Develop the control volume difference equation for one-dimensional steady conduction in a in with variable cross-sectional area A(x) and perimeter P(x). The heat transfer coefficient from the fin to ambient is a constant oh and the fin tip is adiabatic.
GIVEN
• Fin with variable cross-sectional area and perimeter
• Convection coefficient to ambient is constant, ho
FIND (a) Control volume difference equation
SKETCH
Consider a control volume as shown above
An energy balance on this control volume is expressed by
heat conducted into left face =
heat convected out perimeter + heat conducted out right face
which can be rearranged to give
The boundary conditions can be written as
This can be written in the form of a tridiagonal matrix, per Equation (3.11) where the coefficients of
the matrix are
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