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½ ÔÖØÙÖÖÒØÓÚÖØÐÑÒØÒØÖÒÓÖÑÐØÓØÖØÓÒÓ ¾ÌÀÇÊ Ì××ÜÔÖ××ÒÓÙÖÖ³×ÐÛÓØÓÒÙØÓÒº ØØÖÒ×Ö¸ÙØÒÚÖ×ÐÝÔÖÓÔÓÖØÓÒÐØÓØÐÑÒØØÒ××º ÌÖÒØÐÓÖÑ×ÚÒÝÐØØÒ∆Ü→¼ ´½µ ˙Q = −kA ∆T ÏÖk×ØØÖÑÐÓÒÙØÚØÝÓØÑØÖÐº ´½µ ∆x ÌÖ××ØÒ×ÔÓØÛÐÒÔÖÓ×××ÓØÒÑÓÐÐÒÝÐÒÖÐÓÓÖ¹ ÒØ×ÛØ×ÝÑÑØÖÝÖÓÙÒØÞ¹Ü×¸Û×ÖØÓÙØÓØ×Ø ÔÐÒºÀÒØØÓÒÙØÓÒÕÙØÓÒÒÝÐÒÖÐÓÓÖÒØ×ÛØ ÒÒØÖÒÐØÒÖØÓÒÒ×ØÓÙÖÓÑØÓÙÖÖ³×ÐÛÓ ØÓÒÙØÓÒº Þ Ö ÓÒ×ÖÒÒÒØ×ÑÐ×ÓÖØÒØÒÝÐÒÖÐÐÑÒØ×Òº¸ÛØ ÙÖÝÐÒÖÐÐÑÒØ Ò×ØÝ̺¸×ÔØCÒÚÓÐÙÑV¾πÖ∆Ö∆Þº ÌÒÖÝÐÒÓØ×ÐÑÒØÙÖÒ×ÓÖØØÑÒØÖÚÐ∆Ø¹ ÅÓÐÐÒÓØÊ××ØÒËÔÓØÏÐÒÈÖÓ×× ´¾¼µ ÓÑ×∆E element = ∆t ˙Q r − ˙Q r+∆r + ˙Q z − ˙Q z+∆z + Ġelement ˙Q = −kA dT dx ∆r ∆z ˙Q z+∆z ˙Q r ˙Q r+∆r ˙Q z
¾ºÀØÌÖÒ×Ö ´Ö×ØÐÛÓØÖÑÓÝÒÑ×µ¸ÛÖ˙Q×ØØØÖÒ×ÖÖØÒĠ× ½ ØØÒÖØÓÒÖØº ÌÒÒÒÖÝÓØÐÑÒØÒÜÔÖ××ÒØÖÑ×ÓØÑ××m¸ Ø×ÔØC´ØÒÖÝÖÕÙÖØÓÖ×ØØÑÔÖØÙÖÓ×Ô ´¾½µ ÕÙÒØØÝÓ×Ù×ØÒÝ×ÔÑÓÙÒØµÒØÒÒØÑÔÖØÙÖº Ġ = I 2 R ÌØÒÖØÓÒÖØÒÜÔÖ××ÒØÖÑ×ÓØÒÖØÓÒÖØ ÔÖÚÓÐÙÑġº )´¾¾µ ∆E element = E t+∆t −E t = mC(T t+∆t −T t ) = ̺C2πr∆r∆z(T t+∆t −T ÛÖġÒÜÔÖ××× t ´¾¿µ ÆÓÛÛØÕº´¾¾µÒ´¾¿µÒÕº´¾¼µØÒÖÝÐÒÒÛÖØØÒ Ġ element = V element ´¾µ ġ = 2πr∆r∆zġ ġ = J 2 ρ ÚÒÝ¾πÖ∆Ö∆ÞÚ× ġ´¾µ ÐØØÒ∆Ö¸∆Þ¸∆Ø→¼ ÆÓÛ¸ÝÙ×ÒØÒØÓÒÓÖÚØÚ×ÒÓÙÖÖ³×ÐÛÓØÓÒ¹ ÙØÓÒ¸ØØÖÑ×ÒÕº´¾µÒÜÔÖ××ÒØÖÒØÐÓÖÑÝ + ´¾µ ´¾µ ÓÚ lim ∆r→0 lim ∆z→0 ρC2πr∆r∆z T t+∆t − T t ∆t ρC T t+∆t − T t = − 1 ∆t 2πr∆z 1 2πr∆z 1 2πr∆r ˙Q r+∆r − ˙Q r ∆r ˙Q z+∆z − ˙Q z ∆z = ˙Q r − ˙Q r+∆r + ˙Q z − ˙Q z+∆z + 2πr∆r∆zġ´¾µ ˙Q r+∆r − ˙Q r − 1 ∆r 2πr∆r 1 = 2πr∆z = − 1 r = ∂ ˙Q r ∂r = 1 2πr∆z ∂ ∂r (kr∂T ∂r ) 1 ∂ ˙Q z 2πr∆r ∂z = 1 2πr∆r = − ∂ ∂z (k∂T ∂z ) ˙Q z+∆z − ˙Q z ∆z ∂ ∂r (−2πr∆z∂T ∂r ) = ∂ ∂z (−2πr∆r∂T ∂z ) =
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