Acta Polytechnica


doi:10.14311/AP.2015.55.0140
Acta Polytechnica 55(3):140–145, 2015 © Czech Technical University in Prague, 2015

available online at http://ojs.cvut.cz/ojs/index.php/ap

SIMULATION OF HEAT TRANSPORT DURING
THE PROCESS OF COOLING A SUGAR SOLUTION

IN A RECUPERATION EXCHANGER

Tomáš Brestoviča, Mária Čarnogurskáa,∗, Miroslav Příhodab,
Michal Kubíka

a Faculty of Mechanical Engineering, Technical University of Košice, Department of Power Engineering,
Vysokoškolská 4, 042 00 Košice, Slovak Republic

b Faculty of Metallurgy and Materials Engineering, VŠB – Technical University of Ostrava, Department of
Thermal Engineering, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic

∗ corresponding author: maria.carnogurska@tuke.sk

Abstract. The paper describes a mathematical model of the cooling process of a highly concentrated
sugar solution in an exchanger with a specifically shaped heat exchanging surface of the cooling panels.
An analysis of the individual parts of the stum cooling line is made, dealing with the cooling performance
of the cooling panels located in the stum tanks, whose volume is 3230 litres or 1430 litres. One of the
monitored parameters is the cooling performance of the JN30 aggregate. The article also deals with
the appropriateness of the aggregate for cooling the stum with a total volume 78.21m3, from the real
operation temperature to 0 °C during 48 hours.

Keywords: concentrated solution; cooling; numeric simulation..

1. Introduction
The cooling performance of cooling units working on
the principle of coolant compression and expansion
depends on the operating temperature of the evapo-
rator. The heat transport from the aggregate to the
appliance is done via heat transferring medium, which
is usually an anti-freeze fluid. The appliance used for
cooling of various solutions (such as stums) is a heat
exchanger (cooling panel). The heat in the primary
circuit is transferred by force convection; the heat in
the secondary circuit is led away by free convection.
In order to determine the dependence of solution tem-
perature on time during the cooling, it is necessary
to know the cooling performance of the heat trans-
fer surface. The performance is dependent on the
geometric shape, the physical properties of the heat
transfer material, the material characteristics of the
panel, the physical characteristics of the solution from
which the heat flow is led away, and on the tempera-
ture gradient between the panel and the solution. By
increasing the temperature gradient, the performance
of the cooling panel increases, but the performance
demand of the cooling device decreases. The deriva-
tion of the mathematical dependencies describing the
balanced thermodynamic state of this process is the
basic presupposition allowing us to design devices of
this kind.
When determining the cooling performance of the

panel, criteria equations are used from which we can
calculate the HTC (Heat Transfer Coefficient) between
the external heat exchanging surface and the cooled
solution. However, there is a problem with defining
the average panel temperature, as the temperature

arrangement on the heat exchanging panel may be
significantly uneven. For this reason, it is better to
execute the calculation using a numerical method, for
example using the ANSYS CFX program.
The final mathematical model has been used to

verify the thermal performance of a particular device
intended to cool the stum. The cooling aggregate
performance of the JN30 type is 27.5 kW at a cooling
mixture temperature of 13 °C. The aggregate works at
a pressure of 2.5 bar and at a cooling mixture flow of
120 lmin−1. The volume of the cooling mixture tank
is 330 litres. The device cools down 22 tanks with a
volume of 3230 litres (T1400 tank type) and 5 tanks
with a volume of 1430 litres (T930 type), which equals
78210 litres of stum in total.

The mathematical model requires a knowledge of
the functional dependence of the cooling performance
(Pch) on the cooling mixture temperature at the en-
trance to the cooling panel (tv1). Using the data from
the technical documentation of the JN30 device, a
regression equation in the following form was derived:

Pch = atv1 + b = 938.98tv1 + 15380. (1)

At a temperature of approximately 0 °C, the cooling
performance of the installed device falls to approxi-
mately 15 kW. Cooling the stum cooling to the tem-
perature of approximately 0 °C in 2 days requires an
increase in the cooling aggregate performance. That is
why the calculation is focused on the heat transfer in
the cooling panels, as well as on obtaining the cooling
curve for 100% of the volume of the cooled stum, and
on the minimum cooling performance necessary to
meet the requirement of the stum cooling.

140

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vol. 55 no. 3/2015 Simulation of Heat Transport

Figure 1. Dependence of the specific thermal capacity of the 30% sugar solution on the temperature.

Entered values
∆tN (°C) −4 −8 −15 −20
S (m2) 1.1627
tv1 (°C) −4
Qm (kg s−1) 0.0773

Calculated values
t (°C) 0 4 1 16
cstr (J kg−1 K−1) 3699 3702 3704.7 3707.4
tv2 (°C) −3.053 −1.892 0.586 2.704
tstr-panel (°C) −1.491 1.413 7.198 11.812
αpanel (Wm−2 K−1) 159.4 205.4 305.5 406.7
Pch,panel (W) 270.8 603.2 1313.3 1921.2
Pch (W) 6634 14779 32176 47071

Table 1. Entered and calculated values.

2. Heat transfer in the cooling
panels

In order to construct the heat exchanger thermal bal-
ance equation, it is necessary to know the dependence
of the cooling performance of the panel on the tem-
perature difference (∆tN) between the cooling mix-
ture (tv1) and the stum temperature (t). In order
to achieve this, numerical calculations for four tem-
perature states have been made (at the temperature
differences of ∆tN = −4,−8,−15,−20 °C) and for the
nominal flow of the cooling mixture through one panel
of 7.407 · 10−5 m3 s−1 (corresponding to the total flow
of 120 lmin−1 through 27 cooling panels). In order to
ensure the temperature gradient between the coolant
and the stum, the cooling mixture temperature must
be constantly lower than the stum temperature. The
higher the absolute value of the temperature differ-
ence, the higher the panel cooling performance. Ta-
ble 1 shows the values of the input data and the values
of the relevant parameters calculated by analytic pro-
cedure or by numerical simulation.

In Table 1, S is the real total heat exchanging sur-

face of the cooling panel (m2), tv2 is the temperature
of the cooling mixture on the cooling panel output
(°C), tstr-panel is the average temperature of the rust-
less panel surface (°C), cstr is the thermal capacity of
the coolant at its average temperature (J kg−1 K−1),
Qm is the mass flow of the coolant (kg s−1), αpanel
is the heat transfer coefficient on the panel surface
(Wm−2 K−1), Pch,panel is the cooling performance of
one panel (W).
The heat transfer coefficient from the stum to the

cooling panel surface was determined from the crite-
ria (2) and (3) for a free convection at the vertical
panel [1]:

Nu =
(
0.825 + 0.387(Ra f(Pr))1/6)2

, (2)

f(Pr) =
(
1 + (0.492/Pr)9/16)−16/9

. (3)

In order to calculate the Prandtl (Pr) and Rayleigh
(Ra) similarity criteria, it is necessary to know the ma-
terial characteristics of the stum, which consists, ide-
ally, of a mixture of water and sugar with an amount
of 400 g of sugar per 1 litre of the solution [2]. The
volume concentration c, which is calculated as the

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T. Brestovič, M. Čarnogurská, M. Přihoda, M. Kubík Acta Polytechnica

Figure 2. Dependence of the volume expansion coefficient of the 30% sugar solution on temperature.

Figure 3. Temperature field at ∆tN = −8 °C.

ratio of the sugar weight to the total solution weight,
is about 30%. The dynamic viscosity of this solution
has the value of 3.188 ·10−3 Pa s at the temperature of
20 °C and 2.5 · 10−3 Pa s at the temperature of 30 °C.
The temperature dependence of the dynamic viscos-
ity of liquids may be described [2] by exponential
dependence

η = AeB/T . (4)

The value of the following constants, A = 2 · 10−6 Pa s
and B = 2160.4K, has been obtained thanks to a
logarithmic calculation of (4), and by solving a set of
two linear equations.

The thermal conductivity of the solution is linearly
dependent on the concentration c (kg/kg) [3]:

λ = Dc+ E. (5)

The variables D and E are functionally dependent on

Figure 4. Heat flux at ∆tN = −8 °C.

the temperature of the solution, in accordance with

D = 5.466 · 10−8t2 − 1.176 · 10−5t− 3.024 · 10−3,
(6)

E = −7.847 · 10−6t2 + 1.976 · 10−3t+ 0.563. (7)

The thermal capacity of the solution in the range of 0
to 90 °C is shown in Figure 1 [4].
The volume expansion of the sugar solution may

be defined from the known density dependence on
temperature. It is determined by solving the following
equation numerically:

β = −1
%

( ∂%
∂T

)
p
. (8)

The dependence of the coefficient β of the used solu-
tion on temperature in the range from 0 to 70 °C is
plotted in Figure 2.

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vol. 55 no. 3/2015 Simulation of Heat Transport

Figure 5. Dependence of the cooling performance on the temperatures of the cooling mixture and of the stum.

In the T1400 and T930 tanks, cooling panels with
similar channel arrangements are used for the cooling
mixture transport, but the panel of the T930 tank has
half the total dimension and channel length compared
to the T1400 tank panel. Taking these facts into
account, the total cooling performance of 24.5 cooling
panels is considered (22 panels of the T1400 type +
0.5 × 5 panels of the T930 type). The temperature
field of the cooling panel in the axis section is seen
in Figure 3 at ∆tN = −8°C. The distribution of heat
flux on the surface of the panel at above specified
boundary conditions is shown in Figure 4.
The course of another investigated dependence of

the cooling performance of all the panels on the tem-
perature difference of the cooling mixture and the
stum is shown in Figure 5.
The regression line describing the cooling perfor-

mance must cross zero because if the mixture and the
stum have the same temperature, the cooling perfor-
mance is zero. The regression dependence equation
for a performance ranging from 0 to 30 kW is in the
form of

Pch = d∆tN = −2000∆tN. (9)

This equation shows that in order to achieve a cooling
performance of 27.5 kW, it is necessary to bring the
cooling mixture to a temperature which is lower by
13.75 °C than the temperature of the stum (at the
stum temperature of 0 °C, the cooling mixture will
have the temperature of -13.75 °C).

3. JN30 device cooling capacity
calculation

The cooling device takes off the thermal output (Pch)
from the tank, including the heat exchange with the
environment (Pok), and at the same time the heat
taken from the tank material and from the stum (Pak)
– Figure 6.

Figure 6. Stum tank scheme with thermal flows.

The thermal balance of this system may be written
in a time unit as the following equation:

Pok − Pch = Pak (10)

The thermal flow coming from the environment
into the tank is given by the Newtonian relationship
and the accumulated performance is described by a
calorimetric equation concerning the elementary time
change [5–7]:

nαstrS(tok − t) − Pch =
∑

(mc) dt
dτ . (11)

where αstr is the average value of the transfer coeffi-
cient at the surrounding temperature of 8 °C and in
the interval of the tank surface temperatures of 0 to
22 °C (Wm−2 K−1); S is the external heat exchanging
surface of the tank (m2); tok is the environment tem-
perature (°C); t – stum temperature in time τ (°C); n
– number of cooling panels (1); Pch – device cooling
performance (W); m – cooled material weight (kg);

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T. Brestovič, M. Čarnogurská, M. Přihoda, M. Kubík Acta Polytechnica

Figure 7. Course of the stum temperature, depending on time.

Figure 8. Dependence of excessive cooling of the cooling mixture on the stum temperature.

c – specific thermal capacity of the cooled material
(J kg−1 K−1). The item of

∑
(mc) in relationship (11)

represents the total capacity of the stum, the stainless
tank and the cooling panel material.
If the equation tv1 = ∆tN + t applies, the differen-

tial equation which describes the stum temperature
dependence on time, can be obtained by combining
(1) and (11)

As there is the following equation: tv1 = ∆tN + t,
this is the connection of relationships (1) and (11), a
differential equation describing the stum temperature
dependence on time, which may be constructed in the
following form:

dτ =
∑

(mc)
nαstrStok − nαstrSt− at− a∆tN − b

dt. (12)

If the thermal losses in the distributions are omitted,
the thermal performance, taken by all the panels,
must be equal to the performance of the JN30 cooling
device. This shows the uniformity of relationships
(1) and (9), on the basis of which you can define the

functional dependence between ∆tN and the stum
temperature t in the following form:

∆tN = a

d− a
t+ b

d− a
. (13)

After inputting ∆tN from relationship (13) to (12),
we get

dτ =
∑

(mc)
nαstrStok − nαstrSt− at− a a

d−a t+ b
d−a − b

dt.

(14)
By solving the differential equation (14), we get the
resulting relationship for the calculation of the depen-
dence of the stum temperature change on time, whose
solution is in the following form:

τ = 1
nαstrS + a+ a2

d−a

∑
(mc)

ln
(
nαstrStok− ab

d−a −b
)

−
(
nαstrS+a+ a2

d−a

)
t1(

nαstrStok− ab
d−a −b

)
−
(
nαstrS+a+ a2

d−a

)
t
,

(15)

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vol. 55 no. 3/2015 Simulation of Heat Transport

where t1 is the stum temperature at the beginning of
the cooling process (°C).
The stum cooling time according to (15) was cal-

culated for the following parameters: number of
tanks n = 24.5pcs, αstr = 2.853Wm−2 K−1, surface
S = 11.875m2, coefficient of linear regression from
(1) a = 938.98WK−1, coefficient of linear regression
from (1) b = 15380W, coefficient of linear regression
from (9) d = −2000WK−1, stum volume in one tank
V = 3.23m3, stum density % = 1400 kgm−3, steel
weight moc = 148.9 kg (weight of the steel tank coat-
ing and weight of the cooling panel), specific thermal
capacity of steel coc = 465 J kg−1 K−1, temperature
of environment tok = 8°C, stum thermal capacity
c = 3460 J kg−1 K−1, initial temperature of the cooled
stum t1 = 22 °C. The time course of the stum is shown
in Figure 7.

On the basis of the described model, we may calcu-
late the dependence of excessive cooling of the cooling
mixture on the stum temperature (Figure 8).

Figure 8 shows that at a stum temperature of 0 °C,
the temperature of the cooling mixture is -5.2 °C. At
this temperature, frost may form on the heat exchang-
ing surface, which significantly increases the thermal
resistance of the cooling panel and lowers the stum
cooling intensity. An increase of the cooling mixture
temperature can be achieved for example by increasing
the heat exchanging surface of the cooling panel.

4. Conclusion
The methodology for determining cooling parameters
using a combination of a numeric and an analytical
calculation enables us to describe the dynamic be-
haviour of the cooling system precisely. With the help
of the described model, it is possible to determine
the cooling performance, the cooling factor and the
excessive cooling of the cooling medium not only in
relation to the temperature of the cooled solution, but
also to the changes of these parameters over time.
The executed thermal calculations show that the

analysed cooling device cannot cool 78.21m3 of stum
in 48 hours from the initial temperature of 22 °C to

0 °C, but that it makes it possible, within the given
timeframe, to cool the stum to the temperature of
10 °C (arrows in Figure 7). At the moment, the cooling
aggregate is only fully sufficient for the cooling of
the given amount of stum to the temperature of the
managed fermentation, which is approximately 7 °C.

The requirement of cooling the stum to the temper-
ature of approximately 0 °C in 2 days will be fulfilled
if the performance of the cooling aggregate increases
from the current 27.5 kW to approximately 47 kW. Due
to the large temperature difference between the cool-
ing mixture temperature and the stum temperature
at the performance of 47 kW, it might be appropriate
to also increase the total heat exchanging surface of
the cooling panels.

Acknowledgements
This article was written with the support of VEGA
projects 1/0004/2014 MŠ SR, VEGA 1/0686/13 MŠ SR
and SP2015/86-FMMI VŠB TUO.

References
[1] VDI Wärmeatlas, GVC, Berlin, 2006, 10th edition,
ISBN-10 3-540-25504-4.

[2] HEINDRICH J. Vlastnosti tekutín. SNTL Alfa, 1980.
[3] WERNER, M., BAARS, A., WERNER, F., EDER, C.,
DELGADO, A. Thermal Conductivity of Aqueous Sugar
Solutions under High Pressure, International Journal of
Thermophysics (2007) 28, 1161–1180, ISSN 0195-928X.

[4] Material of Nordic Sugar company: The functional
properties of sugar.

[5] HIMMEL, M. On the Solvability of Some Partial
Differential Inequality, Acta Polytechnica, Vol. 53, No 3
(2013).

[6] DURANSKY, P., PAPUCIK, S., JANDACKA, J.,
HOLUBCIK, M., NOSEK, R. Design of Heat Exchanger
for Ericsson-Brayton Piston Engine. The Scientific World
Journal (2014), ID 138254, doi:10.1155/2014/13825

[7] PŘÍHODA, M., MOLÍNEK, J., PYSZKO, R.,
VELIČKA, M., VACULÍK, M. a BURDA, J. Heat
Transfer during Cooling of Hot Surfaces by Water
Nozzles. Metalurgija = Metallurgy. 2009, 48(4), 235-238.

145

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	Acta Polytechnica 55(3):140–145, 2015
	1 Introduction
	2 Heat transfer in the cooling panels
	3 JN30 device cooling capacity calculation
	4 Conclusion
	Acknowledgements
	References