Stationary Condition in a Gravity-Flow Tank

The main equation that governs the tank system is the mass balance over the tank. It results in an ODE, which relates the tank level h to the inflow and outflow q rates as follows  

In addition to this, an algebraic equation foDows on from the pressure balance, equating the hydrostatic pressure at the bottom of the tank with the pressure losses over the valve and the overall tube. This equation reads as follows  

When a turbulent flow regime is assumed, the friction factor is constant Thus, the algebraic equation can be solved analytically in q(h(t), 

We want to calculate the diameter of the tube that leads to the steady state with level h = 10 m and a valve open position v = 0.5. At the steady state 

the problem is brought back to the search for the diameter Dtuhe that gives 

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