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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