Calculating Pipe Diameter for Desired Pressure in Inclined Fluid Flow

[Mario Carcamo]

The question:
There is a 650m long tube on a 1 degree incline. At one end of the tube is a pump that alone has a volumetric flow rate of 750 g/min. The fluid in question is water and its a PVC pipe. What diameter does this tube need to be in order to have 25 psi on the top end of the tube? (Water is at room temperature)

So far:
There is an empirical formula that is only an approximation but i realized that it can't be used in this situation cause the pipe is on an incline and its so long that the mass of the pumped out water will push down on the flow rate. How do you take into consideration friction and the water viscosity? Is this even possible?

 

[vin300]

If the pump delivers constant head and flow and conditions are smooth no viscosity or friction then difference in pressure can be caused in homogeneous condition by varying pipe cross section thus varying kinetic energy density. Otherwise only losses will contribute to difference in pressure. With no loss, multiply with conversion 750 g with velocity and divide area A this gives pressure which is same as long as A is same.

 

[Mario Carcamo]

Here's the thing... That's the flow without the pipping and inwant to take into account friction and viscosity

 

[Chestermiller]

Is the question, "How do I determine the pressure-drop/flow-rate relationship for water flowing through a pipe?"

 

[Mario Carcamo]

Yeah that's the question and with that relationship hopefully determine the diameter of the pipe

 

[Chestermiller]

Yeah that's the question and with that relationship hopefully determine the diameter of the pipe

The pressure drop is related to the shear stress at the wall τ by$$\Delta P=\frac{4L}{D}τ$$
The shear stress at the wall is related to the "friction factor" f by:$$τ=\frac{1}{2}\rho v^2f$$
In turbulent flow, the friction factor f is related to the "Reynolds Number" Re for the flow by:$$f=\frac{0.0791}{Re^{0.25}}$$
The Reynolds Number is given in terms of the viscosity μ by:$$Re=\frac{\rho vD}{\mu}$$
For more details, see Chapter 6 of Transport Phenomena by Bird, Stewart, and Lightfoot