Can this be solved without a knowledge of turbines?

[BigWill]

1) 10 Hydroturbines are in a line, fitted inside a 1m diameter vertical tube its height 1.2km.
The top of the tube is 100m deep underwater in a high mountain saltwater lake.
This 1200m long tube acts like an orthogonal drainpipe down the mountain.

Hdyroturbine-A, hydroturbine-B,...Hydroturbine-J.
These ten hydroturbines are spaced 100m apart and the first turbine-A is located 100m down the tube giving the head of water (200m) on the first turbine. Each turbine is 100m below the one above with a height from the last hydroturbine-J of 100m to the bottom of the tube which is 50m above sea level, falling freely.

2) The flow rate through each turbine is governed by the turbine aperture and allows 1m3/sec flowrate ( tube is 1m in diameter).

3) Is it right to say that the head of water that reaches each turbine-A to J will be the product of the head of water between each turbine (100m) plus the weight of water above it multiplied by gravity minus resistance of turbine blades at each hydroturbine and the resistance of the tube walls? The outlet at the bottom of the tube falls into an open reservoir depressurising the water.

4) Decide on which is the best hydroturbine for this application ( Impulse??),
http://energy.gov/eere/water/types-hydropower-turbines
and calculate the power produced at each turbine (A to J) then the sum of the power produced.
Resistance losses are also to be calculated in each turbine (Rt) and the wall resistance (Rw) of the vertical tube, should be less important.

I hope to have some responses soon to discuss how to solve.
Many Thanks
Will

 

[billy_joule]

3) Is it right to say that the head of water that reaches each turbine-A to J will be the product of the head of water between each turbine (100m) plus the weight of water above it multiplied by gravity minus resistance of turbine blades at each hydroturbine and the resistance of the tube walls? The outlet at the bottom of the tube falls into an open reservoir depressurising the water.

If you assume each turbine captures all available energy (ideal case) then the head for each turbine is 100m (200m for first one).


Total power available = m[dot]gh = 1000kg/s * 9.81m/s^2 * 1200m = 11.8 MW

This is the upper limit. Actual output will depend on how realistic you want (or can) be.

 

[BigWill]

Thanks for your input Billy,
As turbine blade resistance losses could be significant but the resultant of the water head height should add to the total force experienced at each succeeding turbine. What is the way to calculate effective head at each subsequent turbine further down the tube after the losses from energy extraction and electricity production?

 

[billy_joule]

Thanks for your input Billy,
As turbine blade resistance losses could be significant

Ignore anything that happens in the turbine, overall turbine efficiency takes all of that into account.

What is the way to calculate effective head at each subsequent turbine further down the tube after the losses from energy extraction and electricity production?

The Bernoulli equation