Chapitre 7 :Transfert d`énergie par rayonnement
Transcription
Chapitre 7 :Transfert d`énergie par rayonnement
T his work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. " ! # " ! $% ( " & ) ') ! '(' ! # *+ = "ω − , * " " ". * "0 # *+ # ) - - # / + ( ω 1 = π 1λ = π ω # ! *+" $ 3 # # % 2 ) ( * +! ) = ω # ! λ= = ν # $ = ε, = + µ, + Ismaël Bouya http://melusine.eu.org/syracuse/immae/ # 2 4 • 5 • ) # * ν - ' ε = ν 6 = 8!8 # , −.0 7# = ν ) 9 ' ν = : ; 4 ) # = # ) % < = • $ ) ( ' = % # " ) >! ) 9 ' = 4 ' = • + ' + γ " ) ?) 9 ) 3 ' ! " ' @ 3 # 9 ) ! # ( " 4 ( # ! # ! #$ 9 ) % ! # =4 * ! ) ) ) ! ) 9 ) A ! 9 9 !9 ! $ / * # ) ! 9 # ) # # 9 • $ ) # ! =4 B C → B + ν " ν = ε −ε @ B + ν →B C ) + ν ∆ε • # + " ! D = " + ∧ E ( ! ) * # 4 4 # 9 9 ! ) ) % ! > ) ! # . % • 3 ' =4 ϕ" : = ϕ ", "φ = − # <: <3 ! 6ϕ " = ! 9 φ" ! 4 # ⋅ ϕ φ" φ ", 4 " $ 9 δ= 9 # 1 9 9 9# ' ; ϕ =− ϕ !ϕ G! ) ( =− $ F " 6 ϕ ! " − 4 !ϕ 4 9 = ) # "− 9 9 . ! 99 9 9 H ( #= ) ' " 3 9 # : ϕ" $ " ϕ" − = ϕ ", = ϕ ", , − ( ε ( 1 4 ) 3 ε ". 9 9 ϕ ", α ) α = ! " ) ) # ! 6ε = α 4 " ' ' 3# ϕ" ϕ" =( ϕ ", = "0 = ! ε ) ( ! 4 = "ε + ε # ! ) ) # 4 λα ! λβ ! % ! ε "λα = , J • @ ' $ D D ' ! λβ 4 J α β = "ε "λα = "ε "λβ + ε "λα + ε "λβ I ε "λ ! ε "λ β = , # # λα 4 # = +∞ # = ,# 4 ! ∀ !ϕ " = , " δ = , ! ∀ ! ϕ " = ϕ ", " δ = +∞ ! 4 # ' 9 * δ << ϕ 3 E # 0 * δ >> ϕ 3 ' E δ 4 # λ# E ) # ! ' δ # ! 3 9 9 ) 2 4L # 4 9 9# L @ Σ 2 4L L 4 # 2 4 • @ δ .φ λ 4 Ω ' 3 λ+ λ! # λ δ .φ = θ # # λ # Ω 6! λ θ θ " λ = λ "λ ! ! " O λ N = M# − #µ ' 3 3 9 " ! D ) − # − # λ ) 3 9 # K + ) E ) F θ G λ # ) % ! & ! δ φ = λ "λ θ# # ! 4 9 ! λ# λ θ # # λ# Ω . " '(' 3 ( ( 9 ) ) ( # 2 4 • +∞ δ φ = "λ λ λ , θ# # Ω 3 = +∞ " λ λ " O N = M# λ , !δ φ = − − # θ# # Ω # . 2 4 • @ δ φ = θ Ω ? λ "λ # # λ λ θ Ω= ? ! λ • θ# π , λ # ) 3 9 # λ# 4 2 4 # +∞ δφ = θ# θ = π = π λ "λ " = δ φ = λ π λ , # λ # =4 = +∞ , λ # λ = π# : 2 4 J 2 4 Σ 8 δ .φ λ λ+ λ δφ 9 ! 4 ) Ω 4 # ) # 4 4 ' * ' * % 9 9# δ .φ = δ .φ + δ .φ # • 9 δ .φ αλ = . # δ φ αλ ; ) 4 9 λ! " % ! αλ ! αλ ! " #" 6 4 ' 9 + 9 α= δφ δφ 4 4 + 4 J 2 4 $ # "=4 % # "=4 ! ( 4 4 2 4 δφ • " , ≤ αλ ≤ ' • % % ) 9 ! 3 4 9 E 4 $ 9 ( ) # "3 % % ! δ φ = δφ − δφ • $ δ φ = δφ − δφ " δφ 4 4 ! & '(' ( • 4 = + + + ⋅ =ϕ = ⋅ = −λ # • δφ ∂ ∂ ⋅ = " − < 4 δ φ = δ φ "ϕ = ϕ $ δφ − δφ = δφ − δφ ! δφ − δφ = δφ − δφ # ∂ ∂ ∂ −λ = −λ −λ = " − ∂ ,− ∂ ,+ ∂ ,− • ; 4 # 4 4 # ; ϕ ϕ δ →, δ 3 @ # + $ −λ + ∂ ∂ ! =ϕ −λ ,− ∂ ∂ ,+ + P + 9 $ • 3 ) ! 9 9 # R R ( 9 ) % ( " 1 ! ! 9 9 4 • 9 R# ) 4 % ( 9 # • ' 3 ' " = " = ( % E % 9 9 $ # E ! # ) ! = ) • $ 9 τ ! = τ ! τ# != " = ε, " = • + ν τ ∈ τ @ ) ' µ, ) λ ν ν= λ λ δ != ! λ = λ τ " +∞ , λ != δ ! λ λ λ λ Q ' ν δ != ν τ! ν ' + λ ν= λ , = ! $ λ ν λ# λ ν λ +∞ = . 3 − ν , ν − ν= #$ +∞ = λ= +∞ , λ λ# ν ν λ S • =4 ( ν = Pπ ν . ν • ' ' 3 ' ν = " ν 9 ( . − # ν # "ν ! "ν ν ν" " ) ∝ν . ν "ν ν < < . ! ν" T ν" +∞ = ! , ν ν # # S = • ' T ' + ' * " = " ) " " Q,,! ) H( ν = ν "ν ! ! , 1 ) ) ( J <= : " • ' %# # 2 ν 4 Pπ = ν " . " 3 " = + <7 " Q,, ) % +∞ ν , ) ν = +∞ ν << 9 " S ) 4 ' ν = ν M ( " PQ. . − #ν ) ν >> 9 ' λ" = # ν ! = = ! λ = λ Pπ . ! #= # # λ" = .,,U ! λ = 0Pµ = .,,,U ! λ = 0!Pµ 0 3 * • <: R =4 = = +∞ ν , ν #= Pπ 0 . '(' σ = Pπ K K . . <: R # = ν ! 9 − , π0 K Pπ K = K . . 0 9 . +∞ = & ) 0 × 0 =σ * 0 • σ = ,! K8080# , − K 7# −. #U −0 # = .,,U ! = 8# , −8 7# −. . . " R !!= = $! R = ,,, U ! = ! # , − 7# −. # = 8,,,U " ! = 7# . ! = , 7# −. # = , U " = −. . ! 9 K 3 9 • $ δ .φ = , " δ .φ , = ) θ , λ % ( 9 ( # Ω# λ ) 9 "( • =4 V % ) ) ! # Ω θ ' 5 ' 4 ' ) λ λ , = , λ λ "λ ! = 0π λ θ # # Ω# λ # θ# λ 4 ) 9 , δφ , = , σ , θ # ' 4 = θ# Ω = λ 3 =σ = Ω # # 0π λ H ' =4 • λ , δ φ, = , λ 0π + • " ' $ λ % ( ) !δ φ = , Ω ( # . " ) Ω " 0π λ λ * θ # # , , ! π? θ# π# λ , = , λ λ= θ# θ = π# , λ 0 * 0 , #$ * = = σ 0 =σ 0 # 0 0 ! σ = K!8 # , −P M# − #U −0 # = 0 λ 4 • ( δφ = δφ , = 4 # = δφ + δφ = , + δφ ! , $ ≤ , " ) ≥, 9 9 3 ! 8 3 M • = " " = λ , λ ν λ = Pπ λ "λ Pπ . = λ. . − λ λK λ − λ ∆λ λ λ" ) λ λ =,⇔ * " = 9 − = = 0!Q8K λ # K λ" = PQPµ #U # ! λ" ";+! ) 9 ! = P! , ! ∆λ# = . PPµ #U # $ 6 M ! D5 = ! .! λ ! " = ,!P. # ∆λ $ P#λ" λ" λ λ = ,!QP# +∞ , λ λ# • λ" .,,µ • .0 µ + Q80 M 3 ,µ ! µ ,!Kµ R ) .# , − 0 µ ) ) ! ! . " = ! ) U λ λ" = 9 ) ) ) W " 9 '9 4 ) X " Q0P % ) # : 3 $ αλ = ! "αλ = 9 9 # # δφ δ .φ . = ( • 3 δ .φ = 9 θ # # Ω# λ %& λ = δ φ = . λ , θ # # Ω# λ = δ .φ + δ .φ , λ =, $ ' ' 3 • = %& λ , λ # ! 3 9 # 9 9 # ! =4 %& λ • 3 , λ = 0 λ = M %& , λ %& ! λ = = 0 , λ ! =σ 0 ! * ) # λ" = PQPµ #U # . = 9 • V % # 3 • 3 4 '" ( ! =4 9 %& H ) E = , ! %& = , =σ 0 ! λ" = PQPµ #U # # 0 " ! λ" ≈ QPQµ #U " λ" = ,!Kµ "H > 9 < # = = ≈ 8,,,U # ! 0 + ! • • • * 3 # ) 9 # αλ = A 9 # 3 9 9 # $ ' D ! E E 9 9 ) # ' 3 9 # 3 αλ ≠ # = ) • $ δ .φ = = δ .φ , = $ λ ≤ θ # # Ω# λ λ θ # # Ω# λ = δ .φ + δ .φ , λ , λ # ) λ = ελ , λ # λ λ ελ = • , ≤ ελ ≤ ελ % # # K ! ελ = # • =4 λ = π# , = λ " λ = π# $ =4 = λ 3 , λ # = ελ# , # λ λ = ελ λ 9 λ# , λ = ε# ) = ε #σ # ! 0 # # ( ! " 3 , 9 ! ( ) 9 9 # U αλ = ε λ $ % ( ! • : " 9 δφ = δφ " δφ = λ λ# ( δφ = α λ $ 9 H( ) , λ ( = (ε λ# # H ( ελ ) , λ 9 ) λ λ# λ = αλ λ • : ε λ λ, = α λ λ, 9 9 ! ε λ = αλ # • ) # , λ # λ , ( ) E E 9 ! # ( 9 ! . = αλ 9 9 # λ ! ' * αλ = , ! , λ # λ ! =, 4 # 8 , ' * J λ ( =, " 4 % 9H ) 9 ! H ) 9 0 • 5 $X αλ = , ελ = , ! $ 9 ) ! ) 9 # # 9 "αλ = , 9 "ελ = , # ) • * ελ ≈ ! 9 αλ ≈ ! @ 4 3 ( # 9 # B X : ! ' ≠ ' T ' ' , << # = ,! 1 → +∞ ! 9 φ = φ −φ # 9 # → , % ( " " 9 * = 9 9 • φ = φ = ,( = σ • φ '@ φ = " , # 0 × 0π ! φ =φ 9 φ =σ $ " × 0π 0 , =σ , 0 , = , # # × 0π # '@ δ φ = θ × Ω# , α $ δ φ = , * δφ = ,π # = δφ = π # , × θ × π# α# α θ# θ# , = 0π # = • φ =σ " * Y , − 0 0 , # " ! = φ = 0σ #( . , " − 0 = 0 , +0 . , " − , ! , Γ = 0σ #( . , # @ φ 0σ # . , . : • V % 9 " & '(' 9 • ) = "# # ) = −φ # # P $ * " # = −0 ,.σ #(#" − − − 0 σ #( "# = , − • $ 3 • ρ = • , . , , ,,S # ) τ= −. = π # . ρ# 0 ,.σ #0π # 0 . λ = ρ % − ) = ρ# # σ . , # # ! , = ,° 9 # # % ρ # #* . ,σ λ = ,,M# ) #U − ! = τ* # *Z * 0 σ #( . , − ! = Q,,7#S 8, Z .,& # % "/ * τ * << τ ! 3 "# 6τ = ( + O *N = ! − ?τ =" − , ) << * λ << − τ* σ . , τ* Z ! << * # ρ# # σ . , # #U − # << 0 # : =) V % ' ' = ' * 4! # ) # # = 80,,S ! = KP,,U ! = # ,K S = K,# ,8 S # φ = φ −φ = , ' φ = σ # 0 × 0π # ' φ 2 4 = σ# 0 × 0π # φ = σ# 0 # × 0π # × Ω 0π Q π# Ω≈ ) ! φ = σ# 0 × 0π # × = # = P,U = ° ! 9 ) # * φ $ φ =φ 0 ! = −λ = ! − 0 ) 0 × × ( Ω 0π ( λ σ = .,U#S ! − # = Q# ,−. U # ) # &− &= P,U ! % H ) ! −σ# 0 9 # " 0 ) = [ φ = φ −φ = σ# @ . 0 φ =φ ! H ⋅ ! Ω= " [ " * = 9 T = ) = ⋅ = δ ν " ) θ = $ δ ν θ# ν = ) ) + " % ν θ − θ ) − ( ν θ = + 9 , θ ν ν+ ν 9 δ 0& θ δ ν %# ν = θ #δ 0 & # ν θ B 9 & = & "ν ν " ν "ν 9 ) ) $ T ) 9 ! δ 0 & = & "ν δ $ = = & $ ν θ +∞ ν =, θ# # ν× ν Ω= π δ ν+ ν S θ# # θ ( θ ) ν ( ν % θ+ θ Ω 0π × $ θ #& "ν ν × Ω 0π θ# # $ × Ω 0π θ# θ ν "ν ν π? θ , & θ# θ ?. ! = ! × $ . = $ . = . # = = σ 0 . # . = ! = $ = . $ = $σ ! %$ = ∂! ∂ 0 " = 0$σ $ R . # !!= . $ 0 = 0 ) = ! + $ = 0 $ = σ 0$ . ∂) % = ! ∂ # K = =4 ! = $ = 0σ + = 0σ 0 + σ . $ →, =, +σ $ . $+ σ . . . $ $ 0 = σ . $+ . # ".% B 0 = σ . ! . $# $= . ' $ ' * ) 8 = $ 0?. = × = ! 9 # 9 ! • 0 = σ . +=)− • +=− β $ +µ +$ $× =, . ∂+ ∂ #$ µ= $α + µ !α / !α =,# α !α = $ 0 $− σ . 0 α α − α ! !α + ! β = , ! ,=" α − β α + µβ α β ! + β !β = β =,# α = β µβ = , β − β $β + µ β β