Problem:

$(1)$ Derive the entropy balance equation:

$\frac{Ds}{Dt}=+\frac{1}{}\frac{\text{d}\text{e}\text{l}}{del{x}_i}(\frac{k}{T}\frac{\text{d}\text{e}\text{l}\text{T}}{del{x}_i})+\frac{k}{T^2}(\frac{\text{d}\text{e}\text{l}\text{T}}{del{x}_i}{)}^2+\frac{}{T}$ with $-=\frac{1}{}{}_{ij}{S}_{ij}$

and then discuss the meaning of the right$-$hand$-$side terms and their effects on entropy generation.

Hint: apply the Gibb's property relation $Tds=de+pdv$

$(2)$ Derive the following alternative form of the thermal energy balance

$c_p(D\frac{T}{D}t)=T(D\frac{p}{D}t)++(de\frac{l}{d}el{x}_i)(k(del\frac{T}{d}el{x}_i))$

with the thermal expansion coefficient $-=-\frac{1}{}(\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{T}}{)}_p$ and $-=\frac{1}{}{}_{ij}{S}_{ij}$

Hint: consider enthalpy $h=h_o+c_{p}T$