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Journal of Engineering Science and Technology Vol. 14, No. 5 (2019) 2913 - 2927 © School of Engineering, Taylor’s University
2913
CALCULATION OF FREEZING RADIUS AROUND VERTICAL TWO-PHASE THERMOSYPHON IN SUBARCTIC CLIMATE
EVGENIY V. MARKOV*, SERGEY A. PULNIKOV, YURI S. SYSOEV
Industrial University of Tyumen, 38, Volodarskogo, Tyumen, 625000, Russian Federation
*Corresponding Author: [email protected]
Abstract
Permafrost is a significant construction problem, as the defrosting leads to
decrease in volume and uneven subsidence of foundations. Thawing of soils is
associated with thermal influence of the structures and global warming.
Therefore, the problem of temperature stabilization of permafrost remains
relevant. Typically, for the temperature stabilization energy-free two-phase
thermosyphons are used. In the southern subarctic climate, thermosyphon
operates under the influence of increased solar radiation. Therefore, the freezing
radius can differ significantly from the designed value. The article is devoted to
the improving of calculation method of freezing radius around two-phase
thermosyphon in areas with subarctic climate. The authors derived formulas for
calculating the average film thickness of the liquid refrigerant on the inner surface
in all parts of the thermosyphon: condenser, overground pipe, underground
thermal insulated pipe, and evaporator. Numerical study has shown that the
thermal resistance of the refrigerant film in thermosyphon with 3 m evaporator
length is 2 orders less than the thermal resistance of the condenser. Thus, in case
of short evaporator the practical calculations soil-freezing radius can be done
without taking into account refrigerant film thickness. The formula for
calculating absorbed direct, diffuse and reflected solar radiation by the condenser
and overground pipe is derived. Required data for a formula coincide with
meteorological quantities. A comparison of two models of system two-phase
thermosyphon - soil - atmosphere showed that the authors’ model reduces the
maximal distance between thermosyphons by 20%. The result is extremely
important in designing of thermosyphons quantity in permafrost stabilization
system at subarctic climate.
Keywords: Heat transfer in soil, Permafrost, Refrigerant film, Solar radiation.
2914 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
1. Introduction
Construction on the permafrost usually requires to save temperature of soils low
then freezing point. In zone of subarctic climate, it is impossible to save permafrost
using standard solution: Ventilated cellar or piles foundation. The global warming
and thermal influence of structure leads to defrosting and uneven subsidence. In
such situation, two-phase thermosyphons are used to lower the temperature of soils
and save the stability of spatial position of various structures: Buildings and
pipelines [1-5].
The main design problem is the calculation of quantity and density on the
general layout. The distance between thermosyphons depends on the freezing
radius, which depends on the refrigerating power. Thus, the accuracy of the
calculation directly affects the stability of the structure.
There is a large number of researches devoted to the study of vertical
thermosyphons, which are described in review articles of Jafari et al. [6] and Jadhav
and Patil [7]. Lee and Mital [8] investigated the influence of pressure, length of
evaporator and condenser, type of refrigerant (water and Freon) to the efficiency of
two-phase thermosyphons. Terdtoon et al. [9] investigated the corrosion of inner
surface. Park et al. [10] studied the influence of fill charge ratio to speed of heat
exchanging. Zhu et al. [11] examined the work of semi-open thermosyphon. Noie
[12] investigated the work of thermosyphons in the heat recovery systems and
found the optimal filling ratio. Carvajal-Mariscal et al. [13] developed the method
of designing two-phase thermosyphons with optimal parameters for the
temperature up to 250 ºC. Khazaee [14] got the limits of heat transfer due to dry
out in the evaporator. Lotfi et al. [15] estimated the heat losses in thermosyphon.
Gandal and Kale [16] found that optimal inclination for clear water and glycol-
water solution differ by 10º. Gorelik and Seleznev [17] evaluated the optimal length
of short thermosyphon for the stabilization of permafrost. de Haan et al. [18, 19]
developed new methods for thermodynamic calculation of thermosyphons based
on finite element method and division of time-step to enthalpy and entropy change.
Özbaş [20] investigated the influence of working fluid to the performance of
thermosyphon. Ali and Hassan [21], Menni et al. [22] and Pathak et al. [23] studied
the influence of channel shape to heat transmission.
Despite extensive researches of thermosyphon, there is a gap in the study of
their efficiency in permafrost. There are still no estimates of the influence of liquid
refrigerant film on the soil-freezing radius taking into account real climatic
conditions. Similarly, there are no physically valid formulas and studies that allow
to estimate the effect of solar radiation (direct, diffuse, and reflected) on the
freezing radius. This is probably due to the fact that the value of solar radiation in
the arctic climate is low. Gorelik and Seleznev [17] work is the closest work with
calculation of the freezing radius. Therefore, everywhere further, the comparison
of the authors’ model and the model in [17] was made.
In the southern subarctic climate, the sun radiation becomes more intense, heats
the condenser part and reduces the freezing radius. Therefore, the improving of
calculation method requires to research the influence of solar irradiance to the
freezing radius.
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2915
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
2. Problem Statement
In this article, the authors solved the following 4 problems:
Developing a mathematical model of a vertical two-phase thermosyphon for
calculating the thermal regime of the soil;
Developing a method for calculating the absorbed solar radiation by the
condenser and overground pipe taking into account the available irradiance
data;
Estimating the error in the calculation of the soil freezing radius around
thermosyphon without taking into account the thermal resistance of the liquid
refrigerant film on the inner surface;
Estimating the difference in the calculation of soil freezing radius around
thermosyphon taking into account direct, diffuse and reflected solar radiation
absorbed by condenser and overground pipe using the authors’ model and the
model given in the paper of Gorelik and Seleznev [17].
3. Model of Two-Phase Vertical Thermosyphon
The simulation of two-phase thermosyphon was according to the design scheme
in Fig. 1.
Fig. 1. Calculation scheme for the system
two-phase thermosyphon-soil-atmosphere.
The mathematical model of two-phase thermosyphon was developed on the
basis of the following assumptions:
The heat exchange in the thermosyphon is quasi-stationary. This assumption is
explained by the fact that heat transfer by thermal conductivity in the soil is much
slower than convective heat transfer in thermosyphon. As a result, calculation of
soil temperature around thermosyphon can be done using the quasi-stationary
s
h
rR
conRcon
z
evp
aT
T
gT
conl
evpR
conq
ogq
ugq
evpq
ogl
ugl
evplhi
lowz
evpz
0z
upz
r
condenser
overgroundpipe
evaporator
undergroundpipe
sunE
2916 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
model and independent from z temperature of the gaseous phase of refrigerant
inside.
The flow of a liquid film along the inner surface is described by Nusselt theory.
The supercooling of refrigerant vapor is insignificant, which makes it possible
to equate the heat flux to the rate of evaporation-condensation taking into
account the heat of phase transition.
In this case, the following fundamental equation of heat balance can be used for
the mathematical model of a thermosyphon:
0up
low
z
zt sunq dz E (1)
Equation (1) differs from the equation given in work of Gorelik and Seleznev
[17] by the term, which is correspond to the absorbed solar radiation. This term will
be discussed below.
In accordance with the design scheme in Fig. 1, the construction of the
thermosyphon consists of four main parts:
Condenser transfers heat to the atmosphere during condensation of
refrigerant vapor.
Overground pipe transfers a small amount of heat to the atmosphere and
supports the condenser above the snow cover.
Underground pipe provides the transfer of liquid refrigerant to the bottom
without heat losses.
Evaporator takes heat from a soil when the refrigerant evaporates.
Using the Newton's law of cooling equation (1) we can find the temperature of
the gaseous phase of refrigerant:
0 evp
loe wvp
zz
z zcon con og og air ug evp sun
gcon con og og ug ug evp evp
K l K l T K Tdz K Tdz E
TK l K l K l K l
(2)
The heat transfer coefficients Kcon, Kop, Kup, and Kevp is:
13
1
1
2 2 2i
con i
con N con R i i i
l RK ln
R s R hn R
(3)
where ξ=1 for the condenser and ξ=0 for another parts; the summation is: 1 - film
of the refrigerant, 2 - wall of thermosyphon, 3 - thermal insulation.
1 1
0 0
2 2
2 2
r R con r R con
r R con r R con
K hR a I hR b
K hR a I hR b
(4)
1 02 2 2r R r R r R r R ra hI hR I hR (5)
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2917
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
1 02 2 2r R r R r R r R rb hK hR K hR (6)
Expressions (3) - (6) are obtained on the basis of an analytical solution of the
heat equation in a thin ring plate and boundary conditions of first type on the inner
radius of the ring and third kinds on the other surfaces.
Convective heat transfer coefficients αr, αR, αN is calculated according to the
theory of boundary layer.
In Eq. (3), the average thickness of the film of the refrigerant in different parts
of the thermosyphon δrca, δroa, δrua, and δrea are unknowns. It will be shown below
that, the film thickness makes an insignificant contribution to the thermal
resistance. But now, we immediately ignore this value. This will not lead to a
significant error in the evaluation of the film thickness, because, on the contrary, it
increases its thickness. Next, we wrote the Nusselt equation for the film thickness
using explicit solution of Navier-Stokes equation for the free movement of fluid by
gravity on a vertical surface:
3
23
tr
rf rr
ff
qgR
z L
(7)
Then we integrated (7) and got the average film thickness for every sections:
1
333 3
4 2 4
con g air rf c
rf con rf
on
rca rccon
K T T l
g R L
(8)
2
rf con rconroa
og g air
f
rf og
g R L
K T T l
4
33 4
3
2rc rc
rf c
og g air rf og
o on rc n f
K T T l
g R L
(9)
rua roa , (10)
3
4rea rua . (11)
Equation (11) is explained by the fact that the section is insulated and the heat
flow is very small.
Thus, we found all the unknowns in equations (5) - (8). It should be noted that
the calculation sequence is as follows: equation (3) - (6), then (2), then (8) - (11).
After this, the iteration cycle is repeated until the values (9) - (12) stop changing.
Problem No 1 solved.
Next, a method for calculating the absorbed solar radiation Esun is described.
The first thing that needs to be established is the set of source data necessary for
calculating. Consider the physical side of the process. Solar radiation during the
2918 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
passage of the atmosphere is partially scattering in clouds and dense layers. Two
components of solar radiation are measuring at the meteorological stations: direct
and diffuse solar radiation. Direct solar radiation is measuring on a horizontal
surface and the surface normal to the ray. Diffuse solar radiation is measuring only
on a horizontal surface. In addition, albedo of the surface is measuring [24]. All
measurements are given under average cloud conditions.
The overground part of the thermosyphon generally in a vertical position in order
to ensure the maximal flow of liquid refrigerant. Therefore, it is necessary to know
the flux of solar radiation on a vertical surface. However, direct measurements of
these quantities are not systematized. Therefore, the author has developed a method
for calculating Esun, which takes into account the absorption of direct, diffuse and
reflected solar radiation by the vertical overground part of the thermosyphon.
The calculation of direct solar radiation flux on the vertical surface of the
condenser is not difficult, because there is a data of direct irradiance and
horizontal irradiance:
2 22s con r con'sun sunSE R l S (12)
On the contrary, the calculation of the flux of diffuse and reflected solar
radiation is much more complicated. Consider the design scheme in Fig. 2: a
condenser with radius Rr is irradiated by diffuse solar radiation from the height Ha.
At this stage, we will not assume the value of Ha.
Fig. 2. Scheme for calculation of absorbed
sun radiation by condenser and overground pipe.
The area of the infinitesimal element of the cylindrical surface of the condenser
dFcon collects on itself radiation from a source in the atmosphere with an area of
dFa. In accordance with Lambert's law, we can write the density of the energy flow
on a condenser:
2
20 0 0 0
conc
lasun
d con r onon
c
GE d rdr R d d
cos cosl
R
(13)
The result of the integration gives the absorbed energy of diffuse solar radiation:
24
d con sun r conE G R l (14)
kn
an
adF
condF
rR
aH
r
Rcon
ad
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2919
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
A feature of expression (15) is the absence of Ha. Therefore, it is not necessary
to make any assumptions about the dispersion distance Ha.
Solar radiation disperses due to reflection on the not smooth snow surface. In this
case, the mathematical description is similar to dispersion in the clouds. The absorbed
energy by condenser butt include only horizontal irradiance. The summation over the
all components gives us expression for the absorbed solar radiation by thermosyphon:
2 2 22 'susun con sun r con r con og on sun n gcoS S R l RE Q R l
2
sun surf sun r con con con og ogG A Q R l R l
(15)
Expression (15) takes into account the absorption of direct, diffuse and reflected
solar radiation by the overground part of the thermosyphon. Problem No 2 solved.
4. Calculation of Freezing Radius
Next, a numerical study of the soil freezing radius around the thermosyphon was
performed using the classical example (Fig. 3). The ensuring the freezing of the
soil at depths from -4 to -7 m for the object in Western Siberia was the design task.
In accordance with the results of standard strength calculations, it is necessary to
lower the temperature to -0.7 ºС within 5 years before construction. The duration
of the freezing is associated with economic calculation. The main purpose of the
heat transfer calculation is to determining the freezing radius rfr - the distance from
the axis of symmetry to the isotherm -0.7ºC after 5 years at a depth of z = -5.5m.
The calculations were performed using the authors’ model and the model, which is
described in work of Gorelik and Seleznev [17].
The problem was solved in axisymmetric formulation. Thermosyphon is on the
axis r = 0. The characteristics of the thermosyphon are given in Table 1. The
thermosyphon has a total length of 9 m, is made of stainless steel, and the filled up
by ammonia. Soil properties are shown in Tables 2 and 3. Climatic conditions
correspond to the Vorkuta city and are shown in Table 4. The parameters of solar
radiation are shown in Table 5.
Fig. 3. Calculation scheme for determining
the radius of freezing around thermosiphon.
z
conl
ogl
ugl
evpllowz
evpz
upz
0z
8 m
12 m
r
sunQ
sunQ
snow cover
0T n 0T n
0T n
soil inwsurfT n Q
sunE
1Soil
2Soil
evpQ
ugQ
2920 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
The nonlinear heat equation was used for the numerical study [25]:
wsoil soil w soil
Tc L T
T t
(16)
4 401inw surf air surf sun air aiurf rsQ T A QT b T Ta (17)
Markov et al. [26, 27] and Bhaskar and Choudhary [28] described the
method for calculating the thermal conductivity and isobaric heat capacity by
using transit functions.
The soil temperature at the depth of zero annual amplitudes is 0.5 °C. To
achieve the desired temperature, the snow cover thickness coefficient was 1.091,
which slightly increases the snow cover thickness (Table 4). The duration of the
calculation was 5 years. The problem is solved by the finite difference method in
an implicit scheme.
Table 1. Thermosyphon construction parameters.
lcon log lug levp Rr
1.0 1.0 4.0 3.0 0.2
Rcon δcon Revp δevp h
0.01685 0.0035 0.01685 0.0035 0.002
s εcon εog λcon λR
0.01 0.9 0.9 40 40
λevp λhi δhi λrf ρrf
40 0.06 0.025 0.506 639
νrf Lrf
2.73e-7 1.262e+6
Table 2. Thermal properties of soils.
Soil Type cth cfr λth λfr Tbf Tint
1 Sand 1280 960 2,51 2,93 -0,2 0,2
2 Clay loam 1489 1143 1,57 1,86 -0,1 0,8
Table 3. Physical properties of soils.
Soil Type ρsoil ρsk ρw,tot ρw,max ρw,nf
1 Sand 2125 1800 325 325 0
2 Clay loam 2157 1800 357 357 120
Table 4. Climate and show cover.
Month Tair vair δsn ρsn
1 -20,5 6,2 34,6 230
2 -19,6 5,9 40,0 240
3 -16,6 6,2 45,0 250
4 -9,7 5,9 36,0 283
5 -3,1 6,0 5,6 330
6 5,9 5,4 0,0 0
7 12,0 4,7 0,0 0
8 9,6 4,5 0,0 0
9 3,8 5,0 0,0 0
10 -4,5 5,7 7,3 155
11 -13,5 5,8 22,3 190
12 -17,9 6,4 29,7 217
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2921
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
Table 5. Solar radiation parameters.
Month S’sun, J/m2 Ssun, J/m2 Gsun, J/m2 Qsun, J/m2 Asurf, u.f.
1 21 4 11 15 0.80
2 125 24 51 75 0.79
3 341 106 153 259 0.78
4 458 203 241 444 0.69
5 486 243 362 605 0.43
6 475 260 299 559 0.24
7 584 306 276 582 0.24
8 329 154 213 367 0.24
9 181 68 130 198 0.26
10 106 25 73 98 0.62
11 43 8 20 28 0.79
12 8 2 4 6 0.80
5. Results and Discussion
Figure 4 shows the time dependence of the average thickness of the refrigerant film
in the condenser.
Analysis of the dependence in Fig. 4 shows that the maximum thermal
resistance of the condensate film is no more than 1.04e-4 m2∙K/W. This value is
significantly less than the thermal resistance of the condenser, which is about 2.46e-
2 m2∙K/W. The error created by the absence of a film in the calculation is within
the error of the numerical method. Thus, the thermal resistance of the film does not
have a significant effect on the results of calculating of freezing radius. The position
of the isotherm -0.7 ºC in the calculation without taking into account the film are
not shown here, since there are no visible differences compared to the calculation
in which, the film was taken into account. Problem No. 3 solved.
Fig. 4. Time dependence of average
thickness of liquid refrigerant film in condenser.
Figure 5 shows the temperature distribution in the soil around the thermosyphon
after 5 years.
The freezing radius rfr at a depth of z = -5.5 m of the authors’ model of
absorption of solar radiation by thermosyphon is 1.71 m and in the case Esun = 0 the
radius is 2.05 m. Thus, the calculation according to the authors’ model of
thermosyphon leads to reduction in the radius of freezing by 20%. It is obvious that
heat flow from the soil to the evaporator for the authors’ model is significantly less.
This is confirmed by the results in Fig. 6.
0
10
20
30
40
50
60
0 1 2 3 4 5 6
year
,rc m
2922 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
Figure 6 shows the value of the difference in heat exchanging in evaporator
(∆Qevp) between the model of Gorelik and Seleznev [17] and the author’s model.
As can be seen, the maximum difference is up to 50 W, i.e., the authors’ model
reduces heat flow. The maximum difference is in a spring, which is associated
with an increase of absorbed solar energy. The second peak is in an autumn, when
the thermosyphon start to work. However, the autumn peak is much smaller,
which is due to the delay in decrease of air temperature relative to decrease of
solar irradiance: according to the Tables 4 and 5 the air temperature in February
is much less than in October, although the solar irradiance in February, on the
contrary, is more. Therefore, thermosyphon operates in conditions of high solar
irradiance in a spring. This is the main feature of operation of two-phase
thermosyphons in conditions of subarctic climate.
The values Esun and ∆Qevp differ by about 3 to 4 times. Despite the heating by
solar radiation, the heat flow from the soil to the evaporator is reduced by no more
than 35% of Esun due to an increase in the refrigerant and soil temperature. The
reducing of freezing radius due to the influence of solar radiation to can be achieved
by an increase in the heat transfer coefficient in the condenser in accordance with
equation (2). This can be implemented by increasing in the number of fins or the
length of the condenser. Problem No. 4 solved.
(a) Esun according to formula (15). (b) Esun = 0 according to Gorelik
and Seleznev [17].
Fig. 5. Temperature distribution in soil around the thermosyphon.
Fig. 6. Difference in heat flow in the two
models and the heat flux of solar radiation.
0 2 4
,z m0
1
2
3
4
5
6
7
8
9,r m
frr
1.71m
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
,T C
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
,z m,T C
frr
0
1
2
3
4
5
6
7
8
90 2 4
,r m
2.05m
0
50
100
150
200
0 1 2 3 4 5 6
Ряд1
Ряд2
,evpQ W
,sunE W
year
W
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2923
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
6. Conclusions
The authors derived formulas for calculating the average film thickness of the liquid
refrigerant on the inner surface for all parts of the thermosyphon: condenser,
overground pipe, underground thermal insulated pipe, and evaporator (8) - (11).
The authors used the iterative method for solving the Nusselt equation.
Numerical study has shown that the thermal resistance of the refrigerant film is 2
orders less than the thermal resistance of the condenser if the length of evaporator is
3 m. Thus, the practical calculations of the temperature regime of the soil can be done
without taking into account refrigerant film thickness for short evaporator part.
The formula for calculating absorbed direct, diffuse and reflected solar radiation
by the condenser and overground pipe is derived (15). Required data for a formula
coincide with meteorological quantities: direct normal irradiance, direct horizontal
irradiance, diffuse horizontal irradiance, albedo of ground surface.
A comparison of two models of system two-phase thermosyphon - soil -
atmosphere showed that the authors’ model reduces freezing radius by 20%. The
significant difference between the models is in the method of calculation of
absorbed solar radiation by condenser and overground pipe of thermosyphon. Thus,
in a subarctic climate with high solar radiation the safe operation of buildings and
structures can be ensured only if the distance between the heat stabilizers is
significantly reduced due to strong heating of the condenser by solar radiation. The
result is extremely important in designing of buildings and structures in permafrost.
Nomenclatures
Asurf Albedo of ground surface, u.f.
bair Back radiation factor of atmosphere, u.f.
cfr Isobaric heat capacity of frozen soil, J kg-1 K-1
csoil Isobaric heat capacity of soil, J kg-1 K-1
cth Isobaric heat capacity of thawed soil, J kg-1 K-1
Ed Absorbed diffuse solar radiation by condenser, W
Esun Absorbed diffuse solar radiation by thermosyphon, W
Fa Surface area of solar radiation diffusion in the atmosphere, m-2
Fcon Condenser area, m-2
Gsun Diffuse horizontal irradiance, W m-2
g Acceleration of gravity, m s-2
Ha Vertical distance from the surface of the condenser to the surface
of solar radiation diffusion, m
K Overall heat transfer coefficient, W m-1 K-1
Kcon Overall heat transfer coefficient of condenser, W m-1 K-1
Kevp Overall heat transfer coefficient of evaporator, W m-1 K-1
Kog Overall heat transfer coefficient of overground pipe, W m-1 K-1
Kug Overall heat transfer coefficient of underground pipe, W m-1 K-1
Lrf Heat of refrigerant vaporization, J kg-1
Lw Heat of fusion, J kg-1
lcon Length of condenser, W m-1
levp Length of evaporator, W m-1
log Length of overground pipe, W m-1
lug Length of underground pipe, W m-1
2924 E. V. Markov et al.
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
na Inner normal vector of solar radiation diffusion surface
ncon Condenser surface normal vector
n Outward ground-normal vector
Pra Prundtl number of air, d.l.
Qinw Inward heat flux on the surface of the ground, W m-2
Qsun Global horizontal irradiance, W m-2
Qup Inward heat flux in underground pipe, W m-1
Qevp Inward heat flux in evaporator, W m-1
qt Linear heat source in thermosyphon, W m-1
qug Linear heat source in underground pipe, W m-1
R Distance from the surface of the condenser to the surface of solar
radiation diffusion, m
Rcon Condenser radius, m
Revp Evaporator radius, m
Rr Fin radius, m
r Projection of R on the surface of the ground, m
rfr Radius of freezing, m
SN Surface of unfinned area of condenser, m2
Ssun Direct horizontal irradiance, W m-2
S’sun Direct normal irradiance, W m-2
Sv Direct vertical irradiance, W m-2
s Distance between fins, m
T Temperature of soil, K
Tair Temperature of air, K
Tbf Soil freezing point, K
Tg Temperature of refrigerant gaseous phase, K
Tint Soil freezing interval, K
t Time, s
Va Wind speed, m s-1
z0 Ground level, m
zevp Top of evaporator, m
zlow Bottom of thermosyphon, m
zup Top of thermosyphon, m
Greek Symbols
∇∙ Divergence operator
∇ Gradient operator
αN Convective heat transfer coefficient for overground pipe,
W∙m-2∙K-1
αR Convective heat transfer coefficient for side surface of condenser
fin, W∙m-2∙K-1
αr Convective heat transfer coefficient for condenser fin, W∙m-2∙K-1
αsurf Convective heat transfer coefficient on the surface of the ground,
W∙m-2∙K-1
δcon Condenser wall thickness, m
δevp Evaporator wall thickness, m
δr Refrigerant film thickness, m
δrc Refrigerant film thickness in lowest point of condenser, m
δrca Average refrigerant film thickness in condenser, m
δrea Average refrigerant film thickness in evaporator, m
Calculation of Freezing Radius Around Vertical Two-Phase Thermosyphon . . . . 2925
Journal of Engineering Science and Technology October 2019, Vol. 14(5)
δroa Average refrigerant film thickness in overground pipe, m
δrua Average refrigerant film thickness in underground pipe, m
δsnow Snow thickness, m
εcon Emissivity of condenser surface, u.f.
εog Emissivity of overground pipe surface, u.f.
εsurf Emissivity of ground surface, u.f.
θ Angle in a cylindrical coordinate system, rad
λa Thermal conductivity of air, W m-1K-1
λcon Thermal conductivity of condenser material, W m-1K-1
λevp Thermal conductivity of evaporator material, W m-1K-1
λfr Thermal conductivity of frozen soil, W m-1K-1
λhi Thermal conductivity of heat insulation, W m-1K-1
λR Thermal conductivity of material of finning, W m-1K-1
λrf Thermal conductivity of refrigerant, W m-1K-1
λth Thermal conductivity of thawed soil, W m-1K-1
νa Air viscosity, m2 s-1
νrf Liquid refrigerant viscosity, m2 s-1
ρrf Density of liquid refrigerant, kg m-3
ρsk Density of dry soil, kg m-3
ρsnow Density of the snow, kg m-3
ρsoil Density of soil, kg m-3
ρw,max Maximal content of water in the soil, kg m-3
ρw,nf Content of non-freezing water in the soil, kg m-3
ρw,tot Content of water in the soil, kg m-3
σ0 Stefan Boltzmann's constant, W m-2 K-4
φa Angle between R and na
φcon Angle between R and ncon
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