Analytical Solution to One Problem of Cooling Fiber Mass

Mamatov Alisher Zulunovich; Jamshidbek Rakhmonov Turdaliyevich

doi:doi:10.11648/j.ajma.20261304.15

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Analytical Solution to One Problem of Cooling Fiber Mass

Mamatov Alisher Zulunovich

, Jamshidbek Rakhmonov Turdaliyevich^*

Published in American Journal of Mechanics and Applications (Volume 13, Issue 4)

Received: 24 September 2026 Accepted: 8 October 2026 Published: 26 October 2026

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Abstract

To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.

Published in	American Journal of Mechanics and Applications (Volume 13, Issue 4)
DOI	10.11648/j.ajma.20261304.15
Page(s)	84-89
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Heat Transfer, Cooling, Raw Cotton, Fiber Quality, Initial Temperature, Thermal Conductivity, Parabolic Type, Orthogonally of Function, Characteristic Equation

1. Introduction

During cooling, a complex, non-stationary heat and mass transfer process occurs in a cotton sack, determining its external and internal states. External processes are characterized by mass transfer from the sack's surface to the surrounding environment and heat exchange between the fiber mass and the environment.

The rate of heat transfer within the cotton sack is crucial for maintaining the quality of the fiber and seeds during cooling

[1-6]

. Therefore, we examined the problem of heat transfer within a cotton sack during cooling as a sphere.

2. Method of Research

The problem of heat transfer in a fibrous mass can be expressed as follows: A spherical body of radius R is given, possessing the property of isotropy, with a known initial temperature distribution.

Convective heat exchange occurs within the body between the fibrous mass and the air according to Newton's law. It is necessary to find the radial temperature distribution of the fibrous mass and air at any given time.

Then, based on the laws of thermodynamics (Fourier’s and Newton's laws of heat conduction, conservation of energy, etc.), this problem can be formulated as a system of parabolic differential equations

[7-15]

(1)

with initial

and boundary conditions

where

are the heat capacity, thermal conductivity, and density of the fibrous mass and air, respectivly;

is the heat transfer coefficient between the fibrous mass and air;

is the temperature of the fibrous mass, air, and the external environment, respectively; and

is the initial temperature of the fibrous mass

[16-18]

: .

Let's make a change of variables

Then, we obtain the problem

(2)

where

We seek the solution to problem (2) in the form

Let’s substitute them into the system (2)

Considering the orthogonality of the function

We will receive

Because,

So, we have a system of ordinary differential equations in the form

(3)

where

We seek the solution of the homogeneous system (3) in the form

Then, the characteristic equations have the form

From here, we find

Since

, we introduce the notation

The solution of the homogeneous system (3) has the form

where

(4)

3. Research Results

We write the solution of the inhomogeneous system as

(5)

Substituting into the equations, we obtain

Solving this system, we find

(6)

From the initial conditions, we find

Using the orthogonality condition

From here, we find

(7).

Therefore, the solution to problem (1) has the form

where

are calculated using formulas (5) - (7).

Below in Tables 1 and 2 are presented the calculation results obtained for the following values of the parameters of the model under consideration

[15]

For fiber:

;

Table 1 shows the experimental and calculated data on cooling raw cotton relative to the radius of the raw cotton lump at an initial fiber mass temperature of 70°C, lump radii of 200 mm, 150 mm, 30 mm, air temperature of 10°C and initial raw cotton moisture of 20%.

Table 1. Experimental and calculated data on cooling relative to the radius of a raw cotton lump.

R, mm	200		150		30
, a secund	Experimental	Calculations	Experimental	Calculations	Experimental	Calculations
0	70	70	70	70	70	70
50	62	63,36	60	61,81	56	57,71
100	46	47,15	41	41,98	24	25,18
150	30	30.98	27	28,31	38	38,96
200	21	21,81	19	19,95	16	16,92

Table 2 shows the experimental and calculated data on cooling raw cotton relative to the density of raw cotton at an initial fiber mass temperature of 70°C, lump density of 70; 160 va 200 kg/m³ and an initial raw cotton moisture content of 15%. Air temperature 10°C.

Table 2. Experimental and calculated data on cooling relative to the density of raw cotton.

, Density	70 kg/m³		160 kg/m³		200 kg/m³
, Time	Experimental	Calculations	Experimental	Calculations	Experimental	Calculations
0	70	70	70	70	70	70
20	46	46,96	58	59,01	68	69,11
40	28	28,75	40	41,05	57	58,08
60	19	19.91	24	24,61	50	50,82
80			20	19,82	41	41,65
100					38	38,52

An analysis of the results in Tables 1 and 2 shows that the lump diameter significantly influences the process intensity; i.e., as the lump diameter increases, the fiber cooling rate decreases sharply. The temperature difference in the seeds along the lump radius and the temperature difference between fibers and seeds located at equal distances from the lump surface are very large, leading to uneven cooling of the raw cotton components. To increase the seed cooling rate and achieve uniform drying, it is necessary to organize the cooling period for the raw cotton in a loosened state, i.e., to reduce the lump diameter whenever possible.

4. Conclusions

This paper presents an analytical study of the cooling problem of fibrous mass. Based on mathematical modeling of the heat transfer process, analytical solutions were obtained that allow us to describe the temperature distribution in the fibrous medium as a function of time and spatial coordinates.

The main results of the paper include:

1. Formulation of a mathematical model of the cooling process, taking into account the thermophysical properties of the fibrous mass, as well as the boundary and initial conditions.

2. Analytical solution of the problem using mathematical physics methods, such as separation of variables or integral transformations.

3. Analysis of the influence of parameters (thermal conductivity, heat capacity, material density) on cooling dynamics.

4. Practical significance of the obtained solutions, which can be used to optimize industrial processes related to the processing of fibrous materials.

The obtained results demonstrate that the analytical approach allows not only a qualitative description of the cooling process but also a quantitative assessment of the influence of various factors on its dynamics. This opens up opportunities for developing more effective methods for controlling thermal processes in fibrous materials.

In the future, a promising direction of research may be the consideration of nonlinear effects, such as the dependence of thermophysical properties on temperature, as well as numerical modeling for more complex geometric configurations and boundary conditions.

Abbreviations

Temperature

Author Contributions

Mamatov Alisher Zulunovich: Methodology, Project administration, Supervision, Writing – review & editing

Jamshidbek Rakhmonov Turdaliyevich: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[2]	Mamatov A. Z., Usmankulov A. K., Abbazov I. Z., Norboyev U. A., Mukhametshina E. T. Determination of Temperature of Components of Cotton-Raw Material in a Drum Dryer with a Constant. IOP Conference Series: Earth and Environmental Sciencethis link is disabled, 2021, 939 (1), 012052.
[3]	Mamatov A. Z., Pardaev X. N., Mardonov J. S. H., Plekhanov A. F. Determining of the heat-moisture stateof raw cotton in a drum dryer. Izvestiya Vysshikh Uchebnykh Zavedenii, Seriya Teknologiya Tekstil’noi Promyshlennostithis link is disabled, 2021, 391(1), c. 46-49.
[4]	Mamatov A. Z., Pardaev H., Bomuratov Sh. Analytical solution of one problem of cooling seeds in the fluff of raw cotton// Devolment of science. - ISSN 3030-3007, 2025.
[5]	Zhamuratov, K., Umarov, K., & Dodobayev, A. Drainage of a semi-infinite aquifer in the presence of evaporation. In AIP Conference Proceedings (Vol. 3147, No. 1). AIP Publishing. AIP Conf. Proc. 3147, 030036 (2024), https://doi.org/10.1063/5.0210201
[6]	Mamatov A. Z. Modeling of raw cotton drying technology to improve fiber quality. Dissertation for the degree of Doctor of Technical Sciences. Tashkent, 1995.
[7]	Abdullaev H. A., Rakhimov S. R. Thermophysical properties of raw cotton and their influence on fiber quality. - Tashkent: Fan, 1985.
[8]	Ismailov Sh. I., Karimov M. K. Heat transfer in fibrous materials during cotton processing. - Tashkent: Uzbekistan, 1990.
[9]	Nurmatov A. N., Saidov M. S. Mathematical Modeling of Raw Cotton Drying and Cooling Processes. - Dushanbe: Donish, 1988.
[10]	Isachenko V. P., Osipova V. A., Sukomel A. S. Heat Transfer: Textbook for Universities. - Moscow: Energia, 1975.
[11]	Lykov A. V. Theory of Heat Conductivity. - Moscow: Vysshaya shkola, 1967.
[12]	Tikhonov A. N., Samarskii A. A. Equations of Mathematical Physics. - Moscow: Nauka, 1977.
[13]	Kutateladze S. S. Fundamentals of Heat Transfer Theory. - Moscow: Atomizdat, 1979.
[14]	Dulnev G. N., Parfenov V. G. Heat and Mass Transfer in Radioelectronic Equipment. - Moscow: Energy, 1971.
[15]	Baer G. D., Brown R. A. Heat Transfer in Fibrous Materials. - Moscow: Mir, 1985.
[16]	Pavlov K. F., Romankov P. G., Noskov A. A. Examples and Problems in the Course on Processes and Apparatus of Chemical Engineering. - Leningrad: Chemistry, 1981.
[17]	Kreit F., Black W. Fundamentals of Heat Transfer. - Moscow: Mir, 1983.
[18]	Wang H., Li X. Heat and mass transfer in fibrous materials. - Journal of Applied Physics, 2015.

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Zulunovich, M. A., Turdaliyevich, J. R. (2026). Analytical Solution to One Problem of Cooling Fiber Mass. American Journal of Mechanics and Applications, 13(4), 84-89. https://doi.org/10.11648/j.ajma.20261304.15

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Zulunovich, M. A.; Turdaliyevich, J. R. Analytical Solution to One Problem of Cooling Fiber Mass. Am. J. Mech. Appl. 2026, 13(4), 84-89. doi: 10.11648/j.ajma.20261304.15

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AMA Style

Zulunovich MA, Turdaliyevich JR. Analytical Solution to One Problem of Cooling Fiber Mass. Am J Mech Appl. 2026;13(4):84-89. doi: 10.11648/j.ajma.20261304.15

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@article{10.11648/j.ajma.20261304.15,
  author = {Mamatov Alisher Zulunovich and Jamshidbek Rakhmonov Turdaliyevich},
  title = {Analytical Solution to One Problem of Cooling Fiber Mass},
  journal = {American Journal of Mechanics and Applications},
  volume = {13},
  number = {4},
  pages = {84-89},
  doi = {10.11648/j.ajma.20261304.15},
  url = {https://doi.org/10.11648/j.ajma.20261304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20261304.15},
  abstract = {To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.},
 year = {2026}
}

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TY  - JOUR
T1  - Analytical Solution to One Problem of Cooling Fiber Mass
AU  - Mamatov Alisher Zulunovich
AU  - Jamshidbek Rakhmonov Turdaliyevich
Y1  - 2026/10/26
PY  - 2026
N1  - https://doi.org/10.11648/j.ajma.20261304.15
DO  - 10.11648/j.ajma.20261304.15
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 84
EP  - 89
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20261304.15
AB  - To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.
VL  - 13
IS  - 4
ER  -

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Author Information

Mamatov Alisher Zulunovich

Department of Mathematics and Computer Science, Tashkent Institute of Textile and Light Industry, Tashkent, Uzbekistan

Biography: Mamatov Alisher Zulunovich is a Doctor of Technical Sciences (DSc) and Professor at Tashkent Institute of Textile and Light Industry, Uzbekistan. He is associated with the Department of Mathematics and Computer Science. His research interests include mathematical modeling, differential equations, heat and mass transfer, and modeling of cotton drying processes.

Research Fields: Mathematical Analysis, Differential Equations, Fourier Series, Numerical Methods, Mathematical Modeling.

Contact Email

http://orcid.org/0009-0006-2458-7047
Jamshidbek Rakhmonov Turdaliyevich

Department of Mathematics and Computer Science, Gulistan State Pedagogical Institute, Gulistan, Uzbekistan

Biography: Jamshidbek Rakhmonov Turdaliyevich is a mathematics specialist and Senior Lecturer at Gulistan State Pedagogical Institute, Uzbekistan. His research interests include mathematical analysis, differential equations, Fourier series, and numerical methods. He has contributed to research on the Galerkin method and mathematical modeling of differential problems.

Research Fields: Mathematical Modeling, Differential Equations, Heat and Mass Transfer, Cotton Processing Technologies, Drying Processes.

Contact Email

http://orcid.org/0009-0006-2520-188X

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Zulunovich, M. A., Turdaliyevich, J. R. (2026). Analytical Solution to One Problem of Cooling Fiber Mass. American Journal of Mechanics and Applications, 13(4), 84-89. https://doi.org/10.11648/j.ajma.20261304.15

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ACS Style

Zulunovich, M. A.; Turdaliyevich, J. R. Analytical Solution to One Problem of Cooling Fiber Mass. Am. J. Mech. Appl. 2026, 13(4), 84-89. doi: 10.11648/j.ajma.20261304.15

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AMA Style

Zulunovich MA, Turdaliyevich JR. Analytical Solution to One Problem of Cooling Fiber Mass. Am J Mech Appl. 2026;13(4):84-89. doi: 10.11648/j.ajma.20261304.15

Copy | Download

@article{10.11648/j.ajma.20261304.15,
  author = {Mamatov Alisher Zulunovich and Jamshidbek Rakhmonov Turdaliyevich},
  title = {Analytical Solution to One Problem of Cooling Fiber Mass},
  journal = {American Journal of Mechanics and Applications},
  volume = {13},
  number = {4},
  pages = {84-89},
  doi = {10.11648/j.ajma.20261304.15},
  url = {https://doi.org/10.11648/j.ajma.20261304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20261304.15},
  abstract = {To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Analytical Solution to One Problem of Cooling Fiber Mass
AU  - Mamatov Alisher Zulunovich
AU  - Jamshidbek Rakhmonov Turdaliyevich
Y1  - 2026/10/26
PY  - 2026
N1  - https://doi.org/10.11648/j.ajma.20261304.15
DO  - 10.11648/j.ajma.20261304.15
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 84
EP  - 89
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20261304.15
AB  - To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.
VL  - 13
IS  - 4
ER  -

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