Suppose that a solid of indefinite thickness is subjected to a constant heat flux $q_0$ 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?

$=\frac{T-T_{}}{b\sqrt[\phantom{2}]{t}},=\frac{x}{c\sqrt[\phantom{2}]{t}}$

$b$ 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 $b=2q_0\sqrt[\phantom{2}]{?}\frac{}{k}$ and $c=2\sqrt[\phantom{2}]{?}$ and use the similarity method to show the derivation of the ODE

governing $()$

$\frac{d^2}{d{}^2}+2\frac{d}{d}-2=0$

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$={\displaystyle \underset{x}{\overset{}{}}}$dserfcs$=-$xerfcx$+\frac{e^{-x^2}}{\sqrt[\phantom{2}]{}}$

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$.4\mathrm{.}2-1$ with substitutions $CT-T_{},D,$ and $K{C}_0{T}_0-T_{}.$ Make three

plots:

constant flux: $T_{x,t}v{s}_x$ for several $t$

similarity solution: $\frac{}{{}_{{?}_{?}}}(0)$ vs $$ for both constant flux and constant temperature boundary conditions

Use the values $T_{}=0,k=1,=1,T_0=1,q_0=1,$ and $t=0\mathrm{.}01,0\mathrm{.}10,0\mathrm{.}30$ when plotting.