Page 59 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
Table 3.2 Data for Example 3.2.2
Node x (cm) y (cm) T C 51
◦
i 0.0 0.0 50.0
j 4.0 0.0 70.0
k 0.0 2.5 100.0
The temperature at any location within the triangle is given by Equation 3.38
The shape functions are calculated using Equation 3.39 with the x and y coordinates as
given in Table 3.2. The result is
1
N i =
10
5
N j =
10
4
N k = (3.45)
10
The substitution of the nodal temperatures and the above shape function values into
Equation 3.38 results in the temperature of the point (2.0, 1.0) being
1 5 4
◦
T = N i T i + N j T j + N k T k = (50) + (70) + (100) = 80 C (3.46)
10 10 10
The components of heat flux in the x and y directions are calculated as
 
T i 
q x k b i b j b k 2 50
=− T j =− (3.47)
q y 10 200
2A c i c j c k  
T k
The position of the 60 C isotherm may be obtained from Figure 3.8. From the given
◦
temperature values, it is clear that one 60 C point lies on the side ij (point P ) and another
◦
lies on the side ik (point Q). It should be noted that the temperature varies linearly along
these sides, that is, temperature is directly proportional to distance.
y

k (0,2.5)
100°C
60°C
Q(0,0.5) 70°C
x
50°C i P(2,0) j
(0,0) (4,0)
Figure 3.8 Isotherm within a linear triangular element