25: Lecture 2-D Boundary Convection and Line Sources
13.5 Two-Dimensional Finite Element Formulation.
We have completed the derivation of the triangular heat transfer element. The element equations we have are specific to constant distributed energy generation and conductive fluxes across the boundaries. We will look at altering the terms in the element equations so that we can treat problems that include:
Convective conditions at the element (system) boundaries.
This will require an adjustment of the boundary flux term.
Point (line) sources in the 2-D element.
This will require evaluation of the Q residual term in the weighted residuals procedure.
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
Lecture #24 produced:
[ K ]C
1 4S i j m 2 3 0 i j m Sh T+ K yy i j m 2b 1 1 q L QS 2hT S nij ij 1 + 1 2 3 3b 1 1
[ K ]H
1 3 2 3 1 3 1 T= 3 2 3 0 q L 1 nij mi 2 1 1 0 1
i K xx j 0 m
1 q L 1 n jm jm 2 0
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
The conductive stiffness matrix can be put in a closed form:
K xx i2 + K yy i2 4S = K xx i j + K yy i j 4S K xx j2 + K yy 2 j 4S K xx i m + K yy i m 4S K xx j m + K yy j m 4S 2 2 K xx m + K yy m 4S
[ K ]C
If the material is isotropic then Kxx=Kyy and the matrix simplifies.
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
Consider P. 13.9:
P( e )
N T qn dP = N T qnij dP
i
j
N T qn jm dP
j
m
N T qnmi dP lim
m x
i
Over edge ij we have convection. When we treated the boundary flux terms in Lecture #24 we said that a conductive flux must be constant over the edge. Since temperature varies over edge ij, a convective flux must vary as we move along the edge. So, we have to redo our boundary flux term for edge ij.
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
Over the edge ij the boundary term becomes:
T T T N T qnij dP = N ij h Tij T dP = N ij hTij dP + N ij hT dP i i i i j j
(
)
j
j
Tij = N i
Nj
Ti T j Nm ij T m Ti 1 (1 + s ) 0 T j 2 Tm
A constant load applied over the edge ij A move in the direction dP along edge ij is is a move along the s axis of the natural coordinate system.
1 = (1 s ) 2
Using the shape functions derived in Lecture #24 along edge ij, t=-1..
MECH 420: Finite Element Applications
dP =
Lij 2
ds
Lecture 2-D 25: Boundary Convection and Line Sources
The variation of the temperature profile over the edge is linear. We can evaluate the reformed boundary term easily using the natural coordinates. 1
N hTij dP = h
T ij i j
Lij
1 1 2 (1 s) 2
+1
2 (1 s ) Ti 1 1 (1 + s ) 0 (1 + s ) ds T j 2 2 Tm
2 3 hLij 1 = 2 3 0
1 0 3 Ti 2 0 T j 3 Tm 0 0
The convective boundary condition has produced an addendum to the stiffness matrix.
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
Bringing in the final term due to the ambient load over the boundary edge ij:
2 3 j hLij 1 N T qnij dP = 2 3 i 0 1 0 3 Ti 2 T + hT Lij 0 j 3 2 T 0 0 m 1 1 0
Over any edge the temperature variation would be linear. We can expect the same results if we looked at convection on the other two edges
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
Over edge jm:
0 0 m hL jm 0 2 N T qn jm dP = 2 3 j 1 0 3 0 Ti hT L 1 jm Tj + 3 2 T 2 m 3 0 1 1
Over edge mi:
N T qnmi
m i
1 2 0 3 3 Ti hT Lmi hLmi dP = 0 0 0 T j + 2 2 T 1 2 m 0 3 3
1 0 1
MECH 420: Finite Element Applications
Lecture 25: 2-D Boundary Convection and Line Sources
13.6 Line or point sources.
Consider again P. 13.9. The source is contained within a filament that extends into the page. The source is in terms of Btu/(h.ft). In the plan view, the line sou...