Prominent hydraulics consulting company problem

You are the Chief Engineer for a prominent hydraulics consulting company. Your client wants to pay you lots of money to design a system to deliver 1 m3/s from reservoir A at elevation 300 m to reservoir B at 500 m. The distance from A to B is 1000 km and the elevation between the two reservoirs can be approximated as parabolic with a maximum elevation of 660 m reached 600 km from A.

The system is to consist of a pipeline and a series of pumps. Commercial steel pipe with a roughness e = 0.05 cm and a pressure rating of 150 m head is available. The total cost of purchasing pipe and constructing a pipeline is listed below for various pipe sizes. Pumps that can deliver 1 m3/s at 100 m head cost $10,000,000 each (installed). Determine the most economical design (i.e. pipe size and number and location of pumps). Hint: Vary your pipe diameter and consider pumps in series. Pipe diameter (m) Total Cost ($/km)
0.9 80,000
1.0 100,000
1.1 120,000
1.2 150,000
1.4 200,000

Note 1: Theoretically, a pipeline may be designed to allow pressure to fall to the vapour pressure. In practice, however, water usually contains dissolved gasses that will vapourize well before the vapour pressure point is reached. Such gasses dissolve very slowly. They can move with the water in the form of large bubbles that disrupt the flow. Therefore, negative pressures shouldn’t be allowed to exceed about 70% of atmospheric pressure – use this figure in your design calculations.

Note 2: The pressure rating of a pipe is the maximum pressure that it can withstand.

1. first I calculated the length using : -x^2+660 000 where x is 600m and got the elevation= 300 000m. Does this have something to do with the cavitation?

2. I calculated V based on Q and the given D= 0.9m and got 1.57m/s

3. based on the friction factor = 0.017, i used my moody diagram and approximated the reynolds number to be about 127 170, I also used Re= VD/ kinematic viscosity and got about 0.017 also so either methods worked.

4. then I calculated the headloss = ((f*L/D + K)v^2)/ 2g and got about 2379m

5. then i used the energy equation and assumed v1, p1, p2 to be 0 and calculate the pump head energy in metres. I got about 2579m and divided that by 100 to get about 26 pumps.

6. i then calculated the cost of the entire project

these are my doubts:

we have to calculate cavitation so Pcav = P guage- Patm, would Pcav be P2 in the energy equation? I used 2300 as my guage (assuming its at standard condition). How do I use this with the 300 000m I calculated?

i know how to calculate the HGL and EGL but because I have 26 pumps above, I dont know how to draw both lines, where to put the locations of the pumps and how to draw my design.

Another doubt I have is the pressure rating, is that relevant to this problem?

are my steps above correct? I'm not sure if im doing my calculations right and would appreciate any help given to me! i've been doing this problem for 2 days now..i know i also have to apply these steps for the next 4 diameters to figure out which design is cheaper.

 PhysOrg.com engineering news on PhysOrg.com >> Researchers use light projector and single-pixel detectors to create 3-D images>> GPS solution provides 3-minute tsunami alerts>> Single-pixel power: Scientists make 3-D images without a camera