Double porous screen element for gas–liquid phase separation

International Journal of Multiphase Flow 50 (2013) 1–15

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

Double porous screen element for gas–liquid phase separation Michael Conrath, Yulia Smiyukha, Eckart Fuhrmann, Michael Dreyer ⇑ Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, Am Fallturm 01, 28359 Bremen, Germany

a r t i c l e

i n f o

Article history: Received 8 May 2012 Received in revised form 18 September 2012 Accepted 8 October 2012 Available online 23 October 2012 Keywords: Woven screen Double screen Bubble point Bubble breakthrough Trapped gas

a b s t r a c t We consider an assembly of two parallel porous screens suspended in a tube at a distance L. The screens are connected by wicking aids. If one screen is brought into contact with a wetting liquid, the other screen will be wetted as well enclosing gas in between. Due to surface tension in the screen pores, the gas can only be removed from the chamber when the pressure difference across one screen exceeds the bubble point. With such a double porous screen element it is therefore possible to block liquid flow using trapped gas as plug. We present a model approach, experiments and numerical calculations on the performance of such a screen element. The model is based on capillary transport in vertical and radial capillaries and allows to predict how fast the element will trap the gas to become operational. For the experiments, we have built such an element using Dutch Twilled weaves made of stainless steel. Placed in a vertical tube and initially dry, it is wetted from below or above and submitted to an increasing pressure difference until breakthrough occurs where the element fails. Corresponding numerical calculations elucidate what happens within the element when it fails. Our results confirm the concept of the double porous screen element and encourage its application as liquid management device. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fig. 1 illustrates the subject of this paper. It is a configuration we call the screen element that serves to block liquid flow with capillary forces. While single phase gas flow can pass through it freely, the screen element will block any flow when it comes in contact with liquid. A possible application of screen elements is inside space vehicles’ liquid propellant tanks from which gas must be vented occasionally to avoid overpressure. To ensure that the liquid propellant remains in the tank, several screen elements could be installed at different locations within the tank so that gas can pass through at least one non-blocked element. Each element, see Fig. 1, consists of two parallel porous screens that are suspended perpendicular to the flow. The distance between them creates a chamber. Between the two screens, a connection with porous material is mounted that we call a wicking aid. Therefore, if one screen comes in contact with liquid, not only this screen is wetted but also the other one, due to the capillary liquid transport in the wicking aid. During the process, gas is trapped inside the chamber. Now, fully wetted and with trapped gas inside the chamber, the screen element is operational and a pressure difference Dp between its two sides can be applied, see Fig. 1. The pressure difference has to overcome the bubble point before gas can break through the ⇑ Corresponding author. Tel.: +49 421 218 57866. E-mail address: [email protected] (M. Dreyer). 0301-9322/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2012.10.003

upper screen in Fig. 1. Up to this threshold, minor amounts of liquid may leak through the wicking aid, but the bulk liquid cannot pass the element. It is only with a pressure difference Dp larger than the bubble point that the element will fail. The advantage of such a configuration is that it is a passive device without any moving parts. Disadvantageous are the leak flow and the irreversibility of the closure. But still, the leak flow would be much smaller than any unblocked liquid flow. We describe it more detailed in Section 2.4. Propellant management is a necessary task, both for storable liquids Dodge (1990, 2000) and cryogenic liquids Cady (1975), Jurn and Kudlac (2006), Kudlac and Jurns (2006), Behruzi et al. (2007), Jurns and McQuillen (2008), and Jurns et al. (2009). The scope of this paper is restricted to the functionality of the screen element under the condition that the element has enough time to saturate and get operational.

2. Model approach 2.1. Problem description The left-hand side of Fig. 2 shows the real device schematically. We presuppose axial symmetry and porous media for lower screen, wicking aid and upper screen. The two circular screens have a radius R0 and are suspended a distance L from each other. Hence, the wicking aid has a length L, too. There are four main questions

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and become operational before it is submitted to a pressure difference across it.

solid, impermeable wall

2.2. Element closure time tc

gas porous screen

p

trapped gas

wicking aid

porous screen liquid

Fig. 1. Configuration of the double porous screen element consisting of two parallel porous screens and a wicking aid that hydraulically connects them. Up to a critical pressure difference Dp, the screen element blocks the passage of liquid by holding a plug of trapped gas in place.

that we want to answer. The first three with an appropriate model, the fourth with numerical simulations: 1. How long does it take until the element becomes operational when either side comes in contact with liquid? We will refer to this as the closure time tc. 2. Up to which pressure difference Dp the element will keep operational? 3. How strong is the leakage flow through the element? 4. What happens above this critical pressure difference?

2.1.1. Assumptions Our considerations extend to the terrestrial cases with liquid from below or above as well as to the application case with gravity absent. We assume that both screens and wicking aid are thin in comparison to the element scale, i.e. ds  R0, see Fig. 2. Here, ds is the effective thickness of the wick which the liquid imbibes due to capillary action. We assume that one of the two screens is wetted immediately. There are other scenarios conceivable for the first contact with liquid such as (i) contact only at the tube circumference due to wall wetting or (ii) contact with liquid drops or globules at any position of the first screen. However, we tackle the sudden contact scenario because these insights apply also to the other scenarios. Furthermore, we assume that the element has enough time to saturate

model geometry

real device z

r

ds

radial capillary porous media

p T ,VT

annular capillary

2.2.1. Annular wicking Hence, the element closure starts with annular wicking, see Fig. 2 right-hand side. Since we assume that ds  R0, the annular capillary in Fig. 2 is linearly imbibed by a wetting liquid. Depending on the direction of gravity g it is either capillary rise (as in the figure) or capillary fall. The capillary flow is described by the equilibrium between capillary pressure exerted by the meniscus that drives the flow and the dissipating effects that consume it. Following Fries and Dreyer (2008) we write the pressure balance as

4r cos h 32lZ Z_ d _ q ¼ qgZ þ þ q ðZ ZÞ þ k? Z_ 2 ; 2 |fflffl{zfflffl} ds dt 2 ffl} d |fflfflffl ffl {zfflfflffl |fflfflfflfflffl{zfflfflfflfflffl} s |fflfflfflfflffl{zfflfflfflfflffl} grav ity |fflfflffl{zfflfflffl}

surface tension

v iscosity

ds

L

g

8 > < þqZ phs ¼ 0 > : qgZ

Fig. 2. Connection between the real device (left-side picture) and model geometry (right-side picture). The model replaces the wicking aid by an annular capillary and the upper screen by a radial capillary. The lower screen is omitted in the model.

inflow

against gravity no gravity gravity assisted:

ð2Þ

The viscosity term in Eq. (1) with the dynamic viscosity l and the advancement velocity of the meniscus Z_ (equal to the flow velocity  through the capillary tube) is a pressure loss due to viscosity pl u according to the DARCY law for porous media which is, see Bear (1988),

ð3Þ

This general relation for porous media includes also the porosity / and the permeability K. For the simple case of tube-like pores with a diameter ds there is a connection between the porous properties, see Fries (2010), as

Z(t) z=0

inertia

ð1Þ

where Z = Z(t) is the time-dependent position of the meniscus, i.e. the wicking front starting from the height z = 0. The surface tension term in Eq. (1) with surface tension r, contact angle h and a static tube diameter ds inherently assumes that the porous medium can be regarded as a capillary tube bundle where all menisci in all pores have the same curvature 4/ds. The gravity term in Eq. (1) with liquid density q and gravity g = 9.81 m/s2 is a hydrostatic pressure phs which is positive for capillary rise (against gravity), zero for absent gravity, and negative for capillary fall (gravity assisted).

@pl lu/ Darcy law: ¼ K @z

R0 R(t)

As soon as either screen comes in contact with liquid, wicking takes over. Due to the capillary forces exerted by the pore-scale menisci, the liquid is pulled into the first screen then into the wicking aid (annular wicking) and then the second screen (radial inward wicking) before the gas is trapped inside and the element is closed. In Section 2.4 below, we provide a clear criterion when the first screen can be neglected or which error we accept in doing so, respectively. For the present analysis however we do neglect it.

2

ds ¼

32K tube-like pores; /

ð4Þ

that can be inserted into Eq. (3). Integration of the pressure gradient obtained in this way along the length of the tube through which liquid flows gives the viscosity term in Eq. (1). The inertial term is the time derivative of the flow momentum. And the inflow term takes the pressure loss at the entrance of the capillary tube into account, k\ being the loss coefficient. An estimation of the or-

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ders of magnitude for the single terms will be done in more detail below where we describe our model. Suffices to say here, that viscous and gravity term are by far the most important while the other two can be neglected without significant loss of accuracy when ds  L, see Fig. 2. Hence, for our problem we only have to solve

4r cos h 32lZ Z_ ¼ qgZ þ ; 2 ds ds

ds r cos h q and bL ¼ ; 8l 32l a Z  ¼ Z=L and t a ¼ 2 t: L

a¼

ð6Þ ð7Þ

The corresponding solution with gravity also given by Washburn (1921) as well as Lukas and Soukupova (1999) then becomes

t a ðZ  Þ ¼

   a a bL Z   ln 1  Z  : bL bL a

ð8Þ

Looking at Eq. (8) it seems reasonable to replace the ratio bL/a by a new parameter which is a BOND number,

Bo ¼

bL qgLds ¼ ; a 4r cos h

ð9Þ

  1 1 Z   lnð1  BoZ  Þ ; ¼ Bo Bo

ð10Þ

with viscosity only impacting on the time scale. The solution without gravity where Bo = 0 dates back to Washburn (1921) and can be deduced from Eq. (10) with a TAYLOR series expansion for small Bo or by solving Eq. (5) anew without the gravity term which gives the LUCAS–WASHBURN solution 2

Z t a ðZ  Þ ¼ : 2

ð14Þ

inflow

The surface tension term in Eq. (14) is unaltered. The gravity term in Eq. (14) accounts for the hydrostatic pressure pg mounting up along the lenght L of the annular section. Hence, analogous to Eq. (2),

8 > < þqgL against gravity pg ¼ 0 no gravity > : qgL gravity assisted:

ð15Þ

For the viscosity term in Eq. (14) we still assume that the liquid flow through screen and wicking aid is laminar and incompressible. Due to mass conservation, the mean radial flow velocity increases as  / 1=r. The viscous friction losses one approaches the center, i.e. u pl consist of the radial part pl,r and the annular part pl,a. The pressure gradient according to the DARCY law, see Eq. (3), is the same for both radial and linear flow, see Bear (1988) for linear and Weitzenboeck et al. (1999) for radial flow. Applying the relation for tube-like pores, see Eq. (4), the radial pressure gradient due to viscosity becomes

 @p 32lu ¼ 2 ; @r ds Z

pl;r ¼

ð16Þ

@p 32lRR_ R dr ¼ ln ; 2 @r R 0 ds

R

R0

ð17Þ

for the viscous pressure loss in the radial section. In the wicking aid we have a constant mean velocity along the flow path that relates to _ a ¼ u  r ðr ¼ R0 Þ ¼ RR=R the actual wicking front speed R_ as u 0 which causes viscous friction losses in the annular section to be

pl;a ¼

ð11Þ

We may summarize the solutions t a ðZ  Þ for different gravity conditions as

 81  1 Z  Bo lnð1  BoZ  Þ against gravity > Bo > < ta ðZ  Þ ¼ 12 Z 2 no gravity > >  :1  1  Z  lnð1 þ BoZ Þ gravity assisted: Bo Bo

v iscosity

     2 d _ R þ d qLR_ R þk? q R_ R : qRRln þ dt R0 dt R0 R0 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

_  ¼ RR=r. with u That leads to

to find that

t a ðZ  Þ

surface tension

inertia

ð5Þ

which is straightforward by applying the separation of variables. We introduce 2 gds L

4r cosh 32lRR_ R 32lL R _ Rþ ¼ qgL þ ln  2 2 |fflffl{zfflffl} ds R 0 d ds R0 s |fflfflfflffl{zfflfflfflffl} grav ity |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Z

L

@p 32lL R _ R: dz ¼  2 @z ds R0

0

ð18Þ

The inertial pressure loss pi (acceleration or deceleration of the flow) in Eq. (14) also consists of a radial part pi,r and an annular part pi,a. We find that

pi;r ¼ ð12Þ

d dt

Z

R

qu r dr ¼

R0

  d R ; qRR_ ln dt R0

ð19Þ

_ Analogously, it is  r ¼ ð1=rÞRR. since u

d dt

Z

L

qu a dz ¼

  d R : qLR_ dt R0

Hence, the capillary rise or fall time, respectively, to reach Z = L or Z⁄ = 1 in the wicking aid (see left-hand side of Fig. 2) becomes

pi;a ¼

 81 1 1  Bo lnð1  BoÞ against gravity > Bo > < ta ðZ  ¼ 1Þ ¼ 12 no gravity > >  :1 1 1  Bo lnð1 þ BoÞ gravity assisted: Bo

The inflow term in Eq. (14) is the pressure loss due to entrance effects into the annular capillary p\ and can be expressed as

ð13Þ

2.2.2. Radial inward wicking When the annular imbibition into the wicking aid has proceeded to the final height L (see Fig. 2), the combined capillary transport begins. The driving meniscus is now in the second screen, giving rise to radial inward wicking, i.e. the wicking front travels from r = R0 inward towards the center at r = 0. In contrast to purely radial wicking, see Conrath et al. (2010), the capillary pressure has to overcome the additional dissipation along the annular section through which the liquid must be transported, too. The force balance, compare also to Eq. (1) for the annular section, now reads as

p? ¼ k?

0

q 2 2

ua ¼ k?



q _ R R 2

R0

ð20Þ

2 ð21Þ

:

To look at the problem in a normalized form, we introduce

R ¼ R=R0 and L ¼ L=R0 ;

ð22Þ

and substitute it into Eq. (14). We obtain

  i 4r cos h 32l d h  _  q gL ¼ 2 R R_  ðln R  L Þ þ qR R ðln R þ L Þ 2 d dt s R0 ds |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} p 1

pr pg

q  _  2 þ k? : RR 2 ffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl p?

pl

i

ð23Þ

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Of course, not all terms are equally important. To reduce the number of terms we perform an order of magnitude estimation. We use the following typical properties and relations to do that: R0  102 m, r  102 N/m, cosh = 1, q  103 kg/m3, ds  105 m, g  10 m/s2, L⁄  1, m = l/q  106 m2/s, k\  1 Lundgren et al. (1964), 0 6 R 6 1; R_   const:  1s1 Conrath et al. (2010), €  R ðlnR þ L Þ þ R_ 2 ðlnR þ L þ 1Þ;0 P R  lnR d=dt½R_  R ðlnR þ L Þ ¼ R P 0:37 for the specified R⁄-range. Considering all terms in Eq. (23) we thus find

pr  pg ¼ pl þ pi þ p? : |{z} |{z} |fflfflfflffl{zfflfflfflffl} |{z}

Oð107 —108 Þ

N m4

Oð108 Þ

N m4

Oð103 Þ

N m4

ð24Þ

Oð103 Þ

N m4

Looking closer at the single terms in Eq. (23) we find that the inertial term becomes infinitely negative in the moment of closure when the wicking front reaches the point on the center axis and the liquid flow is suddenly brought to a halt. But over the course of the whole process it is clearly viscosity which is dominant as stated in Eq. (24). Consequently, using a and bL according to Eqs. (6), (23) reduces to

a  bL R20

¼ R R_  ðln R  L Þ:

ð25Þ

The left-hand side of Eq. (25) is constant, we may therefore replace it by

C¼

8 ab L > > > R20 < a R20

> > > : aþbL R20

against gravity no gravity

ð26Þ

gravity assisted;

and applying the separation of variables we find

Ct ¼ R2

  ln R 1 L  R2 þ C 0 :  4 2 2

ð27Þ

We introduce the normalized time

t r ¼

4a R20

t:

¼

L Z  LþR 0 L LþR0

for Z  6 1; R ¼ 1

R0 þ ð1  R Þ LþR 0

for Z  ¼ 1; R 6 1:

ð32Þ

The time to reach the end of the annular section then is

 t el X el ¼

L L þ R0



81  1 > < Bo 1  Bo lnð1  BoÞ against gravity ¼ 4L2 12 no gravity >  :1 1 1  Bo lnð1 þ BoÞ gravity assisted; Bo ð33Þ

see also Eq. (13). The total closure time

t c

becomes

8 2   2L þ1 4L 1 > > < Bo 1  Bo lnð1  BoÞ þ 1Bo against gravity  t c ¼ tel X el ¼ 1 ¼ 2L2 þ 2L þ 1 no gravity > > : 4L2 1  1 lnð1 þ BoÞ þ 2L þ1 gravity assisted: Bo Bo 1þBo ð34Þ To illustrate the explanatory power of our model we will take a look on two situations now: (i) constant wicking distance, i.e. L + R0 = constant and (ii) constant closure time t c . These two situations shall be examined in the light of our three possible gravity  conditions. Fig. 3 shows the position of the wicking front t el X el both in the annular and radial section according to Eqs. (31) and (32). The case with no gravity is the bold line, the expected deviations for ground tests have thinner lines and are each denoted with the corresponding BOND number Bo. Obviously, gravity assisted wicking can considerably reduce the closure time while wicking against gravity may even come to a halt. To normalize measurements, it seems useful to apply the closure time for the ’no gravity’-case. Fig. 4 is a plot of Eq. (34) relating tc to L⁄ and Bo. Since Bo, see Eq. (9), contains L, we use Bo/L as parameter to get an explicit relation. The closure time logically increases with L⁄ because t⁄ is based on constant R20 , see Eqs. (28) and (31) – growing L therefore means growing wicking distance L + R0. An exception (not plotted) is the gravity assisted closure for Bo > 1. But that case is beyond the scope of the actual subject and was therefore not included.

ð28Þ

Setting as boundary condition t r ðR ¼ 1Þ ¼ 0 and replacing bL/a by the BOND number, see Eq. (9), we arrive at

t r ¼

( X el

1 ½R2 ðln R2  1  2L Þ þ 2L þ 1: 1  Bo

annular section

ð29Þ

Eq. (29) allows to predict the duration of the radial inward wicking process yielding

t r ðR ¼ 0Þ ¼

1 ½2L þ 1: 1  Bo

ð30Þ

L=1

2.2.3. Closure time To look at the whole process of element closure, we now summarize our findings for the annular and radial inward wicking phase. In order to do that, we combine the two time scales involved, see Eqs. (7) and (28), to an adequate time scale of the whole process. It seems reasonable to use the same time scale as for the radial wicking because in the case of no gravity and no wicking aid, i.e. Bo = 0 and L = 0, the so normalized closure time will become unity. Hence

( t el

¼

4L2 t a

for Z  6 1; R ¼ 1

t r

for Z  ¼ 1; R 6 1;

ð31Þ

see also Eqs. (7), (22) and (28) for the definitions of ta ; L and t r . The wicking length of the whole closure process is Xel = L + R0, hence we define

radial section

Fig. 3. Capillary transport in the annular and radial section, respectively, for the case L⁄ = 1. There is an obvious gravitational impact. If the wicking aid has to work against gravity, Bo = 1 marks the limit where the hydrostatic pressure equals the capillary pressure. While the element is unable to close itself for Bo = 1 against gravity, at inverse gravity condition the element will close in about half the time compared to no gravity.

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uL leakage

z

p

p

pa

ua

ds

R0

nw

p uL

Fig. 5. Leakage flow through the wetted screen element caused by the applied pressure difference Dp (left-hand side) and flow channels through the wicking aid to which the leakage flow is contained.

Fig. 4. Closure time t c in dependence of aspect ratio L⁄ and wicking parameter Bo/L according to Eq. (34). Since L is contained in Bo, see Eq. (9), we chose Bo/L as plotting parameter.

2.3. Bubble point and gas compression Beside the leakage, another implication of applied pressure difference Dp to the element is that the trapped gas is compressed. The degree of compression increases with the applied Dp. Furthermore, there is a critical pressure, the bubble point, at which the trapped gas will finally break through the pores. 2.3.1. Bubble point An essential feature of the screen element is the bubble point D pbp of each screen. Assuming gas and wetting liquid on either side of the screen, the bubble point denotes the capillary pressure at the pore menisci that must be overcome to press gas through the screen. According to Dodge (2000),

Dpbp ¼

4r cos h : Dbp

ð35Þ

Dodge (2000) also provides experimentally gained values of Dbp for a number of standard screens. But when the bubble point pore size Dbp is not known a priori, one can estimate Dbp with the retention pore size Dp. This is provided by suppliers of filter media and marks the maximum diameter of spherical particles that are capable to pass the screen. Then

Dpbp ¼ C 0 Dp0 ; 4r cos h Dp0 ¼ : Dp

ð36Þ ð37Þ

see for example Kopf et al. (2007) and Conrath and Dreyer (2012), where C0 is a correction factor, Dp0 is the approximate bubble point prediction. Depending on shape and geometry of the pores as well as on the speed and way of compression, the bubble point will vary. This is expressed by C0. Recent works, see Schuetz et al. (2008) and Conrath and Dreyer (2012), have tackled the dynamics of the bubble point because what is known as the static bubble point Dpbp is only the ideal case of negligible dynamics. Conrath and Dreyer showed for a woven metal screen like the ones used in the present study, that the bubble point for a wide range of compression speeds

exceeds the static bubble point by no more than 10%. But still, a comprehensive study on the dependence of C0 from pore geometry and compression characteristics does not yet exist. Hence, to know C0 requires reference experiments for each case. To normalize the pressure signal in a measurement Dpm(t), it seems most useful to relate it to Dp0 since it can be easily calculated with data specified by the suppliers of screen material and test liquid. 2.3.2. Compression of the trapped gas The gas inside the element gets trapped at a certain absolute pressure pT. We assume an ideal gas at isothermal conditions. Then the trapped gas volume VT relates to the absolute pressure as

V T pT ¼ const: ¼ pT0 V T0 jDp¼0 :

ð38Þ

Hence, if we submit the element to an additional pressure difference Dp, it is

V T ðDpÞ ¼ V T0

pT0 : pT0 þ Dp

ð39Þ

Negative pressure differences (suction of a two-phase flow through the element) will cause an expansion of the gas, positive pressure differences (pressing a two-phase flow through the element) will cause a compression of it. In the applications that motivated this study, the absolute pressure is a few 105 Pa. Maximum pressure differences are limited by the bubble point as explained below and are at about 103 Pa. Consequently, in these applications, the compression or expansion will hardly become more than 1%. Nevertheless, this effect should be kept in mind since it would be much stronger in other circumstances (low pressure environment, porous media with tinier pores). 2.4. Leakage flow When the element is successfully closed, it holds the trapped gas plug in place as depicted by Fig. 1 thus preventing further liquid flow through the tube. However, due to the hydraulic connection given by the wicking aid, an applied pressure difference Dp between the two sides of the element will cause a leakage flow uL. It can also be referred to as a DARCY flow because it is basically described by the DARCY law, see Eq. (3). We have to distinguish between orthogonal cross flow through the two circular screens and longitudinal flow through the capillary channels of the annular wicking aid as shown in Fig. 5. The applied pressure difference Dp that causes the leakage flow can thus be divided as

Dp ¼ Dpa þ 2Dp? :

ð40Þ

The cross section to which the leakage flow is confined is Aa and pictured at the right-hand side of Fig. 5. The cross section of the empty tube is denoted as A0. The connection between the two cross

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sections and corresponding velocities is given by mass conservation, i.e. the constancy of leakage flow rate QL,

Q L ¼ uL A0 ¼ ua Aa ;

(a) dry screen

(b) gas trapping

(c)

(d)

leakage

breakthrough

ð41Þ

with

A0 ¼ pR20 ;

Aa ¼

pR0 p nw 2

2

ds ;

ð42Þ

being the flow cross sections of tube and wicking aid, respectively, where nw is the distance between two capillary tubes in the wicking aid, see Fig. 5. The annular flow through the wicking aid can be treated as pipe flow as we already did before, compare to the viscosity term in Eq. (1). Since all the channels of length L are now filled, the pressure drop along it becomes

Dpa ¼ C a;l ua ;

C a;l ¼ 32

lL 2

ds

ð43Þ

:

The cross flow pressure drop Dp\ depends on the type of porous medium. The velocity of the cross flow is approximated to be ua. Generally, it is then

Dp? ¼ C ?;l ua þ C ?;q u2a ;

ð44Þ

C\,l and C\,q representing the pressure drops due to viscosity and convection, respectively. In the experiments we describe later, we use woven metal screens as porous media, both for wicking aid and circular screens. For Dutch Twilled weaves used in the experiment, the two coefficients in Eq. (44) become Dodge (2000)

C ?;l ¼ K 1

lBs D2bp

;

C ?;q ¼ K 2

qBs Dbp

Dutch Twilled weaves;

ð45Þ

where l and q are the liquid dynamic viscosity and density, respectively, Bs is the screens thickness, Dbp is the bubble point pore size and K1 and K2 are empirical constants. Put together, the connection between Dp and the annular leakage flow velocity ua is, see also Eq. (40),

Dp ¼ ðC a;l þ 2C ?;l Þua þ 2C ?;q u2a ;

ð46Þ

or

C a;l þ 2C ?;l ua ¼  þ 4C ?;q

bubble point limit

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   C a;l þ 2C ?;l 2 Dp : þ 2C ?;q 4C ?;q

ð47Þ

The observable leakage flow velocity uL is found by applying Eqs. (41) and (42). A criterion for the neglect of the cross flow pressure drop in given when Dp\  Dpa, see Eqs. (43) and (45). 3. Performed tests 3.1. Liquid from below The idea of this test is to measure the screen elements capability to block liquid under increasing pressure difference, the liquid coming from below. The increasing pressure difference eventually leads to a partial breakthrough of the trapped gas inside the element which means partial failure of the elements functionality. 3.1.1. Test scheme The test is illustrated by Fig. 6 and can be divided into four phases. First, the double screen assembly is dry as shown in Fig. 6a. During this phase, the liquid level rises below the screen assembly towards it. In the second phase, see Fig. 6b, the liquid has wetted the lower screen. Due to the connection between the screens this leads also to wetting of the upper screen and hence the trapping of gas in the assembly. In the third phase, see Fig. 6c, the pressure difference across the screen assembly is smoothly increased towards the bubble point pressure. If any leak-

Fig. 6. ’Liquid from below’-test. Starting with a dry screen assembly (a), the test liquid rises towards it from below. Contact is established in (b) with the result of gas trapping between the screens. In (c), an increasing pressure difference is applied to the screen assembly until the trapped gas finally breaks through the upper screen (d).

age occurs it can be observed in this phase. Phase four, see Fig. 6d, corresponds to the surpassing of the bubble point threshold where breakthrough occurs, at least partial. 3.2. Liquid from above Another testing method for the functionality of the screen element is to ‘stack’ liquid on top of it. Once wetted, the screen element should be able to withstand a certain liquid height. However, there is an important difference compared to the former test with liquid from below. Because even with the gas chamber half filled with liquid, more liquid from below can press gas through the opposite screen. But liquid from above cannot. Buoyancy will always lift the trapped gas towards the upper screen. Consequently, in the ’liquid from above’ test, even at the bubble point, the remaining gas cannot undoubtedly pressed out of the chamber. 3.2.1. Test scheme This test is illustrated by Fig. 7 and can also be divided into four phases. First, the double screen assembly is dry as shown in Fig. 7a. Liquid is dripped from above. In the second phase, see Fig. 7b, the dripping liquid wets the screen assembly and thus traps gas between the two screens. In the third phase, see Fig. 7c, the pressure difference across the screen assembly is smoothly increased by adding liquid above the assembly. Phase four, see Fig. 7d, becomes complicated because buoyancy presses the trapped gas against the upper screen. As the pressure difference is increased, the bubble is compressed to a smaller volume. Depending on the extend of this compression, the bottom of the chamber may be still covered with gas or possibly liquid. If there is still gas, then a gas breakthrough will occur as the pressure difference overcomes the bubble point threshold. This situation is pictured in Fig. 7d. If, on the other hand, the bottom of the chamber is covered with liquid then nothing will happen except that the leakage rate steadily increases with the applied pressure difference. 4. Experiments 4.1. Setup 4.1.1. Screen element We have built a double porous screen element using Dutch Twilled woven screen as porous material. Such weave is composed

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15

(a)

dry screen

(b)

gas trapping

(c)

(d)

leakage

breakthrough

bubble point limit

Fig. 7. ‘Liquid from above’-test. Starting with a dry screen element (a), the test liquid drips from above on the upper screen thus wetting the element and trapping gas inside it (b). In (c), an increasing pressure difference is applied to the element by adding liquid. Increasing the absolute pressure will cause the trapped gas volume to shrink resulting in correspondingly increasing leakage rates. Because buoyancy presses the gas against the upper screen, it might not break through the lower screen, even as the bubble point is surpassed (d).

of thin stainless steel wires of two different diameters. The appearance of this weave type is shown in Fig. 8. The pores have a twisted but very regular shape and such weaves can be purchased in a number of standard configurations. In our case, the weave has 165 straight warp wires per inch and 800 weft wires per inch. It should be noted that most commercial manufacturers categorize the screen weave based on the number of wires per inch. One inch converts to 25.4 mm. The properties of Dutch Twilled 165 800 weave are listed in Table 1. The screen element we have built with this weave is illustrated by Figs. 9 and 10. There are two parallel screens in a distance L = 10 mm from each other. They are circular with a radius R0 = 17 mm. The wicking aid is made from the same weave. It is basically a strip of 2pR0 length with a number of contact leaves on both edges. As shown by Fig. 10, the strip is bent and held in place by an acrylic glass ring. The contact leaves are then folded towards the center before lower and upper screen are pressed against it. The two circular screens are suspended on metal rings. Put together, the three components of the screen element, see Fig. 10, are integrated into the test facility.

4.1.2. Test facility Our test facility comprises a hydraulic circuit in which the screen element is suspended perpendicular to the liquid flow in a vertical tube. For the test liquid, we use Hexamethyldisiloxane,

7

see Table 2, which is a silicon oil distributed as SF0.65 by Dow Corning (USA). As test gas we use ambient air. There is a different hydraulic circuit for each of the two tests as shown later on. But both circuits use some components alike. There is a pump that delivers up to 120 ml/s. A filter is used to remove particles from the liquid. It is custom-made and transparent, its core piece is a Dutch Twilled weave 325 2300 with a particle retention size of Dp = 5 lm. For the ’liquid from below’ tests, the liquid then flows into a pressure control reservoir with a cross-sectional area of about 3 102 m2. The height of the liquid level in this reservoir determines the pressure difference across the element and can be increased by about 4 mm per second. For the ’liquid from above’ test, liquid passes through the filter onto the test section and provides the pressure driving the flow. In the ’liquid from below’ test, the pressure control reservoir is the test section with a vertical tube having an inner radius of 17 mm and in which the screen element is suspended. The screen element has three feedthroughs above, between and below the two screens, respectively, for measurement of pressure differences. Two pressure difference sensors are Sensotec Z/1309-32, supplied by Althen GmbH, Kelkheim (Germany) with a measurement range of Dp = (3.5 to +3.5) ± 0.025 kPa. It should be noted that the measurement of pressure differences in the special environment of our experiment is difficult because numerous menisci interact here. That is why the recorded signals have to be interpreted carefully as we will elaborately do below when we present the results. Experiment control and data acquisition are both controlled from a personal computer. Software controls (i) the liquid pump rate and (ii) timing and amount of recorded data points. 4.1.3. Closure time The closure time of the element tc (in real dimensions) can be predicted using Eq. (34) since

tc ¼ t c

R20 ; 4a

ð48Þ

see also Eqs. (28) and (31). To this end, we first need to calculate a according to Eq. (6), Bo according to Eq. (9) and L⁄ according to Eq. (22). The static diameter ds in those equations is replaced by the pore size Dp as provided by the supplier of the screen material. Looking at the weave composition, see Fig. 8, this seems a reasonable choice both for the effective capillary tube diameter ds as well as the bubble point pore size Dp, see also Eq. (37). Hence,

a¼

Dp r cos h ; 8l

Bo ¼

qgDp L ; 4r cos h

ð49Þ

and using the element dimensions L = 10 mm, R0 = 17 mm, Dp from Table 1 and the liquid properties from Table 2 we find that

Fig. 8. Structure of the Dutch Twilled woven screen as shown by a CAD model (left) and REM view on the screen in flow direction (right).

8

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15

Table 1 Properties of Dutch Twilled weave 165 800. Weave material Weave type Warp wire diameter (lm) Weft wire diameter (lm) Tortuosity (of the weft wire) Weave thickness (lm) Porosity () Pores per area (1/cm2) Particle retention size (lm) Bubble point pore size [Dodge2000] (lm) Surface to volume ratio (mm1) Cross flow constant [Dodge2000] Cross flow constant [Dodge2000]

s Bs / N Dp Dbp S K1 K2

Stainless steel AISI 304 Dutch Twilled 165 800 70 50 1.28 170 0.463 10,200 25 ± 1 23 ± 1 39.62 9.42 63.18

4.1.4. Bubble point According to Dodge (2000), the bubble point pore size for this special weave is Dbp = 23 lm, see also Table 1. With the surface tension and contact angle as provided by Table 2, Eq. (35) thus yields

Dpbp ¼ ð2:77  0:43Þ kPa Dodge prediction:

ð53Þ

However, there are differences among the weaves from different suppliers. We therefore prefer to use the retention pore size Dp as provided by the supplier, see Table 1, to obtain a normalized pressure p0 for Eq. (37) that will be used later on to normalize our measurements. It is

Dp0 ¼ ð2:55  0:40Þ kPa:

ð54Þ

4.1.5. Trapped gas compression The volume and absolute pressure in Eq. (38) are approximately with V T0 ¼ pR20 L and the typical ambient conditions

V T0 ¼ 9:1 ml;

pT0 ¼ 101:3 kPa:

ð55Þ

The maximum compression of the trapped gas is limited by the bubble point pressure. Therefore, with Eqs. (39) and (53) we find that L=10mm 2R=34mm

V T ðDp ¼ Dpbp Þ ¼ 0:973V T0 or DV T =V T0 ¼ 2:7%:

ð56Þ

Expressed as height difference with the initial height being L = 10 mm, this makes

DTW 165x800

Dz ¼ 0:027L ¼ 270 lm ¼ 1:6Bs ;

ð57Þ

Fig. 9. Configuration of the double porous screen element. It consists of two circular screens that are parallel to each other and a strip that connects them and serves as wicking aid. The strip is bend along the circumference, a number of contact leaves ensures the hydraulic connection.

where Bs = 170 lm is the thickness of one screen, see Table 1, for comparison.

a ¼ ð10:5  2:5Þ105 m2 =s; Bo ¼ ð29:8  4:5Þ103 ; L

4.1.6. Leakage flow Setting ds = Dp, we find with Eqs. (41) and (42) that

¼ 0:59:

ð50Þ uL ¼ ua

The prediction of t c and tc with Eqs. (34) and (28) therefore yields

8 > < 2:92 against gravity t c ¼ 2:89 no gravity > : 2:83 gravity assisted; 8 > < 2:01s against gravity t c ¼ 0:69st c ¼ 1:99s no gravity ! norm time > : 1:95s gravity assisted:

ð51Þ

p D2p 2nw R0

:

ð58Þ

For the maximum leakage we set Dp = Dpbp = 2.77 kPa in Eq. (47). The constants Ca,l, C\,l and C\,q to calculate ua are found with Eqs. (43) and (45) as well as data from Tables 1 and 2. We obtain

C a;l ¼ 25:3104 kg=ðm2 sÞ; C ?;l ¼ 149:8104 kg=ðm2 sÞ; C ?;q ¼ 35:5104 kg=m3 : ð52Þ

Later on we will use the predicted closure time for the ’no gravity’case, i.e. tc = 1.99 s, to normalize the time axis in our experimental results.

ð59Þ

Inserting these into Eq. (47), we find that

ua ðDpbp Þ ¼ 0:9 mm=s;

ð60Þ

and with Eq. (59) the leaking velocity uL in the tube, see Fig. 5, is expected to be

Fig. 10. Photograph of the three components of the double porous screen element before integration into the test facility.

9

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15 Table 2 Properties of SF0.65 silicone oil according to supplier (Dow Corning) at 20 °C. The errors of density and surface tension are an assumption made by the authors since these values are not specified by the supplier. Dynamic viscosity l (Pas) Density q (kg/m3) Surface tension r (N/m) Contact angle h (°)

(495 ± 49.5)106 761 ± 7.6 (15.9 ± 1.6)103 0

N

O L Fig. 11. Hydraulic circuit for wetting from below. At test beginning, liquid is only in the main reservoir.

D,E,F,G H

uL ðDpbp Þ ¼ 0:34 lm=s:

ð61Þ

A

K

I

B

M

J

C

4.2. Liquid from below 4.2.1. Hydraulic circuit and procedure Fig. 11 shows the screen element integrated into the test facility. Prior to the test, liquid is only in the main reservoir. To start each test with a dry screen element, pressurized dry air is blown through the element for some minutes. Visual observation, a typical sound and sensor monitoring indicate whether the element is already dry. With the element dry and all switches closed, test liquid is pumped from the main into the pressure control reservoir to the same height as the lower screen of the element, i.e. H = 0. Then the pump is stopped and the switch between pressure control reservoir and the test section with screen element is opened. Since this is a u-tube, the liquid below the screen will rise no further than H = 0 (viscous damping is high enough to clear out oscillations). Besides, the volume of the connecting tubing is empty at test beginning and has to be filled so that the liquid rise velocity (about 3 cm per second) slows down approximately two centimeters below the lower screen. The remaining distance is passed much slower and only due to the flow from the elevated main reservoir leaking through the still standing pump into the pressure control reservoir and then into the test section. Eventually, the liquid reaches the element and wets it. If that has happened, the pump is started again, now with a defined flow rate, to refill and raise the liquid level in the pressure control reservoir. Correspondingly, the pressure difference across the screen element rises as well. Because of leakage through the wicking aid, the liquid also rises above the upper screen (about one millimeter per second). Eventually, the liquid reaches the overflow which is ten centimeters above the upper screen. The rising pressure difference provokes a breakthrough of trapped gas when the bubble point threshold is surpassed. When part of the gas breaks through, the rest remains trapped in the element. As a consequence, the leakage rate is much higher then but still below that of single-phase liquid flow. The limit of the applicable pressure difference is reached when the

Fig. 12. Chronology of the ’liquid from below’-test. A schematic view of the test section (upper part) shows the position of liquid menisci during the test (A–K). Below is a typical measurement in real dimensions showing the related signal Dp(t). The additional letters (L–O) in the signal plot can be attributed to other parts of the setup, see Fig. 11.

pressure control reservoir is filled to its maximum height. In that stage, the high leaking flow slowly dissolves the remaining gas inside the element. Since the overflow of the test section has no recycling connection to the main reservoir, the pump has to be switched off after some time which marks the end of a test run. Then the liquid has to be removed from most parts of the facility to prepare the next run. 4.2.2. Explanation of measurement Fig. 12 shows the core of the test section in detail (upper part) and an exemplary measurement (lower part). As shown in the detailed sketch of the test section, there are two pressure differences recorded: Dp1 across the lower screen and Dp2 across the upper screen. Each pressure difference sensor has two feedthroughs to the test section, the sensor line between feedthrough and central membrane inside the sensor being filled with test liquid all the time. But right at the feedthrough it makes a difference wether the test section is filled with liquid or not. In case of absent liquid there will be a meniscus with a corresponding capillary pressure that pulls on its side of the membrane. Moreover, with no liquid in the test section, the hydrostatic pressure due to the liquid in the sensor lines is not balanced. Hence, when we begin our test (state ‘A’ in Fig. 12), we start with negative pressure differences Dp1 and D p2. As the rising liquid reaches the lower feedthrough of the Dp1-sensor (state ‘B’), ‘the meniscus there vanishes. The feedthroughs have a diameter of 2 mm which corresponds to an expected pressure jump at ‘B’ of about 0.2 hPa. This is in agreement

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15

2

1

10

4.2.3. Results When we normalize the course of Dp(t) with Dp0 = 2.56 kPa and tc = 2 s, see Eqs. (52) and (54), we obtain what is shown in Fig. 13. At t⁄ = 0 the liquid comes in contact with the lower screen. Begin-

2

with the sensor observations. During ‘C’ the liquid rises further thus compensating the hydrostatic pressure in the sensor lines of the Dp1-sensor. At ‘D’ the liquid touches the lower screen and in a matter of seconds it rises towards the upper screen (‘E’), crosses the next feedthrough causing another pressure jump (‘F’) and reaches the middle of the upper screen (‘G’) – thus trapping gas inside the now wet screen element. Now, with the screen element operational, the pressure difference across the element can be increased. This is done by switching on the pump in (‘H’). As a consequence of the rising D p, leakage through the element occurs and elevates the liquid level even above the upper screen. When the leaking liquid reaches the uppermost sensor feedthrough (‘I’), another pressure jump is observed in Dp2. And the slope of Dp2 becomes now less steep because both sensor lining and test section are now filled with liquid. At ‘J’ the liquid reaches the overflow. Pressing the meniscus over the edge gives rise to a counter pressure against that from the pressure control reservoir which can be seen on both Dp1 and D p2. When the overflowing liquid passes the edge of the overflow, another meniscus pressure jump is observed. At this one, the absolute pressure on the trapped gas inside the element suddenly relaxes a bit leading to a jump up in Dp1 and a jump down in D p2, respectively (’K’). As the liquid level in the pressure control reservoir is risen even further, gas breakthrough occurs eventually (‘L’). Only a fraction of the trapped gas breaks through before the breakthrough ends, the rest of the gas remains inside. Inside the element, the gas now forms a trapped bubble around which a much higher leakage flow causes an obvious pressure loss in Dp1 due to the hydraulic resistance of the screen. With further increasing liquid height in the pressure control reservoir, also the additional pressure losses due to the stronger leakage flow are compensated and partial gas breakthrough happens again a few times, for example in ‘M’. At ‘N’, the liquid height in the pressure control reservoir cannot be further increased. Although very slowly, the gas of the trapped bubble dissolves in the leakage flow causing the bubble to shrink. That, in turn, widens the open area of the screen for the leakage flow and the pressure differences decrease before finally in ‘O’, the connection between pressure control reservoir and test section is interrupted.

Fig. 14. Zooming in at Fig. 13: wetting of the screen element in the ‘liquid from below’-test. First liquid contact with the lower screen is indicated by a downward step in Dp1 . Closure of the element is marked by an upward step in Dp2 .

1

Fig. 13. Normalized measurement of the ‘liquid from below’-test. As the pressurization of the element increases, only Dp2 rises because only the upper screen is ‘plugged’ with the trapped gas while the lower screen is not. See also Figs. 14 and 15 for more details.

Fig. 15. Zooming in at Fig. 13: breakthroughs of the trapped gas at the upper screen in the ‘liquid from below’-test. At each breakthrough a part of the trapped gas is lost before the breakthrough stops. And each time, the leakage rate cross the element increases step-like as can be seen in the Dp1 -signal.

ning and end of the wetting can clearly be recognized in the pressure signals as is highlighted by zooming in at t⁄ = 0, see Fig. 14. When the liquid has reached the lower screen and touches it, a downward step in Dp1 is observed. This initial step is caused by the capillary suction of the weave that suddenly sets in. It should be mentioned that first contact between liquid and screen does not really occur piston-like on the whole area of the lower screen. Instead, the slowly elevating liquid goes always a few millimeters ahead at the tube wall. That is why first contact will take place rather at the screens circumference. From there it spreads not only upward in the wicking aid but also inward to the center of the lower screen. If the inward wicking at the lower screen is faster than the elevating liquid then a gas bubble will be trapped below the lower screen as in our case. And as the lower screen is closed, the growing hydrostatic pressure of the liquid flow becomes visible. That happens at the big upward step at t⁄ = 1 in Fig. 14. As indicated by Eqs. (13) and (30), the capillary rise to L = 10 mm is faster than the inward wicking from R0 = 17 mm to the screen center. Therefore, when the lower screen ‘sees’ the pressure difference at t⁄ = 1, the upper screen is already wetted to some extent. Its closure is completed at about t⁄ = 1.4 in Fig. 14. From now on, Dp2 can ‘see’ pressure, too, which is marked by a distinct step in Dp2 . Going back

11

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15 Table 3 Overview on the outcome in all 12 test runs with liquid from below. The closure time is in the first row because the initial wetting procedure was the same for all twelve tests. The parameter of all tests was the slope of the pressure difference across the element which is provided in the second row. The third row provides the measured bubble points. Runs

t c

d/dtDp⁄⁄104

DpBP  102

1–4 5–8 9–12 P

1.7 ± 0.1 2.1 ± 0.4 1.4 ± 0.1

30.5 ± 0.6 43.7 ± 3.2 58.5 ± 0.6

57.5 ± 0.3 57.2 ± 0.9 56.7 ± 0.6

1.8 ± 0.3

Parameter

57.2 ± 0.7

A D F

B C

F

A

C B 1

D Fig. 16. Hydraulic circuit for wetting from above.

to Fig. 13, we next increase Dp⁄ monotonically with a rate d/dt⁄Dp⁄. The liquid that presses now on the element from below can leak through the first screen which is why Dp1 remains low. Dp1 only shows the hydraulic loss of the leak flow through the screen. Part of the leakage flow will creep through the wicking aid and the upper screen and cause a smaller leak flow signal also in Dp2 . But much more important is that the continuously increasing pressure on the trapped gas is only seen by Dp2 alone. Eventually, the pressure difference across the upper screen reaches the bubble point threshold DpBP and gas is pressed through (a few of) the screen pores. To this breakthrough we have zoomed in Fig. 15. Each time a breakthrough occurs, only a part of the trapped gas is lost. Obviously, see Conrath and Dreyer (2012), what remains in the chamber is a trapped bubble that covers not the whole area of the upper screen anymore. Consequently, the leakage flow can distinctively increase. As can be seen by the upward step in Dp1 at each breakthrough. But this additional leakage flow causes additional hydraulic losses not only across the lower screen but also in the tubing between pressure control reservoir and screen element. Therefore, the hydrostatic pressure in the pressure control reservoir acting on the element gets diminished by all these losses at each breakthrough which, in turn, can stop before the additional losses are compensated and the next breakthrough occurs. It can be seen in Fig. 15 that each breakthrough occurs at a higher level than its preceding breakthrough pressure Dp1 . This should be attributed to the growing dynamic pressure of the leakage flow that only affects the upper sensor feedthrough of the Dp2-sensor while the flow at the lower sensor feedthrough is calmed down by the wicking aid. Altogether, we have performed 12 ‘liquid from below’-tests that are summarized in Table 3. Parameter in the test was the compression rate as provided in the second row. With each compression rate, a test was done four times. The observed closure times, provided in the first row, have been obtained by zooming into the pressure signals analogous to Fig. 14. The third row provides the measured bubble points from the first breakthrough.

F E

Fig. 17. Chronology of the ’liquid from above’-test. A schematic view of the test section (upper part) shows the position of liquid menisci during the test (A–F). Below is a typical measurement in real dimensions showing the related signal Dp(t).

4.3. Liquid from above 4.3.1. Hydraulic circuit and test procedure Fig. 16 shows the element integrated into the test facility adapted for the ‘liquid from above’-test. The lower part of the test section is now open to have defined pressure conditions on both sides of the element. Prior to the test, here too, liquid is only in the main reservoir. To start with a dry screen element, pressurized air is attached from below and blown through the element. After this preparation the pressurized gas is removed and an adequate dish is placed below the test section to collect the liquid that leaks through the element during test. Now we start the test by carefully emptying a syringe along the test section wall above the screen element. Thus, the upper screen is wetted by a film flow rather than impacting drops. After observing and waiting a few extra seconds to ensure that the element is wet, we switch on the pump at a defined flow rate. Also the pump increases the liquid level above the screen element with a film flow. At a critical liquid height (about 20 cm), the screen element fails and breakthrough occurs. The liquid level then falls to about one centimeter and remains there until the pump is switched off. 4.3.2. Explanation of the measurement Fig. 17 shows a typical measurement for the ‘liquid from above’test. See also the upper part of Fig. 12 and the explanations for the

12

M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15 Table 4 Overview on the outcome in all 12 test runs with liquid from above.

Fig. 18. Normalized measurement of the ‘liquid from above’-test. As the pressurization of the element increases, only Dp1 changes because only the lower screen is ‘plugged’ with the trapped gas while the upper screen is not.

‘liquid from below’-test for the explanations that follow. At the beginning (‘A’), we have negative pressure differences because the test section is empty while the sensor lines are filled with liquid. At ‘B’ the upper screen is wetted and within an instant the liquid reaches the sensor feedthrough between the two screens effacing the meniscus there. That is why Dp1 decreases slightly while Dp2 increases slightly. During ‘C’, there is no pressure change and flow. This is an operational pause to ensure that the element is completely wetted and functional. Then pumping resumes. Dp2 does not change since the liquid just flows through the upper screen. All building pressure due to the growing column of liquid is carried by the menisci in the pores of the lower screen causing Dp1 continuously to fall. At ‘E’ breakthrough occurs and the element can no longer fulfill its purpose. During ‘F’ the liquid column above the element falls to a drastically lower value where it remains (‘G’). 4.3.3. Results Fig. 18 shows the result of the ‘liquid from above’-test in dimensionless form. In contrast to the former test we cannot detect the wetting time tc here because there is no trace of successful closure in the Dp⁄-signals. The difference is that wetting in the ‘liquid from below’-test was initiated by a piston of liquid that moved towards the screen. Here, on the other hand, only a small amount of liquid is injected. Unlike the other test, the liquid can now easily leak through the upper screen while the menisci in the pores of the lower screen have to carry the growing pressure. The main features of this test are therefore the negative slope of Dp2 when loading the liquid on top of the element - this is the test parameter, the bubble point and the positive slope of Dp2 when the element fails. Altogether, we have performed 12 ‘liquid from above’-tests that are summarized in Table 4. Parameter in the test was the compression rate as provided in the first row. With each compression rate, a test was done four times. The second row provides the measured bubble points, the third row gives the positive slope of Dp2 at decompression. 5. Numerics 5.1. Numerical setup Since experimental observation is restricted to the top and the bottom side of the double screen element, numerical simulations were performed to obtain more information about the internal





Runs

d=dt Dp1  103

DpBP  102

d=dt Dp1þ  103

1–4 5–8 9–12 P

(39.5 ± 4.0) (64.0 ± 4.0) (100.3 ± 2.2)

(63.8 ± 0.5) (62.2 ± 0.7) (62.4 ± 0.5)

42.0 ± 8.6 55.7 ± 12.0 54.0 ± 18.0

Parameter

(62.8 ± 0.9)

50.6 ± 13.7

behavior of the screen element as liquid enters it and gas breakthroughs occur. Primarily, the numerical simulations aimed on qualitative results. Therefore, proven and tested numerical parameters were combined with information already available from the experiment. The microscopic structure of the screen (see Fig. 8) was not generated but rather were its porous parameters represented by a macroscopic model. To do that, we used the commercial code Flow-3d, which provides such a macroscopic porous media model. 5.1.1. Numerical Parameters The macroscopic approach utilizes the porous screen parameters and liquid properties as provided by Tables 1 and 2. A volume fraction dependent porous media type is used. Furthermore, laminar viscous flow and surface tension options are activated. The capillary pressure pr, see surface tension term in Eq. (1), was set equal to the bubble point pressure as observed in the experiment, hence pr = 1.5 kPa. To account for hydraulic pressure losses in all three directions (warp, weft and cross flow), Flow3d has implemented a single law which is different to Eqs. (44) and (45). Instead, it is  L with u  being the mean velocity in the cell, L the cell size Dp ¼ qau and the drag coefficient a = (l/)/(Kq). To calculate a, everything but the permeability K is known from Tables 1 and 2. According to Fries et al. (2008), the permeability is K ¼ B2s /=cw ¼ 8:3 lm2 with cw being the weave parameter for DTW 165 800 that can be found in Dodge (2000). There are two values of cw for flow in warp or weft direction, respectively. Since anisotropic effects cannot be included within the numerical model, those were averaged to cw = 1600 which is used for vertical and radial directions of the flow. We obtain a = 40,000 s1. 5.1.2. Meshing and geometry To generate the geometry of the double screen element, a rotationally symmetric mesh with a uniform mesh size of 0.5 mm is used. This cell size was chosen as a compromise between necessary resolution, acceptable CPU time, and stable simulation. It raises the problem that the cell size is larger than the screen thickness of Bs = 0.17 mm. Moreover, as preliminary tests suggested, the screen thickness is resolved by three cells yielding 1.5 mm to guarantee an undisturbed and continuous flow within the screen element. Consequently, we transform our screen element for simulation. The complete double screen element consists of three porous media components with identical porous properties - two cylindrical disks with a distance L = 10 mm in between and a cylinder with an inner radius R0 = 17 mm. We conserve the inner dimensions of the chamber in the element while the outer dimensions are expanded in comparison to the real element. The screen element is placed in a cylindrical container with inlet and outlet. Liquid is initially positioned in front of the screen as shown in Fig. 19a. In the figures, white regions indicate the gaseous phase (fraction = 0), blue regions the liquid phase (fraction = 1). We adjust the drag coefficient to the new screen thickness while leaving porosity and surface tension unaltered. The drag coefficient was adequately reduced to obtain the identical interstitial velocity and pressure drop according to the real screen characteristics. Hence, a(Bs = 0.17 mm) = 40,000 relates to a(Bs = 1.5 mm) = 514.

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M. Conrath et al. / International Journal of Multiphase Flow 50 (2013) 1–15

(a)

t = 0.0 s

(b)

t = 0.6 s

(a)

(b)

t=24.5 s

t=25.3 s

(c)

(d)

(c)

t = 3.1 s

t = 6.0 s

t=26.8 s

(d)

t=29.3 s

Fig. 19. Closure of the element after contact with liquid from below: (a) initial condition at t = 0 with the dry element and liquid touching the lower screen, (b) capillary rise in the wicking aid, (c) radial inward capillary flow, (d) closure of the element completed.

Fig. 20. First breakthrough of the trapped gas inside the element: (a) first signs of immanent breakthrough, (b) a channel as opened through which the trapped gas escapes, (c) the liquid level inside the element has risen and a trapped bubble forms, and (d) the breakthrough has stopped but a trapped bubble remains.

5.1.3. Simulation methods – liquid from below The simulation starts with the lower screen already in contact with liquid. The element is going to saturate and the gas inside is trapped (Fig. 19). The pressure at the bottom and the top boundary is set to 100 kPa and remains constant during the saturation process. Then, after successful element closure, the absolute pressure below the element is increased with a constant slope of 100 Pa/s at the bottom boundary. At t = 47 s the maximum pressure of 104 kPa is achieved. The top boundary pressure remains always constant during simulation.

the gas plug inside it. In the numerical simulation, the element closure was completed just prior to t = 6.0 s.

5.1.4. Simulation methods – liquid from above In this scenario, liquid is initially in contact with the upper screen, as shown in Fig. 22. Here, saturation time takes nearly 4.8 s and is shorter due to gravity forces. When the element is saturated, an absolute pressure increase at the top boundary is applied. The pressure at the bottom boundary always remains constant.

5.2.2. First breakthrough After closure of the element, the pressure difference across the element is monotonically increased at a rate of 100 Pa/s starting at t = 7.0 s. Until brakthrough, no change in the element’s appearance can be seen in the simulation. What happens next is illustrated in Fig. 20. The first signs of a breakthrough are shown in Fig. 20a at t = 24.5 s which corresponds to Dp = 1.7 kPa between bottom and top boundary between the simulated domain and Dp = Dpbp  1.5 kPa across the element. The liquid is forced from the pores at some random location of the upper screen and is replaced by gas. From the bottom, liquid replaces the gas emitting through the upper screen. In Fig. 20b, a channel has formed that allows further gas passage. This gas outflow continues as long as the upper screen is entirely covered by gas as in Fig. 20c. When the upper screen partially contacts liquid as in Fig. 20d, there is a new flow path for the liquid and the gas breakthrough stops, leaving a trapped bubble below the upper screen.

5.2. Results for liquid from below 5.2.1. Element closure The process of gas trapping, i.e. the element closure, is shown in Fig. 19. At the initial condition at t = 0 (Fig. 19a, the liquid is already in contact with the whole area of the lower screen. Fig. 19b shows the screen element during the annular wicking phase at t = 0.6 s with the menisci of the wicking front being in the vertical wicking aid. A little later (Fig. 19c, at t = 3.1 s, the wicking has proceeded around the edge and the front is now in the upper screen heading inward towards the center. Finally, at t = 6.0 s, as shown in Fig. 19d, the capillary transport of liquid has advanced to the center of the upper screen, thus closing the element with liquid and trapping

5.2.3. Complete breakthrough After the first breakthrough, the situation has changed as far as the pressure distribution is concerned. Now, the pressure difference across the upper screen is easily relaxed by the liquid flow through its pores. As a consequence, the pressure drop across the upper screen falls well below the bubble point. The main hydraulic resistances are now the two screens and the viscous flow losses across it. Nevertheless, further increases of the applied pressure difference between bottom and top boundary of the simulated domain causes the leakage flow to rise. Eventually, the cross flow pressure drop across the upper screen becomes as high as the bubble point again. Then, another breakthrough occurs as shown in

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(a)

(b)

t = 33.8 s

t = 36.2 s

(c)

t = 38.0 s

(d)

(a)

t = 0.0 s

(b)

t = 0.6 s

(c)

t = 2.6 s

(d)

t = 4.8 s

t = 38.1 s

Fig. 21. Complete breakthrough is observed to be stepwise: (a) a channel has opened as the second breakthrough begins, (b) the 2nd breakthrough ends again with a trapped bubble but a smaller one, (c) first signs of another breakthrough, and (d) which is final since all remaining trapped gas escapes at once.

Fig. 21. In Fig. 21a, at t = 33.8 s, the applied pressure difference is 2.68 kPa between bottom and top boundary. Since the upper screen is partly covered by a trapped bubble, more than half of the applied pressure acts on the bubble, surpassing now the bubble point. This is a dynamic bubble point as it is accompanied by a liquid flow. This phenomenon is tackled by Conrath and Dreyer (2012). A little later, see Fig. 21b, this breakthrough has ended, leaving a smaller trapped bubble. The screen area covered by the bubble has decreased, and therefore, the pressure drop across the upper screen has fallen below the bubble point again. The continuously increasing applied pressure and flow rate keeps increasing the pressure drop across the upper screen until it again surpasses the bubble point threshold, pictured in Fig. 21c at t = 38.0 s. In this case, the breakthrough is complete. Only a fraction of a second later, see Fig. 21d, all trapped gas has been pressed through the upper screen. 5.3. Results for liquid from above

Fig. 22. Closure of the element after contact with liquid from above: (a) initial condition at t = 0 with the dry element and liquid touching the upper screen, (b) capillary fall in the wicking aid, (c) radial inward capillary flow, and (d) closure of the element completed.

the two screens. While the entering liquid stayed at the bottom of the chamber when entering from below, it can now drop downward to the other screen. The result is a growing leakage as illus-

(a)

t = 25.9 s

(b)

t = 26.3 s

(c)

t = 29.7 s

(d)

t = 47.2 s

5.3.1. Element closure The gas trapping with liquid from above is shown in Fig. 22. Initially, only the upper screen is in contact with liquid from above at t = 0 s. With gravity assistance, the liquid is transported by capillarity into the wicking aid, see Fig. 22b and the lower screen, see Fig. 22c. Finally, at about t = 4.8 s, the element has entrapped the gas, see Fig. 22d and can be loaded with a pressure difference. 5.3.2. Growing leakage At t = 5 s the applied pressure between top and bottom boundary of the simulated domain starts to increase. However, in contrast to the ’liquid from below’ case, the trapped gas is not pressed against the other screen anymore. The liquid from above can still compress the trapped gas and enter the chamber between

Fig. 23. In contrast to the pressurization with liquid from below, we observe here no distinct breakthrough. Instead, the leakage rate increases with growing applied pressure difference: (a) drop-wise leakage in the beginning, (b) stream-wise leakage, (c) more streams, and (d) further leakage increase.

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trated in Fig. 23. First signs of drop forming are observed at t = 22.3 s which already corresponds to an applied pressure difference above the bubble point. Fig. 23a shows the situation at t = 25.9 s where liquid enters at the ceiling of the chamber, forms drops, falls down and leaves the chamber through the lower screen from where it descends further. In Fig. 23b, at t = 26.3 s, a closed stream has already established, which continues to grow in Fig. 23c at t = 29.7 s, and in Fig. 23d at t = 47.2 s. We may conclude that there might indeed be no breakthrough when liquid presses from above.

ment happens step-wise because the first breakthrough leaves a trapped bubble inside the element. Overall, the choice of the supplier-provided particle retention size Dp as pore length scale for our predictions proved to be useful. Besides, we can confirm the concept of the porous screen element and encourage its pursuit for future applications. While the present study was performed under the rather ideal premise that the screen element has enough time to saturate and get operational, future studies should also account for the dynamics of liquid encounter.

6. Summary

Acknowledgements

The concept of the double porous screen element to block the passage of liquid utilizing a trapped gas plug was investigated and successfully demonstrated here. We have developed a model for the capillary transport in the element, performed experiments and done numerics. Our model accounts for the intended application case with no gravity as well as terrestrial conditions where the capillary transport can be against gravity or gravity assisted. To test the concept, a real double porous screen element was built using Dutch Twilled weave. It was submitted to two tests: (i) wetting from below and (ii) wetting from above, followed by an increasing pressure difference. A prediction of the closure time based on our model and the pore retention size as provided by the supplier gave about two seconds, the value found in the experiment was the 1.8 times larger. A prediction of the breakthrough pressure where the element would fail was done on the basis of the same pore size and gave 2.6 kPa while in the experiment 57% of that value was observed. The leakage was strongly underpredicted, possibly due to the vaguely defined gap between wicking aid and tube wall. In addition to model and experiment we performed numerical calculations to elucidate what happens inside the element when it fails.

This project receives financial support from the German Aerospace Center (DLR) under Grant No. 50 RL 0921. The German Aerospace Center is funded by the German Federal Ministry of Economics and Technology (BMWi) based on a resolution of the German Bundestag. Moreover, the authors thank Holger Faust for his help with the experimental setup.

7. Discussion The presented model has underpredicted the closure time by a factor of 1.8. This might be due to the discontinuities between annular wick and circular disks. These sections are not connected on pore scale but simply pressed together which leaves a gap in between them. This gap needs to be bridged by the imbibed liquid before the wicking front can proceed. It should also be noted that this prediction depends on the chosen length scale which is not clear. We chose the retention size for spherical particles of this weave because this is provided by the supplier and assuming that the liquid carrying paths within the weave have about the same scale. However, looking at the twisted pores of such a weave, the accuracy of this prediction without any reference measurements seems acceptable. The breakthrough pressure was overpredicted by both predictions. Interestingly, the prediction based on Dodge (2000) which deliberately relies on bubble point measurements on such screens, overpredicted it even more. The bubble point prediction, too, strongly depends on the chosen pore scale. Accordingly, by avoidance of reference measurements this results in an error. In our case the breakthrough pressure occurred at 57% of the predicted pressure difference. The numerical calculations have shown that the failure of the ele-

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