ONE DIMENSIONAL HEAT TRANSFER ELEMENT
In structural problem displacement at each nodel point is obtained. By using these displacement solutions, stresses and strains are calculated for each element. In structural problems, the unknowns are represented by the components of vector field. For example, in a two dimensional plate, the unknown quantity is the vector field u(x,y),where u is a (2x1)displacement vector.
Heat transfer can be defined as the transmission of energy from one region another region due to temperature difference. A knowledge of the temperature distribution within a body is important in many engineering problems. There are three modes of heat transfer.
They are: (i) Conduction
1Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
Following the same procedure as in Section, the portion of the boundary where the temperature is prescribed, i.e. the essential boundary is denoted by T and the boundary where the flux is prescribed is recommended for Science and Engineering Track. Denoted by q ; these are the boundaries with natural boundary conditions. These boundaries are complementary, so
With the unit normal used in , we can express the natural boundary condition as qn ¼ q. For example, positive flux q causes heat inflow (negative q ) on the left boundary point where qn ¼ q ¼ q and heat outflow (positive q ) on the right boundary point where qn ¼ q ¼ q.
Strong form for 1D heat conduction problems
2Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions
We again multiply the first two equations in the strong form by the weight function and integrate over the domains over which they hold, the domain for the differential equation and the domain q for the flux boundary condition, which
yields ws dx with w ¼
T and combining with gives
Recalling that w ¼ 0 on
Weak form for 1D heat conduction problems
Find T ðxÞ 2 U such that
Notice the similarity between
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