A cylindrical tank of height 2 m and cross sectional area of 0.5 m2 contains hot

Question

A cylindrical tank of height 2 m and cross sectional area of 0.5 m2 contains hot water at 80oC, 125 kPa. It is in a room with temperature To = 20oC, so it slowly losses energy to the room air proportional to the temperature difference as

?Q'=CA'(T-T0)

With the tank surface area, A and C is a constant. For different values of the constant C, estimate the time it takes to bring the water to 50oC. Make enough simplifying assumptions so that you can solve the problem mathematically, to find a formula T(t).

Explanation / Answer

This is a classical problem in Newton's Law of cooling.
The differential equation governing the cooing process (which assumes constant heat transfer coefficients is;
dT/dt = -k.(T-T?)
The general solution (GS) is easily obtained by separation of variables to be :
T = c.e^(-k.t) + T? where, in this case C.A = k
[Note: since heat leaving the tamk represents cooldown you original equation would have been more clearly written as : Q* = -C.A.(T-T?)]
In the above GS obviously c = T(0) - T? = 60

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