Civil Engineering Reference
In-Depth Information
The differential operator for three-dimensional problems is defined as
w
-
½
°
°
w
x
°
°
w
®
¾
w
y
(4.4)
°
°
°
°
w
°
°
¯
¿
w
z
and for two-dimensional problems
w
-
½
°
®
°
¾
w
x
(4.5)
w
°
°
w
y
¯
¿
The conservation of energy condition states that the outflow must be equal to the
inflow, plus any flow per unit volume,
ˆ
, generated by a source.
w
q
z
q
dz
z
w
z
q
x
w
q
x
q
dx
w
q
x
w
x
q
y
y
q
dy
y
w
y
z
dz
y
dx
dy
x
q
z
Figure 4.1
Heat flow in an infinitesimal cube
For the infinitesimal cube in Figure 4.1 this gives the following
w
q
w
q
§
·
w
q
§
·
§
·
y
x
z
q
dx
dydz
¨
©
q
dy
¸
¹
dxdz
q
dz
dxdy
©
¹
©
¹
(4.6)
x
y
z
w
x
w
y
w
z
ˆ
q
dydz
q
dxdz
q
dxdy
Q
dxdydz
x
y
z
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