Gas breakthrough at a porous screen

International Journal of Multiphase Flow 42 (2012) 29–41

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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

Gas breakthrough at a porous screen Michael Conrath, Michael Dreyer ⇑ Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, Am Fallturm, 28359 Bremen, Germany

a r t i c l e

i n f o

Article history: Received 20 July 2010 Received in revised form 22 December 2011 Accepted 16 January 2012 Available online 28 January 2012 Keywords: Woven screen Screen resistance Bubble point Bubble breakthrough Trapped bubble

a b s t r a c t We consider the dynamic breakthrough of a large bubble which is trapped below a porous screen and exposed to a liquid flow. Due to gravity, the bubble is rather flat and blocks part of the screen. Upward liquid flow across the screen leads to a pressure difference between its front and back side. As this pressure difference exceeds the bubble point threshold, the trapped bubble starts to break through the screen causing a number of small bubbles to emerge in chains from the upper side. These emerging bubbles form and detach in their typical manner. They affect the breakthrough and even bring it to a halt as the pressure difference falls below a detachment pressure which is intrinsic to the screen pores. During breakthrough, the trapped bubble shrinks and correspondingly blocks the screen to a lesser extent. The liquid flow around the trapped bubble deforms it and tends to make it flatter. The interplay of all these effects is a complex issue that is addressed in the present article. We develop a model comprising most of the effects and present it along with original experimental data. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fig. 1 illustrates the scope of this paper. A porous screen is suspended in a vertical tube. The screen is wetted by a liquid. Therefore, the pressure difference Dp has to overcome the bubble point before gas can break through the screen. If the complete screen is covered by pressurized gas then the gas will break through when the pressure difference between screen front and back side exceeds the static bubble point. If the gas does not cover the complete screen as in Fig. 1, a trapped bubble forms which cannot be pressurized in the same manner. A breakthrough of this bubble is now triggered by the flow-induced pressure drop across the screen. This is what we call the dynamic bubble point. Such configurations are of interest in a wide field of applications. Our motivation lies in gas-free delivery of propellant in spacecrafts, both for storable liquids (Dodge, 1990, 2000) and cryogenic liquids (Cady, 1973; Jurns and Kudlac, 2006; Kudlac and Jurns, 2006; Behruzi and Dodd, 2007). Other applications include geotextiles (Bhatia and Smith, 1994, 1995), the regenerators of STIRLING engines, paper making and of course filtration (Purchas and Sutherland, 2002). Even the retention of bacteria lies in the realm of applications (Reichelt, 1991). Connections to our problem are also found in plants that are vulnerable to gas intrusion into their xylems (Choat et al., 2003) leading to degradation of liquid transport in the plant.

⇑ Corresponding author. E-mail address: [email protected] (M. Dreyer). 0301-9322/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2012.01.002

The issue addressed in this paper is composed of several subproblems: (i) hydraulic resistance of a porous screen, (ii) static bubble point of a porous screen, (iii) bubble growth and detachment as gas breakthrough occurs, (iv) shape of a trapped bubble and (v) dynamic, i.e. the flow-induced, bubble point of a trapped bubble. The hydraulic resistance of porous screens to single-phase cross flows tackled in numerous articles, for example (Ergun, 1952; Armour and Cannon, 1968; Cady, 1973; Laws and Livesey, 1978; Jones and Krier, 1972; Munson, 1988; Brundrett, 1993; Sodre and Parise, 1997; Wu et al., 2005; Kopf et al., 2007; Valli et al., 2009). Olbricht (1996) tackles the two-phase flow of immiscible drops through the channels of porous media along with the corresponding pressure drop for a number of channel geometries and drop sizes. The static bubble point measurement as method to determine pore sizes was first established by Washburn (1921) although not under this name. Since then it was established as a more or less simple testing method for porous media (Bhatia and Smith, 1994, 1995; ASTM International, 2003; SAE International, 2001; Reichelt, 1991). The design of liquid acquisition devices (LAD) and propellant management devices (PMD) for spacecrafts often involves the bubble point (Cady, 1973; Jurns and Kudlac, 2006; Kudlac and Jurns, 2006; Jurns and McQuillen, 2008; Jurns et al., 2009). An interesting study is presented by Schütz et al. (2008) who focus on the dynamics of static bubble point measurements. The growth and detachment of bubbles on the upside of the porous screen as the static bubble point threshold is surpassed is related to works of Fritz (1936) and Kumar and Kuloor (1970) as well as Oguz and Prosperetti (1993). The pinch-off behavior of

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M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

Rep ¼

trapped bubble

liquid flow

Fig. 1. Configuration for the gas breakthrough at the porous screen.

detaching bubbles was studied by Thoroddsen et al. (2007, 2008), Gordillo (2008), Bolanos-Jimenez et al. (2008, 2009). The shapes of axially symmetric trapped bubbles that form below the screen are identical to those of sessile drops on a substrate. These shapes were first derived by Bashforth and Adams (1883) who used an early numerical approach. Based on laboriously calculated tables of functions they developed and applied a shooting method. Padday (1971) basically used the same procedure but aided by a then available personal computer. He modified and extended the tables. Langbein (2002) reviewed the trapped bubble problem focusing also on analytical approaches. But still today, the YOUNG–LAPLACE equation that describes the axially symmetric shapes of trapped bubbles can only be solved by numerical means. The axially symmetric deformation of bubbles or drops in flow was investigated for example by Payne and Pell (1960), Taylor and Acrivos (1964), and Hetsroni et al. (1970). The collapse of a capillary surface that occurs when surface pressure of the bubble is overcome by dynamic pressure is tackled by Rosendahl et al. (2004) and Grah and Dreyer (2010). However, it seems that a trapped bubble in stagnant flow was not yet examined. Therefore, the dynamic bubble point poses a novel problem. After describing the state of the art in the problems involved in our study we will describe the setup before we present results to the different aspects and discuss it.

2. State of the art 2.1. Hydraulic resistance of the woven screen Liquid or gas flow through a woven porous screen causes a pressure drop Dp between its front and backside. Generally, like on any other obstacle, an EULER number Eu can be defined which relates Dp to the dynamic pressure arising from liquid density q and mean velocity u0 = QL/A0, where QL and A0 are the liquid flow rate and the cross section of the tube that carries it, respectively. The EULER number is

Eu ¼

Dp

qu20

sB /2 Dp

m

:

ð3Þ

2.2. Static bubble point If we have two phases, i.e. gas and liquid, one or more gas–liquid interfaces will form that have a surface tension r. That is why, due to capillary forces, a porous screen is capable of separating the two phases. To illustrate the principle, see Fig. 2. Supposing the screen material is wetted by the liquid with a contact angle h smaller than 90°, then the liquid will imbibe the pores while the gas needs to be pressurized in order to push back the meniscus which means to de-wet the pores. Fig. 2a shows the meniscus in an arbitrary pore. Curvature and surface tension of the meniscus give the capillary pressure pr that has to be balanced by the gas gauge pressure. Assuming an axisymmetric pore of varying diameter dp(z), one would expect a meniscus pressure due to surface tension and curvature (White, 2003) of pr = 4r cos h/dp(z), z pointing vertically upward. Since the curvature of the meniscus is strongest at the bottleneck of the pore where dp(z) = Dp, it will also be there that the highest gas gauge pressure is needed to shift the meniscus further. That situation is shown in Fig. 2b. The pressure difference between front and backside of the screen to achieve that situation is the bubble point pressure Dpbp. The size of Dp is often determined (and provided by suppliers) as maximum diameter of spherical particles that can pass through the pores. However, the pores might not be axisymmetric. In fact, as in our case, they might be irregular and even twisted without a direct through path. Then what matters for the pressure difference at the meniscus is the equivalent radius of the contact line instead of Dp/2. One would expect a bigger value than Dp and therefore an overestimation of the bubble point when Dp is used to predict it. As the pressure difference surpasses the bubble point threshold, bubble breakthrough occurs, see Fig. 2c. In fact, although it is named the static bubble point, there is dynamics involved. It is

Dpbp ¼ C 0 Dp0 ; 4r cos h Dp0 ¼ : Dp

ð4Þ ð5Þ

ð1Þ

:

According to Armour and Cannon (1968), the woven screen can be described by

EuAC ¼

u0

Typical values of Rep are below 200. In Eq. (3), S denotes the surface to volume ratio of the weave based on geometrical considerations (these equations are not repeated here), m is the kinematic viscosity of the fluid. Armour and Cannon derive the viscous C1 and inertial coefficients C2 from the fit on a number of experiments carried out on five different weave types in gaseous Helium and Nitrogen flow. They find that C1,AC = 8.61 and C2,AC = 0.52. Cady (1973) performed numerous experiments on his own and used the same model approach. But he fits the coefficients for each weave separately. For the weave we use in the present study, i.e. Dutch Twilled 200  1400, he finds C1,Cady = 4.2 and C2,Cady = 0.2.

porous screen

1g

1 S2 Dp

 C1 þ C2 ; Rep

EULER

number at a



ð2Þ

sbeing the tortuosity of the wires, B the thickness of the weave, Dp the retention size of spherical particles and / the porosity of the screen. C1 and C2 are two empirical constants accounting for viscous and inertial effects, respectively. Rep is the pore REYNOLDS number as defined by Armour and Cannon,

Fig. 2. Schematic pore arrangement and location of the menisci in the screen arrangement.

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

C0 is here a constant that depends on pore geometry and breakthrough dynamics. Before a bubble breakthrough can be detected, the meniscus has to move through the pores, and of course that builds up additional pressure if it happens too fast. Schütz et al. (2008) develop a more general and dimensionless model that accounts for the effects of gravity (BOND number Bop = qg L2/r), inertia (WEBER number Wep = qu2L/r) and viscosity (CAPILLARY number Cap = lu/r), where g, L and l are gravity constant, characteristic length at the pore scale and dynamic viscosity, respectively. They point out that the bubble point test method inherently assumes that on the pore scale Bop = 0, Wep = 0 and Cap = 0 – which is not always fulfilled. A very useful relation is provided by Hernandez et al. (1996) that connects the bubble point not only with the pore size but also with the pressurization rate. Vice versa, as given there

Dp ¼

4r Dpbp

"

 1=2 # 2B dðDpÞ 1þ l ; dt r

ð6Þ

 is the mean dynamic viscosity of gas and liquid and t is where l time, it allows to predict the pore size more precisely by taking both bubble point pressure and pressurization rate into account.

Neglecting the impact of viscosity, the dynamics of spherical bubble growth obey the RAYLEIGH–PLESSET equation

  3 1 2r Rb R€b þ R_b 2 ¼ pb;in   pb;out ; 2 q Rb

VF ¼

4 3 pR ; 3 F

RF being the

2

Vb ¼

3

2R3b

DP 2 þ þ 2R2b 2

!sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 DP 2 5 R2b  : 2

ð7Þ

The smallest bubble radius occurs when Rb = Dp/2, meaning a semi-spherical bubble that also relates to the highest meniscus pressure and therefore marks the bubble point threshold Dpbp. We normalize the bubble volume and pressure with the values of this threshold bubble

V b ¼

12V b

pD3p

pb ¼ pb

;

Dp Dp ¼ 4r cos h 2Rb

ð8Þ

where pb = pb,in  pb,out = 2rcosh/Rb. The right-hand side of Fig. 3 shows the connection between growing bubble volume and the meniscus pressure. It suggests that, if the bubble point threshold is surpassed at one pore, a bubble will grow there providing a pressure relaxation that hinders the bubble growth at other pores.

Rb Vb

p

b,in

p

b,out

Dp

pc

Fig. 3. Growth of a single bubble at the pore. Left: sketch of the arrangement, right: meniscus pressure at the growing bubble. Here, pb = pb,in  pb,out.

ð10Þ FRITZ

radius

 1=3 3rDp RF ¼ : 4qg

ð11Þ

The bubble cannot detach before the FRITZ volume is reached. As illustrated in Fig. 3, the FRITZ volume corresponds to a bubble detachment pressure. The FRITZ volume is either reached in the inertial time

tin ¼ RF

p4

ð9Þ

see for example Oguz and Prosperetti (1993), where pb,in is the pressure inside the bubble itself and pb,out is the pressure in the surrounding liquid at orifice height. A critical size is reached when the bubble has grown to the so called FRITZ volume, i.e. the volume when buoyancy overcomes surface tension force that holds the bubble at the needle. The FRITZ volume is

2.3. Bubble growth and detachment As breakthrough occurs, bubbles grow and then detach on the screen’s upper side. We assume that the bubbles remain spherical, i.e. that the bubble diameter remains small compared to the capilpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi lary length Lr ¼ r=ðqgÞ. Although the weave has no cylindrical pores, we relate our considerations to such idealized pores that have an orifice of radius Dp/2, see left picture of Fig. 3. Depending on the orifice radius a, the volume Vb of the growing bubble becomes

31

q

!1=2

pc  pb;out

ð12Þ

assuming a given chamber pressure pc (which is slightly higher than pb,in) or in the FRITZ time

tF ¼

VF QG

ð13Þ

assuming a given gas supply rate QG that forces bubble growth. Especially at high gas supply rates, the bubble growth is not driven by the self-expansion of the meniscus but rather is the bubble blown up until it detaches. Therefore, the ratio tF/tin determines the nature of breakthrough. Because the pores of our weave are no idealized cylindrical orifices but rather trumpet-like openings, two effects are expected: (i) The bubble detachment pressure will be lower than the bubble point pressure because the meniscus curvature at a wider opening is smaller and therefore the meniscus pressure, too. (ii) The bubble detachment pressure will not be sharply defined since the contact line of a growing bubble is not pinned. Instead, at a trumpet-like opening, it can move freely which affects the curvature of the meniscus. 2.4. Trapped bubble shape For the dynamic bubble point where a single bubble is trapped below the woven screen one needs to know the bubble shape to judge the coverage of the screen or vice versa the constriction of liquid flow around the bubble. The shapes of a trapped bubble are shown in Fig. 4. They are found by solving the axially symmetric YOUNG–LAPLACE equation, see for example Langbein (2002), applying a RUNGE–KUTTA shooting method of 4th order (RK4), see for example Kreyszig (2006). By shooting the height-normalized contour from two sides (vertex and contact position) the correct pairing of BOND number Bo and vertex curvature for a given contact position cTB and contact angle h is iteratively obtained. Once the correct pairing is known, further properties of the trapped bubble are derived, first of all the r(s) and z(s) coordinates along the height-normalized contour. To display the trapped bubble contours in real dimensions, the r- and z-axes have to be stretched. pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The stretching factor is BoLr with Lr ¼ r=ðqgÞ being the capillary length. The bubble volume is found by application of

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M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

3.2. Flow effects on the bubble As outlined in the former subsection, we know the trapped bubble shapes at rest, but in flow the bubble will experience additional effects: (i) deformation due to the flow around it and (ii) deformation due to bubble-wall interaction.

VTB

z [m]

HTB rTB [m]

cTB RTB

Fig. 4. Shapes of a trapped bubble. Starting with a given aspect ratio cTB/HTB and a contact angle h = 0, the corresponding pairing of BOND number and vertex curvature is found. With the BOND number Bo the real dimensions of the bubble can be calculated.

3.2.1. Flow-induced bubble deformation The degree of deformation can be estimated with a WEBER number. Due to the flow constraint given by the tube, the mean flow velocity around the bubble is QL/(A0  ATB) and the definition of the WEBER number becomes

We ¼ the LEIBNITZ sector formula for axially symmetric bodies. As the volume of the trapped bubble increases, its height converges towards two times the capillary length, HTB ? 2Lr (Myshkis et al., 1987).

3. Model for the dynamic bubble point 3.1. Flow through the partially covered screen Presumably, a trapped bubble in flow will only break through the screen when the flow-induced pressure drop over the screen is beyond the bubble point threshold. This flow-induced pressure drop depends on the mean flow velocity through the screen. Tube and screen have a radius R0. The mean velocity depends on the coverage of the screen caused by the trapped bubble. With the area of the uncovered screen A0 ¼ pR20 and that one which is covered by the bubble ATB ¼ pc2TB we can define a cover ratio

C¼

ATB c2TB ¼ 2 A0 R0

ð14Þ

and with a given liquid flow rate QL, the velocity of liquid flow crossing the uncovered part of the screen becomes

Q 1 u1 ¼ L : A0 1  C

ð15Þ

Index 0 denotes the undisturbed liquid flow while index 1 refers to the constricted flow due to the trapped bubble. The limit case C = 1 would cause an infinite velocity in the remaining gap and therefore generate an infinite pressure difference resulting in sudden breakthrough. To generalize our considerations, we now relate u1 to an empty tube REYNOLDS number Re0 = 2R0u0/m or

Re0 ¼

2R0 Q L : mA0

ð16Þ

Inserting u1 and Re0 into Eqs. (1) and (2), we arrive at an expression for the flow-induced pressure drop across the screen depending on cover ratio C and tube REYNOLDS number Re0.



Dp ¼



Re20

qm sB C1 þ C2 ; 4/2 Dp R20 Rep1 ð1  CÞ2 2

1

Re0 : 1 C 2Dp R0 S 2

rðA0  ATB Þ2

ð19Þ

:

The annular flow around it causes a suction of the trapped bubble rim towards the tube wall. Enlargement of the bubble will lead to higher flow concentration in the annular gap around the bubble and therefore a growing WEBER number, too. If the bubble blocks the tube completely, the WEBER number will consequently be infinity. Hence, up to a critical flow rate, the bubble will remain stationary deformed. Beyond it, the bubble will suddenly reach the tube wall, followed by a breakthrough. We indeed observe this bubble-wall interaction in our experiment. However, we will desist to develop a more detailed model of this phenomenon here. 3.3. Dynamic bubble point model Instead, we want to shed some light on the implications that arise from the aforementioned equations: With a trapped bubble, the pressure drop in Eq. (17) will rise the faster with Re0 the higher the cover ratio C is. But when Dp reaches the bubble point, breakthrough occurs. Hence, there must be a critical cover ratio Ccrit for every empty tube REYNOLDS number that marks the breakthrough threshold.

DpðC; Re0 Þ ¼ C 0 Dp0 ! Ccrit ðRe0 Þ:

ð20Þ

Rearranging terms in Eqs. (17) and (18) to find Ccrit(Re0) is straightforward and yields



Ccrit ðRe0 Þ ¼ 1  Re0 C 3 þ C3 ¼

C 1 lmsBS2 4/2 R0 ðC 0 Dp0 Þ

;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 23 þ C 4 with

C4 ¼

C 2 lmsB 4/2 Dp R20 ðC 0 Dp0 Þ

ð21Þ :

ð22Þ

Since C3 and C4 are constants, the dependence Ccrit(Re0) is linear. It is convenient to define a maximum empty tube REYNOLDS ^ 0 where Dp(C = 0) = Dpbp = CDp0, i.e. where even the number Re last bit of trapped gas would break through the screen pores

^ 0 ¼ Re0 ðCcrit ¼ 0Þ ¼ Re

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : C 3 þ C 23 þ C 4

ð23Þ

The critical cover ratio Ccrit, see Eq. (21), then becomes

Re0 : ^0 Re

ð17Þ

Ccrit ¼ 1 

ð18Þ

As shown in Fig. 4 and consistent with our experiments, the height of the trapped bubble converges pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffito ffi a fixed value of two times the capillary length 2Lr ¼ 4r=ðqgÞ. Thus, except for small volumes of the trapped bubble,

where

Rep1 ¼

qQ 2L RTB

A delicate property in Eqs. (17) and (18) is C because it depends on the shape of the trapped bubble which is affected by the liquid flow.

ð24Þ

C ’ V TB = Vb TB ;

ð25Þ

V TB;crit

ð26Þ

  b TB 1  Re0 : ¼V ^0 Re

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

b TB refers to the maximum possible trapped bubble volHere, V ume. Without the hat, i.e. VTB, it refers to just the trapped bubble volume. Using Eq. (25), we can predict the volume evolution of the trapped bubble versus flow rate, accounting for both the breakthrough and detachment limit at the same time. For the bubble detachment pressure after breakthrough we can apply the same arguments as outlined by Eqs. (20)–(26). Fig. 5 shows the result of such a volume evolution prediction. Instead of bubble volume versus liquid flow rate it depicts the evolution of the cover ratio C versus REYNOLDS number Re0. Starting with a bubble that covers 90% of the screen, i.e. C = 0.9, the REYNOLDS number is steadily increased. At the critical Re0 for C = 0.9, the first breakthrough occurs which makes the bubble shrink and thus C, too. When C has fallen to the detachment limit, the breakthrough stops and Re0 can be further increased before the next breakthrough occurs. This cycle of breakthrough beginning and stop happens between the bubble point and the detachment limit marked as two straight lines in Fig. 5 that accord to Eq. (23). The relation between the both limits is taken from the experimental results that are described below. From the evolution of the trapped bubble volume in flow, we can also predict the breakthrough cycles upon quasi-steady flow rate increase. We have to bear in mind that a breakthrough starts when the bubble point threshold is surpassed and stops when the small bubbles on the upper side of the screen cannot grow and detach anymore. So to say, there is an upper and a lower Ccrit. While the bubble volume in the experiment cannot be directly measured, the pressure drop across the screen can. To predict the trend of Dp(Re0,C), see Eq. (17), we rearrange this equation into

Re0 Re20 ~1 ~2 DpðRe0 ; CÞ ¼ p þp 1C ð1  CÞ2

ð27Þ

with the two constants

~1 ¼ p

lmsBS2 2

2/ R0

C1 ;

~2 ¼ p

lmsB 4/2 Dp R20

C2:

ð28Þ

An important feature of our problem is also the maximum possible flow rate up to which bubbles can exist trapped below the screen.

33

Fig. 6. Pressure drop across the screen with different trapped bubble volumes VTB. In the figure, C = 0;10%; . . . ; 90%. The lowest line where C = 0 therefore coincides with single-phase flow, see Eqs. (1) and (2). All the other lines have a similar appearance but are valid only up to the bubble point limit.

This relation between maximum REYNOLDS number and static bubble point is found by setting Dp = Dpbp and C = 0 in Eq. (27) giving

^0 ¼ Re

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~21 ~1 p ðC 0 Dp0 Þ2 p þ :  ~2 ~22 ~22 2p 4p p

ð29Þ

To get an idea of the trend Dp(Re0,C) and in anticipation of the ~1 ¼ 0:23 Pa, p ~2 ¼ 4:4  106 Pa performed experiments we choose p and Dp0 = 4.89 kPa. Fig. 6 illustrates the prediction of Dp(Re0, C). The bubble point limit in Fig. 6 depicts the measured value from the experiments described later on that was found at Dpbp = 0.65Dp0. Also the detachment limit is taken from our measurements where it was found at Dpd = 0.51Dp0. The maximum REY^ 0 ¼ 5720. NOLDS number becomes Re So in essence, our model predicts the screen resistance with trapped bubble. Unfortunately, such a prediction necessitates (i) two coefficients for the flow resistance of the screen, (ii) bubble point pressure and (iii) bubble detachment pressure – all of them found by experiments. That is why we first have to collect some reference data to gain these basic properties. 4. Experimental setup

Fig. 5. Trapped bubble volume for quasi-steady increase of the liquid flow rate and breakthrough cycles. In the example, an initial screen coverage of C = 90% was chosen. Upon increase of the flow rate, i.e. the REYNOLDS number Re0, breakthrough will eventually occur. Under quasi-steady conditions, i.e. a very slow increase of Re0, the breakthrough takes place at a constant REYNOLDS number (vertical lines) and ends when the flow-induced pressure drop falls below the bubble detachment limit. The detachment limit is chosen in accordance with our experimental.

We have built up an experiment as shown schematically in Fig. 7. It comprises a hydraulic circuit in which a woven screen is suspended perpendicular to the flow in a vertical tube. The screens weave pattern is of Dutch Twilled type. It is woven with stainless steel wires of two different diameters, namely 70 lm for the warp wires and 40 lm for the weft wires. There are 200 straight warp wires per inch and 1400 weft wires per inch contorted crosswise around the warp wires. The appearance of this DTW200  1400 screen and its conditioning for use in the experiment are illustrated by the three pictures in Fig. 8. As shown in the left picture of Fig. 8, the test screen is suspended on an aluminum ring, O-ring seals are used to tightly position it between the two halves of an acrylic glass block. The middle picture of Fig. 8 illustrates the weave’s complex but regular structure. Dutch Twilled weave is a typical filter screen that offers particles no direct path to flow through it, see right picture of Fig. 8. The properties of our test screen are summarized by Table 1. Since some properties are directly taken from the supplier (Spoerl, Germany) and others are calculated from the reference (Armour and Cannon,

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M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41 Table 1 Properties of the screen weave used in all experiments. Weave material Weave type: DTW 200  1400

Warp wire diameter (lm) Weft wire diameter (lm) Tortuosity (of the weft wire) Weave thickness (lm) Porosity (–) Pores per area (1/mm2) Projected area per pore (lm2) Particle retention size (lm) Surface to volume ratio (lm1)

lower valve

Fig. 7. Schematic arrangement of the two-phase flow experiment to study gas separation behavior of a woven screen under static and dynamic conditions.

Fig. 8. Left: Core piece of the test section is the ring on which the woven screen is suspended. Middle: structure of the Dutch Twilled woven screen as shown by a CAD model. Right: REM view on the screen in flow direction.

1968) with equations provided there, the properties in question are marked with a clarifying footnote 1 or.2 As test liquid we use silicon oil SF0.65 (supplier Dow Corning) with the properties listed in Table 2. The pump (supplier Getriebebau Nord, Germany) to circulate it is supplied by threephase alternating current and provides a maximum power of 1.2 kilowatts thereby delivering up to 120 milliliters per second in the actual circuit. Its pumping power is controlled with a personal computer by software. Its momentum is transferred to the flow with a pumping head (supplier Micropump, Germany). The hydraulic tubing between the single components of the circuit have an inner diameter of 6 mm and are made of PVC. A filter is used to clean the test liquid from particles that otherwise are always present and would interfere withhe measurements thus affecting reproducibility. Our filter is self-made and transparent so that the cleaning success can be directly observed. Its core piece is a Dutch Twilled weave 325  2300 (wires per inch in warp direction times wires per inch in weft direction) with a particle retention size of 8 lm. Downstream of the filter the flow rate is measured with a turbine type sensor RCQ02 supplied by Rösler + Cie Instruments GmbH, Offenbach (Germany). The RCQ02 transmits 8000 infrared pulses per liter and covers a measurement range and accuracy of (5 . . . 150) ± 1.5 milliliters per second. The test section is a vertical tube with a length of 600 mm and a diameter of 34 mm, see Fig. 7. The main part of the test section is the aforementioned test screen. Two feedthroughs shortly above and below the test screen allow to measure the pressure difference between its upper and lower side, see Table 3. The pressure difference sensor is a Sensotec Z/1309-32, supplied 1 2

According to supplier (Spoerl, Germany). According to (Armour and Cannon, 1968).

s B / N Ap Dp S

Stainless steel AISI 304 Dutch Twilled with 200 Warp wires/inch  1400 weft wires/inch 701 401 1.282 1501 0.2942 21, 71 4608 13 ± 11 0.0622

Table 2 Properties of SF0.65 silicone oil according to supplier (Dow Corning). The errors of density and surface tension are an assumption made by the authors since these values are not specified by the supplier. Kinematic viscosity, m (mm2/s) Density, q (kg/m3) Surface tension, r (mN/m) Contact angle (on stainless steel), h (°)

0.65 ± 10% 761 ± 1% 15.9 ± 10% 0

by Althen GmbH, Kelkheim (Germany) that can measure in the range Dp = 0. . .20 kPa with an accuracy of 0.25% of full scale, i.e. ±0.05 kPa. An absolute pressure sensor TJE/1256-30TJA, also supplied by Althen, measures the absolute pressure below the screen at the location of the trapped bubble. It has a range and tolerance pabs = (0 . . . 200) ± 2 kPa. At a distance 100 mm below the test screen, air can be injected by a step motor based syringe system (Nemesys, NEM B003-02D), supplied by Cetoni GmbH, Korbussen (Germany). It allows gas injection rates in the range QG = 103 . . . 2;ml/s. Past the test section, the liquid is led via an overflow towards the separator basin where it is freed from the gas bubbles before the cycle starts over again. Two identical high-speed cameras (Kodak Motion Corder Analyzer SR-500) with a Zoom 7000 Navitor lens that can record up to 250 frames per second allow to simultaneously observe what happens below and above the test screen which is important for bubble breakthroughs. Experiment control and data acquisition are both controlled from a personal computer. Via software are adjusted the liquid pump rate, the air flow injection rate as well as timing and amount of recorded data points. Table 3 provides an overview on geometrical properties of the test section. Table 4 summarizes the adjustable parameters and its limits in our setup. And Table 5 summarizes the range and accuracy of the sensors in our setup. 5. Results Table 6 provides an overview on the performed experiments that we are going to present here. Table 3 Overview on geometrical parameters of our test section. Properties except R0 and A0 refer to vertical distance in respect to the screen. Vertical extension Radius of the tube Radius of the test screen Cross flow area of the screen Location of the gas injection Location of Dp-sensor bores Location of pabs-sensor bore Location of T-sensor

R0 R0 A0

500 mm . . . +100 mm (17 ± 0.1) mm (17 ± 0.1) mm (908 ± 0.1) mm2 100 mm 10 mm; +10 mm 10 mm +10 mm

35

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41 Table 4 Range of the adjustable experimental parameters. Trapped gas volume Gas injection rate Liquid flow rate Flow rate increase

VG (ml) QG (ml/s) QL (ml/s) dQL/dt (ml/s2)

0 . . . 100 103 . . . 2 0 . . . 120 0; 1 . . . 10

Table 5 Overview on the sensors used in the experiment. Sensor

Type

Range

QL Dp pabs T Video

RCQ02 Sensotec Z/1309-32 TJE/1256-30TJA PT100 SR-500

(5 . . . 150) ± 1.5 ml/s (0 . . . 20) ± 0.05 kPa (0 . . . 200) ± 2 kPa (0 . . . 100) ± 0.5 °C 25 . . . 250 fps

5.1. Screen resistance The hydraulic resistance of the screen in single-phase flow, i.e. the flow-induced pressure drop Dp over the screen, was measured pointwise under steady flow conditions. No gas was present. To evaluate its accuracy, the measurement was recorded three times. We normalized our data to display EULER number versus pore REYNOLDS number according to Eqs. (1) and (3). Fig. 9 shows our data to be situated right in between that of Armour and Cannon (1968) and that of Cady (1973). Moreover, Fig. 9 shows the range of our measurements: 0.14 6 Rep 6 4. For comparison, the data range of Armour and Cannon was 0.2 6 Rep 6 20 while Cady even collected data for 0.035 6 Rep 6 190. Cady used several different fluids; he covers the lower Rep-range with gaseous Helium and the upper Rep-range with liquid hydrogen. The constants that provide the best fit of our data are C1 = 6.04 ± 1% and C2 = 0.05 ± 150%.

5.2. Static bubble point and bubble breakthrough In this section, we will have a look not only on the bubble point itself but also on what happens afterwards in the breakthrough regime. We have used an open arrangement as illustrated by Fig. 10. There is a gas chamber with a volume of VG = (100 ± 1) ml below the weave. After initial filling, the chamber is pressurized by injection of a constant gas flow rate QG using ambient air as gas, see left Table 6 Overview on the performed experiments presented here. Here, VG is the gas volume which is trapped below the screen, QG is the gas supply rate to that volume, QL is the liquid flow rate and Dtgas is the duration of gas injection.

Section 5.2 Static bubble point – 100 – 100 – 100 – 100

QG (ml/s)

Dtgas (s)

–

–

1 101 102 103

Section 5.3 Dynamic bubble point – different bubbles 0 . . . 120 0 0 0 . . . 95 (1st breakthr.) 0.25 0 0 . . . 78 (1st breakthr.) 0.5 0 0 . . . 55 (1st breakthr.) 0.75 0 0 . . . 41 (1st breakthr.) 1.0 0 0 . . . 31 (1st breakthr.) 1.25 0 0 . . . 20 (1st breakthr.) 1.5 0 0 . . . 12 (1st breakthr.) 1.75 0 0 . . . 120 1.5 0

40 400 4,000 40,000 0 0 0 0 0 0 0 0 0

(a)

(b) p

p

5 cm

VG or VTB at begin (ml)

Section 5.1 Screen resistance 0. . . 120 –

picture of Fig. 10. Eventually, the gas in the chamber breaks through the screen pores, see right picture in Fig. 10. The liquid column above the test screen ensures that the gas must form bubbles at breakthrough. If the screen was only wet without any liquid column on it then the pores would become open channels at breakthrough where the gas just blows out. Furthermore, the column is high enough to conveniently observe emerging bubbles from aside. This is in accordance with recommended practice (ASTM International, 2003). We have recorded the trend of Dp(t) for different gas supply rates QG. The different stages of the bubble point measurements can be seen in Fig. 11. As gas injection begins, the connection between the syringe gas reservoir at ambient pressure with the trapped gas in the chamber is established which leads to a short decompression, followed by an almost linear increase of Dp. Eventually, the first breakthrough occurs at Dpbp and the breakthrough regime is entered. Please note that the dimensionless Dpbp is identical to C0 since C0 = Dpbp/Dp0. In the breakthrough regime, the number of pores carrying the breakthrough may vary resulting in pressure falls and rises. But also steady equilibrium between gas supply rate and gas

5 cm

QL (ml/s)

Fig. 9. EULER number of DTW 200  1400 related to the PORE REYNOLDS number as defined by Armour and Cannon, see Eq. (3). Our own data range is 0.14 6 Rep 6 4. The data range of Armour and Cannon is 0.2 6 Rep 6 20, that of Cady 0.03 6 Rep 6 190. The expected error of the EULER number for each data point is about 1%.

p

VG

pc

p test screen

QG

VG

pc

QG

Fig. 10. Arrangement for static bubble point measurement that is established by closing the lower valve in Fig. 7. Below the screen, a trapped gas volume VG is pressurized by a gas supply rate QG. The gas injection needle exits in VG and above the liquid level to avoid distributions of the Dp-signal caused by bubble detachments at the needle tip. A fixed height of liquid above the test screen followed by ambient atmosphere pressure ensures that the built up pressure pc can only unload in form of tiny bubbles on the upper side. Note that, as breakthrough occurs, the gas exits only through a small number of pores: (a) Pressurization phase. (b) Breakthrough phase.

36

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

breakthrough rate is possible. Upon gas supply switch-off, Dp falls down to a value that relates to the bubble detachment pressure Dpd . But again, the number of open pores determines how far down the pressure relaxes. To predict the static bubble point beforehand, one can apply Eq. (4) using C0 = 1, surface tension and contact angle from Table 2 and the pore diameter Dp (retention size for solid, spherical particles) from Table 1. That prediction yields

Dp0 ¼ ð4:89  1Þ kPa:

To normalize the time axis of our experiments, we make use of the FRITZ time, see Eq. (13). In short, to nondimensionalize our results, we set

Dp ¼

Dp  t ;t ¼ : tF Dp0

ð31Þ

FRITZ radius and FRITZ volume are calculated with Eqs. (10) and (11) using data from Tables 1 and 2. They are

ð30Þ

This is an overestimation because the bubbles which are squeezed through the weave pores are not restricted to spherical shape. They can also invade the pore edges. As a consequence, the effective contact line will be longer and the cross section larger leading to an effectively lower corvature compared to a spherical cap with radius Dp/2. This, in turn, facilitates a lower breakthrough pressure than predicted by Eq. (30). Nevertheless, we will use this estimate to normalize our results since the necessary data is always provided by the suppliers. Another way to obtain more exact predictions are reference measurements. That is what we did here. The result along with a number of other experimental results for the static bubble point of a Dutch Twilled weave 200  1400 found in the literature – also for cryogenic liquids – is provided by Table 7. It shows that the pore diameter obtained in this way varies over a range of about 10 percent.

RF ¼ ð275  17Þ lm;

V F ¼ ð87  16Þ103 mm3 :

ð32Þ

The corresponding FRITZ times for each measurement, see Eq. (13), along with gas supply rate and FRITZ to inertial time ratio are provided by Table 8. The inertial time for bubble growth up to the FRITZ volume is in all experiments identical. We estimate it following Eq. (12) to be

tin ¼ RF



qSF0:65 Dp0

1=2

! tin ¼ ð108  21Þ

ls:

ð33Þ

Here, the static bubble point Dpbp equals the pressure difference pc  pb,out, see also Fig. 2c. The ratio tF/tin tells us which of the two times dictates the bubble growth and detachment (Oguz and Prosperetti, 1993). If t F =t in  O(1) or below, the bubbles are inflated by the forced gas flow up to the FRITZ volume rather than growing self-propelled, see Eqs. (12) and (13). Breakthrough will not be restricted to a single pore. At the limit tF/tin ? 0, violent gas breakthrough at many pores will occur with non-spherical gas blobs forming and coalescing on the upper side. On the other hand, in the limit tF/tin ? 1, quasi-static conditions are ensured. This should be the basis for reliable static bubble point measurements. At each time, only one bubble forms. This consideration provides a simple mean to assess the usefulness of a static bubble point measurement. In our experiments, the range of Fritz to inertial time ratio is 0.8 6 tF/tin 6 800 or O(1)6 tF =tin 6 O(1000) spanning three orders of magnitude. For all experiments presented on the static bubble point, the gas volume in the chamber below the screen was Vgas = (100 ± 1) ml and the height of the liquid column above the screen was Dh = 50 mm.

Table 8 Performed experiments on static bubble point and bubble breakthrough at a Dutch Twilled 200  1400 woven screen. Test liquid was silicon oil SF0.65, test gas was air. Fig. 11. Different stages of a bubble point measurement. First, the pressurization phase increases the pressure below the screen until the first gas breakthrough occurs. In the breakthrough regime thereafter, Dp may rise, fall, or roughly remain on one level depending on the varying number of pores that carry the breakthrough. Upon gas supply switch-off, Dp relaxes down to the bubble detachment pressure where it remains.

Runs Runs Runs Runs

1–4 5–8 9–12 13–16

QG (mm3/s)

tF (ms)

tF/tin (–)

OðtF =tin Þ (–)

1000 100 10 1

0.087 0.87 8.7 87

0.8 8.0 79.6 796.3

1 10 100 1000

Table 7 Experimental results for the static bubble point in Dutch Twilled weave 200  1400 that can be found in the literature. In all experiments, the contact angle is extremely low and can be set h = 0o. The static bubble point pore size Dbp exceeds the retention size for solid particles Dp, see Table 1, because the pores have no cylindrical cross section at the neck. Reference

Liquid and conditions

Dpbp (kPa)

r (mN/m)

Dbp ¼ D4pr (lm)

Setup of present study Cady (1973) Dodge and Bowles (1983) Kudlac and Jurns (2006) Kudlac and Jurns (2006) Kudlac and Jurns (2006) Jurns and McQuillen (2008) Jurns and McQuillen (2008) Jurns and McQuillen (2008)

SF0.65 (295 K) LH2 (25.2 K, 3.45 bar) Ethanol, isopentane (295 K) IPA (isopropyl alcohol) LN2 (liquid nitrogen) LOX (liquid oxygen) LOX, cold (85.6 K) LOX, NBP (89.8 K) LOX, warm (94.1 K)

3.18 0.21 Not spec. 3.86 1.64 2.37 3.04 2.50 2.29

15.9 1.178 Not spec. 22.0 8.75 13.2 14.13 13.08 12.05

20.0 22.17 19 22.8 21.3 22.32 18.57 20.91 21.07

bp

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

Fig. 12. Upper left picture: Bubble point and breakthrough behavior in the open arrangement when tF =t in ¼ O(1). Here, the bubble point is followed by a sharp pressure drop and a subsequent partial recovery. This looks like a step response after which the pressure difference converges to a value below the bubble point pressure Dpbp . The bubble point itself is clearly visible as a sharp peak. In the four runs, its height varies in the range 0:69 6 Dpbp 6 0:71. When the gas injection is finally turned off, the pressure difference falls to the distinctly lower value of the bubble detachment pressure Dpd that is marked by a dashed line in each picture.

Fig. 12 shows one of the first four runs, see Table 8, where FRITZ to inertial time ratio is of order 1, tF =tin ¼ O(1), which is the smallest. In the experiments, 40 ml of gas (air) were injected within 40 s into the gas chamber below the screen, the total recording time was 100 s. As the gas injection begins, the pressure difference Dp⁄(t) rises linearly until breakthrough occurs which is followed by a sharp pressure drop and a subsequent partial recovery. This looks like a step response after which the pressure difference converges to a value below the bubble point pressure Dpbp . The bubble point itself is clearly visible as a sharp peak and marked with a circle. In the four runs, its height varies in the range 0:69 6 Dpbp 6 0:71, see also Table 9, where the bubble points of all 16 runs are summarized. As can be seen in the following figures, the bubble point varies depending on the tF/tin ratio. Therefore, we omitted the static in its denotation. After the bubble point was visible at about 8 pores, a number of pores remained open, about 5. Hence, there is a continuous breakthrough. When the gas injection is finally turned off, the pressure difference falls to a distinctly lower value where it remains. This is the bubble detachment pressure Dpd that is marked by a dashed line. The given values of Dpd are

Table 9 Values of C0 or overview on the bubble point pressure Dpbp in all runs. Due to the choice of the norm pressure difference, C 0 ¼ Dpbp . Oðt F =tin Þ

1 of 4

2 of 4

3 of 4

4 of 4

Mean + error

1 10 100 1000

0.713 0.700 0.649 0.647

0.695 0.699 0.648 0.653

0.710 0.700 0.647 0.649

0.703 0.700 0.646 0.646

0.705 ± 0.007 0.700 ± 0.000 0.648 ± 0.001 0.648 ± 0.003

37

Fig. 13. Upper right picture: Bubble point and breakthrough behavior in the open arrangement when tF =tin ¼ O(10).

calculated by averaging over the last 20 s of each measurement and range 0:44 6 Dpd 6 0:45. Fig. 13 shows one of runs 5–8, see Table 8, where FRITZ to inertial time ratio is of order 10, tF =tin ¼ O(10). In the experiments, 40 ml of gas were injected within 400s into the gas chamber below the screen, the total recording time was 460 s. Again, the pressure difference Dp⁄(t) rises linearly until breakthrough occurs. The following pressure drop is still distinct but not as striking as in Fig. 12. The number of pores that open at the bubble point is about 4 while during the subsequent continuous breakthrough this open pore number varies from 1 to 3. That produces pressure falls, rises or steady appearances in the Dp-signal. As the gas injection is turned off, the pressure difference falls down to the detachment pressure. Interestingly, while the bubble point in Fig. 13 is practically as high as in Fig. 12 where the FRITZ to inertial time ratio is ten times lower, the bubble detachment pressure is obviously higher, being at Dpbp ¼ 0:51. This is attributed to the diminished number of open pores through which the pressure can relax upon switch-off of the gas supply. Fig. 14 shows one of runs 9–12, see Table 8, where the FRITZ to inertial time ratio is of order 100 now, tF =tin ¼ O(100). In the experiments, 40 ml of gas were injected within 4000s into the gas chamber below the screen, the total recording time was 4200 s. That is why, to increase the resolution of the time axis, we changed the aspect ratio of the plots here. On first sight, it looks as though the slope of the pressure increase before the bubble point is now smaller. But due to the chosen time normalization, this slope is the same for all the presented bubble point experiments in this section. In Fig. 14, after the bubble point that opens 1 or 2 pores, the pressure difference Dp⁄ decays in a partially exponential manner and then rises again or remains at an intermediate level. This intermediate level can be found in all four runs and most of the time it marks an upper level for further falls and rises of Dp⁄. Only once, Dp⁄ mounts up above this intermediate level and reaches the initial bubble point again. In the breakthrough regime, the number

Fig. 14. Bubble point and breakthrough behavior in the open arrangement when t F =tin ¼ O(100).

38

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

Fig. 15. Bubble point and breakthrough behavior in the open arrangement when t F =t in ¼ O(1000).

of open pores is 1 or 2, sometimes 0. In all four runs where tF =tin ¼ O(100), the bubble point is now definitely smaller than in the smaller time ratio measurements and is found to be Dpbp ¼ 0:65. This suggests that more pore openings than 1 at breakthrough indicate inertial effects that affect the static bubble point insofar that they lead to an overestimation. The bubble detachment pressure is Dpd ¼ 0:51 except for run 9 where pd ¼ 0:48. Possibly, in that single case 2 pores were open to relax the pressure to a lower level. Fig. 15 shows one of runs 13–16, see Table 8, where the FRITZ to inertial time ratio is now ten times higher again, i.e. tF =tin ¼ O(1000). In the experiments, 40 ml of gas were injected within 40,000 s into the gas chamber below the screen, the total recording time was 40,200 s. Fig. 15 displays some striking features: the pressure difference cyclically falls and rises without interruption between two levels. However, while the upper level is the bubble point, the lower level changes. It is either as low as the detachment pressure or much higher. The number of cycles in the small and big interval seems randomly distributed. Another feature is the lack of a definite bubble detachment pressure: as the gas injection is turned off, the pressure practically remains where it is. In runs 13–16, the number of open pores was only 1 or 0. Therefore, the meniscus of a single pore (out of 197,020 for the whole screen) determines the Dp-signal. Table 9 provides an overview on the measured bubble points in all 16 runs. It is supplemented by Table 10 that uses the relation of Hernandez (see Eq. (6)) to illustrate how the predicted pore size of the screen is modified by the pressurization rate. The properties inserted into Eq. (6) are r = 15.9103 N/m (see Table 2), Dpbp = C0Dp0 (see Table 9 and Eq. (30)), B = 15,0106 m (see Table 1), d/dt(Dp)bp  ¼ 1=2ðlSF0:65 þ lair Þ ¼ 25:6105 as observed, see Table 10 and l Pa s (see Table 2 and lair = 1.8105 Pa s White, 2003) giving "  1=2 # 63:6103 3 5 d Dp ¼ pbp ½Pa 1 þ 18:8710 25:610 : ðDpÞbp ½Pa=s D dt

ð34Þ Basically, Table 10 suggests that the bubble point tests to determine the pore size must always take the pressurization rate into account, too. Analogously, Table 11 provides the measured detachment pressures. And finally, Table 12 lists the most important results of our experiments on static bubble point and breakthrough behavior in a

Table 10 Dependence of the apparent pore size Dp on compression rate. DP (last column) is calculated using the Hernandez-relation in Eq. (6). Oðt F =tin Þ 1 10 100 1000

Dpbp 0.71 0.70 0.65 0.65

Dpbp (Pa) 3472 3423 3178 3178

d=dtðDpÞbp 5

1.24  10 1.2  105 1.0  105 1.0  105

d/dt(Dp)bp (Pa/s)

Dp (lm)

697 67.4 5.62 0.56

18.4 18.6 20.0 20.0

Table 11 Overview on the bubble detachment pressure Dpd in all runs. OðtF =t in Þ

1 of 4

2 of 4

3 of 4

4 of 4

Mean + error

1 10 100 1000

0.452 0.507 0.481 –

0.449 0.509 0.515 –

0.435 0.509 0.513 –

0.448 0.506 0.514 –

0.446 ± 0.007 0.508 ± 0.001 0.506 ± 0.014 –

Table 12 Summary of all results in our the static bubble point experiments. tF/tin

0.8

8.0

79.6

796

C0 Dpd Open pores

0.705 0.446 8–5

0.700 0.508 4–1

0.648 0.506 2–1

0.648 – 1–0

Table 13 Overview on the four tF =tin ¼ O(1000)-runs. The position series of local minima uses ‘d’ for a Dp⁄-value as low as the detachment pressure and ‘m’ for the intermediate level. Run

position series of local minima

C0

Dpd

Dpm

13

2m4d6md5md3md3md3md 3md4md3md4md5mdmd4m

0.65

0.51

0.60

14

3dm2dm2dmd4md8m2d 6md7md5m2d6md

0.65

0.51

0.60

15

d2md4m2d3md3md2mdmd 2mdm3dmdmd3md9md6md

0.65

0.51

0.60

16

d5md2md4md4md16md2m d2md10md16m

0.65

0.51

0.59

condensed manner. Special attention is paid to the latter four runs in Table 13. Since these four runs were performed with the lowest gas supply rate, their results come closest to the ideal of a static bubble point measurement. Therefore, they allow to pinpoint not only the static bubble point but also the detachment pressure. And interestingly, even an intermediate level of detachment pressure is observed here and there is no telling to which of the two Dpd -levels the pressure difference Dp⁄ will relax upon breakthrough. In the second row of Table 13 we list whether the pressure relaxation after breakthrough ended at the intermediate level (‘m’) or the detachment level (‘d’). For example, ‘2m4d. . .’ means that the relaxation ended two subsequent times at the intermediate level, then four subsequent times at detachment level and so on. Apparently, there is no pattern. The characteristic length scales involved with the static bubble point are illustrated by Fig. 16: (i) the always available particle retention size that was used to normalize the Dp-signal, (ii) the bubble point pore size which is obtained in the bubble point measurements, see also Table 7, (iii) the bubble detachment size that corresponds to the pressure below which the bubbles cannot grow and detach anymore and (iv) the size of the FRITZ bubble to which

39

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41 Table 14 Trapped bubble volumes with corresponding contact position and cover ratios. VTB (ml) cTB (mm) C (%)

Fig. 16. Length scales involved in bubble point and breakthrough at the Dutch Twilled weave 200  1400, using silicon oil SF0.65 and air as test liquid and gas, respectively. (1) Size of spherical particles that can pass the pores. (2) Bubbles point equivalent pore size. (3) Bubble detachment equivalent pore size. (4) Size of the FRITZ bubble up to which the bubble grows before it can detach.

0.25 5.0 8.7

0.5 7.2 17.9

0.75 8.8 26.8

1.0 10.2 36.0

1.25 11.4 45.0

1.5 12.5 54.1

1.75 13.6 64.0

t0

t0 + t

t = t0 + 2 t

t0 + 3 t

t0 + 4 t

t0 + 5 t

t0 + 6 t

t0 + 7 t

t0 + 8 t

t0 + 9 t

t0 + 10 t

t0 + 11 t

t0 + 12 t

t0 + 13 t

t0 + 14 t

t0 + 15 t

t0 + 16 t

t0 + 17 t

t0 + 18 t

t0 + 19 t

t0 + 20 t

t0 + 21 t

t0 + 22 t

t0 + 23 t

the bubbles grow upon breakthrough before they can detach. To help association, the background shows an image of DTW200  1400 taken with a scanning electron microscope.

5.3. Dynamic bubble point 5.3.1. The trapped bubbles The volumes of trapped bubbles VTB that can be examined with our experimental setup have been determined before as outlined in the ‘State of the art’ section. A number of other properties have been extracted as well from the contours. Fig. 17 shows the coordinates of contact position cTB, the radial extension RTB along with the aspect ratio cTB/HTB of the trapped bubbles in dependence of VTB. Due to the given tube radius in the setup, RTB,max = 17 mm. That b TB  2:5 ml and yields a maximum volume of the trapped bubble V a maximum contact position cTB,max  16.2 mm. Since the remaining open area for the liquid flow, and thus C, is determined by the contact position, the cover ratio according to its definition C ¼ c2TB =R20 cannot reach unity. Instead, Cmax  0.9. However, if RTB = R0 = 17 mm, the tube is blocked and therefore the screen as well. The chosen test volumes for trapped bubbles in our experiment, along with their corresponding contact positions and cover ratios are presented in Table 14. But RTB = R0 is only possible under static conditions, i.e. QL = 0. If QL > 0 and the bubble radius approaches the wall, the thinning gap gives rise to a suction

Fig. 17. Predicted contact position and radius of the bubble contours depending on the bubble volume VTB. The data points cover a range of contours with a contact position to height ratio cTB/HTB = 0; 0.1; . . . ; 5.4, the integer ratios are highlighted. Naturally, the bubble radius RTB always exceeds the contact position cTB. Shortly below VTB = 2.5 ml, the bubble touches the tube wall at R0 = 17 mm.

Fig. 18. Simultaneous camera view from above and below onto the screen during a linear increase of the liquid flow rate around the trapped bubble and through the screen. As the pictures elucidate, the bubble (i) remains trapped until a critical liquid flow rate is surpassed, (ii) then starts to break through whereby the pressure drop across the screen falls, (iii) eventually stops to break through since the pressure difference has fallen below the detachment pressure and (iv) the process is repeated until the bubble has completely broken through.

pressure that chokes this gap further until RTB = 1 which triggers a sudden gas breakthrough. In the experiment, we indeed observe this effect. Applied trapped bubble volumes in the experiment were VTB = 0;0.25; . . . ; 1.75 ml. Hence, C  0;10%; . . . ; 70%. Larger bubbles could of course be injected but broke through too fast to collect useful data. 5.3.2. Trapped bubble at increasing flow rate Fig. 18 pictures the typical chain of events that can be observed when a trapped bubble is submitted to a increasing liquid flow, starting from quiescent liquid (t = t0 frame). The volume of the trapped bubble was here so big that it blocked the screen completely. Therefore, as soon as the liquid starts to flow, the bubble is squeezed and a first breakthrough is forced as shown in the second frame at t = t0 + Dt. With gas breaking through the screen, the trapped bubble shrinks. And as soon as the bubble does not plug the flow anymore, this first breakthrough stops (t = t0 + 2Dtframe). The liquid flow is now concentrated to just the narrow annular gap around the trapped bubble. But with the increasing flow, the pressure drop across the screen rises and eventually reaches the bubble point pressure which causes a second breakthrough (t = t0 + 3Dt-frame). Combined with the shrinkage of the trapped bubble is a decrease of the screen coverage. Hence, the pressure drop might relax below the bubble detachment pressure and the breakthrough stops again (t = t0 + 4Dt-frame). This cycle of breakthrough and stop, breakthrough and stop is repeated some times. But with every single breakthrough the volume of the

40

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

Fig. 19. Breakthrough cycles of a trapped bubble in increasing liquid flow. The example in the diagram shows a bubble with VTB = 1.5 ml which corresponds to a cover ratio C = 54%. The data points are obtained in steady flow, while the thin lines connecting these data points mark transient flow. It shows that the first breakthroughs cannot be explored with steady flow. Moreover, it shows that the dynamic bubble point level seems independent of the flow rate.

.8

=

36

.0

.0

%

%

%

45 =

=5 4.1

=6 4.0

%

trapped bubble is further diminished. That implies two consequences: (i) the additional pressure drop due to liquid flow rate increase eventually compensates the pressure decrease due to bubble shrinkage which results in longer breakthrough duration. And (ii) in the end, the last bit of gas will be broken through the screen. While Fig. 18 shows camera snapshots of the observed effects, Fig. 19 shows a comparable data record Dp/Dp0 versus Re0 on the dynamic bubble point in ramping liquid flow rate. In single-phase flow, the pore REYNOLDS number is more useful for dimensionless presentation but with the bubble in place it is more appropriate to use the empty tube REYNOLDS number. Please note that the two figures were extracted from two different experiments: while in Fig. 18 we started with a bubble of maximum volume (2.5 ml), in Fig. 19 we started with a better defined bubble of VTB = 1.5 ml. The latter experiment was also repeated three times. In the Fig. 18 experiment, the liquid flow rate QL grew steadily with dQL/dt  1 ml/s2 while the data record in Fig. 19 shows discrete data points. To collect it, the liquid flow rate was stepwise increased and each time held for some seconds to guarantee steady flow conditions. But as QL approaches the breakthrough level for

=

26

% =

17

.9

%

=

8.7

%

=0

static bubble point detachment pressure

an actual bubble, the pressure difference rises in a singular manner due to the interaction between bubble and tube wall. This is evidently an important effect that should be tackled in future, but it is beyond the scope of the present paper. The thin lines between the discrete data points give a clue how fast this bubble-wall interaction happens. Beside that, the upper edges of these thin lines reveal that the dynamic bubble point pressure oversteps the static bubble point (dashed line) to no more than about 15 percent. However, since the bubble-wall interaction event is so transient that we cannot resolve the peak edges (just one point), it cannot be outruled that the real peak value lies even higher. Fig. 19 shows how often the breakthrough cycle repeats itself. For all four runs with VTB = 1.5 ml shown in the figure, 7 or 8 breakthroughs can be distinguished. The pressure difference where the dynamic bubble point occurs appears rather independent of the REYNOLDS number. The pressure difference where the breakthrough stops, on the other hand, depends on it. It seems that, the higher the volume of the trapped bubble is, the lower sinks the pressure difference before breakthrough stops. When all gas of the trapped bubble has broken through the screen, we are in the single-phase flow regime again, see also Fig. 9, which happens here at Re0  6000. The hydraulic resistance of the screen with different trapped bubble volumes is shown by Fig. 20. Combined in the figure are measurements for seven different bubble volumes as well as no bubble (single-phase flow), each measurement was performed three times. VTB = 0;0.25 ml; . . . ; 1.75 ml were chosen as test volumes which, according to Eq. (25), correspond to C = 0;10%; . . . ; 70%. Moreover, the measured data is compared with the model predictions for each bubble volume. This is accomplished by applicaf1 and C f2 being tion of Eq. (27), the necessary constants C calibrated by the outcome of the single-phase flow tests. It shows that the model correctly predicts the trend of Dp vs. Re0 up to say 30% of the bubble point pressure. Only then, the experimental data points deviate from the model predictions. As explanation of this deviation we assume flow-induced deformation of the bubble culminating in bubble-wall interaction. This phenomenon deserves more attention in future. 6. Summary We have presented a model and original experimental data for the separation of gas and liquid phase at a woven metal screen. In our experimental setup we have used a Dutch Twilled 200  1400 weave as test screen and silicon oil SF0.65 as liquid. We have examined the influence of the gas supply rate on breakthrough characteristics like bubble point and bubble detachment. The role of bubble growth and detachment from pores on the upper side of the screen was established for the first time. It is an essential detail of any model to predict the dynamic bubble point. The model we present here accounts for the (i) static shapes of trapped bubbles as solution of the axially symmetric YOUNG–LAPLACE equation, (ii) degree of screen coverage caused by a trapped bubble and corresponding pressure drop across the screen, (iii) bubble point limit and (iv) detachment limit. The model is used to predict the hydraulic resistance of a screen with trapped bubble and to explain a number of experimentally observed effects. The model allows to predict and explain most of the experimental observations. 7. Conclusions

Fig. 20. Hydraulic resistance (slope of pressure difference vs. flow rate) of a woven screen with bubbles trapped at it and comparison with model predictions. While the model prediction coincides with the data for low pressure differences, it deviates at higher pressure differences, say above 30% of the bubble point threshold. This deviation is attributed to flow-induced deformation of the bubble which was not taken into account in the model.

The following conclusions can be drawn from our work: (1) Static bubble point measurements are most reliable when just a single pore opens at breakthrough. Otherwise, the bubble point is overestimated.

M. Conrath, M. Dreyer / International Journal of Multiphase Flow 42 (2012) 29–41

(2) The bottleneck effect upon breakthrough (that a pore remains open once it is opened) can be explained by the decreasing meniscus pressure once the meniscus has moved pass this obstacle. (3) The FRITZ to inertial time ratio is a useful parameter to describe the breakthrough behavior in static bubble point experiments. (4) The bubble detachment pressure marks a lower limit where breakthrough stops. Like the bubble point, the detachment pressure depends on the gas supply rate. (5) The detachment pressure which allows a breakthrough to stop is an essential element in understanding the dynamic bubble point. (6) Flow-induced deformation of a trapped bubble effectuates the bubble point at lower REYNOLDS numbers because the flow tends to tear the bubbles apart what makes them cover more screen area. (7) Sudden extension of the bubble towards the wall followed by breakthrough poses a singular event and can be explained as a positive feedback of flow-induced bubble deformation and wall interaction. (8) Due to flow-induced deformation and bubble-wall interaction, the hydraulic resistance of trapped bubbles in liquid flow is underestimated by our model which assumes static bubble shapes. (9) The bubble point pressure in flow (dynamic bubble point) seems to be independent of the REYNOLDS number and lies about 15% above the static bubble point.

Acknowledgements The funding of the research project by the German Federal Ministry of Economics and Technology (BMWi) through the German Aerospace Center (DLR) under Grant No. 50 RL 0741 is gratefully acknowledged. Moreover, the authors thank Holger Faust for his help with the experimental setup. References Armour, J.C., Cannon, J.N., 1968. Fluid flow through woven screens. AIChE J. 14, 415– 420. ASTM International F316-03, 2003. Standard Test Methods for Pore Size Characteristics of Membrane Filters by Bubble Point and Mean Flow Pore Test, 7p. Bashforth, F., Adams, J.C., 1883. An Attempt to Test the Theories of Capillary Action by comparing the Theoretical and Measured Forms of Drops of Fluid. Cambridge University Press. Behruzi, P., Dodd, C., 2007. Future propellant management device concepts for restartable cryogenic upper stages. In: Proceedings of the 43nd AIAA/ASME/ SAE/ASEE Joint Propulsion Conference, Cincinatti (Ohio, USA), 8–11 July 2007. Bhatia, S.K., Smith, J.L., 1994. Comparative study of bubble point method and mercury intrusion porosimetry techniques for characterizing the pore-size distribution of geotextiles. Geotextiles and Geomembranes 13, 679–702. Bhatia, S.K., Smith, J.L., 1995. Application of the bubble point method to the characterization of the pore-size distribution of geotextiles. ASTM Geotechnical Testing Journal 18, 94–105. Bolanos-Jimenez, R., Sevilla, A., Martinez-Bazan, C., Gordillo, J.M., 2008. Axisymmetric bubble collapse in a quiescent liquid pool. II. Experimental study. Physics of Fluids 20, 112104, 12p.. Bolanos-Jimenez, R., Sevilla, A., Martinez-Bazan, C., van der Meer, D., Gordillo, J.M., 2009. The effect of liquid viscosity on bubble pinch-off. Physics of Fluids 072103, 9p.. Brundrett, E., 1993. Incompressible flow through screens. Journal of Fluids Engineering 115, 239–242. Cady, E.C., 1973. Study of Thermodynamic Vent and Screen Baffle Integration for Orbital Storage and Transfer of Liquid Hydrogen. NASA Report No. CR-134482. . Choat, B., Ball, M., Luly, J., Holtum, J., 2003. Pit membrane porosity and water stressinduced cavitation in four coexisting dry rainforest tree species. Plant Physiology 131, 41–48.

41

Dodge, F.T., 1990. Low-gravity fluid dynamics and transport phenomena. Progress in Astronautics and Aeronautics 130, 3–14 (Chapter 1). Dodge, F.T., 2000. The New Dynamic Behavior of Liquids in Moving Containers. Southwest Research Institute. Dodge, F.T., Bowles, E.B., 1983. Vapor flow into a capillary propellant acquisition device. In: Proceedings of the 19th AIAA/ASME/SAE Joint Propulsion Conference, Seattle (Washington, USA), 27–29 June 1983. Ergun, S., 1952. Fluid flow through packed columns. Chemical Engineering Progress 48, 89–94. Fritz, W., 1936. Berechnung des Maximalvolumens von Dampfblasen. Physikalische Zeitschrift 36, 379–384. Gordillo, J.M., 2008. Axisymmetric bubble collapse in a quiescent liquid pool. I. Theory and numerical simulations. Physics of Fluids 20, 112103, 15p.. Grah, A., Dreyer, M., 2010. Dynamic stability analysis for capillary channel flow: one-dimensional and three-dimensional computations of the equivalent steady state technique. Physics of Fluids 22, 014101. Hernandez, A., Calvo, J.I., Pradanos, P., Tejerina, F., 1996. Pore size distributions in microporous membranes. A critical analysis of the bubble point extended method. Journal of Membrane Science 112, 1–12. Hetsroni, G., Haber, S., Wacholder, E., 1970. The flow fields in and around a droplet moving axially within a tube. Journal of Fluid Mechanics 41, 689–705. Jones, D.P., Krier, H., 1972. Gas flow resistance measurements through packed beds at high Reynolds numbers. Journal of Fluid Engineering 105, 168–173. Jurns, J.M., Kudlac, M.T., 2006. NASA Glenn Research Center Creek Road Complex – cryogenic testing facilities. Cryogenics 46, 98–104. Jurns, J.M., McQuillen, J.B., 2008. Liquid acquisition device testing with sub-cooled liquid oxygen. In: Proceedings of the 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Hartford (Connecticut, USA), 21–23 July 2008. Jurns, J.M., McQuillen, J.B., Gaby, J.D. Jr., Sinacore, S.A. Jr., 2009. Bubble Point Measurements with Liquid Methane of a Screen Channel Capillary Liquid Acquisition Device. NASA Report No. TM-2009-215494. . Kopf, P., Piesche, M., Schtz, S., 2007. Beschreibung des Druckverlusts von Drahtgeweben mit Hilfe von hnlichkeitsgesetzen. Filtration & Separation 21, 330–335. Kreyszig, E., 2006. Advanced Engineering Mathematics, ninth ed. Wiley. Kudlac, M.T., Jurns, J.M., 2006. Screen channel liquid acquisition devices for liquid oxygen. In: Proceedings of the 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Sacramento (California, USA), 9–12 July 2006. Kumar, N., Kuloor, N.R., 1970. The formation of bubbles and drops. Advances in Chemical Engineering 8, 255–369. Langbein, D., 2002. Capillary Surfaces. Shape – Stability – Dynamics, in Particular Under Weightlessness. Springer, Berlin. Laws, E.M., Livesey, J.L., 1978. Flow through screens. Annual Review of Fluid Mechanics 10, 247–266. Munson, B.R., 1988. Very low Reynolds number flow through screens. Journal of Fluids Engineering 110, 462–463. Myshkis, A.D., Babskii, V.G., Kopachowskii, N.D., Slobozhanin, L.A., Tyuptsov, A.D., 1987. Low-Gravity Fluid Mechanics. Springer Verlag, Berlin. Oguz, H.N., Prosperetti, A., 1993. Dynamics of bubble growth and detachment from a needle. Journal of Fluids Mechanics 257, 111–145. Olbricht, W.L., 1996. Pore-scale prototypes of multiphase flow in porous media. Annual Review of Fluid Mechanics 28, 187–213. Padday, J.F., 1971. The profiles of axially symmetric menisci. Philosophical Transactions of the Royal Society of London, Series A 269, 265–293. Payne, L.E., Pell, W.H., 1960. The Stokes flow problem for a class of axially symmetric bodies. Journal of Fluid Mechanics 7, 529–549. Purchas, D.B., Sutherland, K., 2002. Handbook of Filter Media. Elsevier Science Ltd.. Reichelt, G., 1991. Bubble point measurements on large areas of microporous membranes. Journal of Membrane Science 60, 253–259. Rosendahl, U., Ohlhoff, A., Dreyer, M., 2004. Choked flows in capillary channels: theory, experiment and computation. Journal of Fluid Mechanics 518, 187–214. SAE International, 2001. Aerospace Recommended Practice (ARP) 901, Bubble Point Test Method. Schütz, S., Kopf, P., Piesche, M., 2008. Prediction of pore size and pressure drop of porous woven wire cloth filter media on the basis of calculation models. Filtration 8, 259–286. Sodre, J.R., Parise, J.A.R., 1997. Friction factor determination for flow through finite wire-mesh woven-screen materials. Journal of Fluid Engineering 119, 847–851. Taylor, T.D., Acrivos, A., 1964. On the deformation and drag of a falling drop at low Reynolds number. Journal of Fluid Mechanics 18, 466–476. Thoroddsen, S.T., Etoh, T.G., Takehara, K., 2007. Experiments on bubble pinch-off. Physics of Fluids 19, 042101, 29p.. Thoroddsen, S.T., Etoh, T.G., Takehara, K., 2008. High-speed imaging of drops and bubbles. Annual Review of Fluid Mechanics 40, 257–285. Valli, A., Hyväluoma, J., Jäsberg, A., Koponen, A., Timonen, J., 2009. Pressure drop for low Reynolds number flows through regular and random screens. Transport in Porous Media 80, 193–208. Washburn, E.W., 1921. Note on a method of determining the distribution of pore sizes in a porous material. Proceedings of the National Academy of Sciences of the United States of America 7, 115–116. White, F.M., 2003. Fluid Mechanics, fifth ed. McGraw Hill, New York. Wu, W.T., Liu, J.C., Li, W.J., Hsieh, W.H., 2005. Measurement and correlation of hydraulic resistance of flow through woven metal screens. International Journal of Heat and Mass Transfer 48, 3008–3017.