MATHEMATICAL MODELS OF HEAT TRANSFER IN ELEMENTS OF TURBOGENERATORS

2019;
: 22-27
https://doi.org/10.23939/ujit2019.01.022
Received: November 07, 2019
Accepted: November 20, 2019
1
Lviv Polytechnic National University, Lviv, Ukraine
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University, Lviv, Ukraine
4
Lviv Polytechnic National University

Se­pa­ra­te mat­he­ma­ti­cal mo­dels for de­ter­mi­ning the tem­pe­ra­tu­re distri­bu­ti­on in the ele­ments of tur­bo­ge­ne­ra­tors ha­ve be­en de­ve­lo­ped, which are descri­bed ge­omet­ri­cally by an isot­ro­pic half-spa­ce and a he­at-sen­si­ti­ve spa­ce with lo­cally con­centra­ted so­ur­ces of he­ating. For this pur­po­se, using the the­ory of ge­ne­ra­li­zed functi­ons in a con­ve­ni­ent form, we wri­te the ini­ti­al dif­fe­ren­ti­al eq­ua­ti­ons of ther­mal con­duc­ti­vity with bo­un­dary con­di­ti­ons. For ther­mo­sen­si­ti­ve spa­ce (ther­mophysi­cal pa­ra­me­ters are tem­pe­ra­tu­re de­pen­dent), the ori­gi­nal non­li­ne­ar ther­mal con­duc­ti­vity eq­ua­ti­on and the non­li­ne­ar bo­un­dary con­di­ti­ons are li­ne­ari­zed using the Kirchhoff transform, for which a li­ne­ar dif­fe­ren­ti­al eq­ua­ti­on is ob­ta­ined. An in­teg­ral Han­kel transform was used to sol­ve the bo­un­dary val­ue prob­lems of ther­mal con­duc­ti­vity, and as a re­sult analyti­cal so­lu­ti­ons in the ima­ges we­re ob­ta­ined. The­se so­lu­ti­ons we­re appli­ed by the in­ver­ted Han­kel in­teg­ral transfor­ma­ti­on, which ma­de it pos­sib­le to ob­ta­in the fi­nal analyti­cal so­lu­ti­ons of the ori­gi­nal prob­lems. The analyti­cal so­lu­ti­ons ob­ta­ined are pre­sen­ted in the form of non-na­ti­ve con­ver­gent in­teg­rals. For the construc­ti­on ma­te­ri­al of the he­at-sen­si­ti­ve spa­ce, a li­ne­ar de­pen­den­ce of the ther­mal con­duc­ti­vity co­ef­fi­ci­ent on the tem­pe­ra­tu­re was used. The re­sult is a con­ve­ni­ent for­mu­la for de­ter­mi­ning the tem­pe­ra­tu­re fi­eld, which al­lows to analyze tem­pe­ra­tu­re re­gi­mes in a he­at-sen­si­ti­ve en­vi­ron­ment. To de­ter­mi­ne the nu­me­ri­cal val­ues ​​of tem­pe­ra­tu­re in the abo­ve struc­tu­res, as well as to analyze the he­at exchan­ge in the ele­ments of the tur­bo­ge­ne­ra­tors cau­sed by dif­fe­rent tem­pe­ra­tu­re re­gi­mes due to the he­ating of lo­cally con­centra­ted he­at so­ur­ces, com­pu­ta­ti­onal prog­rams ha­ve be­en de­ve­lo­ped. Using the­se prog­rams are graphs that show the be­ha­vi­or of sur­fa­ces construc­ted using nu­me­ri­cal val­ues ​​of the di­men­si­on­less tem­pe­ra­tu­re distri­bu­ti­on de­pen­ding on the spa­ti­al di­men­si­on­less co­or­di­na­tes. The ob­ta­ined nu­me­ri­cal val­ues ​​of tem­pe­ra­tu­re in­di­ca­te that the mat­he­ma­ti­cal mo­dels of de­ter­mi­ning the distri­bu­ti­on of tem­pe­ra­tu­re to the ac­tu­al physi­cal pro­cess are con­sis­tent. The softwa­re al­so al­lows you to analyze lo­cally he­ated en­vi­ron­ments for the­ir he­at re­sis­tan­ce. As a con­seq­uen­ce, it be­co­mes pos­sib­le to ra­ise it, to de­ter­mi­ne the al­lo­wab­le tem­pe­ra­tu­res of nor­mal ope­ra­ti­on of the tur­bo­ge­ne­ra­tors, to pro­tect them from over­he­ating, which can cau­se destruc­ti­on not only of in­di­vid­ual ele­ments, but al­so of the who­le struc­tu­re.

[1]     Ba­yat, A., Mo­osa­vi, H., & Ba­yat, Y. (2015). Ther­mo-mec­ha­ni­cal analysis of functi­onally gra­ded thick sphe­res with li­ne­arly ti­me-de­pen­dent tem­pe­ra­tu­re. Sci­en­tia Ira­ni­ca, 22(5), 1801–1812.

[2]     Car­pin­te­ri, A., & Pag­gi, M. (2008). Ther­mo­elas­tic mis­match in non­ho­mo­ge­ne­ous be­ams. J. Eng. Math, 61, 2–4, 371–384.

[3]     Gavrysh, V. I., & Fe­das­juk, D. V. (2012). Mo­del­ju­vannja tem­pe­ra­turnyh rezhymiv u kus­ko­vo-od­no­ridnyh struk­tu­rah. Lviv: Vyd-vo Nac. un-tu "L'vivs'ka po­li­teh­ni­ka",176–178. [In Uk­ra­ini­an].

[4]     Ghan­nad, M., & Yag­ho­obi, M. P. (2015). A ther­mo­elas­ti­city so­lu­ti­on for thick cylin­ders sub­jec­ted to ther­mo-mec­ha­ni­cal lo­ads un­der va­ri­ous bo­un­dary con­di­ti­ons. Int. Jo­ur­nal of Ad­van­ced De­sign & Ma­nu­fac­tu­ring Techno­logy, 8(4), 1–12.

[5]     Har­ma­tii, H. Yu., Po­povych, V. S., & Krul, M. (2019). Vplyv ter­moc­hutlyvos­ti ma­te­ri­alu na ne­us­ta­len­yi tep­lov­yi stan ba­ha­tos­ha­ro­voi plastyny. Fizyko-khi­michna mek­ha­ni­ka ma­te­ri­aliv, 1, 98–104. [In Uk­ra­ini­an].

[6]     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 Sci­en­se, Control, 3(46), 7–15.

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

[8]     Jab­ba­ri, M., Ka­ram­po­ur, S., & Es­la­mi, M. R. (2011). Ra­di­ally symmet­ric ste­ady sta­te ther­mal and mec­ha­ni­cal stres­ses of a po­ro FGM hol­low sphe­re. In­ter­na­ti­onal Scho­larly Re­se­arch Net­work ISRN Mec­ha­ni­cal En­gi­ne­ering, 3, 1–7. https://doi.org/10.5402/2011/305402

[9]     Ko­li­ano, Iu. M. (1992). Me­tody tep­lop­ro­vod­nos­ti i ter­mo­up­ru­gos­ti ne­od­no­rod­no­go te­la. Kyiv: Nau­ko­va dum­ka, 268 p. [In Rus­si­an].

[10]  Korn, G., & Korn, T. (1977). Spra­vochnik po ma­te­ma­ti­ke dlia na­uchnykh ra­bot­ni­kov i inzhe­ne­rov. Mos­cow: Nau­ka, 650 p. [In Rus­si­an].

[11]  Lu­kas­hevych, A. (2019). Tem­pe­ra­tur­ne po­le u zo­ni kon­tak­tu pid chas ro­tat­si­ino­ho zva­riu­van­nia me­ta­liv ter­ti­am. Fizyko-khi­michna mek­ha­ni­ka ma­te­ri­aliv, 1, 41–46. [In Uk­ra­ini­an].

[12]  Mo­haz­zab, A. H., & Jab­ba­ri, M. (2011). Two-Di­men­si­onal Stres­ses in a Hol­low FG Sphe­re with He­at So­ur­ce. Ad­van­ced Ma­te­ri­als Re­se­arch, 264–265, 700–705. https://doi.org/10.4028/scientific.net/amr.264-265.700

[13]  Podstri­gach, Ia. S., Lo­ma­kin, V. A., & Ko­li­ano, Iu. M. (1984). Ter­mo­up­ru­gostь tel ne­od­no­rod­noi struk­tury. Mos­cow: Nau­ka, 354 p. [In Rus­si­an].

[14]  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, WA­SE. Int. Conf. on In­for­ma. Eng., 2–2, 433–436