That would be shear times inside area of pipe = force. In EGS, lbs/in2 * in2 = lbs... no problem. The trickey part is evaluating the shear at all points along the pipe and integrating from beginning of pipe to end of pipe for time step over which the accelaration takes place, which BTW you also have to do for mass * accelaration. Acceleration will not be the same at beginning of pipe to end of pipe, as there is some compression going on as the velocity at the beginning of the pipe starts to increase, but still remains at zero at the end of the pipe. Of course mass changes as the pressure increases and the fluid compresses as well.
Best to find a good computer program to handle all that bookkeeping for you.
Maybe you want the easier solution,
Calculate the power you need for steady state flow.
Accelaration of fluids in piping is provided for by any excess power available from the pump driver at any given time.
For a simplified example of starting just a water pipeline, the pump has a driver with 100 HP rating, but at 10% of BEP flow when starting, power required for maintaining that low flowrate would be say 5 HP, therefore you would have 95 HP available for accelerating to a higher flowrate. Power required increases with the ^3 of flow velocity, so you would have much less power available for accelerating the flowrate from there to 100% velocity and the acceleration from that velocity to BEP velocity takes relatively a much longer time as you near rated power, if it wern't for the torque delivery characteristics of the driver, which I'll get to in a minute. So, if you need to still have rapid changes of velocity at BEP flowrate, you should provide a driver that gives much higher torque as its speed drops from sycronous speed as it would when being called upon to provide additional acceleration, which BTW most electric drivers do, reaching about 150% of rated torque with relatively little drop in sycronous speed. Any EEs out there, feel free to help ... now.
**********************
"Pumping accounts for 20% of the world’s energy used by electric motors and 25-50% of the total electrical energy usage in certain industrial facilities."-DOE statistic (Note: Make that
99% for pipeline companies)
