1.3 (heat transfer with phase change) (phase change) (evaporation) (boiling) (condensation) (melting) (solidification) 1.4 (thermal radiation)

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1 CAE ( 6 ) 1 1. (heat transfer) (heat conduction) 1.2 (convective heat transfer) (convection) (natural convection) (free convection) (forced convection) 1

2 1.3 (heat transfer with phase change) (phase change) (evaporation) (boiling) (condensation) (melting) (solidification) 1.4 (thermal radiation) (isothermal plane) A T 1, T 2 (T 1 > T 2 ) x Q Q Q (T 1 T 2 ) A (1) x (1) λ x (1) ( ) T Q = λ A (2) x T x t T x x T/ x T/ x < 0 (2) Q (heat flux) q [W/m 2 ] q = Q A = λ T (3) x (3) (Fourier s law) λ (thermal conductivity) [W/(m K)] λ ρ [kg/m 3 ] c [J/(kg K)] α α = λ ρc (4) 2

3 T1 A T2 A Q x 1: 2 α (thermal diffusivity) [m 2 /s] 3. 2 T w T f (T w T f ) (boundary layer) Q A h Q = h(t w T f )A (5) (5) q q = h(t w T f ) (6) h (heat transfer coefficient) [W/(m 2 K)] Q (5) h h h dt solid dt fluid q = λ solid = λ dy fluid (7) surface dy surface 3

4 Tw Q h Tf 2: y solid, fluid, surface (7) q h l T 1 T 2 λ q (3) dt dx = q λ = const. = C 1 (8) x = 0 : T = T 1 x = l : T = T 2 (9) (8) T = C 1 x + C 2 (10) C 1, C 2 T = T 1 (T 1 T 2 ) x (11) l 3 x ( ) T q = λ = T 1 T 2 (12) x l/λ 4

5 T1 T T2 0 l x 3: l/λ (thermal resistance) [ 1] 6 cm 20, m W/(m K) [ ] 912 kj , 2 T f1 T f2 1 2 h 1, h 2 4 x 1 (8) 1 dt 1 dx = q λ 1 = C 1 (13) 2 dt 2 dx = q λ 2 = C 2 (14) T w1, T w2 ( ) dt1 x = 0 : h 1 (T f1 T w1 ) = λ 1 (15) dx x=0 ( ) ( ) dt1 dt2 x = l 1 : λ 1 = λ 2 (16) dx x=l 1 dx x=l 1 x = l 1 : T 1 = T 2 (17) 5

6 1 2 Tw1 Tf1 h1 h2 q λ1 λ2 Tw2 0 l1 l2 x Tf2 4: ( ) dt2 x = l 1 + l 2 : λ 2 = h 2 (T w2 T f2 ) (18) dx x=l 1 +l 2 (13), (14) T 1 = C 1 x + C 2 (19) T 2 = C 3 x + C 4 (20) C 1, C 2, C 3, C 4 (15) (18) T w1, T w2 2 2 (19), (21) (20), (22) C 1, C 2, C 3, C 4 x = 0 : T 1 = T w1 (21) x = l 1 + l 2 : T 2 = T w2 (22) T w1 = C 2 (23) T w2 = C 3 (l 1 + l 2 ) + C 4 (24) C 1 = 1 λ 1 (T f1 T f2 ) ( 1 h1 + l 1 λ 1 + l 2 λ h 2 ) (25) C 2 = T f1 1 h 1 (T f1 T f2 ) ( 1 h1 + l 1 λ 1 + l 2 λ h 2 ) (26) C 3 = 1 λ 2 (T f1 T f2 ) ( 1 h1 + l 1 λ 1 + l 2 λ h 2 ) (27) 6

7 C 4 = T f2 + ( ) l1 λ 1 + l 2 λ h ( )(T l 1 h1 λ 1 + l 2 λ f1 T f2 ) (28) h 2 (19), (20) q ( ) ( ) dt1 dt2 q = λ 1 = λ 2 dx dx = (T f1 T f2 ) ( 1 h1 + l 1 λ 1 + l 2 λ h 2 ) (29) (29) (overall thermal resistance) (29) λ 1, λ 2,, λ n l 1, l 2,, l n 1 q = (T f1 T f2 ) ( 1 + ) = K(T l 1 h1 λ 1 + l 2 λ 2 + ln λ n + 1 f1 T f2 ) (30) h 2 K K = 1 ( 1 + ) (31) l 1 h1 λ 1 + l 2 λ 2 + ln λ n + 1 h 2 K (overall heat transfer coefficient) [W/(m 2 K)] [ 2] 10 mm 100 mm 5 mm 10 W/(m 2 K) 1.2, 0.76, 0.15 W/(m K) [ ] K = 1/( )= 1/0.373 = 2.68 W/(m2 K)

8 6. 3 h h dimensional analysis 6.1 h ρ µ λ c p u l L M T Θ H h λ c p ρ µ u l [ H L 2 T Θ ] [ H LT Θ] [ H MΘ] [ M L 3 ] [ M LT ] [ L T 5 7 H Θ (H/Θ) 1 4 π 7 4 = 3 h h 6 ] [L] h = f(λ, c p, ρ, µ, u, l) (32) f n 1, n 2,, n 7 (33) h n1 λ n2 c n 3 p ρ n4 µ n5 u n6 l n 7 (33) [ ( ) H n1 ( ) H n2 ( ) H n3 ( ) M n4 ( ) M n5 ( ) L n6 (L) 7] n L 2 T Θ LT Θ MΘ L 3 LT T 4 H Θ : n 1 + n 2 + n 3 = 0 L : 2n 1 n 2 3n 4 n 5 + n 6 + n 7 = 0 T : n 1 n 2 n 5 n 6 = 0 M : n 3 + n 4 + n 5 = h, c p, u n 1, n 3, n 6 4 n 2 = n 1 n 3, n 4 = n 6, n 5 = n 3 n 6, n 7 = n 1 + n 6 8

9 (33) (34) h n1 λ n 1 n3 c n 3 p ρ n6 µ n 3 n6 u n6 l n 1+n 6 (34) ( ) n1 hl λ ( ) cp µ n3 λ ( ulρ n 1, n 3, n 6 ( hl ulρ λ = f µ, c ) pµ (35) λ hl λ Nusselt number µ = ) n6 Nu (36) ulρ µ = ul µ/ρ = ul ν = Re (37) Reynolds number 2 c p µ λ = ν α = Pr (38) Prandtl number Pr ν α 1 (35) Nu = f(re, Pr) (39) f (39) Nu = Re f(pr) (40) 9

10 1: Pr Pr 20 C R-12(CF 2 Cl 2 ) u F b F b = g(ρ ρ w ) = ρgβ(t w T ) (41) g, ρ, β w, β = 1/T h (32) u F b h = f(λ, c p, ρ, µ, F b, l) (42) h λ c p ρ µ F b l [ ] [ [ [ ] [ ] [ ] H H H M M ML L 2 T Θ LT Θ] MΘ] [L] L 3 LT L 3 T hl λ = f ( Fb ρl 3, c pµ µ 2 λ 1 F b Gr (Grashof number) Gr ) (43) Gr = F bρl 3 = ρg(ρ ρ w )l 3 = gβ(t w T )l 3 (44) µ 2 µ 2 ν 2 10

11 [ (Rayleigh number) ] (43) Nu = f(gr, Pr) (45) f Nu = 4 Gr f(pr) (46) 7. h Nu 7.1 l Nu : : Nu = hl λ = 0.664Re1/2 Pr 1 3 (Re < ) (47) Nu = hl λ = 0.037Re0.8 Pr 1 3 ( < Re < 10 7 ) (48) 7.2 D : Nu = hd λ = 4.36 (Re < ) (49) Nu = hd λ = 3.66 (Re < ) (50) : Nu = hd λ = 0.023Re0.8 Pr n ( < Re < ) (51) n = 0.4 : n = 0.3 : Re Re = ud/ν 7.3 l Nu : Nu = hd λ = (2Gr) 1 4 : Pr 1 2 [5(1 + 2 Pr Pr)] 1 4 (10 4 < Gr Pr < 10 9 ) (52) Nu = hl λ = 0.120(Gr Pr) 1 3 ( < Gr Pr < ) (53) 11

12 (pool boiling) (flow boiling) (forced convective boiling) (subcooled boiling) (surface boiling) (degree of subcooling) T sub (saturated boiling) (bulk boiling) 8.2 T w q T w T sat T sat = T w T sat q 5 T sat (degree of superheating) T sat A q AE E q (Critical Heat Flux point, CHF) AB (nucleation site) AB (nucleate boiling) E D.N.B. (Departure from Nucleate Boiling) T sat B F BCD T sat B-F (burnout) 12

13 log q E B C D M F G A log Tsat 5: M B B (burnout point) (Minimum Heat Flux point, MHF) D DFG (film boiling) T sat 5 GFD D D BCD (transition boiling region) (Leidenfrost) 5 D 5 (boiling curve) T sat q 9. (condensation) (condensor) 13

14 q [W/(m 2 K)] T=Tsat - Tw [K] 6: (film condensation) (dropwise condensation) 3 mm 6 6 T sat T w T q T T µm µm 1.3 µm

15 7: mm E E a E r E t (absorptivity) : a = E a /E (reflectivity) : r = E r /E (transmissivity) : t r = E t /E : a + r + t r = 1 ( ) : t r = 0, a + r = 1 : r + t r = 0, a = 1 : r = 0, a + t r = (black body) 7 15

16 K Wien's displacement law K Ebλ [W/m 3 ] K 500 K 300 K 200 K K Visible 50 K Wavelength, λ [µm] 8: 10.2 (Planck s law) (M. Planck, ) 1900 T λ E bλ (spectrum) E bλ = C 1λ 5 e C 2/λT 1 (54) C 1 = W m 2 C 2 = m K (54) (Wien s displacement law) 8 λ m (54) λ m T = µm K (55) (55) 8 (55) λ m E bλ,max E bλ,max = C 3 T 5 (56) C 3 = W/(m 3 K 5 ) 16

17 0 Ag, Al, Cu, Cr, Au Ni, Pt Cu 9: ε 10.4 (Stefan-Boltzmann s law) T E b 8 (54) λ = 0 E b = 0 C 1 λ 5 e C 2/λT 1 dλ = σt 4 (57) σ = W/(m 2 K 4 ) E b T 4 σ [ 3] W 1.8 m 2 0 W [ ] 943 W [Q = AσT 4 ] 376 W [Q = Aσ(T 4 T 4 a )] a < 1 E (57) E = εe b = εσt 4 (58) ε (emissivity) ε

18 Eλ ε=1/ λ 1 10: (gray body) 10.6 (Kirchhoff s law) a ε a = ε (59) (59) ε a 10 1., (1989), 2., (1983), 3., (1995), 4. JSME, (2005), 18

E F = q b E (2) E q a r q a q b N/C q a (electric flux line) q a E r r r E 4πr 2 E 4πr 2 = k q a r 2 4πr2 = 4πkq a (3) 4πkq a 1835 4πk 1 ɛ 0 ɛ 0 (perm

E F = q b E (2) E q a r q a q b N/C q a (electric flux line) q a E r r r E 4πr 2 E 4πr 2 = k q a r 2 4πr2 = 4πkq a (3) 4πkq a 1835 4πk 1 ɛ 0 ɛ 0 (perm 1 1.1 18 (static electricity) 20 (electric charge) A,B q a, q b r F F = k q aq b r 2 (1) k q b F F q a r?? 18 (Coulomb) 1 N C r 1m 9 10 9 N 1C k 9 10 9 Nm 2 /C 2 1 k q a r 2 (Electric Field) 1 E F = q

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