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A Numerical Study of Acoustic Cavitation in Cryogenic Liquids

Saleh E. Najim Abbas Z. AL-Asady Ahmad A. K AL-Sahlanee

Department of Mechanical Eng., College of Engineering, University of Basrah, Iraq E-mail: ahmad_sahlanee@yahoo.com

Keywords: Acoustic Cavitation, bubble dynamics, Cryogenic Liquids

Abstract

In this paper, behavior of Helium gas bubble in liquid of Nitrogen and Oxygen in acoustic field under ambient pressure and low temperature is investigated theoretically. Initially, it is introduced the derivation of equation of moving the liquid around the bubble (radius bubble equation) beginning from the Keller–Kolodener equation (which derived from the continuity, momentum, and energy equations). Also two cases are suggested to solve this equation numerically. In the first case, it is considered that, heat is transferred across the bubble wall. And bubble wall temperature is assumed to be constant. In the second case, it is considered that the bubble is adiabatic and its internal pressure obeyed the adiabatic law. Radius, velocity of the bubble wall, internal pressure and temperature at the bubble center are calculated according to the two cases. Also, comparison is made between the cases. It is showed that the physical quantities in the adiabatic bubble case have higher values than the other cases. In each case, the effects of evaporation and condensation process are neglected.

Nomenclature

Unite Definition Symbol m/s Sound speed in the liquid at infinity c

J/kg.K Heat capacity at constant pressure for the liquid CpL

Hz Acoustic field frequency F J/kg Enthalpy function H

W/m.K Thermal conductivity of the gas kg

W/m.K Thermal conductivity of the liquid kL

Variable are defined in eqs. (17-21 ) L1, L2, L3, L4, L5 N/m2 Pressure of acoustic field PA

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N/m2 Pressure gas in the bubble Pg N/m2 The surface tension effect Ps.t N/m2 The viscous effect Pv N/m2 Hydrostatic pressure Po N/m2 Acoustic pressure amplitude Pa N/m2 Pressure liquid near the bubble wall PL N/m2 Pressure liquid far from the bubble wall P∞ N/m2 Initial pressure inside the bubble Pgo

m Radius distance from the bubble center r m Bubble radius R m Maximum and minimum bubble radius Rmax, Rmin m Initial bubble radius Ro

m/s Bubble wall velocity R� m/s2 Second derivative of the bubble radius R� K Gas temperature Tg

K Initial gas temperature To K Gas temperature far from the bubble T∞ K Temperature of the bubble wall TLw

s Time t Variables are defined in eqs. (13-16) W1, W2, W3, W4,

Greek symbols m2/s Velocity potential in the liquid

Ratio of specific heats of Helium γ rad/s Angular frequency ω

N.s/m2 Viscosity of the liquid µ N/m Surface tension σ

kg/m3 Density of the liquid far from the bubble ρ∞

1-Introduction

Cavitation is the formation of vapor or gas cavities within a given liquid due to pressure drop. It may be observed in various engineering systems such as hydraulic constructions، aeronautics, aerospace, power systems and turbo machinery [1]. Cavitation can be caused by surface tension effects or by energy deposit. In 1917, Lord Rayleigh investigated this mechanical cavitation problem and confirmed that the vibrations were caused by the enormous turbulence, heat and pressure of imploding cavities. He derived a simple equation, which described the collapse of spherical cavity and he neglected the effect of viscosity, surface tension and compressibility. Plesset & Blake In 1949, Noltingk & Neppiras in 1950 and Poritsky in

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1952, modified the Rayleigh equation to as bellow, which are often called Rayleigh-Plesset equation (RPNNP) [2]:

RR� + 1.5 R� = 1ρ�

� P� + 2σR�� R�R ��γ − 2σR − 4µR�R − P∞(t)� … (1)

In acoustic cavitation, the ambient pressure can be varied by sending sound waves through the liquid. If the amplitude of the pressure variation is great enough to bring the pressure locally down to, or below, the vapour pressure in the negative parts of the sound cycle traversing the liquid, any minute cavities or bubbles will grow.

The content of the bubble is vapor or other gases. Physical properties of the fluids are important factors to understand acoustic cavitation. Cryogenic materials are very cold substances (gases, solids and liquids at or below 200 K). There are used in a wide variety of processes. Cryogenic liquids have boiling points below (183 K). Common cryogenic liquids of concern include Nitrogen, Helium, Argon, Hydrogen, Methane and Carbon dioxide [3].

Table (1) Parameter of some cryogenic liquids at 0.103 MPa [4][5][6]

Properties Helium Hydrogen Nitrogen Oxygen Melting temperature (K) ---- 13.95 63.15 54.35 Boiling temperature (K) 4.21 20.38 77.35 90.18

Critical temperature (K) 5.19 33.23 126.25 154.77 Critical pressure (MPa) 0.226 1.316 3.345 50.23 Specific heat at boiling (J/kg.K) 4923 9816 2046 1699 Liquid density at boiling (kg/m3) 125.2 71 809 1141 Thermal conductivity at boiling(W/mK)

32 100 135 152

The melting temperature of helium is not presented in the Table (1) as under normal conditions, it doesn’t solidify if the temperature decreases to absolute zero. One can obtain a solid helium phase only at pressure above 25 atm [5]. The boiling temperature is given for normal atmospheric pressure 1.0 atm.

The operating point of cryogenic liquids is generally quite close to the critical point unlike water [5]. Furthermore, the vapor pressure of liquids such as liquid Hydrogen and liquid Nitrogen is expected to show great sensitivity to small temperature drops [7]. There can no gas cavitation in liquid Helium at all because at substances which under common temperature are in gaseous state (Hydrogen, Neon, Nitrogen, Oxygen, ...etc) became solid (see Table(1)). In liquid Hydrogen, only Helium can be available in the form of a gas.

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2- Mathematical Formulation

When a gas bubble is oscillating in liquid under acoustic field, its dynamic of a bubble is strongly dependent on the pressure of the gas.

The present model assumes a spherical shape bubble whose physical quantities are a functions of radial distance (r). Mathematically, a model of spherical bubble oscillations can be understand as a basic differential equation that describes the flow of the liquid outside of the bubble, and a set of boundary conditions and correction terms that describes the gas pressure, surface tension, viscosity and other. There is a spherical bubble of initial radius Ro containing non-condensable gas in liquid. Schematic diagram depicting a model illustrated in Figure A. To introduce an equation which describes the oscillation some, assumptions are mode:

1. The bubble is stable and spherically symmetric. 2. The effects of gravity and diffusion are negligible. 3. The evaporation and condensation at bubble wall are negligible. 4. The non-condensable gas (Helium gas) inside the bubble obeys the perfect-gas law. Several forces act on the bubble wall during acoustic cavitation. The total pressure in the bubble (Pg) equals the sum of pressures. Dominant pressures in the liquid just near the bubble wall are the surface tension (Ps.t), the viscous effect pressure (Pv), the hydrostatic pressure (Po) and time-dependant acoustic pressure (Pa(t)).[22]

P� = P�.� + P� + P� + P�(t) = 2σR + 4µR�R + P� + P�(t) … (2)

Here (R) represent the bubble radius and (R� ) it's time-derivative. The dynamic viscosity is denoted by (µ) and (σ) is the surface tension. The acoustic pressure PA(t) can be assumed :

P�(t) = P� sinωt … (3) Where ω denotes the radial frequency and its value determined from (2πF), in which (F) denotes the ultrasound frequency. (Pa) is pressure amplitude.

The pressure near the bubble wall is denoted as (PL) and the pressure far from the bubble as (P∞) [8][9].

P∞ = P� − P� sinωt … (4)

P# = P� − 2σR − 4µR�R … (5)

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Figure A: Forces act on the bubble wall during acoustic cavitation

From the continuity, momentum, and energy equation, Keller and Kolodener obtained a partial differential equation (eq.(6)) concerning the velocity potential (ϕ) of the liquid [10].

1r∂∂r r ∂ϕ∂r� − 1c

∂ϕ∂t = 1c �∂ϕ∂t ∂ϕ∂r� + 2 ∂ϕ∂r ∂ϕ∂r ∂t� … (6)

Where:

c: speed of sound in liquid, t: time and r: the radial distance from the center of bubble.

This equation can accomplish and solved with the following boundary and initial conditions [11]:

0.5R� + )∂ϕ∂r*+ = −H … (7) R� = )∂ϕ∂r*+ … (8)

Where: H: is the enthalpy function and R� : is the bubble wall velocity

Viscous force (Pv)

Surface tension (Pst)

Gas pressure (Pg)

Liquid

Bubble

Hydrostatic pressure (Po )

Acoustic pressure (PA)

Ro

R

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H = (P# − P∞)ρ∞

… (9) Where:

PL: the pressure near the bubble wall; P∞: the pressure far from the bubble; and ρ∞: the density of the liquid far from the wall of bubble.

The initial conditions are:

R(t = 0) = R� … (10)

R� (t = 0) = 0 … (11)

Then the solution of the equation (6) is :

;� <= = <> + ? <@ + <A B?BC … (12)

Where:

WE = R �F1 − R�cG F1 + LL�R�

c G� … (13)

W = − LIc − R� F1.5 − R�2cG … (14)

W� = J1 + R�c �1 − LL�c F1 − R�

cG�K … (15)

WL = Rc �1 − LL�c F1 − R�cG� … (16)

LE = −H − 12 R� … (17)

L = − Hc − 12c R� + R� … (18)

L� = �1 − 2Hc − R�c + R� Hc� − R� �

2c��ME … (19)

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LL = N1 − 2c H + 12 R� �OME … (20)

LI = 2LELLLR� + LE F1 − R�cG − LL�R� F1 − R�

cG � LEc � − R� 2 � … (21)

3- Model Solution

Equation (12) can be solved numerically by Range-Kutta method. In the next sections, we prepare the parameter to the solving by considering the effect of heat transfer with variable wall temperature Case1, and adiabatic case (Case2).

Case1: Heat is transferred across the bubble wall and internal pressure (Pg) is assumed spatially uniform in a bubble and founded by integrating the following equation [12]:

dP�dt = 3R �(γ − 1)k� )∂T�∂r S+

− γP�R� � … (22)

Where γ is the ratio of the specific heats of the gas, kg is the gas thermal conductivity, Tg is the gas temperature {T�(r, t)} which is obtained by solving the energy equation [12]:

P�T� �∂T�∂t + 1γP� V(γ − 1)k� ∂T�∂r − 13 r dP�dt W ∂T�∂r �

+γ(γ − 1) − dP�dt = 1r ∂∂r Fk�r ∂T�∂r G … (23)

These two equations (22, 23), shall be solved numerically to get Pg and Tg.

The liquid temperature on the external side of the bubble wall is assumed constant (constant wall temperature) during oscillations.

T�(R, t) = T∞ … (24)

Where T∞ is the temperature of the liquid far from the bubble. Eq.(23) is solved using finite difference method .

Case2: Adiabatic Case, In this case we assumed no heat is transferred across the bubble wall. Pressure inside the bubble (Pg) can be represented by adiabatic law of compression by [13].

P� = P�� R�R ��γ … (25)

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T�X = T� R�R ��(γME) … (26) Where Ro is the initial radius and

P�� = P� + 2σR�� … (27)

4- Results and Discussions

For the two cases which are studied, calculations are performed under conditions, listed in Table (2), for two acoustic field cycles. The frequency and the amplitude of the acoustic field are chosen to be (50 kHz) and (1 bar), respectively. Ambient pressure (1 bar) and initial temperature of all liquids is taken between the melting and evaporating point (70 K for liquid Nitrogen). The bubble gas is Helium only [14], and its physical properties are { γ =1.6 , Kg=0.05641(W/m.K) at

70K}[6]. Calculation is started with the initial condition that R=Ro , R� = 0 .

Table (2) Physical quantities used in the calculations of the various cases

Physical quantities

Liquid Nitrogen

Liquid Oxygen

Ro (µm) 25 25 Po (MN/m2) 0.1 0.1 To (K) 70 70

Pa (MN/m2) 0.1 0.1 F (kHz) 50 50 c (m/s) 1092 1272 σ (mN/m) 10.52 18.268 ρo (kg/m3) 840.9 1236 µ (mN.s/m2) 0.2167 0.368 KL (W/m.K) 0.1402 0.1894 Cp (kJ/kg.K) 2.024 1.668

A periodic solution is obtained by numerical calculation under physical conditions which are described in Table (3). All figures show the physical quantities for the liquid Nitrogen and Oxygen.

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Table (3) calculated results at the max. and min. bubble radius

Liquids Cases Radius (µm) Tg (K)

at center Pg (MPa)

Liquid Nitrogen

Case1 Rmax=63.7748 49.106 0.00530

Rmin=9.9778 303.789 20.2896

Case2 Rmax= 49.249 20.464 0.00379

Rmin= 9.680 394.257 10.1134

Liquid Oxygen

Case1 Rmax= 56.122 50.419 0.007502

Rmin= 9.7457 265.541 16.3926

Case2 Rmax= 62.956 13.278 0.001205

Rmin=6.2931 443.246 27.372

As shown in Figs. (1, 2), oscillations of the bubble in liquids Oxygen and Nitrogen are similar

in the form. High Rmax is recorded in adiabatic case (Case2) in liquid Oxygen, but it is recorded in Case1 in liquids Nitrogen. Low Rmin is recorded in adiabatic case (Case2) in all liquids.

Figs. (3, 4) show variations of the bubble wall velocity with time. High maximum velocity is recorded in adiabatic case (Case2) in all liquids but the low minimum velocity is recorded in Case1.

As shown in Figs. (5, 6), High maximum temperature and low minimum temperature inside the bubble are recorded in adiabatic case (Case2). High increase in the bubble temperature center is recorded in liquid Oxygen (679K), but in liquids Nitrogen it is 331K. High decrease in the bubble temperature at center is recorded also in liquid Oxygen (about 66.732 K), but in liquids Nitrogen being 49.746 K.

As shown in Figs. (7, 8), High maximum pressure is recorded in Case1 in liquid Nitrogen, but it is recorded in Case2 in liquid Oxygen. Low minimum pressure (Pg) is recorded in adiabatic case (Case2) in all liquids. High increase in internal pressure is recorded in liquid Oxygen (about 27 MPa), but in liquids Nitrogen being about 10.013 MPa. High decrease in internal pressure is recorded also in liquid Oxygen.

5- Conclusions

The bubble wall motion equation derived from the potential velocity equation, the solution of this equation for liquids Nitrogen and Oxygen (under ambient pressure and 70 K ) can draw that the values of radius, velocity, pressure and temperature inside the bubble. In adiabatic case (Case2) are higher than from the values in the variable bubble wall temperature case (Case1) in all liquids except the radius in liquid Nitrogen. Also can be note that the oscillations of the bubble in liquids Oxygen and Nitrogen are similar in the form. High temperature and pressure occur in the bubble at bubble collapses.

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Rad

ius

(µm

)

Time (µs)

Fig. (1) The bubble radius as a function of time for liquid Nitrogen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

Rad

ius

(µm

)

Time (µs)

Fig. (2) The bubble radius as a function of time for liquid Oxygen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

0 5 10 15 20 25 30 35 40 45

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45

0

10

20

30

40

50

60

70

Case1 ــــــــــ

Case2 ------

Case1 ــــــــــ

Case2 ------

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Vel

ocity

(m

/s)

Time (µs)

Fig. (3) The bubble wall velocity as a function of time for liquid Nitrogen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

Vel

ocity

(m

/s)

Time (µs)

Fig. (4) The bubble wall velocity as a function of time for liquid Oxygen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

0 5 10 15 20 25 30 35 40 45

-80

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30 35 40 45

-150

-100

-50

0

50

100

Case1 ــــــــــ

Case2 ------

Case1 ــــــــــ

Case2 ------

12

Tem

pera

ture

(K

)

Time (µs)

Fig. (5) The Temperature inside the bubble as a function of time for liquid Nitrogen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

Tem

pera

ture

(K

)

Time (µs)

Fig. (6) The Temperature inside the bubble as a function of time for liquid Oxygen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

0 5 10 15 20 25 30 35 40 45

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35 40 45

0

100

200

300

400

500

600

700

800

900

Case1 ــــــــــ

Case2 ------

Case1 ــــــــــ

Case2 ------

13

Pre

ssur

e (M

Pa)

Time (µs)

Fig. (7) The Pressure inside the bubble as a function of time with logarithmic vertical axis for in liquid Nitrogen

(Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

Pre

ssur

e (M

Pa)

Time (µs)

Fig. (8) The Pressure inside the bubble as a function of time with logarithmic vertical axis for liquid Oxygen (Ro=25 µm , F=50kHz , pa=1bar, Po=1bar)

0 5 10 15 20 25 30 35 40 45

0.00

0.01

0.10

1.00

10.00

100.00

0 5 10 15 20 25 30 35 40 45

0

0

0

1

10

100

Case1 ــــــــــ

Case2 ------

Case1 ــــــــــ

Case2 ------

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[1]

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[6]

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[7]

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]