Mathematical Models of Heat Transfer in Elements of Turbogenerators

Abstract

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.