Short: The principle that in flowing fluid, an increase in velocity happens simultaneously with decrease in pressure.
Long: You have a lengthy pipe with a change in cross-sectional area, as stated in your query, and you need to figure out the change in speed. Because Bernoulli's equation is not a function of area, you would not utilize it. Just apply the continuity equation, please. What goes in must come out, in other words, according to the continuity equation for steady flow. Therefore, at any two sites along the pipe, the mass flow rate must be constant (as long as there are no leaks).
ρ1Q1=ρ2Q2
or ρ1v1A1=ρ2v2A2
Since liquids are considered incompressible, ρ1=ρ2
So the continuity equation for liquids becomes:
v1A1=v2A2
This is the equation you would use to solve your problem.
The total energy in the flow is generally tracked by Bernoulli's equation. Bernoulli's equation maintains track of how energy changes between potential energy, pressure energy, and velocity energy throughout the flow since energy cannot be generated or destroyed.
P1γ+Z1+v212g=P2γ+Z2+v222g
In order to calculate the amount of pressure variations caused by changes in elevation, for instance, you may use this equation (Z). Keep in mind that the continuity equation requires the mass flow rate to be maintained. Therefore, energy must be withdrawn from the pressure term in order for a pipeline with constant diameter to go up a slope, raising the potential energy term. Due to continuity, velocity must be constant (for constant diameter), hence energy cannot be taken from the velocity component.
Thank you,
Eddie