Problem $1.$ Conceptual$/$multiple choice questions:

A$.(2$pts$)$ Simplify the heat equation below by applying ALL the assumptions listed:

Two$-$dimensional conduction $2)$ Steady state heat transfer $3)$ no internal heat generation $4)$ constant thermal conductivity $k_y=4k_x$

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(k_x\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(k_y\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{y}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(k_z\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}})+q^{}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$

Circle the correct solution below:

a$).\frac{de{l}^{2}T}{del{x}^2}+4\frac{de{l}^{2}T}{del{y}^2}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$

b$).4\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{y}^2}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$

c$).\frac{de{l}^{2}T}{del{x}^2}+4\frac{de{l}^{2}T}{del{y}^2}=0$

d$).4\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{y}^2}=0$

B$.$ Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as:

$T_{\text{l}\text{e}\text{f}\text{t}}+T_{\text{t}\text{o}\text{p}}+T_{\text{r}\text{i}\text{g}\text{h}\text{t}}+T_{\text{b}\text{o}\text{t}\text{t}\text{o}\text{m}}-4T_{\text{n}\text{o}\text{d}\text{e}}+q^{}\frac{(x{)}^2}{k}=0$

$(1)(2$pts$)$ Is the heat transfer process in this medium steady or transient?

$(2)(2$pts$)$ Is the heat transfer process one$-,$ two$-,$ and three$-$dimensional?

$(3)(2pts)$ Is there heat generation within the medium?

$(4)(2$pts$)$ Is the nodal spacing constant or variable?

$(5)(2$pts$)$ Is the thermal conductivity of the medium constant or variable?

C$.(3$pts$)$ Explain the difference between Biot number and Nusselt number.