# Writing correct physical equation for mixing process

## Main Question or Discussion Point

Hello to all,

I want to analyze and make model of the water mixing process as shown in file in attacment. Basically, there are two input variables: input hot water flow and cold water flow. There is constant flow FL3. I need to write equation that describe how level and tank's temperature is changing. Equation for the level is:
dh/dt=(FL1(t)+FL2(t)-FL3(t))/Area;

And I need to figure out what equation to use to describe temperature.
At first I thought that this is correct equation:
V*dT/dt = Fl1*T1+Fl2*T2-(FL1+FL2)*T
But I have saw different equation:
V*dT/dt = Fl1*T1+Fl2*T2-FL3*T

Which of these is correct?
I know that these equations should be derived from heat equations m*C*(Ta-Tb), but not sure how to do this.
Can you help?

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I've been doing some research and think I got it right, but need confirmation.
Mass must be conserved, so dm1/dt+dm2/dt-dm3/dt = mass accumulation in
the tank.
Now from the heat point of view, there is a heat going into tank through
flows F1 and F2, there is heat coming out of the tank through flow F3,
and there is heat that is contained in the tank.
Sincefluid is water, I assume same density and specific heat, so I
have:
V1*T1+V2*T2-V3*T3 = V*T(T is temperature in the tank, and V is
current volume in the tank, V1, V2 are volumes coming into tank).
Now, since I need to know how T3 is changed, I need to find derivation
of above's equation:
F1*T1+F2*T2-F3*T = d(V*T)/dt
Further, I have:
F1*T1+F2*T2-F3*T = T*dV/dt+ V*dT/dt
Since tanks is assumed to be cylinder, cross section area A is
constanst:
F1*T1+F2*T2-F3*T = A*T*dLevel/dt+A*Level*dT/dt
So final equation is:

dT3/dt = (F1*T1+F2*T2-F3*T-A*T*dLevel/dt)/(A*level)

Maybe, I missed something

In the attachment you can see how process looks like. I hope my temperature equation is good:
dT/dt = (F1*T1+F2*T2-F3*T-A*T*dLevel/dt)/(A*level)
If we write differential equation for level:
A*dLevel/dt = F1+F2-F3 now temperature equation becomes even simpler:
dT/dt = (F1*(T1-T) + F2*(T2-T)) / (A*level)
Am I doing something wrong?

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