- #1

dRic2

Gold Member

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## Main Question or Discussion Point

Hi, this problem should be very simple: I have a tank containing water. (going in and out). I want to write the differential equations to see what happen to the temperature and the level of the vessel if the flow of water is not constant.

the system should be:

$$A \frac {dl} {dt} = F_{in} - F_{out} $$

$$ \frac {dE} {dt} = H_{in} - H_{out} - Q_{out} $$

##A## is the section of the tank

##l## is the level of water inside the vessel

##F## are the volumetric flow rates

##Q_{out}## represents the heat loss

I have a problem with the second equation (energy balance). How do I handle ##H_{in}## and ##H_{out}## ?

I tried to do this:

$$ \frac {dE} {dt} = ρ F_{in} \left [ h(T_r) + \int_{T_r}^{T_{in}} cp \right ] - ρ F_{out} \left [ h(T_r) + \int_{T_r}^{T_{out}} cp \right ] - Q_{out}$$

##\rho## can be assumed to be constant, but ##F_{in} ≠ F_{out}##, so how to I handle ##h(T_r)##? I can arbitrary set ##h(T_r)## to be ##0##, but then what about ##T_r## ?

Thanks

Ric

the system should be:

$$A \frac {dl} {dt} = F_{in} - F_{out} $$

$$ \frac {dE} {dt} = H_{in} - H_{out} - Q_{out} $$

##A## is the section of the tank

##l## is the level of water inside the vessel

##F## are the volumetric flow rates

##Q_{out}## represents the heat loss

I have a problem with the second equation (energy balance). How do I handle ##H_{in}## and ##H_{out}## ?

I tried to do this:

$$ \frac {dE} {dt} = ρ F_{in} \left [ h(T_r) + \int_{T_r}^{T_{in}} cp \right ] - ρ F_{out} \left [ h(T_r) + \int_{T_r}^{T_{out}} cp \right ] - Q_{out}$$

##\rho## can be assumed to be constant, but ##F_{in} ≠ F_{out}##, so how to I handle ##h(T_r)##? I can arbitrary set ##h(T_r)## to be ##0##, but then what about ##T_r## ?

Thanks

Ric