Two-dimensional Steady State Heat Consuction
4
7/23/2019 Two-dimensional Steady State Heat Consuction http://slidepdf.com/reader/full/two-dimensional-steady-state-heat-consuction 1/4 Two-Dimensional Steady State Heat Conduction in a Rectangular Plate G.E : B.C Solution : Method of Separation of Variables Assume Solution of the form T 1 , θ= 0 T 1 , θ= 0 x a b T 2 , θ= 1 T 1 , θ= 0 2 2 2 2 0 x y θ θ ∂ ∂ + = ∂ ∂ (0, ) 0; ( , ) 0 ( ,0) 0; ( , ) 0 y a y x x b θ θ θ θ = = = = y 2 (, ) () ( ) "( ) "( ) G.E where constant () ( ) x y F x G y F x G y F x G y θ λ λ = ∴ ⇒ − = = =
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Transcript of Two-dimensional Steady State Heat Consuction
7/23/2019 Two-dimensional Steady State Heat Consuction
http://slidepdf.com/reader/full/two-dimensional-steady-state-heat-consuction 1/4
Two-Dimensional Steady State Heat
Conduction in a Rectangular Plate
G.E :
B.C
Solution : Method of Separation of Variables Assume Solution of the form
T1 ,
θ= 0
T1 , θ= 0 x
a
b
T2 , θ= 1
T1 , θ= 0
2 2
2 20
x y
θ θ ∂ ∂+ =
∂ ∂
(0 , ) 0 ; ( , ) 0( , 0 ) 0 ; ( , ) 0
y a y x x b
θ θ
θ θ
= =
= =
y
2
( , ) ( ) ( )
"( ) "( )G.E where constant( ) ( )
x y F x G y
F x G y
F x G y
θ
λ λ
=
∴ ⇒ − = = =
7/23/2019 Two-dimensional Steady State Heat Consuction
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Two-Dimensional Steady State Heat
Conduction in a Rectangular Plate
( )( )
( )
( )
1 2
3 4
1 2 3 4
1
2 3 4
3 4
2 4
2
cos sin
cos sin
(0, ) 0 0
( ,0) 0 sin 0
( , ) 0 sin 0
sin 0 / for 1,2,3,
y y
y y
y y
F C x C x
G C e C e
C x C x C e C e
y C
x C x C C
C C
a y C C a e e
a n a
C
λ λ
λ λ
λ λ
λ λ
θ λ λ
θ
θ λ
θ λ
λ λ π λ
θ
−
−
−
= +
= +
= + +
= ⇒ =
= ⇒ + =
⇒ = −
= ⇒ − =
⇒ = ⇒ = =
=…
( )/ /
4 sin / sin( / )sinh( / )n y a n y a
nC n x a e e C n x a n y aπ π π π π
−− =
7/23/2019 Two-dimensional Steady State Heat Consuction
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Two-Dimensional Steady State Heat
Conduction in a Rectangular Plate
( )
( )( )
1
1
1
2
0 0
sin( / ) sinh( / )
( , ) 1 sin sinh 1
2
0 0
sin ; sin 1 cos2
1 cos2sinh /
sin sinh sin
n
n
nn
n
a a
n
C n x a n y a
n x n b x b
a a
a a
n
n x a n x adx dx n
a a n
nC n n b a
C
n x n b n xC dx
a a a
θ π π
π π θ
π π π
π
π π π
π π π
∞
=
∞
=
∞
=
=
= ⇒ =
= = −
−=
∑
∑
=∑ ∫ ∫
∫ ∫
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w ( )
( )
( ) ( )
( )
( ) ( )
( )
( )( ) ( )( )
1
0
1 cos2sin sinhsinh /
sin / sinh / sin 3 / sinh 3 /4
sinh / 3sinh 3 /
cosh / 1 cosh 3 / 14 1..........
sinh / 3 sinh 3 /
n
b
x a
n n x n y
n n b a a a
x a y a x a y a
b a b a
T q k dy x
b a b ak q
b a b a
π π π
θ π π
π π π π
θ π π π
π π
π π π
∞
=
=
−
=
= + +
∂ = − ∂
− −
= + +
∑
∫
…
Two-Dimensional Steady State Heat
Conduction in a Rectangular Plate
