I have three equations governing pump behaviour and I am trying to obtain an expression for pump speed at timestep t. The variables are $Q$ = flowrate, $H$ = Head and $N$ = pump speed. For a given pump, Duty Head, $H_D$, and Duty Flow, $Q_D$ are constants.
$Eq. 25$ & $Eq. 26$ and $Eq. 17$ are shown in photos 1 and 2, respectively. Eq. 25 & Eq. 26 Eq. 17
- $Eq. 25$: $\frac{Q_1}{Q_2} = \frac{N_1}{N_2}$
- $Eq. 26$: $\frac{H_1}{H_2} = \left(\frac{N_1}{N_2}\right)^2$
- $Eq. 17$: $H = \frac{4}{3}H_D - \frac{H_DQ^2}{3Q_D^2}$
My thinking is to let $N_2 = 1$ such that $N(t)$ becomes a ratio of the pumps original speed. i.e. $N(t) = 2.0$ or $0.5$ as shown in the graph in the first photo.
My next step would be to substitute $Eq. 17$ as $Q_1$ and rearrange to obtain an expression for $N(t)$ as a function of $H(t)$. But alas, I am absolutely stuck!
I have somehow done this before and ended up with the expression, Eq. 27. But I cannot replicate this with logical workings.
Note that $Q_2 = Q_{avg}$ are interchangeable and if $Q_2 = Q_{avg}$, the equation further simplifies to Eq. 28
Can you please help?
Thanks in advance!