A viscous liquid is pumped through a pipe network in a chemical plant. The annual pumping cost per unit length of pipe is given by $$C_{\text{pump}} = \dfrac{48.13\: q^2 \mu}{D^4}$$ The annual cost of the installed piping system per unit length of pipe is given by $$C_{\text{piping}}=45.92D$$
Here, $D$ is the inner diameter of the pipe inmeter, $q$ is the volumetric flowrate of the liquid in $m^3s^{-1}$ and $\mu$ is the viscosity of the liquid in $Pa.s$.
If the viscosity of the liquid is $20 \times 10^{-3} Pa.s$ and the volumetric flow rate of the liquid is $10^{-4} m^3s^{-1}$, the economic inner diameter of the pipe is _______ meter (round off to $3$ decimal places).