# Relative improvement in flow from diameter change with sub-milliimeter tubing

I want to calculate the improvement in flow rate that can be achieved with changing the tubing size when the tubing is sub-millimeter size.

I originally approached this using a simplification of Poiseuille's law which says $$\Delta Q=\Delta r^4$$

However, lets say I'm changing from 0.5 to 1.5 millimeter tubing. That gives $$\Delta r=0.050$$, which when raised to the 4th power is a tiny number (6.25E-05). This seems wrong intuitively, based on the relative increase in size (i.e. new tube is 300% of the original tube)

If I convert to micrometers first, the result jumps to 6.25E06, which seems really really large.

What am I missing here, because this doesn't seem like the right approach.

Thanks!

$$Q_{final}= Q_{initial}\Delta r^4 \rightarrow \Delta Q =(\frac{r_{final}}{r_{itial}})^4$$
$$Q_{final}=Q_{initial} (0.75/0.25)^4 \\ =Q_{initial}3^4=81Q_{initial}$$