Boundary Conditions and Other Important Issues Notes | EduRev

Created by: Renu Garg

: Boundary Conditions and Other Important Issues Notes | EduRev

 Page 1


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_1.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
The Lecture deals with:
Boundary Conditions and Other Important Issues
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_1.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
The Lecture deals with:
Boundary Conditions and Other Important Issues
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_2.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
Let us, for the present, consider a problem with no radiating boundary. Thus, the nodal
equation for any of the nodes of the FEM can be constructed by appropriately summing up
the contributions form the heat conduction term  heat generation term 
convective boundary terms  and the heat flux boundary  if the point lies on the 
 boundary. If the point lies on a known temperature boundary, the nodal equation is
very simple, it takes the form  for the node  where  is the prescribed boundary
temperature.
The process of assembling all the contributions from the element in terms of nodal
temperatures is done as follows. We try to obtain a global matrix equation in terms of
nodal temperatures:
(22.1)
where  is conduction matrix and  is the heat load vector.
The  matrix is formed from the contribution of conduction resistances between the
nodes of the triangular elements[Eq. (20.7)] and the convective resistance of the surface 
 where the convective losses take place [Eq. (21.5)]. The heat load vector on the right
hand side receives contributions from [Eq. (21.1) for each triangular element and heat flux
through  boundary [Eq. (21.4)]. The nodes falling on radiative boundary (term  in
Eq. (19.1 ) requires special attention.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_1.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
The Lecture deals with:
Boundary Conditions and Other Important Issues
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_2.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
Let us, for the present, consider a problem with no radiating boundary. Thus, the nodal
equation for any of the nodes of the FEM can be constructed by appropriately summing up
the contributions form the heat conduction term  heat generation term 
convective boundary terms  and the heat flux boundary  if the point lies on the 
 boundary. If the point lies on a known temperature boundary, the nodal equation is
very simple, it takes the form  for the node  where  is the prescribed boundary
temperature.
The process of assembling all the contributions from the element in terms of nodal
temperatures is done as follows. We try to obtain a global matrix equation in terms of
nodal temperatures:
(22.1)
where  is conduction matrix and  is the heat load vector.
The  matrix is formed from the contribution of conduction resistances between the
nodes of the triangular elements[Eq. (20.7)] and the convective resistance of the surface 
 where the convective losses take place [Eq. (21.5)]. The heat load vector on the right
hand side receives contributions from [Eq. (21.1) for each triangular element and heat flux
through  boundary [Eq. (21.4)]. The nodes falling on radiative boundary (term  in
Eq. (19.1 ) requires special attention.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2022/22_3.htm[6/20/2012 4:51:12 PM]
 Module 3: Introduction to Finite Element Method
 Lecture 22:
 
The radiation heat transfer process can be modeled (using the procedure of quasi-
linearization) as
(22.2)
where  with  denoting the guess value of  calculated
from the previous iteration or the specified guess. Therefore, the radiation integral
becomes:
(22.3)
which can be handled like a convection term. Although  varies over the length of the line
element  due to variation in the value of  it may not be worth the trouble if the
element sizes are not too large. In such situations, one can set
(22.4)
where  and  are the nodes which make up the concerned boundary element.
A similar averaging procedure is often done for non-linear effects like temperature
dependent properties, variable heat transfer coefficient at the boundary and variable heat
flux at the boundary.
 
Congratulations, you have finished Lecture 22 To view the next lecture select it from the
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