The air specific impulse $I_{sp,\text{a}\text{i}\text{r}}$ and the specific fuel consumption SFC for the turbofan engine $($ideal cycle$)$ are given below:

$I_{sp,\text{a}\text{i}\text{r}}=\frac{a_0{M}_0}{g}(\frac{1}{1+}(\frac{u_9}{u_0}-1)+\frac{}{+1}(\frac{u_9^{\text{'}}}{u_0}-1))$

$SFC=\frac{c_p{T}_0}{Q_R}(\frac{{}_{}-{}_r{}_c}{(1+)I_{sp,\text{a}\text{i}\text{r}}})$

where

$(\frac{u_9^{\text{'}}}{u_0}{)}^2=\frac{{}_r{}_c^{\text{'}}-1}{{}_r-1}$

$(\frac{u_9}{u_0}{)}^2=\frac{{}_{}}{{}_r-1}[{}_t-\frac{1}{{}_r{}_c}]$

${}_t=1-\frac{{}_r}{{}_{}}[({}_c-1)+({}_c^{\text{'}}-1)]$

and where $$ is the bypass ratio.

a$)$ For fixed values of all system parameters ${}_c,{}_c^{\text{'}},M_0,{}_{},$ etc., show that the expression for ${}^{**},$ the value of the bypass ratio which gives the minimum fuel consumption $($SFC$)$ is:

${}^{**}=[(\frac{u_9^{\text{'}}}{u_0}{)}^2-1{]}^{-1}[\frac{{}_r}{{}_r-1}(\frac{{}_{}}{{}_r{}_c}-1)({}_c-1)+\frac{{}_{}}{{}_r{}_c}-\frac{1}{4}(\frac{u_9^{\text{'}}}{u_0}+1{)}^2]$