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Suppose that a solid of indefinite thickness is subjected to a constant heat flux q 0 at x = 0 , beginning at t =

Suppose that a solid of indefinite thickness is subjected to a constant heat flux q0 at x=0, beginning
at t=0. It occupies the space x>0 and its initial temperature is uniform at T. The solid's thermal
conductivity is k and its thermal diffusivity is .
(a) State the partial differential equation, boundary conditions, and initial conditions that govern T??(x,t).
Use order-of-magnitude reasoning to infer that both the penetration depth (t) and the surface temperature.
transformations when seeking a similarity solution?
=T-Tbt2,=xct2
b and c are positive constants. (Hint: You have two unknowns, the characteristic T and , and two
equations, the PDE for T and the flux boundary condition on which you can order-of-magnitude estimate.)
(b) Set b=2q0?2k and c=2?2 and use the similarity method to show the derivation of the ODE
governing ()
d2d2+2dd-2=0
State the boundary conditions on ().
(c) The general solution to this ODE is
()=Aierfc()+Bierfc(-)
where ierfc is the integral of the complementary error function
ierfcx=xdserfcs=-xerfcx+e-x22
Use the boundary conditions to complete the solution for ().
(d) How does T??(0,t) from the similarity solution in part (b) compare with the order-of-magnitude estimate
in part (a)?
(e) Plot your similarity solution and compare the results to the similarity solution for transient conduction
with a constant temperate boundary condition. Transient conduction is identical to transient diffusion, so
use the solution from Ex.4.2-1 with substitutions CT-T,D, and KC0T0-T. Make three
plots:
constant flux: Tx,tvsx for several t
similarity solution: ??(0) vs for both constant flux and constant temperature boundary conditions
Use the values T=0,k=1,=1,T0=1,q0=1, and t=0.01,0.10,0.30 when plotting.
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