The local heat transfer coefficient determination during boiling a subcooled liquid on a superheated surface by the gradient heatmetry

. The article presents a comparison of the experimental local heat transfer coefficient and the heat transfer coefficient obtained by well-known empirical formulas. To calculate the heat transfer coefficient for stable pool film boiling according to the formulas given in the classical literature, it is necessary to know exactly the thermophysical parameters of the liquid and vapor at a given pressure and temperature. This approach satisfactorily describes the heat transfer coefficient during the boiling of homogeneous liquids. When calculating the heat transfer coefficient during the boiling of a suspension of particles in a liquid such an approach is impossible, because the exact values of thermophysical parameters for such a system are unknown. A new approach is proposed for determining the local heat transfer coefficient during boiling using gradient heatmetery. The method based on the use of heterogeneous gradient heat flux sensors allows the direct measurement of the heat flux per unit area. And with a known temperature difference, using thermometry, according to Newton's law of cooling, it is possible to determine the local heat transfer coefficient without reference to the thermophysical parameters of the liquid and vapor.


Introduction
The main task in the study of heat transfer during boiling is the determination of the local or surface-averaged heat transfer coefficient (HTC).There is an extensive database of experimental data that makes it possible to estimate the average heat flux and HTC at steady film and nucleate boiling [1,2].
Most of the formulas presented in the literature [3 -6] include the thermophysical parameters of liquid and vapor, as well as the heat flux (HF), the estimation of which has a high indeterminacy.If it is possible to determine these parameters for homogeneous liquids, then this approach is problematic for liquids with a suspension of particles.
A new method for determining HTC is based on gradient heatmetery.It allows you to directly estimate the local heat flux and then calculate the local HTC at boiling.

Gradient heatmtery
Gradient heatmtery is based on the use of heterogeneous gradient heat flux sensors (HGHFS) that implement the transverse Seebeck effect in media with anisotropy of thermal conductivity, electrical conductivity, and thermopower coefficient.In an anisotropic plate through which the heat flux passes (Fig. 1a), a transverse thermoEMF component appears, which is proportional to the heat flux, where E0 is transverse thermoelectric power, mV; S0 is volt-watt sensitivity of HGHFS, mV/W; A is HGHFS' cross-area, m 2 ; q is heat flux per unit area, W/m 2 [7].
HGHFS were created at the Peter the Great St. Petersburg Polytechnic University and consist of cross-layered metal components.The sensor used in this study (Fig. 1b) have a heat stability about 1500 K [8].In E3S Web of Conferences 459, 05007 (2023) https://doi.org/10.1051/e3sconf/202345905007XXXIX Siberian Thermophysical Seminar the paper [9] using this approach the boundaries of the regimes of film, transition, and nucleate boiling were determined.It is shown how the local heat flux changes in these regimes.
Using the gradient heatmtery it was possible to obtain the heat transfer coefficient and increase the local heat flux, which is especially important for achieving single-phase and two-phase flows.In article [10], during flow around heated finned cylinders, an HTC distribution over the fin' surface was obtained.Using a combination of the PIV (Particle Image Velocimetry), gradient heatmetry, and thermometry the fin effectiveness was obtained experimentally.Additionally, the local HTC during condensation on the outer surface of the tube was examined.[11].The authors found that at an angle of inclination of the tube from the vertical ψ = 20°, the HTC during condensation increases by 14.9%.In the experiment, four GHFS based on a single crystal of bismuth were used, installed along the length of a steel tube.The data of the authors [10] and [11] indicate the applicability of this technique for determining the local HTC in problems of applied heat transfer.

Experimental facility and model
The scheme of the experimental facility is shown in Fig. 2.
The model 1 under study is placed in a through-type furnace 2 and fixed in it with a holder 3. The temperature of the model and the uniformity of heating are monitored by two thermocouples, the signals from which are output to the measuring computer complex (MCC) National Instruments (NI-9261) 5.When the required temperature is reached, the holder 3 releases the model, and it 1 fall into the vessel 4 with a volume of 10 l; the recording of thermocouple and HGHFS signals on MCC 5 begins.The temperature in the vessel is controlled by the Fluke 289 device with a thermocouple, and the required temperature of the liquid is maintained by an electric heater.Data from HGHFS and thermocouples were recorded at a frequency of 5000 measurements/s.
The model used in the experiments (Fig. 3) is a cylinder with a diameter of 34 mm and a height of 22.4 mm made of VT22 titanium alloy.The HGHFS with dimensions of 3 × 3 × 0.3 mm is made on the basis of a copper + nickel composition.
The temperature of the model was controlled by two thermocouples.The junction of the thermocouple 2 was installed near the HGHFS 1, and junction of the thermocouple 3 was installed in the centre of cylinder.A mica layer provided electrical isolation of the HGHFS from the model, and a high-temperature compound provided the fixation of the HGHFS on the cylinder.The working surface of the HGHFS was mounted flush with the surface of the model.Note, that with this HGHFS installation heat monitoring did not reveal significant distortions of the temperature field in the installation area.

Experimental results
In the study the local heat flux, the temperature on the surface and the temperature in the centre of the model were measured synchronously.HTC was determined by dependence: where  is heat flux per unit area, W/m 2 ,   ,   are model surface temperature and liquid temperature, respectively °С.Fig. 4 shows an example of the integrated use of heatmetry and thermometry.
Boiling is divided into 3 classical sections: I is the film regime, II is the transitional regime, III is the bubble regime.For the convenience of displaying temperature  and local heat flux, on the one graph the result is presented in a dimensionless form.Here q is the actual local heat flux at a given time, qmax is the maximum heat flux in this process, t is the actual temperature at a given time, and tmax is the initial temperature of the model before the experiment.
The results for the film boiling region were compared with those calculated from the following data: 1. D. A. Labuntsov's [3] at stable film boiling: where λ, ,  are thermal conductivity of steam, kinematic viscosity of steam, thermal diffusivity of steam; ρ  and ρ  are liquid and vapor density, respectively;  is acceleration of gravity.The index m means that the thermophysical parameters are related to the average vapor temperature in the film calculated as   = 0,5 • (  −   ) ˚ С.Here   is wall temperature, °С and   is liquid saturation temperature, °С [4].2. M.A. Mikheev's [4] for stable film boiling: on a vertical surface: where  * =  + 0,5  ( The physical properties in formulas ( 4) and (5) (except for the liquid density ρ  ) refer to the vapor phase and are taken at an average vapor temperature.
The comparison results are shown in Fig. 5.The initial temperature of the model is   = 464 °С.The values of water subcooling Δt, K: 36, 27, 20 and 0.
In the monograph [4], a deviation of the HTC calculated by the formulas from the experimental values by 35% is allowed.The HTC determined by the method of gradient heatmetry is consistent with the calculation formulas everywhere, except in cases of significant subcooling of the liquid.The discrepancy is due to the instability of the film boiling regime at a low temperature of the liquid.In this area, the presented approach provides an advantage.
Uncertainty calculations were carried out according to ISO/IEC GUIDE 98-1:2009 -Uncertainty of Measurement [12], according to which uncertainty of measurement is the expression of the statistical dispersion of the values attributed to a measured quantity.The combined standard uncertainty of  = ( 1 ,  2 …   ) is calculated by the following formula: where       is the dispersion of  .The values of    were assumed to be known and were determined by the characteristics of the devices.
Factors contributing to uncertainty in heat flux measurements are: • error in the HGHFS volt-watt sensitivity; • error in measuring the HGHFS area; • error in measurement using ADC.These errors should be evaluated as type B uncertainties, as follows: where q is counted according to formula (1) and the standard uncertainty of the HGHFS cross-area, that is   , is equal to 5 × 10 -8 m 2 .Here, the standard uncertainty of the HGHFS volt-watt sensitivity   0 was assumed to be equal to 0.117 mV/W.The detailed of the volt-watt sensitivity of the HGHFS uncertainty calculation is given in [13].In the experiments, a measuring computer complex (MCC) National Instruments (NI-9261) was used to register the signal and  E was equal to 0.004%.
The standard expanded uncertainty of measuring the local heat flux for our experiments did not exceed 6.3%.
As for the calculation of the uncertainty of the local heat transfer coefficient, the factors affecting the uncertainty of the HTC are: • error in measuring the heat flux with HGHFS; • error in measuring the temperature difference.These errors also should be evaluated as type B uncertainties, as follows: where α is counted according to Formula (2) and the standard uncertainty of the heat flux, that is   , counted according to Formula (7).The standard uncertainty of the temperature difference, that   , is equal 2˚ С.
The standard expanded uncertainty of measuring the local HTC for our experiments did not exceed 7.4 %.
The integrated use of heatmetery and thermometry made it possible to fix the local heat flux and the surface temperature at the moment of complete destruction of the vapor film and transition to developed nucleate boiling, which are located on the border of regions II and III (Fig. 4).This allows us to estimate the local HTC (Fig. 6).
Gradient heatmetry makes it possible to measure local HTC and heat flux at any moment of model cooling both in pure water and in liquids with suspensions.On Fig. 6 shows the dependence of the maximum HTC on water subcooling and the presence in it of a suspension of micro-(1 μm) and nanoparticles (54 nm) Al2O3 with a mass concentration of 0.32%.These values of HTC, heat flux and temperature make it possible to evaluate the effect of particles and subcooling on heat transfer.The values presented in Fig. 6 clearly indicate the intensity of the process at the moment of contact of the liquid with the heating surface.
From Fig. 6 and 5, it follows that the HTC during film boiling and the maximum HTC synchronously increase with an increase in liquid subcooling [5].
The addition of Al2O3 microparticles made it possible to increase the HTC and accelerate the cooling of the model [7].With an increase in the subcooling of the liquid, the effect of the addition of microparticles increases (by a maximum of 42%).This is probably caused by the deposition of particles on the surface of the model, which increases the heat transfer surface.
The addition of Al2O3 nanoparticles reduced the HTC (by a maximum of 32%) and slowed down the cooling of the model.When these particles settle on the surface, they form a heat-insulating layer, which reduces the intensity of heat transfer.

Conclusion
The integrated use of gradient heatmetry and thermometry made it possible to evaluate the local HTC at the boiling of subcooled pure water and a suspension of Al2O3 particles.In the traditional settings of the experiment, the HTC obtained by gradient heatmetery are in good agreement with the works of other authors [14].
Determining the thermophysical parameters of a liquid is the weak point in the any technique for determining heat removal during phase transition.When a subcooled liquid boils on an overheated surface, a large temperature gradient arises near it.As a result, there is a change in the parameters of the coolant.It makes difficult to calculate processes.Consideration of a suspension of solid particles in a liquid leads to impossibility of use the average values of the thermophysical properties of a suspension.
The proposed gradient heatmetry eliminates the need to determine the parameters of a liquid (suspension), which makes it a unique method for studying heat transfer during boiling.

2 3 1 Fig. 4
Fig.4 Dimensionless heat flux graph and temperature graph with boiling regimes: I is film boiling, II is transition regime, III is bubble boiling regime.

Fig. 5
Fig.5 Comparison of experimental and calculated HTC at film boiling for various water subcoolings.