Laminar flow over pipes with injection and suction through the porous wall at low Reynolds number

Journal of Membrane Science 327 (2009) 104–107

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Laminar flow over pipes with injection and suction through the porous wall at low Reynolds number Y. Moussy a,∗ , A.D. Snider b a b

Brunel Institute for Bioengineering, Brunel University, Uxbridge UB8 3PH, UK Department of Electrical Engineering, University of South Florida, Tampa, FL 33620, USA

a r t i c l e

i n f o

Article history: Received 25 September 2008 Received in revised form 6 November 2008 Accepted 8 November 2008 Available online 18 November 2008 Keywords: Low Reynolds number flows Porous pipes Navier-Stokes equations Hollow fibers Analytic solution

a b s t r a c t Analytic expressions describing two-dimensional steady-state laminar flow over an array of porous pipes were developed from the solution of the Navier-Stokes equations for the case of low wall Reynolds number. For flow in the shell (the space separating the porous pipes), the stream function formulation was used to find the radial and axial velocity profiles and pressure distribution. The Navier-Stokes equations in cylindrical co-ordinates reduced to a fourth-order nonlinear differential equation, which was solved for flows through the porous wall using a zeroth- and first-order perturbation method. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The study of flow within and over porous pipe structures is important in fields such as biomedical engineering [1–6], drinking water treatment [7], and processing of beverages [8,9]. The description of the performance in hollow fiber modules has been reported widely, usually on the basis of the particular application for which they are used [10,11]. Most mathematical models only predict the filtrate flux and not the velocity profiles and pressure profiles [12,13]. The velocity profiles and pressure profiles can be obtained by solving the Navier-Stokes equation in the area of interest. From the solution to the Navier-Stokes equation not only can the filtration flux be calculated, but the velocity gradient and shear stress as well. Numerical solutions for the Navier-Stokes equation can be obtained [12], however analytical solutions are few. The nature of the flows in porous pipes is characterized by the wall (or radial) Reynolds number  = vo R/, where R is a characteristic radius, vo is a characteristic velocity in the radial direction, and  is the viscosity. The classical approximate analytic solution for lumenal flow in a porous pipe at both low and high  has been found by Berman [14]. Berman used a perturbation method to solve the Navier-Stokes equation. An analytic solution of the Navier-Stokes equation has also been found for flow on the exterior of a porous pipe for  ≈ 0 [3]. This analytical solution was also determined using

∗ Corresponding author. Tel.: +44 1895265331; fax: +44 1985274608. E-mail address: [email protected] (Y. Moussy). 0376-7388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2008.11.018

the perturbation technique. Since the range for  is approximately 10−1 for water treatment to 10−5 for biomedical applications, the primary objective of this present study was to find an approximate analytical solution by solving the Navier-Stokes equation for flow over a porous pipe with larger Reynolds number, but still in the range  ≤ 1. 2. Analytic solution In this analysis, the pipe arrangement first suggested by Waterland, Robertson & Michaels [15] was employed. The porous pipes are assumed to be arranged in an equilateral triangular array so that each pipe is equidistant from six other pipes (Fig. 1). The outer radius of the pipe is denoted by rm , and the radius of the shell region defining the boundary between adjacent pipes by rs . For further simplification, the triangular voids between the shell regions were neglected [15]. This problem is an approximation of flow in an annulus with injection or suction at the lower boundary and symmetry conditions at the upper boundary (Fig. 1). The assumptions made in the present study were as follows: (1) the fluid was an incompressible Newtonian fluid; (2) the flow was assumed to be laminar; and (3) the fluid flowing through the porous wall was presumed uniform throughout, i.e., the velocity at the wall vw is independent of the axial coordinate x. The flow from one shell region to another was neglected [16]. For two-dimensional incompressible flow in cylindrical coordinates (x,r), the axial (u) and radial (v) velocities can be expressed in

Y. Moussy, A.D. Snider / Journal of Membrane Science 327 (2009) 104–107

105

Fig. 1. Schematic representation of a typical multi-porous pipe unit (upper left). Porous pipes in an equilateral triangular array (upper right). Longitudinal view of a single tube and shell arrangement (lower right).

terms of a stream function ru(x, r) =

∂ ∂r

and

as follows:

− r v(x, r) =

∂ ∂x

The Navier-Stokes equations in the axial and radial directions are: (1)

In 1953, Berman [17] correctly proposed the form of the stream function for a fully porous channel:

 (x, ) = (A + Bx) · f

r2 2 rm

 ≡ (A + Bx) · f ()

(2)

where A and B are constants, f is a dimensionless function, rm is the outer radius of the fiber, and  = (r/rm )2 is the similarity variable [14]. A is determined by the volume rate at an arbitrary inlet (x = 0). The expression of this stream function in terms of the flow rate at the inlet of the shell Qs , the outer radius of the pipe rm , and the radial velocity at the outer radius of the pipe vm was previously derived [3]. (x, r) =

Q

s





+ 2vm rm x · f ()

(3)

With this particular choice of the radial velocity v = −[∂ /∂x]/r becomes a function of  and totally independent of x. Only the dimensionless function f has to be determined.

2.1. Shell flow From the equations above, the velocity components in the axial and radial directions can be expressed in terms of f:

 u(x, r) =

2Qs 2 rm



4vm x  + f () rm f () 

v(x, r) = −2vm √



1 ∂u ∂u ∂2 u 1 ∂P ∂2 u ∂u +v + =− + + 2 u  ∂x r ∂r ∂x ∂r ∂r 2 ∂x

(5)

(6)



u

∂v ∂2 v 1 ∂v ∂v ∂2 v v 1 ∂P + 2 + +v =− + − 2 2  ∂r r ∂r ∂x ∂r ∂x ∂r r

 (7)

Substitution of (4) and (5) into (6) and (7) results in 4 1 ∂P =− 2 −  ∂x rm 1 ∂P 2vm =  ∂ rm





2Qs 2 rm

vm rm

4vm x + rm





f2 2 − ff   2

f  + f  −





vm rm 



(f  − ff  ) 2

− 2f 

(8)

(9)

The right-hand side of Eq. (9) is a function of  only; its xderivative is zero ∂2 P = 0, ∂x∂

(10)

implying that P(x,) has the simple form g(x) + h(). Consequently the derivative of (8) with respect to  is also zero, and cancellation of 2 )[(2Q /r 2 ) + (4v x/r )] results in the ordinary the factor −(4/rm s m m m fourth-order differential equation for f d  2 [f  + f  − (f  − ff  )] = 0, d

(11)

with the Reynolds number  = vm rm /. The four boundary conditions for this flow are as follows: at r = rm , at r = rs ,

(4)



u = 0, v = vm ∂u = 0, v = 0 ∂r

(12)

where rs is the shell radius. Using (4) and (5) to express (12) in terms of f, the boundary conditions may be expressed: f  (1) = f (H) = f  (H) = 0,

f (1) = −

1 2

(13)

106

Y. Moussy, A.D. Snider / Journal of Membrane Science 327 (2009) 104–107

where rs2

H=

(14)

2 rm

Eq. (11) implies f

 

+ f  − (f  − ff  ) = c 2

(15)

where c is a constant. Thus (8) can be integrated to express the variation in pressure in the axial direction: P(x, ) = P(0, ) +

4 2 rm



2Qs x 2 rm

+

2vm x2 rm



c

(16)

The value of c will result from the calculations to be described. 2.2. Perturbation analysis for small Reynolds number If small values of  are treated as a perturbation parameter, a solution of Eqs. (11) and (13) can be obtained by a perturbation series developed near  = 0 f = f0 + f1 + 2 f2 + . . .

The mathematical details of the calculations of f0 and f1 are given in Appendix A. The results are: 1 ( − 1)2 f0 () = − + C0 − C0 H( ln  −  + 1) 2 2

(18)

where C0 =

1/2 2

(H − 1) /2 − H(H ln H − H + 1)

(19)

in agreement with [3]; and f1 () = C1

C0 (−1)2 − C1 H( ln − + 1) + × {(−1 + )[(−1 + ) 2 72

× (18 + C0 (9 + (−4 + ))) + C0 H(176 + 5(−35 + )) + 36C0 H 2 (−13 + 12) + 300C0 H 3 ] + 6H[ ln (−C0 (12 − 12 + 2 + 6(−7 + 6)H + 50H 2 ) + 3 ln (−1 + C0 + 2C0 H(−1 + ))) + 6(1 + C0 (−1 + H(−2 + 7H)))(1 −  +  ln ) × ln H − 12C0 H 2 (ln H)2 (1 −  +  ln )]}

Fig. 2. Velocity profiles versus length in radial direction (Re = 100, x/rm = 10).

(17) 3. Results and discussion The velocity distribution for the main flow in the annular region as calculated from Eqs. (4), (5) and (18)–(21) for  = 0, ±1 is shown in Fig. 2. For the case of fluid being injected through the wall,  = 1, the axial velocity increases and the velocity gradient at the wall (r/rm = 1) increases compared to the  = 0 case. For the case of fluid being withdrawn through the wall,  = −1, the axial velocity and velocity gradient at the wall decrease compared to the  = 0 case. In Fig. 3, the pressure drop in the main flow direction is shown. The pressure drop increases for fluid injection at the wall compared to the  = 0 case. The pressure drop decreased for the case of suction at the wall compared to the  = 0 case. The pressure drop in the shell in the radial direction can be neglected in most practical applications [14]. The third assumption stated in a our analytic solution was, “(3) the fluid flowing through the porous wall was presumed uniform throughout, i.e., the velocity at the wall vw is independent of the axial coordinate x.” Obviously, this condition will not exist in reality as vw is dependent on x. The axial variation can be taken care of using a numerical scheme suggested by Ma et al. [18] or Moussy [3] and is beyond the scope of the present manuscript.

(20)

where C1 = −

C0 {−2 + 4H − 2H 2 − 93C0 H 2 4[−1 − 3H 2 + 4H + 2H 2 ln H]

+ 154C0 H 3 − 82C0 H 4 + 22C0 H − C0 + 4H 2 ln H − 4H ln H − 2H 2 (ln H)2 + 8C0 H 2 ln H − 36C0 H 3 ln H + 58C0 H 4 ln H − 2C0 H 2 (ln H)2 − 12C0 H 4 (ln H)2 + 12C0 H 3 (ln H)2 + 8C0 H 4 (ln H)3 − 4C0 (−1 − 2H + 7H 2 )H 2 (ln H)2 + 4C0 (−1 − 2H + 7H 2 )H 2 ln H − 4C0 (−1 − 2H + 7H 2 )H ln H} (21)

The velocity components in the axial and radial directions are obtained by substituting these expressions for f() = f0 + f1 into Eqs. (4) and (5), respectively. The value of the constant c in the pressure distribution formula (16) is simply c = C0 + C1 .

(22)

Fig. 3. Axial pressure drop versus length in flow direction (Re = 250).

Y. Moussy, A.D. Snider / Journal of Membrane Science 327 (2009) 104–107

The equations that we found here are important in determining the efficiency of porous pipe-type systems in which the fluid flow occurs in the shell region between multiple porous pipes. For example, one can optimize filtration fraction by adjusting the surface area. One can also calculate pressure drop and energy requirements. The determination of these factors will be very important given the growing need for seawater desalination plants.

f0 () = −

1 + 2

As suggested by Eq. (15), the analysis of the differential equation and boundary conditions d  2 [f  + f  − (f  − ff  )] = 0; d f  (1) = f (H) = f  (H) = 0,

1 2

(11,13 repeated)

f  + f  − (f  − ff  ) = c 2

(15 repeated)

or (A.23)

With f expressed via the perturbation series (17) and the constant c expressed via c = C0 + C1 + 2 C2 + . . . ,

(A.24)

equating coefficients of like powers of  in (A.23) yields d [f  ()] = C0 d 0

(A.25)

d 2 [f  ()] = C1 + f0 − f0 f0 d 1

(A.26)

d [f  ()] = C2 − f0 f1 − f0 f1 + 2f0 f1 d 2

(A.27)

(and so on). The boundary conditions to be satisfied from Eq. (13) are for all n,

and

1 f0 (1) = − , fn (1) = 0 2

for n ≥ 1.

(A.28)

Each of these differential equations d/d[f  ()] = F() can be solved via 3 nested integrals with the limits chosen to fit the boundary conditions. For (A.25),



f0 () = C0



d = C0 ( − H), H



f0 () = C0 1 −

 f0 () = C0



1

(so f0 (1) = 0)



H 



1−

(so f0 (H) = 0)

,

H 

(A.29)

(A.30)



(A.32)





3

= C1

1

1 2



2

[C1 + f0 (1 ) − f0 (1 )f0 (1 )] d1 d2 d3 2

H

( − 1)2 − C1 H( ln  −  + 1) 2







+ 1

1

3

1 2



2

H

[f0 (1 ) − f0 (1 )f0 (1 )] d1 d2 d3 2

(A.33)

and (20) follows, with C1 given by (21) to enforce f1 (H) = 0. For convenience in expressing the velocities we display C0 × {−22C0 + 36C0  − 36 − 18C0 2 72

+ 279C0 H − 288C0 H − 648C0 H 2 + 36 + 4C0 3 + 9C0 H2

− 300C0 H 3 ln +72C0 H 2 (ln )2 +144C0 H ln  − 18H(ln )2 + 18C0 H(ln )2 − 36C0 H 2 (ln )2 − 36H ln  − 18C0 H2 ln  + 36H ln H ln  − 36C0 H ln H ln  − 72C0 H 2 ln H ln 

d 2 [f  ()] = c + [f  − ff  ]. d

fn (1) = fn (H) = fn (H) = 0

1 2

+ 648C0 H 2  − 360C0 H 2  ln  − 36C0 H ln  + 180C0 H 2 ln 

can be facilitated through the observation 



f1 = C1 ( − 1) − C1 H ln  +

f (1) = −

1 ( − 1)2 + C0 2 2

in agreement with (18), with C0 chosen via (19) so that f0 (H) = 0. For the first-order perturbation (A.26),

1

Appendix A. Calculation of the perturbation series to order O()

f0 () d = −

1

f1 () =

The authors would like to thank Dr. Art Price for valuable mathematical assistance, and Eric Guegan for assistance with the Figures.



− C0 H( ln  −  + 1), so f0 (1) = −



Acknowledgements



107

d = C0 ( − 1) − C0 H ln , (A.31)

+ 252C0 H 3 ln H ln  − 72C0 H 3 (ln H)2 ln }.

(A.34)

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