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 left hand side menu of the page or click the next button.Read More

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