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Background

I am trying to assess the Power output of a Pumped Hydro project based on this website. I have all the required inputs except the flow rate (m3/s).

Problem

However, this is where I think I'm getting confused,

  1. When I looked up how to calculate the flow rate (here), I need to first know the water velocity,
  2. When I looked up how to calculate the water velocity (here), I first need to know the flow rate

How would I go about calculating this, when both are unknowns?

Known Information

What I do know is

  1. Pipeline Diameter (762 mm)
  2. Pipeline length (38,000 m)
  3. Pipeline hydraulic head / drop (176 m)
  4. Pipeline Roughness (concrete lined, operating 0.3mm)
  5. Upper Reservoir volume (78.8m m3)
  6. Water density (1000 kg/m3)
  7. Gravity (9.81 m/s^2)

is that sufficient information to be able to calculate either water velocity or flow rate?

Note

This question has been asked before (here) but it previously went unanswered.

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This question cannot be answered as it is (without numerical values). The main reason is that you need to determine the pressure Losses due to the pipe length. However the pressure losses are dependent on the type of flow (Laminar or turbulent), which will be dependent on the actual velocity and the roughness in the pipe.

Having said the above, the process to calculate the velocity by (using the Bernoulli equation and finally arriving at):

$$\Delta P = \frac{1}{2}\rho v^2$$

where:

  • $\Delta P = \Delta P_{hydro} - \Delta P_{losses} $, is the total pressure differential between the top surface and the exit.
  • $\Delta P_{hydro}$ is the hydrostatic pressure (a first simplification is $\Delta P_{hydro}=\rho g H$
  • $\Delta P_{losses} $: is the pressure drop due to losses in the pipes. See this question for details.
  • $H$ is the Pipeline hydraulic head / drop (m)
  • $\rho$ is the water density in $kg/m^3$
  • $v$ is the exit velocity of the water in $m/s$

approach

You'd need to perform an iterative approach.

  1. More specifically, you'd need to start with assuming the losses being zero,
  2. With that assumption for the losses you calculate the velocity (let's say that is velocity $v_i$, i being the iteration index)
  3. then you'd need to plug that value and calculate the losses (first the type of flow, then the losses).
  4. With that value for the losses, you can calculate another velocity $v_{i+1}$
  5. then go back to two until $v_{i+1}- v_{i}$ is sufficiently small.

Then you can calculate the flow rate by:

$$\dot{q} = \rho A v_{i}$$


Corrections on calculations

There are some empirical correction coefficients that can be used for the exit velocity and the flow rate. e.g.:

  • $C_v$ = velocity coefficient
  • $C_d$ = Discharge coefficient

Have a look at engineering toolbox for more details

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  • $\begingroup$ @bobbyheyer were you able to obtain a result? $\endgroup$
    – NMech
    May 20 at 13:26

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