A forward feed double effect evaporator is to be used to concentrate sugar solution. The schematic of a double effect evapopXor is

$V_1$

shown below. The vapor from the first effect is used to heat the second effect. The concentrated product from the first effect is fed to the second effect. The feed flow rate $F=10000K\frac{g}{h}$;$x_y=0\mathrm{.}1,x_2=0\mathrm{.}5.T_y=50C.$ Saturated steam at $115C$ is used to heat the first effect. Pressure of $13\mathrm{.}65$kPa is maintained in the second effect,i$.$e $P_2=13\mathrm{.}65$kPa with a saturation temperature of $51\mathrm{.}67C.$ The boiling point rise of sugar solution in $C$ is given by$,BPR=1\mathrm{.}78x+6\mathrm{.}22x^2.$ The heat capacity of sugar solution is given by $c_p=4\mathrm{.}19-2\mathrm{.}35x$ in $k\frac{J}{k}gK.$ The overall heat transfer coefficients for the two effects are given by$,U_1=2000\frac{W}{m^2}K$ and $U_2=1500\frac{W}{m^2}K.$

a$.$ From overall mass balance, calculate $L_2$

b$.$ Assuming equal vapor generation, calculate $V_1,V_2,L_1$ and $x_1.$

c$.$ Calculate $T_{\text{m}\text{a}\text{n}\text{d}}$ and assuming equal areas for both effects, estimate $T_1$ and $T_2.$

d$.$ Calculate $T_1,T_2$ and $P_1.$

e$.$ Write energy balances for effects $1$ and $2$ and recalculate $V_1$ and $V_2.$

f$.$ Calculate the areas of the two effects.

g$.$ Evaluate the assumptions of equal vapor generation and equal areas.