The features of rewetting dynamics of the overheated ... - Springer Link

Thermophysics and Aeromechanics, 2012, Vol. 19, No. 2

The features of rewetting dynamics of the overheated surface by a falling film of cryogenic liquid* I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev Kutateladze Institute of Thermophysics SB RAS, Novosibirsk, Russia E-mail: [email protected] (Received February 7, 2011) The dynamic process of rewetting of the overheated surface by gravitationally falling film of cryogenic liquid was firstly modeled numerically with consideration of local distribution of heat transfer coefficient in the wetting zone along the 2D front. The front shape corresponding to self-organizing regular structures observed in experiments was obtained in the numerical experiment. Evolution of the front shape was studied. It was shown that local motion velocities of different areas of the 2D wetting front differed significantly. Total time of transitional process was determined by the minimal velocity of evaporating liquid boundaries in the front zones between boiling jets. This model allows quantitative determination for the wetting front velocity, variable in time and space, and temperature fields in the heater. Reliability of calculation results was proved by direct comparison with experimental data. Key words: transitional processes, film flow, liquid nitrogen, numerical modeling.

Introduction The liquid film flows are widely used in various modern heat exchanging devices for intensification of heat and mass transfer processes. Evaporation and boiling in thin films of liquid ensure high intensity of heat transfer at low heat-carrier flow rates and low temperature differences. Development of the compact film systems for cooling of microelectronic equipment, efficient processors, whose performance and service life depend significantly on efficiency of dissipated power withdrawal, is the topical problem. Application of cryogenic liquid films as a coolant in technologies using hightemperature superconductive ceramic elements seems to be promising. It is shown in [1] on the basis of experimental investigations of crisis phenomena in the falling cryogenic liquid films (liquid nitrogen) that for heat load varying periodically, the parameters of appearing metastable regular structures, critical parameters of heating surface drying, and reverse transitions to efficient heat removal are *

The work was financially supported by the Russian Foundation for Basic Research (Grant No. 09-08-00118-а), Integration Project of SB RAS together with UB RAS (No. 68) and Federal Target Program “Investigations and developments in the priority fields of S&T complex of Russia for 2007 − 2012”. (Action 1.8).

© I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev, 2012

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I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev

determined by dynamics of the movable wetting boundaries in the process of selforganization of the system. The transitional process develops after heat removal under the critical conditions; as a result an absolutely dry surface is again cooled by the falling film of liquid nitrogen. This phenomenon is called the process of repeated wetting or rewetting. In nuclear power engineering, when studying emergency conditions in active zones this phenomenon is called reflooding, and it has been studied for many years [2−11]. Numerical simulation of possible emergency is an important grounds element for reactor plant safety. To analyze the behavior of reactor at operation and emergency, the thermal-hydraulic, strength, neutron-physical and other codes are used. The problems of heat transfer at reflooding of the active zone of nuclear power plant with different methods of cooling water supply are illustrated in detail in the monograph [12]. Similar questions arise at development of the system for magnet protection in transition from the superconductive to normal state [13, 14]. At operation of charged-particle accelerator there can be situations, when some superconducting magnets or sets of magnets turn to the normal state (the quench-effect). At this, the task of the cryogenic system is ensuring of integrity of magnet cryostats in the process of quenching; thus, the problems of repeated cooling arise. Development of fuel supply systems for cryogenic rocket engines is also the important application of results on rewetting. In the pipelines of these systems cooled by rocket fuel, the temperature changes from T ~ 300 K to the cryogenic value with successive change of heat transfer regimes from post-CHF heat transfer to nucleate boiling and forced convection of the single-phase liquid. Investigation results on dynamics of a change in heat transfer regimes with regard to the conditions of cryogenic system cooling, including the conditions of microgravitation, are presented in [15]. A typical fragment of high-speed recording of rewetting process on the overheated −6

surface (constantan foil with the thickness of δ = 25⋅10 m) after pulsed heat release is shown in Fig. 1 [16]. The complicated 2D character of the wetting boundary, whose consideration is required for detailed simulation of front dynamics, is shown on the basis of visualization of the transitional process. Visualization allowed the revealing of this process features. It was found that the rewetting front is not flat crosswise. The regular liquid jets are formed even at the inlet of overheated surface, and intensive boiling develops in their lower part (zones 1 in Fig. 1). Liquid rolls are formed between the jets at the boundary of dry surface (zones 2 in Fig. 1), heat transfer in these zones is determined predominantly by evaporation. According to experimental data treatment, at rewetting of the overheated surface by the falling liquid film, the local velocities of different zones of the 2D front differ significantly. Local velocities of the wetting front in the jet zone are significantly higher than those in the zones between the jets. Therefore, the rivulet flows are formed at transition, and the total time of reflooding is determined by the velocity of motion of film flow boundaries in the zones between the jets. Fig. 1. Fragment of high-speed video of rewetting process after the heat release pulse. q = 21.23⋅10 W/m , Re = 1690, τ = 2.5 s. 4

308

2


According to [17, 18], heat transfer at boiling in the liquid film is characterized by considerably higher heat transfer coefficients than that at evaporation. Boiling development in the bottom part of jets leads to a fundamental change in the structure of the two-phase layer in the wetting front. Fast growing bubbles break the roll shape of interface observed between the jets and at lateral boundaries of jets. This leads to the fact that heat transfer in the bottom zone of jets in the wetting front occurs with very high averaged heat transfer coefficient at boiling in the film with local zones with a thin liquid microlayer. As was shown in [18], the zone of efficient heat transfer at boiling in the front of heat transfer regime change on the thin-wall heater at ε < 1 is limited by the temperatures of heat-releasing surface corresponding to the temperature of limit overheating of liquid at homogeneous nucleation Tbound = Tlim. In papers [9−11], which dealt with experimental and theoretical studies of rewetting at irrigation of the overheated horizontal surface [9] by a single jet and at flooding of vertical channels from the bottom [11] at development of calculation models the heat transfer coefficient in the wetted zone was not changed, the wetting front was taken flat with the same propagation velocity in transverse direction. Here is the known calculation relationship for velocity of wetting front propagation in one-dimensional statement derived in [2]: V

−1

=

ρ heat cheat 2

δ heat heat αλ

⎛ ⎡ 2(T − T ) ⎤ 2 ⎞ ⎜⎢ 0 + 1⎥ − 1⎟ ⎜ ⎣ T − Tsаt ⎟ ⎦ ⎝ ⎠

0.5

,

(1)

is the temperature at the boundary of wetting front, T is the initial temperature where Ф 0 of the overheated wall. In the framework of this model, heat transfer coefficient α in the wetted zone was taken equal to some constant value, and in the dry zone it was taken zero. In the current study we have modeled numerically the observed transitional phenomena with consideration of essential 2D character of the front and spatial maldistribution of heat transfer coefficients in the wetted zone determined experimentally. The goal of this modeling is explanation and quantitative description of complicated 2D shape of rewetting front obtained at transitions under the considered conditions. Modeling and experiment The mathematical model assumes the thermal nature of development of rewetting dynamics. The spatial and time change in temperature fields for a thin heater (Biot number Bi = qδheat /(λheat(Theat – Tsat)) < 1) is described by 2D nonstationary equation of heat conductivity with corresponding initial and boundary conditions:

λheat ∂Theat = cheat ρ heat ∂τ

⎛ ∂ 2Theat ∂ 2Theat + ⎜⎜ 2 x ∂ ∂y 2 ⎝

⎞ 1 ( q + − q − (Theat )). (2) ⎟⎟ + δ ρ c heat heat heat ⎠

In the framework of the current study modeling was carried out in 2D calculation domain. Here х, y are transverse and longitudinal coordinates of the heater, ( x , y ) ∈ G ,

{

where G = 0 ≤ x ≤ Lx , 0 ≤ y ≤ Ly

} is the rectangle with sides Lx, Ly. The ordinate axis

is directed along the vertical wall, where the liquid flows down, upstream the film flow. The density of heat flux removed into liquid is q− = q−(Theat). The density of heat release q+ is taken constant q+(х, y) = const. Equation (2) supplemented by initial and boundary 309

I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev

conditions allows modeling of temperature field evolution in time within 2D calculation domain and obtaining of the dynamic pattern of front propagation as a sequence. Initial temperature field T0(x, y) shown in Fig. 2 corresponds to experimental data obtained for the moment of completion of the heat-release pulse according to [1]. The initial temperature of surface dried by pulse heat release at the moment of rewetting development was T0 = 693 K, in the top part of the heater in the wetted zone T0 = Tsat, at this, in calculations, the temperature jump was smoothed exponentially. Boundary conditions ∂Тheat /∂y = 0 for y = 0, х = 0 ÷ Lx; ∂Тheat /∂x = 0 for x = 0; Lx, y = 0 ÷ Ly were set from the point of symmetry. For the upper edge of the heater y = Ly, х = 0 ÷ Lx, we take boundary condition Theat = Tsat. Heat transfer intensity q−(ΔTh) is modeled by heat transfer curves, where experimental data of [17, 19] are used. Linearized heat transfer coefficient α changed abruptly at the front boundary in different zones: at points ΔTbound.1 = ΔTlim = 26 K at the lower boundary of jets with developed boiling, ΔTbound.2 = 11 K at the boundary of film flow with evaporation. Respectively, at Theat ≤ Tbound.1, α = α nucl.boil (nucleate boiling in the film) or at Theat ≤ Tbound.2, α = α eva (evaporation in the film) depending on the regime in the local film zone. At Theat > Tbound α = α d.s , what corresponds to heat transfer in the zone of dry spots at turbulent free convection in the vapor phase. In calculations we took thermal-physical properties and geometrical parameters of the heatreleasing wall, corresponding to these properties of constantan foil used in experiments 3 −6 of [1]: λheat = 18 W/(m⋅K), сheat = 245 J/(kg⋅K), ρheat = 8850 kg/m , δheat = 25·10 m, −3

−3

Ly = 32⋅10 m, and Lx = 20⋅10 m. Data on heat transfer were also taken from [1]: 4 2 2 α = αeva = 6000 W/(m2⋅K), α = α nucl.boil = 4.7·10 W/(m ⋅K), α = α d.s = 50 W/(m ⋅K) at Тsat = 77.4 K. Thus, with consideration of process features, determined in experiments, on the lateral surface of jets and between them within the wetted zone, heat transfer with evaporation is set with corresponding boundary conditions for the roll shape of the wetting front. It is necessary to note that the strict physical grounding of existence of a stable meniscus between the jets and on lateral curvilinear edges of falling liquid jets observed experimentally under the conditions of intensive evaporation is an object of

Fig. 2. Initial distribution of temperature, τ = 0. 310


separate theoretical examination. The rivulet flow is formed at rewetting by the falling liquid film. According to visualization by the high-speed digital camera, the falling 3D waves affect mainly dynamics of jet formation and their transverse size in the rewetting front. The process of jet formation is not described in the current paper; this is an object of separate study. Simultaneously, it becomes evident that falling 3D waves of high amplitude transferring an increased local liquid flow rate are the intensive perturbations influencing stability and shape of the wetting boundary at the initial stage of jet formation. In the zone of forming jets with an increased thickness of liquid layer, there can be the conditions for development of boiling in the lower part, where required overheating of the heat-releasing surface is achieved in the wetted zone. According to analysis of video fragments, boiling develops exactly in the zones of increased film thickness in jets. It is assumed in the first approximation that transverse size of jets with boiling in their lower part δboil equals the typical transverse size of the incident 3D waves

δ ⊥w , for liquid nitrogen film in the studied range of Reynolds number this value is

δ ⊥w ∼ 4 mm. The minimal transverse size of liquid jets δboil with nucleate boiling in the lower part should be set in calculations from the conditions of front shape stabilization with minimal value of δ ⊥cr = 2 R0cr , which is determined at analysis of behavior dynamics of the local wetted zones of a smaller radius, formed at its boundary. Therefore, in case of δ ⊥cr > δ ⊥w , value δboil will be determined from condition δ boil = δ ⊥cr . Analysis of estimate for δ ⊥cr = 2R0cr is shown in the final part of this paper. For the conditions of experiment, where experimental data are compared with calculation results value δ ⊥w is significantly higher than the value of critical size δ ⊥cr , which determines minimally possible transverse size in the lower part of liquid jets from the conditions of front shape stabilization. The case of δ ⊥cr > δ ⊥w is not considered in this paper, it is an object of the following studies both in the framework of numerical modeling and in experiments. The problem was solved numerically with application of the scheme of alternating direction method [20], which combines the best qualities of the explicit and implicit schemes: economy and stability, respectively. Together with main values Tijk and Tijk +1 we introduce intermediate value Tijk +1/ 2 , which can be formally considered as the value

at τ = τ k +1/ 2 = τ k + 1/ 2. For the 2D situation the scheme of alternating direction method

takes form:

Tik, j+1/ 2 − Ti k, j

τ /2 Tik, j+1 − Tik, j+1/ 2

τ /2

=

=

a hx2 a hx2

(Ti k+1,+1/j 2 − 2Ti k, j+1/ 2 + Ti k−1,+1/j 2 ) +

(Ti k+1,+1/j 2 − 2Tik, j+1/ 2 + Ti k−1,+1/j 2 ) +

a hy2 a hy2

(Tik, j +1 − 2Ti k, j + Ti k, j −1 ) + fi k, j+1/ 2 , (3)

(Tik, j++11 − 2Ti k, j+1 + Tik, j+−11 ) + fi k, j+1/ 2 . (4)

Here a = λheat /(cheat ρ heat ), fi k, j = 1/(δ heat сheat ρ heat )(q+ − q− (Ti k, j )), hx, hy are grid steps in x and y directions, respectively. On every fractional time layer, one of spatial differential operators is approximated implicitly (scalar sweeps is performed along 311

I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev Fig. 3. Evolution of the film flow boundary. δboil = 4 mm, in the lower part of boiling jet ΔTbound.1 = 26 K, in the zone with evaporating film ΔTbound.2 = 11 K, gap between the curves Δτ = 0.5 s.

the corresponding coordinate direction), other operators are approximated explicitly. At the next fractional step the following differential operator is approximated implicitly and the rest operators are approximated explicitly, and so on. The solution to the problem in this case is reduced to solution of two systems with three-diagonal matrices, what allows application of one-dimensional sweep. The result of solution to two equation systems is k

nonstationary temperature field Ti , j obtained in 2D calculation domain (or, in other words, temperature matrix located on different time layers). Analyzing the values of temperature at the grid nodes obtained via solution and using the known temperatures at the boundary (Тbound.1 is for boiling film, Тbound.2 is for evaporating film), we find coordinates of the surface corresponding to instantaneous front position. To achieve the pattern of wetting front propagation in dynamics, the procedure is repeated on the next time layer, etc. The instantaneous propagation velocity of the front local zone is determined by the time derivative of longitudinal coordinate of the front boundary k +1 k − ybound ) /(τ k +1 − τ k ). in corresponding local zone: V ( ybound

Results of numerical modeling of propagation dynamics of the wetting front boundary are shown in Fig. 3. The front shape changing in time was obtained, as it is described above, under the condition that at the initial time moment the film in the local zone of the upper part of heat-releasing surface, limited by sizes δboil,x, δboil,y , is under the regime of boiling. The temperature at the boundary of evaporating film is Tbound.2 = = 88.4 K, at the boundary of boiling jet it is Tbound.1 = 103.4 K. According to results of numerical experiment, the instantaneous velocity of boundary changes non-linearly: a drastic increase in velocity occurs at front propagation down along the heater. This is caused by the fact that in the dry zone, the temperature of heat-releasing surface without heat release decreases significantly at transition because of heat transfer at free convection in the vapor phase. The average velocity of motion of boiling jet boundary V1 exceeds significantly the average velocity of evaporating film boundary V2. Evidently, the time of complete wetting of the whole heat-releasing surface is determined by the minimal velocity of boundaries of evaporating film between the jets. Results of numerical modeling of temperature field evolution in the heatreleasing wall at propagation of the film flow front are shown in Fig. 4. Comparison of calculation results with experimental data on velocities of front propagation in the zone of film boiling and evaporation and on evolution of rewetting front is shown in Figs. 5 and 6. The dashed lines in Fig. 6 show the boundaries of film flow obtained in numerical experiment for corresponding time moments. 312

Thermophysics and Aeromechanics, 2012, Vol. 19, No. 2 Fig. 4. Change of temperature fields in time. τ = 1.5 (а), 2.0 (b), 2.5 (c) s.

Evolution of the front shape with the complicated 2D boundary of film boiling in transitional process obtained in numerical modeling agrees well with that obtained in experiments. Results of solution to the model problem of thermal stability of local temperature perturbation are shown below. This model describes development of local sites of nucleate or film pool boiling on a flat horizontal surface. Results of this calculation are presented for explanation of existence of the critical magnitude of perturbations in the rewetting front. If perturbation magnitude is less than the critical one, the local front velocity in the zone of perturbations will decrease drastically; according to estimate calculations, this will cause stabilization of the front shape within the jet zone. To reveal the mechanism of boiling jet stabilization, evolution of axisymmetric temperature perturbations with negative (Fig. 7а) and positive (Fig. 7b) deviations was modeled numerically. The first of them corresponds to the local zone with boiling liquid on the dry surface overheated up to the temperature of ~700 K. Evolution of the second describes development dynamics of a “dry” spot with the temperature in the center of ~ 700 K on the surface cooled by boiling liquid nitrogen. The front radius in the lower part of boiling jets, observed in experiments with −6

constantan foil (δheat = 25⋅10 m), is Rtyp ≈ ≈ 3 ÷ 5 mm. According to results of numerical experiments, small-scale perturbations with negative deviation (local sites of nucleate

Fig. 5. Change in velocity of rewetting front in time. 1 ⎯ in the zone of evaporating film, 2 ⎯ in the zone of boiling jets. 1′ ⎯ experiment, 2′ ⎯ calculation, 3′ ⎯ calculation [2].

313

I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev Fig. 6. Fragments of high-speed video of the transitional process after a single pulse of heat release with duration of Δτ = 0.2 s. q = 21.23⋅10 W/m , Re = 1690, τ = 1.5 (а), 2.5 (b), 3 (с) s. 4

2

boiling on the dry surface) with magnitudes below the critical one R ≤ R0cr ≈ Rtyp collapse (Fig. 8) or their velocity is considerably less, what, probably, leads to front leveling in the zone of boiling jets until its stabilization in the shape with the typical radius curvature Rtyp ≥ Rcr . With a rise of plate thickness, the critical size of local dry axisymmetric spots decreases (curve 1 in Fig. 9). On the contrary, the critical magnitude of perturbations with negative deviation, corresponding to local axisymmetric zones of boiling liquid on the dry surface, increases. Calculation results shown in Fig. 9 allow the assumption that under certain conditions, an increase in thickness and heat conductivity coefficient of the plate will lead to a corresponding rise (curve 2) of transverse size of boiling jets in their lower zone. This hypothesis will be checked by the following experiments on investigation of rewetting of thick-wall and more high-heat-conductive surfaces.

Fig. 7. Temperature perturbations with negative (а) and positive (b) deviation. 314


Fig. 8. Time evolution of temperature perturbations with negative deviations of different initial amplitudes.

Fig. 9. Critical magnitude of temperature perturbation with positive (1) and negative (2) deviation depending on plate thickness.

Conclusion Rewetting of the vertical overheated surface by the falling film of cryogenic liquid with allowance for locally distributed heat transfer coefficient in the wetted zone along the 2D front was investigated for the first time. According to results of numerical experiments, the total time of rewetting is determined by the minimal velocities of evaporating film boundaries in the front zones between boiling jets. In the numerical experiment, we have achieved the shape of front in dynamics, which correlates with the regular structures observed in experiment. The model allows us to quantify the wetting front velocity and temperature fields in the heater, variable in space and time. Reliability of results obtained by numerical methods is proved by direct comparison with experimental data. The developed numerical model ensured good qualitative and quantitative description of the total pattern of rewetting front propagation. The results obtained are important for determination of the main development regularities of transitional processes and crises at boiling and evaporation, including the falling liquid films, for development of new approaches to description of crisis phenomena under different laws of heat release. Information about the total time of transitional rewetting, velocities of different zones of the front, and temperature fields in hot rods is required for solutions to practical tasks of safe operation of nuclear reactors. Understanding of mechanisms, causing formation of jet flows in the heated film, allows control of jet formation, regulation of dry spot development, and improvement of heat transfer intensity. To determine the typical sizes of regular structures, such as the transverse jet size and distance between the jets, the additional studies are required. Nomenclature q ⎯ heat flux density, W/m , 2 g ⎯ acceleration of gravity, m/s , Т ⎯ temperature, K, V ⎯ velocity, m/s, 2

ε=

λheatδ heat g (ρ ′ − ρ ′′) ⎯ dimensionless parameter α nucl.boilσ

characterizing the ratio of the width of temperature gradient along the heater in the front of intensive heat transfer to the typical action scale of capillary forces Λ, σ ⎯ surface tension coefficient, N/m,

Λ ⎯ Laplace constant, ΛR−T = 2 3 πΛ ⎯ most dangerous wavelength of Rayleigh ⎯ Tailor instability, 2 α ⎯ linearized heat transfer coefficient, W/(m ⋅K), δ ⎯ thickness, typical size,

δ ⊥cr , R0cr ⎯ typical critical values of linear perturbation scale in the rewetting front,

λ ⎯ coefficient of heat conductivity, W/(m⋅K), 3 ρ ⎯ density, kg/m , τ ⎯ time, s.

315

I.P. Starodubtseva, A.N. Pavlenko, O.A. Volodin, and A.S. Surtaev Indices ′ ⎯ liquid, ′′ ⎯ vapor, lim ⎯ limit overheating, bound ⎯ boundary value in the front, d.s ⎯ attributed to dry spots, heat ⎯ heater, sat ⎯ at the line of saturation,

cr ⎯ critical, typ ⎯ typical, nucl.boil, boil. ⎯ boiling regime, eva ⎯ evaporation regime, 0 ⎯ initial conditions, ┴ ⎯ attributed to the typical transverse size of 3D waves.

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