$5\mathrm{.}2$ Consider a spherical vessel $($constant volume$)$ having a radius of $10cm.$

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

is initially at temperature $T_i.$ The heat losses to the surroundings per unit

volume of the vessel are $q_L^{}{?}^{\text{'}\text{'}\text{'}}=\frac{A_s}{V}$tilde$(h)(T-T_{}),$ where $T$ is the temperature, $V$ is

the volume of the vessel, $A_S$ is its surface area, tilde$(h)$ is the heat transfer coefficient

$(15\frac{W}{m^2}-K),$ and $T_{}$ is the ambient temperature $(300K).$ The rate of heat

generation per unit volume is $q_G^{}{?}^{\text{'}\text{'}\text{'}}=Q_c$hat$(r^{})$ where $Q_c$ is the heat of combustion

$(M\frac{J}{m}ol)$ and hat$(r)$ is the fuel consumption rate $mo\frac{l}{m^3-s}.$

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

b$.$ For tilde$(h)=15\frac{W}{m^2}-K,$ plot $q_L^{}{?}^{\text{'}\text{'}\text{'}}$ and $q_G^{}{?}^{\text{'}\text{'}\text{'}}$ as a function of the system's initial

temperature $T_i$ for $T_{i}300K.$ You do not have to calculate how the system

evolves in time, focus only on its initial state.

c$.$ For tilde$(h)=15\frac{W}{m^2}-K,$ 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 $(T_c).$