Fluid Dynamics Question - Pressure Difference in Horizontal Pipe

[fazer2014]

Homework Statement

The Trans-Alaska Pipeline System (TAPS) carries around 100,000m3 of oil per day from the Northern Alaskan oil fields to the nearest ice-free port of Valdez, around 1300km away. The pipe has an outer diameter of 1.22m and a wall thickness of 12mm. Eleven pumping stations are used along the total length of the pipeline to transport the oil.

Note: Assume that the pumping stations are equally spaced along the pipeline, that the pipe is roughly straight and horizontal, and that the flow with the pipe is laminar, Newtonian and steady-state. Also assume that the pumps are 100% efficient so that all energy consumed by the pumps is dissipated by the fluid. The density and kinematic viscosity of the oil are (rho)=890 kg/m3 and (nu)= 7.17 x 10-4 m2 /s respectively.


(a) Starting from first principles, estimate the pressure increase that must be generated by each of the eleven pumping stations to maintain the flow.


(b) The rate that energy is dissipated D(W) by a fluid when it flows through a horizontal pipe under the influence of a pressure difference is given by

D = (delta)P x Q

where (delta)P is the difference in pressure between the inlet and outlet to the pipe (Pa), and Q is the volumetric flowrate through the pipe (m3/s). How much power (rate of energy use) is required to maintain the flowrate of oil through the entire pipeline?

If oil is burned to power the pumps, and 3.6 x 104 MJ of energy can be harnessed from burning 1 m3 of oil, what percentage of the total flowrate needs to be burnt to maintain the flow?

Homework Equations

Momentum balance over CV in the direction of flow:
rate of change of momentum = rate of momentum in - rate of momentum out + weight + pressure forces + viscous stresses

The Attempt at a Solution

We have been taught a certain way of solving these problems, I ended up with a momentum balance that only contained pressure forces and viscous stresses (everything else either equalled zero or canceled out).
Using this momentum balance over the control volume, and Newtons Law of Viscosity, I derived an equation for shear stress (tau) and velocity in the horizontal direction where v_z = v_z(r) where r is the axis of the pipe radius.

I got:

v_z(r) = [ΔP / 4*μ*L][R² - r²]

I assume I need to find ΔP to estimate the pressure difference but I'm not sure how to do this? I subbed in all the known constants but still ended up with only:

ΔP = v_z,max / 1.12x10^-7

Can anyone guide me on how to finish up part a? Or perhaps tell me if I've done something terribly wrong along the way?

Thanks for your help!

 

[Chestermiller]

Homework Statement

The Trans-Alaska Pipeline System (TAPS) carries around 100,000m3 of oil per day from the Northern Alaskan oil fields to the nearest ice-free port of Valdez, around 1300km away. The pipe has an outer diameter of 1.22m and a wall thickness of 12mm. Eleven pumping stations are used along the total length of the pipeline to transport the oil.

Note: Assume that the pumping stations are equally spaced along the pipeline, that the pipe is roughly straight and horizontal, and that the flow with the pipe is laminar, Newtonian and steady-state. Also assume that the pumps are 100% efficient so that all energy consumed by the pumps is dissipated by the fluid. The density and kinematic viscosity of the oil are (rho)=890 kg/m3 and (nu)= 7.17 x 10-4 m2 /s respectively.


(a) Starting from first principles, estimate the pressure increase that must be generated by each of the eleven pumping stations to maintain the flow.


(b) The rate that energy is dissipated D(W) by a fluid when it flows through a horizontal pipe under the influence of a pressure difference is given by

D = (delta)P x Q

where (delta)P is the difference in pressure between the inlet and outlet to the pipe (Pa), and Q is the volumetric flowrate through the pipe (m3/s). How much power (rate of energy use) is required to maintain the flowrate of oil through the entire pipeline?

If oil is burned to power the pumps, and 3.6 x 104 MJ of energy can be harnessed from burning 1 m3 of oil, what percentage of the total flowrate needs to be burnt to maintain the flow?

Homework Equations

Momentum balance over CV in the direction of flow:
rate of change of momentum = rate of momentum in - rate of momentum out + weight + pressure forces + viscous stresses

The Attempt at a Solution

We have been taught a certain way of solving these problems, I ended up with a momentum balance that only contained pressure forces and viscous stresses (everything else either equalled zero or canceled out).
Using this momentum balance over the control volume, and Newtons Law of Viscosity, I derived an equation for shear stress (tau) and velocity in the horizontal direction where v_z = v_z(r) where r is the axis of the pipe radius.

I got:

v_z(r) = [ΔP / 4*μ*L][R² - r²]

I assume I need to find ΔP to estimate the pressure difference but I'm not sure how to do this? I subbed in all the known constants but still ended up with only:

ΔP = v_z,max / 1.12x10^-7

Can anyone guide me on how to finish up part a? Or perhaps tell me if I've done something terribly wrong along the way?

Thanks for your help!

Integrate the velocity equation over the cross sectional area to get the relationship between the volumetric throughput rate and the pressure drop. I.E., multiply both sides of the equation by 2∏rdr, and integrate with respect to r from r= 0 to r=R.

Chet