2.47
KNOWN: PROBLEM Plane wall, initially at a uniform temperature Ti, is suddenly exposed to convection with a fluid at T at one surface, while the other surface is exposed to a constant heat flux q . o FIND: (a) Temperature distributions, T(x,t), for initial, steady-state and two intermediate times, (b) Corresponding heat fluxes on q x coordinates, (c) Heat flux at locations x = 0 and x = L as a x function of time, (d) Expression for the steady-state temperature of the heater, T(0,), in terms of
q , T , k, h and L. o
SCHEMATIC:
ASSUMPTIONS: (1) One-dimensional conduction, (2) No heat generation, (3) Constant properties. ANALYSIS: (a) For Ti < T , the temperature distributions are
Note the constant gradient at x = 0 since q ( 0 ) = q . x o (b) The heat flux distribution, q ( x,t ) , is determined from knowledge of the temperature gradients, evident x from Part (a), and Fouriers law.
(c) On q ( x,t ) t coordinates, the heat fluxes at the boundaries are shown above. x
(d) Perform a surface energy balance at x = L and an energy balance on the wall:
q cond = q conv = h T ( L, ) T (1), q cond = q . (2) o
For the wall, under steady-state conditions, Fouriers law gives q = k o T ( 0, ) T ( L, ) dT =k . dx L
q o . 1/h + L/k
(3)
Combine Eqs. (1), (2), (3) to find:
T ( 0, ) = T +
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