Direct (Classical) Method of Calculation of the Temperature Field in a Hollow Multilayer Cylinder

Tatsii, R. M.; Pazen, David

doi:10.1007/s10891-018-1871-3

Cited by 15 publications

(13 citation statements)

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“…As it is known [12,18], the roots of the characteristic equa tion (32), which are the eigenvalues of problem (24-26), are positive and different.…”

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confidence: 99%

“…(18) for arbitrary k > 0 by the method of mathematical induction on the index k we obtainp l, m+k = B l (r m+k , r m ) ⋅ p l,m .Encharging m = 0, we obtainp l, k = B l (r k , r 0 ) ⋅ p l, 0 .Using boundary conditions (8), we obtain an expression for the calculation p l, 0 ≡ u l, 0 (r 0 )…”

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confidence: 99%

“…To construct the solution of the problem (10-13), we apply the method of eigenfunctions [17,18], which consists in the fact that we will look for the solution of this problem in the form…”

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Direct method of studying heat exchange in multilayered bodies of basic geometric forms with imperfect heat contact

Tatsiy¹,

Pazen²,

Vovk³

et al. 2021

Nauk. visn. nat. hirn. univ.

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Purpose. Characteristics of heat transfer processes in multilayer bodies of basic geometric shapes simultaneously under conditions of convective heat transfer on its surfaces and taking into account imperfect thermal contact between the layers. Methodology. A direct method was applied to solve a one-parameter family of boundary value problems in the theory of heat conduction. This method is based on the reduction method, the concept of quasiderivatives, a system of differential equations with impulse action, the method of separation of variables, and the modified method of eigenfunctions of Fourier. It is worth noting that the application of the concept of quasiderivatives allows you to circumvent the well-known problem of multiplication of generalized functions, which arises when using the differentiation procedure of the coefficients of a differential equation. Such a procedure, in our opinion, casts doubt on the equivalence of the transition to the differential equation obtained in this way with generalized coefficients. Findings. The solution to the problem is obtained in a closed form. The proposed algorithm does not contain a solution to volume conjugation problems. It includes only: a) finding the roots of the corresponding characteristic equations; b) the multiplication of a finite number of known (2 2) matrices; c) the calculation of certain integrals; d) summing the required number of members of the series to obtain the specified accuracy. As an illustration, we consider model examples of heating eight-layer structures in a fire. Originality. For the first time, the direct method has been applied to solving the problem of the distribution of an unsteady temperature field over the thickness of multilayer structures of basic geometric shapes simultaneously, in the presence of imperfect thermal contact between the layers. Practical value. The implementation of the research results allows us to effectively study the heat transfer processes in multilayer structures, which are found in a number of applied problems.

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“…As it is known [12,18], the roots of the characteristic equa tion (32), which are the eigenvalues of problem (24-26), are positive and different.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Direct method of studying heat exchange in multilayered bodies of basic geometric forms with imperfect heat contact

Tatsiy¹,

Pazen²,

Vovk³

et al. 2021

Nauk. visn. nat. hirn. univ.

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“…Згадані проблеми можна легко оминути, застосовуючи так званий прямий метод, що детально розроблений і описаний в роботах [2][3][4][5].…”

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Determination of Temperature Field in a Solid Circular Cylinder in the Fire

2020

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The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a solid homogeneous cylindrical structure in the presence of convective heat exchange with the environment, that is, the boundary conditions of the third. It is assumed that the law of change of ambient temperature, which washes the surface layer of the structure, is an arbitrary function of time. This function is uniformly distributed over the surface such that the isotherms inside the structure are coaxial cylindrical surfaces, that is, the temperature field inside the cylinder depends only on the radius r and the time τ.This problem is solved by applying the reduction method, when the original problem is represented by the sum of two unknown but related functions. It is found that the solution of the corresponding quasistationary problem does not depend on the radius r and is equal to the law of change of the ambient temperature.In the following, the corresponding inhomogeneous problem is solved, the main stage of which is the eigenvalue problem obtained after applying the Fourier variable method. As a standard procedure, a characteristic equation is obtained and the eigenvalues are determined and their own functions are constructed. The Fourier method was used to construct a nonuniform problem solution.It is found that when at the time τ = 0 the initial distribution of the structural temperature field and the ambient temperature coincide, the solution of the original problem is greatly simplified.To illustrate the proposed method, a model example of finding the distribution of the temperature field in a reinforced concrete homogeneous column of a circular cross section under the conditions of the temperature of the hydrocarbon fire is solved. The results of the calculations are presented as a graph of temperature change as a function of time. The numerical implementation of the method was performed using the Maple 13. computer algebra system. It should be noted that the first 30 roots of the characteristic equation were used to achieve the result with the given accuracy. The results obtained are directly applicable in a number of applications.

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“…Вступ. Застосування прямого методу до розв'язування задач теплообміну в багатошарових порожнистих циліндричних та сферичних конструкціях описано в численних публікаціях [1][2][3]. Актуальними задачами сьогодення є знаходження розподілу температурного поля в циліндричних та сферичних конструкціях типу «куля в сферичній оболонці» або «суцільний циліндр всередині циліндричної оболонки».…”

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Determination of the Non-Stationary Temperature Field in the System of Two Cylindrical Shell Under the Fire Conditions

Tatsii

Pazen

Shypot

2019

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The proposed work is devoted to the application of the direct method to the study of heat transfer processes in the system "solid cylinder inside a cylindrical shell". It is assumed that there is an ideal thermal contact between them, and the law of changing the ambient temperature, which rinses the surface of the structure, is an arbitrary function of time, and evenly distributed over the surface. Consequently, isotherms inside this construction are concentric circles, that is, the problem is symmetric and is solved for the first time in such a statement. To solve such a problem, the auxiliary problem of determining the distribution of a non-stationary temperature field in a two-layer hollow cylindrical structure with a "withdrawn" cylinder of sufficiently small radius is raised in parallel. In this case the symmetry condition of the original problem is replaced by the condition of the second kind on the inner surface of this construction. The implementation of the solution of the auxiliary problem is carried out by applying a reduction method using the concept of quasi-derivatives. In the future, the Fourier scheme is used with the use of the modified eigenfunctions method. To find the solution of the original problem, the idea of the boundary transition is used by passing the radius of the withdrawn cylinder to zero. It is established that in this approach all the eigenfunctions of the corresponding problem on the eigenvalues have no singularities at zero, which means that the solutions of the original problem are constrained throughout the design. In order to illustrate the proposed method, a model example of finding the temperature field distribution in a column of a circular cross-section (concrete in a steel shell) is solved under the influence of the standard temperature regime of the fire. The results of the calculations are presented in a bulk schedule of temperature changes, depending on time and spatial coordinates. The generalization of the results obtained in the case of any finite number of cylindrical shells is a purely technical problem, and not a fundamental one. Note that while changing the boundary condition of the third kind to any other boundary condition (for example, the first kind) does not affect the scheme of solving similar tasks. Since the general scheme of studying the distribution of temperature fields in multi-layered structures with an arbitrary number of layers in the presence of internal sources of heat is studied in detail, the setting and solving of such problems for the system of "solid cylinder inside a cylindrical shell" is not without difficulty.

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Direct (Classical) Method of Calculation of the Temperature Field in a Hollow Multilayer Cylinder

Cited by 15 publications

References 2 publications

Direct method of studying heat exchange in multilayered bodies of basic geometric forms with imperfect heat contact

Direct method of studying heat exchange in multilayered bodies of basic geometric forms with imperfect heat contact

Determination of Temperature Field in a Solid Circular Cylinder in the Fire

Determination of the Non-Stationary Temperature Field in the System of Two Cylindrical Shell Under the Fire Conditions

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