{"id":52280,"date":"2020-02-18T09:56:55","date_gmt":"2020-02-18T08:56:55","guid":{"rendered":"https:\/\/www.thermal-engineering.org\/exemple-dequation-thermique-probleme-de-solution-definition\/"},"modified":"2021-08-05T07:15:10","modified_gmt":"2021-08-05T06:15:10","slug":"exemple-dequation-thermique-probleme-de-solution-definition","status":"publish","type":"post","link":"https:\/\/www.thermal-engineering.org\/fr\/exemple-dequation-thermique-probleme-de-solution-definition\/","title":{"rendered":"Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution &#8211; D\u00e9finition"},"content":{"rendered":"<div class=\"su-quote su-quote-style-default\">\n<div class=\"su-quote-inner su-clearfix\">Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution.\u00a0Dans cet article, il y a deux exemples d&#8217;\u00e9quation de solution de chaleur.\u00a0Les deux exemples sont avec solution.\u00a0G\u00e9nie thermique<\/div>\n<\/div>\n<div class=\"su-divider su-divider-style-dotted\"><\/div>\n<div class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights lgc-first lgc-last\">\n<div class=\"inside-grid-column\">\n<div class=\"su-spacer\"><\/div>\n<h2>Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution<\/h2>\n<\/div>\n<\/div>\n<div class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights lgc-first lgc-last\">\n<div class=\"inside-grid-column\">\n<div class=\"su-spacer\"><\/div>\n<h2>Conduction thermique dans un grand mur plan<\/h2>\n<p><strong>Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution<\/strong><\/p>\n<p>Consid\u00e9rons la paroi plane d&#8217;\u00e9paisseur 2L, dans laquelle il existe\u00a0<strong>une g\u00e9n\u00e9ration de chaleur uniforme et constante<\/strong>\u00a0par unit\u00e9 de volume,\u00a0<strong>q\u00a0<sub>V<\/sub>\u00a0[W \/ m\u00a0<sup>3<\/sup>\u00a0]<\/strong>\u00a0.\u00a0Le plan central est pris comme origine pour x et la dalle s&#8217;\u00e9tend de + L \u00e0 droite et &#8211; L de gauche.\u00a0Pour une\u00a0<a title=\"Conductivit\u00e9 thermique\" href=\"https:\/\/www.thermal-engineering.org\/what-is-thermal-conductivity-definition\/\">conductivit\u00e9 thermique<\/a>\u00a0constante\u00a0k, la forme appropri\u00e9e de l&#8217;\u00e9quation de la chaleur est la suivante:<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-heat-generation-equation.png\"><img loading=\"lazy\" class=\"size-medium wp-image-20229 aligncenter lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-heat-generation-equation-300x96.png\" alt=\"\u00e9quation de chaleur - g\u00e9n\u00e9ration de chaleur - \u00e9quation\" width=\"300\" height=\"96\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-heat-generation-equation-300x96.png\" \/><\/a><\/p>\n<p>La solution g\u00e9n\u00e9rale de cette \u00e9quation est la suivante:<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-general-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20230 lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-general-solution.png\" alt=\"\u00e9quation de conduction thermique - solution g\u00e9n\u00e9rale\" width=\"290\" height=\"150\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-general-solution.png\" \/><\/a><\/p>\n<p>o\u00f9 C\u00a0<sub>1<\/sub>\u00a0et C\u00a0<sub>2<\/sub>\u00a0sont les constantes d&#8217;int\u00e9gration.<\/p>\n<p>1)<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Heat-Conduction-in-a-Large-Plane-Wall-figure.png\"><img loading=\"lazy\" class=\"alignright size-medium wp-image-20228 lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Heat-Conduction-in-a-Large-Plane-Wall-figure-300x276.png\" alt=\"Conduction thermique dans un grand mur plan\" width=\"300\" height=\"276\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Heat-Conduction-in-a-Large-Plane-Wall-figure-300x276.png\" \/><\/a><\/p>\n<div class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights lgc-first lgc-last\">\n<div class=\"inside-grid-column\">\n<p>Calculez la distribution de temp\u00e9rature, T (x), \u00e0 travers cette \u00e9paisse paroi plane, si:<\/p>\n<ul>\n<li>les temp\u00e9ratures sur les deux surfaces sont de 15,0 \u00b0 C<\/li>\n<li>l&#8217;\u00e9paisseur de cette paroi est de 2L = 10 mm.<\/li>\n<li>la conductivit\u00e9 des mat\u00e9riaux est k = 2,8 W \/ mK (correspond au dioxyde d&#8217;uranium \u00e0 1000 \u00b0 C)<\/li>\n<li>le taux de chaleur volum\u00e9trique est q\u00a0<sub>V<\/sub>\u00a0= 10\u00a0<sup>6<\/sup>\u00a0W \/ m\u00a0<sup>3<\/sup><\/li>\n<\/ul>\n<p>Dans ce cas, les surfaces sont maintenues \u00e0 des temp\u00e9ratures donn\u00e9es T\u00a0<sub>s, 1<\/sub>\u00a0et T\u00a0<sub>s, 2<\/sub>\u00a0.\u00a0Cela correspond \u00e0 la\u00a0<a title=\"Conditions aux limites de Dirichlet - Conditions aux limites de type I\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/thermal-conduction\/heat-conduction-equation\/dirichlet-boundary-condition-type-i-boundary-condition\/\">condition aux limites de Dirichlet<\/a>\u00a0.\u00a0De plus, ce probl\u00e8me est thermiquement sym\u00e9trique et, par cons\u00e9quent, nous pouvons \u00e9galement utiliser\u00a0<a title=\"Limite adiabatique - Sym\u00e9trie thermique\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/thermal-conduction\/heat-conduction-equation\/adiabatic-boundary-thermal-symmetry\/\">la condition aux limites de sym\u00e9trie thermique<\/a>\u00a0.\u00a0Les constantes peuvent \u00eatre \u00e9valu\u00e9es en utilisant la substitution dans la solution g\u00e9n\u00e9rale et sont de la forme:<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-boundary-conditions.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20231 lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-boundary-conditions.png\" alt=\"\u00e9quation de conduction thermique - conditions aux limites\" width=\"428\" height=\"110\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-boundary-conditions.png\" \/><\/a><\/p>\n<p>La distribution de temp\u00e9rature r\u00e9sultante et la temp\u00e9rature de la ligne centrale (x = 0) (maximale) dans cette paroi plane \u00e0 ces conditions aux limites sp\u00e9cifiques seront:<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20232 lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-solution.png\" alt=\"\u00e9quation de conduction thermique - solution\" width=\"608\" height=\"210\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-conduction-equation-solution.png\" \/><\/a><\/p>\n<p>Le\u00a0<a title=\"Densit\u00e9 de flux thermique - Flux thermique\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/introduction-to-heat-transfer\/heat-flux-density-thermal-flux\/\">flux de chaleur<\/a>\u00a0en tout point, q\u00a0<sub>x<\/sub>\u00a0[Wm\u00a0<sup>-2<\/sup>\u00a0], dans la paroi peut, bien s\u00fbr, \u00eatre d\u00e9termin\u00e9 en utilisant la distribution de temp\u00e9rature et avec la\u00a0<a title=\"Loi de Fourier de la conduction thermique\" href=\"https:\/\/www.thermal-engineering.org\/what-is-fouriers-law-of-thermal-conduction-definition\/\"><strong>loi de Fourier<\/strong><\/a>\u00a0.\u00a0Notez qu&#8217;avec la g\u00e9n\u00e9ration de chaleur, le flux de chaleur n&#8217;est plus ind\u00e9pendant de x, donc:<\/p>\n<p><a href=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-condution-through-wall.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20233 lazy-loaded\" src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-condution-through-wall.png\" alt=\"conduction de chaleur \u00e0 travers le mur\" width=\"315\" height=\"178\" data-lazy-type=\"image\" data-src=\"https:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-condution-through-wall.png\" \/><\/a><\/p>\n<p><strong>Exemple d&#8217;\u00e9quation de chaleur &#8211; probl\u00e8me avec la solution<\/strong><\/p>\n<\/div>\n<\/div>\n<div class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights lgc-first lgc-last\">\n<div class=\"inside-grid-column\">\n<div class=\"su-spacer\"><\/div>\n<h2>Conduction de chaleur dans une barre de combustible<\/h2>\n<p>La plupart des REP utilisent le\u00a0<strong>combustible d&#8217;uranium<\/strong>\u00a0, qui est sous forme de\u00a0<strong>dioxyde d&#8217;uranium<\/strong>\u00a0.\u00a0Le dioxyde d&#8217;uranium est un solide semi-conducteur noir \u00e0 tr\u00e8s faible conductivit\u00e9 thermique.\u00a0En revanche, le dioxyde d&#8217;uranium a un point de fusion tr\u00e8s \u00e9lev\u00e9 et a un comportement bien connu.\u00a0L&#8217;UO\u00a0<sub>2<\/sub>\u00a0est press\u00e9 en\u00a0<strong>pastilles cylindriques<\/strong>\u00a0, ces pastilles sont ensuite fritt\u00e9es dans le solide.<\/p>\n<p>Ces\u00a0<strong>pastilles cylindriques<\/strong>\u00a0sont ensuite charg\u00e9s et encapsul\u00e9s dans une barre de combustible (ou aiguille de combustible), qui est constitu\u00e9 d&#8217;alliages de zirconium en\u00a0raison de sa tr\u00e8s faible absorption\u00a0<a href=\"https:\/\/www.nuclear-power.net\/neutron-cross-section\/\">section transversale<\/a>\u00a0(contrairement \u00e0\u00a0l&#8217;acier inoxydable).\u00a0La surface du tube, qui recouvre les pastilles, est appel\u00e9e\u00a0<strong>gaine de combustible<\/strong>\u00a0.<\/p>\n<p>Voir aussi:\u00a0\u00a0<a title=\"Conductivit\u00e9 thermique du dioxyde d'uranium\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/thermal-conduction\/thermal-conductivity\/thermal-conductivity-of-uranium-dioxide\/\">Conduction thermique du dioxyde d&#8217;uranium<\/a><\/p>\n<p><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Le<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0comportement\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">thermique<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0et\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">m\u00e9canique<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0des\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">pastilles<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0de\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">combustible<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0\u00a0et\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">des crayons combustibles<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0constitue l&#8217;une des trois principales disciplines de conception de base.\u00a0<\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">Le combustible nucl\u00e9aire<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0est exploit\u00e9 dans des conditions tr\u00e8s inhospitali\u00e8res (thermique, rayonnement, m\u00e9canique) et doit r\u00e9sister \u00e0 des conditions de fonctionnement plus \u00e9lev\u00e9es que la normale. <\/span><span style=\"vertical-align: inherit;\">Par exemple, les temp\u00e9ratures au centre des pastilles de combustible atteint plus de\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">1000 \u00b0C<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">\u00a0(1832\u00b0F) accompagn\u00e9s de rejets de gaz de fission. <\/span><span style=\"vertical-align: inherit;\">Par cons\u00e9quent, une connaissance d\u00e9taill\u00e9e de la distribution de la temp\u00e9rature dans un seul crayon de combustible est essentielle pour un fonctionnement s\u00fbr du combustible nucl\u00e9aire. <\/span><span style=\"vertical-align: inherit;\">Dans cette section, nous \u00e9tudions\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">l&#8217;\u00e9quation de conduction thermique<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> \u00a0en\u00a0 <\/span><\/span><strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">coordonn\u00e9es cylindriques<\/span><\/span><\/strong><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\"> en utilisant la condition aux limites de Dirichlet avec une temp\u00e9rature de surface donn\u00e9e (c&#8217;est-\u00e0-dire en la condition aux limites de Dirichlet). <\/span><span style=\"vertical-align: inherit;\">Une analyse compl\u00e8te du profil de temp\u00e9rature des crayons de combustible sera \u00e9tudi\u00e9e dans une section distincte.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><strong><span>Temp\u00e9rature dans l&#8217;axe d&#8217;une pastille de combustible<\/span><\/strong><\/h2>\n<p><span>Consid\u00e9rons la pastille de combustible de rayon\u00a0<\/span><strong><span>r\u00a0<\/span><sub><span>U<\/span><\/sub><span>\u00a0= 0,40 cm<\/span><\/strong><span>\u00a0, dans laquelle il y a une g\u00e9n\u00e9ration de chaleur uniforme et constante par unit\u00e9 de volume,\u00a0<\/span><strong><span>q\u00a0<\/span><sub><span>V<\/span><\/sub><span>\u00a0[W\/m\u00a0<\/span><sup><span>3<\/span><\/sup><span>\u00a0]<\/span><\/strong><span>\u00a0.\u00a0Au lieu du taux de chaleur volum\u00e9trique q\u00a0<\/span><sub><span>V<\/span><\/sub><span>\u00a0[W\/m\u00a0<\/span><sup><span>3<\/span><\/sup><span>\u00a0], les ing\u00e9nieurs utilisent souvent le\u00a0<\/span><strong><span>taux de chaleur lin\u00e9aire, q\u00a0<\/span><sub><span>L<\/span><\/sub><span>\u00a0[W\/m]<\/span><\/strong><span>\u00a0, qui repr\u00e9sente le taux de chaleur d&#8217;un m\u00e8tre de crayon combustible.\u00a0Le\u00a0<\/span><strong><span>taux de chaleur lin\u00e9aire<\/span><\/strong><span>\u00a0peut \u00eatre calcul\u00e9 \u00e0 partir du taux de chaleur volum\u00e9trique par :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/linear-heat-rate-vs-volumetric-heat-rate.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20236\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/linear-heat-rate-vs-volumetric-heat-rate.png\" alt=\"taux de chaleur lin\u00e9aire vs taux de chaleur volum\u00e9trique\" width=\"125\" height=\"49\" \/><\/a><\/p>\n<p><span>La ligne m\u00e9diane est prise comme origine pour la coordonn\u00e9e r.\u00a0En raison de la sym\u00e9trie dans la direction z et dans la direction azimutale, nous pouvons s\u00e9parer des variables et simplifier ce probl\u00e8me en\u00a0<\/span><strong><span>un probl\u00e8me unidimensionnel<\/span><\/strong><span>\u00a0.\u00a0Ainsi, nous allons r\u00e9soudre pour la temp\u00e9rature en fonction du rayon,\u00a0<\/span><strong><span>T(r)<\/span><\/strong><span>\u00a0, uniquement.\u00a0Pour une\u00a0<\/span><a title=\"Conductivit\u00e9 thermique\" href=\"https:\/\/www.thermal-engineering.org\/what-is-thermal-conductivity-definition\/\"><span>conductivit\u00e9 thermique<\/span><\/a><span>\u00a0constante\u00a0, k, la forme appropri\u00e9e de l&#8217;\u00a0<\/span><strong><span>\u00e9quation de la chaleur cylindrique<\/span><\/strong><span>\u00a0, est\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-2.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20237\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-2.png\" alt=\"\u00e9quation de la chaleur - cylindrique - 2\" width=\"503\" height=\"178\" \/><\/a><\/p>\n<p><span>La solution g\u00e9n\u00e9rale de cette \u00e9quation est :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-general-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20238\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-general-solution.png\" alt=\"\u00e9quation de la chaleur - cylindrique - solution g\u00e9n\u00e9rale\" width=\"323\" height=\"149\" \/><\/a><\/p>\n<p><span>o\u00f9 C\u00a0<\/span><sub><span>1<\/span><\/sub><span>\u00a0et C\u00a0<\/span><sub><span>2<\/span><\/sub><span>\u00a0sont les constantes d&#8217;int\u00e9gration.<\/span><\/p>\n<p><strong><a href=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Thermal-conduction-fuel-pellet.png\"><img loading=\"lazy\" class=\"alignright size-medium wp-image-20223\" src=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Thermal-conduction-fuel-pellet-244x300.png\" alt=\"Conduction thermique - pastille combustible\" width=\"244\" height=\"300\" \/><\/a><span>Calculer la distribution de temp\u00e9rature, T(r)<\/span><\/strong><span>\u00a0, dans cette pastille de combustible, si :<\/span><\/p>\n<ul>\n<li><span>la temp\u00e9rature \u00e0 la surface de la pastille de combustible est\u00a0<\/span><strong><span>T\u00a0<\/span><sub><span>U<\/span><\/sub><span>\u00a0= 420\u00b0C<\/span><\/strong><\/li>\n<li><span>le rayon des pastilles de combustible\u00a0<\/span><strong><span>r\u00a0<\/span><sub><span>U<\/span><\/sub><span>\u00a0= 4 mm<\/span><\/strong><span>\u00a0.<\/span><\/li>\n<li><span>la conductivit\u00e9 moyenne du mat\u00e9riau est\u00a0<\/span><strong><span>k = 2,8 W\/mK<\/span><\/strong><span>\u00a0(correspond au dioxyde d&#8217;uranium \u00e0 1000\u00b0C)<\/span><\/li>\n<li><span>le taux de chaleur lin\u00e9aire est\u00a0<\/span><strong><span>q\u00a0<\/span><sub><span>L<\/span><\/sub><span>\u00a0= 300 W\/cm<\/span><\/strong><span>\u00a0et donc le taux de chaleur volum\u00e9trique est q\u00a0<\/span><sub><span>V<\/span><\/sub><span>\u00a0= 597 x 10\u00a0<\/span><sup><span>6<\/span><\/sup><span>\u00a0W\/m\u00a0<\/span><sup><span>3<\/span><\/sup><\/li>\n<\/ul>\n<p><span>Dans ce cas, la surface est maintenue \u00e0 des temp\u00e9ratures donn\u00e9es T\u00a0<\/span><sub><span>U<\/span><\/sub><span>\u00a0.\u00a0Cela correspond \u00e0 la\u00a0<\/span><a title=\"Condition aux limites de Dirichlet \u2013 Condition aux limites de type I\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/thermal-conduction\/heat-conduction-equation\/dirichlet-boundary-condition-type-i-boundary-condition\/\"><span>condition aux limites de Dirichlet<\/span><\/a><span>\u00a0.\u00a0De plus, ce probl\u00e8me est thermiquement sym\u00e9trique et nous pouvons donc utiliser \u00e9galement\u00a0<\/span><a title=\"Limite adiabatique \u2013 Sym\u00e9trie thermique\" href=\"https:\/\/www.nuclear-power.net\/nuclear-engineering\/heat-transfer\/thermal-conduction\/heat-conduction-equation\/adiabatic-boundary-thermal-symmetry\/\"><span>la condition aux limites de sym\u00e9trie thermique<\/span><\/a><span>\u00a0.\u00a0Les constantes peuvent \u00eatre \u00e9valu\u00e9es par substitution dans la solution g\u00e9n\u00e9rale et sont de la forme :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-boundary-conditions.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20239\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-boundary-conditions.png\" alt=\"\u00e9quation de la chaleur - cylindrique - conditions aux limites\" width=\"337\" height=\"140\" \/><\/a><\/p>\n<p><span>La distribution de temp\u00e9rature r\u00e9sultante et la temp\u00e9rature centrale (r = 0) (maximum) dans cette pastille de combustible cylindrique \u00e0 ces conditions limites sp\u00e9cifiques seront\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20240\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cylindrical-solution.png\" alt=\"\u00e9quation de la chaleur - cylindrique - solution\" width=\"322\" height=\"162\" \/><\/a><\/p>\n<p><span>Le\u00a0<\/span><strong><span>flux de chaleur radial<\/span><\/strong><span>\u00a0\u00e0 n&#8217;importe quel rayon, q\u00a0<\/span><sub><span>r<\/span><\/sub><span>\u00a0[Wm\u00a0<\/span><sup><span>-1<\/span><\/sup><span>\u00a0], dans le cylindre peut, bien s\u00fbr, \u00eatre d\u00e9termin\u00e9 en utilisant la distribution de temp\u00e9rature et avec la\u00a0<\/span><a title=\"Loi de Fourier de la conduction thermique\" href=\"https:\/\/www.thermal-engineering.org\/what-is-fouriers-law-of-thermal-conduction-definition\/\"><span>loi de Fourier<\/span><\/a><span>\u00a0.\u00a0Notez que, avec la g\u00e9n\u00e9ration de chaleur, le flux de chaleur n&#8217;est plus ind\u00e9pendant de r.<\/span><\/p>\n<p><span>La figure suivante montre la r\u00e9partition de la temp\u00e9rature dans la pastille de combustible \u00e0 diff\u00e9rents niveaux de puissance.<\/span><\/p>\n<p><a href=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Temperature-distribution-nuclear-fuel.png\"><img loading=\"lazy\" class=\"aligncenter size-large wp-image-20224\" src=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Temperature-distribution-nuclear-fuel-1024x465.png\" alt=\"Distribution de temp\u00e9rature - combustible nucl\u00e9aire\" width=\"669\" height=\"304\" \/><\/a><\/p>\n<p><span>______<\/span><\/p>\n<p><span>La temp\u00e9rature dans un r\u00e9acteur en fonctionnement varie d&#8217;un point \u00e0 un autre dans le syst\u00e8me.\u00a0En cons\u00e9quence, il y a toujours\u00a0<\/span><strong><span>un crayon combustible<\/span><\/strong><span>\u00a0et\u00a0<\/span><strong><span>un volume local<\/span><\/strong><span>\u00a0, qui sont\u00a0<\/span><strong><span>plus chauds\u00a0<\/span><\/strong><span>\u00a0que tous les autres.\u00a0Afin de limiter ces\u00a0<\/span><strong><span>points chauds,<\/span><\/strong><span>\u00a0des\u00a0<\/span><a title=\"Facteurs de canal chaud - Facteurs de cr\u00eate\" href=\"https:\/\/www.nuclear-power.net\/nuclear-power\/reactor-physics\/reactor-operation\/normal-operation-reactor-control\/hot-channel-factors-peaking-factors\/\"><strong><span>limites de puissance cr\u00eate<\/span><\/strong><\/a><span>\u00a0doivent \u00eatre introduites.\u00a0Les limites de puissance de pointe sont associ\u00e9es \u00e0 une\u00a0<\/span><strong><span>crise d&#8217;\u00e9bullition<\/span><\/strong><span>\u00a0et aux conditions pouvant provoquer la fonte des pastilles de combustible.\u00a0Cependant, des consid\u00e9rations m\u00e9tallurgiques imposent une\u00a0<\/span><strong><span>limite sup\u00e9rieure<\/span><\/strong><span>\u00a0\u00e0 la temp\u00e9rature de la gaine du combustible et de la pastille de combustible.\u00a0Au dessus de ces\u00a0<\/span><a title=\"Qu'est-ce que la temp\u00e9rature - Physique\" href=\"https:\/\/www.thermal-engineering.org\/what-is-temperature-physics-definition\/\"><span>temp\u00e9ratures<\/span><\/a><span>il y a un risque que le carburant soit endommag\u00e9.\u00a0L&#8217;un des objectifs majeurs dans la conception d&#8217;un r\u00e9acteur nucl\u00e9aire est de pr\u00e9voir l&#8217;\u00e9vacuation de la chaleur produite au niveau de puissance souhait\u00e9, tout en garantissant que la temp\u00e9rature maximale du combustible et la temp\u00e9rature maximale de la gaine soient toujours inf\u00e9rieures \u00e0 ces valeurs pr\u00e9d\u00e9termin\u00e9es.<\/span><\/p>\n<h2><span>R\u00e9partition de la temp\u00e9rature dans la gaine de combustible<\/span><\/h2>\n<p><strong><span>La gaine<\/span><\/strong><span>\u00a0\u00a0est la couche externe des barres de combustible, situ\u00e9e entre le\u00a0<\/span><strong><span>liquide de refroidissement<\/span><\/strong><span>\u00a0du\u00a0\u00a0<strong>r\u00e9acteur<\/strong>\u00a0\u00a0et le\u00a0\u00a0<\/span><strong><span>combustible nucl\u00e9aire<\/span><\/strong><span>\u00a0\u00a0(c&#8217;est-\u00e0-dire\u00a0\u00a0<\/span><strong><span>les pastilles de combustible<\/span><\/strong><span>\u00a0).\u00a0Il est constitu\u00e9 d&#8217;un mat\u00e9riau r\u00e9sistant \u00e0 la corrosion avec une faible section efficace d&#8217;absorption pour les neutrons thermiques, g\u00e9n\u00e9ralement en\u00a0\u00a0<\/span><strong><span>alliage de zirconium<\/span><\/strong><span>\u00a0.\u00a0<\/span><strong><span>Les gaines<\/span><\/strong><span>\u00a0\u00a0emp\u00eachent les produits de fission radioactifs de s&#8217;\u00e9chapper de la matrice combustible dans le caloporteur du r\u00e9acteur et de le contaminer.\u00a0Les bardages constituent l&#8217;une des barri\u00e8res de\u00a0l&#8217;approche\u00a0\u00ab\u00a0<\/span><strong><span>d\u00e9fense en profondeur<\/span><\/strong><span>\u00a0\u00bb.<\/span><\/p>\n<p><span>Consid\u00e9rons la gaine de combustible de rayon int\u00e9rieur\u00a0\u00a0<\/span><strong><span>r\u00a0<\/span><sub><span>Zr,2<\/span><\/sub><span>\u00a0\u00a0= 0,408 cm<\/span><\/strong><span>\u00a0\u00a0et de rayon ext\u00e9rieur\u00a0\u00a0<\/span><strong><span>r\u00a0<\/span><sub><span>Zr,1<\/span><\/sub><span>\u00a0\u00a0= 0,465 cm<\/span><\/strong><span>\u00a0.\u00a0Par rapport aux pastilles de combustible, il n&#8217;y a pratiquement pas de g\u00e9n\u00e9ration de chaleur dans la gaine du combustible (la gaine est\u00a0\u00a0<\/span><a title=\"Lib\u00e9ration d'\u00e9nergie de la fission\" href=\"https:\/\/www.nuclear-power.net\/nuclear-power\/fission\/energy-release-from-fission\/\"><span>l\u00e9g\u00e8rement chauff\u00e9e par rayonnement<\/span><\/a><span>\u00a0).\u00a0Toute la chaleur g\u00e9n\u00e9r\u00e9e dans le carburant doit \u00eatre transf\u00e9r\u00e9e par\u00a0\u00a0<\/span><strong><span>conduction \u00e0<\/span><\/strong><span>\u00a0\u00a0travers le rev\u00eatement et, par cons\u00e9quent, la surface int\u00e9rieure est plus chaude que la surface ext\u00e9rieure.<\/span><\/p>\n<p><span>Pour trouver la distribution de temp\u00e9rature \u00e0 travers le rev\u00eatement, nous devons r\u00e9soudre l&#8217;\u00a0\u00a0<\/span><strong><span>\u00e9quation de conduction thermique<\/span><\/strong><span>\u00a0.\u00a0En raison de la sym\u00e9trie dans la direction z et dans la direction azimutale, nous pouvons s\u00e9parer les variables et simplifier ce probl\u00e8me en un probl\u00e8me unidimensionnel.\u00a0Ainsi, nous allons r\u00e9soudre pour la temp\u00e9rature en fonction du rayon, T(r), uniquement.\u00a0Dans cet exemple, nous supposerons qu&#8217;il n&#8217;y a strictement aucune g\u00e9n\u00e9ration de chaleur \u00e0 l&#8217;int\u00e9rieur du rev\u00eatement.\u00a0Pour une conductivit\u00e9 thermique constante, k, la forme appropri\u00e9e de l&#8217;\u00e9quation de la chaleur cylindrique, est\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20241\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding.png\" alt=\"\u00e9quation de la chaleur - rev\u00eatement\" width=\"190\" height=\"72\" \/><\/a><\/p>\n<p><span>La solution g\u00e9n\u00e9rale de cette \u00e9quation est :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding-general-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20242\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding-general-solution.png\" alt=\"\u00e9quation de la chaleur - gaine - solution g\u00e9n\u00e9rale\" width=\"245\" height=\"133\" \/><\/a><\/p>\n<p><span>o\u00f9 C\u00a0<\/span><sub><span>1<\/span><\/sub><span>\u00a0\u00a0et C\u00a0<\/span><sub><span>2<\/span><\/sub><span>\u00a0\u00a0sont les constantes d&#8217;int\u00e9gration.<\/span><\/p>\n<p><span>1)<\/span><\/p>\n<p><a href=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Thermal-conduction-fuel-pellet.png\"><img loading=\"lazy\" class=\"alignright size-medium wp-image-20223\" src=\"http:\/\/thermal-engineering.org\/wp-content\/uploads\/2019\/05\/Thermal-conduction-fuel-pellet-244x300.png\" alt=\"Conduction thermique - pastille combustible\" width=\"244\" height=\"300\" \/><\/a><span>Calculer la distribution de temp\u00e9rature, T(r), dans cette gaine de combustible, si :<\/span><\/p>\n<ul>\n<li><span>la temp\u00e9rature \u00e0 la surface int\u00e9rieure du rev\u00eatement est T\u00a0<\/span><sub><span>Zr,2<\/span><\/sub><span>\u00a0\u00a0= 360\u00b0C<\/span><\/li>\n<li><span>la temp\u00e9rature du fluide caloporteur \u00e0 cette coordonn\u00e9e axiale est T\u00a0<\/span><sub><span>bulk<\/span><\/sub><span>\u00a0\u00a0= 300\u00b0C<\/span><\/li>\n<li><span>le coefficient de transfert thermique (convection ; \u00e9coulement turbulent) est h = 41 kW\/m\u00a0<\/span><sup><span>2<\/span><\/sup><span>\u00a0.K.<\/span><\/li>\n<li><span>la conductivit\u00e9 moyenne du mat\u00e9riau est k = 18 W\/mK<\/span><\/li>\n<li><span>le taux de chaleur lin\u00e9aire du combustible est q\u00a0<\/span><sub><span>L<\/span><\/sub><span>\u00a0\u00a0= 300 W\/cm et donc le taux de chaleur volum\u00e9trique est q\u00a0<\/span><sub><span>V<\/span><\/sub><span>\u00a0\u00a0= 597 x 10\u00a0<\/span><sup><span>6<\/span><\/sup><span>\u00a0\u00a0W\/m\u00a0<\/span><sup><span>3<\/span><\/sup><\/li>\n<\/ul>\n<p><span>A partir de la relation de base pour le transfert de chaleur par convection, nous pouvons calculer la surface ext\u00e9rieure du rev\u00eatement comme suit\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/newtons-law-cladding.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20269\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/newtons-law-cladding.png\" alt=\"loi de newtons - bardage\" width=\"423\" height=\"299\" \/><\/a><\/p>\n<p><span>Comme on peut le voir, \u00e9galement dans ce cas, nous avons donn\u00e9 les temp\u00e9ratures de surface T\u00a0<\/span><sub><span>Zr,1<\/span><\/sub><span>\u00a0\u00a0et T\u00a0<\/span><sub><span>Zr,2<\/span><\/sub><span>\u00a0.\u00a0Cela correspond \u00e0 la condition aux limites de Dirichlet.\u00a0Les constantes peuvent \u00eatre \u00e9valu\u00e9es par substitution dans la solution g\u00e9n\u00e9rale et sont de la forme :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-general-solution-cladding.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20264\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-general-solution-cladding.png\" alt=\"\u00e9quation de la chaleur - solution g\u00e9n\u00e9rale - gaine\" width=\"352\" height=\"53\" \/><\/a><\/p>\n<p><span>En r\u00e9solvant C\u00a0<\/span><sub><span>1<\/span><\/sub><span>\u00a0\u00a0et C\u00a0<\/span><sub><span>2<\/span><\/sub><span>\u00a0\u00a0et en substituant dans la solution g\u00e9n\u00e9rale, on obtient alors :<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding-solution.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20265\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/heat-equation-cladding-solution.png\" alt=\"\u00e9quation de la chaleur - gaine - solution\" width=\"381\" height=\"84\" \/><\/a><\/p>\n<p><strong><span>\u2206T &#8211; surface de rev\u00eatement &#8211; liquide de refroidissement<\/span><\/strong><\/p>\n<p><span>Une connaissance d\u00e9taill\u00e9e de la g\u00e9om\u00e9trie, du rayon ext\u00e9rieur de la gaine, du taux de chaleur lin\u00e9aire, du coefficient de transfert de chaleur par convection et de la temp\u00e9rature du liquide de refroidissement d\u00e9termine\u00a0\u00a0<\/span><strong><span>\u2206T\u00a0<\/span><\/strong><span>\u00a0entre le liquide de refroidissement (T\u00a0<\/span><sub><span>vrac<\/span><\/sub><span>\u00a0) et la surface de la gaine (T\u00a0<\/span><sub><span>Zr,1<\/span><\/sub><span>\u00a0).\u00a0On peut donc calculer la temp\u00e9rature de surface de la gaine (T\u00a0<\/span><sub><span>Zr,1<\/span><\/sub><span>\u00a0) simplement en utilisant la\u00a0\u00a0<\/span><strong><span>loi de Newton<\/span><\/strong><span>\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/\u2206T-cladding-surface-coolant-newtons-law.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20266\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/\u2206T-cladding-surface-coolant-newtons-law.png\" alt=\"\u2206T - surface de gaine - r\u00e9frig\u00e9rant - loi de Newtons\" width=\"425\" height=\"306\" \/><\/a><\/p>\n<p><strong><span>\u2206T dans la gaine du combustible<\/span><\/strong><\/p>\n<p><span>Une connaissance d\u00e9taill\u00e9e de la g\u00e9om\u00e9trie, du rayon ext\u00e9rieur et int\u00e9rieur du rev\u00eatement, du taux de chaleur lin\u00e9aire et de la temp\u00e9rature de surface du rev\u00eatement (T\u00a0<\/span><sub><span>Zr,1<\/span><\/sub><span>\u00a0) d\u00e9termine\u00a0\u00a0<\/span><strong><span>\u2206T\u00a0<\/span><\/strong><span>\u00a0entre les surfaces ext\u00e9rieure et int\u00e9rieure du rev\u00eatement.\u00a0Par cons\u00e9quent, nous pouvons calculer la temp\u00e9rature de surface de la gaine int\u00e9rieure (T\u00a0<\/span><sub><span>Zr,2<\/span><\/sub><span>\u00a0) simplement en utilisant la\u00a0\u00a0<\/span><strong><span>loi de Fourier<\/span><\/strong><span>\u00a0:<\/span><\/p>\n<p><a href=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/\u2206T-in-fuel-cladding-fouriers-law.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-20267\" src=\"https:\/\/www.thermal-engineering.org\/wp-content\/uploads\/2019\/05\/\u2206T-in-fuel-cladding-fouriers-law.png\" alt=\"\u2206T en gaine combustible - loi de fouriers\" width=\"370\" height=\"460\" \/><\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution.\u00a0Dans cet article, il y a deux exemples d&#8217;\u00e9quation de solution de chaleur.\u00a0Les deux exemples sont avec solution.\u00a0G\u00e9nie thermique Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution Conduction thermique dans un grand mur plan Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution Consid\u00e9rons la paroi plane d&#8217;\u00e9paisseur 2L, dans laquelle il &#8230; <a title=\"Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution &#8211; D\u00e9finition\" class=\"read-more\" href=\"https:\/\/www.thermal-engineering.org\/fr\/exemple-dequation-thermique-probleme-de-solution-definition\/\" aria-label=\"En savoir plus sur Exemple d&#8217;\u00e9quation thermique &#8211; Probl\u00e8me de solution &#8211; D\u00e9finition\">Lire la suite<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[8],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v15.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Exemple d&#039;\u00e9quation thermique - Probl\u00e8me de solution - D\u00e9finition<\/title>\n<meta name=\"description\" content=\"Exemple d&#039;\u00e9quation thermique - 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