A premixed laminar flame is stabilized in a one$-$dimensional gas flow where the vertical

velocity, $v(c\frac{m}{s}),$ of the unburned mixture varies with the horizontal coordinate, $x(cm),$

according to the following equation. A in the equation is used to assess different multiples of the

laminar flame speed. In this equation $L$ is half of the burner horizontal opening. For this

problem, consider $L=1cm.$

$v=S_L+(\frac{A{S}_{I_{.}}-S_{I_{.}}}{L})x$

Consider propane $($at $T_u=400K,P=2$atm $)$ combustion in air at an equivalence ratio of $1\mathrm{.}1.$ Use

the experimentally determined laminar flame speeds correlations determined by Metghalchi and

Keck as follows:

$S_L=S_{L\text{r}\text{e}\text{f}}(\frac{T_u}{T_{u\text{r}\text{e}\text{f}}}{)}^{}(\frac{P}{P_{\text{r}\text{e}\text{f}}}{)}^{}(1-2\mathrm{.}1Y_{\text{d}\text{i}\text{l}})$

$S_{L\text{r}\text{e}\text{f}}=B_M+B_2(\frac{O}{-}\frac{O}{{?}_M}{)}^2$

$=2\mathrm{.}18-0\mathrm{.}8(\frac{O}{-}1)$

$=-0\mathrm{.}16+0\mathrm{.}22(\frac{O}{-}1)$

These relations were developed relative to $298K$ and $1$atm. For propane, $\frac{O}{{?}_M}=1\mathrm{.}08,B_M=$

$34\mathrm{.}22c\frac{m}{s},$ and $B_2=-138\mathrm{.}65c\frac{m}{s}.$ Note, $Y_{\text{d}\text{i}\text{l}}$ is the mass fraction of diluent present $($if any$)$

in the air$-$fuel mixture. Complete the following steps.

A$)$ Determine the laminar flame speed for propane at the given conditions. $(6$ points$)$

B$)$ Determine the distribution of the local angle of the flame surface from vertical for $A=$

$1\mathrm{.}1,2,3,4,$ and $5.$ Display your results by plotting the local angle vs$.$ horizontal distance,

x$.(6$ points$)$

C$)$ Assuming the flame speed is independent of position and constant for all cases, determine

the flame shape. Display your results by plotting the flame shape $($z$)$ vs$.$ horizontal

distance $($x$),$ scaling your figures so that $1mm$ on the $x-$axis is the same size as $1mm$ on

the z$-$axis. $(9$ points$)$

D$)$ Describe the results of part B and C in your own words, referring to the plots developed.

What is the impact of increasing the flow rate? $(4$ points$)$