Page 38 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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For the furnace wall,
dT w 4 4 SOME BASIC DISCRETE SYSTEMS
c w = p σA p (T − T ) + h g A g (T g − T w )
p
w
dt
4
4
− h w A w (T w − T a ) − w σA w (T − T ) (2.41)
w a
The above three equations can be recast in matrix form as
·

[C] T + [K]{T}= {f} (2.42)
where
 
c p 0.00.0
[C] =  0.0 c g 0.0  (2.43)
0.00.0 c w
 
dT p
 
 
 
 dt 
 
 
· dT g
 
T = (2.44)
 dt 
 
 
 
 
dT w 
 
dt
 
T p
{T} = T g (2.45)
 
T w
 
h p A p −h p A p 0.0
[K] =  −h p A p h p A p + h g A g −h g A g  (2.46)
0.0 −h g A g h g A g + h w A w
and
 
0.0
 
{f} = 0.0 (2.47)
h w A w T a + p σA p (T − T ) − w σA w (T − T )
 4 4 4 4 
p
a
w
w
where h p is the heat transfer coefficient from the metallic part to the gas; A p , the surface
area of the metallic part in contact with the gas; h g , the heat transfer coefficient of the
gas to the wall; A g , the surface area of the gas in contact with the wall; h w , the heat
transfer coefficient from the wall to the atmosphere; A w , the wall area in contact with the
atmosphere; p and w , the emissivity values of the metallic part and the wall respectively
and σ the Stefan–Boltzmann constant (Chapter 1).
Although we follow the SI system of units, it is essential to reiterate here that the
temperatures T p , T g , T w and T a should be in K (Kelvin) as radiation heat transfer is
involved in the given problem. In view of the radiation terms appearing in the governing
equations (i.e., temperature to the power of 4), the problem is highly nonlinear and an