Three tanks in series are used to prebeat a multicomponent oil solution before it is fod to a diatillation column for separation as shown in the figure below. Each tank is imitially filled with $Mkg$ of ail at $T_n"$C$.$ Saturated steam at a tempenture of $T_m=="$C condenses within coils imumersed in each tank.

Figure: Series of for oil heating

The oil is fed into the fint tank at the rate of $W_{1}k\frac{g}{m}in$ and overllows into the second and the third tunis at the same flow rate. The temperature of the eil fed to the first tank is $T_{0}C.$ The tanks afe well mised so that the temperature inside the tamks cat be assumed to be uniform, and the outlet stream temperature is sume as the temperature within the tunk. For a particular tank, the rute at which heat is transferred to the oil from the stam coil is given by the expresaion:

$Q=UA(T_{}-D)$

where

$Q=$ rate of heat transferred in $\frac{K}{m}in.$

$\frac{U}{A}=10k\frac{J}{m}in-{?}^{-2}C$ is the product of the heat transfer coefficient and the area of the cuil, and $T=$ temperature of the oil in the tank in $("$C$),$ and

$T_{\text{a}\text{a}\text{m}}=$ tempenatare of $).$

Energy balances can be made on each of the individual tanks. In these balances, the mass flow rate to each tank will remain at the same fixed value. Thus $W-W_l-H_2-W_3($kgimin$).$ The mass in each tank will be assumed conshant as the tank volume and oil density are assumed to be constant. Thus $M-M_t-M_2-M_1(kg).$ For the finst tanik$,$ the energy balance can be expreased by:

Accumelation $=$ Input $-$ Output

$1$

where $C_p(\frac{k}{k}g)$ is heat capacity of the oil.

Note that the enstesuly state mass balunce is not needed for tank $1$ so any other tunks since the mase in each tank does not change with time. The above differential equation cant be rearmanged and explicitly solved for the derivative which is the usual format for numerical solution:

$\frac{\text{:}\frac{T_1}{dt}}{M}=[U{C}_0(T_0-T_3)+UA{T}_2-T_0)C_r$

Similarly, for the second tank:

For the third tank:

$\frac{U_1}{dt}=\frac{H{C}_{,}(T_2-T_0)+CU(T_{-}-T_1)}{M}{C}_{,}$

Given the following given data:

$T_n=20C,T_{\text{m}\text{a}\text{n}}=250C,M=1000kg$:$W=100$kgimin, und $Cp=2\mathrm{.}0(k\frac{J}{k}q)$

You can use time interval of $0\mathrm{.}5$min

Solve equations $(4)-(6)$ simultaneously and determine the steady state temperatures in all three taka i$.$c$.T_1,T_2$ and $T_2.$

What time interval will be requined for $T$ to reach $995$ of this steady state value?

On a single graph, make a plot of temperature verses time for $T_1,T_2$ and $T_2.$