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We all know the formula for volume flow rate is, Q=Vavg·A, and that it is constant through a pipe. Now, when a fluid flows in a pipe, a pressure drop will happen thus velocity will decrease and, again to the formula, even if the area is constant the velocity will decrease because of the friction and thus affecting the flow rate.

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Hold on, the velocity is not decreasing through the pipe. Who told you that?

Actually the answer is in your question, when the friction occurs, a pressure drop takes place reducing the energy content of the fluid flow.

You actually ignored the fact that the fluid's energy contain its flow energy, which is basically its pressure energy. This pressure will decrease in order to keep the velocity constant in the pipe.

See this photo, you can see that the velocity which is the gap between the hydraulic grade line and energy grade line is constant, while the pressure is not.

When dealing with fluids you should take in consideration that there is a kinetic energy + pressure energy. It is not a rigid body to own only a kinetic energy. The friction has another way to steal energy from the fluid in form of pressure energy.

enter image description here

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