I've found formulas to solve the pressure drop, but not necessarily the back pressure that would be seen if a gage were on the pipe. A combination of equations perhaps?
Unfortunately, I don't think there is any way to calculate the back pressure without more information. Specifically, you need to know what the pressure is somewhere along the pipe. Once you have that, then you can just use Bernoulli's equation.
As Kate Yoder said before, pressure must be known in at least on position of the system, otherwise the problem is ill-posed. If for instance flow rate is fixed at pipe inlet and pressure at outlet, pressure at inlet (along with the whole pressure profile along the line) can be calculated; the opposite conditions (fixed inlet pressure and outlet flow rate) can be solved also, as well as the condition in which both inlet and outlet pressures are fixed and flow rate is calculated. Actually for a complete hydraulic calculation of a pipe of significant length, the heat transfer to external environment must be taken into account, hence the Whole system of conservation equations (mass, momentum and energy) must be solved. For pipelines, you need also to take into account the elevation profile and the fluid flow regime (laminar or turbulent, single or multiphase flow).
if the flow is incompressible and irrotational, meaning a slow velocity of Mach less than .3 you can use Bernoulli equation and momentum theory. Depending on what is your application you can ignore the friction based on material properties. You still need to know the pressure at some point and go from there. I recommend using 1 atmosphere as your starting point (very dependent of your current altitude)
mdot = density*Area*Velocity (start from here, good luck)
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