Moscow, г. Москва и Московская область, Россия
Moscow, г. Москва и Московская область, Россия
Moscow, г. Москва и Московская область, Россия
Moscow, г. Москва и Московская область, Россия
A number of sectors in the food industry practice cooling substances of biological origin. This contributes to the maintenance of their biological properties, as well as prevents microflora growth in the product. One of the ways to intensify production processes and maintain the quality of raw materials and finished products is their accelerated cooling with the help of low-energy cooling equipment. The use of physical bodies cooled to low temperatures is a promising way to accelerate liquid cooling. We used balls with frozen eutectic solution. In our research, the problem of cooling a liquid system is formulated and solved within the framework of classical linear boundary value problem for the equation of a stationary convective heat transfer. In the area of the actual values of the process parameters on the study object, the solution obtained is used as the basis for numerical experiment on the modelling of the cooling liquid flow with the cooling agent system, namely balls filled with eutectic solution. By calculation, the efficiency of the proposed method for cooling liquid was justified based on such factors as temperature, the number of balls in a two-phase liquid system, and the duration of low-temperature treatment. The presented results of the numerical experiment complied with real heat transfer processes during liquid cooling.
Low-temperature treatment, cooling, liquid system, heat transfer, eutectic solution
INTRODUCTION
An important issue of the intensification of
production processes and rational use of raw materials
is the possible rapid decrease in the temperature of the
liquid system. For example, in processing industries and
other sectors of the economy, the cooling of biological
origin substances contributes to the maintenance of
their biological properties, as well as prevents microflora
growth in the product [1, 2].
Currently, the cooling of liquids by using frozen
solids is one of the methods for lowering the temperature
in a liquid medium [3–7, 11, 14, 15]. This technology is
useful and in refrigeration engineering, where frozen
physical bodies are balls filled with eutectic solution
[8–10, 12]. This is confirmed by results of theoretical
and experimental studies conducted by reserchers
of Moscow State University of Food Production and
Razumovsky Moscow State University of technology
and management. The results confirm the advantages
of cooling water with frozen balls over other methods,
providing a high intensity of the process and reduced
energy consumption.
To accelerate the heat transfer process based on
water treatment using the technology of enrichment of
the working volume of cooling liquid with frozen bodies
(cooling agent), it is advisable to carry out this process
in the mode of flow of the liquid through a container
with frozen balls. At the same time, it should be noted
that there are no theoretically based calculation methods
for predicting and controlling the heat transfer process,
including in the flow, when the cooling process develops
in a heterogeneous liquid “water-frozen balls” system.
Bases on the law of conservation of mass and energy,
we presented the results of the analytical and numerical
studies on the of cooling a liquid flow moving through
a heat exchanger filled with balls with a frozen eutectic
coolant to justify the intensification of the heat transfer
process between a coolant and liquid.
172
Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176
The setting of the problem. Let a liquid flow (for
example, water) be supplied to a certain container filled
with balls with frozen eutectic solution in a stationary
mode, with a small productivity Q and a low flow rate v0
(Fig. 1). We will consider the selected motion model of a
two-phase liquid system as the filtration flow of a fluid
through a porous medium formed by balls (coolant) [6].
It is assumed that the balls are statistically uniformly
distributed inside a cylinder of length h and radius R
containing a liquid (water) and solid (balls) phases.
To simplify the process for quantitative analysis of
the heat removal from the liquid to the balls, we replaced
the pore volume occupied by the liquid (preserving
the values of porosity) with a set of cylindrical tubes
(conditionally, capillaries). Each of them had an axis
parallel to the axis of the vessel, length h and reduced
radius r0 (Fig. 2).
For the selected geometric model of the liquid
volume, we introduce the following notations: db is the
ball diameter and m is the surface porosity of the system
(m < 1), numerically equal to the ratio of the volume of
liquid filled pores to the volume of the system [8].
Since the volume and surface porosities of the
working volume of the capacitance are quantitatively the
same [8], the approximate ratio, based on the porosity
definition can be written as follows:
πr0
2 /m = πdb
2/[4(1 – m)]
which yields the relationship:
r0 = 0,5d[m/(1 – m)]1/2 (1)
henceforward, r0 = r0 (d,m), d = db is the diameter of the
ball.Thus, as can be seen, the problem of calculation
reduces to the quantitative analysis of temperature in the
isolated capillary.
The solution of the problem. Due to the possible
axisymmetric nature of heat transfer from the walls of
the channel to the liquid, the quantitative modeling
of this process will be carried out in a cylindrical
coordinate system, in the meridional section of the
channel rOz (Fig. 2).
For the selected simulation geometric model, we
use the equation of stationary convective thermal
conductivity related to cylindrical coordinates as a
basic differential equation describing the heat transfer
phenomenon in the flow [9]:
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i AJ r r 1
0 0 ( / ) exp( 0 10 100
200
300
400
500
Frozen ball Liquid
Flow of Flow of cooled uncooled
liquid
v0 v0
h
r
R
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, (2)
where r, z – radial and axial coordinate, respectively;
T – the temperature of water; vz – axial component
of fluid flow rate in the capillary; а = λ/(сρ), а – is the
coefficient of thermal diffusivity of water, λ – thermal
conductivity coefficient, c – specific conductivity
coefficient, and ρ – is the density of water.
In practice, fluid rate vz can be replaced with its
averaged value over the cross section of the channel,
with a small error. Then a simplified form of Eq. (2) can
be written as follows:
( 1 ) 2
2
r
T
r r
T
z
T
∂
∂
+
∂
∂
=
∂
∂
β (3)
where
β = а/vz (4)
vz = v0/m, v0 = Q/S – volume rate of fluid flow in the
capacitance (filtration rate [8]), S = πR2 – cross-sectional
area of the capacitance; β is the specific coefficient of
thermal diffusivity calculated taking into consideration
axial velocity vz.
For simplicity, assume inlet temperature to be
constant:
Т(r,z) = Т0 at 0 ≤ r ≤ r0, z = 0 (5)
The condition of symmetry of the temperature
distribution along the channel diameter corresponds to
the condition of the maximum temperature in the middle
the capillary walls.
∂∂ÒÒT//∂/ ∂rr a==t 0r0 = 0, 0 < z ≤ h (6)
Taking into account the fact that heat energy
develops from the liquid to the capillary wall, the
boundary condition on the surface of this channel is:
Н[Т(r0,z) – Тk] – ∂T(r0,z)/∂r = 0 (0 < z ≤ h) (7)
Figure 1 Scheme of liquid cooling with frozen balls
Figure 2 Scheme to calculate the cooling process of liquid
with frozen balls
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 30
100
200
300
400
500
T
Frozen ball Liquid
Flow of Flow of cooled liquid
uncooled
liquid
v0 v0
h
z
r
R
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, С
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 30
100
200
300
400
500
T
Frozen ball Liquid
Flow of Flow of cooled liquid
uncooled
liquid
v0 v0
h
z
r
R
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, С
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 30
100
200
300
400
500
T
Frozen ball Liquid
Flow of Flow of cooled liquid
uncooled
liquid
v0 v0
h
z
r
R
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, С
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 100
200
300
400
500
T
v0 v0
h
z
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, С
173
Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176
where Н = α /λ, α i s heat transfer coefficient from the
liquid to the capillary wall, Тk is the eutectic temperature
of frozen balls (Fig. 1).
According to (5)–(7), in the framework of the
terminology adopted in the theory of heat transfer, we
have a problem with boundary conditions of the third
kind for differential equation (3).
From the point of view of quantitative analysis of
the thermal regime in a steady stream of fluid inside the
capillary, the considered problem is formally equivalent
(provided that the diameter of the capillary dk is much
less than the capacitance height h) to the problem of the
temperature distribution over time in an unsteady mode
in an unlimited cylinder. What is more, in Eqs. (3)–(7)
the duration of the heat transfer process is displayed on
the axial coordinate z.
Thus, the solution to the problem with an unsteady
mode of heat transfer in an unbounded cylinder can be
adapted to the boundary value problem (3)–(7) of the
temperature distribution in the convective fluid stream in
the capillary and is formulated as a dependence:
dк
1 )
2
2
r
T
r r
T
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 30
100
200
300
400
500
T
Frozen ball Liquid
Flow of cooled liquid
v0
h
z
R
vz
Т/r
r
r0
h
z
, W/(m2К)
Т, С
, n → ∞ (8)
where
θ = θ(d,m,r,z) = (Т – Т0)/(Тк – Т0) > 0 (9)
is specific value reflecting the differential temperature of
the ball Тk and the initial temperature of the liquid Т0, as
well as the current differential temperature of the liquid
Т(d,m,r,z) and the coolant temperature Т0.
Аi = 2J1(νi)/{νi[J0(νi)2 + J1(νi)2]} (10)
where J0, J1 – Bessel function of the first kind of
zero and first order respectively; positive roots of
transcendental equations.
J0(ν)/J1(ν) = ν/Bi (11)
Bi = Bi(α,d,m) = αr0/λ – the Biot number (12)
Fo* = Fo*(z,d,m) = βz/r0
2 – the modified Fourier number (13)
RESULTS AND DISCUSSION
Quantitative modeling of the heat transfer process
was carried out based on relations (8)–(13) using the
Mathcad medium.
We used the following process parameters: the
length of the capacitance h = 0.5 m; the diameter of
the capacitance D = 0.1 m; equipment productivity (by
water) Q = 2×10–4 m3/s; kinematic viscosity coefficient
ν = 10–6 m2/s; ball eutectic temperature Тk = – 10°С;
thermal c onductivity c oefficient λ = 0 .58 W /(m·К);
thermal diffusivity coefficient a = 13.8×10–8 m2/s; and
ball diameters d = 0.0375, 0.04, and 0.0425 m. The
calculation of the current temperature of the liquid
was carried out according to two variants of porosity:
m = 0.5 and m = 0.35.
In accordance with the selected parameter values,
the volume rate of flow (filtration rate) for all calculation
options was v0 = 4Q/(πD2) = 4×2×10–4/(3.14×0.12) =
0.0254 m/s.
As a calculated value of the temperatu∫re 0
2 0
0
2 r ( , , , ) , d m r z rdr
r
θ given over
the radius r inside the capillary, we used its valu∫e 0
2 0
0
2 r ( , , , ) , d m r z rdr
r
θ av,
averaged over the channel cross-sectional area:
∫ 0
2 0
0
2 r ( , , , ) , d m r z rdr
r
θ av(d,m,z) = ∫ 0
2 0
0
2 r ( , , , ) , d m r z rdr
r
θ (14)
The following dependence was used as a calculated
dependence for the liquid temperature based on the
operating parameters of the axis coordinate z and the
time of the process τ:
T(z) = T0 + (Тк – Т0)θ, (15)
To calculate the number of balls N in the capacitance,
we used the formula:
N(m,d) = 1.5D2h×(1 – m)/d3 (16)
where m is the porosity if the liquid system, d is
the diameter of the ball, D is the diameter of the
capacitance, and h is the length of the capacitance.
Thus, according to the geometrical parameters,
the number of balls of diameter d = 0.04 m was 58 for
m = 0.5 and 76 for m = 0.35.
Previously, to assess the convergence of series (8),
we performed test calculations using formula (14).
Temperature T0 was 36°С and heat transfer coefficient
α was 440 W/(m2·К) ( Fig. 3 ). B ased o n B i = α r0/λ =
440×0.02/0.58 ≈ 15, we found partial sums of this series,
from the first to the sixth sum inclusively.
Since determination of the roots of transcendental
equation (11) when varying the parameters of the Bi
criterion involves laborious calculations, we used tabular
data. As in all calculations a slight difference in the
value of partial sums was noted only starting from the
sixth sum (Fig. 4), the sum of six members of this series
was used in the calculations (8).
dк
О
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i i AJ r r Fo
1
2
0 0 ( / ) exp( ),
0 10 20 30
100
200
300
400
500
T
Frozen ball Liquid
Flow of Flow of cooled liquid
uncooled
liquid
v0 v0
h
z
r
R
vz
Т/r
r
r0
h
О z
r
, W/(m2К)
Т, С
Figure 3 Heat-transfer coeffici ent as a function of the
temperature of water for the “water-ice” system
× ×
( ) ( 1 )
2
2
r
T
r r
a T
z
Tvz
= 1
n
i
i i AJ r r 1
0 0 ( / ) exp( 0 10 100
200
300
400
500
, W/(m2К)
Т,
174
Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176
In addition, to find the dependence of temperatures
in the capillary on the diameter of the balls for each of
the options, we considered the situation when balls
with a diameter of 0.0375 or 0.0425 m (closed to the
diameter of the test ball d = 0.04 m) acted as a coolant.
This made it possible, with some approximation, to use
tabular data [13] on the roots of equation (11) based on
Bi = αr0/λ = 15 which corresponded to d = 0.04 m.
From relations (8)–(13) and (15), in the range of
variation of the parameters of water cooled by frozen
balls, 7 calculation results were obtained and graphs
were plotted (Figs. 5–7). Based on the analysis of
the graphs, the dependence of the variables of the
considered problem on the operating parameters was
revealed.
Since the calculated temperature of the coolant
reduced along the channel (Fig. 4), the specific
temperature of the liqui∫d 0
2 0
0
2 r ( , , , ) , d m r z rdr
r
θ increased in the same
direction.
In turn, the rate of change in the temperature of the
liquid decreased with time (Figs. 5–7). This is because
of the reduction of the specific surface area of the ball,
which is the ratio of the surface area of the ball to its
volume. This resulted in decreasing heat exchange on
the border between solid and liquid phases in the liquid
system (for example, in Fig. 5 graph 3 corresponding to
the diameter of the ball d = 0.0425 m is above graph 2
for the ball diameter d = 0.04 m).
In addition, as it can be seen from Fig. 7, in the
case of water productivity Q fixed for all variants, the
increase in the rate of flow of the coolant along with the
decrease in the porosity of the liquid system naturally
leads to a decrease in the rate of cooling of the liquid.
Thus, for example, in Fig. 7 graph 4 corresponding to
porosity m = 0.35 and the diameter of the ball d = 0.04 m
is above graph 1 for porosity m = 0.35 and the ball with
the same diameter.
However, it should be noted that, despite the
qualitative consistency of the calculated results with
Figure 5 Relation between the outlet temperature of liquid Т
and time τ at different values of parameters (z = 0.5 m;
m = 0.5; Т0 = 36°С; α = 440 W/(m2·К); Bi = 15: 1 – d = 0.0375
m, 2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 = 20°С; α = 230 W/
(m2·К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m,
6 – d = 0.0425 m)
0 2 4 6 8
0
10
20
30
40
T11()
T12()
T13()
T14()
T15()
T16()
Relation between the outlet temperature of liquid Т and time at different values of parameters (z =
Т0 = 36С; = 440 W/(m2К); Bi 15: 1 – d = 0.0375 m, 2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 =
W/(m2К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m, 6 – d = 0.0425 m)
the calculated temperature of the coolant reduced along the channel (Fig. 4), the specific temperature
increased in the same direction.
the rate of change in the temperature of the liquid decreased with time (Figs. 5–7). This is because
of the specific surface area of the ball, which is the ratio of the surface area of the ball to its volume.
decreasing heat exchange on the border between solid and liquid phases in the liquid system (for
Fig. 5 graph 3 corresponding to the diameter of the ball d = 0.0425 m is above graph 2 for the ball
0.04 m).
, s
1
Т, С
2
3
4 5 6
Figure 6 Relation between the outlet temperature of liquid Т
and time τ at different values of parameters (z = 0.5 m; m =
0.35; Т0 = 36°С; α = 440 W/(m2·К); Bi = 15: 1 – d = 0.0375 m,
2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 = 20°С;
α = 230 W/(m2·К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m,
6 – d = 0.0425 m)
of the liquid Ɵ increased in the same direction.
In turn, the rate of change in the temperature of the liquid decreased with time (Figs. 5–7). This of the reduction of the specific surface area of the ball, which is the ratio of the surface area of the ball to This resulted in decreasing heat exchange on the border between solid and liquid phases in the liquid example, in Fig. 5 graph 3 corresponding to the diameter of the ball d = 0.0425 m is above graph 2 diameter d = 0.04 m).
0 2 4 6 8
0
10
20
30
40
T11()
T12()
T13()
T14()
T15()
T16()
, s
4 5 6
1 2 3
Т, С
determination of the roots of transcendental equation (11) when varying the parameters of the Bi
laborious calculations, we used tabular data. As in all calculations a slight difference in the value of
noted only starting from the sixth sum (Fig. 4), the sum of six members of this series was used in the
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
1(z)
2(z)
3(z)
4(z)
5(z)
6(z)
z
between the radius mean specific temperature of liquid and the axial coordinate z for partial sums of
0.04 m; m = 0.5; 1, 2, and 3 are numbers of summands from 1 to 3; and 4–6 from 4 to 6.
addition, to find the dependence of temperatures in the capillary on the diameter of the balls for each of
considered the situation when balls with a diameter of 0.0375 or 0.0425 m (closed to the diameter of
0.04 m) acted as a coolant. This made it possible, with some approximation, to use tabular data [13]
equation (11) based on Bi = r0/ = 15 which corresponded to d = 0.04 m.
relations (8)-(13) and (15), in the range of variation of the parameters of water cooled by frozen balls,
results were obtained and graphs were plotted (Figs. 5–7). Based on the analysis of the graphs, the
variables of the considered problem on the operating parameters was revealed.
1
2
3, 4–6
z, m
Figure 4 Relation between the radius mean specific
temperature of liquid and the axial coordinate z for partial
sums of the series (8). d = 0.04 m; m = 0.5; 1, 2, and 3 are
numbers of summands from 1 to 3; and 4–6 from 4 to 6
175
Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176
the physical meaning of the process under study, the
quantitative assessment that characterizes the course of
this process needs additional comments.
First of all, this relates to the question of formalizing
the boundary condition at the interface, which is typical
for many works on the theory of heat conductivity,
where, when setting the problem, the temperature
is assumed unchanged, while for the heat exchange
between phases (for example, between liquid and
coolant) phase temperature tends to level off. This leads
to a decrease in the intensity of heat transfer from the
liquid to the ball as naturally as in the studied problem.
Therefore, the results of the the cooling rate of water
presented in Figs. 5–7 on, in fact, are overestimated
compared to real data.
At the same time, despite the simplifications
based on the theory of heat conduction and used in the
formulation of the problem, physical and mathematical
modeling of processes makes it possible to predict and
control their development. To the same extent, this is
also applied to the complex problem of justifying the
rate of liquid cooling due to the accumulation of frozen
balls with a developed heat exchange surface that is
analyzed in this paper.
CONCLUSION
To justify heat transfer from the fluid flow to balls
with eutectic frozen solution, we applied an analytical
tool forecasting the course of this process in the
innovative technology for cooling this liquid.
In the quantitative analysis of this problem kinetic
aspects of filtration fluid motion was used, namely, when
the working volume occupied by the liquid between the
balls was simulated by equivalent plurality of ordered
cylindrical capillary channels. This allowed us, from
the point of view of analytical and numerical analysis
of the thermal regime in a steady fluid flow inside the
working volume of the capacitance, to adapt the solution
of this problem to the study of this regime in an isolated
capillary.
The accepted conditions, namely the size of the
capacitance and balls and the volume fraction of balls in
the capacitance created the preconditions for conducting
quantitative modeling of the process under study based
on the calculated dependences of the temperature
distribution in an unlimited cylinder under an unsteady
regime.
To assess the efficiency of the cooling process
of fluid flow in a heat exchanger with a frozen solid
phase and a developed heat exchange surface in the
field of the real values of the process parameters, the
obtained temperature dependences were used to carry
out a numerical modeling of the cooling process of this
medium.
Based on the results of the analytical and numerical
study of the problem, an acceptable region for varying
the mechanical and thermotechnical parameters of these
processes was determined. This region is of importance
for engineering calculations of the low-temperature
processing of raw materials and finished products of
biological origin.
CONTRIBUTION
The authors were equally involved in writing the
manuscript and are equally responsible for plagiarism.
CONFLICT OF INTEREST
The authors declare that there is no conflict of
interest related to the publication of this article.
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