Question $2-$ hints

For this problem, remember that

Mhat$(C{)}_p=\frac{U_2-U_1}{T_2-T_1}$

Where $M$ represents the mass of a system. We can also use the equation above by defining the internal energy $U$ of a system at temperature

$T$ relative to some reference state with internal energy $U_{\text{r}\text{e}\text{f}}$ and temperature $T_{\text{r}\text{e}\text{f}}.$ This gives:

Mhat$(C{)}_p=\frac{U-U_{\text{r}\text{e}\text{f}}}{T-T_{\text{r}\text{e}\text{f}}}$

For systems with constant volume, we can use a similar approach for enthalpy $H.$

Remember from our definition of enthalpy that:

$H=U+$PVA mixing tank initially contains $100$ kg of water at $20\backslash$deg C$.$ Suddenly, two inlet valves and one outlet valve are opened. Once the valves open, there are two inlet streams of water each having a flow of $10$ kg$/$min into the tank; and there is one outlet stream of water at a flow of $20$ kg$/$min leaving the tank.

The temperature of one inlet stream is $70\backslash$deg C$,$ and the temperature of the other is $40\backslash$deg C$.$ The tank is well$-$mixed, so that the temperature of the outlet stream is always the same as the temperature of the water in the tank. Assume that the stirring in the tank does not significantly affect the energy of the water in the tank.

a$)$ Compute the steady$-$sate temperature that will finally be obtained in the tank.

b$)$ Develop an expression for the temperature of the fluid in the tank at any time. Note: this expression must be of the form T$=$f$($t$).$