Question

This question is worth 12 marks.

A lake high up in the mountains supplies water to a hydroelectric power station similar to Trawsfynydd Lake shown in Figure 1.

Figure 1 Trawsfynydd Lake that feeds water to Maentwrog hydroelectric power station, Wales

The surface of the lake is 1000m above sea-level and the deepest point in the lake is 180m below its surface. Assume that the water has a uniform density of $1012kg m^{-3}$ and that $g = 9.81ms^{-2}$ .

• a.Figure 2 shows the cross-section of a typical type of dam called a gravity dam.

  

  Figure 2 Cross-section through a gravity dam

  • i.Explain why the dam needs to have a thicker profile at its base than at the level of the top of the water. To do this consider the force on the unit area of the dam wall near the surface and at the bottom of the lake. Use some simple calculations to support your argument, showing all your workings. (4 marks)

  • ii.Sketch a graph of how the pressure varies with depth below the surface of the lake. (2 marks)

• b.When the lake overflows $1.21 x10^{8} 1 s^{-1}$ ,  of water exits the lake by flowing over the top of an overflow spillway such as that shown in Figure 3.

  

  Figure 3 Overflow spillway on Claerwen Dam, Wales

  • i.Show that the mass of water leaving the lake in 1 second is $1.22 x10^{8}$ kg (to 3 s.f.). (1 mark)
  • ii.The water from the reservoir flows down a river that can be modelled as an open channel. If the hydraulic diameter is 4.80m , the wetted perimeter is 16.0m, the slope is 0.008 and the Chézy coefficient is $5.00m^{0.5} s^{-1}$ , calculate the volumetric flow rate for this open channel flow (to 3 s.f.). (5 marks)

Answer 1

The pressure increases with depth.

Suppose we move a depth h below the surface

Then the pressure difference between the surface and depth=dgh

g=acceleration due to gravity

d=density

Now as we go deep the pressure increases resultant the force also increases

In order to withstand stronger forces wall at bottom are thicker.

By simple calculation pressure at lower point will be higher by 1012*9.8*180=1785168Pa

which will cause an extra force of 1785168 N per unit area at bottom compared to top.

ii)The graph will be a straight line having slope dg(density*g).

The slope will be positive.It it have a positive intercept.(pressure at a reference is not negative)

iii)mass flow rate=vol flow rate*density

=1.21*10^8*1012=1.22*10^kg/s