$2$ Consider a spherical vessel $($constant volume$)$ having a radius of $10$ cm$.$

It contains a stoichiometric mixture of methane and air at $1$ atm. The system

is initially at temperature Ti

$.$ The heat losses to the surroundings per unit

volume of the vessel are q$\_$

$000$

L $$

AS

V

~h T$$ T$1,$ where T is the temperature, V is

the volume of the vessel, AS

is its surface area, h~is the heat transfer coefficient

$(15$ W$/$m$2$

$-$K$),$ and T$1$ is the ambient temperature $(300$ K$).$ The rate of heat

generation per unit volume is q$\_$

$000$

G $$ Qc

$^$r$\_$ where Qc is the heat of combustion

$($MJ$/$mol$)$ and $^$r$\_$ is the fuel consumption rate $[$mol$/($m$3$

$-$s$)].$

a$.$ Calculate the heat of combustion of the mixture Qc

$.$

b$.$ For h~ $15$ W$/$m$2$

$-$ K$,$ plot q$\_$

$000$

L

and q$\_$

$000$

G

as a function of the system$$s initial

temperature Ti

for Tir$300$ K$.$ You do not have to calculate how the system

evolves in time, focus only on its initial state.

c$.$ For h~$15$ W$/$m$2$

$-$ K$,$ what is the lowest initial temperature at which the rate

of heat production by combustion offsets the heat losses?

d$.$ Calculate the autoignition temperature of the system $($Tc

$).$