$6\mathrm{.}15$ Consider airflow over a flat plate of length $L=1m$ under conditions for which transition occurs at $x_c=0\mathrm{.}5m$ based on the critical Reynolds number, $R{e}_{x,c}=5{10}^5.$

$($a$)$ Evaluating the thermophysical properties of air at $350$ K $,$ determine the air velocity.

$($b$)$ In the laminar and turbulent regions, the local convection coefficients are, respectively,

$h_{\text{l}\text{a}\text{m}}(x)=C_{\text{l}\text{a}\text{m}}{x}^{-0\mathrm{.}5}$ and $h_{\text{w}\text{u}\text{r}\text{b}}=C_{\text{t}\text{u}\text{r}\text{b}}{x}^{-0\mathrm{.}2}$

where, at $T=350K,C_{\text{l}\text{a}\text{m}}=8\mathrm{.}845\frac{W}{m^{\frac{3}{2}}}*K,C_{\text{t}\text{u}\text{r}\text{b}}=49\mathrm{.}75$ W$/$m $m^{1\mathrm{.}8}*K,$ and $x$ has units of m $.$ Develop an expression for the average convection coefficient, $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{l}\text{a}\text{m}}(x),$ as a function of distance from the leading edge, $x,$ for the laminar region, $0x{x}_c.$

$($c$)$ Develop an expression for the average convection coefficient, $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_{\text{u}\text{r}\text{b}}(x),$ as a function of distance from the leading edge, $x,$ for the turbulent region, $x_{c}xL.$

$($d$)$ On the same coordinates, plot the local and average convection coefficients, $h_x$ and $\frac{?}{\text{b}\text{a}\text{r}}(h{)}_x,$ respectively, as a function of $x$ for $0xL.$