Dynamic Pressure Calculation at Hydroelectric Power Station Valve

[grscott_2000]

A hydroelectric power station is supplied with water from a reservoir. A pipeline connects the reservoir to the turbine hall.

The flow of water through the pipeline is controlled by a valve which is located 500 metres below the surface of the water in the reservoir. The lower end of the pipeline is 0.30m in diameter where it enters the valve.



Q1 If the valve is opened to give a flow rate of 5m^3 per second, calculate the pressure at the valve.


Answer. I have previously calculated the pressure at the valuve when it is closed by adding atmospheric pressure with pgh to give a static pressure of 5 x 10^6 Pa.

I am given a flow rate of 5 m^3/s and i also know the diameter of the pipe, so to obtain velocity I have said

v = flow rate / area = 5 / (pi x (0.15)^2) = 70.736 m/s (is this correct?)

Now I can use Bernoullis equation which would be

P (which is static pressure at valve) + 1/2pv^2

I am assuming that pgh is not needed here as we are no longer considering a change in height. Thus the pressure should be

(5 x 10^6 Pa) + (1/2 * 1.00 x 10^3 * (70.736)^2) = 7.5 x 10^6 Pa

Although the magnitude looks ok, I was expecting the pressure at the valve to drop here, but its gone up... Have I gone wrong somewhere?

 

[anathema]

Answer: Your calculations and approach are correct. The pressure at the valve should indeed decrease when the valve is opened, as the flow rate increases. However, in this case, the increase in velocity causes the pressure to increase as well. This is known as the Bernoulli effect, where an increase in fluid velocity leads to a decrease in pressure. In this case, the increase in velocity is significant enough to offset the decrease in pressure due to the increased flow rate, resulting in a net increase in pressure at the valve. So your final answer of 7.5 x 10^6 Pa is correct.

 

[m2003]

Your calculation of the velocity appears to be correct. However, your use of Bernoulli's equation may not be appropriate in this scenario. Bernoulli's equation assumes that the flow is steady, incompressible, and there are no energy losses. In reality, there may be energy losses due to friction and turbulence in the pipeline, which would result in a decrease in pressure along the pipeline. Additionally, the valve itself may cause a drop in pressure due to its design and the flow rate. Therefore, the pressure at the valve may not necessarily be equal to the static pressure calculated when the valve is closed. To accurately calculate the pressure at the valve, a more detailed analysis of the pipeline and valve design would be necessary.