Theory of isobaric pressure exchanger for desalination

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Theory of isobaric pressure exchanger fordesalination

Mei, Chiang C.; Liu, Ying‑Hung.; Law, Adrian Wing‑Keung

2012

Mei, C. C., Liu, Y.‑H., & Law, A. W.‑K. (2012). Theory of isobaric pressure exchanger fordesalination. Desalination and Water Treatment, 39(1‑3), 112‑122.

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Theory of isobaric pressure exchanger for desalinationChiang C. Mei a , Ying-Hung Liu b & Adrian W-K. Law ca Department of Civil and Environmental Engineering, Massachusetts Institute of Technology,Cambridge, MA, 02139, USA Phone: Tel. +1 617 253 2994 Fax: Tel. +1 617 253 2994b Department of Civil and Environmental Engineering, Massachusetts Institute of Technology,Cambridge, MA, 02139, USAc School 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 fordesalination, Desalination and Water Treatment, 39:1-3, 112-122

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Desalination and Water Treatment www.deswater.com1944-3994/1944-3986 © 2012 Desalination Publications. All rights reserveddoi: 10/5004/dwt.2012.2962

*Corresponding author.

39 (2012) 112–122February

1. Introduction

Desalination plants have been in operation for years at many coastal sites not only for producing drinking water but also for treating waste water generated from oil production fi elds [1]. In the technology of desalination by sea water reverse osmosis (SWRO), salt is removed by applying very high pressure up to 80 bars to force sea water against a semi-permeable membrane. Energy needed 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: [email protected] 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 AC T

A theory is developed to predict the time of sustained operation of a rotary pressure exchanger used for energy recovery in seawater reverse osmosis system. Based on past experiments for oscillating pipe fl ows, it is found that the existing plug fl ow velocity in the ducts is not high enough to induce turbulence in the wall boundary layer. Modeling the time series of the fl ow velocity in the inviscid core as a periodic series of rectangular pulses, the structure of the laminar momentum boundary layer is fi rst derived. The mass boundary layer induced by the oscillating velocity is then solved in order to obtain the slow diffusion of the averaged brine concentration 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 to the small viscosity and mass diffusivity in the very thin boundary layers, however it is still augmented 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 the duct is predicted for large Schmidt number appropriate for salt in water. After transient mixing is 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 energy recovery devices (ERD) have been designed to reclaim the high pressure remaining in the brine reject. Among these the isobaric pressure exchanger (PX) seems to be the most effi cient [3,4]. The central part of the design is a rotor with several ducts of small radius distributed along a circle of large radius, as shown in Fig. 1. At any instant the right half of a duct is fi lled with brine reject and the left half with fresh seawater. During one half of the rota-tion cycle, several ducts are open to the inlet A and the outlet B. A fi xed volume of brine reject enters the right end and instantly passes the high pressure to the fresh

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C.C. Mei et al. / Desalination and Water Treatment 39 (2012) 112–122 113

sea water. The same volume of seawater is pushed out of B, and joins the fl ow in the outgoing pipe towards the membrane. This is followed by a short period of block-age during which there is no fl ow in the duct. In the next half cycle after passing a blocked sector, the same duct is open to the feeder pipe at the left end and to the brine outfall at low pressure at the right end. Fresh sea water enters C and discharges the brine reject at D. After pass-ing another blocked sector the same duct is open at the inlet A again to receive new brine reject for the second cycle, etc.

Because there are many ducts in the same rotor, the brine reject and the seawater fl ow steadily through the sys-tem but at opposite ends of each duct, recycling the high pressure continuously.

At the moment of switch-on there is a pulsating interface near the middle of the duct separating the fresh sea water from the brine reject. Diffusion will change the interface 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]. The rotor speed lies between 500 to 2000 rpm. An array of 40 rotor units can be installed at a plant. Such units are attractive not only for on-land installations but also on desalination vessels for serving offshore sites.

An isobaric pressure exchanger should be designed to avoid as much as possible adding more salt to the fl ow reentering the membrane from the high-pressure outlet B of the system. While the design can be guided by laboratory tests, mathematical models can be useful as economical alternatives. Based on the assumption of full turbulence in the ducts, a numerical analysis based on k-ε model has been reported by [5]. This assumption is however at variance with laboratory studies for oscil-latory fl ows in pipes of comparable radius. Motivated by other reasons, Hino and Ohmi et al. have shown for a pipe fl ow under a sinusoidal pressure gradient that transition to turbulence takes place when the Reynolds number Reδ = δ/ν can be higher than 760, where is

the characteristic velocity, ν the fl uid kinematic viscosity and δ ν ω2 /ν is the Stokes boundary layer thickness and ω the oscillation frequency [6,7]. By numerical simu-lation the critical Reynolds number is found by Ahn & Ibrahim to be

Re ./

δ δ= ⎛

⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞336 75

1 7/ao (1)

which shows a weak dependence on the ratio a0/δ, where a0 is the pipe radius [8]. Motivated by fl ows in blood vessels, Akahaven et al. have carried out extensive laboratory and numerical studies in simple-harmonic pipe fl ows, and concluded that transition to turbulence occurs roughly when Reδ= 500−550 [9,10]. Similar exper-iments by Eckman & Grotberg also found that transition to turbulence in a tube of diameter 1.25 inches occurs during the decelerating phase when 500< Reδ <854 for 9< a0/ /ω/ <33 [11].

In the duct of a pressure exchanger, the plug fl ow is not simple harmonic but a periodic series of intermit-tent pulses of alternating signs. Let T0 be the duration of blockage per half rotation and T=2π/ω the rotation period. Their ratio depends on the inlet design. If only one duct is blocked during each half cycle, the ratio can be estimated by T0/T = O(a0/πr0) where r0 is the radial distance from the duct center to the rotor axis. Using a generous estimate of the duct velocity 5 m/s, we get

~δ 700. Hence duct fl ow is likely laminar under many operating conditions. A theory of convective dif-fusion in laminar oscillatory fl ow is called for.

Following the pioneering work of G.I. Taylor [12] on convective diffusion (i.e., dispersion) in steady channel fl ows, theories for sinusoidal laminar pipe fl ows have been advanced in [12−15]. In this article we extend their theories to intermittent fl ows in a duct with a view to predict the longitudinal dispersivity, i.e., the effective diffusivity, of the averaged concentration. The dura-tion of transient diffusion for the mixing zone to spread across the full length of the duct will be fi rst predicted for a range of duct and rotor dimensions, and rotor speed. To assess the effi ciency of the pressure exchanger, the amount of salt transfer during the fi nal steady state will also be calculated. Details of transient evolution of the averaged concentration as well as its quasi-periodic fl uctuation will be examined.

2. Velocity in the duct

Because the drum is forced to rotate like a hydraulicturbine by high-pressure infl ow through specially shaped inlets, detailed prediction of the velocity near the duct ends is a complex task of numerical simulation

Fig. 1. SWRO system and isobaric pressure exchanger. From [3].

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depending on the inlet/outlet geometry. It is only known that the oscillating fl ow velocity is roughly proportional to the rotation frequency so that the fl uid plug is kept away from the ends of the duct [3]. Because the duct radius is typically much smaller than the duct length, we shall ignore the end effects and assume the velocity to be uniform along the entire length, and its maximum amplitude is known. For generality T0/T is assumed to range from moderate to very small values. Although the radial profi le of the longitudinal velocity u r t’( ’, ’) can be found exactly in terms of Bessel functions by extending Womersley from simple-harmonic to multi-harmonic fl ows, it is suffi cient for present purposes to employ the boundary layer approximation because of the small viscosity [16].

Using primes to denote physical variables, we let the velocity profi le be the sum of the inviscid core velocity W t’( ’) (plug-fl ow velocity) and the correction in the vis-cous boundary-layer V' = (r',t')

′ ′ ′( ) ′ ′( ) ′ ′ ′( )u r′ ( t W′ ) = t V′ ) + r t′, ,) ( ) (t W) t V) + r (2)

2.1. Inviscid core

The plug-fl ow velocity W'(t') is a periodic series of intermittent pulses of alternating signs. Let all dimen-sionless variables be without primes. Defi ning the dimensionless time by t = ωt', we expand the series of pulses as an odd Fourier series in –π < t < π,

′ ′( ) = ( )= −∞

∞∑

= ( )=

∞∑

−W t′ ( Wn

W n( tn

u u( ) =W t(

u

nWW

nWW

e int

s21

(3)

then

WW t nt dt

nt dtW t nt dtnWW =

( ) ( )( )

= ( ) ( )∫∫

∫sin

sinsin0∫∫

20∫∫

0∫∫2

1π

ππ

π (4)

The crudest model of the plug-fl ow velocity is a series of rectangular pulses

Wt c

c t ct

=≤ t

≤t≤

⎧⎨⎪⎧⎧⎨⎨⎩⎪⎨⎨⎩⎩

0 010

, ;t c≤ t0, ;c t c≤t −, ;

ππ πtc ≤c t− c ≤

(5)

where 2c/π = 2T0/T is the fraction of blockage time in a half-period. It is easy to fi nd

W nt dt nc

n W n

nWWc

nWW

( ) ( )=W =

∫c

1 2d

c ( )−∫

0π πnc

( )∫cs cdtnt( ) = os ,

,odd; even (6)

which converges slowly. Because it takes fi nite time for a circular duct to be fully open to the inlet fl ow, a slightly more realistic model is a series of rectangles with rounded corners

W t

t c

( ) =

≤ t

+

0 0

12

1

, ;t c≤ t0

cosπ

bc t c b

c b t

( )ttt ( )c b+c⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥ ≤t

≤b ≤

, ;c t c b≤t

,1, πππ

π π

− ( )−

( )( )π⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦( )≤

+(π −πb

+ t cπ≤ −

;

cos ,( )

⎝⎜⎝⎝ ⎠

⎟⎠⎠

⎥⎥⎥⎥

b;

,

12

1

0 , π π≤

⎧

⎨

⎪⎧⎧

⎪⎪⎪

⎪⎪⎪

⎪⎨⎨

⎪⎪

⎩

⎪⎨⎨

⎪⎪⎪

⎪⎪⎪

⎪⎩⎩

⎪⎪t≤

(7)

More accurate computation of W(t) accounting the duct/rotor geometry is possible but it will only affect the details of Wn and not the essential physics. In this article the rounded rectangles will be used with the spe-cial choice1 of c = 2b for simplicity. The lengthy expres-sion of Wn is 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′ rr

Vr

υ (8)

subjected to the boundary conditions:

′ ′ ′∂ ′∂ ′

= ′V W′ = − a′Vr

, ;r a′ = ,0 0 0′ =r,and (9)

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

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

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With the normalization:

′ ′ ′( ) ( ) ′ = ′u W′ V u′ ) = ( W V tt

r a′ = ru, ,W , ,W , ,tω 0 (10)

Eq. (8) becomes

∂∂

= ∂∂

∂∂

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞V

t r rr

Vr

ε2 (11)

The typical values of the scales are = 1 m/s, ω = 100 rad/s, a0 = 1.5 cm, ν = 10–2 cm2/s, hence the inverse of the Stokes-Womersley number defi ned by

ε υω

= 11

0a , (12)

is very small. The boundary conditions are

V t WV t

r1

00,W, t

,( ) = −∂ ( )

∂=and (13)

Let

V Vn

nVV= −∞

∞∑ −e int (14)

then

− = ∂∂

∂∂

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ = ∂

∂+ ∂

∂⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

iVn r r

rr n⎠⎠⎠ r r

VrnVV n n⎞⎞⎞ ∂⎛⎛⎛ VV nVVε ∂ ∂⎛⎛⎛ Vn ⎞⎞⎞VV2 2∂ ∂⎛ ⎞ 2

2

1∂⎛ Vn∂⎛ VV∂ ∂⎛⎛⎛ ε⎞Vn ⎞⎞⎞VV 2∂ ∂⎛ ⎞V ε⎞∂ ∂⎛ V 2 (15)

with the boundary conditions:

V WV

rn nV WV W nVV 00( ) ∂ ( )

∂=, (16)

Similar to the simple harmonic case treated by Watson [15], we introduce the boundary layer coordi-nate (see Fig. 3),

zrn

n = −1ε/

(17)

then,

− = ∂∂

−−

∂∂

= ∂∂

+ ⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

iVVz n

z

Vz

Vz n

nVV nVV

nn

nVV

n

nVV

n

2

2

2

2

1

ε

εΟ (18)

The leading-order solution is

V Wn nV WV W izn( ) − −e (19)

which diminishes to zero exponentially outside the boundary layer (zn >> 1).

In summary the dimensionless velocity everywhere is

u r t Un

n, int( )= −∞

∞∑ −e (20)

where

U Wn

n nWW izn−W + ⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

⎛⎝⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

− −1 e Ο ε (21)

to the leading order. The amplitudes of n and u depend on two parameters, c and ε.

3. Effective equation for salt diffusion

We consider diffusion in a duct of fi nite length –L/2 < x' < L/2, and begin with the exact equation for salt concentration C':

∂ ′∂ ′

+ ′ ′ ′( ) ∂ ′∂ ′

= ∂ ′∂ ′

+′

∂∂ ′

′∂ ′∂ ′

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞⎛

⎝⎜⎛⎛⎝⎝

⎞Ct

u r′ ( tCx

DC

x r′ rr

Cr

,2

2

1⎠⎠⎟⎞⎞⎠⎠⎠⎠

(22)

Let

′ ′ ′ ′( ) + ′ ′ ′ ′( )C C′ = x t′ C x′ ( r t′0 1( )) +x t C ,( x(1) +t C , (23)

where

′ ′ ′( ) ′xC′ ( Ct′ ) ≡0 , (24)

is the time and area average defi ned by

fa

f r ta

= ′ ′( ) ′∫ ∫r f r t r fa

′rff ′( )222 0∫∫0∫∫ f f( )2

2ππ

ωπ

π ωt )) ,

/df r t′ ′t )∫r f)) =r) ,

/

(25)

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

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Thus

′ =C1 0 (26)

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

′ ′∂

∂ ′∂

∂ ′C′

x r′ ∂1 0C ′C0C , (27)

and that the time scale of C'0 is much longer than O(2π/ω).

Substituting Eq. (23) in Eq. (22) and taking the area and time average, we get

∂ ′∂ ′

+ ′∂ ′∂ ′

= ∂ ′∂ ′

Ct

uCx

DC

x

2

2 (28)

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

∂ ′∂ ′

+ ∂ ′∂ ′

+ ′∂ ′∂ ′

− ∂ ′∂ ′

= ∂ ′∂ ′

+′

∂∂ ′

′

Ct

uCx

uCx

uCx

DC

xDr r′ ∂

r

1 0+ ′∂

uC 1 1′

∂u

C

212

∂∂ ′∂ ′

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞C

r1

(29)

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

∂ ′∂ ′

+ ∂ ′∂ ′

=′

∂∂ ′

′∂ ′∂ ′

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞C

tu

Cx

Dr r′ ∂

rCr

1 0+ ′∂

uC 1 (30)

From this result we infer that

′ =⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

′CL

Cu

1 0=⎝⎜⎝⎝ ⎠⎟⎠⎠L

CΟω

(31)

Using for estimates = 1 m/s, ω =100 rad/s, l=1 m, it is evident that /ωL << 1, hence C'1 << C'0, as expected.

Now assume

′ ′ ′( ) ∂ ′∂ ′

C B′ = r t′Cx1

0, (32)

then B' is governed by

∂ ′∂ ′

+ ′ =′

∂∂ ′

′∂ ′∂ ′

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞B

tu

Dr r′ ∂

rBr

(33)

with the boundary conditions

∂ ′∂ ′

= ′Br

a0 0′r′ = 0, ,0r (34)

Once B' is solved, the solution can be substituted in Eq. (28) to get the effective diffusion equation for the area- and period-averaged concentration:

∂ ′∂ ′

+ ′ ′∂ ′∂ ′

= ∂ ′∂ ′

Ct

u B′C

xD

Cx

02

02

202 , (35)

namely,

∂ ′∂ ′

= + ′( ) ∂ ′∂ ′

Ct

DC

x0

202 , (36)

where

′ = − ′ ′ u B′ (37)

is the dispersion coeffi cient (or dispersivity).Let us introduce the additional dimensionless vari-

ables

′ ′ = ′C C′ = C CL

C C B B′ = x L′ = xsCL

u u′C C B′0 0C= CsC 1sL1 1= C CC ,C 1CsL

C , ,x Lxω ωL

(38)

where Cs is the concentration of the seawater. The scale of B' is inferred from Eq. (31) and Eq. (32). In normalized variables, B is governed by

∂∂

ε ∂∂

∂∂

Bt

uS r r

rBrc

+ =u ⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞2

0 1r< <r, (39)

where

SDc = υ

(40)

is the Schmidt number. B must also satisfy the boundary conditions

∂∂

=Br

0 0=r 1, ,0r (41)

Finally the dimensional and dimensionless disper-sion coeffi cients are related by

′ = − ′ ′ ′ = u

u B′2

ω, (42)

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4. Boundary layer solution for B

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

B Bm

m= −∞

∞∑ −e i tm , (43)

Then Bm is governed by

− + = ∂∂

∂∂

⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞imB U

S r rr

Brm m+ U

c

mε2 1 (44)

and the boundary conditions:

∂∂

=Brm 0 0=r 1, ,0r . (45)

Note that B0 = 0 since U0 = 0.Using the boundary-layer coordinate zm defi ned for

the velocity profi le, the leading-order approximation of Eq. (44) is governed by

− ∂∂

imS B S+ U m= Bz

mc mB c mU m

m

2

2 0 1< <zm

m,ε

(46)

and

∂∂

= ∞Bz

m

mm0 0=zm, ,m 0zm (47)

Let Bm be the sum of homogeneous and inhomoge-neous parts

B B Bm mBImH+BmBI (48)

It is easy to fi nd by boundary layer approximation

B iWm

SSm

I m cWW S

c

izm+−

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

− −11

e (49)

and

B iWm

SSm

H mWW c

c

iS zc mz

−− −

1e (50)

Hence

B iWm

SS

iWm

SSm

m cWW S

c

iz mWW c

c

iS zm c mz+⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎞⎞⎠⎠

+−

− − − −11m1 Sc− ⎠⎠⎠

ei m ce izm⎞⎠⎟⎞⎞⎠⎠

+

(51)

The solution here is a straightforward extension of of Watson for the simple harmonic case [15]. Note that both Um and Bm are essentially constant across the duct except in the velocity and mass boundary layers. For salt in water, D = 1.62 × 10–9 m2/s, and ν = 10–6 m2/s, hence the Schmidt number Sc= 617.28 is very large. The concentration boundary layer is much thinner than the velocity boundary layer. Eq. (51) can be well approxi-mated simply by

B r iWm S

S

mmWW iz

c

iS z

c

c mz( ) = − − +izm + ( )SC⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠− − − −1

1

1

e e+izm Ο ,

(52)

5. The dimensionless dispersion coeffi cient

The dimensionless dispersion coeffi cient is

= − ′ ′ = −= −∞

∞∑

= −∞

∞∑

= −

− −

=

∞

∑

u B′ Un

Bm

U B

n m∑

n mBn

e e∑ Bm∑ Bint i tm

*Re21

(53)

where asterisks denote the complex conjugates.Let us derive the dispersivity for arbitrary Sc. Trans-

forming to boundary-layer coordinates z nn ( )r /( )εand taking the area average, we get after some algebra,

U B rdrWiW

nS

S

n nB nWW iz nWW

c

n* = −( ) ⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞

× +−

− −∫ 2 1rdrWnWWrdrW (1

1

0∫∫1

e

cc

iz c

c

iS z

n

n c nzSS

niWn

nn

e eiz cn− − − −

−⎛

⎝

⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

=⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

1

22

2εεε

− −−( )

⎡

⎣⎢⎡⎡

⎣⎣⎢⎢⎣⎣⎣⎣

+ ( )− ( )+(2

2 1(

2 ( 1 + ( −

SS

SS i−

c

c

cc ( +(i )) (54)

Hence

=−⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠ − ( )−

=

∞

∑242 11

2

2Re *U Bn

Wn

SSn mB

n

nWW c

c

ε

=42

2εn

Wn

SnWW c⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠ ( )1 Sc ( )S1 cS1

(55)

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

= ( ) ( )=

∞

∑ε 42 +

2

1 nWn

S

) ( +) (nWW c

) () ( +) (n

(56)

The small factor O(ε) arises from the small thickness of the boundary layer, where dispersion is produced by shear. In general depends on ε, c and Sc. For salt in water Sc= 617.28 >> 1. can be approximated by the simple formula

==

∞

∑εS n

Wnc

nWW

n

42

2

1

(57)

This limiting result, which can also be derived quickly by using the approximate formula Eq. (52), will now be employed to predict the performance of the iso-baric pressure exchanger.

6. Dispersivity and the performance of the pressure exchanger

The following values of duct and rotor radii are typi-cal in existing designs : 0.015 < a0 < 0.05 m and 0.20 m < r0 < 0.60 m [3]. In order to have suffi ciently high fl ow rate only a few of the many ducts should be blocked, hence the blockage parameter c is likely a small number. Fig. 4 shows the variation of the dimensionless dispersivity /ε for different blockage parameters and Schmidt numbers. Expectedly /ε decreases with decreasing mass diffusiv-ity, hence with increasing Sc. It is interesting that for salt and water Sc = 617.28, /ε changes very little with the blockage parameter.

Since the rotor is driven as a turbine by the high pres-sure infl ow, the plug fl ow velocity is nearly propor-tional to the frequency ω so that the maximum amplitude

of the interface displacement is nearly a constant equal to a few duct radii [3]. To have some quantitative idea of the dispersivity in physical dimensions, we consider only salt and water mixture and take a typical pres-sure exchanger with duct radius a0 = 0.015 m, and rotor radius r0 = 0.20 m. Let the rotating frequency be ω = 150 rad/s and plug fl ow speed be = 3 m, implying that the order of magnitude of the interface displacement is

X = ω cm. The physical dispersivity is calculated to be

5.72 × 10–7m2/s. Compared with the molecular diffu-sivity of salt in water D = 1.62 × 10–9 m2/s, the physical dispersivity ’ is greater by a factor of O(300). Note that the boundary layer thickness is δ = 1.15 × 10–4 m and the Reynolds number is Reδ = 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 sharp concentration discontinuity to spread from the middle to the ends is roughly,

TD

LLTT =

( )L

+ ′≈

′4

2 2LL ′ 4

(58)

Beyond this time diffusive transfer of salt will take place 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 TL = 121 h or 5 d. Multiply-ing the duct length by N increases TL by N2 times. Longer ducts are clearly better for delaying the steady transfer.

Note that from our theory,

′ = = = ≡u u

aX

o oa

2εω ε ω ω εao υ ε ω

, ,≡X

(59)

and /ε depends only on c for fi xed Sc. Increasing the rotor frequency by a factor N will increase ’ by a factor of N . The operating time TL is reduced by a factor of 1 N . On the other hand ’ is inversely proportional to the duct radius a0, hence can be made smaller by using a larger duct. This likely calls for a larger rotor in order to have a fi xed number of ducts per rotor. The interface dis-placement X should be kept as small as O(a0) in order to keep the mixing zone close to the middle for a long time. Sample predictions on the effects of rotor frequency and duct 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 /ε for different blockage ratios c and Schmidt numbers. (a): Overview for 1 < Sc < 617.28, (b): Enlarged view for 100 < Sc < 617.28. For salt in water Sc= 617.28.

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

FC C

LAb sC

dispFF = ′ (60)

where Cb is the salt concentration in the reject from the membrane, Cs the salt concentration in the fresh seawa-ter, and A = π a0

2 is the cross-sectional area of the duct. Ignoring the thin boundary layer, salt fl ux in the brine reject from the membrane is

F AC W tbu

reFF je dtW ( )∫ ∫W t ACbW t Cudt ACbt ACu

′t′ (( ) ′ωπ π∫

π ω π

0∫ ∫∫ ∫b( )π∫∫ ( ) (61)

The fraction of leakage is

F

F L W t tudispFF

reFF je d=

′ ( )C Cs bC−

( )∫

10∫∫ππ (62)

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

1 21

0ππ

πtW t

c( ) = = ( )∫0d O (63)

In practice, Cs/Cb = 0.6 , = O(1) m/s, L= O(1) m, and ’= O(10–7) m2/s is small, the fraction of leakage is of the order

Ο ′⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

uL 1 (64)

confi rming the high effi ciency 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 in the middle of the duct. Let us defi ne the dimensionless time τ by

τ = + ′ ′ ≈ ′DL

ttTLTT

2LL 4

(65)

So that the dimensionless averaged salt concentra-tion C0 satisfi es

∂∂

= ∂∂

< <C C∂x

x02

02

12

12τ

, ;− < <x2 2

(66)

Consider the initial conditions

C xx

C x0 0

112

0

1 0C12

,, ,x

20

,( ) =

− < <

C <

⎧

⎨⎪⎧⎧

⎨⎨

⎩⎪⎨⎨

⎩⎩ (67)

and the boundary conditions

′ ( ) ′ ( ) =C′ −( C)0 0( )−( 1 0/ / ,2 τ′ (C) = 0) C) = 1 / ,2 (68)

where ΔC is the percentage concentration difference between the brine reject and the fresh seawater. Clearly the fi nal steady state at τ ∼ ∞ is given by

C x C x x0 112

12

12

, ,C x12

∞( ) = +11 +⎛⎝⎝⎝

⎞⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ − < <Δ (69)

The transient state C'(x, τ) defi ned by

′ ( ) ( ) − − +⎛⎝⎝⎝

⎞⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞C x′ ( C ( C x⎛

⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛,(, ) ,τ τ) = C x( ,() x(C0 1

12

Δ (70)

satisfi es Eq. (66), the initial condition

′ ( ) =− +⎛

⎝⎝⎝⎞⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ − < <

− −⎛⎝⎝⎝

⎞⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ < <

⎧

⎨⎪⎧⎧

C x′ (C x⎛

⎝⎛⎛⎝⎝

x

C x⎛⎝⎛⎛⎝⎝

x0 0

12

12

0

12

012

,, ,< <x

20

,

Δ

Δ

⎪⎪⎨⎨⎨⎨⎪⎪⎪⎪

⎩⎪⎨⎨

⎪⎩⎩⎪⎪

(71)

Table 1Effect of rotor frequency ε for fi xed X = 0.02 m, L = 1 m, a0 = 0.015 m and r0 = 0.2 m

(m/s) ω (rad/s) δ (m) Reδ ’ (m2/s) TL(h)

1.0 50 2.00 × 10–4 200.0 3.30 × 10–7 209.4

2.0 150 1.15 × 10–4 346.4 5.72 × 10–7 121.2

3.0 200 8.94 × 10–5 447.2 7.37 × 10–7 93.9

Table 2Effect of duct radius a0 for fi xed c = 2a0/r0 = 0.15, L=1 m, = 3 m/s and ω = 150 rad/s. δ =1.15 × 10−4 m and Reδ = 346.4

a0 (m) ’ (m2/s) TL (h)

0.015 5.72 × 10–7 121.2

0.03 2.58 × 10–7 272.3

0.05 1.16 × 10–7 592.9

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and the boundary conditions

′ ( ) ′ ( ) =C C′ −( ) = t0 0( ) C−( ) = 1 0/ / ,2 (72)

By expanding C'0(x, 0) as a Fourier series it is easy to solve for C'0 and get fi nally

CC

xC

m

m xm

01

12

2 1x

2

= +1 ( ) +

{ }222m( )⎡⎣

⎤⎦ ( )

=

∞

∑Δ ΔC2 1( )

∞

∑ π

πexp s{ }m2m− ( )⎡⎣⎡⎡

⎦) ⎦ (73)

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

in Fig. 5 in physical variables for the typical sea-water concentration of C0 = 40,750 mg/l and brine reject con-centration of 73,110 mg/l. After about 200 h, steady leakage of salt enters the feeder pipe and returns to the membrane section.

In dimensionless form, the concentration fl uctuation from the mean can be obtained from

C x t BCx1

0,( ) ∂∂

(74)

(cf. Eq. (32)). The dimensionless concentration fl uctuation is

C CC L

CL

BCxs

u uC= ′ = =C

∂∂

11

0

ωL (75)

For salt in water, the Schmidt number is so large that the mass boundary layer is extremely thin and Bm is essentially constant across the duct, as seen in Eq. (52). Thus,

C x tCx

iWm

mWW

m

imt1

0, ,t τ( ) = − ∂∂ = −∞

∞−∑ e (76)

Recall that C0(x, t) varies in time slowly through τ. In Fig. 6 we show some sample results of transient fl uc-tuations for two oscillation periods around four dimen-sionless times at τ =0.005, 0.01, 0.03 and τ = 0.05. For the pressure exchanger with L=1 m, a0 = 0.015 m, r0 = 0.2 m, ω = 150 rad/s and = 1.5 m/s, we have /ωL = 0.01. The corresponding physical times are t' = 9.6, 19.2, 57.7 and 96.1 h respectively. After such a long time the mean gradient ∂ ∂x∂/ ’0 approaches constant along the duct. The concentration fl uctuation eventually becomes only periodic in time.

8. Conclusions

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

Fig. 5. Evolution of the averaged concentration C0 in 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 ω = 150 rad/s and the maxi-mum plug fl ow velocity is =1.5 m/s. ’ = 1.43 × 10–7 m2/s. The blockage time fraction is assumed to be c = 0.15.

Fig. 6. Evolution of concentration fl uctuation in C C Cs1’/ in dimensionless variables at (a) τ =0.005 (t' = 9.6 h); (b) τ =0.01 (t' = 19.2 h); (c) τ =0.03 (t' = 57.7 h); (d) τ =0.05 (t' = 96.1 h). For a rotor with L = 1 m, a0 = 0.015 m, r0 = 0.2 m, ω = 150~rad/s and = 1.5 m/s. The blockage time fraction is assumed to be c = 0.15.

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

Inc

nc

nn

ncnc

nc

12

2

2

32

2 2n

32

4=

( ) ⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

−

⎛⎝⎛⎛⎝⎝

⎞⎠⎞⎞⎠⎠ + ( )

−

cos cnc( ) − os co cs ⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ + os

π,,

I

nc

n2

32=

( )n1 1(( ) ⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞cos

,

I

ncnc

n c

ncc

n

3

2

2

32

212

2

2

=

⎛⎝⎛⎛⎝⎝

⎞⎠⎠⎠⎞⎞ ( )

+−( ) ⎛

⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

×+

⎛

cos c⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ − os

cos

cos

π

π⎝⎝⎜⎛⎛⎝⎝⎝⎝

⎞⎠⎟⎞⎞⎠⎠

++⎛⎝⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

−⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

⎧

⎨⎪⎧⎧

⎪⎨⎨⎪⎪

⎩⎪⎨⎨

⎪⎩⎩⎪⎪

− +⎛⎝

cos

cos

32

2

22

2

2

2

ncc

nc

ncc

π

π

π

+⎜⎜⎛⎛⎛⎛⎝⎝⎝⎝

⎞⎠⎟⎞⎞⎠⎠

+ − +⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

+⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

⎫

⎬⎪⎫⎫

⎪⎬⎬⎪⎪

⎭⎪⎬⎬

⎪⎭⎭⎪⎪

−−( )

cos

sin

32

2

22

12

2

2

2

ncc

nc

c

n

π

π

π⎛⎛⎝⎜⎛⎛⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

×− +

⎛⎝⎛⎛⎝⎝

⎞⎠⎞⎞⎠⎠

+−⎛⎝⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

−⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

⎧

⎨si sn +

⎝⎝⎝⎞⎠⎟⎞⎞⎠⎠

− inc c

nc

2 3⎞ ⎛i

2

22

2 2⎞ ⎛ 3⎞ ⎛ 2πnc3⎞ ⎛i

2nc3⎞ ⎛ 2

π

⎪⎪⎧⎧⎧⎧

⎪⎨⎨⎪⎪⎪⎪

⎩⎪⎨⎨

⎪⎩⎩⎪⎪

−+

⎛⎝⎛⎛⎝⎝

⎞⎠⎞⎞⎠⎠

+ +⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

+⎛⎝⎛⎛⎛⎛⎝⎝⎛⎛⎛⎛ ⎞

⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞

⎫

⎬sin s− +

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

+ inc c

nc

3⎛2 ⎞i

2

22

2 2⎞ ⎛ 3⎞ ⎛ 2πnc3⎞ −⎛i

2nc3⎞ ⎛ 2

π

⎪⎪⎫⎫⎫⎫

⎪⎬⎬⎪⎪⎪⎪

⎭⎪⎬⎬

⎪⎭⎭⎪⎪

.

(A.3)

Symbols

F’ — quantity F in physical dimensions.F — normalized quantity F without dimen-

sions.a0 — duct radius.B — concentration normalized for unit

∂ ∂x∂0/ .Bm — m-th harmonic amplitude of B.C — brine concentration.Cm — concentration perturbation at order m.C — time-averaged brine concentration.⟨C⟩ — cross-sectional average of duct con-

centration.ΔC — concentration difference between

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

water.

Hence the high effi ciency of the isobaric pressure exchanger is theoretically confi rmed. Detailed labora-tory measurements are not yet available in the literature and would be very worthwhile. For guiding the design it is worth further investigation to predict accurately the magnitude of the pressure transmitted to the feeder pipe. For this purpose the detailed fl uid mechanics in the inlet, the outlet and the feeder pipe may have to be taken into account. Other design concerns such as leakage and loud noise 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 Earth Systems Initiative. YHL is supported by the Postdoctoral Program of the National Taiwan University, the National Research Council of Republic of China, under Contract No. NSC 098-2811-E-002-112 (PI: Chien C. Chang), and in part by MIT. AWKL acknowledges the fi nancial sup-port by Environment and Water Industry Council (EWI) of Singapore, under Project MEWR C651/06/173.

Appendix

Fourier expansion of the rectangular pulses with rounded corners

In the positive half period, let

W t

t c

( ) =

≤ t

+

0 0

12

1

, ;t c≤ t0

cosπ

bc t c b

c b t

( )ttt ( )c b+c⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥ ≤t

≤b ≤

, ;c t c b≤t

,1, πππ

π π

− ( )−

( )( )π⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦( )≤

+(π −πb

+ t cπ≤ −

;

cos ,( )

⎝⎜⎝⎝ ⎠

⎟⎠⎠

⎥⎥⎥⎥

b;

,

12

1

0 , π π≤

⎧

⎨

⎪⎧⎧

⎪⎪⎪

⎪⎪⎪

⎪⎨⎨

⎪⎪

⎩

⎪⎨⎨

⎪⎪⎪

⎪⎪⎪

⎪⎩⎩

⎪⎪t≤

(A.1)

ππ

Wb

nt dtnWWc

c=

( )t ( )c b+c⎛

⎝⎜⎛⎛

⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥

⎧⎨⎪⎧⎧⎨⎨⎩⎪⎨⎨⎩⎩

⎫⎬⎪⎫⎫⎬⎬⎭⎪⎬⎬⎭⎭

( )+ 12

1 c+ os sinbb

c b

c bnt dt

b

∫c

∫c+ ( )

+ −( )t ( )c⎛

c−( )sin

cos

π

π (t (12

1⎝⎝⎜⎛⎛

⎝⎝⎝⎝

⎞

⎠⎟⎞⎞

⎠⎠

⎡

⎣⎢⎡⎡

⎢⎣⎣⎢⎢

⎤

⎦⎥⎤⎤

⎥⎦⎦⎥⎥

⎧⎨⎪⎧⎧⎨⎨⎩⎪⎨⎨⎩⎩

⎫⎬⎪⎫⎫⎬⎬⎭⎪⎬⎬⎭⎭

( )

+

−( )−

∫ sin nt dt

I I+ I

c b+

c

π∫∫π

= 1 2I+ 33. (A.2)

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[3] R.L. Stover, Development of a fourth generation energy recovery device, A CTO’s Notebook, Desalination, 165 (2004) 313−321.

[4] R.L. Stover, Seawater reverse osmosis with isobaric energy recovery devices, Desalination, 203 (2007) 168−175.

[5] Y-H. Zhou, X-W. Ding, M-W. Mao and Y-Q. Chang, Numerical simulation on a dynamic mixing process in ducts of a rotary pressure exchanger for SWRO, Desalin. Water Treat., 1(2009) 107−113.

[6] M. Hino, M. Sawamoto and S. Takasu, Experiments on transi-tion to turbulence in an oscillatory pipe fl ow, J. Fluid Mech., 75 (1976) 193−207.

[7] M. Ohmi, M. Iguchi, K. Kakehachi and T. Masuda, Transition to turbulence and velocity distrubution in an oscillating pipe fl ow, Bull. Japan Soc. Mech. Engrs., 25 (1982) 365−371.

[8] K.H. Ahn and M.B. Ibrahim, Laminar/turbulent oscillating fl ow in circular pipes, Int. J. Heat Fluid Flow, 13 (1992) 340−346.

[9] R. Akhaven, R.D. Kamm and A.H. Shapiro, An investigation of transition to turbulence in bounded oscillatory Stokes fl ows Part 1. Experiments, J. Fluid Mech., 225 (1991) 395−422.

[10] R. Akhaven, R.D. Kamm and A.H. Shapiro, An investigationof transition to turbulence in bounded oscillatory Stokes fl ows Part 2. Numerical simulations, J. Fluid Mech., 225 (1991) 423−444.

[11] D.M. Ekmann and J.B. Grotberg, Experiments on transition to turbulence in oscillatory pipe fl ow, J. Fluid Mech., 222 (1991) 329−350.

[12] G.I. Taylor, Dispersion of soluble matter in solvent fl owing slowly through a tube, Proc. R. Soc. London, ser. A., 219 (1953) 186−203.

[13] A. Aris, On the dispersion of a solute in pulsating fl ow through a tube, J. Fluid Mech., 259 (1960) 370−376.

[14] P.C. Chatwin, On the longitudinal dispersion of passive con-taminant in oscillatory fl ow in tubes, J. Fluid Mech., 71 (1975) 513−527.

[15] E.J. Watson, Diffusion in oscillatory pipe fl ow, J. Fluid Mech., 133 (1983) 233−244.

[16] J.B. Womersley, Method for the calculation of velocity, rate of glow and viscous drag in the arteries when the pressure gradi-ent is known, J. Physiol., 127 (1955) 553−563.

— dispersion coeffi cient (Eq. (42)).δ ν ω2 /ν — dimensional boundary layer thick-

ness.ε — Stokes-Wormersley number (Eq. (12)).ν — molecular viscosity.L — duct length.ω — rotation frequency.r — radial distance from duct center line.Reδ — Reynolds number (Eq. (1)).Sc — Schmidt number.t — time.Tc — time for concentration interface to

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

layer.Vn — amplitude of the n-th harmonic of V.W(t) — fl ow velocity in the inviscid core.Wm — 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 of produced water from oil production fi elds by membrane pro-cess, Desalination, 222 (2008) 176−186.

[2] C. Fritzmann, J. Lowenberg, T. Wintgens and T. Melin, State-of-the-art of reverse osmosis desalination, Desalination, 216 (2007) 1−76.

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Documents

Theory of isobaric pressure exchanger for desalination