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Volume 2 Issue 12, November 2022 ISSN 2181-2020
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Volume 2
№
8
ISSN: 2181-2020
Volume 2 Issue 12 (2022): EJAR
Volume 2 Issue 12 (2022): EJAR
DIFFERENTIAL EQUATIONS OF HEAT AND MASS
TRANSFER IN THE DRYING PROCESS OF RICE GRAIN
Ergashev Sharibboy Tulanovich
Namangan State Technical University
Otakhanov Bakhrom Sadirdinovich
Namangan State Technical University
Abdumannopov Nasimjon Abdulkhakimovich
Namangan State Technical University
nasimjonabdumannopov@gmail.com
https://doi.org/10.5281/zenodo.15262275
ARTICLE INFO
ABSTRACT
Received: 18
th
April 2025
Accepted: 21
st
April 2025
Online: 22
nd
April 2025
,
The article describes the peculiarities of rice grain, analyzes
differential equations of heat and moisture transfer, and develops
recommendations for calculating heat and moisture transfer
during drying of rice grain.
KEYWORDS
Structure of rice grain,
coefficients and gradients
of heat and moisture
transfer
INTRODUCTION
The unique structure of the rice grain requires many features to be taken into account in
calculating its heat and moisture. According to the structure of the rice grain , there is a rice, a
husk, and an air layer between them (Figure 1). When rice is dried convectively, the heat
provided by the heat transfer agent passes through the husk to the air layer, and then reaches
the rice. To bring the rice grain to the storage moisture content, the moisture content must be
brought to 12-13 percent. To do this, the previously described heat and moisture transfer
process must be carried out in reverse order. Now the amount of heat supplied to the rice grain
must be increased or the amount of heat supplied must be maintained. This situation leads to a
sharp increase in energy and time consumption.
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1-heat and moisture transfer gradient in the rice; 2-heat and moisture transfer gradient
in the air layer; 3-heat and moisture transfer gradient in the shell
Figure 1. Rice grain structure, heat and moisture transfer gradients
If we take into account the differences in heat and moisture transfer of the
aforementioned organizers, it is not difficult to imagine that this process would take a long time.
During the drying process
In the equation for heat transfer in capillary pores, the
convective component is replaced
𝑐
𝑖
𝑗
𝑖
∇𝑇
by the conductive component.
𝑑𝑖𝑣(𝜆∇𝑇)
several times
smaller than.
This result can be reached based on the following analysis. The total
(𝐺𝑟 ∙ Pr) < 1 ∙ 10
3
heat transfer coefficient in a dispersed (two or more phase) medium is is equal to the molecular
heat transfer coefficient, or heat transfer occurs by thermal conductivity.
(𝐺𝑟 ∙ Pr) < 1 ∙ 10
3
The
value
𝑅𝑒
∋
= 22
corresponds to the equivalent criterion. We estimate the value of S h . When
drying is carried out in convective dryers, the moisture transfer rate is approximately
40
𝑘𝑔
𝑚
2
∙𝑠
equal to. Under the most favorable conditions, the equivalent diameter of the capillary
𝑑
∋
is = 3
mm, or the porosity of the div is 70%. Viscosity of water at a temperature of
30° C
𝜂
2
2.88
𝑘𝑔
𝑚∙𝑠
is equal to . Then the criterion is equal to,
𝑅𝑒
∋
=
𝑗
2
𝑑
∋
𝜂
2
𝑃
=
40∙3∙10
−3
0.7∙2.88
≈ 5.2 ∙ 10
−2
,
or several times smaller than 22. For this reason
∑ 𝑐
𝑖
𝑗
𝑖
∇𝑇
𝑖
, it
𝑑𝑖𝑣(𝜆∇𝑇)
can be ignored
when considering .
DISCUSSION AND METHODS
Due to the absence of an overall pressure gradient, the system of differential equations
for heat and moisture transfer takes the following form (Figure 1) :
heat and moisture transfer
𝜕𝑢
𝜕𝜏
= 𝑘
11
∇
2
𝑢 + 𝑘
12
∇
2
𝑇
(1)
𝜕𝑇
𝜕𝜏
= 𝑘
22
∇
2
𝑇 + 𝑘
21
∇
2
𝑢
(2)
this is it odds is defined as follows
𝑘
11
= 𝑎
𝑚
, 𝑘
12
= 𝑎
𝑚
𝑇
= 𝑎
𝑚
∙ 𝛿, 𝑘
22
= 𝑎 + 𝑎
𝑚
1
𝑇
∙
𝑟
12
𝑐
, 𝑘
21
=
𝑎
𝑚1
∙
𝑟
12
𝑐
(1) and (2) is more generalized and is valid not only for the drying processes of wet
materials, but also for arbitrary forms of heat and mass transfer.
The drying process is a typical heat and mass transfer process in motion. The source of
moisture in it
𝐼
2
= −𝐼
1
𝜕𝑢
𝜕𝜏
can be expressed by the local humidity at time.
The total change in div moisture
𝑑
𝑒
𝑢
is equal to du, the change in moisture transfer and
𝑑
𝑖
𝑢
ga occurs due to the conversion of liquid into vapor, or,
𝜕𝑢 = 𝑑
𝑒
𝑢 + 𝑑
𝑖
𝑢
. (3)
In this case, div moisture is equal to the specific gravity of the fluid in the div
(𝑢 =
𝑢
1
+ 𝑢
2
= 𝑢
2
)
, or in the system under consideration
𝑖 = 1.2,𝑢
1
= 0
.
If the moisture transfer process is in motion
(𝑑𝑢 ≠ 0)
, then
𝑑𝑢
𝑖
/𝑑𝑢
the ratio is the final
value that determines the relative change in humidity due to evaporation of moisture at a given
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point on the div. This value is called the phase change coefficient of the transformation of the
liquid into vapor and is defined as:
𝜀 =
𝑑
𝑖
𝑢
𝑑𝑢
(4)
If
𝑑
𝑖
𝑢 = 0
, then the coefficient
𝜀 = 0
, or indicates that moisture transfer occurs only by
liquid transfer; if there is no liquid transfer (
𝑑
𝑒
𝑢 = 0
), or if the change in moisture at an
arbitrary point of the div occurs by evaporation, the coefficient is equal to one. Thus, in the
general case, the coefficient varies from zero to one,
(0 ≤ 𝜀 ≤ 1)
and the differential equations
for heat and mass transfer take the following form:
𝜕𝑢
𝜕𝜏
= 𝑎
𝑚2
∇
2
𝑢 + 𝑎
𝑚2
𝛿∇
2
𝑇 + 𝜀
𝜕𝑢
𝜕𝜏
(5)
𝜕𝑢
𝜕𝜏
= 𝑎∇
2
𝑇 + 𝜀
𝑟
21
𝑐
𝜕𝑢
𝜕𝜏
(6)
Equation (5) can be reduced to the following form by equating it with (6):
𝜕𝑢
𝜕𝜏
=
𝑎
𝑚2
(1−𝜀)
[∇
2
𝑢 + 𝛿
2
∇
2
𝑇],
(7)
with the equation
𝜕𝑢
𝜕𝜏
= 𝑎
𝑚
(∇
2
𝑢 + 𝛿∇
2
𝑇),
(8)
and in the implementation of the following :
𝑎
𝑚
=
𝑎
𝑚2
1−𝜀
𝑣𝑎𝛿 = 𝛿
2
(9)
is correct for the most general case.
From here the phase shift coefficient takes the following form
𝜀 =
𝑎
𝑚1
𝑎
𝑚1
+𝑎
𝑚2
=
𝑎
𝑚1
𝑎
𝑚
. (10)
The system of equations for heat and moisture transfer (5) and (6) or (6) and (8) can be
written in the form of the following system of equations (1) and (2), but in this case
𝑘
𝑖𝑗
(𝑖 =
1,2; 𝑗 = 1,2)
the coefficients will be equal to:
𝑘
11
= 𝑎
𝑚2
(1 − 𝜀) = 𝑎
𝑚
;𝑘
12
=
𝑎
𝑚2
𝛿
2
1−𝜀
= 𝑎
𝑚
𝛿;
(11)
𝑘
22
= 𝑎 + 𝜀
𝑟
21
𝑐
𝑎
𝑚2
𝛿
2
1−𝜀
= 𝑎 + 𝜀
𝑟
21
𝑐
𝑎
𝑚
𝛿;
(12)
𝑘
21
= 𝜀
𝑟
21
𝑐
𝑎
𝑚2
1 − 𝜀
= 𝜀
𝑟
21
𝑐
𝑎
𝑚
Thus , the source of vaporous moisture in the drying process in motion
𝐼
1
is expressed by
the following ratio
𝐼
2
= −𝐼
1
= 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
. (13)
In that case,
𝜀
instead of the evaporation coefficient, the ice coefficient
𝜀
3
is obtained.
(13)
𝜀
allows us to express the ratio coefficient of the liquid flow with
|𝑗
2
|
the vapor flow
|𝑗
1
|
. In practice,
𝜌
0
𝜕𝑢
𝜕𝜏
we substitute the following expressions
𝜌
0
𝜕𝑢
𝜕𝜏
= −𝑑𝑖𝑣𝑗
1
− 𝑑𝑖𝑣𝑗
2
and the following we use equality
𝐼
2
= −𝐼
1
= 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
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and we have the following expression:
𝐼
2
= 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
= −𝜀(𝑑𝑖𝑣𝑗
1
+ 𝑑𝑖𝑣𝑗
2
) = −𝑑𝑖𝑣𝑗
1
(14)
from this
𝜀 =
𝑑𝑖𝑣𝑗
1
𝑑𝑖𝑣𝑗
1
+𝑑𝑖𝑣𝑗
2
= (1 +
𝑑𝑖𝑣𝑗
2
𝑑𝑖𝑣𝑗
1
)
−1
(15)
Vectors in one-dimensional tasks
𝑗
1
𝑣𝑎𝑗
2
parallel or opposite direction , then :
𝑑𝑖𝑣𝑗
2
𝑑𝑖𝑣𝑗
1
=
(
𝜕
𝜕𝑥
)𝑗
2
(
𝜕
𝜕𝑥
)𝑗
1
(16)
If
𝜀 = 𝑐𝑜𝑛𝑠𝑡
we assume that , then, naturally, the ratio (16)
|𝑗
2
|
|𝑗
1
|
= 𝑐𝑜𝑛𝑠𝑡
must be constant
and equal to , from formula (15) we obtain:
𝜀 =
|1|
|1|+|𝑗
2
|
( 17 )
For drying wet materials, the ratio (15) can be written as:
𝜀 =
𝑎
𝑚1
(∇
2
𝑢+𝛿
1
∇
2
𝑇)
𝑎
𝑚
(∇
2
𝑢+𝛿∇
2
𝑇)
(18)
we take the formula (9) as a polynomial
𝛿 = 𝛿
1
= 𝛿
2
, then
𝜀 =
𝑎
𝑚1
𝑎
𝑚
we obtain the equality,
which was previously derived above.
Thus,
𝜀
to derive the coefficient of heat and moisture transfer in the process of drying in
motion, it is necessary to follow the equation (9) or (17) when describing the liquid and vapor
flows. These equations are satisfied in the hygroscopic state of the wet material.
𝜀
Deriving the
coefficient using the equation (4) does not require a number of requirements to be met,
including constancy with respect to the coordinate.
Formula (17) is used as a basis for deriving differential equations for heat and moisture
transfer [L. 38].
From relation (17) the following expression follows
|𝑗
1
| =
𝜀
1−𝜀
|𝑗
2
|
. ( 19 )
As a result, we get the following expressions:
𝑗
1
= 𝐼
𝑛1
|𝑗
1
|;𝑗
2
= 𝐼
𝑛2
|𝑗
2
|
( 20 )
in
𝐼
𝑛1
𝑣𝑎𝐼
𝑛2
these n - separate vectors,
𝑗
1
𝑖𝑗
2
directed along the vectors .
Then from the differential equation
𝜌
0
𝜕𝑢
𝜕𝜏
= −𝑑𝑖𝑣𝑗
1
− 𝑑𝑖𝑣𝑗
2
(21)
we get:
𝜌
0
𝜕𝑢
𝜕𝜏
= −𝑑𝑖𝑣𝐼
𝑛1
|𝑗
1
| − 𝑑𝑖𝑣𝑗
2
= −𝑑𝑖𝑣
𝜀
1 − 𝜀
|𝑗
2
|𝐼
𝑛1
− 𝑑𝑖𝑣𝑗
2
If
𝐼
𝑛1
= 𝐼
𝑛2
we accept that,
𝐼
𝑛1
𝑣𝑎𝐼
𝑛2
indicates that it is directed in one direction, then
we have :
𝜌
0
𝜕𝑢
𝜕𝜏
= −𝑑𝑖𝑣
𝜀
1−𝜀
𝑗
2
− 𝑑𝑖𝑣𝑗
2
(22)
Then ,
𝜀
assuming that the coefficient (
𝜀 = 𝑐𝑜𝑛𝑠𝑡
) does not depend on the coordinate, we
obtain from (22):
𝜌
0
𝜕𝑢
𝜕𝜏
= −𝑑𝑖𝑣𝑗
2
+ 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
(23)
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this n we define the expression for the moisture source:
𝐼
2
= 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
(24)
𝐼
𝑛1
𝑣𝑎𝐼
𝑛2
equality of vectors
𝐼
𝑛1
= 𝐼
𝑛2
is applied to the drying process, then the vectors
are defined by the following expressions,
𝑗
1
= −𝑎
𝑚1
(∇𝑢 + 𝛿
1
∇𝑇)𝜌
0
; (25)
𝑗
1
= −𝑎
𝑚2
(∇𝑢 + 𝛿
2
∇𝑇)𝜌
0
(26)
and have the same direction, for this to happen
𝛿
1
= 𝛿
2
. This equality must be
𝛿
1
=
𝛿
2
takes place in the hygroscopic field .
is valid for moving heat and moisture transfer when
𝜕𝑢
𝜕𝜏
≠ 0
. For stationary heat and
moisture transfer
𝜀 =
𝑑
𝑖
𝑢
𝑑𝑢
→ ∞
, because
𝑑𝑢 = 0
. Therefore, the value of the source
𝐼
2
= ∞ ∙ 0
,
there is uncertainty.
To solve for the uncertainty, we use equation (5), from which we obtain:
𝜀
𝜕𝑢
𝜕𝜏
=
𝜕𝑢
𝜕𝜏
− 𝑎
𝑚2
𝛿
2
∇
2
𝑢 − 𝑎
𝑚2
𝛿
2
∇
2
𝑇
(27)
In the stationary case
𝜕𝑢
𝜕𝜏
= 0
, it follows that,
𝜀
𝜕𝑢
𝜕𝜏
= −𝑎
𝑚2
(∇
2
𝑢 + 𝛿
2
∇
2
𝑇) =
1
𝜌
0
𝑑𝑖𝑣𝑗
2
(28)
On the other hand, from equation (21)
𝜕𝑢
𝜕𝜏
= 0
, we get :
𝑑𝑖𝑣𝑗
2
= −𝑑𝑖𝑣𝑗
1
(29)
Thus , we obtain the expression for the heat source in steady flow
𝐼
2
= 𝜀𝜌
0
𝜕𝑢
𝜕𝜏
= 𝑑𝑖𝑣𝑗
2
= −𝑑𝑖𝑣𝑗
1
(30)
or the expression (15).
Naturally, the equation for the moisture source is
𝐼
2
= −𝑑𝑖𝑣𝑗
1
= 𝑑𝑖𝑣𝜌
0
(𝑎
𝑚1
∇𝑢 + 𝑎
𝑚1
𝑇
∇𝑇)
(31)
It is more general in moisture and heat transfer for moving and stationary states in moist
bodies.
Thus, the system of differential equations (1) and (2) remains the same, only the
coefficient
𝑘
𝑖𝑗
is determined by formulas (11) and (12).
In addition, the coefficient
𝜀
describes
the mode of moisture in motion and heat transfer
during the heating or cooling phase .
Heating intensity value
𝑚 = −(
1
𝑇
𝑐
−𝑇
)
𝜕𝑇
𝜕𝜏
. (32)
m
The value is the last value and is equal to
𝑚 =
𝑎
𝑅
𝑣
2
𝐵𝑖𝜓
. (32)
where n is
𝜓
the unevenness of the temperature field , from 0 to 1
(0 ≤ 𝜓 ≤ 1)
,
𝑅
𝑣
—
hydraulic radius of the div ,
𝐵𝑖
—
Bio criterion . In a stationary state
𝜕𝑇
𝜕𝜏
= 0
, and the value is
(
1
𝑇
𝑐
−𝑇
) = ∞(𝑇 = 𝑇
𝑐
)
. Naturally,
𝜀
the heating rate, like the coefficient, is a characteristic of the
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heat transfer in motion. However, the value of
m
does not mean that the heat transfer in motion
varies from 0
∞
to.
RESULTS AND CONCLUSIONS
It should be noted that the following coefficients are used to calculate the temperature
and humidity of the moving div during the drying process:
𝑎, 𝑎
𝑚
, 𝑎
𝑚
𝑇
, 𝜀
and the
thermodynamic c and r descriptions must be known. However, if the more general ratio (31) is
used, then the thermophysical description is as follows
𝑎, 𝑐, 𝑎
𝑚1
, 𝑎, 𝑎
𝑚
, 𝑎
𝑚
𝑇
, 𝑎
𝑚2
, 𝑎
𝑚
𝑇
serves.
Thus,
𝜀
introducing the coefficient reduces the number of parameters from 7 to 6. The system
of differential equations for moisture and heat transfer (1) and (2) remains the same in all cases,
only the expression of the coefficients when solving this system
𝑘
𝑖𝑗
has different values
depending on the thermophysical characteristics.
References:
1.
Лыков
А.
В.
Михайлов Ю.А., Теория тепло
-
и массопереноса
Госэнергоиздат, 1963
.
2.
Лыков
А.
В.
Теория сушки. М., «Энергия», 1968.
465
с.
3.
Sharibboyto‘Lanovich,
E., Sadriddinovich, O. B., Abdulxakimovich, A. N., & O‘Gli, A. A. A.
(2022). Sholi navlarining fizik-
mexanik xossalari. Механика и технология, 3(8), 86
-90.
4.
To‘Lanovich, E. S., Sadirdinovich, O. B., Rustamovich, Q. A., Abdulxakimovich, A. N., &
O‘G‘Li, S.
M. A. (2024). Sholi donini aerodinamik hususiyatlari.
Строительство и
образование,
3(5), 142-146.
5.
Эргашев, Ш. Т., Отаханов, Б. С., & Абдуманнопов, Н. А. (2021). МАЛОГАБАРИТНАЯ
ЗЕРНОСУШИЛКА ДЛЯ ФЕРМЕРСКИХ ХОЗЯЙСТВ.
Universum: технические науки
, (6-1
(87)), 55-58.
6.
Tolanovich, E. S., Sadirdinovich, O. B., Rustamovich, K. A., & Abdulkhakimovich, A. N.
(2021). New Technology for Drying Grain and Bulk Materials. Academic Journal of Digital
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