Large-scale fluctuations during a boiling crisis and during the boiling up of a superheated liquid jet

. The results of an experimental study of pulsation processes during the transition from nucleate to film boiling on a wire heater and in transient regimes of boiling up of a liquid jet flowing from a high-pressure chamber are presented. An analogy in the behavior of the boiling curve and in the boiling liquid flow rate dependence on the pressure in a thermodynamically strongly nonequilibrium flow regime is shown. The similarity of pulsation processes in a vapor jet during crisis boiling of water on a wire heater and in a jet of a superheated liquid (water and ethanol) under the conditions of the transient boiling regime is noted. Conditions under which the power spectra of local and integral fluctuations changed in inverse proportion to the frequency in the low-frequency region (1/f fluctuations) have been established. A high correlation coefficient between local and integral pulsation characteristics both in a vapor jet on a wire heater and in a jet of a boiling liquid has been revealed.


INTRODUCTION
An important feature of critical regimes of heat and mass transfer with phase transitions is a possibility of largescale temperature fluctuations, heat and mass flows occurring in them.Random pulsations in crisis modes must be taken into account when diagnosing the reliable operation of power equipment in limiting and critical thermal loads.
Large fluctuations spontaneously arise in complex statistical systems that are far from equilibrium.A significant part of the energy of such pulsations is associated with slow processes and accumulates at low frequencies, so large-scale high-energy emissions are possible.Such fluctuations have the property of scale invariance, which manifests itself both in the power-law behavior of the power spectra, which are inversely proportional to the frequency (1/f spectrum), and in the power-law behavior of the amplitude distribution functions.A similar situation exists at the thermodynamic critical point of the liquid-vapor phase transition.The scale invariance of fluctuations of thermodynamic quantities near the critical point is determined by the conditions for the convergence of the properties of different phases and requires precise tuning and long relaxation times.In contrast to the thermodynamic critical point, non-equilibrium processes with large fluctuations show a stable spatial and temporal scale-invariant state without fine tuning of the parameters.Therefore, the occurrence of 1/f fluctuations is often associated with the concept of selforganized criticality [1,2], which describes avalanche dynamics and is used to demonstrate the criticality of behavior in a large number of computer models, including the most famous "sand pile" model.
Random large-scale pulsations with a power-law behavior of the spectral density and amplitude distributions were experimentally found in heat and mass transfer processes with nonequilibrium phase transitions, in particular, in the critical regimes of transition from the bubble boiling regime to the film boiling regime and during the critical outflow of a boiling liquid [3 -6], during acoustic cavitation [7], under electric discharge and in transient combustion regimes [8,9].Such fluctuations were modeled theoretically by a system of nonlinear stochastic equations describing critical fluctuations arising during interacting nonequilibrium phase transitions [10].
A typical example of a critical mode of heat and mass transfer is a boiling crisis that occurs when the heat flux reaches a critical value and there is a transition from the nucleate boiling mode with discrete bubbles on a surface to the film boiling mode in which a vapor layer covers the entire heating surface [11].Such a layer causes a sharp deterioration in the process of heat removal, which leads to a strong increase in the temperature of the heater.Figure 1a shows a typical temperature dependence of the heat flux on the heater during the transition from bubble to film boiling [12].The boiling crisis is a well-studied phenomenon.In most works, the influence of various factors on the critical heat flux density is studied and possible mechanisms of transition to film boiling are analyzed.It is shown in [3,4] that the boiling crisis can be considered as a critical process with the property of scale invariance, in which large-scale fluctuations arise.Another example of the crisis behavior is the explosive boiling-up of superheated liquid jets flowing out of a high-pressure vessel [13,14].Figure 1b shows the dependence of the flow rate on pressure during explosive boiling in a nozzle when a highly superheated liquid flows out of a high-pressure vessel through a short nozzle [15].This dependence is similar to the boiling curve at a heat transfer crisis shown in Figure 1a.The process of a hot liquid outflow from the high-pressure chamber through a short nozzle into the atmosphere is characterized by a deep penetration of the liquid into the region of metastable (superheated) phase states.The lifetime of a liquid in a superheated state depends on the magnitude of supersaturation and is determined by the kinetics of the birth of vapor bubble nuclei.During the outflow, transitional regimes associated with a change in boiling mechanisms are observed: vaporization at individual non-interacting centers, intense heterogeneous nucleation of vapor bubbles, in which a chain avalanche-like activation of boiling centers is observed, and homogeneous fluctuation nucleation.The processes of transition from bubble to film boiling on a heater surface and the outflow of a highly superheated liquid through a nozzle are nonequilibrium phase transitions and have much in common, in particular, large fluctuations of the process parameters can occur in crisis regimes of boiling and outflow which can be comparable to the average values or even exceed them.In the present work, attention is drawn to common features in the dynamics of the transition from nucleate to film boiling on the surface of a wire heater and during the outflow of a boiling jet of a superheated liquid.An experimental study of local and integral characteristics of pulsations and determination of their mutual correlations has been carried out.

MEASURING TECHNIQUE AND DATA ANALYSIS
Along with the existing analogy between the boiling curve of a liquid (Figure 1a) and the flow characteristic of a boiling liquid in a thermodynamically nonequilibrium flow regime (Figure 1b) similar patterns of behavior of pulsation processes both during film boiling on a vertical wire heater (details in [3,4]) and intense boiling up of a jet of a superheated liquid flowing from the high-pressure chamber through a short nozzle in the horizontal direction were noted.Figure 2 shows the images of a vapor film of boiling water on a vertical wire heater (Figure 2a) and a jet of boiling water flowing through a short nozzle into the atmosphere (Figure 2b).Observations of the behavior of a vapor film on a wire show the presence of significant fluctuation processes.In particular, the length of the hot zone of the wire which is under the vapor film at a temperature of about 6000 C fluctuates and the angle of the cone of the vapor jet near the wire changes significantly.These pulsations are reflected in the behavior of the electric current flowing through the heater.Noticeable fluctuations in the length of the non-boiled core of the superheated jet ahead of the rather intense boiling front were also noted in the jet of boiling liquid.A comparison of these two processes shows their common features.
Experiments were carried out to find the correlation of local and integral fluctuations both on a wire with a transient regime of liquid boiling and in a jet of boiling liquid in order to identify a possible source of 1/f fluctuations.To fix the pulsations on the wire, two independent measurement lines were organized: an electric line that monitors the current oscillations (integral fluctuations) on the heater caused by changes in the film size on it and a photometric line that fixes local oscillations of the vapor film by measuring the intensity of the laser radiation passing through it.The measured time series are used to determine the power spectra of large-scale fluctuations.
Figure 3 shows the time series of pulsation processes on the wire for the resistive method (Figure 3b) and the photometric one (Figure 3c) in cross section 1 of the vapor film (the sections are shown in Figure 3a) and their power spectra (Figure 3d1, Figure 3d2, Figure 3d3).Experimental data show that the power spectrum corresponding to the resistive method (spectra of higher intensity in Figure 3d1, 3d2, 3d3) always changed in the low frequency region according to the 1/f law.To obtain time series by the photometric method, a linear laser beam 10 mm long was used so that this length exceeded the transverse dimension of the vapor cone near the wire.As a rule, the transverse size of the cone did not exceed 6 mm.The linear beam was placed horizontally so that it crossed the top of the vapor cone (lower point of the vapor jet, section 1 in Figure 3a), then the middle of the cone (section 2 in Figure 3a) and section 3 in Figure 3a.It was found that for cross section 1 the spectrum also obeyed the 1/f dependence at low frequencies, and the correlation coefficient between the resistive and photometric realizations in their coarsened form (Figs. 3b and 3c) reached 0.85.Since the vast majority of the energy in 1/f fluctuations is concentrated in low-frequency oscillations, we carried out the comparison for scale-transformed time series.A largescale coarsening of the measured realizations of pulsations is carried out by averaging over a certain time scale in accordance with the expression where Ij is the measured variable, N is the number of points in the measured time series, the parameter τ is called the scaling factor.For the first scale (τ = 1), the new time series I (1) is simply the original one.The length of each subsequent coarse realization decreases by τ factor, i.e., contains N/τ points.This scaling transformation does not change the low-frequency behavior of the power spectrum, and the probability density functions starting from a certain value τ cease to change.This indicates that the measured fluctuations have the property of scale invariance [16,17].The course time series were used to find the correlation coefficient.
The power spectrum for the photometric time series in section 2 in Figs.3a at low frequencies tends to a constant value -white noise (Figure 3d2).And the correlation coefficient with the resistive time series is significantly reduced to a value of 0.25.Fundamentally, the same picture is observed for the spectrum in section 3 in Figure 3a, and the pulsation correlation coefficient decreases to 0.1.
The high correlation of local fluctuations at the top of the vapor jet cone with integral resistive fluctuations makes it possible to associate the latter with the processes occurring at the top of the jet cone, i.e. to localize the source of integral fluctuations.
Figure 2b shows noticeable fluctuations in the length of the non-boiled jet core behind the exit from the short nozzle under the conditions of the transition mode of vaporization (transition from boiling at single centers to boiling at intense heterogeneous nucleation).Previously, we constructed the dependence of the average length of the non-boiling jet core on liquid superheats in the case of low superheats and boiling up at single centers [18].The dependence obtained was close to a linear one.In the case under consideration, under conditions of high superheats of the liquid, the length fluctuations were concentrated within 5 mm near the exit of the nozzle.This made it possible to use a point laser source focused on the jet core ahead of the intense jet boiling front and to achieve a frequent crossing of the laser beam by the boiling front.In this way, the experimental realizations of local pulsations in the jet were obtained.The integral pulsations of the jet were recorded by a linear laser beam about 10 cm long.The laser line was located at a distance of 20 cm from the point source further downstream of the jet.This line crossed only the upper boundary of the jet, tracking the dynamics of changes in the jet opening angle (integral parameter).Figure 4 shows the coarse time series of local (a) and integral (b) fluctuations in the jet and the power spectra of initial local fluctuations (c) and initial integral fluctuations (d).The high correlation of integral and local pulsations in the jet (correlation coefficient ~0.85), both of which had 1/f power spectra in the lowfrequency region, makes it possible to relate the highintensity low-frequency pulsations of the jet to oscillations of the jet boiling front.

CONCLUSIONS
A comparison of pulsation processes during the transition from nucleate to film boiling on a wire heater and during transient regimes of boiling up of a liquid jet flowing from a high-pressure chamber has been made.The presence of an analogy in the behavior of a vapor jet on a vertical wire heater and fluctuations in the length of the superheated liquid jet core ahead of the intense boiling front has been shown.The existence of integral and local high-energy low-frequency pulsations with a power spectrum inversely proportional to frequency (1/f fluctuations) in the behavior of a vapor jet on a vertical wire heater has been established.High correlation coefficients (0.80 -0.96) between the current on the wire heater (integral pulsations) and local pulsations of the top of the vapor jet cone during water boiling crisis indicate a possible local source of extreme current pulsations (1/f fluctuations).A connection between the pulsations of the angle of the jet cone (integral pulsations with the 1/f spectrum) of a superheated liquid flowing from a high-pressure vessel into the atmosphere in the transient boiling mode with local pulsations of a non-boiled jet core has been found.

Fig. 1 .
Fig. 1.Boiling curve of water (a) and flow rate characteristic of boiling water jet (b) under initial parameters of the liquid (temperature and pressure) in the chamber corresponded to the saturation line.

Fig. 2 .
Fig. 2. Photographs of a vapor jet of boiling water on a vertical wire heater (a) and a jet of boiling water at temperature of 1800 C (b).

Fig. 3 .
Fig. 3. Schematic representation of a vapor jet on a vertical wire heater in the film boiling mode (a), coarsened time series of pulsations measured by resistive (b) and photometric methods (c), and power spectra of electric current pulsations in the heater and photometric pulsations in various sections of the vapor jet (d1, d2, d3).The dashed lines on the spectrum plots correspond to 1/f dependence.

Fig. 4 .
Fig. 4. Time series of pulsation processes (local coarse pulsations (a), integral coarse pulsations (b)), and power spectra of local pulsations (c) and integral pulsations (e) in a jet of boiling liquid.The dashed lines on the spectrum plots correspond to 1/f dependence.