Criterion Demarcating Cavitation and Boiling - Springer Link

Theoretical Foundations of Chemical Engineering, Vol. 36, No. 6, 2002, pp. 546–550. Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 36, No. 6, 2002, pp. 599–603. Original Russian Text Copyright © 2002 by Yudaev.

Criterion Demarcating Cavitation and Boiling V. F. Yudaev Moscow State University of Environmental Engineering, Staraya Basmannaya ul. 21/4, Moscow, 107866 Russia Received May 16, 2000

Abstract—The equation of radial–spherical nonlinear bubble oscillations in a liquid–gas mixture under the action of pressure pulses and/or a local hydrodynamic pressure drop and the thermal conductivity equation for the liquid/bubble interface were used to derive a cavitation-to-boiling transition criterion. The line demarcating the cavitation and boiling regions is calculated for water, methanol, and nitrogen.

Both cavitation and boiling bring about minute discontinuities and cavities in a liquid. The difference between these processes is that the vapor in cavitation bubbles may be unsaturated. Hydrodynamic cavitation results from a pressure drop in the liquid flow as a consequence of high local velocities. Pressure waves in the liquid, which are described by the pressure-disturbance propagation equation, may give rise to acoustic or pulsed cavitation (depending on the nature of the waves). The purpose of this work is to establish a boundary between cavitation and boiling using the modified equation of radial–spherical oscillations of a gas–vapor bubble in Herring’s approximation (in relative quantities) [1]: 2

4 d R 3 dR R ( 1 – 2Ma ) --------2- + ---  1 – --- Ma  -------    2 3 dt  dt = [ 1 + Ma ( 1 – Ma ) ]β ( R )

2

∆T ( 1 – ∆T ) + P v + ------J

3n v

4 dR +  We – ---------- ------- [ 1 + Ma ( 1 – Ma ) ]  ReR d t 

(1)

4Ma d R/d t ϕ ( t ) f ( t ) – ----------- ( 1 – Ma ) ------------------- – ----------- – ---------- – 1. Re χh χa d R/d t 2

2

Here, f( t ) is the law of variation of the external pressure generating acoustic cavitation, R = R/R0, t = t/tm, tm = 2πR0(ρl/Ps)1/2 is the period of radial–spherical oscillations of the bubble, and ϕ( t ) is the law of variation of the relative bubble velocity along the liquid-flow line in hydrodynamic cavitation. The speed of sound c in a gas–liquid mixture with a variable gas content 3

λ0 R λ = --------------------------------3 1 – λ0 ( 1 – R )

(2)

depends on R(t ): ρ λ 2 1 – λ - 1 – λ  1 – -----v-  c =  ----- + ----------2   cv  ρl   cl ρ 1–λ + λ ρ l  1 – -----v- ----------- ρ l  nP s

– 1/2

(3)

.

The initial gas content of the liquid is equal to the ratio of the volume of bubbles in a reference volume to this reference volume: 3

4πR 0 λ 0 = ---------------------------, 3 3V l + 4πR 0

(4)

where Vl is the volume of incompressible liquid in a cell containing one bubble. We will assume that the equilibrium bubble radius is slightly scattered about the stable mean radius R0 under working conditions. The initial gas content defined by formula (4) is equivalent to the cavitation index introduced by Rozenberg [2] (note that Rozenberg used the maximum bubble radius in formula (4)). The cavitation index [2] is an empirical constant. Under the assumption that the free-gas content at any time point is given by Eq. (2), the variation of the cavitation index during the period of radial oscillations also obeys Eq. (2). In this model of free-gas-content variation in the cavitation region, the relative bubble concentration in the liquid varies according to the formula 3λ 0 N = --------------------------------------------------3 3 4πR 0 [ 1 – λ 0 ( 1 – R ) ]

(5)

and depends on the oscillation phase of the bubble. This dependence has hitherto been ignored. In large-amplitude nonlinear oscillations of the bubble radius, N may vary between 1 × 10–4 and 1.0 during the oscillation period. In the equation of the nonlinear radial oscillation of a bubble (Eq. (1)), the first-order phase transition is

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CRITERION DEMARCATING CAVITATION AND BOILING

accounted for by the number J. At high temperatures (near the critical value), the variation of the bubble radius is governed by the properties of the liquid under minimal external acoustic or local hydrodynamic actions. In Eq. (1), these actions are represented by the parameters χh and χa. The number J is particularly significant for cryogenic liquids [3, 4]. Cavitation and boiling should be taken into account when optimizing the working temperature in the production of disperse heterogeneous materials in apparatuses operated in the controlled-cavitation mode. Furthermore, these phenomena must be allowed for in intensifying evaporation processes and the separation of liquids differing in volatility [5]. The problem of liquid boiling under the action of sound was solved while developing bubble chambers [6], in which the bubbles were disturbed by harmonic ultrasonic waves. The dynamics of the radial oscillations of bubbles in cryogenic liquids flowing in pipelines and pumps with varied cross sections, including propulsion pumps, is considered in [4]. In this case, cavitation or boiling is caused by a local hydrodynamic pressure drop. Accordingly, the discontinuities and cavities in the liquid are of hydrodynamic nature and are accounted for by the term ϕ( t )/χh in Eq. (1). This equation enables one to carry out a quantitative analysis of the cavitation-to-boiling transition caused by simultaneous drops in the hydrodynamic or/and pulsed acoustic and thermal pressures (terms ϕ( t )/χh, f( t )/χa and ∆T /J, respectively). As the static pressure P∞ grows or the bubble radius or the liquid temperature decreases, the vapor pressure P v and heat and mass transfer play a progressively less significant role in the bubble dynamics. Reducing the static pressure or elevating the liquid temperature causes a cavitation-to-boiling transition. Equation (1) provides an explanation [1] for the disagreement in the relevant experimental data and substantiates the recommendations concerning the optimization of the volumetric gas content of the medium (liquid, emulsion, suspension) in order to attain the maximum pressure pulse amplitude and the highest process rates. As is demonstrated by calculations using Eq. (1) and experimental data [1, 5], the optimum gas content depends on the shape and amplitude |Pmax| of the pressure pulse causing cavitation. As the negative amplitude |Pmax| is increased or, more exactly, as χ is decreased, the optimum free-gas content of the liquid to be treated decreases; that is, other conditions being opt equal, λ 0 = f(χ). Smorodov [7] proposed intensifying cavitation effects in viscous liquids by raising the freegas content. If the gap between the rotor and stator in the rotary apparatus [7] is relatively wide, a sufficient |Pmax| value cannot be attained and, accordingly, χa exceeds the optimum. Small cavitation pressure pulses

547

are compensated for by introducing a large number of bubbles, which raise the small-amplitude integral cavitation noise from a single bubble. At high liquid flow rates in a modulated-flow rotary apparatus (when the radial liquid velocity is high, while the gap between the rotor and stator is narrow), blocking the stator holes causes large negative accelerations in the liquid flow. Under these conditions, we observed an increase in the cavitation-pressure pulse amplitude and in the number of radial–spherical oscillations per exciting negative pressure pulse; that is, there was a decrease in the pressure-pulsation damping coefficient [1, 8]. A similar dependence of the damping coefficient of gas–vapor cavity oscillations on the gas content β is observed in the case of an electric discharge in liquid. Note that, at small cavitation numbers (i.e., at high pressures disturbing the bubble), viscosity has only a slight damping effect on the radial oscillations of the cavitation region over a wide range of Re ≥ 1. The most significant factor in oscillation damping is the gas content [1, 8]. Let us locate the boundary between cavitation and boiling in the case of a simultaneous pressure drop along the bubble trajectory (hydrodynamic cavitation) and propagation of a negative pressure pulse (acoustic pulsed cavitation). Under the assumption that the inertial terms, specifically, the terms taking into account the viscous forces, surface tension, and gas and vapor pressure in the bubble, are small, Eq. (1) takes the form ∆T ϕ ( t ) f ( t ) ------- – ----------- – ---------- – 1 = 0 χa J χh

(1‡)

ϕ( t ) f ( t ) ∆T = J  ----------- + ---------- + 1 .  χh  χa

(6)

or

Calculating the temperature difference by Plesset’s formula, Brennen [4] derived an expression for the cavitation-to-boiling transition temperature T ∞* , which, after modification, appears as χ v ------------h  L  3

1/2

2 2r  ρ *v ∞ 2 - -------- . Ⰶ ------------------1/2 * a c l T *∞  ρ l∞

(7)

Here, L = vt is the characteristic length related to the pressure pulse duration, irrespective of the nature of the pressure pulse. Let us substitute L = c∆t = const, which is the duration of the negative pressure pulse of any nature (hydrodynamic or acoustic), into Eq. (7). The modified cavitation criterion χ will then appear as χa ϕ ( t ) + χh f ( t ) 1 ϕ( t ) f ( t ) -. --- = ----------- + ---------- = -------------------------------------χa χh χa χ χh

(8)

Since the terms of Eq. (1) that describe the trajectories of the bubble and pressure pulse (ϕ( t ) and f( t )) are

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itation mode at the optimum value of the cavitation criterion (which is determined by the free-gas content). Equation (1), depending on the values of its terms, describes boiling up or condensation in cavitation, acoustic or hydrodynamic cavitation, the outgassing of a liquid, the vacuum separation of homogeneous mixtures, or evaporation [9]. By substituting L = c∆t and v ≡ c in Eq. (7), we obtain

log∆t II

3

–1 1

2

–2 –3 I –4 –0.2

–0.1

0

∆T

Pressure pulse duration as a function of ∆ T for (1) nitrogen, (2) methanol, and (3) water at χa = 0.2: (I) region of cavitation and (II) region of boiling.

less than unity by definition, the mixed cavitation criterion can be expressed as χh χa -. χ = ---------------χh + χa Various combinations of χh and χa values define at least the following five operation modes differing in efficiency: (1) cavitation-free mode (χh > 1, χa > 1, χ > 1), in which there is no cavitation, even though the bubbles may execute linear radial–spherical oscillations with a small amplitude of cavitation pressure pulses; (2) hydrodynamic cavitation mode (χh < 1, χa > 1, χ < 1), in which the local dynamic pressure drop is sufficient to bring about nonlinear radial–spherical bubble oscillations even if there are no acoustic pressure pulses or they are weak; (3) acoustic cavitation mode (χh > 1, χa < 1, χ < 1), in which the acoustic-pressure amplitude |Pmax| is sufficiently large to cause nonlinear radial–spherical oscillations even if the hydrodynamic pressure is invariable or varies little along the liquid-flow line; (4) mixed cavitation mode (χh < 1, χa < 1, χ < 1), in which both hydrodynamic and acoustic cavitation take place; and (5) supercavitation mode (χa Ⰶ 1, χh Ⰶ 1, χ Ⰶ 1), in which bubble cavitation gives way to developed cavitation producing macrocavities and the cavitation pressure pulses and cavitation effects are considerably reduced. The boundaries between the supercavitation and cavitation modes and between the cavitation and cavitation-free modes are indicated by inflection points in the pressure pulse amplitude (or process rate) versus χ curve [1]. Apparatuses in which the process is intensified by cavitation are best operated in the acoustic pulsed cav-

2 2r ρ *v ∞ 2 ∆t T *∞ = -------  -------------. c l c  ρ *l∞  χa

(9)

Let us introduce a cavitation-to-boiling transition criterion: 2 cT *c χa ∞ l  ρ* l∞  -------(10) Γ = ------------------. 2  ρ*  ∆t v∞ 2r At Γ > 1, applying a variable pressure to the liquid causes it to boil. At Γ < 1, cavitation dominates. At Γ = 1, there is a transition from cavitation to boiling. In the case of hydrodynamic cavitation, it is possible to substitute χ = χh and v ≡ c into Eq. (10). The cavitation-toboiling transition criterion was qualitatively estimated in our earlier work for the first time [10]. At temperatures and pressures at which pure boiling occurs, ρv and ρl are very similar. Since criterion (10) is rough, we assumed that the vapor and liquid pressures are equal when calculating the transition temperature T ∞* by formula (9). This assumption allows one to estimate T *∞ and χ. The figure shows the line demarcating the cavitation and boiling regions in the cooling of nitrogen, methanol, and water at the optimum value of χ = 0.2. Thermodynamic constants were borrowed from [11]. Lengthening the pulse duration ∆t increases the ∆T/T∞ value corresponding to the cavitation-toboiling transition. Shortening ∆T may lead to cavitation dominating over boiling even in an overheated liquid, for which ∆T > 0 (figure, curves 2, 3). Thus, bubble pulsation is governed by the short negative pressure pulse and, accordingly, inertial effects rather than the low vapor flux through the gas/liquid interface. Note that the log ∆t versus ∆T curve straightens with increasing volatility of the liquid. It was demonstrated experimentally [12] that, in the chamber of a modulated-flow rotary apparatus, pulsed acoustic cavitation dominates in water even at 99°C if the pressure pulses are sufficiently strong. The above dynamic model of the cavitation bubble provides an estimate for the boundary separating the cavitation and boiling regions. The cavitation-to-boiling transition criterion depends on the physical properties of the liquid and the duration and strength of the mechanical action upon the bubble (parameter χ). A possible way of increasing the resolving capacity of a bubble chamber is by generating cavitation by negative pressure pulses with a sufficient amplitude (energy), as


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in the case of the Wilson chamber, rather than by harmonic ultrasonic oscillations. It is noteworthy that, at a sufficiently high P∞and optimal pulsed cavitation, the vapor pressure in the compressed bubble may rise to 1012 Pa and the vapor temperature may reach 106 K. These conditions are sufficient for the fusion of light nuclei with a positive energy balance. NOTATION a—thermal diffusivity, m2/s; c—speed of sound, m/s; cl, cv—constant-pressure specific heats of the liquid and vapor, respectively, J/(kg K); ρ *l∞ – ρ *v ∞ - P —ratio of the initial vapor presJ–1 = --------------------r ∞ ρ *l∞ ρ *v ∞ s sure in the bubble to the equilibrium phase-transition vapor pressure calculated by the Clausius–Clapeyron equation; L—length, m; N—bubble concentration, m–3; nv, n—polytropic index of the free gas in the bubble and in the liquid–gas mixture through which sound propagates, respectively; P—pressure, Pa; Ps = P∞ – Pv∞ + 2σ/R0—liquid pressure at the liquid/gas interface disregarding the external disturbance, Pa; P v = Pv∞ /Ps; R—bubble radius, m; R0—initial bubble radius, m; R = R/R0; r—specific heat of evaporation, J/kg; T—temperature, K; ∆ T = (T – T∞)/T∞; t—time, s; ∆t—pressure pulse duration, s; v—velocity, m/s;

v (t) = v max ϕ( t ); β = Pg0 /Ps—initial gas content of the bubble; Γ—criterion establishing the boundary between cavitation and bubbling; χh χa - —mixed cavitation criterion; χ = ---------------χh + χa χa = Ps /|Pmax|—acoustic cavitation criterion; 2

2

χh = 2Ps /(ρl∞ v max )—hydrodynamic cavitation criterion; 2

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λ—volumetric gas content of the liquid–gas mixture; ν—kinematic viscosity of the liquid, m2/s; ρ—density, kg/m3; σ—surface tension coefficient of the liquid, N/m; 1dR Ma = --- ------- —Mach number; c dt R P Re = -----0 ------s- —Reynolds number; ν ρ l∞ 2σ We = ----------- —Weber number. R0 Ps SUBSCRIPTS AND SUPERSCRIPTS 0—initial value; ∞—infinitely far from the bubble; g—gas; l—liquid; max—maximum value; opt—optimum value; v—vapor; *—cavitation-to-boiling transition. REFERENCES 1. Yudaev, V.F., Hydromechanical Processes in Rotary Apparatuses with a Modulated Flow Area of the Treated Medium, Teor. Osn. Khim. Tekhnol., 1994, vol. 28, no. 6, p. 581. 2. Rozenberg, L.D., Cavitation Region, Fizika i tekhnika moshchnogo ul’trazvuka. Moshchnye ul’trazvukovye polya (Intense Ultrasound Physics and Engineering: Strong Ultrasound Fields), Moscow: Nauka, 1968, p. 221. 3. Filin, N.V. and Bulanov, A.B., Zhidkostnye kriogennye sistemy (Liquid Cryogenic Systems), Moscow: Mashinostroenie, 1985. 4. Brennen, S., Cavitation Flow Dynamics and Compliance, Trans. ASME, Ser. D, 1973, no. 4, p. 121. 5. Bazadze, L.G. and Yudaev, V.F., Intensifying Distillation by Means of Hydrodynamic Cavitation, V Vsesoyuznaya konferentsiya po teorii i praktike rektifikatsii (V AllUnion Conf. on the Theory and Practice of Distillation), Severodonetsk, 1984, part 2, p. 277. 6. Akulichev, V.A., Kavitatsiya v kriogennykh i kipyashchikh zhidkostyakh (Cavitation in Cryogenic and Boiling Liquids), Moscow: Nauka, 1978. 7. Smorodov, E.A., Experimental Study of Cavitation in Viscous Liquids, Abstract of Cand. Sci. (Phys.–Math.) Dissertation, Moscow: Andreev Inst. of Acoustics, 1987. 8. Bigler, V.I. and Yudaev, V.F., Pulsed Acoustic Cavitation in Hydrodynamic Siren Type Apparatuses, Akust. Zh., 1989, vol. 35, no. 3, p. 409.


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9. Balabyshko, A.M. and Yudaev, V.F., Rotornye apparaty s modulyatsiei potoka i ikh primenenie v promyshlennosti (Rotary Modulated-Flow Apparatuses and Their Industrial Applications), Moscow: Nedra, 1992. 10. Yudaev, V.F. and Sokolov, A.F., Cavitation and Boiling, Tezisy dokladov Vsesoyuznogo nauchno-tekhnicheskogo soveshchaniya “Puti sovershenstvovaniya, intensifikatsii i povysheniya nadezhnosti apparatov v osnovnoi khimii” (Proc. All-Union Conf. on the Improvement and

Raising the Efficiency and Reliability of Apparatuses in the Chemical Industry), Sumy, 1980, part 1, p. 43. 11. Vargaftik, N.B., Spravochnik po teplofizicheskim svoistvam gazov i zhidkostei (Handbook of the Thermal Properties of Gases and Liquids), Moscow: Nauka, 1972. 12. Yudaev, V.F., Acoustic Cavitation in Hydrodynamic Sirens, Akustika i ul’trazvukovaya tekhnika (Acoustics and Ultrasonic Engineering), Kiev: Naukova Dumka, 1983, issue 18, p. 9.


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