Page 147 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN MULTI-DIMENSIONS
2
a
l h = 1.2 w/cm °C, T = 30°C k 139
100°C
q = 2 w/cm 2
5 cm

i j
5 cm

Figure 5.12 Heat conduction in a square plate. Approximated using a rectangular (square)
element

again, on simplifying we obtain
 
 5.7
 
8.3
 
{f}= (5.45)
97.7
 
93.3
 
Therefore, the final form of the set of simultaneous equations can be written as
     
8.0 −2.0 −4.0 −2.0
T 1  5.7
   
−2.0 8.0 −2.0 −4.0    8.3 
1   T 2
  = (5.46)
6  −4.0 −2.0 20.0 4.0  T 3 97.7
   
−2.0 −4.0 4.0 20.0    93.3 
T 4
The temperatures at points 2 and 3 are known. Substitution into the above system results
in the following simultaneous equations,
8T 1 − 2T 4 = 634.2
−2T 1 + 20T 4 = 559.8 (5.47)
◦
◦
The solution of the above simultaneous equation gives T 4 = 36.85 C and T 1 = 88.48 C.

5.4 Plate with Variable Thickness

The conduction heat transfer in a plate with variable thickness is essentially a three-
dimensional problem. However, if the thickness variation is small, it is possible to express