Influence of liquid layer height on evaporation/boiling heat transfer under low pressure conditions

. The experimental data of the effect of the height of a horizontal liquid layer on the heat transfer coefficient during evaporation/boiling under low pressure are present on this paper. In addition, data the influence of the height of the liquid layer on the temperature head are also given. It was found that the decreasing of the height of the liquid layer led to increase, the heat transfer coefficient and had a maximum, after which a further decreasing of the height of the liquid layer leads to a decrease in the heat transfer coefficient.


Introduction
Currently, electronics is one of the rapid growing area in the world, the main development of which is associated with increase of power and decrease in the overall dimensions of devices, which leads to an increase in the heat flux that must be removed from the surface.In nowadays, air-cooling systems are widely used to cool electrical appliances, but the technological capabilities of air cooling are approaching their limit and, in some cases cannot provide effective heat removal.The solution to this problem can be the liquid cooling.The most efficient ways of removing heat from the heating surface with the help of a liquid are the processes associated with the use of the latent heat of the phase transition.The ever-increasing demands on the weight, overall dimensions of cooling devices and the improvement of temperature control conditions at high heat fluxes lead to the need to develop more and more efficient technique for intensification of the heat transfer during evaporation/boiling.One of the most effective methods for improving heat transfer during boiling is surface modification.This method includes microribs [1,2], microchannels [3][4][5], micropores [5][6][7], nanowires [8,9], carbon nanotubes [10,11], and hierarchical surfaces [12,13].
However, experimental data obtained on smooth surfaces serve as a comparative basis for estimating the increase in heat transfer coefficients due to the intensification of heat transfer.
Mostly, the experimental works were carried out at boiling in a large volume at atmospheric pressure.
One of the first works on the influence of the liquid layer height is the work of Nishikawa [14].He investigated the influence of the height of the liquid layer on heat transfer during nucleate boiling.In his work, it was found that with a decrease in the thickness of the liquid layer, the heat transfer coefficient increases.
In studies [15][16][17], it was also found that during the nucleate boiling of water, an increase in heat transfer coefficients was observed with a decrease in the thickness of the liquid layer.The experiments were carried out at atmospheric pressure.
To date, studies aimed to investigation of the effect of the liquid layer height on heat transfer during boiling and evaporation are poor, especially at low pressures.

Experiments
The investigation was carried out on an experimental heat-exchange vacuum installation.The working chamber is a thermosyphon, detailed descripted in [18].The pressure and temperature were recorded using National Instruments equipment and processed in the LabVIEW application environment.The working chamber made in the form of a stainless steel cylindrical (1) vessel with a wall thickness of 1 mm (Fig. 1).The internal diameter of the chamber is 120 mm, the height is 300 mm.The working chamber was cooled by water flowing through a coil located on the outer surface of the upper part of the working chamber.A cooling coil (8) is located on the outer surface of the upper part of the working chamber.The chamber was cooled with water flowing through the coil.To reduce heat losses due to overflows along the walls of the chamber from the bottom to the cooling coil, as well as for a more uniform temperature distribution along the bottom of the chamber, a coil for heating (7) the side vertical wall of the chamber is located above the viewing window (6).The bottom (2) of the chamber is also made of steel 12Cr18Ni10Ti.There are five holes in the bottom with a pitch of 2 mm for installing thermocouples.Between the bottom and the electric heater (4), there is a brass plate (3) for a more even distribution of temperatures from the electric heater to the bottom.The electric heater is fixed with a bracket (5).For visual observation from above and from the side, there were observation windows on the working chamber (6).The pressure in the working chamber was measured with a Setra-730 ionization-deformation pressure sensor (11) and maintained by constant regulation using a valve (12).The measurement error of this sensor is ± 0.5% of the current reading.The total uncertainty of heating surface temperature measurement did not exceed ±0.6°С.The relative uncertainty of heat flux measurement Δq/q не did not exceed ±16%, the relative uncertainty of heat transfer coefficient measurement Δα/α did not exceed ±15% with increasing heat flux the uncertainty of heat transfer coefficient and heat flux measurement decreases.N-dodecane was used as the working fluid because of its thermophysical properties.N-dodecane has a low saturated vapor pressure at room temperature.This made it possible to perform experiments at very low reduced pressures using cold tap water in the condenser.The latent heat of vaporization of n-dodecane is much lower than that of water, which is often used as a working body.This made it possible to create a working chamber with a small power heater (2 kW).In the course of the experiments, stationary heat transfer modes were obtained, at which temperatures were recorded along the thickness of the heated wall, pressure above the liquid layer in the working chamber.The pressure in the working chamber varied over a wide range of values from 33 Pa to 2•10 4 Pa.The experiments were carried out at liquid layer heights h (h/l σ ) from 1.7 mm (0.99) to 40.0 mm (23.22).The dimensionless height (h/lσ) was determined at a pressure of 133 Pa (at this pressure, the temperature of the saturated vapors of n-dodecane is 52.07°С, and the capillary constant l σ = 1.78 mm).
The dependence of the saturated vapor pressure of ndodecane on temperature is shown in Fig. 2. Table 1 presents the thermophysical properties of n-dodecane at some pressures given in the experiments [19].

Results and discussion
In this paper, we consider the influence of the height of the liquid layer during evaporation/boiling under reduced pressure conditions.Decreasing of the pressure in the working chamber led to, increasing the critical radius of the bubble.This fact effect to the decreasing of the number of active centers of vaporization, resulting in the increasing of temperature head.It was shown in [18,[20][21][22] that with a significant decreasing in pressure, the heat transfer mechanism changed, at pressures below 10 3 Pa no nucleate boiling is observed, and structures in the form of "funnel" and "craters" are formed in the liquid layers."Funnels" are depressions in a thin layer of liquid with a hemispherical bottom."Craters", in contrast with "funnels", have an extended flat liquid microlayer of finite dimensions in the center of the depression [18].The reason of the formation of structures in the form of "funnel" and "craters" are thermal plumes [20].In these modes, heat transfer is carried out because of the natural convection and evaporation.At pressures below 10 3 Pa, when there was no nucleate boiling, with an increase in the layer height from 4 mm to 20 mm, a significant increase in the temperature head (by about 50 K) was observed, and after 20 mm the temperature head remained practically unchanged (Fig. 3(a)).At pressures above 10 3 Pa, when nucleate boiling is observed in the layer, as the height of the liquid layer increases, the temperature head first decreases (by about 10 K) and has a minimum of its value at a layer height of 10 mm (Fig. 3(b)).A further increasing in the height of the liquid layer leads to a slight increasing in the temperature head.
Figure 4 shows the dependence of the heat transfer coefficient on the height of the liquid layer at a constant heat flux q = 2.5•10 4 W/m 2 for pressures less than 10 3 Pa.The maximum value of the heat transfer coefficient was achieved at a liquid layer height of 2.5 mm (h/l σ = 1.45) (Fig. 4).Then, with an increasing in the height of the liquid layer, the heat transfer coefficient decreased.This is caused by a decreasing in heat removal due to the evaporative component (Fig. 5).At the same time, the convective component, on the contrary, increased with an increase in the height of the liquid layer (Fig. 5).For layer heights of 20 mm and 40 mm, the heat transfer coefficient remains almost constant, since the main mechanism of heat transfer is natural convection in the turbulent regime.When the heat transfer coefficient does not depend on the characteristic linear size.For example, at 133 Pa (Prandtl criterion Pr = 15.4) the Rayleigh criterion (Ra) were about 8.8•10 7 for a liquid layer height of 20 mm and 7.5•10 8 for 40 mm.According to the literature data [23], the values 10 5 < Ra < 10 9 for horizontal liquid layers correspond to the turbulent regime.
Figure 5 shows the dependence of the change in the value of the heat flux removed due to natural convection and due to evaporation on the height of the liquid layer.With a total constant heat flux q = 2.5•10 4 W/m 2 and a pressure of 133 Pa.It can be seen that with an increasing in the height of the liquid layer, the heat flux removed due to evaporation decreases.At the same time, the heat flow removed due to natural convection increases (Fig. 5).To estimate the contribution of heat transfer due to natural convection, we used the formula proposed in [23] for horizontal liquid layers with two rigid boundaries: ) where C and n are empirical coefficients (depending on the value of Ra, shown in Table 2); Ra is the Rayleigh criterion.
The heat flux removed by natural convection was estimated from the formula: where q is heat flow, W/m 2 ; h is liquid layer height, m; λ is thermal conductivity, W/(m‧K); ΔT is temperature head, K: ) where T W is heating surface temperature, K; T S is saturated vapor temperature of n-dodecane, K.The Rayleigh criterion was calculated using the following formula: where Gr is Grashof criterion; Pr is Prandtl criterion; g is acceleration of gravity, m/s 2 ; β is the volumetric expansion coefficient of the liquid, 1/K; ν is kinematic viscosity, m 2 /s; a is thermal diffusivity, m 2 /s.The physical properties (λ, а, ν) of the liquid were determined at an average temperature T m calculated by the formula: The data obtained at pressures of 10 4 Pa and 2•10 4 Pa, correspond to the bubble boiling regime.With an increase in the height of the liquid layer, the heat transfer coefficient increases and reaches its maximum value at a layer height of 10 mm (Fig. 6) with a further increase in the layer height, the heat transfer coefficient practically does not change.Boiling at layer heights of 20 mm and 40 mm correspond to the regime of nucleate boiling in a large volume.Similar results were obtained for water in [16,17], where the difference in the intensity of heat transfer at different heights of the liquid layer is explained by the difference in the movement and detachment of vapor bubbles from the heating surface.

Conclusion
At pressures less than 10 3 Pa, when structures in the form of "funnel" and "craters" were formed in the liquid layers, the temperature head increases with the height of the liquid layer.It was found that at a liquid layer height of 2.5 mm (h/l σ = 1.45), the heat transfer coefficient has a maximum value.When heat exchange is carried out mainly due to the evaporative component.With a further increase in the height of the liquid layer, the heat transfer coefficient decreases.For liquid layer heights of more than 20 mm (h/l σ = 11.61),there is practically no change in the heat transfer coefficient.Since heat transfer is mainly carried out due to natural convection in a turbulent regime.
At pressures above 10 3 Pa, when nucleate boiling was observed, the temperature head decreases sharply with an increase in the height of the liquid layer and reaches a minimum at a layer height of 10 mm (h/l σ = 5.81).A further increase in the height of the liquid layer leads to a slight increase in the temperature head.The heat transfer coefficient sharply increases with increasing liquid layer height and reaches a maximum at 10 mm (h/l σ = 5.81), a further increase in the liquid layer height leads to a slight decrease in the heat transfer coefficient.
The research was carried out within the framework of the state assignment IT SB RAS (№ 121031800216-1).

Fig. 3 .
Fig.3.Change in temperature head from the height of the liquid layer at q = 2.5•10 4 W/m 2 : a) at pressures P < 10 3 Pa; b) at pressures P > 10 3 Pa.

Figure 3 (
Figure 3(a, b) shows a graph of the temperature head versus the height of the liquid layer.At pressures below 10 3 Pa, when there was no nucleate boiling, with an increase in the layer height from 4 mm to 20 mm, a significant increase in the temperature head (by about 50 K) was observed, and after 20 mm the temperature head remained practically unchanged (Fig.3(a)).At pressures above 10 3 Pa, when

Fig. 4 .
Fig. 4. Dependence of the heat transfer coefficient on the height of the liquid layer at q = 2.5•10 4 W/m 2 at pressures P < 10 3 Pa.

Fig. 5 .
Fig. 5.The dependence of the heat flux on the height of the liquid layer at a pressure of 133 Pa:1) total heat flux; 2) heat flow removed due to evaporation; 3) heat flow removed due to natural convection.

Fig. 6 .
Fig. 6.Dependence of the heat transfer coefficient on the height of the liquid layer at q = 2.5•10 4 W/m 2 at pressures P > 10 3 Pa.