##### Аннотация

Sucrose crystallization depends on various thermal phenomena, which makes them an important scientific issue for the sugar industry. However, the rationale and theory of sucrose crystallization still remain understudied. Among the least described problems is the effect of time and temperature on the condensation rate of sucrose molecules on crystallization nuclei in a supersaturated sugar solution. This article introduces a physical and mathematical heat transfer model for this process, as well as its numerical analysis.The research featured a supersaturated sugar solution during sucrose crystallization and focused on the condensation of sucrose molecules on crystallization nuclei. The study involved the method of physical and mathematical modeling of molecular mass transfer, which was subjected to a numerical analysis.

While crystallizing in a vacuum boiling pan, a metastable solution went through an exothermal reaction. In a supersaturated solution, this reaction triggered a transient crystallization of solid phase molecules and a thermal release from the crystallization nuclei into the liquid phase. This exogenous heat reached 39.24 kJ/kg and affected the mass transfer kinetics. As a result, the temperature rose sharply from 80 to 86 °C.

The research revealed the effect of temperature and time on the condensation of solids dissolved during crystalline sugar production. The model involved the endogenous heat factor. The numerical experiment proved that the model reflected the actual process of sucrose crystallization. The obtained correlations can solve a number of problems that the modern sugar industry faces.

##### Ключевые слова

Vacuum boiling pan, sucrose, phase, metastable solution, heat, dissolution, condensation, crystallization##### ВВЕДЕНИЕ

Vacuum boiling pans are an essential component of sugar and starch production. A vacuum pan is a crystallizer filled with a liquid solution of sucrose, salts, or other substances.

A metastable liquid solution behaves like a homogeneous liquid. If it is oversaturated, a thin suspension or a solid phase introduced into the crystallization nuclei can trigger a rapid and powerful thermal reaction. This reaction turns the homogeneous solution into a heterogeneous liquid system called massecuite.

The thermal release during crystal formation is caused by two factors. On the one hand, the force of attraction accelerates the flow of sucrose molecules to the crystallization nuclei. On the other hand, when the molecule clusters stop on the surface of the crystallization nucleus, the accumulated kinetic energy is spent on embedding the molecules into the crystal lattice, as well as on internal energy. As a result, molecules get accumulated on the crystallization nucleus, and this process is known as crystallization of sucrose in a vacuum pan.

In the sugar industry, energy production relies on all physical forms of thermal energy of water, be it liquid or vaporized. Thermal equipment turns water into steam, which acts as the main heat generator to obtain sugar or sugar products. After that, the steam serves as a heater and evaporates moisture from another heterogeneous liquid system, e.g., beet juice. The steam can also go into a new physical state: it settles on the cooled solid walls of the equipment, turns into a liquid, and releases the heat.

This phenomenon illustrates the law of energy conservation. Water molecules move in this gaseous medium and settle down on the equipment walls. As vapor transforms into liquid, it releases thermal energy, which is a powerful and efficient reaction. As a result, the temperature inside the environment rises, which makes steam the main source of thermal energy in sugar production.

The same law of energy conservation is responsible for crystallization, which occurs in a supersaturated sugar solution when the distance between the crystallization nuclei becomes small enough to trigger the forces of attraction between sucrose molecules. Hence, crystallization happens when sucrose molecules concentrate on the surface of the crystallization nuclei.

A lot of studies concentrate on the scientific and technical issues of metastable and supersaturated solutions because these phenomena are crucial for sugar production technology [1–19].

For instance, Saifutdinov *et al*. focused on the effect
of various organic solvents on the molar changes in
the Gibbs energy, enthalpy, and entropy during adsorption
[1]. They established the role of intermolecular
interactions in the solution and at the phase boundary.

In another article, Saifutdinov *et al*. reported the
adsorption thermodynamics for some 1,3,4-oxadiazoles
and 1,2,4,5-tetrazines from water-acetonitrile and watermethanol
solutions on the surface of porous graphitized
carbon at 313–333 K [2]. The absolute values of the
change in the Gibbs energy and enthalpy increased
during the adsorption from water-organic solutions as
the surface area of adsorbate molecules became larger,
the absolute values of the change in entropy decreased,
and the Van der Waals volume of molecules increased.

Makhmudov *et al*. calculated the thermodynamic
parameters for the phenol and sulfonol sorption from
wastewater on activated carbon and anion exchanger [3].

Sagitova *et al*. described the sorption of cobalt
ions by native and modified organic pharmacophores
of pectins [4]. They determined the effect of acidity,
temperature, and solution/sorbent module on the
distribution of cobalt ions in the heterophase system
of polysaccharide sorbent and aqueous solution. This
research also revealed the effect of various biosorbents
on the thermodynamics of cobalt ions.

Sharma *et al*. used the method of isothermal
microcalorimetry to determine the dilution enthalpy of
fluorosiloxane rubber and polychloroprene solutions in
various organic liquids [5]. The dissolution processes
of polychloroprene were accompanied by exothermic
processes, while those of fluorosiloxane rubber – by
endothermic ones.

Sayfutdinov and Buryak applied liquid chromatography to study the adsorption of isomeric dipyridyls and their derivatives from aqueous acetonitrile, aqueous methanol, and aqueous isopropanol solutions on a graphite-like carbon [6].

Fedoseeva and Fedoseev proved that size changes the state and physicochemical properties of dispersed systems in small (nano-, pico-, femtoliter) volumes [7]. The scientists used digital optical microscopy to interpret the concepts of chemical thermodynamics. Their experiments established the effect of such geometric parameters as radius and contact angle on the kinetics of phase and chemical transformations. The research featured polydisperse accumulations of droplets in organic and water-organic mixes that interacted with volatile gaseous reagents.

Other publications reported on the kinetics, mechanism, and heat of crystallization processes [8–15]. Some of them [8–10] focused on phase thermal effects in the sugar industry based on the laws of thermodynamics and the Gibbs theory.

Jamali *et al*. studied such independent kinetic
factors as thermodynamics and sucrose crystal transfer
that occur in an aqueous sugar solution during
crystallization [16]. They used high-precision tools and
scaling to prove that the experimental results confirmed
the precalculated fluid densities, thermodynamic factors,
shear viscosity, self-diffusion coefficients, and the Fick
diffusion coefficients.

Li *et al*. described a modern view on crystal
nucleation [17]. Traditional physical organic chemistry
always combined kinetics and thermodynamics to
study crystallization. The authors studied sucrose and
p-aminobenzoic acid to show how solution chemistry,
crystallography, and kinetics complement each other to
provide a complete picture of all nucleation processes.

Kumagai proved the effect of the water sorption isotherm on the interaction of water and solids in food products [18]. In thermodynamics, the Gibbs free energy (ΔGs) describes the interaction of a solid substance and water. Therefore, the plasticizing effect of water on food products can be evaluated by applying the Gibbs free energy.

Ebrahimi *et al*. studied a mix of 1-butanol + water
with or without sugars and their effect on clouding [19].
This experiment established that 1-butanol + water
solution fortified with sucrose or alcohol reduced
clouding.

These publications give a thorough account of phase transition of liquid to vapor and back, but they provide a poor quantitative assessment of the heat released or absorbed in each case.

The present paper introduces the thermal problem of heat propagation in the intercrystal solution volume adjacent to crystallization nuclei (instantaneous heat source).

##### ОБЪЕКТЫ И МЕТОДЫ ИССЛЕДОВАНИЯ

The research featured a supersaturated sugar solution in a vacuum boiling pan under the conditions of industrial sugar production.

The methods included physical and mathematical modeling of heat and mass transfer in heterogeneous liquid systems.

**Modeling.** Heat transfer in a vacuum pan is a
difficult task for physical and mathematical modeling, while its numerical calculation provides a scheme that
reflects the actual process [13].

The modeling relied on the assumption that
crystallization nuclei are uniformly distributed in the
vacuum pan. Therefore, the calculations relied on the
spherical symmetry of the liquid + solid mix relative to
center *O* in the region of 0 < *r* < *R*, where *r* is the radius
of the model sphere and *R* is the average radial distance
between the spheres (Fig. 1).

The boundary value problem was based on the theory
of thermal conductivity for an isolated model particle
of sucrose near the crystallization nucleus. A certain
volume of intercrystalline solution was represented as
a spherical region with radius *R* and center point *O* at
saturation temperature *Т*_{s}. The volume included a model
sucrose particle represented as a sphere with radius
*r* = *r*_{1} and center *O*. The initial instantaneous heat source
distributed over spherical surface *r* = *r*_{1} (Fig. 1) with
force *Q*_{1} (J). Heat exchange occurred in accordance with
the boundary condition of the third kind between sphere
surface *r* = *R* and its environment. The task was to find
the temperature field in the region of 0 < *r* < *R* and the
average temperature of the medium over time.

The heat transfer equation looks as follows:

where *Т*(*r*,*τ*) is the temperature, K; *τ* is the time, s; and
*а* is the thermal diffusion coefficient, m^{2}/s.

The initial data include:

where

temperature difference between sphere surface *r* = *r*_{1}
and the environment, K; *Q*_{sp} is the specific heat of
crystallization, J/kg; and *с*_{0} is the heat capacity of the
solution, J/(kg⋅ K).

Boundary conditions:

where *T*_{0} and *T*_{1} are the initial temperature (*К*) of the
environment (massecuite) and the temperature on
sphere surface *r* = *R*, m, respectively; *Н* = *α*/*λ*, α is
the thermal diffusion coefficient, Vt/(m^{2}⋅K); and *λ* is
the thermal conduction coefficient, Vt/(m⋅K).

If we introduce the following substitution

the boundary problem (1)–(5) looks as follows:

where *t*(*r*,*τ*) is the reduced temperature, Δ*t* = *T*_{0} – *T*_{1}, and
*δT* is defined according to (3).

Boundary problems (7)–(10) are based on the following correlation [20]:

*b* = *V***t*(*r*, 0), *V* = 4*π**r*_{1}^{3}/3, sucrose crystal volume, m^{3};
*t*(*r*, 0) as in (8), K; and *μ*_{1} and *μ*_{2}, are the roots of the
characteristic equation:

where *Вi* = *αh*/*λ* – Biot number (thermal), *F _{o}* =

*ατ*/

*R*

^{2}– Fourier number [20]. According to (11),

where *А _{n}* is the table coefficients [20].

Formula (6) provides the following solution for (7)–(10):

where *t*(*r*,*τ*) is the calculated according to (13).

Mean temperature 0 < *r* < *R* is calculated as follows:

where function *T*(*r*,*τ*) under the integral depends on
correlation (14).

The temperature and the mean temperature in
the vacuum pan depend on the processing time and
are calculated based on correlations (14) and (15).
As follows from the assumption about the uniform
distribution of the crystallization nuclei, the calculated
thermal characteristics for the selected elementary
volume with radius *R* are also valid for the entire volume
of the vacuum pan.

##### РЕЗУЛЬТАТЫ И ИХ ОБСУЖДЕНИЕ

The initial data included: crystal radius
*r*_{1} = 1×10^{–5} and 2×10^{–5} m; volume concentration *с* = 40
and 50% (*с* = 0.4, 0.5); density of intercrystalline
solution (massecuite) *ρ* = 1450 kg/m^{3}; thermal
conduction and diffusion coefficient (for water at
80°С), respectively, *λ* = 0.56 Vt/(m⋅°С), *с*sub>0 = 1250 J/(kg⋅K),
heat transfer coefficient *α* = 240 Vt/(m^{2}⋅°C) [10, 21].

The resulting thermal diffusion coefficient is
*а* = *λ*/(*с*sub>0⋅*ρ*) = 3.09×10^{–7} m^{2}/s. The equivalent radius of
elementary volume was calculated as follows:

Biot number *Bi* = *α*⋅*r*_{1}/(*λ*⋅*с*^{1/3}).

The specific heat of sucrose crystallization was
as in [13]: *Q*_{sp} = 13.42 kJ/mol (39.24 kJ/kg).

The numerical simulation was based on MATHCAD software.

Sum (13) was calculated based on (12)–(16) with
the same four additive components, while the
parameters of *A*_{n} and *μ*_{n} in (13) were based on the tables
published in [20].

Temperatures *T*_{0} and *T*_{1} were 80°С all the time,
which means that Δ*T* (9) = 0.

Figures 2 and 3 show the calculation results at
the accepted values of the thermal process: volume
concentration c of the solid phase in the solution,
time *τ*, and temperature *T* on surface *r*_{1} for
model sucrose particle and mean massecuite
temperature *T*.

Figures 2 and 3 show that the heat transfer into the sugar solution during crystallization of the model sucrose particle proceeded very quickly and took some thousandths of a second. That was why the thermal regime in the intercrystalline solution stabilized so quickly.

Figures 2 and 3 also demonstrate the same
gradual exponential decrease in temperature, which
is typical for heat transfer problems. If particles
differed in radius by a factor of two, smaller particles
with a larger specific surface area and a greater
heat transfer cooled faster than particles with a
larger radius. For curves 1 and 2, the temperature
rise rate of the particles with radius *r*_{1} = 1×10^{–5} m
exceeded curves 3 and 4 for particles with a radius
twice as large. The accumulation and release
of heat for crystals with radius *r*_{1} = 2×10^{–5} m
was eight times bigger than those for crystals with a
radius two times smaller. Figure 3 clearly demonstrates
that curves 3 and 4 are much higher than
curves 1 and 2.

##### ВЫВОДЫ

The equation of non-stationary Fourier diffusion with initial and boundary conditions of the third kind was applied to calculate the endogenous heat released into the solution during the condensation of sucrose molecules on a spherical particle of a sucrose crystal in a supersaturated sugar solution.

The numerical study involved conditions close to
the actual sucrose crystallization process in a vacuum
boiling pan. It revealed an increase in temperature as a result of the phase transition from 80 to 86°С in 2×10^{–3} s,
which means the process was almost instantaneous.
The calculations were confirmed in practice.
The results can facilitate calculating the effect of
temperature on massecuite viscosity, wash water
temperature, and other characteristics of massecuite
vacuum processing in the sugar and starch
industries.

##### Вклад авторов

E.V. Semenov and A.A. Slavyanskiy supervised the project. D.P. Mitroshina and N.N. Lebedeva performed the experiments.##### КОНФЛИКТ ИНТЕРЕСОВ

The authors declare that there is no conflict of interests regarding the publication of this article.##### СПИСОК ЛИТЕРАТУРЫ

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