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The gauge pressure between the two sections is desired to be 0 and the pumps are centrifugal pumps. There is also head loss due to friction.
What I want to know is how velocity and gauge pressure vary along the suction and discharge lengths of the pipe. And also how the gauge pressure can be made zero at the intersection after exiting the first pump
The application of Bernoulie's equation solves this problem at suction point and delvivery point of the pump. Fluid always flow from higher energy to lower energy.
You need to use the energy equation on the suction and the delivery pipes and take into accopunt headlosses taking place using standrad euqations. So pressure will continously decrease from inlet of the suction pipe to the pump inlet. Pump will provide energy to fluid and its total eenrgy will increase. If the flow is in constact diameter delivery pipe the pressure will again decrease because of friction in delivery pipe.
As for velocity, if density remains constant (the case of a liquid, or of a slow moving gas) then the mass conservation principle dictates that velocity does not change along the pipe. For a fast moving gas, pressure and velocity should be computed with the Fanno equations.
The velocity and pressure can vary due to the viscosity effect of the fluid that causes the head loss of the pipe.
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