$11\mathrm{.}24$ An agitated tank, shown schematically in

Fig. P$11\mathrm{.}24,$ contains $6200kg$ of a dilute agueous

solution. The inside heat transfer coefficient due

to agitation is $5861\frac{W}{m^2}*C.$ The st$$l walls

Condensate out

Figure P$11\mathrm{.}24$

of the tank are $10mm$ thick. Steam at $110C$

condenses on the outside of the steel surface and

gives a heat transfer coefficient of $10k\frac{W}{m^2}*C.$

The heat transfer area is $14m^2.$ How long does

it take to heat the vessel contents from $20C$

to $60C?$

In this problem, assume the thermal conductivity of the wall is k $=16$

W$/($m$*$K$).$ The aqueous solution has density $=1000$ kg$/$m$3$ and heat capacity

Cp $=4\mathrm{.}2*103$ J$/($kg $*$K$)$

$($a$)$ Draw the thermal circuit from the wall to the inside of the tank. Write

an expression for the heat flux going into the tank if the tank is at a

given temperature T$.$

$($b$)$ Perform a macroscopic energy balance over the tank to determine the

differential equation for its temperature T$.$ Solve the differential

equation to find out how long it takes the temperature to increase from

$20$C to $60$C$.$