I put some water in a container with initial temperature $T_0$ in a room, and the room's initial temperature is $T_a$. Now the container is filled to the maximum, so any more water coming in will result in an overflow.
I want to keep the water at initial temperature for a long time, preferably for inifinite amount of time. Also water entering the container flows on a constant rate and constant temperature.
Sorry for being not specific. So far I came up with the differential equation on the temperature of water at any point :
$$T(t)=T_a+(T_0-T_a)e^{-kt}$$ That is just Newton's law of cooling. Then I have found the following equation: The volumetric flow rate in a heating system can be expressed as
$$q=\frac{h}{c_\rho*\rho*(T_{new}-T_t)}$$
where
$q$ = volumetric flow rate
$h$ = heat flow rate
$c_\rho$ = specific heat capacity
$\rho$ = density
$T_{new}-T_t$ = temperature difference
so I thought I would solve it for $h$ and add to the equatio for $T(t)$ Hence $$T(t)=T_a+(T_0-T_a)e^{-kt}+q*c_\rho*\rho*(T_{new}-T_t)$$ From then I made $T_t$ the subject and then set LHS to $T_0$ because that is the temperature I want to keep. I have no idea if that is correct, probably I missed out a lot of crucial variables so I seek help. Is this correct approach?
If not, can you tell me what is? I asked this on physics stack exchange but nobody answered. Regards, Patrick