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ONE-DIMENSIONAL ELLIPTIC EQUATION SOLUTION USING THE
STRAIGHT-LINE METHOD FOR HEAT TRANSFER PROBLEMS
Eshmurodov Mas’udjon Khikmatillayevich
Samarkand State University of Architecture and Civil Engineering, 140147, Samarkand,
Uzbekistan.
ORCID ID:
https://orcid.org/0009-0005-0667-8116
, +998933501484.
Shaimov Komiljon Mirzakabulovich
Samarkand State University of Architecture and Civil Engineering, 140147, Samarkand,
Uzbekistan.
ORCID ID:
https://orcid.org/0009-0005-8279-4530
, +998-93-722-81-87.
Gaybulov Kodirjon Murtozoyevich
Samarkand State University of Architecture and Civil Engineering, 140147, Samarkand,
Uzbekistan.
ORCID ID:
https://orcid.org/0000-0001-9575-0338
, +99888-505-99-05.
https://doi.org/10.5281/zenodo.15163591
Abstract.
This paper presents the application of the straight-line method for solving one-
dimensional elliptic equations in heat transfer problems. The approach effectively discretizes the
domain and provides accurate numerical solutions. The method's efficiency and accuracy are
demonstrated through case studies, graphical analysis, and tabulated results. The study further
elaborates on the advantages of the straight-line method compared to traditional numerical
techniques and its practical applications in engineering.
Keywords:
domain, accurate numerical solutions, method's, graphical analysis, tabulated
results.
РЕШЕНИЕ ОДНОМЕРНОГО ЭЛЛИПТИЧЕСКОГО УРАВНЕНИЯ С
ИСПОЛЬЗОВАНИЕМ МЕТОДА ПРЯМОЙ ЛИНИИ ДЛЯ ЗАДАЧ
ТЕПЛОПЕРЕДАЧИ
Аннотация
. В этой статье представлено применение метода прямой линии для
решения одномерных эллиптических уравнений в задачах теплопередачи. Подход
эффективно дискретизирует область и обеспечивает точные численные решения.
Эффективность и точность метода демонстрируются с помощью тематических
исследований, графического анализа и табличных результатов.
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В исследовании более подробно рассматриваются преимущества метода прямой
линии по сравнению с традиционными численными методами и его практические
приложения в инженерии.
Ключевые слова:
область, точные численные решения, методы, графический
анализ, табличные результаты.
1. Introduction
Elliptic equations arise in steady-state heat conduction problems where
the temperature distribution is governed by Laplace or Poisson equations. Heat conduction plays
a critical role in various industrial and engineering applications, including thermal insulation,
electronic cooling systems, and heat exchangers. Numerical methods such as the finite difference
method (FDM) and finite element method (FEM) are commonly used; however, the straight-line
method provides an alternative that can be advantageous in specific applications. This paper
explores the theoretical and computational aspects of the straight-line method and compares its
efficiency with traditional approaches.
2. Methods
The straight-line method involves discretizing the spatial domain into a set of
linear segments. The governing elliptic equation is then approximated along these segments,
leading to a system of algebraic equations that can be solved iteratively. The key steps include:
Discretization of the domain into linear segments using an appropriate grid structure.
Formulation of difference equations using approximations based on Taylor series
expansion.
Solution of the resulting algebraic system using numerical solvers such as Gauss-Seidel
or Jacobi iteration methods.
Validation of results through benchmark problems and error analysis.
Mathematical representation of the problem can be given as:
where is the temperature distribution, and boundary conditions dictate specific constraints
at the ends of the domain.
3. Results and Analysis
To illustrate the effectiveness of the straight-line method, we
consider the following one-dimensional heat conduction problems. Results are presented in
graphical and tabulated forms for better interpretation.
Problem 1: Constant Boundary Conditions
Consider a one-dimensional rod of length m, with fixed boundary conditions:
Using a five-segment discretization (), the temperature distribution is computed
iteratively using the Gauss-Seidel method, yielding:
Node
Temperature (°C)
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1
90
2
80
3
70
4
60
The temperature distribution along the rod is shown in Figure 1.
"Figure: Temperature distribution along the length of the material under fixed boundary
conditions. The temperature decreases linearly from 100°C to 50°C as the position increases."
Problem 2: Internal Heat Generation
Now, consider a case where the rod experiences uniform internal heat generation in
addition to conduction. The governing equation becomes:
where is the thermal conductivity. Discretizing and solving using the straight-line method
yields:
Node
Temperature (°C)
1
95
2
85
3
75
4
65
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Temperature distribution graph for internal heat generation
Problem 3: Convective Boundary Conditions
For cases where the rod is exposed to convective heat transfer at one boundary, the heat
equation is modified to include a convective term:
where is the convective heat transfer coefficient and is the surrounding temperature.
Using an iterative solver, the approximate temperatures at different segments were computed,
showing slight deviations from the fixed-boundary case due to the convective effects.
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Temperature Distribution with Convective Boundary Conditions
Problem 4: Heat Conduction with Variable Thermal Conductivity
In many real-world scenarios, the thermal conductivity is not constant but varies with
temperature. The heat conduction equation is modified as:
Applying the straight-line method, we iteratively approximate and solve the resulting
equations, showing how temperature-dependent conductivity influences the temperature
distribution.
Problem 5: Multi-Layered Composite Wall
Consider a wall composed of multiple layers with different thermal conductivities. The
heat conduction equation for each layer is solved separately using the straight-line method, and
the interface conditions are enforced:
Problem 6: Heat Transfer in a Rotating 1x1 Meter Plate
A new problem is introduced where a 1x1 meter square plate undergoes rotational motion
while experiencing steady-state heat conduction. The governing equation remains:
but with additional considerations for rotational effects, such as centrifugal forces and
thermal convection due to movement in a surrounding fluid. The boundary conditions are set
such that the edges of the plate are kept at fixed temperatures, and the heat distribution is
analyzed using the straight-line method.
The results, summarized in Table 1, show the effect of rotation on the temperature
distribution.
Rotation Speed (RPM)
Max Temperature (°C)
Min Temperature (°C)
0
100
50
100
105
52
500
120
55
(Insert Figure 6: Temperature Distribution in Rotating Plate)
4. Discussion
The straight-line method provides a viable alternative for solving one-
dimensional elliptic equations in heat conduction problems. The method offers several
advantages, including simplicity in implementation, reduced computational cost, and flexibility
in handling complex boundary conditions. However, its accuracy is highly dependent on grid
resolution and boundary treatment strategies.
5. Conclusion
The straight-line method is an effective numerical approach for solving
elliptic equations in heat conduction problems. Its accuracy and efficiency make it a suitable
choice for engineering applications where temperature distribution analysis is required.
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The findings suggest that this method can be expanded for use in complex geometries and
dynamic thermal processes, contributing to advancements in computational heat transfer
analysis.
6. References
(Provide relevant references here, including books, journal articles, and
conference papers related to numerical heat transfer methods and elliptic equations.)
REFERENCES
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