Laminar flow and heat transfer in a quasi-counter flow parallel-plate membrane channel (QCPMC)

International Journal of Heat and Mass Transfer 86 (2015) 890–897

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Laminar flow and heat transfer in a quasi-counter flow parallel-plate membrane channel (QCPMC) Si-Min Huang ⇑, Minlin Yang, Baiman Chen, Runhua Jiang, Frank G.F. Qin, Xiaoxi Yang Key Laboratory of Distributed Energy Systems of Guangdong Province, Department of Energy and Chemical Engineering, Dongguan University of Technology, Dongguan 523808, China

a r t i c l e

i n f o

Article history: Received 16 June 2014 Received in revised form 29 January 2015 Accepted 9 March 2015 Available online 10 April 2015 Keywords: Laminar flow Heat transfer Quasi-counter flow Parallel-plate membrane channel Air humidity control

a b s t r a c t A parallel-plate membrane contactor with side inlets and side outlets has been designed and used for air humidity control. The contactor is comprised of a series of quasi-counter flow parallel-plate membrane channels (QCPMC). The laminar flow and heat transfer in the QCPMC are investigated based on a unit cell including one membrane and half of the channel. The equations governing the momentum and heat transports are established together with a uniform wall temperature boundary condition and numerically solved by a finite volume difference approach. The mean product of friction factor and Reynolds number (fRe) and Nusselt number (Num) are then obtained and numerically validated by those taken from other well-known publications. Influences of the Reynolds numbers (Re), entrance ratios (xin/x0), aspect ratios (x0/H), and various fluids with different Prandtl numbers (Pr) on the (fRe)m and Num are obtained and analyzed. It can be found that a distinct vortex is generated for the water stream flowing through the QCPMC when the Re is greater than 100. Further, for the same x0/H, the Num of the LiCl stream is the largest, and next the water stream, the least the air stream. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Recently, parallel-plate membrane contactors have been extensively employed for air humidity control (dehumidification/humidification) to avoid the distinct drawback of liquid droplet entrainments encountered in the traditional direct-contacting air humidity adjustment process [1–5]. Whether dehumidification or humidification for the novel membrane-based air humidity control technology, the processing air stream and the liquid stream (liquid desiccant or liquid water) are separated from each other by semipermeable membranes, which only allow the transports of heat and water vapor between the air and the liquid streams through the membranes [1–5]. Therefore the liquid droplets are prevented from escaping into the processing air. In the parallel-plate membrane contactors, a series of plate-type membranes are parallelly stacked together to form the channels. The air and the liquid streams, which flow alternately through the channels, are commonly in a cross-flow arrangement for reasons of convenient duct sealing [1–3]. To improve the performances of the membrane contactors, a quasi-counter flow parallel-plate membrane contactor, as shown in Fig. 1, has been

⇑ Corresponding author. Tel./fax: +86 0769 22862155. E-mail address: [email protected] (S.-M. Huang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.028 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

firstly proposed by Vali et al., which was used for sensible heat exchanger [6]. Then it has been extensively employed for energy recovery from humid air [7–10]. Further, the structure has been combined with other components in HVAC systems [11–13]. As seen from Fig. 1, the contactor is comprised of a number of side in and side out parallel-plate membrane channels (QCPMC). The channels for the air stream and the liquid stream (desiccant/water) are the same in structure and dimension. In other words, both the air and the liquid streams enter from the right headers of the channels and leave them from the left headers. Obviously, the flowing arrangement between the air and the liquid streams is similar to a combination of counter and cross flow, which can also be called quasi-counter flow. The performance analysis and evaluation of the quasi-counter flow parallel-plate membrane contactor are of vital importance in the practical application of air humidity control. The friction factor and heat mass transfer coefficients in the QCPMC are necessary for the performance analysis. However they are still not available from the open literatures. The transport data in the QCPMC [4–8] were simply approximated as those in rectangular channels, which have been thoroughly investigated in some classical books [14,15]. However they are not suitable for the QCPMC because of the different channel structure. Further, the transport data for various fluids with different Prandtl numbers (Pr) cannot be taken from open publications. Therefore the fluid flow and heat transfer in the

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Nomenclature specific heat (J kg1 K1) hydrodynamic diameter (m) friction factor product of friction factor (f) and Reynolds number (Re) channel spacing height (m) Nusselt number pressure (Pa) Prandtl number Reynolds number Sherwood number temperature (K) velocities in x-axial, y- axial, and z- axial directions (m/ s), respectively x0 channel length (m) xin entrance length (m) x, y, z coordinates (m) y0 channel width (m) Greek letters q density (kg/m3) cp Dh f fRe H Nu p Pr Re Sh T u, v, w

QCPMC should be studied. Further, the influences of the structural dimensions and fluid properties on the transport phenomena should be disclosed. The results will provide fundamentals for the structural design and energy analysis in the membrane contactors formed by the QCPMC employed for air humidity control. This is the objective of the present study. 2. Mathematical model 2.1. Governing equations In the aforementioned membrane contactor, as shown in Fig. 1, it is comprised of many identical and individual elements, which are the QCPMC. The air and the liquid streams flow alternately through the QCPMC. Owing to the symmetry and simplicity in calculation, a unit cell including one membrane and half of the channel are selected as the calculating domain. The coordinate system of the unit cell is depicted in Fig. 2. As seen, the upper and lower planes are the symmetric mid-plane and the plate-type membrane, respectively. The fluids (air, water, or desiccant) flow from the right hand corner (inlet header) with a uniform velocity uin and a uniform temperature Tin into the channel and out from the left hand corner (outlet header). Both the inlet and the outlet

l

dynamic viscosity (Pa s) heat conductivity (W m1 K1)

k

Superscripts ⁄ dimensionless Subscripts a air b bulk log logarithmic mean in inlet m average (mean) mem membrane out outlet v water vapor x, y, z x-axial, y-axial and z-axial directions, respectively

have the same length of xin, which is less than the channel length (x0), as shown in Fig. 1. In the practical air humidity control process, both the air and the liquid streams are laminar since the Reynolds numbers for the flows are much less than 2000. The fluids are Newtonian with constant thermo-physical properties (density, thermal conductivity, viscosity and specific heat capacity). Additionally, a uniform temperature boundary condition is imposed on the membrane surface. For the fluid flow, the equations governing momentum and thermal transports are [1,16,17]: Conservation of mass

Fig. 2. Coordinate system of the unit cell showing the QCPMC containing one platetype membrane and a symmetric mid-plane.

Table 1 Comparisons of mean (fRe) and Nusselt number (Num) for the air flowing through the rectangular ducts obtained from present model and those from references, xin = x0 = y0 = 0.1 m, Re = 500.

Fig. 1. Structure of the quasi-counter flow parallel-plate membrane contactor used for air humidity control.

x0/ H

(fRe)m Present

Refs. [14,15]

Error (%)

Present

Num Refs. [14,15]

Error (%)

1.0 2.0 4.0 10.0 20.0 40.0

327.51 269.91 210.43 150.18 123.10 109.08

327.14 269.09 210.83 150.56 123.66 109.44

0.11 0.30 0.19 0.25 0.45 0.33

15.64 13.48 11.37 9.26 8.36 7.84

15.77 13.40 11.31 9.35 8.32 7.75

0.82 0.60 0.53 0.96 0.48 1.16

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Table 2 Mean (fRe) and Nusselt number (Num) for the water stream flowing through the QCPMC under various entrance ratios (xin/x0) and Reynolds numbers (Re), x0 = y0 = 0.1 m, H = 0.005 m (x0/H = 20). xin/x0;

(fRe)m

Re ? 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Num

(fRe)m

50

Num

(fRe)m

100

45.01 58.02 71.02 76.37 81.72 83.61 85.49 87.23 88.96 93.35

5.78 6.16 6.48 6.78 7.08 7.28 7.47 7.75 8.03 8.09

51.46 68.05 79.28 84.75 90.22 91.46 92.69 93.16 93.63 96.92

Num

(fRe)m

200 6.14 6.64 6.95 7.35 7.74 7.96 8.18 8.62 9.02 8.97

64.94 83.57 96.25 101.95 107.65 107.05 106.45 104.17 101.88 103.99

Num

(fRe)m

300 6.20 6.88 7.35 7.99 8.62 8.93 9.23 9.95 10.66 10.38

78.47 98.96 113.67 119.24 124.81 122.87 118.92 114.23 109.54 110.87

Num

(fRe)m

400 6.25 7.01 7.57 8.51 9.44 9.78 10.11 11.07 12.02 11.49

91.53 114.69 131.11 136.22 141.33 136.87 130.41 123.71 117.01 117.59

Num 500

6.43 7.05 7.84 9.02 10.19 10.55 10.91 12.07 13.22 12.39

105.33 130.36 148.05 152.36 156.66 149.24 141.82 133.09 124.35 124.17

6.61 7.26 8.18 9.57 10.95 11.32 11.65 13.56 14.31 13.17

Table 3 Mean (fRe) and Nusselt number (Num) for various fluids flowing through the QCPMC under various aspect ratios (x0/H), x0 = y0 = 0.1 m, xin/x0 = 0.2, Re = 500. x0/H;

(fRe)m

Fluids?

Num

(fRe)m

Air stream (Pr = 0.71)

1.0 2.0 4.0 10.0 16.0 20.0 50.0 100.0

819.22 581.49 381.82 203.04 149.01 130.32 83.84 68.57

Num

(fRe)m

Water stream (Pr = 7.1) 9.04 7.67 6.01 4.44 4.34 4.44 5.48 6.12

819.06 581.37 381.78 202.97 149.05 130.36 83.92 68.52

Num

LiCl solution (Pr = 28.36)

24.25 19.65 14.04 8.91 7.36 7.26 6.99 6.97

819.05 581.36 381.88 203.02 148.99 130.26 83.94 68.56

40.76 36.09 26.17 14.68 13.77 13.56 10.02 9.44

Outlet

1

y*

0.8

0.6

0.4

Outlet

1

0.2

0.8 0

0.2

0.4

0.6 x*

0.8

1

Zone A Inlet

y*

0

(a) Re=100

0.4

Outlet

1

0.2

0.8

y*

0.6

0

0

0.2

0.4

0.6 x*

0.6

0.8

1

Zone A Inlet

(c) Re=300

0.4

0.2

0

0

0.2

0.4

0.6 x*

0.8

1

Zone A Inlet

(b) Re=200 Fig. 3. Velocity vectors of the water flowing through the QCPMC under various Reynolds numbers (Re), xin/x0 = 0.2, x0 = y0 = 0.1 m, H = 0.005 m.

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ð1Þ

where x, y and z are coordinates in spanwise, streamwise, and normal directions, respectively; u, v and w are velocities in x-, y- and zdirections (m/s), respectively; Superscript ‘‘⁄’’ represents dimensionless form. Conservation of momentum in the three coordinates

!



ð2Þ !

@v  @v  @v  @p @2v  @2v  @2v  þ v   þ w  ¼   þ þ þ  @x @y @z @y @x2 @y2 @z2

u

@w @w @w @p @ 2 w @ 2 w @ 2 w þ v   þ w  ¼   þ þ 2 þ 2  @x @y @z @z @x2 @y @z

where p is pressure (Pa). Conservation of heat 

u



2 



2 

2 

@T @T @T 1 @ T @ T @ T þ v   þ w  ¼  þ þ Pr @x @y @z @x2 @y2 @z2

ð3Þ ! ð4Þ

x ; x0

y ¼

y ; x0

z ¼

z x0

ð5Þ

ð6Þ

quDh qv Dh qwDh ; v ¼ ; w ¼ l l l

ð7Þ

where q is density (kg/m3); l is dynamic viscosity (Pa s); Dh is the hydraulic diameter of the channel, which can be calculated by

Dh ¼

4x0 H 2ðx0 þ HÞ

where subscript ‘‘b’’ and ‘‘log’’ stand for bulk value and logarithmic mean difference between channel wall and fluid, respectively. The dimensionless bulk temperature and the log-mean temperature difference can be written as

RR   u T dA T b ¼ R R  T dA

DT log ¼



   T w  T out  T w  T in ln

ð8Þ

In the unit cell, as shown in Fig. 2, the coordinate system of the unit including one membrane and half of the channel is depicted. The fluids flow from the side inlet and out from the side outlet. For the channel used for air dehumidification or humidification, the membrane surface is neither an ideal uniform temperature nor an ideal uniform heat flux boundary condition. However the temperature difference on the membrane surface is relatively small compared with fluid temperature variations between the inlet and the outlet [20]. Therefore a uniform wall temperature

1

ð9Þ

y*

306

0.2

0.4

Inlet

x*

318310

2 30

cp l k

31 320 316 2

0

0.6

x*

(a) Re=100

0.8

1

Inlet

(b) Re=200

Outlet

1

30 3086

304

0.8 y*

0.6

0

32 31 4 2

1360086 331 30

ð12Þ

0

1

302 31 0

30 4

30 360 8 316

320

0

328

0.2

0.4

0.6

x*

3013222 3

T  Tw T in  T w

0.8

0.2

where Tw is membrane surface (wall) temperature (K). The Pandtl number is defined by

Pr ¼

0.6

0.4

The dimensionless temperature is defined by

T ¼

320

0.4

 ð11Þ

0.2

4 30

322

A qDh uin l

0.4

314316 3 12 30 8

306 8 30

2



0.2

0.6

314 318322 26 3248 3 32

qu2in

1

0

3310 12

310 312 316 320

0

0

0.8

4 31 4 318 32 322 326

ðfReÞm ¼ @

out Dh pin p y

0.2

320

308 313612 320

0

0.4

8 30

326

where uin is inlet average velocity for the channel (m/s). The mean friction factor and heat transfer coefficient of the whole channel are necessary for performance analysis of the momentum and heat transports, respectively. The features of the fluid flow in the channel are commonly represented by a product of the friction factor and the Reynolds number, which is denoted as (fRe). The mean (fRe) of the channel can be calculated by the inlet and outlet pressure drops, which can be written as [18,19]

320

318 31106314306 3

ð10Þ

1

3 32628 32 3224

312 324 328

quin Dh l

0.6

322

0.8

The Reynolds number is given by

Re ¼

Outlet

Outlet 326

qpD2h l2

ð16Þ

T w T out T w T in

2.2. Boundary conditions

where H is channel height (m). The dimensionless pressure is defined by

p ¼

ð15Þ

where A is area (m2).

where x0 is channel length (m). The dimensionless velocities are defined by

u ¼

ð14Þ

and

!

The dimensionless coordinates are defined by

x ¼



xin H T b;out  T b;in  x0 y0 DT log

30 4

u

Num ¼ Re Pr 

302



318 316 314

@u @u @u @p @ 2 u @ 2 u @ 2 u u þ v   þ w  ¼   þ þ þ @x @y @z @x @x2 @y2 @z2

where cp is specific heat (J kg1 K1); k is thermal conductivity (W m1 K1). For the fluid flow in the channel, the mean heat transfer coefficient is basic data for energy analysis. Therefore the mean Nusselt number in the channel is used. Considering the energy balance between the inlet and the outlet of the channel, the mean Nusselt number can be calculated by [16,17]

y*

@u @ v  @w þ þ  ¼0 @x @y @z

0.8

1

Inlet

(c) Re=300

ð13Þ

Fig. 4. Temperature contours of the water flowing through the QCPMC under various Reynolds numbers (Re), xin/x0 = 0.2, x0 = y0 = 0.1 m, H = 0.005 m.

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S.-M. Huang et al. / International Journal of Heat and Mass Transfer 86 (2015) 890–897

Outlet

1

y*

0.8

0.6 Outlet

0.4 1

0.2 0.8

0

0

0.2

0.4

0.6

0.8

1

Zone A Inlet

y*

x* (a) xin/x0=0.2

0.6

0.4 Outlet

1

0.2

0.8

0

0

0.2

0.4

y*

Zone A

0.6

0.6 x*

0.8

1

Inlet

(c) xin/x0=0.9

0.4

0.2

0

0

0.2

0.4

0.6 x*

Zone A

0.8

1

Inlet

(b) xin/x0=0.5 24

Fig. 5. Velocity vectors of the water flowing through the QCPMC under various entrance ratios (xin/x0), x0 = y0 = 0.1 m, H = 0.005 m, Re = 200.

boundary condition is imposed on the membrane surface (Tw = const), which can also be expressed as

0 6 x 6 1;

0 6 y 6 y0 =x0 ;

z ¼ 0 : T  ¼ 0

ð17Þ

x ¼ 0 and x ¼ 1 : y ¼ 0;

Wall velocity boundary conditions (no slip)

On all the walls of the channel : ux ¼ 0;

@T  ¼0 @x

0 6 x 6 ð1  xin =x0 Þ and y ¼ y0 =x0 ;

ð22Þ xin =x0 6 x



uy ¼ 0;

and uz ¼ 0

61:

@T ¼0 @y

ð23Þ

ð18Þ Inlet velocity and temperature conditions, 

y ¼ 0;



ð1  xin =x0 Þ 6 x 6 1;

uy ¼ uin ; uz ¼ 0;

2.3. Solution procedure



0 6 z 6 H=2x0 : ux ¼ 0;

and T  ¼ 1

ð19Þ

and those for the outlet (outflow boundary condition)

y ¼ y0 =x0 ; ¼

0 6 x 6 xin =x0 ;

0 6 z 6 H=2x0 :

@ux @uy ¼ @y @y

@uz @T  ¼ ¼0 @y @y

ð20Þ

Symmetric boundary conditions on the mid-plane

0 6 x 6 1; 0 6 y 6 y0 =x0 ; z ¼ H=2x0 :

@ux @uy @uz @T  ¼ ¼ ¼ ¼0 @z @z @z @z ð21Þ

Adiabatic boundary conditions for the side walls (channel sealing)

The differential equations governing the laminar fluid flow and heat transfer in the QCPMC are numerically solved together with the boundary conditions by means of a finite volume method [21]. The unit cell, as shown in Fig. 2, is actually a hexahedron with the side inlet and the side outlet. Therefore hexahedral meshes are generated in the calculating domain. Velocity and pressure are coupled in the momentum transport equations. So a SIMPLEC algorithm is employed for the pressure correction [22]. The difference scheme is hybrid upstream scheme. It has an accuracy between first order and second order. The convergence criterions are that the normalized residuals are less than 107 for the fluid flow equations and 108 for the energy equation. To assure the accuracy of the numerically calculated results, numerical tests have been conducted to check the influences of the grid size. It indicates that 52  52 grids in x–y plane and 41

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S.-M. Huang et al. / International Journal of Heat and Mass Transfer 86 (2015) 890–897

Outlet

Outlet

1

0

0.2

0.4

0.6

0.8

x*

0

1

304 302

30 2

8 30 320

0.2

320 316312

316

0

0.2

312 3 14

0.4

Inlet

(a) xin/x0=0.2

310306

0.4

304 306 314 318 310 316312

0.6

2 30

0

0.8

306 308

0.2

314 318322 26 3 324 328

0.4

4 30

306

0.6

1

314316 3 12 30 8

318310

0.8

320 310 312 316 320

0.6

x*

0.8

1

Inlet

(b) xin/x0=0.5

Outlet 3164 30

1

302

0.8 0.6 0.4 0.2 0

4 30

The mean (fRe) and Nusselt number (Num) for the QCPMC are necessary for pressure drop and energy evaluations, respectively. The (fRe)m and Num for the water stream flowing through the QCPMC under various entrance ratios (xin/x0) and Reynolds numbers (Re) are listed in Table 2. As seen, the larger the Reynolds numbers (Re) are, the higher both the (fRe)m and Num are. When the Reynolds numbers (Re) are less than 100, both the (fRe)m and Num increase with the entrance ratios (xin/x0) varying from 0.1 to 0.9. Further, the (fRe)m and Num for the rectangular channels (xin/ x0 = 1.0) are about 1.05–2.07 times and 1.01–1.40 times of those in the QCPMC (xin/x0 < 1.0), respectively. When the Reynolds numbers (Re) are equal to or greater than 100, the Num increases with the entrance ratios (xin/x0) increasing. The Num for xin/x0 = 1.0 is about 0.55–7.96% less than those for xin/x0 = 0.9. However the (fRe)m first increases and then decreases when the entrance ratios (xin/x0) are ranging from 0.1 to 0.9. Further, the (fRe)m reaches its maximum when the entrance ratio (xin/x0) is equal to 0.5. Further, the (fRe)m for the rectangular channel is less than the maximum but larger than the minimum for the QCPMC. To further illustrate the effects of the entrance ratios (xin/x0) and the Reynolds numbers (Re) on the (fRe)m and Num in the QCPMC, velocity vectors and temperature contours of the water flowing through the QCPMC under various Reynolds numbers (Re) are shown in Figs. 3 and 4, respectively. Further, the velocity vectors and temperature contours of the water flowing through the QCPMC under various entrance ratios (xin/x0) are plotted in Figs. 5 and 6, respectively. As seen from Fig. 3, there is a rather inconspicuous vortex appeared in the zone A when the Reynolds number (Re) is equal to 100. In other words, the water fluid flows in the QCPMC and out from directly without obvious backflow. However a distinct vortex is generated in each zone A when the Reynolds numbers (Re) are equal to 200 and 300, which may be caused by the negative pressures in the zone A because of the large mainly flowing velocities. In other words, there is an obvious vortex appeared in the zone A when the Reynolds numbers (Re) are equal to or greater than 200. As seen from Fig. 5, these velocity vectors are plotted for the Reynolds numbers (Re) of 200. There are vortexes generated in the zone A. Further, the vortex is the most obvious when the entrance ratio (xin/x0) is equal to 0.5. Therefore the (fRe)m increases with the entrance ratios (xin/x0) increasing without the influences of the backflows when the Reynolds numbers (Re) are less than 100. When the Reynolds numbers (Re) are

y*

3.2. Friction factors and Nusselt numbers

304

To validate the established model and the solution procedure, the mean (fRe) and Nusselt number (Num) for the air flowing through the rectangular channels obtained from present model and those from references for xin = x0 = y0 = 0.1 m, Re = 500 are listed in Table 1. It is noteworthy that the numerical methods for the QCPMC (xin/x0 < 1.0) and the rectangular channels (xin/x0 = 1.0) are the same. As seen from Table 1, the values of (fRe)m and Num based on present method and those from other references are rather close. The maximum discrepancy is below 1.2%, indicating that the basic data from present numerical method are in good agreement with those from other well-known publications. Next results are the numerical investigations with the verified model.

302

3.1. Model numerical validation

y*

3. Results and discussion

larger than100 and the entrance ratio (xin/x0) is equal to 0.5, the (fRe)m has the largest value. As seen from Figs. 4 and 6, the temperature variations between the inlet and the outlet of the QCPMC increase with an increase in the Reynolds numbers (Re). Therefore the greater the entrance ratios (xin/x0) are, the larger the Num is. The (fRe)m and Num for various fluids flowing through the QCPMC under various aspect ratios (x0/H) are listed in Table 3. For the air humidity control process, the moist air is the processing air. The water stream is used for air humidification, while lithium chloride (LiCl) solution is commonly used for air dehumidification. Therefore the (fRe)m and Num for the three fluids of the air, water, and LiCl are obtained. As seen from Table 3, for the air, water, and LiCl streams flowing through the QCPMC, the larger the aspect ratios (x0/H) are, the larger the (fRe)m is. Further, the (fRe)m for the aspect ratio (x0/H) of 1.0 is as high as 11.98 times of that for the aspect ratio (x0/H) of 100.0. The (fRe)m for the air, water, and LiCl streams are nearly the same with only numerical deviations. It is because the values of the (fRe)m are independent of the Prandtl number (Pr). For the air flowing through the QCPMC, the Num first decreases and then increases. Further, it reaches its smallest value when the aspect ratio (x0/H) is equal to about 16.0. However for the water and the LiCl solutions flowing through the QCPMC, the Num decreases with an increase in the aspect ratios (x0/H). For the same aspect ratio (x0/H), the Num for the LiCl solution is about 35.12–69.54% larger than that for the water stream. Further, the Num for the water stream is approximately 13.89– 168.25% larger than that for the air stream. The velocity vectors and temperature contours of the various fluids flowing through the QCPMC are plotted in Figs. 7 and 8, respectively. As seen from Fig. 7, although the Num are different for the air, water, and LiCl fluids flowing through the QCPMC, the shapes and variations of the velocity vectors are nearly the same. As seen from Fig. 8, due to the different velocity distributions in the channel and different Prandtl numbers (Pr), the temperature

y*

in z axis are enough adequate, which is less than 0.8% difference compared with the grids with 102  102  82 for the grids in the unit cell. The final numerical uncertainty is 0.8%.

30 30 86

0

0.2

0.4

0.6

x*

0.8

1

Inlet

(c) xin/x0=0.9

Fig. 6. Temperature contours of the water flowing through the QCPMC under various entrance ratios (xin/x0), x0 = y0 = 0.1 m, H = 0.005 m, Re = 200.

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S.-M. Huang et al. / International Journal of Heat and Mass Transfer 86 (2015) 890–897 Outlet

1.2 1 0.8 y*

0.6 0.4

1.2

0.2

Outlet

1

0

0

0.2

0.4

0.6

0.8

1 y*

(a) Air fluid

1.2

0.8

Zone A Inlet

x*

0.6 0.4

Outlet

1

0.2

0.8

0

0

0.2

0.4

0.6 x*

y*

0.6

Zone A

0.8

1

Inlet

(c) LiCl solution fluid

0.4 0.2 0

0

0.2

0.4

0.6 x*

0.8

1

Inlet

Zone A

(b) Water fluid

Fig. 7. Velocity vectors of the various fluids flowing through the QCPMC, xin/x0 = 0.2, x0 = y0 = 0.1 m, H = 0.005 m, Re = 500.

Outlet

0.2

0.4

0.6

x*

y*

0.8

0.2 0

1

Inlet

318

314 320 326

0

0.2

(a) Air fluid

0.4

8 31 0 316

4. Conclusions

328

0.6

x*

0.8

1

Inlet

(b) Water fluid

Outlet

1

0.6

30 2

4 30

y*

0.8

0.4

31 2

320

0

316

0.2

310 314

0.4

0.6

x*

30320084 3

0

306 8 30 312

0.2

0.8

contours of the air, the water, and the LiCl fluids flowing through the QCPMC differ from one another. In order of the largest temperature difference between the inlet and the outlet, it goes: the LiCl fluid, the water fluid, and the air fluid. Therefore the Num for these three fluids flowing through the QCPMC have the same changing trend.

30

312

0

30 4

302 6 30

0

4 30

631 31 3018430 3 323320286

0.2

0.4

8 31 2 24 32 3

316 2 32

0.4

0.6

312

312

30 2

0.8 8 32

4 32

0.6

1

322 32031 3046 331162302 4 30

320 318

06 3310

0.8

4 33222

326 6 32 328

y*

Outlet

1

1

Inlet

(c) LiCl solution fluid

Fig. 8. Temperature contours of the various fluids flowing through the QCPMC, xin/x0 = 0.2, x0 = y0 = 0.1 m, H = 0.005 m, Re = 500.

The laminar flow and heat transfer in the QCPMC used for air humidity control are investigated. The three-dimensional mathematical model is built based on the unit cell, which containing a plate-type membrane and half of the channel. The equations governing the momentum and heat transports are established and solved together with the uniform wall temperature boundary condition. The mean (fRe) and Nusselt number (Num) are then obtained and numerically validated. Following results can be found: (1) For the water stream flowing through the QCPMC, the larger the Reynolds number (Re) is, the higher the (fRe) m and Num are. When the Re is less than 100, both the (fRe)m and Num increase with the entrance ratios (xin/x0) increasing. When the Re is greater than 100, distinct vortexes are generated. The (fRe)m first increases and then decreases. Further, the (fRe)m reaches its maximum when the entrance ratio (xin/x0) is equal to 0.5. (2) For the various streams, the larger the aspect ratios (x0/H) are, the larger the (fRe)m is. For the air stream, the Num first decreases and then increases. Further, it reaches its smallest

S.-M. Huang et al. / International Journal of Heat and Mass Transfer 86 (2015) 890–897

value with the x0/H of 16.0. However for the water and the LiCl streams, the Num decreases with the x0/H increasing. For the same x0/H, in order of the largest Num, it goes: the LiCl stream, the water stream, and the air stream. Conflict of interest None declared.

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Acknowledgement The project is supported by National Natural Science Foundation of China (NSFC), No. 51306038, and jointly supported by Dongguan Science and Technology Bureau, No. 2013108101008.

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