I'm currently working on modeling a vacuum system in which a pump is present. the pump is driven at a varying speed $\omega$. The manufacturer has only given the characteristic curve at a single speed $\omega_{ref}$, which is well-approximated by a 3rd-order polynomial in the mass flow, that is:
$$\Delta p_{\omega=\omega_{ref}}(\dot{m}) = c_3\dot{m}^3 + c_2\dot{m}^2+c_1\dot{m}+c_0.$$
My model however requires this characteristic at arbitrary blower speeds. My current solution is to use the pump affinity laws, i.e.
$$\Delta p(\dot{m}, \omega) = \left(\frac{\omega}{\omega_{ref}}\right)^2 \Delta p_{\omega=\omega_{ref}}(\dot{m}),$$
however this leaves the curve rather "squished", almost like a straight line. Does anyone know of a different way of scaling the characteristic for different speeds?
I also have access to an experimental setup. Here I can impose the pump speed $\omega$ and measure back the pressure difference $\Delta p$, though the mass flow cannot be measured.
