: 21-28
Received: May 10, 2022
Accepted: May 19, 2022

Цитування за ДСТУ: Гавриш В. І. Математичні просторові моделі визначення температурного поля із локально зосередженим тепловим нагріванням. Український журнал інформаційних технологій. 2022, т. 4, № 1. С. 21–28.

Citation APA: Havrysh, V. I. (2022). Mathematical spatial models of determination of temperature field from locally concentrated thermal heating. Ukrainian Journal of Information Technology, 4(1), 21–28.

Lviv Polytechnic National University, Lviv, Ukraine

Linear and nonlinear mathematical models for determining the temperature field, and later the analysis of temperature regimes in isotropic spatial inhomogeneous media exposed to internal and external thermal loads have been developed. To do this, the thermal conductivity for such structures is described as a whole using symmetric unit functions, which allows us to consider boundary thermal conductivity problems with one linear and nonlinear differential equation of thermal conductivity with discontinuous coefficients and linear and nonlinear boundary conditions on boundary surfaces. In the case of a nonlinear boundary value problem, the Kirchhoff transform is applied, which linearizes the initial nonlinear equation of thermal conductivity and nonlinear boundary conditions and results in a second-order linear differential equation with partial derivatives and singular coefficients with respect to the Kirchhoff function with linear conditions. To solve the obtained linear boundary value problem, the method of integral Fourier transform was used, as a result of which an analytical solution was obtained, which determines the Kirchhoff linearizing function. As an example, the linear and cubic dependences of the thermal conductivity of structural materials on the structure, which are often used in many practical problems, are chosen. As a result, analytical relations in the form of quadratic and biquadratic equations are obtained to determine the temperature distribution in the thermosensitive layer with foreign inclusion at external local heating. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of foreign inclusion on the temperature distribution was studied if the material of the medium was selected ceramics VK94-I, and the inclusion – silver, aluminum and silicon. To determine the numerical values of temperature in these structures, as well as the analysis of heat transfer processes in the middle of these structures due to internal and external heat loads, developed software that uses a geometric representation of temperature distribution depending on spatial coordinates. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in spatial inhomogeneous media with internal and external heating to the real physical process. Software also allows you to analyze this type of environment, which are exposed to internal and external heat loads, in terms of their heat resistance. As a result, it becomes possible to increase it and protect it from overheating, which can lead to the destruction of not only individual elements but also the entire structure.

[1]     Aza­ren­kov, V. I. (2012). Issle­do­va­nie i raz­ra­bot­ka tep­lo­voi mo­de­li i me­to­dov ana­li­za tem­pe­ra­tur­nikh po­lei konstruktcii ra­dioelektron­noi ap­pa­ra­tu­ri. Techno­logy au­dit and pro­duc­ti­on re­ser­ves, 3/1(5), 39–40. [In Rus­si­an].

[2]     Car­pin­te­ri, A., & Pag­gi, M. (2008). Ther­mo­elas­tic mis­match in non­ho­mo­ge­ne­ous be­ams. Jo­ur­nal of En­gi­ne­ering Mat­he­ma­tics, 61(2–4), 371–384.

[3]     Dovbnia, K. M., & Dun­dar, O. D. (2016). Stat­si­onarnyi tep­lo­ob­min tonkykh po­lohykh izot­ropnykh obo­lo­nok, ya­ki znak­ho­di­at­sia pid diieiu dzhe­rel tep­la, zo­se­redzhenykh po dvovymir­nii ob­las­ti. Visnyk Don­NU. Ser. A: Pryrodnychi na­uky, 1–2, 107–112. [In Uk­ra­ini­an].

[4]     Havrysh, V. I., & Fe­das­juk, D. V. (2012). Mo­del­ling of tem­pe­ra­tu­re re­gi­mes in pi­ece­wi­se-ho­mo­ge­ne­ous struc­tu­res. Lviv: Pub­lis­hing hou­se of Lviv Po­li­technic Na­ti­onal Uni­ver­sity, 176 p.

[5]     Havrysh, V. I., Ba­ra­netskiy, Ya. O., & Kol­ya­sa, L. I. (2018). In­ves­ti­ga­ti­on of tem­pe­ra­tu­re mo­des in ther­mo­sen­si­ti­ve non-uni­form ele­ments of ra­dioelectro­nic de­vi­ces. Ra­dio electro­nics, com­pu­ter sci­en­ce, ma­na­ge­ment, 3(46), 7–15.

[6]     Havrysh, V. I., Kol­ya­sa, L. I., & Uk­hanska, O. M. (2019). De­ter­mi­na­ti­on of tem­pe­ra­tu­re fi­eld in ther­mally sen­si­ti­ve la­ye­red me­di­um with inclu­si­ons. Nau­kov­yi Visnyk NHU, 1, 94–100.

[7]     Hrytsi­uk, Yu. I., & Andrushcha­kevych, O. T. (2018). Me­ans for de­ter­mi­ning softwa­re qua­lity by met­ric analysis met­hods. Sci­en­ti­fic Bul­le­tin of UN­FU, 28(6), 159–171.

[8]     Hrytsi­uk, Yu. I., & Buchkovska, A. Yu. (2018). Vis­ua­li­za­ti­on of the re­sults of ex­pert eval­ua­ti­on of softwa­re qua­lity using po­lar di­ag­rams. Sci­en­ti­fic Bul­le­tin of UN­FU, 27(10), 137–145.

[9]     Hrytsi­uk, Yu. I., & Dal­yavskyy, V. S. (2018). Using Pe­tal Di­ag­ram for Vis­ua­li­zing the Re­sults of Ex­pert Eval­ua­ti­on of Softwa­re Qua­lity. Sci­en­ti­fic Bul­le­tin of UN­FU, 28(9), 97–106.

[10]  No­da, N. (1991). Ther­mal stres­ses in ma­te­ri­als with tem­pe­ra­tu­re-de­pen­dent pro­per­ti­es. Appli­ed Mec­ha­nics Re­vi­ews, 44, 383–397.

[11]  Otao, Y., Ta­ni­ga­wa, O., & Is­hi­ma­ru, O. (2000). Op­ti­mi­za­ti­on of ma­te­ri­al com­po­si­ti­on of functi­ona­lity gra­ded pla­te for ther­mal stress re­la­xa­ti­on using a ge­ne­tic al­go­rithm. Jo­ur­nal of Ther­mal Stres­ses, 23, 257–271.

[12]  Podstri­gach, Ia. S., Lo­ma­kin, V. A., & Ko­li­ano, Iu. M. (1984). Ter­mo­up­ru­gost tel ne­od­no­rod­noi struk­tury. Mos­cow: Nau­ka, 368 p. [In Rus­si­an].

[13]  Ta­ni­ga­wa, Y., & Otao, Y. (2002). Tran­si­ent ther­mo­elas­tic analysis of functi­onally gra­ded pla­te with tem­pe­ra­tu­re-de­pen­dent ma­te­ri­al pro­per­ti­es ta­king in­to ac­co­unt the ther­mal ra­di­ati­on. Ni­hon Ki­kai Gak­kai Nen­ji Ta­ikai Ko­en Ron­bunshu, 2, 133–134.

[14]  Ta­ni­ga­wa, Y., Akai, T., & Ka­wa­mu­ra, R. (1996). Tran­si­ent he­at con­duc­ti­on and ther­mal stress prob­lems of a non­ho­mo­ge­ne­ous pla­te with tem­pe­ra­tu­re-de­pen­dent ma­te­ri­al pro­per­ti­es. Jo­ur­nal of Ther­mal Stres­ses, 19(1), 77–102.

[15]  Yan­gi­an, Xu, & Da­ih­ui, Tu. (2009). Analysis of ste­ady ther­mal stress in a ZrO2/FGM/Ti-6Al-4V com­po­si­te ECBF pla­te with tem­pe­ra­tu­re-de­pen­dent ma­te­ri­al pro­per­ti­es by NFEM. 2009-WA­SE Int. Conf. on In­for­ma. Eng, Vol. 2, 433–436.

[16]  Ko­li­ano, Iu. M. (1992). Me­tody tep­lop­ro­vod­nos­ti i ter­mo­up­ru­gos­ti ne­od­no­rod­no­go te­la. Kyiv: Nau­ko­va dum­ka, 280 p.

[17]  Korn, G., & Korn, T. (1977). Spra­vochnik po ma­te­ma­ti­ke dlia na­uchnykh ra­bot­ni­kov i inzhe­ne­rov. Mos­cow: Nau­ka, 720 p. [In Rus­si­an].

[18]  Ki­ko­ina, I. K. (1976). Tab­litcy fi­zic­hes­kikh ve­lic­hin. Spra­vochnik. Mos­cow: Ato­miz­dat, 1008 p. [In Rus­si­an].