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Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
Chapter 5
Partial Differential Equations and
Finite Difference Methods
5.2 PDE of 1D Heat Transfer Problem
5.2.1 Governing Equation and Analytical Solution
Consider an element (materials) of width
dx
, density
ρ
, mass
dm
=
dx
, heat capacity
c
v
,
conductivity
k
, internal source intensity (per unit volume)
( )
,
qxt
&
, temperature
distribution
T(x,t)
.
Define
x
T
k
Q
∂
∂
−
=
as heat flux. The heat flux changes into
Q+dQ
after passing though
the materials with a width of
dx
.
Figure 2. Energy balance in a 1D element
Energy conservation leads to the following heat transfer equations which are typical
parabolic PDEs.
()
(
)
22
//
0
0
0
dx
vv
dm
dx
Q
k T x
steady state
t
v
source free q
Td
Q
T
Q
Q
dQ
qdx
dm c
q
c
td
x
t
TT
T
kq
c
xt
x
÷
=
=− ∂ ∂
∂ ∂ =
=
∂
∂
−+
+
=
→
−
+
=
∂
∂
∂∂
∂
→
+ =
→=
∂
&
&&
&
dx
T
Qk
x
∂
=−
∂
2
2
dQ
T
T
Qd
x
k
kd
x
dx
x
x
+=
−
+
−
q
&
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View Full DocumentIn 3D, the general equation is
222
22
,
v
T
kTqc
tx
y
z
ρ
∂
∂∂∂
∇+
=
∇
=
+
+
∂
&
For
steady state
(temperature does not change with time) without source (
0
=
•
q
), the
equation for heat transfer (temperature distribution) is
2
2
2
3D :
0,
1D :
0,
T
dT
Ta
xb
dx
∇=
=
=+
,
where
a
and
b
are determined by
boundary conditions
.
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This note was uploaded on 08/27/2010 for the course ME ME4906 taught by Professor Dr.g.p.zheng during the Spring '10 term at NYU Poly.
 Spring '10
 Dr.G.P.Zheng
 Heat Transfer

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