ITP-09-11

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Michael Conrath. ZARM ... The hydraulic resistance of porous screens was tackled by .... the detachment. Before a bubble detaches a neck forms that pinches. ... current and provides a maximum power of 1.2 kilowatts. The ..... J. Fluid Mech.
Proceedings of ITP2009 Interdisciplinary Transport Phenomena VI: Fluid, Thermal, Biological, Materials and Space Sciences October 4-9, 2009, Volterra, Italy

ITP-09-11 GAS SEPARATION AND BUBBLE BEHAVIOUR AT A WOVEN SCREEN Michael Conrath

ZARM, University of Bremen, Germany

ABSTRACT Gas-liquid two phase flows are widespread. In many applications the separation of both phases is necessary. Chemical reactors, water treatment devices or gas-free delivery of fluid and fuel are only some of them. We study the suitability and performance of woven metal screens for space applications. To this end, we have built an experiment that consists of a Dutch Twilled woven screen made of stainless steel in a vertical acrylic glass tube that can carry upward as well as downward flow. The screen is suspended perpendicular to the flow which is forced through it at variable strength. Controlled injection of air bubbles allows us to examine the ability of the screen to separate air and liquid. We present experimental data on different methods to measure the static bubble point, i.e. the pressure difference when the air breaks through the weave. Moreover, the behaviour of a single bubble that is trapped under the weave in upward flow is scrutinized. The relation of its breakthrough behaviour (dynamic bubble point) to the static bubble point is addressed. NOMENCLATURE (e.g.) a geometry factor/ orifice radius B thickness of the weave Dpore pore diameter f friction factor K curvature p pressure Q flow rate R radius s arc length t time u velocity V volume Re Reynolds number

Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

Michael Dreyer

ZARM, University of Bremen, Germany

Bo

φ ρ ν θ σ τ

Bond number porosity density kinematic viscosity contact angle surface tension tortuosity

1. INTRODUCTION Figure 1 illustrates the scope of this paper.

Figure 1: Configurations for static bubble point (left picture and dynamic bubble point (right picture). A porous test screen is suspended in a vertical tube. The screen is wetted by a liquid, therefore gas needs to be pressurized up to the bubble point to enter the screen pores and hence to break through the screen. If the complete screen is covered by pressurized gas, see left picture of Figure 1, then the gas will break through when the pressure drop over the screen exceeds the static bubble point. If the gas does not cover the complete screen, see right picture of Figure 1, a trapped bubble forms which cannot be pressurized. A breakthrough of this bubble is now triggered by the flow-induced pressure drop over the screen. Such configurations are of interest for example in gas-

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Paper No: ITP-09-11

liquid separation in spacecrafts to ensure the gas-free delivery of propellant (Dodge 1990, Dreyer 2007). The problem addressed in this paper is composed of several sub-problems: i) hydraulic resistance of a porous screen, ii) static bubble point of a porous screen, iii) bubble formation 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 was tackled by Armour & Cannon 1968, Cady 1973, Wu et al. 2005, Kopf et al. 2007. The static bubble point was tackled for example by Washburn 1921 and Schütz et al. 2008. The formation of bubbles on the upside of the porous screen as the static bubble point threshold is surpassed is tackled by Kumar & Kuloor 1970 and Oguz & Prosperetti 1993. The shape of axially symmetric bubbles that form below the screen was first derived by Bashforth & Adams 1883. Padday 1971 basically applied the same procedure but aided by a personal computer. He modified and extended the tables. Langbein 2002 reviews the trapped bubble problem focusing also on analytical approaches. It seems that the very special geometry of a trapped bubble in stagnant flow was not yet examined. Therefore, also the dynamic bubble point is a novel problem. After expounding the theoretical basis of our problem we will describe the setup and results of the performed experiments and discuss the results. 2. THEORETICAL BASIS Hydraulic resistance of the woven screen Liquid or gas flow through the woven screen causes a pressure drop between its front side and backside. According to Armour & Cannon [3], the pressure drop at a woven screen can be described by

∆p = f AC

τB D pore

u ρ  φ 

2

with

f AC =

Cν + Ci Re AC

.

(1) Hereby, the Reynolds number Rep in the pores is defined by

Re AC =

1 u . a D pore ν

(2)

2

γ

fWu

D poreu  1−φ  1−φ =α +β  and ReWu = (1 − φ ) ν ReWu  ReWu 

(4) Kopf et al. [6] correlate the pressure drop in terms of Euler number over Reynolds number for different weave types but also need experiments to fit the model coefficients. Static bubble point Supposing the screen is wetted by a liquid, its pores are to some extend blocked for gas flow through it. Therefore, the gas needs to be pressurized to push back the meniscus in the pores. The static bubble point pSBP denotes the pressure difference between front side and back side of the screen at which gas will break through the screen. Typically, one considers a capillary tube equivalent pore radius and sets 2σ . (5) pSBP = C (cos θ , geometry ) R pore

Figure 2: Schematic pore arrangement and location of the menisci in the screen arrangement. The bubble point only gives the largest pore. To obtain a pore spectrum, the gas flow rate and thus the pressure difference after breakthrough is continuously increased to open also the smaller pores. In the end, all pores are opened and the pressure difference follows the hydraulic resistance line in single phase flow. Bubble growth and detachment As breakthrough occurs, a bubble grows before it detaches. We assume that it remains spherical, i.e. that the bubble diameter remains small compared to the capillary length.

Armour & Cannon derive the viscous and inertial coefficients Cν and Ci from the fit on a number of experiments carried out on four different weave types in Helium flow. Cady [4] uses the same model approach but fits the coefficients for each weave separately. Wu et al. [5] use the same experimental data as Armour & Cannon but apply the different relations

∆p = fWu

ρu 2 L  1 − φ    D pore  φ 2 

with

(3)

Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

Figure 3: Growth of a single bubble at the pore. Left: sketch of the arrangement, right: meniscus pressure at the growing bubble.

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Depending on the pore diameter a, the volume Vb of the growing bubble that can be decomposed to a cone and spherical sector becomes

Vb =

π 3 2R + ( a2 + 2R2 ) R2 − a2  3



.

(6)

The smallest bubble radius occurs when R = a, meaning a semispherical bubble that also relates to the pressure threshold commonly named bubble point. We normalize the bubble volume and pressure with the values of this threshold bubble *

Vb =

a Vb * , pσ = pσ 3 2σ ( 2 / 3) π a

(7)

where pσ = R/(2σ). Figure 4 shows the connection between growing bubble volume and the meniscus pressure. Apparently, if the bubble point threshold was surpassed at one pore, a bubble will grow there providing a pressure relaxation that hinders the bubble growth at other pores. Neglecting the impact of viscosity, the dynamics of bubble growth obeys the Rayleigh-Plesset equation

 + 3 R 2 = 1  p − 2σ − p  , RR B ∞ 2 R ρ  

(8)

see for example Oguz & Prosperetti [10]. It shows that an important point is reached when the bubble has grown to the Fritz volume, i.e. the volume where buoyancy balances the surface tension force. The Fritz volume is

 3σ a  VF =    2ρ g 

1

3

.

(9)

The bubble cannot detach before the Fritz volume is reached which happens in the Fritz time

RF3 µ g l VF 32 . = tF = 3 a 4 ( pC − p∞ ) Q

(10)

On the other hand, strong gas flow through the pores can delay the detachment. Before a bubble detaches a neck forms that pinches. After the moment of breakoff, a bubble rises and at the orifice remains a meniscus whose curvature has abruptly increased. This after-breakoff curvature facilitates another pressure threshold that must be overcome for a lasting gas breakthrough. 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 constriction of liquid flow around the bubble. That shape of a trapped bubble is identical to that of a sessile drop. It is found by a solving the axially symmetric YoungLaplace equation, see for example Langbein [13],

  r ′′ 1   p( z = 0) − p∞ + ∆ρ gz = σ + 3 1   2 2 2 2 ′ r (1 + r ′ )   (1 + r )

Figure 4: Shapes of trapped bubbles with different volumes found by applying a shooting method. which can be transformed into curvilinear coordinates

dα sin α ρ a gz p( z = 0) − p∞ + = + . σ σ ds r

(12)

After normalizing this equation its structure simplifies to

dα sin α + = Bo z + K 0 . ds r

(13)

This first order ODE can now be solved aided by a shooting method. Although the above equations stand for a bubble at rest, the results give also a clue to the bubble behaviour in flow. Dynamic bubble point Presumably, a trapped bubble in flow will only break through when the flow-induced pressure drop over the screen is beyond the bubble point threshold. The flow-induced pressure drop depends on the flow velocity u through the screen. But this velocity, in turn, depends on the cross section of the tube A0 which is constricted by the bubble. If the bubbles maximum radius is denoted rmax and the liquid flow rate is dV/dt then liquid flow velocity is

u=

dV / dt 2 A0 − π rmax

(14)

which goes into Equation (1) or (3), respectively. Hence, as the bubble radius increases, the pressure drop over the screen increases. This is also interesting because the flow streams around the bubble and produces a radial lift. Consequently, as the dynamic pressure exceeds the capillary pressure at the bubble surface, the bubble radius will suddenly grow until breakthrough occurs. And since each breakthrough is accompanied by a pressure relaxation, the breakthrough will eventually stop to start the process again.

(11)

Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

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3. EXPERIMENTAL SETUP

section is shown on the left picture in Figure 6. In its center is the test screen located, see right picture of Figure 6.

We have built up an experiment as shown in Figure 5. It comprises a closed hydraulic circuit in which a woven screen is suspended in the flow. As liquid we use silicone oil SF0.65 (supplier Wacker, Germany) with the properties as given in Table 1. T/[K] ν/[mm²/s] µ/[mPas] ρ/[kg/m³] σ/[mN/m] 293 0.65 0.6 760 15.9 Table 1: Properties of silicone oil SF0.65 The pump in our setup is supplied by 3 phase alternating current and provides a maximum power of 1.2 kilowatts. The pump is controlled from a personal computer by software. Its momentum is transferred to the flow with a pumping head. Figure 6: Core of the test section (left picture) and test screen woven of stainless steel wires (right picture) 100 millimeters below or above the test screen, air can be injected by a step motor based syringe system (supplier Nemesys). Two feedthroughs shortly above and below the test screen allow to measure the pressure difference between its upper and lower side. The test screen itself is a Dutch Twilled weave 200x1400 (supplier Spoerl, Germany) which has a structure as illustrated by Fig.7.

Figure 7: Structure of the Dutch Twilled weave used as test screen. Left picture: CAD model, right picture: REM of a probe that also shows traces from the production process. Figure 5: Setup of the two-phase flow experiment to investigate gas separation at a woven screen The hydraulic lines have an inner diameter of 6 millimeters and are made of PVC. A filter is used to clean the fluid from particles that otherwise are always present and would falsify the 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 2300 x 325 (wires per inch in weft direction x wires per inch in warp direction) with a particle retention size of 2 micrometers. Pass the filter the flow rate is measured before a switch network directs the liquid either upwards or downwards through the vertical test section. After the test section the liquid enters the separator basin to free it from the gas bubbles that where dragged in the test section. The vertical test section is 1200 millimeters long and mirror symmetric in respect to the test screen to guarantee equal flow conditions upon flow reversal. The main component of the test Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

It is a densely woven pattern consisting of rather straight warp wires and strongly deformed weft wires that meander between both sides of the weave. There is no direct flow path through the pores of this weave type. Instead, at each pore the flow splits in two branches that wind to the other side. The retention size for solid spherical particles is about 12 micrometers (according to supplier). Two high-speed cameras (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 i) the liquid flow pump rate, ii) the air flow injection rate and iii) the time and amount of recorded data points.

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4. RESULTS – SCREEN RESISTANCE The hydraulic resistance of the screen, i.e. the flow-induced pressure drop over the screen, was measured by applying a ramp field of the liquid flow rate. No gas was present and different ramp slopes were tested to rule out inertial effects. The obtained hydraulic resistance curve is shown in Figure 8.

The static bubble point traditionally depicts the pressure where gas is pressed through a wetted pore of the screen. In general, the static bubble point pressure is regarded a constant. But we will show here that the level of this bubble point is hardly a constant. Beside that an interesting behavior is observed after breakthrough. Figure 10 illustrates the setup.

Figure 8: Hydraulic resistance of the Dutch Twilled weave In Figure 8, three identical measurements were performed to check for reproducibility. Obviously, the ration between induced pressure drop and flow velocity increases with the flow. But this increase is not governed by a power law as Figure 9 reveals.

Figure 10: Experimental arrangement to measure the gas flow rate dependent static bubble point. A gas volume of 100 milliliters is trapped below the test screen, liquid flow is blocked. Gas is supplied to this trapped volume at different rates. Therefore, the gas pressure will steadily increase until breakthrough occurs. Starting with a rather strong gas supply rate of 1 milliliter per second, see Figure 11, the pressure difference over the screen increases rapidly until breakthrough occurs. The breakthrough limits the pressure difference. But due to the interaction of pressure relaxation and bubble detachments after breakthrough, there is no smooth plateau. Instead, a wild

Figure 9: Logarithmic representation of hydraulic resistance. Figure 9 also displays the data in dimensionless form, i.e. in terms of friction factor versus pore Reynolds number. As the pore Reynolds number decreases, the friction factor increases. By trend, for a pore Reynolds number zero, the friction factor of the weave would be infinity. Because the pore Reynolds number relates to the pore size, this can be explained by assuming a pore size of zero which is a closed plate with infinite hydraulic resistance. 5. RESULTS – STATIC BUBBLE POINT

Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

Figure 11: Static bubble point measurements at a gas supply rate of 1 ml/s.

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fidget is observed. Also when the gas supply is switched off, something interesting is observed: there is no complete pressure relaxation down to zero but only down to a lower threshold. Presumably, that threshold marks the bubble detachment pressure.

The results of another decrease of the gas supply rate by a factor of ten are shown in Figure 13. While the bubble point pressure in Figure 11 was almost 35 mbar and in Figure 12 only slightly below that value, it is now rather at 32 mbar. Besides, the pressure signal now includes clear exponential pressure relaxations as well as recompressions. Curiously, there seems to be a pressure meta-level somewhere in between bubble point pressure and bubble detachment pressure. Looking at Figures 11-13, the bubble detachment pressure, too, is not really a constant property.

Figure 12: Static bubble point measurements at a gas supply rate of 0.1 ml/s. Now we decrease the gas supply rate by a factor of ten, see Figure 12. Again, the pressure rises linearly until breakthrough occurs. Again, there is a fidget instead of a smooth plateau. After some time, the pressure relaxation outgoes the gas resupply and a periodic pressure signal is observed. The reason why this happens only after some time and not at once could be the lack of liquid flow in the circuit. Because then the gas bubbles that passed the screen will collect in the upper part of the test section, see Figures 5 and 10, and therefore vary the absolute pressure.

Figure 14: Static bubble point measurements at a gas supply rate of 0.001 ml/s. If the gas supply rate is again decreased by a factor of ten, see Figure 14, each breakthrough leads to a complete pressure relaxation down to the bubble detachment pressure. Then, as soon as the pores are closed, the pressure rises again. Moreover, the gas supply rate is so low now, that another effect comes into play, namely the gas dissolving ability of the liquid. This gas dissolving was also observed in downward flow experiments with trapped gas below the screen, but then much stronger. Due to the gas dissolving, trapped and injected gas (1 cubic millimeter per second) are absorbed by the liquid which makes the pressure fall. 6. RESULTS – DYNAMIC BUBBLE POINT

Figure 13: Static bubble point measurements at a gas supply rate of 0.01 ml/s. Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

For the dynamic bubble point measurements, several effects are important: i) flow-induced pressure drop over the screen, ii) flow constriction by the trapped bubble alters flow velocity, iii) bubble radius depends on bubble volume, iv) bubble volume changes at breakthrough, v) bubble is deformed by the flow and vi) bubble is compressed by the absolute pressure that is build up by the flow losses in the rest of the circuit. The latter effect, i.e. the compression due to absolute pressure, can be derived from Figure 15. It shows the absolute pressure

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below the test screen when the closed hydraulic circuit is used. This corresponds to a standard situation.

arise from many spots on the upper side of the weave. The breakthrough leads to a decompression. Eventually the pores

Figure 15: Absolute pressure below the screen in the closed circuit arrangement.

Figure 17: Bubble breakthrough at the dynamic bubble point.

At the bubble point around 35 mbar the absolute pressure has double height compared to no liquid flow. Thus the bubble will be squeezed to half its volume at rest. We have therefore considered two arrangements, see Figure 16. One arrangement is the closed hydraulic circuit that corresponds to the standard case. The second arrangement has an overflow shortly above the test screen which effectively limits the absolute pressure below the screen where the trapped bubble is located.

Figure 16: left) closed circuit arrangement and right) arrangement with overflow. In both cases the experiment is to continuously increase the liquid flow rate. The flow passes the bubble and causes a pressure drop over the screen. Up to the bubble point pressure the pressure drop looks very similar to Figure 8. Shortly before breakthrough occurs, the bubble is observed to suddenly expand which is contributed to the radial lift. Due to the radial expansion, the flow constriction is amplified and along with it the radial lift. In the end, the pressure drop exponentially rises to the bubble point pressure and the bubble breaks through. Figure 17 shows the simultaneous view on the bubble from below and above during breakthrough. While the trapped bubble below the screen starts to shrink, small bubble Interdisciplinary Transport Phenomena VI, Volterra, Italy, 2009

of the weaves shut again and the process starts anew with a trapped bubble of reduced volume. Figure 18 presents the experimental findings for the first case, i.e. the closed circuit.

Figure 18: Dynamic bubble point measurement in the closed hydraulic circuit. Apparently, the described cycle is repeated many times. In the displayed experiment it was eight times. However, the breakthrough always occurs at the static bubble point, see Figures 11-14. Also the ramping time is a parameter. Smaller ramping times of the liquid flow rate lead to a more disturbances in the signal although basically the same happens. Higher ramping times can falsify the measurements because the long exposure to the liquid flow leads to dissolving of the trapped bubble as is shown by Figure 19.

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filtering and aerating in chemical processes to soil behavior in natural gas exploitation. We have shown that i) the static bubble point depends on the gas supply rate, ii) the interplay of decompression, bubble snap-off and gas supply is responsible for the unsteady pressure signal, iii) the necessary pressure for bubble detachment provides a lower limit for the decompression pressure, iv) long exposure of gas to liquid leads to dissolving effects, v) the breakthrough pressures are identical for static and dynamic configuration, vi) radial lift leads to sudden breakthrough of the trapped bubble, vii) breakthrough leads to decompression until the pores are shut again and viii) regarding the dynamic bubble point there is a difference between closed hydraulic circuit and open circuit with overflow. As a consequence of our investigations, construction criteria for some applications might be rethought. Figure 19: Gas dissolving during the dynamic bubble point due to long exposure times of the trapped bubble to the flow.

ACKNOWLEDGMENTS

In Figure 20, the second case, i.e. the dynamic bubble point measurement with an overflow device, is presented. Here, the absolute pressure above the weave is fixed, hence below the weave it varies only slightly. Therefore, the pressure signal is

The funding of the research project by the German Federal Ministry of Economics and Technology (BMWi) through the German Aerospace Center (DLR) under grant number 50 RL 0741 is gratefully acknowledged. REFERENCES [1] F.T. Dodge, “Low-gravity fluid dynamics and transport phenomena.” Chapter 1, Aeronautics and Astronautics. 130, 314 (1990) [2] M.E. Dreyer, “Free surface flows under compensated gravity conditions.” Springer (2007) [3] J.C. Armour & J.N. Cannon, “Fluid flow through woven screens.” AIChE Journal. 14,3: 415-420 (1968). [4] E.C. Cady, “Study of thermodynamic vent and screen baffle integration for orbital storage and transfer of liquid hydrogen.” NASA Report No. CR-134482 (1973).

Figure 20: Dynamic bubble point measurement with overflow. more clear. Besides, due to the omission of the flow compression the trapped bubble below the screen undergoes the breakthrough cycle more often, here it is 10 times. 7. SUMMARY

[5] W.T. Wu, J.C. Liu, W.J. Li & W.H. Hsieh. “Measurement and correlation of hydraulic resistance of flow through woven metal screens.” Int. J. Heat Mass Tran. 48: 3008-3017 (2005). [6] P. Kopf, M. Piesche & S. Schütz. „Beschreibung des Druckverlusts von Drahtgeweben mit Hilfe von Ähnlichkeitsgesetzen.“ Filtrieren und Separieren 21,6: 330335 (2007)

We have presented a study on the behavior of gas at a woven screen. The performance of such screens is important in a number of applications that range from phase separation in propellant management devices used in space crafts over

[7] E.W. Washburn. „Note on a method of determining the distribution of pore sizes in a porous material.” Physics 7: 115-116 (1921)

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[8] S. Schütz, P. Kopf, C. Winkler & M. Piesche. „Untersuchungen zur Anwendung der Kapillardruckmethode bei der Bestimmung maximaler Porengrößen in Filtermedien aus Metalldrahtgeweben.“ Chemie Ingenieur Technik 80,7: 975-985 (2008) [9] N. Kumar & N.R. Kuloor, “The formation of bubbles and drops.“ Adv. Chem. Eng. 8: 255-369 (1970) [10] H.N. Oguz & A. Prosperetti, “Dynamics of bubble growth and detachment from a needle.” J. Fluid Mech. 257: 111-145 (1993) [11] F. Bashforth & J.C. Adams, “An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid.” Cambridge University Press (1883) [12] J.F. Padday, “The profiles of axially symmetric menisci.” Philosophical transactions of the Royal Society of London A 269: 265-293 (1971) [13] D. Langbein, “Capillary surfaces. Shape – stability – dynamics, in particular under weightlessness.” Springer, Berlin (2002)

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