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APhO 2008, Теория — Задача 1. Tea Ceremony and Physics of Bubbles

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THEORETICAL COMPETITION. 9th Asian Physics Olympiad, Ulaanbaatar, Mongolia (April 22, 2008 ) · 20 април 2008 г. · 10 т.

Условие

The tea ceremony is traditional in Asia. One of the important steps in preparation of tea is the boiling of fresh water when bubbles appear inside. Bubbles are familiar from daily life and occupy an important role in physics, chemistry, medicine and technology. Nevertheless, their behavior is often surprising and unexpected - and, in many cases, still not understood.

At room temperature the pure water is saturated with gas. With increasing temperature the excess pressure of dissolved gas $P_{ab}$ increases, the dissolved air is liberated and air bubbles (ABs) appear at the bottom and walls of teakettle (Fig.2). For pure water the wettability is sufficient and an AB represents a truncated sphere with radius $R_{ab}$ and with unwetted foundation with radius $r_{ab} \ll R_{ab}$ . At more heating ABs expand and by reaching certain sizes can detach from the bottom (Fig.3), flow up to the water surface and burst there. The vapor bubbles (VBs) appear when the water temperature at the bottom reaches the critical value $T_w^0 \approx T_{crit}^0 = 100^{0}C$ at which the pressure of the saturated vapor exceeds the external pressure. The vapor production increases tens times, VBs expand and detach from the bottom. VB may be considered consisting only of vapor. If the water is heated sufficiently, the uprising VB continue to swell, reach the surface and burst. Else, water is not heated enough in the higher layers and there exits a vertical strong temperature gradient. By reaching relatively cold layers of water VB collapse in the volume of water (Fig.4). This causes the induced degassing - strong oscillations and a considerable amount of dissolved air is released in the form of microscopic air bubbles (MAB). This can generate ultrasonic vibrations.

The main stages of the bubble evolution during the boiling process are:

the appearance and growth of AB at the bottom and walls, their transmutation into VB;
the detachment and uprising of VB, their disappearance in the water volume or at the surface;
the appearance of MAB in the water volume and their uprising to the surface.

This theoretical description is in good agreement with modern experiments. Particularly, an interesting noise analysis experiment (NAE, Ural State University, Ekaterinburg) for the boiling water was performed. Highly sensitive microphones attached to wide-band amplifiers and brought to an electric teakettle have detected three main origins of noises:

AB's detachments from the bottom before boiling (generate oscillations with $\nu_1 \sim 100$ Hz, );
VB's collapses in the volume of water (generate oscillations with $\nu_2 \sim 1$ kHz );
MAB's appearances under the water surface (generate oscillations with $\nu_3$ ~35kHz to 60kHz).

Hints:

1) It is well known that a small bubble rises along a rectilinear path and a laminar flow is observed - water flows easy and layer-wise (see Fig.1). Then, the Stokes formula describes the dissipative force for a particle moving with slow velocity $v_{lam}$ :

F_A = 6\pi\eta_w R_b v_{lam}

In contrast to this picture, when relatively large bubbles lift to the surface, it disturbs the surrounding water, cavitation hollows appear behind and the turbulent flow is observed (see Fig.1). In this case a part of the kinetic energy of an uprising bubble transfers into the dissipative work.

Two panels of dashed streamlines around a circular bubble: on the left smooth symmetric laminar flow lines around the bubble; on the right the lines are disturbed with small vortices (eddies) behind the bubble, showing turbulent flow. — Fig 1. Laminar and turbulent types of flow for rising air bubbles in water

2) When the surface of liquid has a convex (concave) form there appears a surface tension force due to molecular interaction near the edge. This pressure can be given by formula

\Delta P = \frac{2\sigma}{R}

where $\sigma$ - is the surface tension coefficient (unit=N/m), the force coming to unit length of surface, R – is the radius of surface curvity.

3) When dealing with a short process with characteristic duration time "t", its inverse value may be considered as a characteristic frequency $\nu = \frac{1}{t}$ . Use this definition for calculating the noise frequencies.

Useful data:

$P_0 = 1.016\cdot 10^{5}\,[\mathrm{Pa}]$ - atmospheric pressure,

$\rho_w = 10^{3}\,[\mathrm{kg/m^3}]$ - water density,

$\rho_{vapor} = 0.017\,[\mathrm{kg/m^3}]$ - vapor density at T=293K; ( $= 0.596\,[\mathrm{kg/m^3}]$ at T=373K)

$P_{vapor} = 0.023\cdot 10^{5}\,[\mathrm{Pa}]$ - vapor pressure at T=293K; ( $= 1.016\cdot 10^{5}\,[\mathrm{Pa}]$ at T=373K)

Four stacked blue panels showing stages of a vapor bubble collapse: a bright flash, arrows converging on a centre, a bubble, and a tiny remaining bubble. — Fig. 4. Vapor bubble collapsing

$g = 9.81\,[\mathrm{m/s^2}]$ - acceleration of gravity,

$\mu_{air} = 0.029\,[\mathrm{kg/mole}]$ - molecular weight of air

$R = 8.31\,[\mathrm{J/mole/K}]$ - the gas universal constant

$\sigma = 0.0725\,[\mathrm{N/m}]$ - surface tension coefficient of water,

$\eta_w = 0.3\cdot 10^{-3}\,[\mathrm{Pa\cdot s}]$ - coefficient of viscosity of water

Top: sketch of water (cyan) with a truncated-sphere bubble of radius R_ab sitting on the bottom on a foundation of width 2r_ab, and a detached spherical bubble raised by a distance R_ab above the bottom. Bottom: four photographs of a bubble detaching from a surface through a narrow neck. — Fig. 3. Air bubble detaching from the bottom

H=10cm – Water attitude in teakettle

Photograph of a glass jug of water on a stove with many small bubbles on its bottom and walls. — Fig. 2. Bubbles in teakettle

Questions (total 10 points):

Consider water boiling in a flat-bottomed cylinder glass teakettle at normal atmospheric pressure. The bottom of the kettle heats up uniformly and a vertical temperature gradient exists, bubbles appear and evolute (Fig.2).

Q1 Write the pressure condition of the growth of an AB in the water volume at height $h<H$ , where H is the water surface level in the teakettle. Take into account the inequality $2\pi\sigma r_{ab} \gg P_{extern}\pi r_{ab}^2$ . [in terms of $P_{ab}, P_0, R_{ab}, \rho_w, g, \sigma, h, H$ ] [1 т.]

Q2 Write for an AB the condition of the detachment from the bottom of the teakettle (Fig.3). Take into account the relation $r_{ab} \ll R_{ab}$ . [in terms of $r_{ab}, R_{ab}, \rho_w, \sigma$ ] [1,5 т.]

Q3 Consider an AB with radius $R_b$ at the bottom of the teakettle. As water is boiled, the bubble is saturated with vapor and enlarges its radius. Write the ratio $\xi \equiv m_{air}/m_{vapor}$ of the masses of the air and saturated vapor inside the bubble at given temperature T. Calculate the ratio at room temperature T=20 $^{\circ}$ C ( $R_b = 0.5\,\mathrm{mm}$ ) and at boiling point at T=100 $^{\circ}$ C ( $R_b = 1\,\mathrm{mm}$ ). [in terms of $\mu_{air}, T, P_0, P_{vapor}, R_b, \rho_w, \rho_v, \sigma, H$ ] [1,5 т.]

Q4 By using the NAE data and Newton's Law estimate the radius of the AB detached from the bottom and uprised in distance $R_{ab}$ (Fig.3). Assume, that the added-mass (taking into account surrounding water layer) of AB is a half of the analogous water bubble. [1 т.]

Q5 Write the radius of the foundation of an AB just before the uprising, when the connecting "neck" is very narrow (see Fig.3). [in terms of $R_{ab}, \rho_w, \sigma$ ]. Calculate it by using the radius found in Q4. [1,5 т.]

Q6 By using the NAE data estimate the radius of collapsing VB (Fig.4) by assuming that the radial pressure is about 3kPa during this process. [1,2 т.]

Q7 By using previous results for VB calculate the radius of the MAB produced during induced degassing. [0,5 т.]

Q8 Write the uprising velocity for typical AB by using the Stokes law of a laminar flow. [in terms of $R_{ab}, \rho_w, \eta_w$ ]. Estimate the uprising time for H=10cm. [0,6 т.]

Q9 Write the average velocity of the elevation of VB with turbulent type of flow [in terms of $R_{ab}, \rho_w, \eta_w$ ]. Estimate the uprising time for H=10cm. [1,2 т.]

Отговори

Решение

Solutions:

S1. The condition of the survival and growth for AB appeared in the water volume at height $h < H$ is the competitiveness of the pressures acting inside and outside (atmospheric, hydrostatic and surface tension) on the bubble surface:

P_{ab} = P_{in} \ge P_{out} \equiv P_0 + g\rho_w\cdot(H-h) + \frac{2\sigma}{R_{ab}},

S2. The Archimedes lifting force is

F_A = g\cdot(\rho_w - \rho_{ab})\frac{4\pi R_{ab}^3}{3} \approx g\cdot\rho_w\frac{4\pi R_{ab}^3}{3}

where $\rho_{ab} \ll \rho_w$ - the air density in bubble.

For $r_{ab} \ll R_{ab}$ the Laplace surface tension force plays main role in holding down the bubble:

F_{down} = \sigma\cdot 2\pi r_{ab}

The stability of the AB at the bottom means:

F_A = F_{down}

At more heating the lifting force overbalances the holding force:

g\frac{4\pi R_{ab}^3}{3}\rho_w > \sigma\cdot 2\pi r_{ab}

and AB is detached from the bottom and floats (Fig.3).

S3. The air and vapor pressures inside compensate the outer pressure for a stable bubble.

P_{vapor} + P_{air} = P_0 + g\rho_w H + \frac{2\sigma}{R_b}

The mass of air inside AB may be found from the Clayperon-Mendeleev equation

m_{air} = \frac{\mu_{air}P_{air}V}{RT}

The vapor mass is

m_{vapor} = \rho_{vapor}\cdot V

Then the ratio is

\xi \equiv \frac{m_{air}}{m_{vapor}} = \frac{\mu_{air}P_{air}}{RT\rho_v} = \frac{\mu_{air}}{RT\rho_v}\left(P_0 + g\rho_w H + \frac{2\sigma}{R_b} - P_{vapor}\right)

For $T = 20^{0}C = 293K$ and $R_b = 0.5mm$ :

\xi = \frac{0.029}{8.31\cdot 293\cdot 0.0173}\left(1.016\cdot 10^{5} + 9.81\cdot 10^{3}\cdot 0.1 + \frac{2\cdot 0.0727}{0.5\cdot 10^{-3}} - 2.3\cdot 10^{3}\right) \approx 69.2

This bubble consists mostly of air.

For $T = 100^{0}C = 373K$ and $R_b = 1mm$ :

\xi = \frac{0.029}{8.31\cdot 373\cdot 0.596}\left(1.016\cdot 10^{5} + 9.81\cdot 10^{3}\cdot 0.1 + \frac{2\cdot 0.0588}{1\cdot 10^{-3}} - 1.016\cdot 10^{5}\right) \approx 0.017

This bubble consists mostly of saturated vapor.

S4. Consider AB detached from the bottom and uprising in distance $R_{ab}$ (see Fig.3). During the rise, bubbles induce a displacement of the surrounding fluid in their vicinity, which leads to an added-mass force. The added-mass of a bubble is

M_{invol} \approx \frac{V_{ab}\rho_w}{2}

The acceleration at the detachment moment is

a_{\det} = \frac{F_A}{M_{invol}} = \frac{gV_{ab}\rho_w}{V_{ab}\rho_w/2} = 2g

The characteristic "detachment time"

t_1 = \sqrt{\frac{2R_{ab}}{a_{\det}}}

defines the duration of the impact to the liquid during the detachment. The liquid begins to vibrate with characteristic frequency

\nu_1 = \frac{1}{t_1} = \sqrt{\frac{g}{R_{ab}}}

Substituting the date from NAE we estimate the characteristic radius of detaching AB

\nu_1 \approx 100[Hz] \quad \Rightarrow \quad R_{ab} \approx 1\cdot 10^{-3}[m]

S5. The balance of the lifting and confining forces reads

gV_{ab}\rho_w \approx \sigma\cdot 2\pi r_{ab}

Find the typical foundation radius:

r_{ab} \approx \frac{2g\rho_w R_{ab}^3}{3\sigma}

For AB with radius ~1mm we calculate

r_{ab} \approx \frac{2\cdot 9.81\cdot 10^{3}\cdot 10^{-9}}{3\cdot 0.0725}[m] \approx 9.02\cdot 10^{-5}[m]

S6. Consider collapsing VB during time piece $t_2$ (Fig.4). Let's estimate the characteristic frequency of the ultrasonic shock waves. The surrounding water flood the collapsed VB with acceleration a and the Newton equation reads

F_{collap} \approx a\cdot\frac{4\pi R_{cb}^3}{3}\rho_w

where the acceleration of the water front converging in the center of VB is

a = \frac{2R_{cb}}{t_2^2} = 2R_{cb}\nu_2^2

The radial pressure on the VB surface is given

\Delta P = \frac{F_{collap}}{4\pi R_{cb}^2} = \frac{2R_{cb}^2}{3}\rho_w\nu_2^2

Particularly, for the obtained data of the NAE experiment we obtain

R_{cb} = \frac{1}{\nu_2}\sqrt{\frac{3\Delta P}{2\rho_w}}

Particularly,

\nu_2 \approx 1[kHz]\ ,\ \Delta P = 3[kPa] \quad \Rightarrow \quad R_{cb} \approx 3[mm]

This result is in good agreement with another experiment (see Fig.5) where the radius is found about 3.5mm and the collapsing time is about 1ms (i.e., ~1kHz noise).

Graph (panel b) of bubble radius in mm (left axis, −1 to 4, blue points and solid curve) and pressure P(t) in 10^5 Pa (right axis, green dotted curve) versus time in ms from −1.0 to 0.0; the radius grows to about 3.5 mm near −0.3 ms and collapses to zero at 0.0 ms. — Fig.5. A vapor bubble collapsing evolution. [M. P. Brenner, S. Hilgenfeldt, D. Lohse, Rev. Mod. Phys. 74, 425 (2002)]

S7. Obviously, the physical nature of MAB is the same as for VB. Then,

\frac{\nu_3}{\nu_2} = \frac{R_{cb}}{R_{mab}} \quad \Rightarrow \quad R_{mab} = R_{cb}\frac{1[kHz]}{35\div 60[kHz]} \approx \frac{3[mm]}{35\div 60} \approx 0.05\div 0.086[mm]

S8. A bubble detached from the bottom is hoisted under the influence of Archimedes force. The water resistant force depends on the nature of the streamline flow (Figs.1,6,7).

Two grey photographs side by side separated by a black band, each showing a straight vertical trail of a rising bubble. — Fig. 6 The rectilinear air bubble trajectory (R= 0.69mm) rising from the bottom. On the left the XZ view and on the right the YZ view. The black areas are part of the reference system outside the water tank. [Benjamin(1987), A.de Vries (2001)]

But for a bubble with radius about 1mm the emersion laminar velocity becomes too fast

v_{lam} \approx 2.2\,[m/s]

and the bubble passes the 10cm distance to the surface for 0.045 second !!!. It is obviously wrong. Therefore, the Stokes formula is not applicable for AB and VB rising from the bottom.

S9. As a path instability sets in, a bubble can uprise by either zigzag or spiral (Fig.8).

Two grey photographs side by side separated by a black band, each showing a curved (spiralling) bubble trail; a time stamp is printed at the bottom. — Fig.7 The trajectory of a spiralling bubble in two perpendicular views, XZ and YZ (R=1.1mm) [Benjamin(1987), Antoine de Vries (2001)].

Obviously, the lifting force is

F_A = \frac{4\pi R_{vb}^3}{3}g\rho_w

During its uprising in distance 'h' the bubble removes a water portion with mass:

M_{turb} = \rho_w h\cdot\pi R_{vb}^2

and performs a work (transfers kinetic energy)

W_{dissip} = \frac{M_{turb}v_{turb}^2}{2}

Then, we estimate the dissipative force

F_{dissip} = \frac{W_{dissip}}{h} = \frac{M_{turb}v_{turb}^2}{2h}

Since VB flows steady without any acceleration, the dissipative (braking) force balances the Archimedes force:

F_{dissip} = F_A

Then, the average (for spiral motion) turbulent velocity is

v_{turb} \approx \sqrt{\frac{8gR_{vb}}{3}}

For a typical radius of uprising AB (~1mm) or collapsing VB (~3mm) we calculate the velocity

v_{turb} \approx \sqrt{\frac{8\cdot 10\cdot(1\div 3)\cdot 10^{-3}}{3}}[m/s] \approx (0.16\div 0.28)[m/s]

The time required to pass a distance H ~10cm:

t_{turb} > \frac{H}{v_{turb}} \approx \frac{0.1}{(0.16\div 0.28)}[s] \approx (0.35\div 0.62)[s]

This is a quite reasonable result and the bubbles mostly elevate under the turbulence flow law.

[Marking Scheme] Tea Ceremony and Physics of Bubbles

Q	Item	Answer	Points
1	condition of growth	$P_{ab} = P_{in} \ge P_{out} \equiv P_0 + g\rho_w\cdot(H-h) + \frac{2\sigma}{R_{ab}}$	1.0
2	condition of detachment	$g\frac{4\pi R_{ab}^3}{3}(\rho_w - \rho_{ab}) > \sigma\cdot 2\pi r_{ab} + (P_0 + g\rho_w H)\cdot\pi r_{ab}^2$	1.5
3	the ratio	$\xi \equiv \frac{m_{air}}{m_{vapor}} = \frac{\mu_{air}P_{air}}{RT\rho_v} = \frac{\mu_{air}}{RT\rho_v}\left(P_0 + g\rho_w H + \frac{2\sigma}{R_b} - P_{vapor}\right)$	1.1
	the ratio at T=20C	$\xi = \frac{0.029}{8.31\cdot 293\cdot 0.0173}\left(1.016\cdot 10^{5} + 9.81\cdot 10^{3}\cdot 0.1 + \frac{2\cdot 0.0727}{0.5\cdot 10^{-3}} - 2.3\cdot 10^{3}\right) \approx 69.2$	0.2
	the ratio at T=100C	$\xi = \frac{0.029}{8.31\cdot 373\cdot 0.596}\left(1.016\cdot 10^{5} + 9.81\cdot 10^{3}\cdot 0.1 + \frac{2\cdot 0.0588}{1\cdot 10^{-3}} - 1.016\cdot 10^{5}\right) \approx 0.017$	0.2
4	characteristic frequency	$\nu_1 = \frac{1}{t_1} = \sqrt{\frac{g}{R_{ab}}}$	0.8
	radius of detaching	$R_{ab} \approx 1\cdot 10^{-3}[m]$	0.2
5	foundation radius:	$r_{ab} \approx \frac{2g\rho_w R_{ab}^3}{3\sigma}$	1.2
	For radius 1mm	$r_{ab} \approx \frac{2\cdot 9.81\cdot 10^{3}\cdot 10^{-9}}{3\cdot 0.0725}[m] \approx 9.02\cdot 10^{-5}[m]$	0.3
6	Radius of collapsing bubble	$R_{cb} = \frac{1}{\nu_2}\sqrt{\frac{3\Delta P}{2\rho_w}}$	1.0
	Numerical value	$\nu_2 \approx 1[kHz]\ ,\ \Delta P = 3[kPa] \ \Rightarrow\ R_{cb} \approx 3[mm]$	0.2
7	radius	$R_{mab} = R_{cb}\frac{1[kHz]}{35\div 60[kHz]} \approx \frac{3[mm]}{35\div 60} \approx 0.05\div 0.086[mm]$	0.5
8	laminar velocity	$v_{lam} \approx 2.2\,[m/s]$	0.6
9	turbulent velocity	$v_{turb} \approx \sqrt{\frac{8gR_{vb}}{3}}$	1.0
	Numerical speed	$v_{turb} \approx \sqrt{\frac{8\cdot 10\cdot(1\div 3)\cdot 10^{-3}}{3}}[m/s] \approx (0.16\div 0.28)[m/s]$	0.1
	Ascending time	$t_{turb} > \frac{H}{v_{turb}} \approx \frac{0.1}{(0.16\div 0.28)}[s] \approx (0.35\div 0.62)[s]$	0.1
	TOTAL		10.0

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