HW2 solution

HW2 solution - Homework #2 Name_solution_ Problem 1....

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Homework #2 Name___ solution ____ Problem 1 . Starting with an energy balance on a volume element, derive the two dimensional transient heat conduction equation in rectangular coordinates for T((x, y, t) for the case of constant thermal conductivity with heat generation. = + + + element the of content energy the of change of Rate element in generation heat of Rate y y and x at x conduction heat of Rate y and at x conduction heat of Rate t E G Q Q Q Q element element y y x x y x = + + + + yH x g V g G element element = = H is the constant height of element in z direction ) ( ) ( t t t t t t t t t element T T yH x C T T mC E E E = = = + + + ρ substituting in first equation, t T T yH x C yH x g Q Q Q Q t t t y y x x y x = + + + + + ) ( divide by yH x t T T C g y Q Q xH x Q Q yH t t t y y y x x x = + + + + ) ( 1 1 which reduces to x T C g y T k y x T k x = + + because for each of the first two terms we can write
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= = = + x T k x x T yH k x yH x Q yH x Q Q yH x x x x 1 1 1 and for constant thermal conductivity, the equation reduces to. .. t
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HW2 solution - Homework #2 Name_solution_ Problem 1....

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