INTRODUCTION
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.
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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).
STUDY OBJECTS AND METHODS
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,
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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 Тs. The volume included a model
sucrose particle represented as a sphere with radius
r = r1 and center O. The initial instantaneous heat source
distributed over spherical surface r = r1 (Fig. 1) with
force Q1 (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; а is
the thermal diffusion coefficient, m2/s.
The initial data include:
where
temperature difference between sphere surface r = r1
and the environment, K; Qsp is the specific heat
of crystallization, J/kg; с0 is the heat capacity of the
solution, J/(kg⋅ K).
Boundary conditions:
where Т0 and Т1 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/(m2⋅K); λ is the
thermal conduction coefficient, Vt/(m⋅K).
If we introduce the following substitution
(6)
the boundary problem (1)–(5) looks as follows:
(7)
(8)
(9)
(10)
where t(r,τ) is the reduced temperature, Δt = Т0 – Т1, and
δТ is defined according to (3).
Boundary problems (7)–(10) are based on the
following correlation [20]:
(11)
– sucrose crystal volume, m3;
and t(r,0) as in (8), K; μ1 and μ2, are the roots of the
characteristic equation,
(12)
Вi = αh/λ – Biot number (thermal), Fo = ατ/R2 –
Fourier number [20].
According to (11),
(13)
where Аn is the table coefficients [20].
Formula (6) provides the following solution for
(7)–(10):
(14)
where t(r,τ) is the calculated according to (13).
Mean temperature 0 < r < R is calculated as follows:
(15)
where function T(r,τ) under the integral depends on
correlation (14).
Figure 1 Heat and mass transfer for sucrose crystallization in
a vacuum boiling pan
Sucrose crystal
Intercrystal solutian
Intercrystalline
solution
(1)
(2)
(3)
(4)
(5)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟
, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟
⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) 3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ 𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 ⁄ 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟
< 𝑅𝑅𝑅𝑅)
𝑟𝑟𝑟𝑟, 0) 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑉𝑉𝑉𝑉 = 4𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟1
3⁄3
𝑡𝑡𝑡𝑡g𝜇𝜇𝜇𝜇 = −𝜇𝜇𝜇𝜇⁄(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)
𝑉𝑉𝑉𝑉 = 4𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟1
3⁄3
𝑡𝑡𝑡𝑡g𝜇𝜇𝜇𝜇 = −𝜇𝜇𝜇𝜇⁄(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=
1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1
𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏
= 𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜
(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1
Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0
𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)
𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟
+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟1
∙ sin
𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅
𝑟𝑟𝑟𝑟
∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),
𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)
𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =
𝑏𝑏𝑏𝑏
4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅
∙ 􀷍
1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1
∞
𝑛𝑛𝑛𝑛=1
∙
[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵
∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2
𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)
𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =
3
𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟
𝑅𝑅𝑅𝑅
0
307
Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309
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.
RESULTS AND DISCUSSION
The initial data included: crystal radius
r1 = 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/m3 [10]; thermal
conduction and diffusion coefficient (for water at
80°С), respectively, λ = 0.56 Vt/(m⋅°С), с0 = 1250 J/(rg⋅K),
heat transfer coefficient α = 240 Vt/(m2⋅°C) [21].
The resulting thermal diffusion coefficient is
а = λ/(с0⋅ρ) = 3.09×10–7 m2/s. The equivalent radius of
elementary volume was calculated as follows:
R = r1⋅с–1/3 (16)
Biot number Bi = α⋅r1/(λ⋅с1/3).
The specific heat of sucrose crystallization was
as in [13]: Qsp = 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 Аn and μn in (13) were based on the tables
published in [20].
Temperatures Т0 and Т1 were 80°С all the time,
which means that ΔТ (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 Т on surface r1 for
model sucrose particle and mean massecuite
temperature Т.
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 r1 = 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 r1 = 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.
CONCLUSIONS
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
Figure 3 Correlation of mean massecuite temperature
Т with volume concentration с of the solid phase in the
solution and crystallization time τ (r1 = 1×10–5 m:
1 – с = 40%, 2 – с = 50%; r1 = 2×10–5 m: 3 – с = 40%,
4 – с = 50%)
Figure 2 Correlation of temperature Т on surface r1 of the
model sucrose particle with volume concentration с of the
solid phase in the solution and crystallization time τ
(r1 = 1×10–5 m: 1 – с = 40%, 2 – с = 50%; r1 = 2×10–5 m:
3 – с = 40%, 4 – с = 50%)
80
81
82
83
84
85
86
87
88
1 2 3 4 5
τ×10–3, s
T,°C
1 2 3 4
80
81
82
83
84
85
86
87
88
1 2 3 4 5
τ×10–3, s
T,°C
1 2 3 4
81
82
83
84
85
86
0.5 1.0 1.5
T,°C
τ×10–3, s
1 2 3 4
80
81
82
83
84
85
86
87
88
1 2 3 4 5
T,°C
1 2 3 4
81
82
83
84
85
86
0.5 1.0 1.5
T,°C
τ×10–3, s
1 2 3 4
308
Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309
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.
CONTRIBUTION
E.V. Semenov and A.A. Slavyanskiy supervised the
project. D.P. Mitroshina and N.N. Lebedeva performed
the experiments.
CONFLICT OF INTEREST
The authors declare that there is no conflict of
interests regarding the publication of this article.

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