ЖАНУ ПРОЦЕСІН  ЫҒЫСТЫРҒАН ПИРОЛИЗ ОТТЫҚ ЖАҒДАЙЫНДА

Доклады Национальной академии наук Республики Казахстан  
 
 
   
78  
REPORTS OF THE NATIONAL ACADEMY OF SCIENCES  
OF THE REPUBLIC OF KAZAKHSTAN 
ISSN 2224-5227 
Volume 3, Number 307 (2016), 78 – 86 
 
 
МЕТОД РЕЛАКСАЦИОННЫХ ЯДЕР В МАТЕМАТИЧЕСКОЙ 
МОДЕЛИ ПРОЦЕССОВ ПЕРЕНОСА И АГРЕГАЦИИ 
 
А.С. Муратов, А.М. Бренер, Л. Ташимов 
 
Южно-Казахстанский государственный университет, Шымкент, Казахстан 
Amb_52@mail.ru, brener@fromru.com, asm_59@mail.ru 
 
Аннотация.  В  статье  дан  обзорный  анализ  математических  моделей  для  описания  тепло-  и 
массопереноса  и  агрегационных  процессов  с  помощью  метода  релаксационных  ядер  переноса,  который 
открывает  новые  возможности  для  детального  изучения  влияния  иерархии  времен  релаксации  на 
интенсивность высокоскоростных и нано-масштабных технологических процессов.  
Ключевые  слова:  массо-  и  теплоперенос,  времена  релаксации,  релаксационные  ядра,  перекрестные 
эффекты, агрегация.  
 
THE RELAXATION KERNELS APPROACH  
TO MATHEMATICAL MODELS OF TRANSFER  
AND AGGREGATION PROCESSES 
 
A.S. Muratov, A. M. Brener, L. Tashimov 
 
South Kazakhstan State University, 
5, Tauke Khan, Shymkent, 160012, Kazakhstan 
 
Key words: Mass and Heat Transfer; Relaxation times; Relaxation kernels; Cross effects; Aggregation. 
Abstract.  The paper deals with mathematical models describing heat and mass transfer and aggregation 
processes with the help of relaxation transfer kernels approach, which opens up fresh opportunities for detailed study 
of influence of relaxation times hierarchy on the intensity of high rate and nano-scale technological processes. 
 
1. Introduction 
Consideration of relaxation times and long-range interaction of structural components of a medium is 
a great practical and theoretical problem [1–3] that is relevant in cases of high rate or nano-scale 
technological processes. The operation cycle of these processes is short, and the entire process may go on 
under the transient regime. In this connection, resources of effective controlling such processes are 
limited, and it is important to calculate correctly and select the best values of governing parameters.  
Problems of modeling both high rate and nano-scale processes are in touch with construction of 
equations with retarded or divergent arguments that reflects the actual mechanism of transfer phenomena 
in the medium modeled as a system of interacting oscillators with a set of partial frequencies and 
interaction potentials [2, 3]. 
At the same time, though realization of that investigation program is very tempting, it’s unlikely to 
promise near creation of the reliable engineering methodology for calculating heat and mass transfer 
processes. The alternative approach to the problem is the methodology of relaxation transfer kernels, 
which can be calculated from model evolution equations [3].  In a few articles before we elaborated upon 
this approach to modeling heat and mass transfer in high rate processes [3-8].  
 In this article we summarize briefly our results in the area of description of time nonlocality applied 
to heat and mass transfer and try to develop this approach for describing time nonlocality in aggregation 
processes.  We concentrate our attention upon a problem of equations structure, touching on the problem 

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79 
of analytical solutions of government equations in the lesser degree.   
 
2.  Mass and heat transfer equations  
2.1. Main concept 
Relaxation transfer kernels are the kernels of integral transformations that, in the statistical theory of 
dissipation processes, relate fluxes with thermodynamic forces [1]. The general structure of these relations 
for components fluxes in a multicomponent system according this methodology is like that [6] 
      
)
,
(
)
,
,
,
(
)
,
(
)
,
(
1
1
1
0
t
R
F
t
t
R
R
N
R
d
dt
t
R
J
t
R
J
k
ik
n
k
i
i
′
′
′
+
=
∑∫∫
=
.  
(1) 
Limiting one self to the time nonlocality in the multicomponent system, one can write expressions 
for the 
n
 linearly independent mass fluxes 
i
J
 of components and the heat flux 
h
J
as 
 
 
2
1
0
1
1
1
1
0
1
)
,
(
)
,
(
)
,
(
T
T
t
t
R
N
dt
T
t
R
t
t
R
N
dt
J
iT
t
n
k
k
ik
t
i
∇
−
−
⎟
⎠
⎞
⎜
⎝
⎛
∇
−
−
=
∫
∑∫
=
ν
, (2) 
 
2
2
0
2
1
2
2
0
2
)
,
(
)
,
(
)
,
(
T
T
t
t
R
N
dt
T
t
R
t
t
R
N
dt
J
TT
t
n
k
k
Tk
t
T
∇
−
−
⎟
⎠
⎞
⎜
⎝
⎛
∇
−
−
=
∫
∑∫
=
ν
,  
 (3) 
 
where 
i
ν
 is a chemical potential; 
R
- space coordinates; 
T
 - temperature; 
t
 - time. 
For a more compact description, let’s assume
1
1
−
≡
+
n
ν
. Then, in expressions (2), (3), one can 
replace the subscript 
h
 by  
1
+
n
 and write a unified form for the mass fluxes and heat flux in the 
multicomponent system. 
 
 
∑∫
+
=
⎟
⎠
⎞
⎜
⎝
⎛
∇
−
−
=
1
1
1
1
0
1
)
,
(
)
,
(
n
k
k
ik
t
i
T
t
R
t
t
R
N
dt
J
ν
.  
(4) 
 Let’s also introduce notation for the integral terms 
 
⎟
⎠
⎞
⎜
⎝
⎛
∇
−
=
∫
T
t
R
t
t
R
N
dt
I
k
ik
t
ik
)
,
(
)
,
(
1
1
0
1
ν
.   
(5) 
Now, instead of equations (2), (3) we get 
 
∑
+
=
−
=
1
1
n
k
ik
i
I
J
   
(6) 
 
For calculating the relaxation transfer kernels we can use various approximations which are based on 
information about the physical mechanism of the processes [1, 2].  However, the analyses of various data 
[1, 2] as well as our own experience [4-8] allow us to submit the heuristic unified model equation for 
relaxation kernels   
   
  
1
1
1
−
≠
=
−
∑
+
−
=
ik
n
i
k
k
k
ii
i
i
N
N
t
N
τ
τ
∂
∂
,      
(7) 
where, in order to be in agreement with the Onsager principle, it is assumed that 
ki
ik
τ
τ
=
.  
Of course, it’s impossible to warrant that form (7) is actually universal. But we shall consider 
equation (7) as the base model for our further constructions.   
The matrix of system (7) is symmetrical; therefore, all its eigenvalues are real. In this connection, 
solution (7) can be represented as the sum of the forward and cross terms of the transfer kernels [6]:  
   
∑
=
=
n
k
ik
i
N
N
1
,   
(8) 
where all items are real exponents and 
ki
ik
N
N
=
. 

Доклады Национальной академии наук Республики Казахстан  
 
 
   
80  
 As it is shown in [6] with the help of the above model we can infer the following relations for 
integrals (5):  
 
ik
ik
k
ik
ik
I
T
t
I
τ
ν
η
−
⎟
⎠
⎞
⎜
⎝
⎛
∇
=
∂
∂
,  
(9) 
 
where for isotropic medium we suppose [3] 
 
0
=
R
ik
∂
∂η
   
 (10) 
 
So then, as a result of the repeated differentiation of (4) up to derivatives of the (
1
+
n
)-th order, the 
following relationships are obtained (where for any function
Z
t
Z ≡
∂
∂
0
0
):  
 
 
∑
∑
∑
−
=
+
=
+
+
=
−
−
−
−
+
−
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∇
∂
∂
−
=
∂
∂
1
0
1
1
1
1
1
1
1
1
)
1
(
)
(
)
1
(
m
s
n
k
m
ik
ik
m
n
k
s
ik
k
ik
s
m
s
m
s
m
i
m
I
T
t
t
J
τ
τ
ν
η
 ,   
 (11)  
 
Thus, for each of the components we obtain a system which consists of  
)
1
(
+
n
 equations connecting 
the component flux with its derivatives up to 
)
1
(
+
n
 order inclusive. 
The matrices 
i
M
 of the obtained systems are not degenerate  
 
0
)
1
(
det
det
1
≠
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
−
m
ik
m
i
M
τ
.      
(12) 
In this connection, from the 
)
1
(
+
n
  equations that are linear relative to integrals 
ik
I
 , one can 
express all these integrals through the derivatives of fluxes 
i
J
 and then substitute the obtained 
expressions into equation (6).  
As a result, one can come to the linear differential equation of the 
)
1
(
+
n
th order for the fluxes of 
each of the components [6] 
 
0
,...,
;
,...
,
1
1
1
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
+
+
n
i
n
i
n
n
i
n
J
t
J
t
J
L
ν
ν
.     
       (13) 
where 
L
 is the linear operator.  
The succeeding deduction is based on the conservation laws: 
 
 
0
=
⋅
∇
+
∂
∂
i
i
J
t
ν
.   
 (14) 
Acting on expression (13) by the nabla operator and using equation (14), we can obtain the 
differential equation of the 
)
2
(
+
n
th time-order for the potential of each of the components 
 
 
0
,
,
;
,..,
;
,...,
)
(
,
)
(
2
1
2
1
1
1
2
2
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∇
∇
∂
∂
∂
∂
∂
∂
+
+
+
+
n
n
i
n
i
n
n
i
n
t
t
t
L
ν
ν
ν
ν
ν
ν
ν
K
.       
 (15) 
 
The nonlinear generalization of equation (4) can be represented in a nonlocal quadratic form with 
tensor kernels [6, 8] 

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81 
 
 
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∇
⎟
⎠
⎞
⎜
⎝
⎛
∇
−
−
−
−
⎟
⎠
⎞
⎜
⎝
⎛
∇
⋅
−
−
=
∑∑∫∫
∑∫
+
=
+
=
+
=
T
t
R
T
t
R
t
t
t
t
R
N
dt
dt
T
t
R
t
t
R
N
dt
J
p
n
k
k
ikp
n
p
t t
n
k
k
ik
t
i
)
,
(
)
,
(
:
)
,
,
(
)
,
(
)
,
(
2
1
1
1
2
1
)
2
(
1
1 0 0
2
1
1
1
1
1
)
1
(
0
1
ν
ν
ν
 ,  
(16) 
In the weakly nonlinear approximation, we can assume [8]  
 
 
)
1
(
)
1
(
)
2
(
ip
ik
ikp
N
N
N
ε
=
,        
 (17) 
 
where ε is the series expansion parameter.  
One can evaluate the small parameter 
ε
 as the ratio of the two Knudsen numbers that are calculated 
by two characteristic spatial scales for the elastic and inelastic molecular collisions, respectively [8]   
 
2.2. Examples  
As the first example let’s consider mass and heat transfer in two-component systems like high dilute 
solutions. In this case the cross fluxes may be disregarded [3, 4].  
Thus we use simplest form of relaxation kernels 
                                                 
)
)
(
exp(
)
,
(
)
,
(
1
1
i
i
i
t
t
t
R
t
t
R
N
τ
η
−
−
=
−
.                              (18) 
 
Relations for mass and heat fluxes read 
                                                    
)
,
(
)
,
(
1
1
1
0
1
1
t
R
t
t
R
N
dt
J
t
ν
∇
−
=
∫
,                                       (19) 
                                     
)
,
(
)
,
(
2
2
2
0
2
2
t
R
t
t
R
N
dt
J
t
β
∇
−
=
∫
.       
   (20) 
 
Thus, operating under the above methods applied to an isotropic media we are led to the following 
transfer equations of a hyperbolic type [3]: 
                                
 
ν
η
∂
∂ν
η
∂
∂
τ
∂
ν
∂
τ
2
1
1
1
2
2
1
ln
1
∇
+
⎟
⎠
⎞
⎜
⎝
⎛ −
=
t
t
t
,      
 (21) 
                            
    
β
η
∂
∂β
η
∂
∂
τ
∂
β
∂
τ
2
2
2
2
2
2
2
ln
1
∇
+
⎟
⎠
⎞
⎜
⎝
⎛ −
=
t
t
t
.        
  (22) 
 
The case of a non-isotropic media is also considered in [3]. 
Equations (21), (22) closely resemble transfer equations for media with memory that are presented in 
[2]. It’s easy to check also that under the exponential relaxation kernel the heat transfer equation (22) 
corresponds with the Maxwell – Kattaneo law:     
 
  
T
t
q
q
∇
−
=
+
λ
∂
∂
τ
,   
   (23) 
where 
                                             
2
2
2
1
η
∂
∂
τ
τ
τ
t
−
=
,  
)
1
(
1
2
2
2
2
T
t
η
∂
∂
τ
η
λ
−
=
.    
(24)   
 
Under the relaxation times that are far less than the observation time the above equations can be 
considered by the methods of singular perturbations.   
The next example we consider is mass and heat transfer in diluted two-component systems with 
allowing for cross effects like the thermal diffusion and the Soret effect [4]. Model system for the 

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82  
relaxation kernels reads 
 
  
1
1
−
×
−
+
−
=
τ
τ
∂
∂
h
m
m
m
N
N
t
N
,     
(25) 
                              
1
1
−
−
×
−
=
h
h
m
h
N
N
t
N
τ
τ
∂
∂
.    
 (26) 
 
The choice of signs in equations (25) and (26) is determined by the conditions of coupling distortions 
of the temperature and concentration fields [4].  
The solution of equations (25) and (26) can be written in the form 
                                                
mh
mm
m
N
N
N
+
=
, 
    (27) 
 
hm
hh
h
N
N
N
+
=
,       
  (28) 
Here: 
     
     
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
=
)
exp(
1
)
exp(
1
2
1
1
2
1
2
s
s
N
m
m
m
mm
λ
τ
λ
λ
τ
λ
λ
λ
η
,    
(29)     
 
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
=
)
exp(
1
)
exp(
1
2
1
1
2
1
2
s
s
N
h
h
h
hh
λ
τ
λ
λ
τ
λ
λ
λ
η
,  
(30) 
 
)]
exp(
)
[exp(
)
(
2
1
1
2
s
s
N
N
hm
mh
λ
λ
λ
λ
τ
η
−
−
=
=
×
×
,  
 (31) 
were 
2
1
 ,
λ
λ
 are eigenvalues of the system (25), (26). 
From the condition of damping the perturbations in quasi-equilibrium systems it is readily available 
that both eigenvalues should be negative. From this, one can obtain the inequality: 
                                          
   
h
m
τ
τ
τ
>
×
    
(32) 
 
The time dependence of cross transfer kernels has a maximum. This phenomenon is caused by the 
influence of thermal diffusion or the Soret effect [4, 8]. The peak of the time dependence of cross transfer 
kernels determines the period of increasing initial perturbations of the temperature and concentration 
fields.  This period is easily evaluated [8]: 
                          
    
2
1
1
2
)
ln(
λ
λ
λ
λ
τ
−
=
=
↑
cr
s
    
  (33) 
As it follows from (15) we can obtain equations of the 3rd time-order for heat and mass transfer in 
the considered case.  

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