Transf de calor
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❯❊❘❏ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❞♦ ❊st❛❞♦ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦
Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼♦❞❡❧❛❣❡♠ ❈♦♠♣✉t❛❝✐♦♥❛❧
❉✐s❝✐♣❧✐♥❛✿ ❚r❛♥s❢❡rê♥❝✐❛ ❞❡ ❈❛❧♦r ❈♦♠♣✉t❛❝✐♦♥❛❧
Pr♦❢❡ss♦r✿ ❆♥tô♥✐♦ ❏♦sé ❙✐❧✈❛ ◆❡t♦
❆❧✉♥♦✿ ▼❛rq✉❡s ❋r❡❞♠❛♥ ▼❡s❝♦❧✐♥
✶❛ ▲✐st❛ ❞❡ ❊①❡r❝í❝✐♦s
✶✳ ■♥tr♦❞✉çã♦
❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s r❛③õ❡s ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s é q✉❡ ♠❡s♠♦ ❛s
❡q✉❛çõ❡s ♠❛✐s s✐♠♣❧❡s sã♦ ❝❛♣❛③❡s ❞❡ r❡♣r❡s❡♥t❛r s✐st❡♠❛s út❡✐s ❡ ❛ss✐♠✱ ♦ ❡st✉❞♦ ❞❛
♠♦❞❡❧❛❣❡♠ ❞❡st❛s t❡♠ ♣❛♣❡❧ r❡❧❡✈❛♥t❡ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ♣r♦❜❧❡♠❛s r❡❛✐s✳
❖ ♣r♦❜❧❡♠❛ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡ ❡♠ s✉♣❡r❢í❝✐❡s ❡st❡♥❞✐❞❛s
✭❛❧❡t❛s✮ ✉♥✐❞✐♠❡♥s✐♦♥❛✐s✱ ❞✐s❝✉t✐❞♦ ❡♠ ❙✐❧✈❛ ◆❡t♦ ❡t ❛❧✳✱ ✭✷✵✵✸✮ q✉❛♥❞♦ ❛ ár❡❛ tr❛♥s✈❡rs❛❧ é ❝♦♥st❛♥t❡✱ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ d2 T (x)
− m(T − Tamb ) = 0, dx2 m=
hP
,
kA
✭✶✮
s❡♥❞♦ T (x) ❛ t❡♠♣❡r❛t✉r❛ ❛♦ ❧♦♥❣♦ ❞❛ ❛❧❡t❛✱ Tamb ❛ t❡♠♣❡r❛t✉r❛ ❞♦ ❛♠❜✐❡♥t❡✱ h ♦
❝♦❡✜❝✐❡♥t❡ ❞❡ tr♦❝❛ ❞❡ ❝❛❧♦r ♣♦r ❝♦♥✈❡❝çã♦✱ k ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ❞♦ ♠❛t❡r✐❛❧✱ P
♦ ♣❡rí♠❡tr♦ ❡ A ❛ ár❡❛ tr❛♥s✈❡rs❛❧ ❞❡st❛✳ ◆❡st❡ tr❛❜❛❧❤♦✱ s❡rá ❞✐s❝✉t✐❞❛ ✉♠❛ s♦❧✉çã♦
♥✉♠ér✐❝❛ ♣❛r❛ ❛ r❡❢❡r✐❞❛ ❡q✉❛çã♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❞❛❞♦s ✐♥✐❝✐❛✐s ♣ré✲❢♦r♥❡❝✐❞♦s✱ ✉t✐❧✐③❛♥❞♦
♦ ♠ét♦❞♦ ❞❛s ❞✐❢❡r♥❡ç❛s ✜♥✐t❛s✳
✷✳ ❋♦r♠✉❧❛çã♦ ▼❛t❡♠át✐❝❛
❈♦♥s✐❞❡r❡ ✉♠❛ ❛❧❡t❛ ❝♦♠ s❡❝çã♦ tr❛♥s✈❡rs❛❧ ❝✐r❝✉❧❛r✱ r❡♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳
πD2
❚ê♠✲s❡ A =
❡ P = πD✱ s❡♥❞♦ D ♦ ❞✐â♠❡tr♦✳ ❈♦♥s✐❞❡r❡ ❛✐♥❞❛ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦
4
❞❡ ❉✐r✐❝❤❧❡t✱ T (x)|x=0 = T0 ❡ T (x)|x=L = TN ✳
❈♦♠ ♦s ✈❛❧♦r❡s T0 = 100o C ✱ TN = 25o C ✱ Tamb = 20◦ C ✱ h = 50W/m2 k ✱ D = 5×10−2 m✱
L = 0, 1m ❡ k = 200W/mK ✱ ❛ ❡q✉❛çã♦ ✭✶✮ s❡rá r❡❡s❝r✐t❛ d2 T
− 20T (x) = 400. dx2 ✶
✭✷✮
❆ ❡①♣❛♥sã♦ ❞❡ T ❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦ x0 ❝♦♠ ✐♥❝r❡♠❡♥t♦ h✱ ❛tr❛✈és ❞❛ ❙ér✐❡ ❞❡ ❚❛②❧♦r
♣❛r❛ s♦♠❛ ❡ ❞✐❢❡r❡♥ç❛✱ ❣❡r❛ ❛s ❡q✉❛çõ❡s
T (x0 + h) = T (x0 ) +
T (x0 − h) = T (x0 ) −
dT dx dT dx h+ x0 h+ x0 d2 T dx2 d2 T dx2 d3 T h2 +
2!
dx3
x0
x0
d3 T h2 −
2!
dx3
d4 T h3 +
3!
dx4
x0
x0
d4