Since \(\displaystyle u_d= \left(\frac{d_{nozzle}}{d_{duct}}\right)^2u_0\)

\(\displaystyle u_d^2= \left(\frac{d_{nozzle}}{d_{duct}}\right)^4u_0^2\)

Replace the "\(\displaystyle u_d^2\)" in the second equation,

\(\displaystyle p_{tot}= \frac{1}{2}\rho u_0^2+ \frac{1}{2} \rho u_d^2 \kappa\)

with that:

\(\displaystyle p_{tot}= \frac{1}{2}\rho u_0^2+ \frac{1}{2} \rho \left(\frac{d_{nozzle}}{d_{duct}}\right)^4u_0^2 \kappa\)

And now, replace \(\displaystyle p_{tot}\) in

\(\displaystyle p_{pump}= \frac{p_{tot}n(0.785)d_{nozzle}u_0(60000)}{600 \eta_p}\)

with that:

\(\displaystyle p_{pump}= \frac{\left(\frac{1}{2}\rho u_0^2+ \frac{1}{2} \rho \left(\frac{d_{nozzle}}{d_{duct}}\right)^4u_0^2 \kappa\right)n(0.785)d_{nozzle}u_0(60000)}{600 \eta_p}\)

I think I would also be inclined to note that 60000/600= 100 and that (0.785)(100)= 78.5.