Derivation for equation 4, 5 and 6

and heat flux is given by Overview qs(t)=tk(TsTi) In this section, the heat conduction problem for the semi-infinite solid, with convection boundary Ts is the constant surface temperature, Ti is condition, will be solved. an initial wall temperature, and is the therA semi-infinite solid is an idealized body that mal diffusivity of the wall. Thermal diffusivity has a single plane surface and extends to infinity is related to the thermal conductivity k and is in all directions except one. If a sudden change is the measure of how quickly heat is dissipated imposed to this surface, transient one- in a material. The function erf is called the dimensional conduction will occur within the Gaussian Error Function. Function erf(w) is solid. defined as We can use the general heat equation, which has the following form [1]: x22T=1tT (1) Values for erf(w) are often tabulated or available in graphical form for convenience [2]. Equation (1) is a second order in displacement 2. Constant surface heat flux, qs=q0 and first order in time; therefore, we need an The solution for the temperature is initial condition and two boundary conditions in defined as order to solve it. dimensional heat transfer without heat generation in the can specify that For the initial condition, we terature inside the solid is uniform at Ti : temperatur T(x,0)=Ti kq0xerfc(2tx) (2) 3. Surface convection, kxTx=0=h[TT(0,t)] The solution for the temperature is defined The interior boundary condition is as T(x,t)=Ti (3) TTiT(x,t)Ti=erfc(2tx) For the above initial and interior boundary conditions, three closed form solutions have been obtained for the surface conditions, which are applied suddenly to the surface at [exp(khx+k2h2t)][erfc(2tx+kht)] t=0