Intensification of Heat Transfer Processes in the Low Temperature Short Heat Pipes with Laval Nozzle Formed Vapour Channel

The results of flow studies of moist vapour in Laval-liked vapour channels of short linear heat pipes (HPs) are presented. The increase in heat transfer coefficient of short linear HPs, intended for creation the cooling systems of heatstressed designs of spacecraft is carried out by making the HPs vapour channel forms of the Laval-liked nozzle. Comparison of the heat transfer coefficients of short HPs with the standard cylindrical vapour channel and the channel, made in the Laval nozzle form with the equality of all dimensions, flat evaporator shows that the HPs with the Laval-liked nozzle vapour channel exceeds the heat transfer characteristics of the standard HPs with a cylindrical vapour channel under high thermal loads. The study of the flow and condensation in such shaped vapour channels of the short HPs at high thermal loads gives an opportunity to analyze in detail the advantages of using such HPs. Capacitive sensors are additionally installed in cooled top covers of the HPs, and electromagnetic pulses with a frequency of 100 kHz were supplied to them from the external generator. At heating the HPs evaporator, starting from a certain thermal power threshold value, electromagnetic pulses became modulated. It is related with the formations of the boiling process in the capillary-porous evaporator and large amount of vapour over it and its discontinuous distribution. An analytical and numerical evaluation are applied to study the duration of the occurring pulsations, and the analytical results are compared with numerical and experimentally obtained values of the pulsations periods.


Introduction
The increase in heat transfer coefficient of short linear HPs is a very important task when designing a spacecraft with a rigid regulation of take-off weight. The application of the linear HPs is justified in the case of the structural impossibility of placing a loop (contoured) HPs and to improve long-term reliability of the cooling systems, which are distributed inlet and outlet closed lines of the loop (contour) HPs.
Pulsating (or oscillating) loop HPs -siphons, was invented in 1990 by Akachi [1]. When the temperature difference between evaporator and condenser exceeds a certain threshold, Tong et al [2], R. Naik, et al [3], the vapour bubbles and condensate liquid plugs begin to oscillate spontaneously back and forth. The amplitude of oscillations is quite strong and the liquid plugs penetrate into both condenser and evaporator. The heat is thus transferred not only by the latent heat transfer liquid-vapour-liquid like in other types of heat pipes, but also by conductive heat transfer of the hot oscillating liquid plugs for cold walls in the condensation zone of loop HPs. This phenomenon is the reason of high heat transfer coefficient of loop HPs in comparison with other types of HPs [4][5][6][7][8][9][10][11].
The increase of the heat transfer processes in short linear open HPs with capillary porous insert is achieved using inner vapour channel made as a Laval-liked nozzle. There is an availability to use a jet vapour nozzle, similar to the Laval nozzle, surrounded by a capillary-porous insert along the full length of the HPs, in case if the heat flow is axially directed, which is typical for short HPs. The shape of this vapour Laval-liked channel, is defined by the inner capillary porous insert. The implementation of the nozzle increases flow velocity [12,20], pulse rate of two-phase vapour flow and heat-transfer coefficient in the HPs. Pulsation of the flow in short linear HPs with the capillary porous insert represents more complex phenomena, associated with boiling of the working fluid in the capillary porous evaporator.
Heat-transfer coefficient and operational effectiveness of short linear HPs with capillary-porous inserts and Laval-liked formed vapour channel are defined by closed circulation motion of the working fluid, which undergoes liquid-vapour transition and heat absorption in the evaporation region of the capillary porous insert, vapour phase transfer through a convergent-divergent vapour channel, vapour-liquid transition with heat liberation in the condensation region, and liquid return to the evaporation region through capillaryporous inserts of the HP. Improvement of the heat-transfer processes in HPs is a complex task, concerning the device's detailed mechanical design and gas dynamics. Flowing vapour, with microdrops of condensate, appears as a nonlinear flow, with attendant internal processes, such as interphasic heat and mass exchange, and power dissipation, Gupta et all [13]. Static regain of supersaturated vapour flow in the diffuser condensation region of the HPs with turbulator, causes flow stagnation, strong vorticity production near the condensation surface and reverse vapour flow [14][15][16][17]. Thermal loadrelated processes of vorticity formation and density and pressure fluctuation, proceeding in the convergent-divergent vapour channel of HPs, are of the utmost interest. For a field research of these processes, stainless steel HPs have been manufactured with diethyl ether as the working fluid [18][19].

Materials and Methods
The diethyl ether C 4 H 10 O is selected as the main working fluid, which has the boiling temperature under the atmospheric pressure of T B =308.65K (35.5°C), freezing temperature T F = 156.95K (−116.2°C) and critical parameters T C = 466.55K (193.4°C), P C =3.61MPa. Successful use of diethyl ether as the working fluid for Wilson chambers, on long liquid phase at 413.15K (140°C), shows its heat resistance, and allows its use as the HP working fluid. The volume of the working fluid in the made of a metal mesh capillary-porous insert and evaporator is not less than 18 cm 3 . The developed sensor enables us to perform measurements of local characteristics of the film flows primarily, the film thickness and temperature, without making any major disturbances in the flow. To measure the fluid condensate film thickness, short HPs were used whose vapour channel was made in the form of a Laval-liked nozzle [20]. The HPs are described in details below.
The capillary-porous evaporator 7 is constructed of 43 circular layers of thin stainless steel net. Each layer is 0.07 mm thick with the mesh size of 0.04 mm and the aggregate thickness of 3 mm; all layers are spot-welded to the flat bottom lid 5.  The capillary-porous evaporator 7 is constructed of 43 circular layers of thin stainless steel net. Each layer is 0.07 mm thick with the mesh size of 0.04 mm and the aggregate thickness of 3 mm; all layers are spot-welded to the flat bottom lid 5. Outer diameters of the circular layers gradually decrease in such a manner that the lateral side of the assembled evaporator forms a pointed cone with a 45° total point angle that aligns with the angle of the end part of the capillary-porous insert 4. The capillary-porous insert 4 is assembled on a dedicated profiled frame made of layers of thin stainless steel net same as the evaporator. Each layer is 0.07 mm thick with mesh size of 0.04 mm, the total thickness of the insert is 1.5 mm at the edges and 7.5 mm at the critical diameter of the inner nozzle. All layers of the insert are radially stitched together along the forming insert with a thin wire 0.05 mm in diameter and stitch spacing of 5-7 mm. Eight lines of the wire stitches that radially fix the net layers together are tilted by 45° towards each other and form a rigid structure of the capillary-porous insert with an inner vapour channel made in the form of a nozzle, similar to the Laval nozzle. At the edge of insert 4 that is adjacent to the evaporator 7, an end portion is placed; its total angle is Pipes with Laval Nozzle Formed Vapour Channel somewhat wider than that of the diffuser portion of the vapour channel and equals 45°. The longitudinal length of the end portion is 3 mm and matches the thickness of the evaporator 7 exactly. The 45° angle of the end portion of insert 4 is formed by the gradually decreasing net layers' lengths, from the outer layer to the inner one, and matches the angle of the evaporator 7 exactly. The multilayered evaporator with a tapering lateral side is tightly inserted into the rigid capillary-porous insert 4 at the full length of the end portion and is spot-welded to it in 8 spots along the perimeter of the insert. After being cooled down at the boiling temperature of fluid nitrogen 77K (-196°C), the assembled capillary-porous insert, together with the evaporator on the flat bottom lid 5, is tightly inserted in the stainless steel cylindrical shell 2 of the HP.
Porosity of the insert and evaporator is 72% and together they form one hydraulic system designed to deliver the service fluid to the evaporator when the HP is operating. The HP length is 100 mm, its diameter is 20 mm, the max. diameter of the vapour nozzle in the convergent and divergent regions is 16 mm, the critical nozzle diameter is 4 mm, the length of the nozzle convergent region is 13 mm, the total angle of the convergent region is 41°, the length of the nozzle divergent region is 81 mm, the total angle of the divergent region is 8.5°, and the length of the cylindrical region in the nozzle throat section is 1 mm.

Capacitance Sensor
The compact open capacitance sensor constitutes the following device whose appearance is shown in Figure 2. The sensor's body is made in the form of a flat circular flange 5.5 mm in diameter and 1 mm thick, with two cylindrical through bores 1 mm in diameter each that are arranged symmetrically with respect to the longitudinal axis of the flange, 2 mm from each other. Each through bore has vacuum-tight glass-to-metal seals 1 mm in diameter. Two gauging electrodes are tightly fixed in the vacuum-tight glass-to-metal seals, each of them 0.5 mm in diameter and 10 mm in length. The space between the gauging electrodes' axes is a=2 mm. A ground electrode 0.5 mm in diameter and 10 mm in length is additionally welded to the external surface of the flange. All details of the calculations, design, fabrication and calibration of capacitive sensors and a high-frequency capacitive measurement devices are given in [14][15][16][17][18][19]. The vacuum-tight glass-to-metal seals of capacitance condensation sensors are characterized by high mechanical stability that provides means to sustain the excessive pressure up to 30 bar, and by high air-tightness or low gas permeability (gas leakage) that does not exceed 10 -9 mbar·l·s -1 (10 -5 Pa·m 3 ·s -1 ). The lack of aging materials (plastic masses, adhesives, organic materials, etc.) in the structure of our open capacitance sensors results in the high stability of their features supported by the value permanence of insulation resistance and self-capacitance of the first sensor specimens produced since 1990. The micro-thermistor is 0.2 mm in diameter and is electrically insulated from the HP lid. The HP condensing surface of the top lid 1 and radially oriented end faces of the gauging electrodes of the capacitance sensor are also shown here.

Temperature Measurement by Using a Thermistor
It is well -known used in english-language literature polynomial equation of the third degree by J. S. Steinhart, S. R. Hart [21][22], for a temperature T SH calculation by the thermistor and containing linear and cubic in the logarithm of the resistance lnR C parts: where A, B, C -coefficients of decomposition. The range of application of this equation for temperature measurements with one set of coefficients does not exceed 50 K, and the methodical error of the temperature determination is not less than 5·10 -2 K -10 -1 K [21][22]. To improve the accuracy and extend the range of temperature measurement with thermistors we developed polynomial equation of the fourth degree [23][24][25][26] with a reference temperature to the inflection point of the functional dependence lnR C from the T of the thermistors CT3-19. We obtained a quartic polynomial equation [23][24][25][26] that connected the temperature K and the logarithm of the thermistor resistance as follows: The maximum random error in the measuring of the thermistor resistance is the 5·10 -4 Ohm, the error of determination of temperature (difference of temperature) of the thermistor CT3-19 using the equation (2) not more than (3÷5)·10 -4 K. The range of temperature measurement with thermistors expanded up to 200 K. Condensation zones of the HPs are provided with insulated thermocouples and set to a depth of the 1.5 diameter into the vortex continuous-flow calorimeter, shown in Figure 4, with stabilized water flow. To ensure accurate measuring of thermal power and heat removal augmentation in the HPs, jet flow of input water is swirled, values of flow velocity and Pipes with Laval Nozzle Formed Vapour Channel vorticity due to air bubbles are recorded. The HPs evaporators, also equipped with thermocouples, is heated using a resistance heater, and the temperature is maintained at δТ, K higher than the diethyl ether boiling temperature T B of 308.65K (35.5°C) under atmospheric pressure. The heater temperature is stabilized and HPs evaporator s overheat value is set in the range of δТ = Tev-T B = 0÷20К, herewith thermal power of single HP does not exceed 120W.

Measurement of the Heat Transfer Coefficient
The main HP, called measuring, is filled with diethyl ether and the reference one, which is completely identical to the main HP, is filled with dehumidified air with dew point temperature lower than 233.15К (-40°C). The heat transfer coefficient of the second HP does not exceed 0.15% from the first one (measuring one) and is not taken into account. The second HP, completely identical to the first one, and performing the function of the reference one is designed for measuring the condensate film thickness in the first, measuring one, filled by diethyl ether [14][15][16][17].
To reduce heat loss when working with HPs, they are placed in a stainless steel vacuum chamber 22, where they are further surrounded by thin-walled copper adiabatic screen 23, the inner surface of which is covered with a layer of nickel, and on the outer surface placed 3 section guard heaters 24. The value of nonadiabaticity near the middle of the HP does not exceed ~2·10 -2 K, in the field of the resistive heater H 2 and HPs evaporators nonadiabaticity is about ~10 -1 K.
Measurement were performed to disclose variability of the heat transfer coefficients in Laval-liked HPs and with standard cylindrical vapor channel. The heat transfer coefficients in HPs, Faghri [27] is defined by the formula: where К HP -heat transfer coefficient of the HPs, W/K; Ethermal power, supplied in the evaporator of the HPs, W; F (z) -surface area of the evaporator inside the vapour channel of the HPs, m 2 ; T ev -temperature of the evaporator, К; T condtemperature of the condensation surface, К. Temperature of evaporator surface and condensation surface of HPs were measured using differential thermocouples copper-constantan, heat power E, W; transferred by condensations zones of HPs to the calorimeter, was defined by the formula: where G -water flow in a vortex flow calorimeter, kg/s; C H2O -the specific heat of water, J/kg·K; C K -the heat capacity of the calorimeter, J/K; ρ H2O -the density of water in the calorimeter, kg /m 3 ; V 0 -the internal volume of the calorimeter, m 3 ; T K0 (τ), T K (τ) -the water temperature at the inlet and outlet of the calorimeter, K. Input temperature of water at the calorimeter, temperature differences (heat value) of the flowing water at entrance and exit of the calorimeter were also measured using differential thermocouples copper-constantan, comparator Р3003 and digital voltmeter V7-34A. Water flow rate was measured using ultrasonic flowmeter. Total measuring inaccuracy of the thermal power E of the HPs does not exceed 1.7%.
Comparison of heat transfer coefficients in Laval-liked HPs and HPs with standard cylindrical vapour channel, with the equality of all dimensions showed following results, presented in Figure 5.
The increase of the heat transfer coefficient in the HP with a Laval -liked vapour channel appears to be related to the emergence of a vortex ring near the condensation surface of the HP.

Numerical Model Details. The Vortex Ring Formation
Numerical simulations of the vortex flows inside a vapour channel of the Laval-liked HPs have been performed in finite element modeling in CFD 10.0 code Fluent 6.3.26 under 2D, double precision axis-symmetric conditions. Navier-Stokes equations with measured boundary conditions were solved, i.e. using fixed temperature values of heat source and heat outlet. In the construction of the design model about 457233 finite elements were used, with increased meshing at injection capillary channels sections, nozzle throat section and turbulence element. In Figure 6 and Figure 7 shows a diagrams of the vortex obtained by numerical simulation of CFD 10.0 code.  The numerical analysis of flow in the Laval-liked HPs condensation zone shows that the vortex structure has a spatial nature, at the same time the flow asymmetry becomes apparent being determined by non-linear friction against the underlying surface and the two-dimensional compressibility of the wet condensing vapour as well. Overheating of the evaporator in reference to the boiling temperature of diethyl ether at atmospheric pressure is 15K, the temperature difference between the evaporator and the condensation surface is equal to 25K.

The Distribution of the Axial Component of Velocity
In the Figure 8 Figure 8, confirms the fact that the vortex ring is a zone of sharp velocity gradients, and as a consequence is also a pressure gradient. In the central part the vortex ring has a noticeable positive dynamic pressure, and it has a negative dynamic pressure in the peripheral part of the vortex ring. This means that lower static pressure occurs in the central part of the vortex ring and this leads to additional absorption of moist vapor in the condensation zone of Laval-liked HPs. Apparently, this means that the static pressure in the vortex center in the Laval -liked HP is less than the pressure in the axial part of the condensation zone of the HP with standard cylindrical vapor channel. The appearance of high static pressure in the peripheral part of the vortex ring, where there is a negative dynamic pressure leads to intensification of the condensation process near the top cap of the Laval-liked HPs, which leads, apparently, to an increased value of heat transfer coefficient in the Laval-liked HPs. This explains the reason why the heat transfer coefficient of Laval -like HPs becomes greater than HPs with the cylindrical vapor channel at boiling point in evaporator HP and there is formation of large amounts of vapor and increase in the efficiency of the nozzle in comparison with a cylindrical channel.

Measurement of Pulse Characteristics of HPs
The pulsation characteristics of HPs were measured in the following way [14][15]. An overheating was imposed on the evaporator by δΤ = Т -Т B relative to the boiling temperature of diethyl ether at atmospheric pressure with the aid of resistive heater 6 and high-precision temperature controller increased in discrete steps of 1K. Electrical pulses of 10-100 kHz and amplitude 5V are applied to the electrodes of the capacitive sensor 10, Figure 4. The modulation rate is measured via a gain and filtering circuit 13, whereupon the signal goes to measuring input of digital oscilloscope 14. The oscilloscope is connected to the computer 15 via USBinterface. Started from the specific overheating δT S = (Т -Т B ) of HPs evaporator, electrical pulses become modulated. The modulation rate is measured using gain and filtering circuit 13, digital oscilloscope 14 and computer 15, and are shown in Figure 4. Measuring inaccuracy of modulation rate does not exceed 3-5 Hz. Both HPs have equal outer diameters 20 mm and equal sectional areas of capillary-porous inserts near the condensation region. As the overheating of the evaporator δT~ 20К increases, pulsation frequency in the Laval-liked nozzle goes up to ~ 502 Hz, derivative of the relationship between pulsation frequency and temperature is approximately 10.5 Hz/K. While investigating HPs with standard cylindrical vapour channel and equal outer diameter 20 mm, length 100 mm and thickness of the evaporator and capillary-porous insert 3 mm, initial pulsed flows occur in event of overheating of the evaporator δT~11К, frequency (modulation frequency of electromagnetic pulsation) f ~406 Hz.  As the overheating of the evaporator δT~20К increases, pulsation frequency in the cylindrical vapour channel goes up to 474 Hz, derivative of the relationship between pulsation frequency and temperature is approximately 7.5 Hz/K.
The insensitivity zone of the capacity sensors in the cylindrical channel, defined by the initial convective nature of the vapour flow, is greater than in the vapour channel of Laval-liked HPs. Dynamic range of pulsations in the Lavalliked HPs vapour channel, is a little greater in frequency, in comparison to dynamic range of pulsations 406 Hz -474 Hz in cylindrical vapour channel, and equals to 386 Hz -502 Hz. Measuring inaccuracy does not exceeds 3-5 Hz.

Numerical Model Details
Numerical simulations of the vortex pulsation flows inside a vapour channel of the Laval -liked HP have been performed in finite element modeling in CFD 10.0 code Fluent 6.3.26 under 2D, double precision axi-symmetric conditions. Navier-Stokes equations with measured boundary conditions were solved, i.e. using fixed temperature values of heat source and heat outlet. The model was studied as a longitudinal section along the axes of the two injector channels, which helps to preserve all the specific features of whirling instability under the conditions of continuous circulation motion of the working fluid during liquid and vapour phases. In the construction of the design model about 457233 finite elements were used, with increased meshing at injection capillary channels sections, nozzle throat section and turbulence element. The model size is a compromise between available computer resources and computational investigation error.
Is clearly visible transition from stationary convective flow regime in the vapour channel to pulsatile flow regime. At high heat capacity, received by the HPs, and boiling in the grid evaporator, excessive vapour occurs in the convergent part of the nozzle, thus leads to pressure increase up to the value, at which the average temperature of layers of the flat grid evaporator becomes lower than the boiling temperature working fluid, and boiling in the evaporator stops. Excessive vapour spreads through the divergent part of the vapour channel to the cooled area of the HP and is partially condensed. Due to the condensation effect, the pressure in the vapour channel decreases and the boiling process in the flat evaporator proceeds. Pressure increase period, vapour wave spreading in the condensation area of the HP and return expansion wave define the pulsation period in the vapour channel.

Validation of Numerical Scheme
Evident conversion from convection mode to convectionvortex mode and then to pulsation mode of the vapour flow inside of the HP, is recorded at evaporator overheating value δT~9К.
Maximum value of velocity of diethyl ether moist vapour flow, obtained using calculation method in the throat section of the vapour channel using by the colour indication of the software CFD Design 10.0, reaches 100 -110 m/s when pulsation occurs. This fact gives the opportunity to evaluate the Reynolds number Re of the vapour channel during pulsation, which is defined by the formula:

Analytical Evaluation
Evaporation cycle in HPs exist at low heat load of the evaporator, up to 4-6 W/cm 2 , and is characterized by convective flow in the vapour channel. Evaporation cycle of short HPs, when evaporator heating power is constant and maximum value is limited to prevent the development of bubble boiling in the flat grid evaporator, is defined in the following way: where ∆Q -heat energy, absorbed in the evaporator over a period of ∆G, J; ∆τ -time, sec; E B -thermal power, wherein the process of bubble boiling begins in the grid evaporator W. Rate of evaporation of dry monomolecular vapour over the evaporator is defined by the following equation: where M R -the amount of the dry vapour, generated over the evaporator per unit time, kg/s; nR ST -growth velocity of the number of vapour molecules over the evaporator per unit time, s -1 ; m vp -mass of molecule of diethyl ether, kg; r (T B )specific vaporization heat of the diethyl ether in HPs, depend upon temperature and pressure, J/kg; T B (P) -boiling temperature of the diethyl ether, К. Mass flow of saturated dry monomolecular vapour over the evaporator is defined by the following equation: where G VP -mass flow of dry saturated vapour over the evaporator, kg/s; n vp (T ev ) -average number of dry vapour molecules in the unit volume over the evaporator, m -3 ; u vpaverage velocity of the vapour phase, m/s; ρ VP (T ev ) -vapour density of diethyl ether over the evaporator, kg/m 3 ; T evtemperature of the evaporator, К. Growth rate of the number of molecules of vapour over the evaporator in the convergent region of the nozzle of the vapour channel, which defines excess pressure over the evaporator and mass flow of vapour in HPs, is calculated from the equation: where N A -Avogadro constant, mol -1 ; µ vp -molar mass of diethyl ether vapour, kg/mol. Assuming approximately equal velocities and without considering the jet type nature of the flow pattern, the linear velocity of the hydrodynamic flotation of the vapour flow over the evaporator surface is calculated from the equation: Insert expressions (9) and (8) in the equation (7) to obtain the formula for calculation of molecular flow of the dry vapour over the evaporator: Hydrodynamic flow of saturated dry vapour is defined by pressure difference value between the evaporator and condensation region in HP, according to the formula: where Anondimensional constant about a unit; ηcoefficient of dynamic viscosity of dry vapour, Pa·s; Lthe length of the vapour channel of the HPs, m; P (T ev ) -vapour pressure under the HPs evaporator, Pa; P (T cond ) -vapour pressure near the condensation surface of the HPs, Pa; ∆P vpvapour pressure difference over the evaporator and near the condensation surface in the vapour channel of the HPs, Pa; T cond -temperature of the condensation surface, К. Equate mass and hydrodynamic flow of saturated dry vapour, and obtain the following equation: Excess pressure over the evaporator defines transfer of the vapour flow in the vapour channel of the HPs, and is calculated in linear approximation according to the equation: With respect to the Clapeyron-Clausius equation, pressure derivative of the vapour by temperature is calculated in a conventional manner, nevertheless considering the fact that for liquid the specific volume ratio is small, ν L / ν VP < 10 -2 -10 -3 , hence in the Clapeyron-Clausius equation the value of the specific volume of fluid ν L is ignored, and in ideal gas state the following equation is obtained: where ν VP -specific volume of the saturated vapour, m 3 /kg; ν L -specific volume of the diethyl ether on the liquid-Pipes with Laval Nozzle Formed Vapour Channel vapour coexistence line, m 3 /kg. Substitute expression (10) in (8), and obtain the equation to calculate vapour quantity in the HP: Vapour temperature over the evaporator's surface with low evaporation and without boiling is defined by the equation (16): Steady-state evaporation conditions in HP mean that the temperature in the evaporator does not exceed boiling temperature of the working fluid. Heating capacity of the HP, W, is defined according to the following equation: where n vp (T cond) -average number of dry vapour molecules in the unit volume near the condenser, m -3 ; Heat transfer coefficient K HP at cross-section of the vapour channel of the HP is defined according to the following expression: When the rate of heat supply into the thin evaporator is high, its average temperature is greater than the boiling temperature of the working fluid during bubble boiling or vaporization. The hydrodynamic vapour flow at the converging region of the nozzle has no time to carry out heating capacity generated through boiling in the evaporator.
Vapour density and pressure increase and boiling temperature of the working fluid goes up until it exceeds average temperature of the evaporator. Upon pressure increase the boiling in the evaporator stops (slows), and an overpressure wave spreads in the vapour channel to the condensation region of the HPs, where the vapour becomes supersaturated, and condenses. The condensation process is not instant, and when the evaporator's activity is slowed (stopped), condensation lasts until the pressure decreases to the vapour saturation pressure at the condensation temperature, thereafter the condensation process stops.
The slow process of saturated vapour pressure decrease, due to condensation, feeds back through the vapour channel of the HP back to the evaporator, and the boiling process there is resumed.

Discussion
Pressure pulsations in the vapour channel of the short HPs result from boiling and intensive vapour generation in the evaporator, non-instantaneous mass-transfer through the vapour channel to the condensation region of the HP, and the slow condensation process, which provides pressure decrease in the condensation region at first and then in the evaporator of the HPs.
Following all described processes the next pulsed evaporation cycle is resumed. The heating capacity, entering to the flat grid evaporator of the short HP, when the evaporator's temperature exceeds the boiling temperature T B (p) of the working fluid, is defined according to the next equation: It is considered that moist vapour stream is formed of two subsystems: microdrops stream system G dr and dry vapour stream system, G vp . Rate of vaporization of the moist droplet vapour flow is defined in the standard way: For the purpose of simplification of the analytical model construction, the real droplet vapour flow over the evaporator -having microdrops dimensioned by complicated doublehumped distribution function, Lee, Reges, Almenas, [30], is reported in terms of a mono-dispersal system of spherical microdrops with arithmetic middling radius r a , which is frequently used while analyzing two-phase droplet vapour flows: where n dr -total number of microdrops of all sizes per unit volume of the droplet vapour flow over the evaporator, 1/m 3 ; n dri -number of microdrops of the diethyl ether with radius r dri per unit volume of the droplet vapour medium, 1/m 3 . Considering the accepted assumption over spherical shape of microdrops, the expression for absolute moisture γ of the droplet vapour flow is defined in the standard way: where М dr -total mass of microdrops per unit volume of the droplet vapour flow over the evaporator, kg; М vp -vapour mass per unit volume of the dry vapour flow over the evaporator, kg; n dr -total number of microdrops of all sizes per unit volume of the droplet vapour flow over the evaporator, 1/m 3 ; n dr -total number of microdrops of all sizes per unit volume of the droplet vapour flow over the evaporator, 1/m 3 ; ρ dr -density of microdrops of the diethyl ether, kg/m 3 . The density of moisture vapour ρ ST ]wx T [S without taking into account the relative motion phases is: Synergies between consumable and absolute mass concentrations of microdrops γ G or consumable and absolute moisture are defined according to the following relation: The value γ G represents the relation between quantity of the condensed droplet phase and total quantity of the twophase droplet vapour, and, considering the expression (22) is as follows: Mass flow rate of microdrops on the evaporator's surface are considered to be proportional to vaporization velocity and mass flow rate of the vapour: where a and bnumerical coefficients; Bcoefficient with account for thermophysical properties of the diethyl ether and structural parameters of the evaporator (porosity, typical dimension of channels and pores).
To evaluate the vapour quantity transported from the evaporator to the condensation region of the HP, equation (31) is used, where the microdrops subsystem contribution to the generation of overpressure of the two-phase droplet mixture over the evaporator P (T ev ) in the vapour channel is neglected: As a result of the boiling process in the evaporator, the pressure of the vapour over the evaporator increases up to P * , whereby the boiling process in the surface layers and further in the whole of the thin evaporator (3mm thick) is slowed (stopped) due to the fact that average temperature of the evaporator T [S becomes lower than the boiling temperature T B of the working fluid in the evaporator under the increased pressure and the confined spaces: Furthermore, pulsing of the overpressured vapour begins to spread to the condensation region through the vapour channel. Cessation of boiling and retarding of vaporization in the evaporator of the HP lead to significant reduction (cessation) of the heat release and decrease in moist vapour transportation along the vapour channel of the HP to the condensation region of the HP. The time period ∆τ ev of pressure increase up to P * and cessation of boiling in the capillary-porous evaporator is estimated in linear approximation in ideal gas state and laminar heat transfer inside of the vapour channel of the HP, according to the formula: Surplus energy (increased pressure) release time ∆τ HP in the evaporating region, by means of vapour flow transfer to the condensation region of the HP, is estimated according to the following formula: As a result of the liquid phase formation, pressure near the cooling condensation surface reduces to: This leads to the slowing of heat transfer through the vapour channel, the rarefaction wave propagation from the condensation zone to the evaporator and the start of the next cycle of pulsed increase of the pressure near the evaporator's surface. Thus, cycle pulse duration ∆τ 0 in the vapour channel of the HP: ∆τ ≃ ∆τ [S + ∆τ + ∆τ jkb + ∆τ -k=b The analysis of the obtained measured data of heat transfer over the evaporator shows that the designed HPs work in the boiling mode. This fact is confirmed by influences of operating parameters of the vaporization process (heat flow density q and pressure P (T ev )) on the heat transfer coefficient α. The influence of these parameters is related to the similar influence of heat flow density and pressure over the bubble boiling in the substantial volume. Performed numerical estimation of parts of pulsation duration ∆τ 0 from the expression (33) show the following.
Pressure increase duration ∆τ ev over the HPs evaporator leads to increased values ∆τ ev ~ 10 -1 -10 -2 s, which means incomplete adequate flat model for estimation of evaporability in the evaporator. It is necessary to consider nonequilibrium volumetric vaporization with regard to the structure and porosity of the evaporator, temperature and moisture gradient in the boiling two-phase working fluid.
Moving duration ∆τ HP of the vapour cluster along the convergent-divergent HPs vapour channel at vapour flow velocity in the channel equal ~ 100 m/s ÷ 300 m/s, reaches ∆τ HP ~ (4.8÷1.6)· 10 -3 s, which is acceptable only in the case Pipes with Laval Nozzle Formed Vapour Channel when sonic velocity in dry vapour is ~ 300 m/s and correspondingly large temperature gradients in the vapour channel.
Maximum duration of condensation period of moist vapour is also excessive and reaches ∆τ cond ~ 10 -1 -10 -2 s Duration of expansion wave motion from condensation region to the HP evaporator at sonic speed in moist vapour about 100 m/s, doesn't exceed ∆τ sound ~ 1·10 -3 s.
Comparison of experimental measured results of pulsation period duration in the convergent-divergent vapour channel of short low temperature HPs ∆τ exp = (2÷2.5)·10 -3 s with the valued ∆τ 0 ~ (0.75 -1.5)·10 -3 s obtained using numerical method and the software CFD Design 10.0, shows that they have good fit values.
Comparison of experimental measured results of pulsation period duration in the convergent-divergent vapour channel of short low temperature HPs ∆τ 0 = (2÷2.5)·10 -3 with estimated values obtained using analytical method, shows excessive values of analytical results. Suggested simple theoretical model of the flat surface evaporation gives excessive values of pulsation period duration and is not fully correspond to acutely nonequilibrium evaporation processes in the grid capillary porous evaporator with boiling working fluid.

Conclusions
1. Heat transfer coefficient of the short linear HPs with the Laval-liked vapour channel, reaches the value of (22±1)·10 3 W/m 2 K; heat transfer coefficient of the HPs with the cylindrical vapour channel does not exceed (15±1)·10 3 W/m 2 K, if thickness of the capillary-porous insert layer is 3 mm, diameter 20 mm and length 100 mm. Comparison of heat transfer coefficients in short HPs shows sufficient advantages of HP with the Laval-liked vapour channel in contrast with standard cylindrical channel.
2. The pulsatile flow regime in the vapour channel of short linear HPs occurs when the working fluid in the evaporator starts to boil. Vapour density and pressure increase and boiling temperature of the working fluid goes up until it exceeds average temperature of the evaporator. Upon pressure increase the boiling in the evaporator stops (slows), and an overpressure wave spreads in the convergentdivergent vapour channel to the condensation zone of the HPs, where the vapour becomes supersaturated, and condenses. Condensation of moist vapour results in pressure decrease in condensation region, and after arrival of the expansion wave at the HPs evaporator, pressure over the evaporator also decreases, the boiling process in the evaporator proceeds and the pulsation cycle in the vapour channel is repeated.
3. Flow stagnation leads to vortex formation, and their interaction causes pulse-coupled vortex decay, static pressure boost and complex reverse flows. Calculation results show that flow stagnation during the pulsation cycle leads to an enlargement of the recirculating region and augmentation of condensation, and this effect is significant.
With all the operating parameters of HPs, two effects were observed: pulsating flow regime of two-phase vapour flow and film-type condensation.