As a simplified application of the principles of control theory to a mass balanc

Question

As a simplified application of the principles of control theory to a mass balance, consider the filling of a tank to a desired height hf. The fluid volume is the product of the cross-sectional area multiplied by the height h(t) and the fluid density p is constant. Fluid flows in at flow rate Qin and leaves at Qout. Crudely, this example can model the flow of blood in and from the heart.

a) Use this information to differential equation that describes the mass balance. Write the equation in terms of the fluid volume in the tank V(t) and the flow rates. Note that accumulation of fluid in volume V= dm/dt.
This is what I was thinking:
V= dm/dt=Q_in-Q_out
?V(t)= dm/dt-Q?_in+Q_out

b) Write the differential relation in terms of the height of the volume of fluid, h.
??

c) For the case in which Qin and Qout are constant and the initial condition that h=h0 at t=0, solve and determine whether hf can be obtained.

Explanation / Answer

)Q_in - Q_out =(dm/dt)/? where ? is density dm =?Adh b)hence Q_in - Q_out =Adh/dt. c) if Q_in and Q_out are constant we can write above equation as ( Q_in - Q_out)?dt =A?dh (Q_in - Q_out )t =Ah +c ...........1 where c is the constant at t=0 ,h=h0 putting this information in equation 1 we get c=-Ah0 hence (Q_in - Q_out )t =Ah -Ah0 hence h at any time t is given as ,.. h =(Q_in - Q_out )/A+h0 and if u are looking for hf at time t-infinity then for Qin>Qout hf will be infinite for Qin =Qout hf =h0 for Qin

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