Turbulent heat and mass transfer across a hollow fiber membrane bundle considering interactions between neighboring fibers

International Journal of Heat and Mass Transfer 64 (2013) 162–172

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Turbulent heat and mass transfer across a hollow fiber membrane bundle considering interactions between neighboring fibers Li-Zhi Zhang a,b,⇑, Si-Min Huang b, Wei-Bing Zhang b a

State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China b

a r t i c l e

i n f o

Article history: Received 4 January 2012 Received in revised form 7 November 2012 Accepted 15 April 2013 Available online 13 May 2013 Keywords: Turbulence Heat and mass transfer Hollow fiber membrane bundle Cross-flow In-line Staggered

a b s t r a c t A cross flow hollow fiber membrane bundle is used for liquid desiccant air dehumidification. The turbulent fluid flow and conjugate heat and mass transfer across the bundle considering interactions between neighboring fibers are investigated. In the bundle, the process air flows across the fiber bundle and salt solution flows inside the fibers packed in the shell. Heat and moisture are exchanged through the membranes. Two structured arrangements: in-line and staggered, are considered. Due to the periodicity of the fluid flow and heat and mass transfer across the bundle, two representative periodic unit cells which include 2–3 neighboring fibers simultaneously, are selected as the calculation domains. Turbulence in the shell side is considered. The governing equations for fluid flow and heat and mass transfer in tube side, membrane side, and shell side are coupled together and solved numerically with a self-built code. The fundamental data of friction factor, Nusselt and Sherwood numbers on both the tube and the shell sides are then obtained and experimentally validated. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Dehumidification of moist air is a key issue in HVAC (Heating, ventilating, and Air conditioning). Too much humidity can cause water condensation on walls, damage of materials like wood furniture, and growth of molds [1]. Indeed, molds can cause allergic reactions and make breathing difficult for some asthmatics [2]. Liquid desiccant air dehumidification has drawn much attention in these years [3–7]. However the traditional methods of packed columns have the problem of liquid droplets crossover, which has greatly limited the use of this technology. Recently hollow fiber membrane contactors have been used for liquid desiccant air dehumidification to replace the packed columns [8–12], to address this crossover problem. Such contactors can overcome the problem of solution droplets cross-over, since the solution and the air are separated apart by the membranes. The concept is like a cross-flow shell-and-tube heat exchanger where a bundle of fibers are packed in the shell. The solution flows in the fiber tubes, while the air flows across the fiber bundle. Heat and moisture are exchanged through the membranes.

⇑ Corresponding author at: State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China. Tel./fax: +86 20 87114264. E-mail address: [email protected] (L.-Z. Zhang). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.04.035

The fibers arrangements can be either in-line or staggered, as shown in Fig. 1. As seen, the liquid desiccant flows inside the fiber tubes, while the process air flows across the fiber bank. The membrane is permeable to water vapor, but impermeable to salt solution. The air is dehumidified by the desiccant in the fibers, but solution is prevented from leaking to the air. Since the packing density can be very high and the air side heat and mass transfer is further intensified by the continuous disturbances from the numerous fibers, the dehumidification effectiveness with this cross flow module is very encouraging [8–11]. Heat and mass transfer in such a membrane bundle have been investigated. However previous studies were limited to free surface models with purely laminar flow assumptions [13–15]. The interactions between the neighboring fibers and the turbulent flow nature in shell side were not considered. In this research, to account for these interactions between the neighboring fibers, the fluid in the fibers is modeled in combination with the neighboring fibers. The calculating domains are selected as the periodic area surrounded by the neighboring fibers, as shown in Fig. 1 the area surrounded by the dashed lines. On these representative cells, the fluid flow and the conjugate heat and mass transfer between the fluid and these surrounding fibers are investigated. To account for the turbulence in shell side generated by the impinging fibers, a low-Re k–e turbulence model is used to describe the air flow in shell side. The Nusselt and Sherwood numbers in the bundle are obtained and analyzed. A hollow

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 64 (2013) 162–172

163

Nomenclature a A Am b D Dh f habs kai L mv Nu P Pr r Re Sc Sh SL ST T U u Vai x, y, z Xs

contactor shell width (m) area (m2) fiber membrane area in the air stream side (m2) contactor shell height (m) diffusivity (m2/s) hydrodynamic diameter (m) friction factor absorption heat (kJ/kg) turbulent kinetic energy for the approaching air stream (m2/s2) contactor length (m) moisture flux (kg m2 s1) Nusselt number pressure (Pa) Prandtl number radius (m) Reynolds number Schmidt number Sherwood number longitudinal pitch transverse pitch time averaged temperature for the air stream, temperature for the solution stream (K) dimensionless velocity coefficient time averaged velocity for the air stream, velocity for the solution stream (m/s) approaching velocity for the air stream (m/s) coordinates in physical plane (m) mass fraction of water in solution (kg water/kg solution)

Greek letters q density (kg/m3) l dynamic viscosity (Pa s) d membrane thickness (m) u packing fraction h dimensionless temperature

fiber membrane-based liquid desiccant air dehumidification experiment is performed to validate the results. 2. Mathematical model 2.1. Governing equations In the hollow fiber membrane bundle, the two streams flow in a cross-flow arrangement. The air flow and the heat and mass transfer across the bundle show periodic features [16,17]. For reasons of symmetry and simplicity in calculations, two unit cells, as shown in Fig. 1(a) and (b) for the in-line and the staggered respectively, are selected as the calculation domains. The packing fraction of the whole bundle is equal to that of the unit cell, which can be calculated by

u¼

pr2o ST SL

ð1Þ

where ro is fiber outer radius (m); SL and ST are longitudinal and transverse pitches (m) as shown in Fig. 2, respectively. Due to the complex geometry of the unit cell, a boundary-fitted coordinate transformation technology is used in calculations. The physical planes shown in Fig. 2(a) and (c) are transformed to the computational planes as shown in Fig. 2(b) and (d), respectively. The fibers are oriented normal to the air flow. The solution stream

n

w H U K

x

eai e, g

dimensionless humidity variable dimensionless mass fraction of solution diffusion coefficient heat conductivity (W m1 K1) time averaged humidity for the air stream, humidity for the solution stream (kg moisture/kg dry air) turbulence dissipation rate for the approaching air stream (m2/s3) transversal coordinates in computational plane

Superscripts  dimensionless 0 fluctuation Subscripts a air ave average (mean) b bulk C fully developed under naturally formed boundaries e equilibrium H uniform heat flux (mass flux) condition h heat i inlet, inner L axially local Lat latent, moisture m mass, membrane o outlet, outer s solution T uniform temperature (concentration) condition, heat dissipation rate t turbulent v vapor w wall, water vapor x x axis direction y y axis direction

flows along the z axis in the circular fiber tubes (two round channels for the in-line, three round channels for the staggered), while the air stream flows over the fibers. Heat and moisture can be exchanged through the membrane between the air and the solution streams. When water vapor is absorbed by the solution, absorption heat is released on the interface between the solution and the membrane. In practical applications, Reynolds number for the solution stream in the inner circular channel is below 10 (much less than 2300), so laminar flow is assumed. However, for the shell side air stream, the flow conditions are different. Though the Reynolds number for the air stream is still below 2300 (around 200–600), it has been found that the flow tends to become turbulent due to the continuous disturbances from the numerous fine fibers [15,18]. The laminar model is not suitable for the air stream when the Reynolds number is larger than 300 [15,18]. Certainly a turbulence model is required. However, the local turbulent Reynolds numbers (Ret = qk2/le) are less than 150. In this case, the near wall flow cannot be accurately modeled by a standard k–e turbulence model [19,20]. To address this problem, a low-Re k–e model [19,20] is used for the air stream. Other assumptions are: (1). The air and the solution streams are Newtonian with constant thermal properties. (2). The air stream is two-dimensional [16,17], meaning the velocities are functions of x and y only.

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Air in

Membrane

Vai, Tai, ω ai

Air in

Vai, T ai, ωai

L

y

y

z

z x

Solution in

x

Solution in

(a)

(b)

Fig. 1. Schematic of a cross flow hollow fiber membrane bundle: (a) In-line; (b) Staggered. The area surrounded by the dash line is the representative calculation domain.

(3). The solution stream is hydrodynamically fully developed, but developing both thermally and in concentration. This assumption means that the cross-sectional velocity field does not change with channel length, while temperature and concentration fields vary with the channel length. (4). Axial heat and mass diffusions for the air and the solution streams are negligible. This is valid since Peclet numbers for both the two streams are greater than 10 [14]. For the air flow over the fibers, the turbulent kinetic energy and the dissipation rate are calculated via transport equations. They are solved simultaneously with the conservation equations for the fluid flow. The normalized governing momentum, heat and mass equations for the air stream are described as following [19,20]:

@ux @uy þ ¼0 @x @y @ðux ux Þ @x

ð2Þ

    @ðuy ux Þ @ @ux @ @ux 1 @pa þ  þ ¼ C C x x      @x @y 2 @x @x @y @y      @uy @ @u @ þ  Cx x x þ  Cx  @x @y @ @x

    @ðux uy Þ @ðuy uy Þ @uy @uy @ @ 1 @pa þ ¼  Cy  þ  Cy     @x @y 2 @y @x @y @x @y      @uy @ @u @ þ  Cy x þ  Cy  @x @y @y @y

ð3Þ

ð4Þ

      @ðux ka Þ @ðuy ka Þ @ @ka @ @ka l V ai  þ þ t þ ¼ C C G k k @x @x @y @x @y qa SL kai k @y 

SL eai  e  E V ai kai a

ð5Þ

    @ðux ea Þ @ðuy ea Þ @ @ ea @ @ ea c1 lt V ai þ þ þ ¼ C C e e @x @x @y @x @y qa SL kai @y 

ea Gk  ka

2



c2 SL eai ea   þF V ai kai ka

ð6Þ

where subscripts ‘‘x’’, ‘‘y’’ and ‘‘a’’ refer to x axis, y axis and the air stream, respectively; Superscript ‘‘{⁄}’’ represents dimensionless form; u is time averaged velocity (m/s); p is time averaged pressure (Pa); k, e, h and n are turbulent kinetic energy, turbulent dissipation rate, dimensionless time averaged temperature and humidity, respectively. For the solution flow in the inner circular channels, laminar flow is assumed. The normalized governing equations for the fluid flow, heat and mass conservation are given as following [21]:

@ 2 us @ 2 us S2 þ 2 ¼  2L 2  @x @y Dh;s

ð9Þ

@ 2 hs @ 2 hs @hs þ ¼ Us  @zh;s @x2 @y2

ð10Þ

@ 2 Hs @ 2 Hs @ Hs þ 2 ¼ U s  @zm;s @x2 @y

ð11Þ

where subscript ‘‘s’’ represents the solution stream; u, h and H is velocity, dimensionless temperature and mass fraction of solution, respectively. The dimensionless coordinates are defined by

x ¼

x SL

ð12Þ

y ¼

y SL

ð13Þ

zh;s ¼

z Res Pr s Dh;s

ð14Þ

zm;s ¼

z Res Scs Dh;s

ð15Þ

where Dh,s (=2ri) is the hydraulic diameter for the solution flow channel (m). Re, Pr and Sc are Reynolds, Prandtl and Schmidt numbers, respectively. The generation term and two extra terms appearing in Eqs. (5) and (6) are [19,20]:

" 2   2 #   2 @uy @ux @ux @uy þ þ þ     @x @y @y @x

    @ðux ha Þ @ðuy ha Þ @ @ha @ @ha þ þ ¼ C C T T @x @x @y @x @y @y

ð7Þ

Gk ¼ 2

    @ðux na Þ @ðuy na Þ @ @na @ @na þ þ ¼ C C x x @x @x @y @x @y @y

ð8Þ

E ¼



2la @ðka Þ1=2 qa SL V ai @y

ð16Þ

!2 ð17Þ

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y

η

SL

N

S T /2

E F 0 A

ro

ri x GH I

B CD

M

M

N

L K

E F

D C

J

0 A

B

G H I

(a) η QR V

ε

U TO

M

V

N

R Q

E F

D C

0 A

B

U W P

T O

M

W

SL

P

E F 0 A B CD

J

(b)

y N

L K

S T/2

L K

GH I

x

J

(c)

G H

I

L K

J

ε

(d)

Fig. 2. The coordinate system of the calculated domains. (a) The physical plane for the in-line arrangement; (b) The computational plane for the in-line arrangement; (c) The physical plane for the staggered arrangement; (d) The computational plane for the staggered arrangement.

2 2lt la V ai @ uy F ¼ q2a S3L eai @y2

!2



ð18Þ

where term E⁄ is added due to the anisotropy of the kinetic energy in the viscous layer [19,20]; Term F⁄ is presented for according with the experimental values [19,20]. In Eqs. (3)–(8), the associated diffusion coefficients are [19,20]:

Cx ¼ Cy ¼

lt 2r o 1 þ la rk SL Rea

Ck ¼

Ce ¼

lt 2ro 1 þ la SL Rea

lt 2r o 1 þ la re

CT ¼

SL 2r o SL

2r Cx ¼ o SL

Rea þ llrt T

1 Pr a

a

Rea 1 Sca

þ l lrt x a

Rea

ð19Þ

ð20Þ

ea

Other coefficients are: cl = 0.09, c1 = 1.44, rk = 1.0, re = 1.3, rT = 0.95, rx = 1.0 The dimensionless time averaged velocity for the air stream in the x axis direction is defined by

ux ¼

Ret ¼

q la ea

uy V ai

ð30Þ

ð22Þ

ka ¼

ð23Þ

ð24Þ

ð25Þ



ka kai

ð31Þ

where kai is the turbulent kinetic energy for the approaching air stream (m2/s2), which is obtained by [22]

kai ¼ 0:01ðV 2ai =2Þ

ð32Þ

The dimensionless turbulent dissipation rate for the air stream is defined by

ea ¼

ea eai

ð33Þ

where eai is the turbulence dissipation rate for the approaching air stream (m2/s3), which is equal to [22] 2

eai ¼ ð26Þ

500cl kai V ai Dh;a

ð34Þ

The dimensionless pressure for the air stream is defined by

The Reynolds number for the air stream is defined by [15]

q V ai Dh;a Rea ¼ a la

ð29Þ

and that for the air stream in the y axis direction is defined by

The turbulent Reynolds number 2 a ka

ux V ai

The dimensionless turbulent kinetic energy for the air stream is defined by

where

ct ¼ expð2:5=ð1 þ Ret =50ÞÞ

ð28Þ

ð21Þ

2

ka

c2 ¼ 1:92ð1  0:3 expðRe2t ÞÞ

uy ¼

where la represents air molecular viscosity (Pa s). The turbulent viscosity is equal to [19,20]

lt ¼ cl ct qa

where Dh,a (=2ro) is the hydraulic diameter for the air flow channel (m).In Eq. (6), c2 is determined by [19,20]

pa ¼ 1 ð27Þ

2

pa

qa V 2ai

The dimensionless temperature is defined by

ð35Þ

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h¼

T  T ai T si  T ai

ð36Þ

where Tsi is the inlet temperature of solution (K). The dimensionless humidity is defined by

n¼

x  xai xsi  xai

X  X ei X si  X ei

ð38Þ

where Xsi is the inlet mass fraction of solution (kg water/kg solution), and Xei is the equilibrium mass fraction of the solution with the air stream at the inlet temperature (Tai) and the inlet humidity (xai). The dimensionless velocity for the solution stream is defined by

us ¼ 

ls u s

ð39Þ

s D2h;s dp dz

In Eqs. (10) and (11), the dimensionless velocity coefficient Us is defined by

RR  udA ¼ RR dA

ð41Þ

The characteristic of the solution fluid flow in the circular channel is represented by the product of the friction factor and the Reynolds number as

! dp

Dh dz qu2ave =2

s

qDh uave l

 s



2 ¼ uave



ð43Þ

where subscripts ‘‘ML’’ and ‘‘NE’’ mean the outlet and the inlet plane as shown in Fig. 2, respectively; N is the number of rows in the flow direction; Va,max is equal to [13,14]

ST V ai ST  2r o

ð44Þ

Dimensionless bulk temperature

RR  u hdA hb ¼ R R  u dA

ð45Þ

Dimensionless bulk humidity

RR  u ndA nb ¼ R R  u dA

where Am means the membrane fibers area in the air stream side, which is proL and 2proL for the in-line and the staggered arrangements, respectively; Dha and Dna are the log mean temperature and humidity differences between membrane surface and the air stream, respectively. Dha and Dna are equal to

Dha ¼

ðhw;a  hb;a ÞML  ðhw;a  hb;a ÞNE ln½ðhw;a  hb;a ÞML =ðhw;a  hb;a ÞNE 

ð50Þ

Dna ¼

ðnw;a  nb;a ÞML  ðnw;a  nb;a ÞNE ln½ðnw;a  nb;a ÞML =ðnw;a  nb;a ÞNE 

ð51Þ

where subscripts ‘‘w’’ and ‘‘b’’ mean ‘‘wall mean’’ and ‘‘bulk’’, respectively.The peripherally local Nusselt and Sherwood numbers for the solution stream are [13,14]:

1 dhb;s 4ðhw;s  hb;s Þ dzh;s

ð52Þ

u HdA u dA

ShL;s ¼ 

1 dHb;s 4ðHw;s  Hb;s Þ dzm;s

ð53Þ

For the solution stream, the overall mean values over the entire fiber length are:

1 zh;s

Nuave;s ¼

1 zm;s

Z

zh;s

0

Z 0

zm;s

NuL;s dzh;s

ð54Þ

ShL;s dzm;s

ð55Þ

2.2. Boundary conditions As mentioned, a boundary-fitted coordinates transformation method is employed to transfer the physical domains to rectangular calculating domains [23]. They are shown in Fig. 2. As seen, for the in-line arrangement, planes ABCF, CDEF, HIJK, GHKL and NEDGLM correspond to the solution stream in the left fiber, membrane for the left fiber, the solution stream in the right fiber, membrane for the right fiber and the air stream in cross flow, respectively. For the staggered arrangement, planes ABCF, CDEF, HIJK, GHKL, RWTUV, QPOTWR and NEDGLMOPQ correspond to the solution stream in the left fiber, membrane for the left fiber, the solution stream in the right fiber, membrane for the right fiber, the solution stream in the middle fiber, membrane for the middle fiber and the air stream, respectively. The wall boundary conditions of velocity for the air stream

DE; GL; OPQ : ux ¼ 0 and uy ¼ 0 ð46Þ

ð47Þ

where A is cross sectional area normal to the flow.The transversely mean Nusselt and Sherwood values for the air stream are useful for

ð56Þ

The periodic boundary conditions of velocity for the air stream

ðux ÞNE

Dimensionless bulk mass fraction of solution

RR

ð49Þ

s

0:5Nqa V 2a;max

Hb ¼ R R

ðnb;a ÞML  ðnb;a ÞNE ST L Rea Sca 2Am Dn a

ð42Þ

ðpa ÞML  ðpa ÞNE

V a;max ¼

Shave;a ¼

Shave;s ¼

The friction factor for the air flow across the bundle can be calculated by [13,14]

fa ¼

ð48Þ

ð40Þ

where the average dimensionless velocity on a cross-section is equal to

ðfReÞs ¼

ST L ðhb;a ÞML  ðhb;a ÞNE Rea Pr a 2Am Dha

NuL;s ¼ 

u  S2 U s ¼ s 2L uave Dh;s

uave

Nuave;a ¼ ð37Þ

where xsi is the equilibrium air humidity with the solution at the inlet temperature (Tsi) and inlet mass fraction (Xsi). The dimensionless mass fraction of solution is defined by

H¼

energy and mass conservation analysis along the air flow. The mean Nusselt and Sherwood numbers for the air stream are [13,14]:

¼ ðux ÞML and ðuy ÞNE ¼ ðuy ÞML

ð57Þ

The inlet boundary conditions of the turbulent kinetic energy and dissipation rate for the air stream [22] 

NE : ka ¼

0:01ðu2f =2Þ and kai



ea ¼

500cl ðka Þ2 V ai Dh;a eai

ð58Þ

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 64 (2013) 162–172

where uf is resultant velocity. The inlet turbulence kinetic energy and dissipation rate for the air stream are updated by the new velocity at each iteration according to Eq. (57). The wall boundary conditions of velocity for the solution stream

CF; HK; RWT : us ¼ 0

ð59Þ

To satisfy the periodicity in boundary conditions for the air stream, two dimensionless parameters, i.e., the dimensionless temperature and humidity, are defined as [13,14]:

U ¼ ðh  hw Þ=ðhb  hw Þ

ð60Þ

X ¼ ðn  nw Þ=ðnb  nw Þ

ð61Þ

When the heat and mass transfer boundary layers are fully developed, the dimensionless temperature and humidity at the inlet (NE) are equal to those at the outlet (ML). Therefore the periodic boundary conditions of temperature and humidity for the air stream can be described by

ðUa ÞNE ¼ ðUa ÞML and ðXa ÞNE ¼ ðXa ÞML

ð62Þ

The inlet conditions for the solution stream

167

Air side membrane surfaces; qm ¼ mv

ð70Þ

Solution side membrane surfaces; qm ¼ mv

ð71Þ

where mv is the moisture flux through the membrane, which is determined by diffusion equation in membrane as

mv ¼ qa Dvm

xm;a  xm;s d

ð72Þ

where Dvm is moisture diffusivity in membrane (m2/s); d is membrane thickness (m). On the other hand, moisture flux on membrane surface on the air side is

qm ¼ qa Dva

 @ xa  @n surface;a

ð73Þ

Correspondingly, moisture flux on membrane surface on the solution side is

qm ¼ qs Dws

 @X s  @n surface;s

ð74Þ

where Dws is water diffusivity in solution (m2/s).

zh;s ¼ 0;

hs ¼ 1

ð63Þ

zm;s ¼ 0;

Hs ¼ 1

ð64Þ

2.3. Solution transport properties The symmetric boundary conditions

MN; DG; AB; BC; AF; HI; IJ; JK; RV; UV; and UT :

@w ¼0 @n

ð65Þ

where w stands for variables like pressure, velocity, turbulent kinetic energy, dissipation rate, temperature, humidity or mass fraction. 2.2.1. Heat boundary conditions on membrane surfaces Since the membrane is rather thin (150 lm), trans-membrane temperature differences are so small that they can be neglected [21,24]. When water vapor is absorbed by the liquid desiccant, absorption heat is released on the membrane surface on the solution side. Heat flux on the solution side is comprised of heat flux on the air side and latent or absorption heat flux. Therefore the normalized heat balance equation on the membrane surfaces between the air and the solution streams is

k 

   @ha  @n  @hs   þ habs  a  ¼  @n surface;a @n surface;a @n surface;s

ð66Þ

where the dimensionless absorption heat and the dimensionless heat conductivity are defined as: 

habs ¼ k ¼



qa Dva habs xsi  xai ks



T si  T ai

ka ks

ð67Þ

ð68Þ

where habs is absorption heat (kJ/kg); Dva is moisture diffusivity in air (m2/s). Heat flux on membrane surfaces on the air and the solution sides is

qh ¼ k

 @T  @nsurface

ð69Þ

where subscript ‘‘surface’’ represents the membrane surfaces in the air and the solution sides. 2.2.2. Mass boundary conditions on membrane surfaces Mass boundary conditions on membrane surfaces

Equilibrium air humidity of the liquid desiccant solution is a function of temperature and mass fraction. In order to calculate the equilibrium humidity of solution, water vapor partial pressure should be known. Water vapor partial pressure, temperature, and LiCl solute mass fraction are governed by a set of thermodynamic equations as described in [25]. Then the equilibrium air humidity of the solution can be calculated from the water vapor partial pressure. 2.4. Solution procedure Available commercial software packages are difficult to solve the problem. The purpose here is to develop a software package for the calculations of ducts of arbitrary shapes. So the boundary-fitted coordinate systems are used to convert the physical planes onto square numerical planes [23]. The physical plane for the in-line arrangement, as shown in Fig. 2(a), is comprised of two one-fourth circles and the surrounded air flow channel. The computational plane is shown in Fig. 2(b). FABC, HIJK and NEDGLM on the physical plane are transformed to three rectangles FABC, HIJK and NEDGLM, respectively. The air and the solution streams are conjugated on FC, DE, HK and GL. In contrast, for the staggered arrangement, the physical plane as shown in Fig. 2(c), is comprised of two one-fourth circles, a half circle and the surrounded air flow channel. The computational plane is shown in Fig. 2(d). FABC, HIJK, RWTUV and NEDGLMOPQ on the physical plane are transformed to four rectangles FABC, HIJK, RWTUV and NEDGLMOPQ, respectively. The air and the solution streams are conjugated on FC, DE, HK, GL, OPQ and RWT. So this is a conjugate heat and mass transfer problem. Eqs. (2)–(11) are discretized by means of a finite volume method [23] on the computational domains. The convective terms in Eqs. (3)–(8) are discretized by a power law scheme [13,14]. Since the air and the solution streams and the membrane are interacted, and temperature and concentration are also related to each other, ADI techniques are used to solve these equations. The solution scheme is: The momentum equations, Eqs. (2)–(6), and (9) for the two streams are solved. The Navier–Stokes equations for the air stream together with the turbulent kinetic energy and the dissipation rate equations (Eqs. (2)–(6)) are solved using the SIMPLE algorithm [16,17]. The periodic velocity boundary conditions (Eq. (57)) are realized using mutual replacements method [16,17] The velocity

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at the inlet (NE) is updated by the velocity at the outlet (ML) after certain iterations. It should be noted that the normal outlet velocity should be revised to satisfy the overall mass conservation at each iteration [16,17]. The inlet turbulent kinetic energy and the dissipation rate for the air stream are updated by the new velocity according to Eq. (57). Then the velocity fields and resistance data for the two streams are obtained. The calculations of temperature and humidity fields are performed after the calculation of velocity fields. After these procedures, the couplings between the two streams and the periodicity are satisfied simultaneously. The temperature and concentration profiles on membrane surfaces are the naturally formed boundary conditions. It is a conjugate problem. The heat and mass transfer are strongly coupled. To assure the accuracy of the results calculated, numerical tests were performed to determine the effects of the grid size. It indicates that grids with 21  61 for the in-line and 21  81 for the staggered, and 21  21 for the solution stream and 40 grids in z axis are adequate, which is less than 1.0% difference compared with the grids with 41  81 for the in-line and 41  101 for the staggered, and 41  41 for the solution stream and 60 in z axis. The final numerical uncertainty is 1.0%.

3. Experimental work A hollow fiber membrane bundle based air dehumidification test is performed. The fibers bundle is placed in a shell to form a contactor. The concept is like a cross-flow shell-and-tube heat exchanger, as shown in Fig. 3(a). The fibers can be fabricated to be in-line or staggered arrangements. The voids between the fibers form the shell side flow. Liquid desiccant flows inside the tubes. Air stream flows in the shell side in a cross-flow arrangement. The fibers separate the liquid desiccant from the process air. The fibers membrane selectively permits the permeation of moisture through it while prevents the permeation of unwanted gases and liquid solution form penetration. The hollow fiber membrane contactor with above features is used for liquid desiccant air dehumidification. In order to study the heat and mass transfer in the hollow fiber membrane contactor, a continuous air dehumidification system driven by a heat pump has been designed. The whole test set-up is schematically depicted in Fig. 3(b). As seen, there exists two flowing cycles, namely, the refrigerant cycle in compressor, condenser, expansion valve and evaporator, and the liquid desiccant cycle in dehumidifier, condenser, regenerator and evaporator. The evaporator and the condenser are made with stainless steel to prevent

Air in Diffuser

b Solution in

Solution out

a Hollow fiber membranes Air out

(a) Diluted solution

Condenser

T

T Sampling valve Regenerator

T H

T and Humidity sensors

Expansion valve

T

T H Compressor

Regeneration air

Exhaust air

Concentrated solution T

Dehumidifier

H

T H

Fresh air

Dehumidified air

Evaporator T

Heat exchanger Flow meter Pump

Exhaust water

Cooling water

Solution container

(b) Fig. 3. The experimental set-up for the air dehumidification. (a) The shell and the tube structure of the dehumidifier; (b) Schematic of the experimental set-up.

169

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 64 (2013) 162–172 Table 1 Membrane properties and some transport properties. Symbol

Unit

Value

Symbol

a b L

cm cm cm kg/m3 lm lm m2/s m2/s m2/s lm W m1 K1 W m1 K1 kg/m3

20.0 20.0 30.0 1215 600 750 1.2  106 2.82  105 3.0  109 150 0.0263 0.5 1.205

Rea Res Pra Prs Sca Scs Tai Tsi

qs ri ro Dvm Dva Dws d ka ks

qa

°C °C kg/kg kg/kg kg/kg kg/kg

300 3 0.71 28.36 0.564 1390 30.0 25.0 0.02 0.0055 0.65 0.82

in-line staggered

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

100

200

300

400

500

600

700

Rea Fig. 4. The friction factors for the air stream, SL/(2ro) = ST/(2ro) = 2.0, u = 0.196, Res = 3. The solid and the dash lines represent the data obtained by the low Re k–e turbulent model and the laminar model, respectively. The discrete points are the measured values with error bars.

Rea 0

100

200

300

400

500

600

22

20 15

14

Shave,a

Nuave,a

16

700 25

Nu (in-line) Nu (staggered) Sh (in-line) Sh (staggered)

20 18

The experimental results are used to validate the calculated values. The test set-up is operated under different operating conditions. The friction factors, the mean Nusselt and Sherwood numbers for the air flow across the hollow fiber membrane bundle are shown in Figs. 4 and 5, respectively. Error bars are also plotted. Besides the low Re k–e turbulence model, a laminar model for the air flow is also used for comparison. As seen, the two models fit the tested data well when Rea is less than 300. The maximum difference is less than 7.0% for the three indices. However the low Re k–e turbulence model is better when Rea is higher than 300. Generally the low Re k–e model is successful in modeling the fluid flow, heat and mass transfer across the bundle. For the solution flow in the round channels, the mean Nusselt and Sherwood numbers (Nuave,s and Shave,s) and (fRe)s measured are 4.44, 5.45 and 65.02, respectively. They are in accordance with the calculated values: 4.56, 5.32 and 63.67 respectively. So the

Xsi Xei

Value

0.7

4. Results and discussion 4.1. Model validation

xai xsi

Unit

fa

corrosion. The novelties in this system are: (1) The regenerated solution can be stored for future use when there is no electricity on peak hours. (2) The desiccant solution to the regenerator is heated by the condenser directly, and the solution to the dehumidifier is cooled by the evaporator directly. Heat exchange effectiveness is high due to the direct heating and cooling driven by the heat pump. (3) The packing density of hollow fibers is large. Therefore heat and mass transfer capabilities are high. (4) The traditional dehumidifier and regenerator, packed beds, are replaced by membrane contactors. There are two cross-flow hollow fiber membrane contactors. One is for air dehumidification and the other for solution regeneration. This study is focused on the heat and mass transfer in the dehumidifier. The membrane fibers are made with a layer of modified PVDF (polyvinylidene fluoride). The tested physical properties of the membrane are summarized in Table 1. Some transport properties under design operating conditions are also listed. Two modules are made with these membranes and are experimented for air dehumidification. Module A: in-line arrangement; module length (L = 30.0 cm); width (a = 20.0 cm); height (b = 20.0 cm); number of fibers (nfiber = 4500) and packing fraction (u = 0.196), transfer area (Am = 6.4 m2), longitudinal pitch (SL = 3 mm), transverse pitch (ST = 3 mm). Module B: staggered arrangement; module length (L = 30.0 cm); width (a = 20.0 cm); height (b = 20.0 cm); number of fibers (nfiber = 4433) and packing fraction (u = 0.194), transfer area (Am = 6.3 m2), longitudinal pitch (SL = 3 mm), transverse pitch (ST = 3 mm). Temperature and humidity of inlet air can be adjusted, representing varying ambient air states. In this study, LiCl solution is used as the liquid desiccant. The pressure drop, temperature, humidity and volumetric flow rate to and from the dehumidifier are measured. Solution temperature and volumetric flow rates are measured at the inlet and the outlet of the dehumidifier as well. Mass fraction of solution to and from the dehumidifier is obtained by titration method using silver nitrate solution. Heat and moisture balances are checked for the dehumidifier. After the measurements of inlet and outlet parameters such as pressures, temperatures and humidity, the friction factor and the mean Nusselt and Sherwood numbers for the two streams can be estimated based on Eqs. (42), (43), (48)– (51), with temperature difference substituted by the log mean temperature and humidity difference between the inlets and the outlets. The uncertainties for measurements are: pressure ±0.1 Pa; temperature ±0.1 °C; humidity ±2.0%; mass fraction ±1.0%; volumetric flow rate ±1.0%. The final uncertainties for the tested friction factor, Nusselt and Sherwood numbers are 3.5%, 7.5% and 7.9%, respectively.

12 10 8 5 6 4 2

0

100

200

300

400

500

600

0 700

Rea Fig. 5. The mean Nusselt and Sherwood numbers for the air stream, SL/(2ro) = ST/ (2ro) = 2.0, u = 0.196, Res = 3. The solid and the dash lines represent the data obtained by the low Re k–e turbulent model and the laminar models, respectively. The discrete points are the measured values with error bars.

simulated data are in good agreement with the experimentally obtained values, meaning the model is successful in modeling the

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L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 64 (2013) 162–172

should be taken into account. Therefore the turbulence model employed here is reasonable and necessary to account for the disturbances from the fibers.

0.5 0.07

0.25 00.0.0170

0.08

0.09 .004 .1 3

0.1

01.06 200..11 012.0 30. 0. 1

y*

Air in

0

0

03.1 0.60 0.

4.3. Nusselt numbers

0.1 0.09

0.25

01

0.5

0.75

1

x*

(a)

0.25

0.5

0 0. 0 1.0908

0.75

0. 0

. 05 00.1

1

1.25

1.5

8

00.0.0730 .00.6 0.07 08

19 ..0 0.70.0601 0.00

0

0.0 9

002 .081

0.09 0.09

08

0.

0.25 0.11

0

0 0.1 0.0.70360.0 4

y*

0.5 9 .00.08 0.10

Air in

1.75

2

x*

(b) Fig. 6. Contours of the turbulence intensity for the air stream, SL/(2ro) = ST/ (2ro) = 2.0, u = 0.196, Rea = 300. (a) In-line arrangement; (b) Staggered arrangement.

heat and mass transfer in the cross-flow hollow fiber membrane based liquid desiccant air dehumidification. After the model validation, numerical work will be performed with the low Re k–e turbulence model. The packing fractions (u) range from 0.125 to 0.502 and Rea ranges from 300 to 600. Other parameters and nominal operating conditions are the same as listed in Table 1. 4.2. Turbulence for the air flow Turbulence intensities (u0a /ua) in the air flow for the in-line and the staggered arrangements are shown in Fig. 6(a) and (b), respectively. As seen, the turbulence intensities are higher where velocities are higher. It has been found that vertexes appear behind the fibers [13,14]. The turbulence intensities behind the fibers where vortexes appear are higher than other regions. For both the in-line and the staggered arrangements, the turbulence intensities are higher than 0.07, meaning that the fluctuations of velocities for the air flow in the bundle are relatively large [13,14]. In other words, the turbulent behavior of the air flow across the bundle

It has been found that the boundary conditions on the membrane surfaces are neither uniform temperature (concentration) nor uniform heat flux (mass flux) boundary conditions [13,14]. They are naturally formed by the close coupling between the two flows. Therefore the Nusselt and Sherwood numbers calculated under uniform temperature or uniform heat flux boundary conditions are not appropriate to use here. The values will be obtained by the model established here. Air side mean Nusselt numbers (Nuave,a) and the friction factors (fa) are listed in Tables 2 and 3 for the in-line and the staggered arrangements, respectively. The ranges of the packing fraction (u) and the Reynolds numbers are those commonly used in engineering applications. Also listed are the mean values under uniform temperature (Nuave,T) and uniform heat flux boundary conditions (Nuave,H). In such two cases, the fiber membrane surfaces are set manually either to uniform temperature (T) or to uniform heat flux (H) conditions. As seen, the air side Nuave,a and fa rise with increasing packing fractions (u), but fa decreases with rising Reynolds numbers. For the in-line arrangement, generally the air side Nuave,a is between the (Nuave,T) and the (Nuave,H). There are some exceptions. When Rea is smaller than 400 and u is less than 0.16, Nuave,a is higher than Nuave,H. For the staggered arrangement, the air side Nuave,a is somewhat less than Nuave,T. The values of Nuave,T, Nuave,H, Nuave,a and fa for the staggered arrangement are larger than those for the in-line arrangement. For the solution flow in the circular channels (solution side), it has been known that the local Nusselt numbers decrease sharply from large values at the inlet to fully developed values after the entry length. The channel is long enough (L = 30 cm) for the solution flow to get thermally fully developed (z = 5 mm). The fully developed local Nusselt number (NuC,s) and the overall mean Nusselt number (Nuave,s) under the naturally formed conditions are invariant with different arrangements and different packing fractions. They are 4.48 and 4.56, respectively. 4.4. Sherwood numbers Air side mean Sherwood numbers (Shave,a) for the in-line and the staggered arrangements have been listed in Tables 2 and 3, respectively. For the in-line arrangement, the air side Shave,a is

Table 2 The friction factor and the mean Nusselt and Sherwood numbers for the air stream across the hollow fiber membrane bundle, with in-line arrangement and ST = SL. SL/(2ro)

u

Rea = 300

Rea = 400

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

19.12 10.96 8.52 7.32 6.22 5.68

11.75 7.92 6.84 6.51 6.30 6.03

18.88 10.56 7.90 6.50 5.53 4.75

0.362 0.227 0.172 0.135 0.106 0.0884

15.23 10.07 8.34 7.45 6.76 5.84

21.00 12.52 10.20 8.93 7.71 6.72 Rea = 600

15.49 8.98 7.67 7.03 6.95 6.62

20.21 11.72 8.89 7.36 6.27 5.32

0.328 0.211 0.164 0.131 0.104 0.0875

1.25 1.5 1.75 2.0 2.25 2.5

0.502 0.349 0.256 0.196 0.155 0.125

13.86 8.85 7.18 6.30 5.62 4.89 Rea = 500 Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

1.25 1.5 1.75 2.0 2.25 2.5

0.502 0.349 0.256 0.196 0.155 0.125

16.70 11.33 9.52 8.62 7.95 6.53

23.06 14.44 12.03 10.60 9.24 7.82

17.98 12.66 8.48 7.78 7.55 7.37

21.97 12.98 9.92 8.26 7.02 5.95

0.310 0.203 0.160 0.129 0.103 0.0868

18.19 12.57 10.70 9.78 9.14 8.50

25.31 16.48 13.86 12.19 10.70 9.91

20.66 14.93 12.28 8.59 8.30 8.12

23.91 14.31 11.02 9.15 7.71 6.38

0.299 0.198 0.156 0.128 0.102 0.0852

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L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 64 (2013) 162–172 Table 3 The friction factor and the mean Nusselt and Sherwood numbers for the air stream across the hollow fiber membrane bundle, with staggered arrangement and ST = SL. SL/(2ro)

u

Rea = 300 Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

1.25 1.5 1.75 2.0 2.25 2.5

0.502 0.349 0.256 0.196 0.155 0.125

26.60 18.75 15.39 13.31 12.38 11.58

31.86 21.88 18.33 17.62 15.82 15.23

25.14 17.78 14.44 12.29 11.22 10.31

30.46 21.22 17.82 17.07 15.40 14.85

0.690 0.518 0.431 0.362 0.317 0.276

30.68 21.62 17.84 15.21 14.51 13.62

35.72 24.57 21.69 20.42 18.07 17.51

29.46 20.88 17.12 14.53 13.57 12.56

34.19 23.74 20.01 19.63 17.47 16.94

0.645 0.487 0.408 0.340 0.300 0.262

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

Nuave,T

Nuave,H

Nuave,a

Shave,a

fa

34.58 24.39 20.22 16.91 16.55 15.55

40.02 27.06 22.88 23.11 20.20 19.72

33.52 23.80 19.67 16.57 15.87 14.74

37.52 26.06 22.03 22.10 19.43 18.86

0.611 0.471 0.395 0.325 0.291 0.253

38.37 27.10 22.53 18.42 18.51 17.39

43.92 29.40 24.94 25.65 22.26 21.85

37.33 26.59 22.13 18.43 18.05 16.89

40.58 28.23 23.95 24.45 21.31 20.92

0.592 0.460 0.386 0.313 0.284 0.247

Rea = 400

Rea = 500

1.25 1.5 1.75 2.0 2.25 2.5

0.502 0.349 0.256 0.196 0.155 0.125

Rea = 600

mostly between Nuave,T and Nuave,H. However it becomes less than Nuave,T when u is less than 0.2. For the staggered arrangement, the air side Shave,a is between Nuave,T and Nuave,H all the time. Further, it is more close to Nuave,H. For both the in-line and staggered arrangements, the air side Shave,a increases with increasing packing fraction (u) and/or Reynolds numbers. For the solution flow in the circular channel (liquid side), the variations of the local and mean Sherwood numbers along the z axis are similar to common round tubes [13,14]. However, the concentration boundary layer (zm;s = 0.05, 25 cm) develops much slower than the thermal boundary layer (zh;s ¼ 0:3; 3 cmÞ: It is because that the Schmidt number of LiCl solution is rather large (Scs = 1390). Nevertheless, the channel is still long enough (L = 30 cm) for the solution flow to get fully developed. The fully developed Sherwood numbers (ShC,s) and the overall mean Sherwood numbers (Shave,s) are the same for different fiber tubes. This feature is similar to the solution side NuC,s and Nuave,s. The solution side ShC,s and Shave,s under the naturally formed boundary conditions are 4.35 and 5.32, respectively. The solution side ShC,s is a bit less than the solution side NuC,s (=4.48).

5. Conclusions Turbulent fluid flow and heat and mass transfer in the crossflow hollow fiber membrane bundle for liquid desiccant air dehumidification are solved by considering interactions between the neighboring fibers. Simultaneous solution of the continuity, momentum, energy and mass equations for the two streams are performed on two periodic representative cells. Following results can be found: (1) Compared to the available experimental data, the low Re k–e turbulence model is better than a laminar model when the air side Reynolds numbers are higher than 300. (2) The mean Nusselt numbers rise with increasing packing fractions, but the friction factors decrease when rising Reynolds numbers. Generally the air side Nuave,a is between Nuave,T and Nuave,H. For in line arrangement, when Rea is less than 400 and u is less than 0.16, Nuave,a is higher than Nuave,H. For staggered arrangement, the air side Nuave,a is somewhat less than Nuave,T. The Nusselt and Sherwood numbers for the staggered arrangement are larger than those for the in-line arrangement.

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