( ) { ( )θ

93
⇕ఏᑟ᪉⛬ᘧ䛾↓ḟඖ໬ 㼜㻚㻞㻣㻙㻞㻤
⇕ఏᑟ᪉⛬ᘧ䠄┤஺ᗙᶆ⣔䠅䠖 ⇕ఏᑟ⋡䛜୍ᐃ䛾ሙྜ
↓ḟඖ ᗘ䛸ᗙᶆ䠄ኚᩘኚ᥮䠅
T
T Tf
,
Ti Tf
x
, Y
L
X
y
, Z
L
z
L
wT
wt
(2.31)
Ti Tf 䠖௦⾲ ᗘᕪ䠈 L 䠖≀య䛾௦⾲ᑍἲ
wT
wFo
1 wT
D wt
L2
D (Ti Tf ) § w 2T
w 2T w 2T · qv
¨
¸
© wX 2 wY 2 wZ 2 ¹ U c
L2
2
· qv
¸ Uc
¹
w 2T w 2T w 2T
G
wX 2 wY 2 wZ 2
(2.32)
2
Fo
G
L2
䠖䝣䞊䝸䜶ᩘ 䠄Fouriernumber䠅
95
⇕ఏᑟ᪉⛬ᘧ䛾↓ḟඖ໬ 㼜㻚㻞㻤
↓ḟඖ໬䛾ព⩏䠖ゎ䛾୍⯡ᛶ
➨䠏✀ቃ⏺᮲௳䛾↓ḟඖ໬
h ^ Tf Ts (t ) `
T
(2.27)
T Tf
,
Ti Tf
Bi
hL
k
≀య⾲㠃ᑐὶ⇕ఏ㐩䛸≀యෆ⇕ఏᑟ䛾ఏ⇕䛾Ⰻ䛥䛾ẚ⋡
䠄ᑐὶ⇕ఏ㐩䛾⇕᢬ᢠ䛻ᑐ䛩䜛≀యෆ⇕ఏᑟ䛾᢬ᢠ䛾ẚ⋡䠅
䛭䜜䛮䜜䛾ఏ⇕䛾 ᗘኚ໬䛾㐪䛔䛻䛴䛺䛜䜛䝟䝷䝯䞊䝍
౛䛘䜀䠈ᑐὶఏ⇕㠃䛷䛾⇕཰ᨭ䛻㛵୚䛧䛶
Ts ,1 Ts ,2
(2.34)
Ts ,2 Tf
s
䛣䛾䛸䛝䛾↓ḟඖᩘ
96
⇕ఏᑟ᪉⛬ᘧ䛾↓ḟඖ໬ 㼜㻚㻞㻤
k
(Ts ,1 Ts ,2 ) A
L
x 0
BiT s
Thermal Energy Conversion Lab. / Prof. H. Mori
Biotᩘ䛾ព࿡
≀య䛾ᗄఱᙧ≧䠈≀ᛶ䠈ቃ⏺᮲௳䛜┦ᑐ໬䛷䛝䠈⇕ఏᑟ⌧㇟
䛾┦ఝ๎䛜㐺⏝䛷䛝䜛䠊䛯䛸䛘䜀䠈≀య䛾ᙧ≧䛜┦ఝ䛾ሙྜ䠈
↓ḟඖ᫬㛫䛻┦ᙜ䛩䜛䝣䞊䝸䜶ᩘ䛸ቃ⏺᮲௳䛾↓ḟඖᩘ䛜➼
䛧䛔䛸䛝䠈≀యෆ䛾↓ḟඖ ᗘศᕸ䛿ྠ䛨䛷䛒䜛䠊䠄8❶2⠇䜢ཧ↷䠅
wT
wx
wT
wX
(2.33)
qv L2
k (Ti Tf )
Thermal Energy Conversion Lab. / Prof. H. Mori
k
(2.17)
↓ḟඖ᫬㛫䠖 ⇕ఏᑟ䛻䛚䛡䜛᫬㛫䛾↓ḟඖᩘ
Dt
wT wT wT
qv
2 2
2
(Ti Tf )D
wX
wY
wZ
Uc
L2
2
§ w 2T w 2T w 2T
2 2
2
wy
wz
© wx
D¨
↓ḟඖ⇕ఏᑟ᪉⛬ᘧ
⇕ఏᑟ᪉⛬ᘧ䛾↓ḟඖ໬
wT
(Ti Tf )
wt
94
⇕ఏᑟ᪉⛬ᘧ䠄heatconductionequation䠅㼜㻚㻞㻢
➨3✀ቃ⏺᮲௳
1 (k L)
䠖䝡䜸ᩘ 䠄Biotnumber䠅
1h
(2.35)
Thermal Energy Conversion Lab. / Prof. H. Mori
h (Ts ,2 Tf ) A
( L kA)
(1 hA)
Rcond
Rconv
ᐃᖖ䛷䜒㠀ᐃᖖ䛷䜒᭷⏝
hLc
௦⾲㛗䛥 Lc Bi
k
hL
{ Bi
k
Lc
V
S
˳ᆢ
ሁ̖ᧈƞ
ᘙ᩿ᆢ
(2.112)
Thermal Energy Conversion Lab. / Prof. H. Mori
98
㠀ᐃᖖ⇕ఏᑟ䠄unsteadyͲstateconduction䠅㼜㻚㻟㻤
䝡䜸ᩘ䛾኱䛝䛥䛻䜘䜛ᖹᯈෆ㐣Ώ ᗘศᕸ䛾㐪䛔
Ti
T
T (0, x)
Ti
T (0, x)
T
Bi 1
ึᮇ ᗘ
ưƸ
t
ưƸ
T (t , r L) | ɟ‫ܭ‬
Tf
Tf
L
L
Bi 1
L
T | T (t )
T
L
Bi
1
T (t , x)
Tf
⪃䛘᪉䛿䠈
T = T(x) 䛸䛧䛶䠈
ෆ㒊Ⓨ⇕䛾䛒䜛
ᐃᖖ䛷䜒ྠ䛨
L
L
Bi !! 1
T
ὶయ ᗘ
T (t , x)
100
༙↓㝈ᅛయ
0.6
0.4
0.2
0.0
0
1
2
FoBi=hAt/(cUV )
Fo
Dt
ᅗ㻞㻚㻞㻢
Thermal Energy Conversion Lab. / Prof. H. Mori
Bi
L2c
3
hLc
k
Thermal Energy Conversion Lab. / Prof. H. Mori
101
༙↓㝈ᅛయ䠄semiͲinfinitesolid䠅㼜㻚㻠㻜
ቃ⏺᮲௳ 䠖 ୍⯡䛻䠈䠎䛴ᚲせ
ቃ⏺㠃 (x=0) 䛛䜙 x ᪉ྥ䛻↓㝈䛾ᗈ䛜䜚䜢䜒䛴䜘䛖䛺≀య
≀య䛾኱䛝䛥䛻ẚ䜉䛶䠈ᑐ㇟䛸䛩䜛㐣Ώ⇕ఏᑟ⌧㇟䛾᫬㛫
䛜▷䛟䠈 ᗘ䛾᫬㛫ኚ໬䛜≀య඲య䛻ཬ䜀䛺䛔≧ἣ
↓㝈㐲᪉䠖
ᖖ䛻 ᗘ୍ᐃ䠄䠙ึᮇ ᗘ䠅 t ! 0, x o f : T
⾲㠃
Ⓨ⇕䛺䛧䠈ึᮇ ᗘ୍ᵝ䠄Ti 䠅䛾ሙྜ
wT
w § wT · w § wT
k
¨k
¸
wt
wx © wx ¹ wy ¨© wy
⇕ఏᑟ᪉⛬ᘧ
wT
w 2T
D 2
(2.115)
wt
wx
Uc
ึᮇ᮲௳
x ! 0, t
ĺ ࿘ᅖὶయ ᗘ Tf
0.8
≀య䛾⇕཰ᨭ
dT
cU V
hS (T Tf ) (2.113)
dt
ᗘኚ໬
hS ·
T Tf
t¸
T
exp ¨§ Ti Tf
© cU V ¹
exp Fo Bi (2.114)
ᅗ㻞㻚㻞㻡㻌㻌䝡䜸ᩘ䛾኱䛝䛥䛻䜘䜛ᖹᯈෆ㐣Ώ ᗘศᕸ䛾㐪䛔
༙↓㝈ᅛయ䠄semiͲinfinitesolid䠅㼜㻚㻠㻜
⾲㠃✚ S
ẚ⇕ c
ึᮇ ᗘ Ti
Bi 1
t
t
Tf
య✚ V
ᐦᗘ U
1.0
ቨ ୍ᐃ
㞟୰⇕ᐜ㔞吹吞呂
T | T (t )
㞟୰⇕ᐜ㔞䝰䝕䝹䠄lumpedcapacitancemodel䠅
T=(T-Tf)/(Ti-Tf)
T
hLc
k
h
k Lc
Bi
99
㞟୰⇕ᐜ㔞䝰䝕䝹䠄lumpedcapacitancemodel䠅㼜㻚㻟㻥
· w § wT ·
¸ wz ¨ k wz ¸ qv
©
¹
¹
D
Ts
k
Uc
0: T
Ti
Ti
ᅗ㻞㻚㻞㻣
Thermal Energy Conversion Lab. / Prof. H. Mori
s
T (t , x )
T (t , x )
x
x
Tf , h
T (t , x )
x
Tf
t
(2.116)
(2.118)
ᗘ୍ᐃ 䠄➨䠍✀ቃ⏺᮲௳䠅
⇕ὶ᮰୍ᐃ 䠄➨䠎✀ቃ⏺᮲௳䠅
ᑐὶ⇕ఏ㐩⋡୍ᐃ 䠄➨䠏✀ቃ⏺᮲௳䠅
q
T
Ts
⇕ᣑᩓ⋡䠄 ᗘఏᑟ⋡䠅
Ti
t
x Ti
t
x Ti
(a)➨㸯✀ቃ⏺᮲௳ (b)➨㸰✀ቃ⏺᮲௳ x
(c)➨㸱✀ቃ⏺᮲௳
Thermal Energy Conversion Lab. / Prof. H. Mori
102
༙↓㝈ᅛయ䠄semiͲinfinitesolid䠅㼜㻘㻠㻜
⾲㠃 ᗘ୍ᐃ䛾ሙྜ
ึᮇ᮲௳
t 0, x t 0 : T
ቃ⏺᮲௳
t ! 0, x 0 : T
t ! 0, x o f : T
erf ([ )
ᗘศᕸ
T
Ti
T0
T0
Ti
↓ḟඖ ᗘ T (T Ti ) / (T0
ᇶ♏ᘧ
wT
w 2T
D 2
wt
wx
ึᮇ᮲௳䠈ቃ⏺᮲௳
t 0, ࠉx t 0 : T 0
t ! 0, ࠉx 0 : T 1
t ! 0, ࠉx o f : T 0
(2.117)
(2.118)
U , c, k
T (t , x )
t
Ti )
x
0
(2.119)
ᅗ㻞㻚㻞㻤㻌㻌➨㻝✀ቃ⏺᮲௳䛻䛚䛡
䜛↓㝈≀యෆ䛾㐣Ώ
ᗘศᕸ
1 ·
1 erf §¨
¸
© 2 Fo ¹
Fo
104
➨2✀ቃ⏺᮲௳ 䠖⇕ὶ᮰ qS ୍ᐃ䛾ሙྜ
Ts
Ti x
§ x ·
§ x2 ·
exp ¨ erfc ¨
¸
¸
S
Dt
© 4D t ¹
© 2 Dt ¹
2
1 ·
1
1 ·
exp §¨ erfc §¨
¸
¸
Fo
S
© 4 Fo ¹
© 2 Fo ¹
2
2 qs D t
k S
[
0
e y dy
2
erfc([ ) 1 erf ([ )
x2
⇕ὶ᮰
q(t )
k
wT
wx
k (T0 Ti )
x 0
SD t
d
2
exp([ 2 ) 䜢㐺⏝
^erf ([ )` =
dx
S
ᅗ㻞㻚㻞㻥
Thermal Energy Conversion Lab. / Prof. H. Mori
༙↓㝈ᅛయ䛾ィ⟬౛ 㻝㻌䛸㻞
105
㼜㻚㻠㻜䡬㻠㻝
ึᮇ ᗘ 20 ºC 䛾༙↓㝈ᅛయ
㖡
䠖 D= 117u10-6 m2/s, k = 398 W/(m䡡K) ⇕ᣑᩓ⋡ ኱
䝇䝔䞁䝺䝇㗰䠖 D = 3.36 u10-6 m2/s, k = 12.8
ᑠ
ᗘศᕸ
k (T Ti )
qs D t
S³
Dt
Thermal Energy Conversion Lab. / Prof. H. Mori
༙↓㝈ᅛయ䠄semiͲinfinitesolid䠅㼜㻚㻠㻝
2
x ·
Ti T0 Ti erfc §¨
¸
© 2 Dt ¹
T
Ti
(2.120)
(2.121)
(2.122)
·
1 erf §¨
¸
© 2 Dt ¹
x ·
erfc §¨
¸
© 2 Dt ¹
x
T
q
103
༙↓㝈ᅛయ䠄semiͲinfinitesolid䠅㼜㻚㻠㻜㻙㼜㻚㻠㻝
(2.127)
⾲㠃 ᗘ䜢100 ºC䛻✺ኚ䛥䛫䛯
᫬䛾 ᗘศᕸ䛾ኚ໬
䠄➨䠍✀BC䠅
⾲㠃䜢୍ᐃ䛾⇕ὶ᮰ 5000 W/m2
䛷ຍ⇕䛧䛯᫬䛾 ᗘศᕸ䛾ኚ໬
䠄➨2✀BC䠅
(2.128)
➨3✀ቃ⏺᮲௳ 䠖 ᑐὶ⇕ఏ㐩䛾ሙྜ
h Dt·
§ x
·
¸ erfc ¨
k ¸¹
¹
© 2 Dt
1 ·
§ 1 Bi Fo ·
2
erfc §¨
(2.129)
¸ exp Bi Bi Fo erfc ¨
¸
2
Fo
©
¹
© 2 Fo
¹
T
T Ti
Tf Ti
§ x
erfc ¨
©2 Dt
·
§ hx h 2D t
2
exp
¸
¨
k
© k
¹
Thermal Energy Conversion Lab. / Prof. H. Mori
Thermal Energy Conversion Lab. / Prof. H. Mori
106
༙↓㝈ᅛయ䛾ィ⟬౛ 㻝㻌䛸㻞
㼜㻚㻠㻜䡬㻠㻝
ึᮇ ᗘ 20 ºC 䛾༙↓㝈ᅛయ
㖡
䠖 D= 117u10-6 m2/s, k = 398 W/(m䡡K) ⇕ᣑᩓ⋡ ኱
䝇䝔䞁䝺䝇㗰䠖 D = 3.36 u10-6 m2/s, k = 12.8
ᑠ
107
㻞䛴䛾༙↓㝈ᅛయ䛾᥋ゐ㼜㻚㻠㻞
ึᮇ ᗘ T1,i , T2,i 䛾༙↓㝈≀య䜢᥋ゐ䛥䛫䛯ሙྜ
≀య䠍䠖
⇕ఏ㐩⋡୍ᐃ 1000 W/(m2·K) 䛷䠈
⾲㠃䜢୍ᐃ䛾⇕ὶ᮰ 5000 W/m2
100 ºC 䛾ὶయ䛻䠈ゐ䜜䛥䛫䛯᫬䛾
ᗘศᕸ䛾ኚ໬䠄➨䠏✀BC䠅
䛷ຍ⇕䛧䛯᫬䛾 ᗘศᕸ䛾ኚ໬
䠄➨2✀BC䠅
≀య䠎䠖
T1
T2
§ x1 ·
T1,i TS T1,i erfc ¨
¸
© 2 D1t ¹
§ x2 ·
T2,i TS T2,i erfc ¨
¸
© 2 D 2t ¹
(2.140)
(2.141)
᥋ゐ㠃 (x1 = x2 = 0) 䛷 ᗘ䛸⇕ὶ᮰䛜ᖖ䛻➼䛧䛔䠊
T1
T2 ,
᥋ゐ㠃䛾 ᗘ
TS
k1
wT1
wx1
k2
wT2
wx2
(2.142)
U1c 1 k1 T1,i U2c2 k2 T2,i
U1c 1 k1 U2c2 k2
᫬㛫䛻౫Ꮡ䛫䛪䠈≀య䛾⇕≀ᛶ್䛸ึᮇ ᗘ䛰䛡
䛷Ỵ䜎䜛䠄U ·c·k 䛾✚䛾኱䛝䛔᪉䛻㏆䛔 ᗘ䛻䛺䜛䠅
Thermal Energy Conversion Lab. / Prof. H. Mori
㠀ᐃᖖ⇕ఏᑟ䠄unsteadyͲstateconduction䠅㼜㻚㻟㻤
䝡䜸ᩘ䛾኱䛝䛥䛻䜘䜛ᖹᯈෆ㐣Ώ ᗘศᕸ䛾㐪䛔
T
Ti
T (0, x)
T
Ti
T (0, x)
T
ึᮇ ᗘ
t
Bi
108
hLc
k
h
k Lc
t
t
Tf
Tf
Tf
L
L
Bi 1
L
T | T (t )
T
L
Bi
1
T (t , x)
Tf
ὶయ ᗘ
L
L
Bi !! 1
T
T (t , x)
ᅗ㻞㻚㻟㻜
(2.143)
Thermal Energy Conversion Lab. / Prof. H. Mori
109
ᖹᯈ䠄planewall䠅㼜㻚㻠㻟
䠍ḟඖ㠀ᐃᖖ⇕ఏᑟ
ᇶ♏ᘧ䠈ึᮇ᮲௳䠈ቃ⏺᮲௳
wT
w 2T
D
wt
wx2
t 0, L t x t 0 : T Ti
wT
t ! 0, ࠉx 0 :
0
wx
wT
t ! 0, ࠉx L : k
h T Tf wx
ゎ
f
4sin E n T Tf
E 2 Fo
T
e n cos E n X ¦
Ti Tf n 1 ^ sin 2E n 2 E n `
En
䛿ḟᘧ䜢‶㊊䛩䜛ゎ
cot( E n )
En
Bi
X
x
, Fo
L
ᅗ㻞㻚㻟㻜
Dt
2
L
, Bi
hL
k
ᅗ㻞㻚㻞㻡㻌㻌䝡䜸ᩘ䛾኱䛝䛥䛻䜘䜛ᖹᯈෆ㐣Ώ ᗘศᕸ䛾㐪䛔
Thermal Energy Conversion Lab. / Prof. H. Mori
Thermal Energy Conversion Lab. / Prof. H. Mori
110
ᖹᯈ䠄planewall䠅㼜㻚㻠㻠
Bi 䛻䜘䜛㐣Ώ ᗘศᕸ䛾㐪䛔
Fo
Bi
䝡䜸ᩘ Bi 䛻䜘䜛㐣Ώ ᗘศᕸ䛾㐪䛔
Dt
hL
h
Dt
L2
hL
k
Bi
h
k L
k
k L
Bi
0.1
Bi
Bi 10
Bi 1
ᅗ㻞㻚㻟㻟
0.1
112
⡆᫆᥎⟬ἲ 㼜㻚㻠㻣
㐣Ώ⇕ఏᑟ䛻䛚䛡䜛ྛ✀ᙧ≧䛾୰ᚰ ᗘ
T c Tf
T i Tf
Dt
k
1
f ( Fo { 2 ,
)
Tc
{
Bi
hL
L
ᖹᯈ䠖䝝䜲䝇䝷䞊⥺ᅗ䠄Heislerchart䠅
0
-1
50
8 9
6 7
40 45
30
5
1.4
0.2
35
1/Bi=100
90
25
2.5
0.3
0.05 0.1
4
㻌
㻌
-2
10
3
㻌
10
0.8
0.7
0.6
0.4 0.5
㻌
1.0
㻌
10
12
1.2
2.0 1.8
1.6
Tc =(Tc-Tf)/(Ti-Tf)
10
16
14
80
20
70
18
60
-3
1
2
L2
3
46
12
18
24
30
70
2
Fo=Dt / L
110
150
300
500
700
ᅗ㻞㻚㻟㻡
ᖹᯈ䠈෇ᰕ䠈⌫䠄Fo>0.2䠅䠖⾲2.3+ᘧ(2.178)
ྛ✀ᙧ≧䠄Biĺf䠅䠖 ᅗ2.36
Thermal Energy Conversion Lab. / Prof. H. Mori
Bi 10
Bi 1
Thermal Energy Conversion Lab. / Prof. H. Mori
0
Fo
Biĺ 䛷䛿䠈ቨ㠃 ᗘ୍ᐃ
ᅗ㻞㻚㻟㻞
10
111
↓㝈ᖹᯈ䠄planewall䠅䛾ィ⟬౛ 㼜㻚㻠㻠
Thermal Energy Conversion Lab. / Prof. H. Mori
113
㠀ᐃᖖ⇕ఏᑟ䛻㛵䛩䜛ၥ㢟㼇㻞㻚㻝㻜㼉
ෆ㒊Ⓨ⇕䛾䛺䛔୍ᵝึᮇ ᗘT0䛾ᅛ
య䛻䠈䛒䜛᫬้䜘䜚䠈እ㒊䛛䜙⾲㠃䛻୍
ᐃ䛾⇕ὶ᮰ qw 䜢㈇Ⲵ䛩䜛䠊⇕䛾౵ධ䛻
䜘䜛ᅛయෆ㒊䛾 ᗘኚ໬䛜䜎䛰༑ศዟ
䜎䛷ᒆ䛛䛺䛔䜘䛖䛺䠄༙↓㝈ᅛయ䛸䛧䛶
⪃䛘䜙䜜䜛䠅⤒㐣᫬㛫䛻䛚䛔䛶䠈ᅛయෆ
㒊ዟ⾜ x ᪉ྥ䛾 ᗘ T 䛾ศᕸ䜢䠈⇕ఏ
ᑟ⋡䛜኱䛝䛔㖡 (Cu) 䛸⇕ఏᑟ⋡䛜ᑠ䛥
䛔䝇䝔䞁䝺䝇㗰 (SS) 䛾ሙྜ䛻䛴䛔䛶䠈
ẚ㍑䛩䜛䠊ྠ㔞䛾⇕ὶ᮰ qw 䛜㈇Ⲵ䛥䜜
䜛䛸䛧䛶䠈ྠ䛨⤒㐣᫬㛫䛻䛚䛡䜛୧⪅䛾
ᅛయෆ㒊䛾 ᗘศᕸ䜢䠈୧⪅䛾㐪䛔䛜
䜟䛛䜛䜘䛖䛻䠈ྑᅗ䛻ᥥ䛡䠊
T
ᅛ
య
⾲
㠃
qw
T㻜
x
Thermal Energy Conversion Lab. / Prof. H. Mori

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