  1 Q ma t N 

(35)

The error analysis of the experimental results on the basis of the uncertainties in the primary measurements are performed using the Kline and McClintock relationship [31] as followed: 1  2 2 2 2    f   f   f   2 2 2    y   x   x  ...  x    n    1    2   x1    x2   xn    1 2 2 2  2 2 2 2    y   f   x1    f   x2   ...  f   xn        y  x1   y   x2   y   xn   y    

(36)

Based on the error of sensors, the maximum relative uncertainties of air humidity Y a, adsorbed/desorbed moisture amount per cycle D and cooling/heating capacity per cycle Q calculated from the experimental data are ±11.5%, ±2.0% and ±1.91% respectively. Furthermore, the mass and energy balance verification are performed in Fig. 8. Fig. 8a reveals that, energy balance can be roughly achieved within the error of 25%, although the relative uncertainty of the cooling/heating capacity Q is no larger than 2%. Meanwhile, despite of the little uncertainty of adsorbed/desorbed moisture amount D , only 80% of data are in mass balance within the error of 10% (shown in Fig. 8b). Such deviations from the balance are mainly attributed to the mass and heat loss caused by the switchover, the leakage and the system heat capacity.

(a)

(b)

Fig. 8 (a) Energy balance (b) Mass balance for different conditions of the SDHP

4.2 Validation with the experimental data Comparing the simulation results with the experimental data [32], the validation of the mathematical model could be verified. It is noteworthy that the evaporation temperature used in the calculation is

approximated by the measured refrigerant temperature at the DCE inlet. Also, the measured discharge temperature and pressure of the compressor roughly equal to the condensation inlet temperature and condensation pressure, respectively. Transient mass flow rate of the refrigerant is also available, yielded based on the compressor suction/exhaust condition and its electricity consumption data. Other properties and operation parameters of significance can be found in Table 2. During the calculation, the DCHE is imposed by alternative cooling and heating boundary conditions as suggested by the experimental data. Table 2 Thermodynamic properties and Operating parameters Desiccant (dry) specific heat capacity (J/(kg∙K))

General parameters Atmospheric pressure (pa) 2

Acceleration of gravity (m/s )

3

921

101325

Air density (kg/m )

1.2

9.8

Air conductivity (W/(m∙K))

0.0321

Air (dry) specific heat capacity (J/(kg∙K))

1035

Air (dry) dynamic viscosity (Pa∙s)

1.845e-4

Material properties Critical pressure of R410a (pa)

4902000

Water (vapor) specific heat capacity (J/(kg∙K))

1864

Relative molecular mass of R410a

72.58

Latent heat of water vaporation (J/(kg∙K))

2260000

0.05

Specific sorption heat of water (J/(kg∙K))

2700000

Inlet quality of the refrigerant

*

3

Aluminum fin density (kg/m )

2700

Aluminum fin specific heat capacity (J/(kg∙K))

880

Operating parameters

Copper tube density (kg/m )

8900

Air velocity of the OA(m/s)

0.868

Copper tube conductivity (W/(m∙K))

407

Air velocity of the RA(m/s)

0.95-1.3

390

Desiccant coating amount (kg)

1.8

3

Copper tube specific heat capacity (J/(kg∙K)) 3

Desiccant (including pores) density (kg/m )

(a)

(c)

1000

(b)

(d)

Fig. 9 Comparison between the simulation results and experimental data (a)air-side conditions under Shanghai Summer Conditions (b) air-side conditions under ARI Winter (c) refrigerant-side temperature (d) errors between the simulation and experiment

Fig. 9(a)-(b) illustrate that, for both summer and winter conditions, experimental SA humidity and temperature can be well predicted except the points right after the switchover (switchover period: 3min). The disparities of air temperature in preliminary phase are considerable since the two curves exhibit distinct tendencies. Such errors are probably attributed to the volume and heat capacity of the air ducts. To be specific, as shown in Fig. 7, air conditions are measured at some distance from the DCHEs, therefore the residue air and the heat capacity of the intermediate air ducts would result in a delayed and subdued response at the measure points to the temperature variation at the DCHE outlets. Further interpretation can be found in Appendix C. The refrigerant conditions are also compared between the simulation and experiments. Notably, the experimental data are measured at the outer surface of tubes, which lags behind the temperature variation of the inside refrigerant. This explains the initial divergence of the curves in Fig. 9(c). Fig. 9(d) indicates, apart from the data in the early stage of the switchover period, the errors between the simulation and experiment are within 20%. Taking the measurement error and the heat loss in the experiments into consideration, the simulation results are regarded as reasonable. 5.

Results and Discussion

In the following, a parametric study is conducted to analyze the transfer characteristics of the DCHE in the evaporation phase (namely, its performance under summer conditions). The flow rate of the refrigerant and the air at evaporator/condenser inlet in this section are respectively 450kg/m2s and 1/1.2m/s. In addition, several operating parameters are summarized in Table 3, while the unlisted inputs still comply with Table 2. According to the simulation, the DCHE displays periodical performance corresponding to the periodical boundary conditions. The outputs in the adsorption phase are then logged. The indexes of concern including the time-averaged outlet air humidity Ta ,out ,

Ya ,out

and temperature

as defined below.

 nsample  Ya ,out    Ya ,out j  / nsample  j 1 

(37)

 nsample  Ta ,out    Ta ,out j  / nsample  j 1 

(38)

Table 3 Operating parameters in the parametric study Parameters

Baseline values

Parametric variations

Inlet air temperature (oC)

35

27, 30, 33, 36, 39

Inlet air relative humidity (%)

60

15, 30, 45, 60, 75, 90

Evaporative Temperature (oC)

15

5, 10, 15, 20

Evaporation-Condensation Temperature Difference (oC)

30

20, 25, 30, 35, 40

Switchover Period (min)

3

1, 1.5, 2, 2.5, 3, 4, 6, 8, 10

Coating Amount (μm)

200

150, 200, 250, 300, 350, 400

Fig. 10(a) depicts the averaged air humidity and temperature at the evaporator outlet corresponding to different inlet air conditions. Despite the same inlet temperature, high latent load (inlet humidity) impairs the sensible heat handling capacity of the evaporator, and thus leads to high outlet air temperature. On the other hand, the coincidence of the outlet humidity curves in Fig. 10(a) reveals that the dehumidification capacity of DCHEs is independent of its sensible heat load (inlet temperature). However, at a really high temperature and humidity, the performance of DCHEs deviates from the above-mentioned rules since the presupposed refrigerant flow rate is then incompatible with the heat load. The effects imposed by the varying evaporation/condensation temperature are demonstrated in the Fig. 10(b). In general, outlet air temperature/humidity of the DCHE changes proportionally with the evaporation temperature. To be specific, under the same evaporation-condensation temperature difference, the outlet air dew point is raised by no more than 3.5 oC when the evaporation temperature increases from 5oC to 20oC. Noteworthy, the disparity of the outlet air dew point due to the evaporation temperature is even slighter (≤1.2oC per 10oC) once the adsorption temperature is low (≤15oC) and the adsorption-desorption temperature difference is small (≤30oC). Namely, from the perspective of humidity manipulation, it is unnecessary to employ low evaporation temperature in DCHEs. According to the experiments, a large evaporation-condensation temperature difference has both positive and negative influences on DCHEs. That is, it facilitates the dehumidification via relatively complete regeneration, but impairs the process due to the enlarged heat offset after the switchover. To compromise between the two, a moderate value is expected. Fig. 10(b) suggests, when the evaporation temperature is particularly low (5oC), the outlet air humidity continuously decreases with the increasing temperature difference (20oC-40oC). While the temperature difference of 30oC seems to be the optimal value for the higher evaporation temperatures (10 oC-20oC), and further increasing temperature difference in such cases would either make positive but subtle difference or even impose negative effect. Besides, the sensible heat handling capacity in the Fig. 10(b) is hindered by two factors, namely, high evaporation temperature and high heat offset accompanying the high evaporation-condensation temperature difference. Based on the simulation results, the former one is more determinant.

(a)

(b)

(c) Fig. 10 The variation of outlet air conditions corresponding to different (a) inlet air conditions (b) evaporation and condensation temperature (c) coating thickness and switchover period

Finally, the coupled effect of the switchover period and desiccant coating thickness is discussed. Empirically, greater amount of desiccant can accommodate more moisture and maintain the effect of dehumidification for a longer time span. Moreover, thicker desiccant coating occupies part of the air duct and thus intensifies the convective phenomena via speeding up the air. On the contrary, the desiccant possesses mass transfer resistance so that the dehumidification capacity is somewhat impeded by itself. Based on the Fig. 10(c), the positive influences of desiccant coating on the dehumidification performance (0.1-0.5g/kg per 50μm) overwhelm its negative effect. Also, the impact is weakened for already thick coating. The enhanced mass transfer impairs the sensible heat removal. As shown in Fig. 10(c), thicker coating leads to higher outlet temperature (~0.3 oC per 50μm) regardless of the reinforced heat convection. The temperature/humidity manipulation, however, can be realized not only via component structure but also operation parameters, such as the switchover period. On the basis of the former studies, within one switchover period, the supply air humidity of the DCHE first descends and then ascends. The time-averaged outlet humidity, therefore, displays similar tendency with the elongating switchover period. That is, for a specific coating thickness, there is a critical switchover period, defined as critical time, where the best performance of dehumidification is obtain. Fig. 10(c) reveals that, the critical time varies in the range of 1.5-3 minutes for different coating thickness, and the thicker one exhibits greater value. On the other hand, the cooling effect is monotonously augmented by increasing switchover period. To sum up, the switchover period shorter than the critical time should be avoided, since both the sensible and latent heat handling capacity of the system are then below the optimal values. Once exceeding the critical point, the heat transfer of DCHEs ascends monotonically with the gradually deteriorating mass transfer. That is, the prolonging switchover cycle is expected to help balance the system competence of the sensible (~6oC per 10min) and latent heat handling (~1.5g/kg per 8min). 6

Conclusion

In this paper, a mathematical model is built on the DCHE adopting R410a as refrigerant, followed by the validation and discussion over the model. Afterwards, we conclude that: 1.

The three-dimensional model on DCEs/DCCs overcomes the challenges aroused by the two-phase phenomena, the multi-dimensional of flow transport, the periodical switchover, the solid-side resistance and the coupled heat and mass transfer. It is independent on experimental data and

validated to be accurate and robust. 2.

With carefully designed space and time step, our calculation algorithm yields a quick and mesh-independent solution of the mathematical model.

3.

In DCHEs, high latent load (inlet humidity) impairs the sensible heat handling capacity of the evaporator, and thus leads to high outlet air temperature. One the other hand, the dehumidification capacity of DCHEs is insusceptible to its sensible heat load (inlet temperature).

4.

DCHEs possess satisfying dehumidification capacity even above the dew point, thus it is unnecessary to employ low evaporation temperature. Meanwhile, the adsorption-desorption temperature difference of 30oC seems to be the optimal value for the commonly adopted evaporation temperatures (10oC-20oC). Further increasing temperature difference would either make positive but subtle difference or impose negative effect.

5.

For a specific coating thickness, there is a critical switchover period where the best performance of dehumidification is obtained. The switchover period shorter than the critical time should be avoided, and elongating the switchover cycle helps balance the system competence of the sensible and latent heat handling.

Acknowledgements This research work was founded by Key Program of National Natural Science Foundation of China [No.51336004] and also the Foundation for Innovative Research Groups of the National Natural Science Foundation of China [No. 51521004]

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Appendix A Governing equation derivation General Equation To implement the simulation, equations from fluid mechanism (A.1-A.3) are applied to the cuboid region (Fig. 1) of the heat exchangers.      u  qm t

 

  

 u t

    e  Ek  t

  uu   p  23     u      2 S    f

 

     u  e  Ek   qh      u   u  f      T 

(A.1)

(A.2)

(A.3)

Above equations are valid even with mass transfer or mass source. Notably, moving particles possess mass, velocity and energy. Thus, the mass transfer through boundary or the internal mass source is inevitably accompanied by the momentum and enthalpy input or output, which should be transformed into equivalent force  f and heat source qh imposing on the control volume, respectively. In practical appliance, three-dimension model including all the minor details is redundant. However, once dimensionality is reduced, corresponding boundary conditions are omitted. For example, if ignoring radial temperature variation in a circular tube, heat convection between the tube and the flow disappears from the boundary conditions. This term should be supplemented by an equivalent heat source. Governing equations of the DCHE can be derived from Eq. (A.1-A.3) based on the above instructions and following assumptions.

Air-side equation Eq. (A.1) is firstly adopted in the air side. Then, the complication of the equation can be eased based on the following assumptions: 1.

The density of the dry air can be regarded as a constant both in time and in space.

2.

During the operation, practical fluids will be disturbed by the corrugated fin-tube heat exchangers. However, only velocity variation along the x1 -axis is considered ( va  wa  0 ) in the simulation.

3.

As part of the gas mixture, the moisture of air-side is flowing along with the dry air ( uw  ua

4.

Along the flow, vapor is adsorbed or discharged by the surrounding desiccant, which may induce

umix

).

velocity parallel with the y -axis or z -axis. However, in our study, the detailed analysis of the velocity variation is omitted ( vw  ww  0 ). Instead, the moisture transfer along the boundary (namely, along the copper tubes and the aluminum fins) can be regarded as an internal mass source



( qm  it , f K y Yda,i

ntd



 Ya la ,i , la ,i  Va Aa.i ). Then, all the complexity is lumped into the mass transfer

coefficient K y . Eq. (A.4) and (A.5) are simplified formulas of Eq. (A.1) when it is applied to the dry air and air-side moisture, respectively.

a   a ua    0  ua  const t x w   wuw  1   K y Yda ,i t x1 l i t , f a ,i



n  td



 Ya 

  aYa  t



  a uaYa  x1

(A.4)



l

i t , f

1



K y Yda ,i

n  td

 Ya



a ,i

(A.5) The temperature distribution can be derived via energy conservation. Assumptions are imperative to eliminate unnecessary complexity. 5.

Only energy variation along the

x1 -axis

is considered.

6.

Assuming the pressure field of the humid air is independent of time ( pw t  0， pa t  0 ).

7.

The gravity, viscosity (friction) and heat conduction of the fluid are negligible. Also, the equivalent force (namely, momentum) induced by the absorbed or desorbed moisture is negligible.

8.

The humid air transfers heat by convection with the DCHE. But, in the mathematical model, the temperature and the velocity gradient in the yz -plate, resulting from the convection, are omitted due to the above simplification. Instead, an internal heat source is introduced ( it , f 1 la,i  ha Ti  Ta  ), in which the heat transfer coefficient ha is a general parameter influenced both by the flow configuration and the temperature distribution.

9.

As part of the gas mixture, the temperature of the moisture in air-side is the same with the dry air ( Tw  Ta ). The air enthalpy is calculated by the approximation



j a , w

 j i j  ac paTa +aYa  c pvTa +ilg 0  ,

where specific heat capacity of the dry air and water vapor can be regarded as constant both in time and space. Meanwhile, the equivalent enthalpy input due to the moisture mass source is in the form of

 1 l  K i t , f

a ,i

y

Y

da ,i ntd



 Ya  c pvTs ,i +ilg 0  .

Based on the former assumptions, Eq. (A.6) is simplified formula of Eq. (A.3) when it is applied to the humid air with constant velocity and density (thus constant specific kinetic energy). Combining Eq. (A.6), (A.7) and (A.8), Eq. (A.9) is derived to illustrate the temperature change of the air mixture (including the water vapor).

    e  Ek   t

 

     u  e  Ek    qh      u

   p     e   Ek        p p          u  e   Ek     qh    u p      u t    t  

 



  i  t

 

(A.6)

 

    ui  qh

    j i j     j u j i j     a  c pa +c pvYa  Ta    a ua  c pa +c pvYa  Ta      i    aYa     a uaYa         lg 0  t x1 t x1 x1  j a,w   t   (A.7)

qh 

1



  l

i t , f



K y Yda ,i

n  td



 Ya  c pvTs ,i +ilg 0  +

a ,i

  a  c pa +c pvYa  Ta  t

 Ta , adsorption ha Ti  Ta , Ts ,i   la ,i  Ti , desorption 1

  a ua  c pa +c pvYa  Ta  1      K y c pv Yda ,i x1 i t , f   la ,i



n  td



 Ya Ts ,i +

(A.8)

 Ta , adsorption ha Ti  Ta , Ts.i   la ,i Ti , desorption  1

(A.9)

Desiccant-side Equation The water vapor and adsorbed liquid phase coexisting in the desiccant pores can be governed by Eq. (A.1). Then, the dimensionality of the equation can be reduced based on the following assumptions: 1.

Desiccant is impregnated as an even coating.

2.

Only mass transfer velocity in the thickness direction of the desiccant, which is defined as n -axis, is considered. To be specific, the axial origin moves at the interface between the desiccant and the DCHE metal, with its orientation normal to the surface. Fick law then can be applied to evaluate the

water

vapor

and

adsorbed

water

velocity,

namely,

dg udg   DG  dg n 

and

dl udl  DS  dl n  , attributed to the bulk and surface diffusion.

3.

The density of the gas mixture existing in the desiccant pore can be approximated by the air density ( da

4.

da =a . Then terms of relative small order of magnitude can be neglected

d，Yda

Wd

).

Porosity and density of the desiccant keeps constant despite the increase of the temperature.    da aYda  t



  d Wd  t



  da aYda       d Wd      DG    DS  n  n n   n 

  d Wd  t



    d Wd    DS 0 n  n 

(A.10)

(A.11)

DCHE-side Equation Eq. (A.3) can be degenerated to the heat conduction equation. The reduced formula is then used to describe the heat transfer of the DCHE consisting of different material (fins of aluminum, tubes of copper, desiccant of composite silica gel and moisture inside the desiccant). To get a general equation, reasonable assumptions are still vital: 1.

Moisture transfer velocity is neglected

udl  0

, along with the pressure and density variation of the

adsorbed liquid ( pdl  const, dl  const ). 2.

Temperature throughout material thickness is unchanged. Also, the desiccant coating possesses the same temperature with the local metal ( Td , f  Tf , Td ,t  Tt ). Only temperature variation and heat

conduction by copper tube along the y -axis is considered. Heat conductivity t is regarded as a constant regardless of the temperature change. 3.

Specific internal energy of the materials and the adsorbed water are expressed in Eq. (A.13). They are valid with the constant specific heat capacity ccu ,

4.

cal

, cd and

c pv

.

The heat resulting from the air convection, refrigerant convection and mass transfer can be regarded as three internal heat sources. Specifically, the adsorbed or desorbed moisture with enthalpy ( iw  c pvTs  ilg 0 ) brings energy into or out of the system.

Eq. (A.12) is a combination of Eq. (A.11) and the boundary condition listed in equation (7). Taking the DCHE as a whole, Eq. (A.13) can be derived by applying the reduced formula of Eq. (A.3). Eq. (A.16) then emerges via precise reasoning.

W

* d ,i

t





Wd ,i  n, t 

td

t td

0

dn





td

0

   Wd ,i   DS n  n  td

Wd ,i    dn DS n  

n  td

 DS td

Wd ,i n

n 0





1 K y Ya  Yda ,i td  d

n  td



(A.12)

         j  j ,i c jTi +d  d ,iWd*edl ,i      HX eHX    dl edl    Tt  Tt      i t , f  j cu , al , d      cu cu   qh    cu cu   qh t t y2  y2  t y2  y2  (A.13) Wd*  p  Wd*edl ,i  Wd*  idl ,i  dl   Wd*  c pvTi  ilg 0  const    qst W * , T dW * 0  dl  



 Wd*edl ,i  t



 Wd*  c pvTi  ilg 0   t

*    qst W * , Ti dW *  * * 0   c  Wd Ti   i Wd  q W * , T Wd     pv lg 0 st d i t t t t Wd*

(A.14) qh 

 1

  l

i t , f





K y Ya  Yda ,i

n  td

 c

T  ilg 0  

pv s ,i

aHX , i

1 laHX ,i

 1 Ta , adsorption ha Ta  Ti    hr Tr  THX  , Ts ,i    lrHX Ti , desorption

(A.15)           j  j ,i c j   d  d ,i c pvWd*  Ti      i t , f  j  cu , al , d  t y2  1 =  K y Ya  Yda ,i i t , f   laHX ,i



n  td

 q W ,T  +c st

* d

i

 Tt    cu cu   y2  

T   l

1

pv s ,i

aHX , i

 Ta , adsorption 1 ha Ta  Ti    hr Tr  Tt  , Ts ,i   l Ti , desorption  rHX

(A.16)

Appendix B: The equation discretization method The discrete equation for air-side mass conservation iX  0 : k+1   K y  Ta i  1    X 2  a ua k+1  2  a     Ya i X x1 la  t   

(B.1)

k+1     K y  Ta i  1     2 u X  k+1 k k  2  a a a =    Ya i 1  a Ya i  Ya i 1   X X X  x1 la t   t   



j t , f

k+1   2 K y  Ta i  1  X k+1  2  Yda, j iX  1 la , j 2

iN  I N 1

The discrete equation for air-side energy conservation iX  0 : k+1  k+1    k+1 K y  Ta i  1  c pv   a c pa +c pv Ya iX  12  2  a ua c pa +c pv Ya iX  h  X k+1  k+1   k+1  2  a     j Yda , j  1 iN I N  Ya i  1   Ta i    i  X X X t x1 la j t , f la , j 2   2    k+1 k+1    k+1  c +c Y  1  K y  Ta i  1  c pv    X k+1  k+1   k+1  a  pa pv a iX  2  2  a ua c pa +c pv Ya iX 1  ha  2  =      j Y  Y 1  da , j iX  1 iN I N  a iX    Ta iX 1  t  x l l 2 j  t , f  2  1 a a , j     k k+1    a c pa +c pv Ya i  1  K y  Ta i  1  c pv  X X k+1 k+1  k k k+1    ha  2   2    T  T  2   1   j  Yda , j iX  1 iN I N  Ya iX  1   T j iX  1  a iX  a iX 1  j t l l 2 t , f  2  2 a, j  a, j   

(B.2) 1, Y  da , j 1  Where,  j   k 0, Yda , j   1 k

 Ya 1

k

iN  I N

 Ya 1

k

iN  I N

, j  t, f

The discrete equation for desiccant-side mass conservation iN  0 : 3 Wd 0

k * 1



 4 Wd 1

k * 1

 Wd 2

k * 1

0

2n

iN  1, 2,..., I N  1:













* * 1 1 1 k * 1 k * 1 k * 1  1 *k * 1 k * 1  *k * 1 k * 1 Wd iN 1  2 DS Wd*k 1 , THX k 1 Wd iN 1  t *  n 2 DS Wd , THX  Wd iN  2n 2 DS Wd , THX 2n * * * * * 1 1 1 k* k* k*  1  =  *  2 DS Wd*k , THX kN  Wd i  DS Wd*k , THX k Wd i 1  DS Wd*k 1 , THX k Wd i 1 2 2 N N N  t  n 2  n 2  n  













iN  I N :



 DS Wd*k 1 , THX k 1 *

*



3 Wd  I

k * 1 N

 4 Wd  I

k * 1

2n

N 1

 Wd I

k * 1 N 2





i t , f

Aai Aa





* * k 1 K y Ta k 1 Yda ,i  I  Ya k 1    N *

d

(B.3)

The discrete equation for DCHE-side energy conservation iY  0 :

Tt 0

k 1

9 Tt 1

k 1



8

Tt 2

k 1



8

iY  1: k 1   K y 1 t  Y k 1  Y k 1      c    c W * k 1  1    c  a 1  da , j 1 iN I N  d d ,t pv d 1 t pv  j cu ,al ,d j j ,t j laHX ,t    k 1  K y 1 t  Y k 1  Y k 1  k 1  * k 1  a 1  da , j 1 1    j  j , f c j   d  d , f c pv Wd 1  1   f  c pv laHX , f   j al ,d 

iN  I N

 k 1    h t  1  1   a  laHX ,t laHX , f 

k 1    K y 1 t  Y k 1  Y k 1 k 1  1   k 1    j j , f c j   d  d , f c pv Wd* 1  1   f  c pv  a 1  da , j 1   1  laHX , f   j al ,d   k 1  k 1    K y 1 t  Y k 1  Y k 1   ha t  1  1     j c pv  a 1  da , j 1 iN I N  l   laHX , j   aHX ,t laHX , f  j t , f 

 hr 1

k 1

t

lrHX

Tr 1

k 1





i t , f

K 

k

y 1

t

laHX ,i

 Y  k   Y  k da ,i 1  a 1

iN  I N



k  q W * k ， d 1 Ti 1  st

      T k 1  a 1    

iN  I N

 k  k 4  t  k     j j ,t c j   d  d ,t c pv Wd*   cu cu2  Tt 1     j  j , f c j  d  d , f c pv Wd*   T f 1 1  y 2  j cu ,al ,d   j al ,d  

   k 1   Tt 1 k 1   hr 1 t     lrHX   



k

 cu cu t  8



1

y2 2

4 k k  3 Tt 0  3 Tt 2 



iY  2,3,..., IY  1: k 1   K y iY t  k 1 k 1 * k 1    c    c W  1   c   Ya iY  Yda , j iY iN I N    ,al ,d j j ,t j d d ,t pv d iY t pv  j cu laHX ,t   k 1    k 1  K y iY t  k 1   k 1  k 1 * k 1    h t  1  iY Y  Y     iY    j  j , f c j   d  d , f c pv Wd i  1   f  c pv a iY da , j i iN  I N  a   Y   Y laHX , f   j al ,d   laHX ,t laHX , f    k 1     K y iY t  k 1 k 1   1   k 1  * k 1   c    c W  1   c Ya i  Yda , j  iN I N       j j, f j iY d d , f pv  d i f  pv   i Y Y   Y l j  al , d aHX , f      T k 1  a iY k 1   k 1  K y iY t  k 1 k 1  h t  1  iY     c Y  Y  a l    j pv laHX , j  a iY  da , j iY iN I N   aHX ,t laHX , f  j t , f  





 k  k  2  t  k     j j ,t c j   d  d ,t c pv Wd*   cu cu2  Tt i     j  j , f c j   d  d , f c pv Wd*   T f iY iY Y y2  j cu ,al ,d   j al ,d 

 hr i

k 1



   k 1   Tt iY k 1   hr iY t     lrHX   

t

Y

lrHX

Tr i

k 1



Y



i t , f

K 

k

y i Y

t

laHX ,i

 Y  k   Y  k da ,i i Y  a iY

iN  I N



k  q W * k ， d i Ti iY  st Y



k



iY

 cu cu t  y2 2



Tt i

k Y 1

k  Tt i 1  Y 



iY  IY : k 1   K y IY t  k 1 k 1 * k 1   ,al ,d  j j ,t c j  d  d ,t c pv Wd IY  1   t  c pv l Ya  IY  Yda , j  IY iN I N   j cu  aHX ,t  k 1    K y IY t  k 1  k 1  k 1 * k 1  IY    j  j , f c j   d  d , f c pv Wd  I  1   f  c pv Ya  IY  Yda , j  IY Y l j  al , d aHX , f    

iN  I N

 k 1     h t  1   IY a   laHX ,t laHX , f   

k 1    K y IY t  k 1 k 1  1   k 1  * k 1 Ya  IY  Yda , j  IY    j j , f c j   d  d , f c pv Wd  IY  1   f  c pv IY    l aHX , f  j al ,d    k 1  k 1  K y IY t  k 1 k 1  h t  1   IY    j c pv Ya  IY  Yda , j  IY iN I N  a    laHX ,t laHX , f  j  l aHX , j   t , f 





iN  I N

 k  k  4  t  k     j j ,t c j   d  d ,t c pv Wd*   cu cu2  Tt  I     j  j , f c j   d  d , f c pv Wd*   T f IY IY Y  y 2  j cu ,al ,d   j al ,d 

 hr  I

k 1



Y

lrHX

t

Tr  I

k 1 Y





i t , f

K 

k

y I Y

laHX ,i

t

 Y  k   Y  k da ,i I  a IY Y

iN  I N



k  q W * k ， Ti IY d I  st Y





   k 1   Tt  IY k 1   hr  IY t     lrHX   

      T k 1 a IY     k IY



 cu cu t  4 y2 2

8 k k   3 Tt  IY 1  3 Tt  IY 1 

iY  IY +1: 9 THX  I

k 1

THX I

k 1 Y

 1

8

Y

THX I 1 k 1



(B.4)

Y

8

1, Y k da , j 1  Where,  j   k 0, Yda , j   1

 Ya 1

k

iN  I N

 Ya 1

k

iN  I N

, j  t, f

The discrete equation for refrigerant-side mass conservation iY  1:

 r 1

k 1



2t 2t k 1 k k 1  r vr 1   r 1   r vr 0 y2 y2



5t t  k 1 k k 1 k 1 9  r vr 1  4  r vr 0   r vr 2   r 2   3y2 3y2 

iY  2 :

  r 2

k 1

iY  3, 4,..., IY :

 r i

k 1



Y

3t t  k 1 k k 1 k 1 4  r vr i 1   r vr i 2   r vr iY   r iY  Y Y  2y2 2y2 

(B.5)

The discrete equation for refrigerant-side momentum conservation iY  1:

 r vr 1

k 1



2t 2t k 1 k k 1 k 1  pr  momr 1   r vr 1   pr  momr 0   fr 1 t y2 y2



5t t  k 1 k k 1 k 1 k +1 9  pr  momr 1  4  pr  momr 0    f r 2 t  pr  momr 2   r vr 2   3y2 3y2 

iY  2 :

 r vr 2

k 1

iY  3, 4,..., IY :

 r vr i

k 1



Y

3t t  k 1 k k 1 k 1 k +1 4  pr  momr i 1   pr  momr i 2    fr i t  pr  momr iY   r vr iY  Y Y Y  2y2 2y2 

(B.6)

 1  xr 2 x2  2  r  Where, momr =  r vr     L 1   r  G r 

The discrete equation for refrigerant-side energy conservation iY  1:

 r ir 1

k 1



2t 2t 1 k 1 k k 1 k 1 k k 1 k 1 k 1  r vr ir 1   r ir 1   r vr ir 0   pr 1   pr 1    hr 1 THX 1  Tr 1  t y2 y2 lr

iY  2 :

 r ir 2

k 1



5t t  1 k 1 k k 1 k 1 k 1 k k +1 k 1 k 1 9  r vr ir 1  4  r vr ir 0    pr 2   pr 2    hr 2 THX 2  Tr 2  t  r vr ir 2   r ir 2     lr   3y2 3y2 

iY  3, 4,..., IY :

 r ir i

k 1 Y



3t t  1 k 1 k k 1 k 1 k 1 k k +1 k 1 k 1 4  r vr ir i 1   r vr ir i 2    pr i   pr i    hr i THX i  Tr i  t  r vr ir iY   r ir iY  Y Y Y Y  Y Y Y     2y2 2y2  lr

(B.7)

Appendix C: The interpretation for air-side temperature prediction error

Fig. C.1 Measure points along the air-side in SDHPs Taking summer conditions as an example, the measure points of temperature at air-side are located at 5&6 as shown in the Fig. C.1. Assuming negligible volume and heat capacity of the air ducts, the temperature at point 5 would be the same with point 2 before the switchover and the same with point 3 after the switchover. That is, it would exhibit monotonously decreasing tendency periodically. However, practical systems would inevitably possess volume and heat capacity. Thus, the measured temperature (Ta5) lags behind the changes at the DCE outlet since air would mix with residue gas and exchange the heat with the wall in the air duct. The mixture and heat exchange process can be summarized by the following equations.

T T h  c pa  a  a +  c pa a  ua a  a TD  Ta     t  x la    c  t TD  h T  T  vD D  D a  a D  t 

(C.39)

If the simulated temperature at DCE outlet is adopted as the inlet boundary conditions of the Eq. (C.1), the outlet temperature at position 5 then can be derived via numerical calculation, with one segment of the periodical process shown in Fig. C.2. The existence of the heat capacity and air convection in the air duct alters the concavity and convexity of the temperature curve. When adopting different parameters (like heat capacity cvD and convective heat transfer coefficient ha), the simulated curves exhibit different shapes in the preliminary but are with the same tendency in the later period (after around 40s).

Fig. C.2 Outputs at position 5 corresponding to different heat capacity and transfer coefficient

Fig. C.3. Simulation results when considering air duct volume and heat capacity

By assuming reasonable inputs parameters, the tendency of the curves now accords with the experimental data as indicated in Fig. C.3.