The problem I have at hand is the following: I have a container C with a given volume V. I need to fill it with water, fast, up to a given pressure P. I have a number of high pressure hydropneumatic pumps to choose from, but they are expensive and take a while to get here. I don't even have such pump readily at hand to test the equations. The flow rate for these pumps is easily calculated as:

$$ Q = Q_{max} - K_{b} P $$ $K_{b}$ is a pump parameter per datasheet. Obviously pressure $P$ won't go higher than $Q_{max}/K_{b}$, the pump stops working. Now I start simplifying things: I consider no thermal factors, no sudden pipe diameter change, no turbulence, no back flow, constant ambient pressure, always the same fluid; basically no hydrodynamics. And the longest shot, I consider the pressure per volume rate of the container constant: $$ \frac{P}{V} = K_{t} $$ I treat $K_{t}$ as some sort of elasticity constant which in reality comprises the actual container elasticity,water compressibility, container's volume, etc. It's the only empiric data I have: how many liters of water I stuffed in the container to get the target pressure. Taking differentials, and considering t=0 the moment the container is full with P = 0, I get: $$ P(t) = \frac{Q_{max}}{K_{b}}(1-e^{-K_{t}K_{b}t}) $$ Which makes sense for a single pump. But I can't find a way to adjust it to multiple pumps in parallel with different parameters. I need an equation to later iterate and get the best combination of pumps possible. Trying to get an "equivalent" pump, the flow rate is the sum of flow rates as in: $Q(t) = Q_{1}(t) + Q_{2}(t) ...$, easily resolving to an equivalent pump having parameters $[\sum{Q_{{max}_i}},\sum{K_{{b}_{i}}}] $, but that does not make sense with the equation above. Keep in mind this is for pressures up to 2000 Bar.


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Well, I did a numerical approach on this using a backward euler form, and it turns out the equation above is actually correct. My mistake was that I was not considering anywhere in the formulation that indeed the low pressure pumps shut off.

The optimization algorithm must take this into consideration and recalculate the equivalent pump ruling out the lowest pressure pump as soon as that pressure is reached (or at least considered so, at about 70% of the pressure).


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