[2012] theory of isobaric pressure exchanger for desalination

5/26/2018 [2012] Theory of Isobaric Pressure Exchanger for Desalination

http:///reader/full/2012-theory-of-isobaric-pressure-exchanger-for-desalinatio

This article was downloaded by: [Universidad de Chile]On: 15 May 2014, At: 07:55Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer Hou37-41 Mortimer Street, London W1T 3JH, UK

Desalination and Water TreatmentPublication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/tdwt20

Theory of isobaric pressure exchanger for desalinatiChiang C. Mei

a, Ying-Hung Liu

b& Adrian W-K. Law

c

aDepartment of Civil and Environmental Engineering , Massachusetts Institute of

Technology , Cambridge , MA , 02139 , USA Phone: Tel. +1 617 253 2994 Fax: Tel. +1 61

2994bDepartment of Civil and Environmental Engineering , Massachusetts Institute of

Technology , Cambridge , MA , 02139 , USAcSchool of Civil and Environmental Engineering, Nanyang Technological University ,

SingaporePublished online: 28 Feb 2012.

To cite this article:Chiang C. Mei , Ying-Hung Liu & Adrian W-K. Law (2012) Theory of isobaric pressure exchanger for

desalination, Desalination and Water Treatment, 39:1-3, 112-122, DOI: 10.1080/19443994.2012.669166

To link to this article: http://dx.doi.org/10.1080/19443994.2012.669166

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose oContent. Any opinions and views expressed in this publication are the opinions and views of the authors, anare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not berelied uponshould be independently verified with primary sources of information. Taylor and Francis shall not be liable any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeor howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
http://dx.doi.org/10.1080/19443994.2012.669166http://www.tandfonline.com/action/showCitFormats?doi=10.1080/19443994.2012.669166http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditionshttp://dx.doi.org/10.1080/19443994.2012.669166http://www.tandfonline.com/action/showCitFormats?doi=10.1080/19443994.2012.669166http://www.tandfonline.com/loi/tdwt20



Desalination and Water Treatment

www.deswater.com

1944-3994/1944-3986 2012 Desalination Publications. All rights reserveddoi: 10/5004/dwt.2012.2962

*Corresponding author.

39 (2012) 112122

February

1. IntroductionDesalination plants have been in operation for years

at many coastal sites not only for producing drinkingwater but also for treating waste water generated fromoil production fields [1]. In the technology of desalinationby sea water reverse osmosis (SWRO), salt is removedby applying very high pressure up to 80 bars to forcesea water against a semi-permeable membrane. Energyneeded in such a process can consume as high as 70%

Theory of isobaric pressure exchanger for desalination

Chiang C. Meia,*, Ying-Hung Liub, Adrian W-K. Lawc

aDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 02139Tel. +1 617 253 2994; Fax: +1 617 253 6300; email: ccmei@ mit.edubDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 02139cSchool of Civil and Environmental Engineering, Nanyang Technological University, Singapore

Received 12 May 2011; Accepted 11 August 2011

A B S T R A C T

A theory is developed to predict the time of sustained operation of a rotary pressure exchangerused for energy recovery in seawater reverse osmosis system. Based on past experiments foroscillating pipe flows, it is found that the existing plug flow velocity in the ducts is not highenough to induce turbulence in the wall boundary layer. Modeling the time series of the flowvelocity in the inviscid core as a periodic series of rectangular pulses, the structure of thelaminar momentum boundary layer is first derived. The mass boundary layer induced by theoscillating velocity is then solved in order to obtain the slow diffusion of the averaged brineconcentration along the duct. With the result the effective longitudinal diffusivity (dispersivity)is found explicitly for arbitrary Schmidt number. The dispersivity is found to be small due tothe small viscosity and mass diffusivity in the very thin boundary layers, however it is stillaugmented to hundreds times of the molecular diffusivity. For a range of duct and rotor dimen-sions and rotor frequencies, the time needed for the mixing zone to spread to the ends of theduct is predicted for large Schmidt number appropriate for salt in water. After transient mixingis over, a certain amount of salt leaks steadily into the fresh seawater reentering the membrane.However the leakage is shown to be small due to the small dispersivity.

Keywords: Pressure exchanger; SWRO system; Energy recovery; Boundary-layer theory;Convective diffusion; Taylor dispersion

of the total cost [2]. As a result several types of energyrecovery devices (ERD) have been designed to reclaimthe high pressure remaining in the brine reject. Amongthese the isobaric pressure exchanger (PX) seems to bethe most efficient [3,4]. The central part of the design is arotor with several ducts of small radius distributed alonga circle of large radius, as shown in Fig. 1. At any instantthe right half of a duct is filled with brine reject and theleft half with fresh seawater. During one half of the rota-tion cycle, several ducts are open to the inlet A and theoutlet B. A fixed volume of brine reject enters the rightend and instantly passes the high pressure to the fresh

Downloadedby[UniversidaddeChile]at07:5515May2014



C.C. Mei et al. / Desalination and Water Treatment 39 (2012) 112122 113

sea water. The same volume of seawater is pushed outof B, and joins the flow in the outgoing pipe towards the

membrane. This is followed by a short period of block-age during which there is no flow in the duct. In the nexthalf cycle after passing a blocked sector, the same ductis open to the feeder pipe at the left end and to the brineoutfall at low pressure at the right end. Fresh sea waterenters C and discharges the brine reject at D. After pass-ing another blocked sector the same duct is open at theinlet A again to receive new brine reject for the secondcycle, etc.

Because there are many ducts in the same rotor, thebrine reject and the seawater flow steadily through the sys-tem but at opposite ends of each duct, recycling the highpressure continuously.

At the moment of switch-on there is a pulsatinginterface near the middle of the duct separating the freshsea water from the brine reject. Diffusion will change theinterface into a broad zone of mixing. In existing sys-tems the typical rotor dimensions are: duct length =1 m,rotor radius = 0.2 m and duct radius = 0.01 m [3]. Therotor speed lies between 500 to 2000 rpm. An array of40 rotor units can be installed at a plant. Such units areattractive not only for on-land installations but also ondesalination vessels for serving offshore sites.

An isobaric pressure exchanger should be designedto avoid as much as possible adding more salt to the

flow reentering the membrane from the high-pressureoutlet B of the system. While the design can be guidedby laboratory tests, mathematical models can be usefulas economical alternatives. Based on the assumption offull turbulence in the ducts, a numerical analysis basedon k-model has been reported by [5]. This assumptionis however at variance with laboratory studies for oscil-latory flows in pipes of comparable radius. Motivatedby other reasons, Hino and Ohmi et al. have shown fora pipe flow under a sinusoidal pressure gradient thattransition to turbulence takes place when the Reynoldsnumber Re /can be higher than 760, whereis

the characteristic velocity,the fluid kinematic viscosityand is the Stokes boundary layer thicknessand the oscillation frequency [6,7]. By numerical simu-lation the critical Reynolds number is found by Ahn &Ibrahim to be

Re .=

336 751 7

a (1)

which shows a weak dependence on the ratio a0/,

where a0 is the pipe radius [8]. Motivated by flows in

blood vessels, Akahaven et al. have carried out extensivelaboratory and numerical studies in simple-harmonicpipe flows, and concluded that transition to turbulenceoccurs roughly when Re= 500550 [9,10]. Similar exper-iments by Eckman & Grotberg also found that transitionto turbulence in a tube of diameter 1.25 inches occursduring the decelerating phase when 500< Re



C.C. Mei et al. / Desalination and Water Treatment 39 (2012) 112122114

depending on the inlet/outlet geometry. It is only knownthat the oscillating flow velocity is roughly proportionalto the rotation frequency so that the fluid plug is keptaway from the ends of the duct [3]. Because the ductradius is typically much smaller than the duct length,

we shall ignore the end effects and assume the velocityto be uniform along the entire length, and its maximumamplitude is known. For generality T

0/Tis assumed to

range from moderate to very small values. Although theradial profile of the longitudinal velocity u r , can befound exactly in terms of Bessel functions by extendingWomersley from simple-harmonic to multi-harmonicflows, it is sufficient for present purposes to employthe boundary layer approximation because of the smallviscosity [16].

Using primes to denote physical variables, we let thevelocity profile be the sum of the inviscid core velocityW t) (plug-flow velocity) and the correction in the vis-cous boundary-layer V'=(r',t')

)u t W V + (2)

2.1. Inviscid core

The plug-flow velocity W'(t') is a periodic series ofintermittent pulses of alternating signs. Let all dimen-sionless variables be without primes. Defining thedimensionless time by t= t', we expand the series ofpulses as an odd Fourier series in < t< ,

=

W Wn

W nn

u uW t

un

in

(3)

then

WW t nt t

nt tW t nt t=

(

(= (

in

sinsin

2

(4)

The crudest model of the plug-flow velocity is aseries of rectangular pulses

W= 0

0, ;c

(5)

where 2c/= 2T0/Tis the fraction of blockage time in a

half-period. It is easy to find

W nc

n nn

(

= =

2c

ncn = os ,

, even (6)

which converges slowly. Because it takes finite timefor a circular duct to be fully open to the inlet flow, aslightly more realistic model is a series of rectangleswith rounded corners

W t )

+

0

co

c

( ))+c

,

)

))

+

c ,

2

0

(7)

More accurate computation of W(t) accounting the

duct/rotor geometry is possible but it will only affectthe details of W

nand not the essential physics. In this

article the rounded rectangles will be used with the spe-cial choice1of c= 2bfor simplicity. The lengthy expres-sion of W

nis given in Appendix A. A sample time series

of W(t)is shown in Fig. 2.

2.2. Boundary layer correction

The boundary-layer correction is governed by

V

t r r

V

r

(8)

subjected to the boundary conditions:

V WV

r0 0r (9)

Fig. 2. Model of velocity pulses in the inviscid core assmoothened rectangles with c= 2b= 0.15.

1It has been found that the numerical results computed from the two models are quite close.





With the normalization:

( ) = u V u= Vt

r a= r, , , , 0

(10)

Eq. (8) becomes

V

rr

V2 (11)

The typical values of the scales are= 1 m/s, = 100rad/s, a

0= 1.5 cm,= 102cm2/s, hence the inverse of the

Stokes-Womersley number defined by

0 (12)

is very small. The boundary conditions are

V t WV t

r (13)

Let

V Vn=

e in (14)

then

+ n r r n r r Vn

2 1 (15)

with the boundary conditions:

VV

r

(

(16)

Similar to the simple harmonic case treated byWatson [15], we introduce the boundary layer coordi-nate (seeFig. 3),

zn

n =/

(17)

then,

+

V

z nz

V

z

V

z nn

n

n2

2

2

2

(18)

The leading-order solution is

V ize (19)

which diminishes to zero exponentially outside theboundary layer (z

n>> 1).

In summary the dimensionless velocity everywhere is

u r t U n

nin

=

e (20)

where

niz

e

(21)

to the leading order. The amplitudes ofnand udepend

on two parameters, cand .

3. Effective equation for salt diffusion

We consider diffusion in a duct of finite lengthL/2 < x'< L/2, and begin with the exact equation forsalt concentration C':

+

=

+

Cu r t

C

xD

C

x r rr

C

r,

2

(22)

Let

C C r (23)

where

xC Ct , (24)

is the time and area average defined by

a r2

2

2

r

(25)

Fig. 3. Geometry in dimensionless coordinates. (a) The duct.(b) The boundary layer magnified.





Thus

=C1 (26)

and C'(x', r', t') is the deviation from the average. Weexpect that

C (27)

and that the time scale of C'0 is much longer than

O(2/).Substituting Eq. (23) in Eq. (22) and taking the area

and time average, we get

+

=

C

uC

xD

C

x

2

2 (28)

since u'= 0. The difference of Eq. (22) and Eq. (28) is

+

+

C

xu

C

xu

x

DC

x

D

r rr

1 0 1 1

12

C

r1

(29)

Because of Eq. (27), we have, at the leading order

+

=

C

tu

x

D

r rr

C

r

1 0 1 (30)

From this result we infer that

C

u1 0

(31)

Using for estimates= 1 m/s, =100 rad/s, l=1 m,it is evident that/L




4. Boundary layer solution for B

In view of Eq. (20) and Eq. (21), we assume

B B

m

e

i tm , (43)

Then Bmis governed by

=

S r r

rB

rm

m2 (44)

and the boundary conditions:

=B

rm = , . (45)

Note that B0= 0 since U

0= 0.

Using the boundary-layer coordinate zmdefined for

the velocity profile, the leading-order approximation ofEq. (44) is governed by

S+ m=B

z

mm2

20<




which is positive. This positiveness can be proven fromEq. (44) and Eq. (45) without resorting to their explicitsolution. Finally

=( )=

4

+

2

1 n

W Sc

n

(56)

The small factor O() arises from the small thicknessof the boundary layer, where dispersion is produced byshear. In general depends on , c and S

c. For salt in

water Sc= 617.28 >> 1. can be approximated by the

simple formula

==

S n

W

nc

n

n 1

(57)

This limiting result, which can also be derived

quickly by using the approximate formula Eq. (52), willnow be employed to predict the performance of the iso-baric pressure exchanger.

6. Dispersivity and the performance of the pressureexchanger

The following values of duct and rotor radii are typi-cal in existing designs : 0.015 < a

0< 0.05 m and 0.20 m < r

0

< 0.60 m [3]. In order to have sufficiently high flow rateonly a few of the many ducts should be blocked, hencethe blockage parameter c is likely a small number. Fig. 4

shows the variation of the dimensionless dispersivity /for different blockage parameters and Schmidt numbers.Expectedly /decreases with decreasing mass diffusiv-ity, hence with increasing S

c. It is interesting that for salt

and water Sc= 617.28, /changes very little with the

blockage parameter.Since the rotor is driven as a turbine by the high pres-

sure inflow, the plug flow velocity is nearly propor-tional to the frequency so that the maximum amplitude

of the interface displacement is nearly a constant equalto a few duct radii [3]. To have some quantitative ideaof the dispersivity in physical dimensions, we consideronly salt and water mixture and take a typical pres-sure exchanger with duct radius a

0= 0.015 m, and rotor

radius r0= 0.20 m. Let the rotating frequency be = 150rad/s and plug flow speed be= 3 m, implying thatthe order of magnitude of the interface displacement is

X=

cm. The physical dispersivity is calculated to be

5.72 107 m2/s. Compared with the molecular diffu-sivity of salt in water D = 1.62 109m2/s, the physicaldispersivity is greater by a factor of O(300). Note thatthe boundary layer thickness is = 1.15 104m and theReynolds number is Re= 346.4, well below the thresh-old of turbulence.

6.1. Duration of transient dispersion

Now for a duct of length L, the time scale for a sharpconcentration discontinuity to spread from the middleto the ends is roughly,

TD

L( )L

2

(58)

Beyond this time diffusive transfer of salt will takeplace steadily from the brine reject into the feeder pipe.As an example, let the length be L=1 m, the time for con-tinuous operation is roughly T

L= 121 h or 5 d. Multiply-

ing the duct length by Nincreases TLby N2times. Longerducts are clearly better for delaying the steady transfer.

Note that from our theory,

= = u u

o a ,

(59)

and /depends only on c for fixed Sc. Increasing the

rotor frequency by a factor Nwill increase by a factorof N. The operating time TL is reduced by a factor of1 N. On the other hand is inversely proportional to

the duct radius a0, hence can be made smaller by using alarger duct. This likely calls for a larger rotor in order tohave a fixed number of ducts per rotor. The interface dis-placement Xshould be kept as small as O(a

0) in order to

keep the mixing zone close to the middle for a long time.Sample predictions on the effects of rotor frequency andduct radius are shown in Tables 1 and 2 respectively.

6.2. Steady leakage

Beyond the stage of transient mixing, a steady gradi-ent of the salt concentration is reached so that there is

Fig. 4. The ratio / for different blockage ratios c andSchmidt numbers. (a): Overview for 1 < S

c< 617.28, (b): Enlarged

view for 100 < Sc< 617.28. For salt in water S

c= 617.28.





a constant leakage of salt into the feeder pipe of freshseawater due to dispersion,

FC

LA

ssp = (60)

where Cbis the salt concentration in the reject from the

membrane, Csthe salt concentration in the fresh seawa-

ter, and A= a02is the cross-sectional area of the duct.

Ignoring the thin boundary layer, salt flux in the brinereject from the membrane is

F AC tW

(61)

The fraction of leakage is

F

F L W t tusp

e

= ( )C b

(

1 (62)

Using Eq. (5), we have the crude estimate

1 2 1

tW t c= = ( ) (63)

In practice, Cs/C

b= 0.6 , = O(1) m/s, L= O(1) m,

and = O(107) m2/s is small, the fraction of leakage isof the order

L1 (64)

confirming the high efficiency claimed by the designers [3].

7. Transient evolution of salt concentration in a duct

For detailed checking of the present theory by labo-ratory experiments, it is useful to know the slow spread-ing of the mixing zone from the initial discontinuity inthe middle of the duct. Let us define the dimensionlesstime by

=

D

L t TL (65)

So that the dimensionless averaged salt concentra-tion C

0satisfies

C C

x2, ; x (66)

Consider the initial conditions

C xx

x

00

2

, )

(67)

and the boundary conditions

)=C (68)

where C is the percentage concentration differencebetween the brine reject and the fresh seawater. Clearlythe final steady state at is given by

C x2

, , = < < (69)

The transient state C'(x, ) defined by

) +

C x C x, ,=

2 (70)

satisfies Eq. (66), the initial condition

) +

C xC x

0 2,, ,

(71)

Table 1Effect of rotor frequency for fixed X = 0.02 m, L= 1 m, a

0= 0.015 m and r

0= 0.2 m

(m/s) (rad/s) (m) Re (m2/s) TL(h)

1.0 50 2.00 104 200.0 3.30 107 209.4

2.0 150 1.15 104 346.4 5.72 107 121.2

3.0 200 8.94 105 447.2 7.37 107 93.9

Table 2Effect of duct radius a

0for fixed c = 2a

0/r

0= 0.15, L=1 m,= 3

m/s and = 150 rad/s. =1.15 104 m and Re= 346.4

a0(m) (m2/s) TL (h)

0.015 5.72107 121.2

0.03 2.58 107 272.3

0.05 1.16 107 592.9





and the boundary conditions

(C t0 1 0,2 (72)

By expanding C'0(x, 0) as a Fourier series it is easy tosolve for C'0and get finally

CC

m

m

m 1

x+

)

=

C

exp m (73)

This series converges quickly for > 0.A sample space/time evolution of C'

0/C

s is plotted

in Fig. 5 in physical variables for the typical sea-waterconcentration of C

0= 40,750 mg/l and brine reject con-

centration of 73,110 mg/l. After about 200 h, steadyleakage of salt enters the feeder pipe and returns to themembrane section.

In dimensionless form, the concentration fluctuationfrom the mean can be obtained from

C x t BC0

(74)

(cf. Eq. (32)). The dimensionless concentration fluctuation is

CC

C LB

C

x

u u= =

1 0

(75)

For salt in water, the Schmidt number is so largethat the mass boundary layer is extremely thin and B

mis

essentially constant across the duct, as seen in Eq. (52).Thus,

C xC iW

mmt0, , =

e (76)

Recall that C0(x, t) varies in time slowly through .

In Fig. 6 we show some sample results of transient fluc-

tuations for two oscillation periods around four dimen-sionless times at =0.005, 0.01, 0.03 and = 0.05. For thepressure exchanger with L=1 m, a

0= 0.015 m, r

0= 0.2 m,

= 150 rad/s and= 1.5 m/s, we have/L = 0.01.The corresponding physical times are t' = 9.6, 19.2, 57.7and 96.1 h respectively. After such a long time the meangradient x approaches constant along the duct.The concentration fluctuation eventually becomes onlyperiodic in time.

8. Conclusions

In this article we have provided a theoretical confir-mation of the efficacy of the isobaric pressure exchanger.The theory predicts the spreading by convective dif-fusion of salt along each duct inside the pressureexchanger. The effective equation for the slow disper-sion along the duct is derived and an explicit formulafor the dispersivity (effective diffusivity) is found. Theanalytical result is used to predict the time scale for thetransient mixing to spread across the entire duct length.Moreover, it is shown that even after the transient stateis passed, the steady transfer of salt from brine rejectto the fresh seawater reentering the membrane is small.

Fig. 5. Evolution of the averaged concentration C0in physical

variables x (m) and t (h)), in a duct of L=1 m, a0= 0.015 m,

r0= 0.2 m. The rotor frequency is = 150 rad/s and the maxi-

mum plug flow velocity is=1.5 m/s. = 1.43 107m2/s.The blockage time fraction is assumed to be c= 0.15.

Fig. 6. Evolution of concentration fluctuation in C C Cs1 in

dimensionless variables at (a) =0.005 (t' = 9.6 h); (b) =0.01(t' = 19.2 h); (c) =0.03 (t' = 57.7 h); (d) =0.05 (t' = 96.1 h). Fora rotor with L= 1 m, a

0= 0.015 m, r

0= 0.2 m, = 150~rad/s

and= 1.5 m/s. The blockage time fraction is assumed tobe c= 0.15.





We only give the explicit results for c= 2b,

I

nc

n

ncnc

n c

12

2

2

2n

3( )o cnc os co cs + os

,,

I

nc

n2

3

2( )n o

,

I

ncnc

n c

ncc

n

3

2

2

3

2

2

1 2

2

( )+

( )

+

o c sco

cos

+

+

o

3

2

2

22

2

2nc

nc

c

+

+

( )

cos

sin

2

22

1

2

2

2

2

nc

c

n

c

+

c c

nc

2

2

2

2

+

sin s + + n

nc

2

22

2

.

(A.3)

Symbols

F quantity Fin physical dimensions.F normalized quantityFwithout dimen-

sions.a

0 duct radius.

B concentration normalized for unit

x.B

m m-th harmonic amplitude of B.

C brine concentration.C

m concentration perturbation at order m.

C time-averaged brine concentration.C cross-sectional average of duct con-

centration.C concentration difference between

brine reject and fresh seawater.D molecular diffusivity of brine in

water.

Hence the high efficiency of the isobaric pressureexchanger is theoretically confirmed. Detailed labora-tory measurements are not yet available in the literatureand would be very worthwhile. For guiding the designit is worth further investigation to predict accurately the

magnitude of the pressure transmitted to the feeder pipe.For this purpose the detailed fluid mechanics in the inlet,the outlet and the feeder pipe may have to be taken intoaccount. Other design concerns such as leakage and loudnoise would require more elaborate effort in computa-tional modeling.

Acknowledgements

CCM acknowledges the support of grants from US-Israel Bi-National Science Foundation and MIT EarthSystems Initiative. YHL is supported by the Postdoctoral

Program of the National Taiwan University, the NationalResearch Council of Republic of China, under ContractNo. NSC 098-2811-E-002-112 (PI: Chien C. Chang), andin part by MIT. AWKL acknowledges the financial sup-port by Environment and Water Industry Council (EWI)of Singapore, under Project MEWR C651/06/173.

Appendix

Fourier expansion of the rectangular pulses withrounded corners

In the positive half period, let

W t ) =

+

;c

cos

b

c t

( )t )b+c

;c c

)

))

+

c

;

o , ;2

(A.1)

Wb

nt dtnc

= ( ))b+c

(

+

2c+ os in

b

bnt dt (

))c

in

cos

( )

+

sin n t

I I

b+

= 33. (A.2)





[3] R.L. Stover, Development of a fourth generation energyrecovery device, A CTOs Notebook, Desalination, 165 (2004)313321.

[4] R.L. Stover, Seawater reverse osmosis with isobaric energyrecovery devices, Desalination, 203 (2007) 168175.

[5] Y-H. Zhou, X-W. Ding, M-W. Mao and Y-Q. Chang, Numericalsimulation on a dynamic mixing process in ducts of a rotarypressure exchanger for SWRO, Desalin. Water Treat., 1(2009)107113.

[6] M. Hino, M. Sawamoto and S. Takasu, Experiments on transi-tion to turbulence in an oscillatory pipe flow, J. Fluid Mech., 75(1976) 193207.

[7] M. Ohmi, M. Iguchi, K. Kakehachi and T. Masuda, Transitionto turbulence and velocity distrubution in an osci llating pipeflow, Bull. Japan Soc. Mech. Engrs., 25 (1982) 365371.

[8] K.H. Ahn and M.B. Ibrahim, Laminar/turbulent oscillatingflow in circular pipes, Int. J. Heat Fluid Flow, 13 (1992) 340346.

[9] R. Akhaven, R.D. Kamm and A.H. Shapiro, An investigationof transition to turbulence in bounded oscillatory Stokes flowsPart 1. Experiments, J. Fluid Mech., 225 (1991) 395422.

[10] R. Akhaven, R.D. Kamm and A.H. Shapiro, An investigationof transition to turbulence in bounded oscillatory Stokesflows Part 2. Numerical simulations, J. Fluid Mech., 225 (1991)423444.

[11] D.M. Ekman n and J.B. Grotberg, Experiments on transit ion toturbulence in oscillatory pipe flow, J. Fluid Mech., 222 (1991)329350.

[12] G.I. Taylor, Dispersion of soluble matter in solvent flowingslowly through a tube, Proc. R. Soc. London, ser. A., 219 (1953)186203.

[13] A. Aris, On the dispersion of a solute in pulsating flow througha tube, J. Fluid Mech., 259 (1960) 370376.

[14] P.C. Chatwin, On the longitudinal dispersion of passive con-taminant in oscillatory flow in tubes, J. Fluid Mech., 71 (1975)513527.

[15] E.J. Watson, Diffusion in oscillatory pipe flow, J. Fluid Mech.,133 (1983) 233244.

[16] J.B. Womersley, Method for the calculation of velocity, rate ofglow and viscous drag in the arteries when the pressure gradi-ent is known, J. Physiol.,127 (1955) 553563.

dispersion coefficient (Eq. (42)). / dimensional boundary layer thick-

ness. Stokes-Wormersley number (Eq. (12)). molecular viscosity.L duct length. rotation frequency.r radial distance from duct center line.Re Reynolds number (Eq. (1)).S

c Schmidt number.

t time.T

c time for concentration interface to

diffuse from center to ends of duct.u(r, t) longitudinal flow velocity. characteristic scale of velocity.V(r, t) velocity correction in the boundary

layer.V

n amplitude of the n-th harmonic of V.

W(t) flow velocity in the inviscid core.W

m m-th harmonic amplitude of W.

X displacement amplitude of concen-tration interface.

zn boundary layer coordinate for the

n-th harmonic (Eq. (17)).

References

[1] M. Cakmaker, N. Kayaalp and I. Koyuncu, Desalination ofproduced water from oil production fields by membrane pro-cess, Desalination, 222 (2008) 176186.

[2] C. Fritzmann, J. Lowenberg, T. Wintgens and T. Melin, State-

of-the-art of reverse osmosis desalination, Desalination, 216(2007) 176.


[2012] theory of isobaric pressure exchanger for desalination

Documents