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Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection
Journal Pre-proofs Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection V.O. Tuz, N.L. Lebed, O.M. Tarasenko PII: DOI: Reference:
S2451-9049(20)30046-9 https://doi.org/10.1016/j.tsep.2020.100527 TSEP 100527
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Thermal Science and Engineering Progress
Received Date: Revised Date: Accepted Date:
19 November 2019 11 March 2020 13 March 2020
Please cite this article as: V.O. Tuz, N.L. Lebed, O.M. Tarasenko, Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection, Thermal Science and Engineering Progress (2020), doi: https://doi.org/10.1016/j.tsep.2020.100527
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Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection V. O. Tuz1, N. L. Lebed1*, O.M. Tarasenko2 1National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute" Ukraine, Kyiv 2National Technical University of Ukraine "Kharkiv Polytechnic Institute" Ukraine, Kharkiv valeriituz56@ gmail.com, [email protected], alextar552@ gmail.com
SUMMARY The paper presents experimental results on heat and mass transfer during evaporative cooling of a film of liquid in vertical slot channels under natural convection. In order to intensify the processes and stabilize the flow of the liquid film at Fr = 0.008…0.058, the channel surface was covered with a capillary-porous coating. The research was performed on round, circular and open channels in the range of numbers Ra = 700…2.2·108 and in a wide range of geometric characteristics. The analysis of the experimental results allowed determining the length of the initial region and the stabilized heat and mass transfer region, which are characterized by different rates of the process intensity change. It was found that the regime parameters of contacting phases have a significant effect on the length of the initial thermal region. Using the experimental data, the authors obtained dependences that allow determining the local and average heat and mass transfer coefficients in the initial thermal region and in the stabilized heat transfer region. The use of the obtained dependences may help improve the existing engineering methods for calculating the evaporative cooling process in the technological procedures implemented in the elements of power and chemical equipment. Key words: evaporative cooling, natural convection, heat and mass transfer Corresponding author at: Heat-and-Power Engineering Department, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37 Peremohy Av., 03056 Kyiv, Ukraine. E-mail address: [email protected] (N. L. Lebed). *
1
intensity, heat transfer coefficient, mass transfer coefficient. ABSTRACT Implementation of resource-saving technologies can be based on improving the qualitative characteristics of heat and mass transfer equipment, which is widely used in power engineering, chemical, petrochemical, gas and food industries. One of the most effective ways of interaction between the coolant and the working medium is the direct contact of a gravitationally flowing liquid film and a gas. A promising method for passive intensification of heat and mass transfer between the liquid film and gas is the use of contact surfaces with regular roughness, with capillary-porous or mesh coating of channel walls. The launch and emergency stop of the heat and mass transfer device may generate a transition in the cooling mode of liquid from forced to natural convection. This leads to a change in the conditions of heat and mass transfer, and, consequently, to a change in the parameters of the coolant at the output of the device. In this case, using the dependences obtained for forced convection in the channel leads to a significant error in determining the parameters of the heat carrier. Therefore, in order to design and manufacture heat-mass-transfer units of contact type, one would need the information on the patterns of heat and mass transfer under different operating conditions. An important condition for ensuring the efficiency of contact-type devices is to know the features of the interaction between the liquid and gas mediums. The development of boundary layers in the film and gas leads to the need to allocate the initial thermal region and the stabilized heat and mass transfer region. These factors are not taken into account in the engineering calculation methods that exist today. NOMENCLATURE 2
Symbols а
temperature conductivity coefficient, m2/s;
с
specific heat capacity, J/(kgK);
d
diameter, m;
D
diffusion coefficient, m2/s;
G
irrigation density, kg/(ms);
g
free fall acceleration, m/s2;
L, l
length of the channel, region, m;
р
pressure, Pa;
q
heat flux density, W/m2;
Qm
bulk irrigation density, m2/s;
r
specific evaporation heat, J/kg;
R
channel radius, m;
S
mesh cell size, mm;
t,
temperature, excess temperature, °С;
w, u
velocity of gas, liquid, m/s;
x, y, z
coordinates, m;
heat transfer coefficient, W/(m2K);
p
mass transfer coefficient, kg/(м2sPa);
b
bulk expansion coefficient, 1/K;
film thickness, m;
dynamic viscosity coefficient, Pas;
viscosity kinematic coefficient of, m2/s;
density, kg/m3. Similarity numbers
g b d 3 Gr = t 2g
Grashof number
3
Qm
Fr
К
g
0,5
r t с р
film Froude number Kutateladze number, phase transition criterion
d L ; ;
Nusselt number
d L ; ; D D D
Nusselt diffusion number
Prandtl number
Prandtl diffusion number
Rа Gr Pr
Raleigh number
Re g
w d g
Reynolds number for gas
Refil
Г f
film Reynolds number
Nu
Nu D Pr
а
PrD
D
Subscripts g
gas;
е
equivalent;
і, х
local value;
c fil,
couple; film;
tr
initial thermal region;
f
liquid;
w
wall;
min
minimum;
max
maximum;
av
average value;
4
input (initial) value;
output (final) value.
1. Introduction The necessity of cooling the heat transfer agent under natural convection may arise in technological processes applied when operating power or chemical equipment, e.g., when cooling elements of equipment of thermal and nuclear power plants, in fuel production, adhering to the operating practices in the production of thermolabile substances, etc. The heat and mass transfer are less studied under natural than under forced convection. The experimental data [1 ‒ 5] are limited and often contradictory, which is caused by the difficulties in measuring the parameters of the flows of coolant and working medium. Investigation of the physical phenomenon of natural convection is mainly based on the use of turbulent viscosity models developed for forced convection processes and on the application of the Reynolds analogy between the transfer of momentum and heat [6, 7]. A large number of studies in the classical formulation is devoted to the investigation of heat transfer processes for plates, cylinders and spheres freely oriented inside large spaces, as well as inside closed volume structures. The authors of [8‒10] obtained local heat transfer coefficients on the vertical wall under natural convection, as well as velocity and temperature profiles in the boundary layer at tw = const. The analysis of the results showed that with the increase of the distance from the wall in the sections located at different distances from the front edge, the local temperature of the air decreases exponentially, and the longitudinal velocity first increases from zero to maximum, and then decreases. As the value of the coordinate x increases, the velocity also significantly increases, and its maximum value shifts along the normal away from the surface of the plate (Fig. 1).
5
Fig. 1. Flow scheme for liquid film and gas in the channel The nature of the flow in the boundary layer varies from laminar to turbulent mode. A few analytical papers [11 ‒ 15, 4, 7] use simplified single-valuedness condition, and the obtained results, though difficult to use, allow estimating the influence of the thermophysical properties of a liquid on the heat transfer intensity. Using the generalized law of temperature and velocity distribution in the boundary layer, as well as the "stability criterion" of the laminar layer, proposed by S.S. Kutateladze [16, 17], dependences were obtained for determining the friction coefficient and the Nusselt number. The obtained results are valid for the isothermal heating surface. Regarding the operating conditions of the channels of the contact device, it is necessary to also take into account the influence of the flow of the vapor mass formed during the evaporation process, the nonlinear nature of the change in the heat flux density along the height of the channel qx, the film flow regime, the channel geometry, etc. The insignificant number of papers [18 ‒ 22] devoted to the study of the heat and mass transfer patterns during the cooling of a gravitating fluid film in flat slot channels under natural convection do not fully reflect the process of evaporative cooling of a liquid film in the channels of the contact device. 6
2. Features of the heat and mass transfer during evaporative cooling of the liquid film under natural convection At present, the studies of the heat and mass transfer in film devices are carried out under the boundary conditions of the 1st and 2nd kind. When studying the evaporative cooling of gravitational liquid films, it is necessary to take into account the nonisothermal nature of the films. The change in the thermophysical parameters of the film and gas during evaporative cooling significantly affects the formation of the temperature and velocity profiles in both the liquid and the gas. Under the forced movement of a gas with constant thermophysical properties, the velocity profile does not significantly depend on the temperature, while under natural convection, the velocity profile is affected by the temperature and density distribution in the channel. This is due to the fact that the lifting force, which is the driving force of natural convection, depends on the difference in the temperatures of the fluid film and the flow. Significant changes in velocity and temperature are observed only in the boundary layer. The evaporation causes the liquid to cool, which results in the transfer of heat from the adiabatic wall to the phase separation surface inside the film. The size of the boundary layer formed in the film increases in the direction of the Ox axis (Fig. 2). As a result, the process of evaporative cooling of the film is characterized by the presence of two consecutively located regions. In the first region, for 0 х хtr (хtr is the final coordinate of the initial thermal region), the temperature on the adiabatic wall is constant, the thickness of the thermal boundary layer is less than the thickness of the film. In the second region, for x > хtr, the thickness of the thermal boundary layer and the liquid film are equal. In this region, the temperature of the liquid in the peripheral layer decreases and asymptotically approaches the saturation temperature at the corresponding values of temperature and partial vapor pressure outside the boundary layer. 7
a) b) c) a – circular channel; b – round cross section channel; c – channel inside a large volume Fig. 2. Flow schemes for liquid film and gas in the channels The problems of heat and mass transfer under natural convection can be solved using the momentum, matter and energy conservation equations. The mathematical model of the heat and mass transfer processes during the evaporative cooling of the film under natural convection in the form of a system of differential equations is complicated not only by the necessity of using transport equations for a two-phase medium, but also by the formulation of boundary conditions at the phase separation boundary. In [23], the equality of local heat fluxes at the phase separation boundary is used as the boundary condition. Such problem statement complicated its solution by adding another transport equation, but allowed describing the process of twophase transfer in adjoint conditions at the phase separation boundary. The problem was solved under the condition that the contact phases were moving at average speeds under the assumptions given in [24]. The averaged coefficients of the turbulent transfer were used. Accepted assumptions are quite correct for the smooth-walled channels of the contact device. 8
Taking into account the complex character of the hydrodynamics of the film flow on vertical surfaces with capillary-porous coating and the nonlinear character of the boundary conditions of the 3rd kind, the formulation of the adjoint problem of evaporative cooling under natural convection becomes considerably more complicated. The above indicates the need for an experimental study of the evaporative cooling of a liquid film in channels with a capillary-porous coating. 3. Experimental study of the heat and mass transfer during evaporative cooling of the film under natural convection
The methods of passive intensification of heat and mass transfer in contacttype devices include the use of contact surfaces with artificial roughness, special knurling, mesh or capillary-porous coating. Works [25 ‒ 30] present the research results on the influence of geometric characteristics of the capillary-porous coating in the form of metal mesh and of the regime parameters of the two-phase flow on the intensity of the heat and mass transfer in contact devices. The analysis of the research results allowed determining the optimal geometry of the coating, namely the size of the mesh cell S = 0,5 mm, at which the heat and mass transfer intensity has the maximum value. The presence of a transverse vapour flow, the frictional force that occurs between the gas flow, the film and the channel walls, the significant temperature gradients in the film and in the gas stream make studying the evaporative cooling of the film much more complicated. The determination of the change in the mean value of the heat transfer coefficient along the length of the channel, which in most cases is the subject of an experimental study, does not make it possible to determine the size of the initial thermal region. Gravity-driven flow of liquid films was studied in a vertical tube with a capillary-porous coating of metal mesh. The experimental setup (Fig. 2) was a hydrodynamic circuit consisting of a working area (1), a collector (2), a retaining 9
tank (3), a pump (4), a network collector (5), a condensate tank (6), a measuring system which in turn consisted of two series-connected rotameters PC-3 (7) and (8) with different flow rate ranges and overlapping measuring scales, thermocouples with secondary devices Щ-300 (10) and A-565 (9), connecting tubes and control valves (11). Such setup made it possible to experimentally study the heat and mass transfer processes and hydrodynamics of liquid films and gas flow in the experimental regions under forced and free convection (the schemes are given in Fig. 1).
1 – working area; 2 – collector; 3 – retaining tank; 4 – pump; 5 – network collector; 6 – condensate tank; 7, 8 – rotameters; 9, 10 – secondary devices Щ-300 (millivoltmeter); 11 – control valves; 12 – contact device; 13 – liquid heater; 14 – gas heater; 15, 16 – compressors; 17, 18 – voltage regulators РНО-250/25; 19, 20, 21 – rotameters; 22 – valves Fig. 3. Schematic drawing of experimental setup 10
To date, different methods for measuring local hydrodynamic parameters of gravity-driven liquid films have been developed and widely tested. Given the features of the experimental region, we chose the contact method. The thickness of the film was determined by the position of the needle tip when it touched the surface of the film. The moment of contact was registered either when an electrical signal was generated on the secondary device Щ-300 as the circuit between the needle and the conductive film closed, or visually — through an optical system with a 25-times magnification. A coordinate mechanism with a scale of 0.01 mm was used to ensure measurement accuracy. The temperature distribution in the cross-sections of the working area was determined using chromel-alumel thermocouples by the readings of the secondary devices. The total error in determining the local and average heat transfer coefficients did not exceed 12%. Therefore, local characteristics of the evaporative cooling process were studied, taking into account the characteristic zones of the heat and mass transfer intensity. Measuring the temperature of the film and the gas flow in different sections of the channel allows estimating the temperature fields and, thus, the widening of the boundary layer along the height of the channel. The influence of the geometric characteristics of the channel (height and diameter) on the process of heat and mass transfer was studied in 1 m long channels with equivalent diameters de = 0,0125...0,088 m, at a temperature of the liquid changing within the limits of tf = 40 ... 80 °С, and the irrigation density of G = 0,332 10-2...8,704 10-2 kg/(ms). Fig. 4 shows the dependence of the excess temperature of the liquid film along the height of the channel f = tfx tg, where tfx is the value of the average temperature of the liquid film in the section. Fig. 5 shows the dependence of the excess temperature of the liquid film along the height of the channel on the 11
irrigation density G. Analysis of the results showed that the temperature of the film changes along the height of the channel nonlinearly. Moreover, the film cooling intensity grows with the increase of the initial excess temperature of the film and the decrease of the irrigation density. It should be noted that the irrigation density range, in which the research was conducted, was chosen so that the liquid film did not go beyond the capillary-porous coating and its thickness remained almost unchanged [26].
1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms); 6 – 0,0235 kg/(ms) open pipe Fig. 4. Excess film temperature along the length of the channel under natural convection depending on irrigation density. The temperature of the liquid and gas, respectively, at the entrance to the channel: tf = 80 0С, tg = 20...220С Apart from determining the average excess temperature of the film gx, the temperature field of the gas stream was also measured, both along the height and the across the channel section. Measurement results on the excess air temperature gx = tgx tg (tgx is local value of air temperature in the section of the channel) are presented in Fig. 6. Analysis of the results shows that the change in the air temperature mostly occurs in the boundary layer.
12
1 – tf = 80 °С ; 2 – 70 °С; 3 – 60 °С; 4 – 50 °С; 5 – 40 °С Fig. 5. Excess film temperature along the length of the channel under natural convection depending on the initial temperature of the film. The temperature of the gas at the entrance to the channel tg = 20...22°С, irrigation density G = 0,0235 kg/(ms)
1 – y = 0 (film temperature along the height of the channel); 2 – y = 2 10-3 m (gas temperature along the height of the channel in the corresponding section); 3 – 510-3 m; 4 – 1010-3 m; 5 – 1510-3 m, 6 – 2010-3 m; 7 – 2510-3 m; 8 – 3010-3 m, 9 –
13
4010-3 m; 10 – 5010-3 m; 11 – 7010-3 m, 12 – 10010-3 m Fig. 6. Temperature field of a gas stream during evaporative cooling of a liquid film under natural convection in an open channel. The temperature of the liquid and gas, respectively, at the entrance to the channel: tf = 80 °С, tg = 20...22 °С, irrigation density G = 0,01105 kg/(ms) The obtained temperature of the liquid film and air could then be used to calculate the average heat flux density qlx along the height lx at the heat exchange surface areas, and the local heat flux density qx. The research results on the effect of irrigation density G and the initial excess temperature on the distribution of the local (х and рх) and the average ( and p) heat and mass transfer coefficients along the height of the channel are presented in Fig. 7, 8.
а)
14
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms) Fig. 7. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the irrigation density. The temperature of the liquid and gas at the entrance to the channel: tf = 80 °С, tg = 20...22°С
15
а)
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – tf = 80 °С ; 2 – 70 °С; 3 – 60 °С; 4 – 50 °С; 5 – 40 °С Fig. 8. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the temperature of the liquid at the input. Gas temperature at the entrance to the channel tg = 20…22°С, irrigation density G = 0,0235 kg/(ms) 16
The analysis of the results showed that the increase in the initial excess temperature f from 20°С to 60°С leads to an approximately 2.4 times increase in the values of х and , and the change in the irrigation density in the range of G = 0.33210-2... 2.34910-2 kg/(ms) leads to a 2.7 times increase in the intensity of the process. It should be noted that the nature of the change in the heat transfer intensity along the height of the channel confirms the presence of two areas of the initial region and one area of stabilized heat transfer, which differ in values of temperature gradients. Apart from the regime parameters, the size of the initial thermal region is influenced by the geometric characteristics of the channel, in particular, the equivalent diameter de. As noted in [31], the formation of the boundary layer and the movement of the gas medium is determined not only by the conditions of heat exchange on the surface, but also by the geometry of the slot channels. Making the distance between the walls of flat channels smaller reduces the heat exchange intensity [20, 29]. In order to evaluate the effect of the geometric characteristics of the channel on the intensity of the heat and mass transfer during the evaporative cooling of the liquid film, experimental studies were carried out in the circular channels in the range of equivalent diameters de = 0.0125...0.088 m, as well as in an open channel with a diameter d = 0,022 m with a capillary-porous coating on the outer surface (Fig. 2). Analysis of the research results (Fig. 9) in this range of film parameters f and G showed that the nature of the heat and mass transfer intensity in channels with different equivalent diameters does not change. The decrease in the length of the initial thermal region, which is associated with a decrease in the equivalent diameter of the channel, can be explained by the increasing influence of the shear stress on the opposite wall due to shift of the gas velocity maximum away from the irrigated wall, which is associated with the widening of the boundary layer in the slot channels. An increase in the equivalent diameter of the channel in the entire f and G test range leads to a 1,35…1,6 times 17
increase in the local and medium heat and mass transfer intensity (х and, рх andp).
а)
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – dе = 0,0125 m; 2 – 0,02 m; 3 – 0,088 m Fig. 9. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the diameter of the channel. The 18
temperature of the liquid and gas at the entrance to the channel tf = 80°С, tg = 20°С and the irrigation density G = 0,0235 kg/(ms) Taking into account the change in the heat and mass transfer intensity in the contact device channels, the length of the initial thermal region should be determined depending on the regime parameters of the contact phases, provided that the change in the heat transfer coefficient in the initial region does not exceed its average value in this area by more than 1% [32]: x x 1 100 1% , x x 1 2
(1)
where х and х+1 are local heat transfer coefficients at the initial and final cross sections of a particular basic segment of the channel length. Taking into account the presented research results and the dependence (1), the value of the initial thermal region should be determined from the expression 0,2 t r 66, 45 d e Ra 0,1 K x0,42 . x Fr
(2)
4. Generalization of experimental data on the heat and mass transfer during the evaporative cooling of the film under natural convection in the channels with capillary-porous coating When processing the experimental results, the thermophysical properties of the coolants were determined either by the average temperature of the film and air in a particular channel segment — in order to find the average of heat transfer and mass transferp coefficients, or by the average temperature at a particular channel section — in order to determine the local heat transfer x and mass transfer рх coefficients. The temperature pressure for determining the average heat transfer coefficient was determined as a log average between the input and output temperatures of the film in the experimental segment of the channel and the 19
average air temperatures in the corresponding sections, while for the local heat transfer coefficient — as the temperature difference between the film temperature t fx in the test section with the coordinate x and the average air temperature in the
same section. The generalization of experimental data on the evaporative cooling of a gravitational film under natural convection was made for the initial thermal region, the size of which is determined by the formula (2), and the stabilized heat transfer region of the size L ltr. As a result, the following empirical dependencies were obtained:
local heat transfer coefficient 0,3
Nu x 30,31 Ra
0,14 x
Pr
0,33 x
0,2 x
Fr
K
0,42 x
x , de
average heat transfer coefficient 0,3
Nu 36,78 Ra 0,14 Pr 0,33 Fr 0,2 K 0,42 , de ____
(3)
(4)
local mass transfer coefficient 0,3
Nu Dx 23,36 Ra
____
0,14 x
Pr
0,33 Dx
0,2 x
Fr
K
0,42 x
x , de
(5)
average mass transfer coefficient 0,3
Nu D 27,97 Ra
0,14
Pr
0,33 D
Fr
0,2
K
0,42
. de
(6)
The empirical dependences were obtained in the ranges of Fr = 0.008…0.058 and Ra = 700…2.2·108. The approximation was performed by the least square method, the generalization accuracy was ±12%. The research results on the influence of the regime characteristics of the liquid and gas and the geometric characteristics of the channel on the evaporative cooling intensity of the gravitational film under natural convection in the initial thermal region are presented in Fig. 10. The analysis of the research results showed that outside the initial thermal 20
region the values of the local and average heat transfer coefficients ( х and) and mass transfer coefficients (рх іp) continue to decrease asymptotically, approaching a constant value, but the change rate is much smaller than in the initial thermal region. Therefore, in order to calculate the heat and mass transfer intensity in the stabilized heat exchange region, it is recommended to use the dependences (4) and (6), the defining dimension should be the length of the initial thermal region t r , which is determined by the dependence (2).
а)
b)
21
c)
d) а – local heat transfer coefficients; c – average heat transfer coefficients; c – local mass transfer coefficients; d – average mass transfer coefficients; 1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms) (tf = 80°C, tg = 20...22°C); 6 – tf = 70°C; 7 – 60°С; 8 – 50°С; 9 – 40°С (tg = 20...22°С, G = 0,0235 kg/(ms)); 10 – dе = 0,0125 m; 11 – 0.088 m (tf = 80°C, tg = 20°C, G = 0.0235 kg/(ms)) Fig. 10. Heat and mass transfer intensity during evaporative cooling of the film under natural convection at the initial thermal region 5. Conclusions The analysis and processing of the obtained experimental data on the process of evaporative cooling of the film, which gravitates down the surface with a capillary-porous coating under natural convection, allowed making the following conclusions. 22
1.
Under natural convection, during the evaporative cooling of the liquid
film, the velocity field is directly connected to the distribution of temperature and irrigation density in the channel cross-section. Significant changes in velocity and temperature occur only in the boundary layer. 2.
The intensity of evaporative cooling of a gravitating liquid under
natural convection varies with the height of the channel, which allows allocating the initial heat region and the stabilized heat exchange region. The length of the initial thermal region depends on the parameters of the film and the equivalent diameter of the channel, and is determined by the empirical dependence (2). 3.
The heat and mass transfer intensity significantly depends on the
regime parameters of the contact phases. An increase in the initial excess temperature from 20°С to 60°С leads to an approximately 2.4 times increase in heat and mass transfer coefficients, while a change in the irrigation density in the range of 0,332 10-2 ... 2,349 10-2 kg/(ms) contributes to a 2.7 times increase in the intensity of the process. 4.
An increase in the equivalent diameter of the channel, when the excess
liquid temperature and irrigation density change in the test range, leads to a 1.35...1.6 times increase in the heat and mass transfer intensity. 5.
The generalization of experimental data allowed obtaining empirical
dependences (3) and (4) for determining the local and average heat transfer coefficients in the initial heat region, and (5) and (6) — for the local and average mass transfer coefficients. It is recommended to use the dependences (4) and (6) to determine the heat and mass transfer intensity in the stabilized heat exchange region, with the length of the initial thermal region, which is determined according to the dependence (2), used as the determining size. References [1]
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28
CRediT author statement V. O. Tuz: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Project administration N. L. Lebed: Methodology, Validation, Formal analysis, Investigation, Resources, Writing - Original Draft, Writing - Review & Editing, Visualization. O.M. Tarasenko: Validation, Formal analysis, Resources, Writing - Original Draft, Writing - Review & Editing.
29
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
30
S2451-9049(20)30046-9 https://doi.org/10.1016/j.tsep.2020.100527 TSEP 100527
To appear in:
Thermal Science and Engineering Progress
Received Date: Revised Date: Accepted Date:
19 November 2019 11 March 2020 13 March 2020
Please cite this article as: V.O. Tuz, N.L. Lebed, O.M. Tarasenko, Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection, Thermal Science and Engineering Progress (2020), doi: https://doi.org/10.1016/j.tsep.2020.100527
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Evaporative cooling of the liquid film in slot channels with capillary-porous walls under natural convection V. O. Tuz1, N. L. Lebed1*, O.M. Tarasenko2 1National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute" Ukraine, Kyiv 2National Technical University of Ukraine "Kharkiv Polytechnic Institute" Ukraine, Kharkiv valeriituz56@ gmail.com, [email protected], alextar552@ gmail.com
SUMMARY The paper presents experimental results on heat and mass transfer during evaporative cooling of a film of liquid in vertical slot channels under natural convection. In order to intensify the processes and stabilize the flow of the liquid film at Fr = 0.008…0.058, the channel surface was covered with a capillary-porous coating. The research was performed on round, circular and open channels in the range of numbers Ra = 700…2.2·108 and in a wide range of geometric characteristics. The analysis of the experimental results allowed determining the length of the initial region and the stabilized heat and mass transfer region, which are characterized by different rates of the process intensity change. It was found that the regime parameters of contacting phases have a significant effect on the length of the initial thermal region. Using the experimental data, the authors obtained dependences that allow determining the local and average heat and mass transfer coefficients in the initial thermal region and in the stabilized heat transfer region. The use of the obtained dependences may help improve the existing engineering methods for calculating the evaporative cooling process in the technological procedures implemented in the elements of power and chemical equipment. Key words: evaporative cooling, natural convection, heat and mass transfer Corresponding author at: Heat-and-Power Engineering Department, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37 Peremohy Av., 03056 Kyiv, Ukraine. E-mail address: [email protected] (N. L. Lebed). *
1
intensity, heat transfer coefficient, mass transfer coefficient. ABSTRACT Implementation of resource-saving technologies can be based on improving the qualitative characteristics of heat and mass transfer equipment, which is widely used in power engineering, chemical, petrochemical, gas and food industries. One of the most effective ways of interaction between the coolant and the working medium is the direct contact of a gravitationally flowing liquid film and a gas. A promising method for passive intensification of heat and mass transfer between the liquid film and gas is the use of contact surfaces with regular roughness, with capillary-porous or mesh coating of channel walls. The launch and emergency stop of the heat and mass transfer device may generate a transition in the cooling mode of liquid from forced to natural convection. This leads to a change in the conditions of heat and mass transfer, and, consequently, to a change in the parameters of the coolant at the output of the device. In this case, using the dependences obtained for forced convection in the channel leads to a significant error in determining the parameters of the heat carrier. Therefore, in order to design and manufacture heat-mass-transfer units of contact type, one would need the information on the patterns of heat and mass transfer under different operating conditions. An important condition for ensuring the efficiency of contact-type devices is to know the features of the interaction between the liquid and gas mediums. The development of boundary layers in the film and gas leads to the need to allocate the initial thermal region and the stabilized heat and mass transfer region. These factors are not taken into account in the engineering calculation methods that exist today. NOMENCLATURE 2
Symbols а
temperature conductivity coefficient, m2/s;
с
specific heat capacity, J/(kgK);
d
diameter, m;
D
diffusion coefficient, m2/s;
G
irrigation density, kg/(ms);
g
free fall acceleration, m/s2;
L, l
length of the channel, region, m;
р
pressure, Pa;
q
heat flux density, W/m2;
Qm
bulk irrigation density, m2/s;
r
specific evaporation heat, J/kg;
R
channel radius, m;
S
mesh cell size, mm;
t,
temperature, excess temperature, °С;
w, u
velocity of gas, liquid, m/s;
x, y, z
coordinates, m;
heat transfer coefficient, W/(m2K);
p
mass transfer coefficient, kg/(м2sPa);
b
bulk expansion coefficient, 1/K;
film thickness, m;
dynamic viscosity coefficient, Pas;
viscosity kinematic coefficient of, m2/s;
density, kg/m3. Similarity numbers
g b d 3 Gr = t 2g
Grashof number
3
Qm
Fr
К
g
0,5
r t с р
film Froude number Kutateladze number, phase transition criterion
d L ; ;
Nusselt number
d L ; ; D D D
Nusselt diffusion number
Prandtl number
Prandtl diffusion number
Rа Gr Pr
Raleigh number
Re g
w d g
Reynolds number for gas
Refil
Г f
film Reynolds number
Nu
Nu D Pr
а
PrD
D
Subscripts g
gas;
е
equivalent;
і, х
local value;
c fil,
couple; film;
tr
initial thermal region;
f
liquid;
w
wall;
min
minimum;
max
maximum;
av
average value;
4
input (initial) value;
output (final) value.
1. Introduction The necessity of cooling the heat transfer agent under natural convection may arise in technological processes applied when operating power or chemical equipment, e.g., when cooling elements of equipment of thermal and nuclear power plants, in fuel production, adhering to the operating practices in the production of thermolabile substances, etc. The heat and mass transfer are less studied under natural than under forced convection. The experimental data [1 ‒ 5] are limited and often contradictory, which is caused by the difficulties in measuring the parameters of the flows of coolant and working medium. Investigation of the physical phenomenon of natural convection is mainly based on the use of turbulent viscosity models developed for forced convection processes and on the application of the Reynolds analogy between the transfer of momentum and heat [6, 7]. A large number of studies in the classical formulation is devoted to the investigation of heat transfer processes for plates, cylinders and spheres freely oriented inside large spaces, as well as inside closed volume structures. The authors of [8‒10] obtained local heat transfer coefficients on the vertical wall under natural convection, as well as velocity and temperature profiles in the boundary layer at tw = const. The analysis of the results showed that with the increase of the distance from the wall in the sections located at different distances from the front edge, the local temperature of the air decreases exponentially, and the longitudinal velocity first increases from zero to maximum, and then decreases. As the value of the coordinate x increases, the velocity also significantly increases, and its maximum value shifts along the normal away from the surface of the plate (Fig. 1).
5
Fig. 1. Flow scheme for liquid film and gas in the channel The nature of the flow in the boundary layer varies from laminar to turbulent mode. A few analytical papers [11 ‒ 15, 4, 7] use simplified single-valuedness condition, and the obtained results, though difficult to use, allow estimating the influence of the thermophysical properties of a liquid on the heat transfer intensity. Using the generalized law of temperature and velocity distribution in the boundary layer, as well as the "stability criterion" of the laminar layer, proposed by S.S. Kutateladze [16, 17], dependences were obtained for determining the friction coefficient and the Nusselt number. The obtained results are valid for the isothermal heating surface. Regarding the operating conditions of the channels of the contact device, it is necessary to also take into account the influence of the flow of the vapor mass formed during the evaporation process, the nonlinear nature of the change in the heat flux density along the height of the channel qx, the film flow regime, the channel geometry, etc. The insignificant number of papers [18 ‒ 22] devoted to the study of the heat and mass transfer patterns during the cooling of a gravitating fluid film in flat slot channels under natural convection do not fully reflect the process of evaporative cooling of a liquid film in the channels of the contact device. 6
2. Features of the heat and mass transfer during evaporative cooling of the liquid film under natural convection At present, the studies of the heat and mass transfer in film devices are carried out under the boundary conditions of the 1st and 2nd kind. When studying the evaporative cooling of gravitational liquid films, it is necessary to take into account the nonisothermal nature of the films. The change in the thermophysical parameters of the film and gas during evaporative cooling significantly affects the formation of the temperature and velocity profiles in both the liquid and the gas. Under the forced movement of a gas with constant thermophysical properties, the velocity profile does not significantly depend on the temperature, while under natural convection, the velocity profile is affected by the temperature and density distribution in the channel. This is due to the fact that the lifting force, which is the driving force of natural convection, depends on the difference in the temperatures of the fluid film and the flow. Significant changes in velocity and temperature are observed only in the boundary layer. The evaporation causes the liquid to cool, which results in the transfer of heat from the adiabatic wall to the phase separation surface inside the film. The size of the boundary layer formed in the film increases in the direction of the Ox axis (Fig. 2). As a result, the process of evaporative cooling of the film is characterized by the presence of two consecutively located regions. In the first region, for 0 х хtr (хtr is the final coordinate of the initial thermal region), the temperature on the adiabatic wall is constant, the thickness of the thermal boundary layer is less than the thickness of the film. In the second region, for x > хtr, the thickness of the thermal boundary layer and the liquid film are equal. In this region, the temperature of the liquid in the peripheral layer decreases and asymptotically approaches the saturation temperature at the corresponding values of temperature and partial vapor pressure outside the boundary layer. 7
a) b) c) a – circular channel; b – round cross section channel; c – channel inside a large volume Fig. 2. Flow schemes for liquid film and gas in the channels The problems of heat and mass transfer under natural convection can be solved using the momentum, matter and energy conservation equations. The mathematical model of the heat and mass transfer processes during the evaporative cooling of the film under natural convection in the form of a system of differential equations is complicated not only by the necessity of using transport equations for a two-phase medium, but also by the formulation of boundary conditions at the phase separation boundary. In [23], the equality of local heat fluxes at the phase separation boundary is used as the boundary condition. Such problem statement complicated its solution by adding another transport equation, but allowed describing the process of twophase transfer in adjoint conditions at the phase separation boundary. The problem was solved under the condition that the contact phases were moving at average speeds under the assumptions given in [24]. The averaged coefficients of the turbulent transfer were used. Accepted assumptions are quite correct for the smooth-walled channels of the contact device. 8
Taking into account the complex character of the hydrodynamics of the film flow on vertical surfaces with capillary-porous coating and the nonlinear character of the boundary conditions of the 3rd kind, the formulation of the adjoint problem of evaporative cooling under natural convection becomes considerably more complicated. The above indicates the need for an experimental study of the evaporative cooling of a liquid film in channels with a capillary-porous coating. 3. Experimental study of the heat and mass transfer during evaporative cooling of the film under natural convection
The methods of passive intensification of heat and mass transfer in contacttype devices include the use of contact surfaces with artificial roughness, special knurling, mesh or capillary-porous coating. Works [25 ‒ 30] present the research results on the influence of geometric characteristics of the capillary-porous coating in the form of metal mesh and of the regime parameters of the two-phase flow on the intensity of the heat and mass transfer in contact devices. The analysis of the research results allowed determining the optimal geometry of the coating, namely the size of the mesh cell S = 0,5 mm, at which the heat and mass transfer intensity has the maximum value. The presence of a transverse vapour flow, the frictional force that occurs between the gas flow, the film and the channel walls, the significant temperature gradients in the film and in the gas stream make studying the evaporative cooling of the film much more complicated. The determination of the change in the mean value of the heat transfer coefficient along the length of the channel, which in most cases is the subject of an experimental study, does not make it possible to determine the size of the initial thermal region. Gravity-driven flow of liquid films was studied in a vertical tube with a capillary-porous coating of metal mesh. The experimental setup (Fig. 2) was a hydrodynamic circuit consisting of a working area (1), a collector (2), a retaining 9
tank (3), a pump (4), a network collector (5), a condensate tank (6), a measuring system which in turn consisted of two series-connected rotameters PC-3 (7) and (8) with different flow rate ranges and overlapping measuring scales, thermocouples with secondary devices Щ-300 (10) and A-565 (9), connecting tubes and control valves (11). Such setup made it possible to experimentally study the heat and mass transfer processes and hydrodynamics of liquid films and gas flow in the experimental regions under forced and free convection (the schemes are given in Fig. 1).
1 – working area; 2 – collector; 3 – retaining tank; 4 – pump; 5 – network collector; 6 – condensate tank; 7, 8 – rotameters; 9, 10 – secondary devices Щ-300 (millivoltmeter); 11 – control valves; 12 – contact device; 13 – liquid heater; 14 – gas heater; 15, 16 – compressors; 17, 18 – voltage regulators РНО-250/25; 19, 20, 21 – rotameters; 22 – valves Fig. 3. Schematic drawing of experimental setup 10
To date, different methods for measuring local hydrodynamic parameters of gravity-driven liquid films have been developed and widely tested. Given the features of the experimental region, we chose the contact method. The thickness of the film was determined by the position of the needle tip when it touched the surface of the film. The moment of contact was registered either when an electrical signal was generated on the secondary device Щ-300 as the circuit between the needle and the conductive film closed, or visually — through an optical system with a 25-times magnification. A coordinate mechanism with a scale of 0.01 mm was used to ensure measurement accuracy. The temperature distribution in the cross-sections of the working area was determined using chromel-alumel thermocouples by the readings of the secondary devices. The total error in determining the local and average heat transfer coefficients did not exceed 12%. Therefore, local characteristics of the evaporative cooling process were studied, taking into account the characteristic zones of the heat and mass transfer intensity. Measuring the temperature of the film and the gas flow in different sections of the channel allows estimating the temperature fields and, thus, the widening of the boundary layer along the height of the channel. The influence of the geometric characteristics of the channel (height and diameter) on the process of heat and mass transfer was studied in 1 m long channels with equivalent diameters de = 0,0125...0,088 m, at a temperature of the liquid changing within the limits of tf = 40 ... 80 °С, and the irrigation density of G = 0,332 10-2...8,704 10-2 kg/(ms). Fig. 4 shows the dependence of the excess temperature of the liquid film along the height of the channel f = tfx tg, where tfx is the value of the average temperature of the liquid film in the section. Fig. 5 shows the dependence of the excess temperature of the liquid film along the height of the channel on the 11
irrigation density G. Analysis of the results showed that the temperature of the film changes along the height of the channel nonlinearly. Moreover, the film cooling intensity grows with the increase of the initial excess temperature of the film and the decrease of the irrigation density. It should be noted that the irrigation density range, in which the research was conducted, was chosen so that the liquid film did not go beyond the capillary-porous coating and its thickness remained almost unchanged [26].
1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms); 6 – 0,0235 kg/(ms) open pipe Fig. 4. Excess film temperature along the length of the channel under natural convection depending on irrigation density. The temperature of the liquid and gas, respectively, at the entrance to the channel: tf = 80 0С, tg = 20...220С Apart from determining the average excess temperature of the film gx, the temperature field of the gas stream was also measured, both along the height and the across the channel section. Measurement results on the excess air temperature gx = tgx tg (tgx is local value of air temperature in the section of the channel) are presented in Fig. 6. Analysis of the results shows that the change in the air temperature mostly occurs in the boundary layer.
12
1 – tf = 80 °С ; 2 – 70 °С; 3 – 60 °С; 4 – 50 °С; 5 – 40 °С Fig. 5. Excess film temperature along the length of the channel under natural convection depending on the initial temperature of the film. The temperature of the gas at the entrance to the channel tg = 20...22°С, irrigation density G = 0,0235 kg/(ms)
1 – y = 0 (film temperature along the height of the channel); 2 – y = 2 10-3 m (gas temperature along the height of the channel in the corresponding section); 3 – 510-3 m; 4 – 1010-3 m; 5 – 1510-3 m, 6 – 2010-3 m; 7 – 2510-3 m; 8 – 3010-3 m, 9 –
13
4010-3 m; 10 – 5010-3 m; 11 – 7010-3 m, 12 – 10010-3 m Fig. 6. Temperature field of a gas stream during evaporative cooling of a liquid film under natural convection in an open channel. The temperature of the liquid and gas, respectively, at the entrance to the channel: tf = 80 °С, tg = 20...22 °С, irrigation density G = 0,01105 kg/(ms) The obtained temperature of the liquid film and air could then be used to calculate the average heat flux density qlx along the height lx at the heat exchange surface areas, and the local heat flux density qx. The research results on the effect of irrigation density G and the initial excess temperature on the distribution of the local (х and рх) and the average ( and p) heat and mass transfer coefficients along the height of the channel are presented in Fig. 7, 8.
а)
14
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms) Fig. 7. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the irrigation density. The temperature of the liquid and gas at the entrance to the channel: tf = 80 °С, tg = 20...22°С
15
а)
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – tf = 80 °С ; 2 – 70 °С; 3 – 60 °С; 4 – 50 °С; 5 – 40 °С Fig. 8. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the temperature of the liquid at the input. Gas temperature at the entrance to the channel tg = 20…22°С, irrigation density G = 0,0235 kg/(ms) 16
The analysis of the results showed that the increase in the initial excess temperature f from 20°С to 60°С leads to an approximately 2.4 times increase in the values of х and , and the change in the irrigation density in the range of G = 0.33210-2... 2.34910-2 kg/(ms) leads to a 2.7 times increase in the intensity of the process. It should be noted that the nature of the change in the heat transfer intensity along the height of the channel confirms the presence of two areas of the initial region and one area of stabilized heat transfer, which differ in values of temperature gradients. Apart from the regime parameters, the size of the initial thermal region is influenced by the geometric characteristics of the channel, in particular, the equivalent diameter de. As noted in [31], the formation of the boundary layer and the movement of the gas medium is determined not only by the conditions of heat exchange on the surface, but also by the geometry of the slot channels. Making the distance between the walls of flat channels smaller reduces the heat exchange intensity [20, 29]. In order to evaluate the effect of the geometric characteristics of the channel on the intensity of the heat and mass transfer during the evaporative cooling of the liquid film, experimental studies were carried out in the circular channels in the range of equivalent diameters de = 0.0125...0.088 m, as well as in an open channel with a diameter d = 0,022 m with a capillary-porous coating on the outer surface (Fig. 2). Analysis of the research results (Fig. 9) in this range of film parameters f and G showed that the nature of the heat and mass transfer intensity in channels with different equivalent diameters does not change. The decrease in the length of the initial thermal region, which is associated with a decrease in the equivalent diameter of the channel, can be explained by the increasing influence of the shear stress on the opposite wall due to shift of the gas velocity maximum away from the irrigated wall, which is associated with the widening of the boundary layer in the slot channels. An increase in the equivalent diameter of the channel in the entire f and G test range leads to a 1,35…1,6 times 17
increase in the local and medium heat and mass transfer intensity (х and, рх andp).
а)
b) а – local and average heat transfer coefficients; b – local and average mass transfer coefficients; 1 – dе = 0,0125 m; 2 – 0,02 m; 3 – 0,088 m Fig. 9. Heat and mass transfer intensity during the evaporative cooling of the film under natural convection, depending on the diameter of the channel. The 18
temperature of the liquid and gas at the entrance to the channel tf = 80°С, tg = 20°С and the irrigation density G = 0,0235 kg/(ms) Taking into account the change in the heat and mass transfer intensity in the contact device channels, the length of the initial thermal region should be determined depending on the regime parameters of the contact phases, provided that the change in the heat transfer coefficient in the initial region does not exceed its average value in this area by more than 1% [32]: x x 1 100 1% , x x 1 2
(1)
where х and х+1 are local heat transfer coefficients at the initial and final cross sections of a particular basic segment of the channel length. Taking into account the presented research results and the dependence (1), the value of the initial thermal region should be determined from the expression 0,2 t r 66, 45 d e Ra 0,1 K x0,42 . x Fr
(2)
4. Generalization of experimental data on the heat and mass transfer during the evaporative cooling of the film under natural convection in the channels with capillary-porous coating When processing the experimental results, the thermophysical properties of the coolants were determined either by the average temperature of the film and air in a particular channel segment — in order to find the average of heat transfer and mass transferp coefficients, or by the average temperature at a particular channel section — in order to determine the local heat transfer x and mass transfer рх coefficients. The temperature pressure for determining the average heat transfer coefficient was determined as a log average between the input and output temperatures of the film in the experimental segment of the channel and the 19
average air temperatures in the corresponding sections, while for the local heat transfer coefficient — as the temperature difference between the film temperature t fx in the test section with the coordinate x and the average air temperature in the
same section. The generalization of experimental data on the evaporative cooling of a gravitational film under natural convection was made for the initial thermal region, the size of which is determined by the formula (2), and the stabilized heat transfer region of the size L ltr. As a result, the following empirical dependencies were obtained:
local heat transfer coefficient 0,3
Nu x 30,31 Ra
0,14 x
Pr
0,33 x
0,2 x
Fr
K
0,42 x
x , de
average heat transfer coefficient 0,3
Nu 36,78 Ra 0,14 Pr 0,33 Fr 0,2 K 0,42 , de ____
(3)
(4)
local mass transfer coefficient 0,3
Nu Dx 23,36 Ra
____
0,14 x
Pr
0,33 Dx
0,2 x
Fr
K
0,42 x
x , de
(5)
average mass transfer coefficient 0,3
Nu D 27,97 Ra
0,14
Pr
0,33 D
Fr
0,2
K
0,42
. de
(6)
The empirical dependences were obtained in the ranges of Fr = 0.008…0.058 and Ra = 700…2.2·108. The approximation was performed by the least square method, the generalization accuracy was ±12%. The research results on the influence of the regime characteristics of the liquid and gas and the geometric characteristics of the channel on the evaporative cooling intensity of the gravitational film under natural convection in the initial thermal region are presented in Fig. 10. The analysis of the research results showed that outside the initial thermal 20
region the values of the local and average heat transfer coefficients ( х and) and mass transfer coefficients (рх іp) continue to decrease asymptotically, approaching a constant value, but the change rate is much smaller than in the initial thermal region. Therefore, in order to calculate the heat and mass transfer intensity in the stabilized heat exchange region, it is recommended to use the dependences (4) and (6), the defining dimension should be the length of the initial thermal region t r , which is determined by the dependence (2).
а)
b)
21
c)
d) а – local heat transfer coefficients; c – average heat transfer coefficients; c – local mass transfer coefficients; d – average mass transfer coefficients; 1 – G = 0,0235 kg/(ms); 2 – 0.01105 kg/(ms); 3 – 0.00829 kg/(ms); 4 – 0.00580 kg/(ms); 5 – 0.00332 kg/(ms) (tf = 80°C, tg = 20...22°C); 6 – tf = 70°C; 7 – 60°С; 8 – 50°С; 9 – 40°С (tg = 20...22°С, G = 0,0235 kg/(ms)); 10 – dе = 0,0125 m; 11 – 0.088 m (tf = 80°C, tg = 20°C, G = 0.0235 kg/(ms)) Fig. 10. Heat and mass transfer intensity during evaporative cooling of the film under natural convection at the initial thermal region 5. Conclusions The analysis and processing of the obtained experimental data on the process of evaporative cooling of the film, which gravitates down the surface with a capillary-porous coating under natural convection, allowed making the following conclusions. 22
1.
Under natural convection, during the evaporative cooling of the liquid
film, the velocity field is directly connected to the distribution of temperature and irrigation density in the channel cross-section. Significant changes in velocity and temperature occur only in the boundary layer. 2.
The intensity of evaporative cooling of a gravitating liquid under
natural convection varies with the height of the channel, which allows allocating the initial heat region and the stabilized heat exchange region. The length of the initial thermal region depends on the parameters of the film and the equivalent diameter of the channel, and is determined by the empirical dependence (2). 3.
The heat and mass transfer intensity significantly depends on the
regime parameters of the contact phases. An increase in the initial excess temperature from 20°С to 60°С leads to an approximately 2.4 times increase in heat and mass transfer coefficients, while a change in the irrigation density in the range of 0,332 10-2 ... 2,349 10-2 kg/(ms) contributes to a 2.7 times increase in the intensity of the process. 4.
An increase in the equivalent diameter of the channel, when the excess
liquid temperature and irrigation density change in the test range, leads to a 1.35...1.6 times increase in the heat and mass transfer intensity. 5.
The generalization of experimental data allowed obtaining empirical
dependences (3) and (4) for determining the local and average heat transfer coefficients in the initial heat region, and (5) and (6) — for the local and average mass transfer coefficients. It is recommended to use the dependences (4) and (6) to determine the heat and mass transfer intensity in the stabilized heat exchange region, with the length of the initial thermal region, which is determined according to the dependence (2), used as the determining size. References [1]
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CRediT author statement V. O. Tuz: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Project administration N. L. Lebed: Methodology, Validation, Formal analysis, Investigation, Resources, Writing - Original Draft, Writing - Review & Editing, Visualization. O.M. Tarasenko: Validation, Formal analysis, Resources, Writing - Original Draft, Writing - Review & Editing.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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