ENGG2500: A Control Gate Is Located On The Wall Of A Reservoir: Fluid Mechanics For Engineers, SU, Australia

QUESTION 1 

A control gate is located on the wall of a reservoir, as shown schematically in Figure 1(a). Inside view, the gate has the shape of a quarter of a circumference with a radius of 3 m. The width of the gate (in the direction normal to the page) is 8 m. The liquid in the reservoir has a specific gravity (relative density) SG = 1.20 and is at rest with a depth of h = 9 m. For your reference, Figure 1(b) indicates the location of the centroid C of a quarter of a circle.

Figure 1. (a) Schematic representation (not to scale) of the side view of part of the reservoir and gate, as considered for Question 1. (b) Location of the centroid for a quarter of a circle.

Complete the following:

a) Calculate the value of the pressure (relative) at the floor of the reservoir.

b) Calculate the value of the pressure force exerted by the liquid on the gate.

c) Calculate the equivalent position of the pressure force exerted by the liquid on the gate (that is, XP and yp). Use the x and y axes indicated in Figure 1(a).

d) Calculate the moment of the pressure force exerted by the liquid on the gate around the hinge indicated in Figure 1(a).

The weight of a cylindrical water tank open to the atmosphere is balanced by a counterweight of mass M, as shown schematically in Figure 2. The diameter of the tank is D = 60 cm and the water level in the tank is h = 50 cm. At the bottom of the tank, there is an orifice (which can be either open or closed) with a diameter d = 4 cm and a discharge coefficient of 1.00. When the orifice is open, it discharges into the atmosphere, and water is supplied to the tank via a horizontal inlet pipe, in order to compensate for the discharge Q leaving through the orifice and keep the water level in the tank steady at h = 50 cm. For your calculations, you can neglect the weights of the inlet pipe and of the water in the inlet pipe, and neglect energy losses in the tank and orifice. Figure 2. Schematic representation (not to scale) of water tank balanced by the counterweight, as considered for Question 2.

Complete the following:

a) If the orifice is closed and the water level in the tank is steady at h = 50 cm (that is, no flow entering or leaving the tank), calculate the mass M of the counterweight to balance the water tank.

b) If the orifice is fully open and the water level in the tank is kept steady at h = 50 cm (by supplying water via the inlet pipe), calculate the discharge Q through the orifice.

c) If the orifice is fully open and the water level in the tank is kept steady at h = 50 cm (by supplying water via the inlet pipe), calculate the mass M of the counterweight to balance the water tank.

The Figure below (Figure 3) shows a pipe system carrying water at Q = 1 L/s. Details of the pipe system are provided between a pressurized location (1) and the end of a nozzle at location (2) where the water discharges into the atmosphere. At three sections the pipe has bends with a local loss coefficient of kL_Bend = 0.4 for each bend. The pipe has a valve with a loss coefficient kL_Valve = 0.5. The roughness of the pipe is ks = 0.25 mm. The pipe has a
diameter D = 5 cm and the total pipe length between (1) and (2) is L = 60 m. The nozzle has a local loss coefficient kL_Nozzle = 1 and the nozzle exit diameter is 2 cm. The difference in elevation between (1) and (2) is ∆h = 5 m. Further details are provided in Figure 3.

Complete the following:

a) In a first approximation, calculate the pressure at location (1) neglecting all losses in the system.

b) In a more detailed assessment, calculate the pressure at location (1) including all losses in the system.

c) In your exam booklet (with the white pages), sketch the hydraulic grade line (HGL) and total headline (THL) along with the pipe system with losses between (1) and (2)

d) Assuming an increase in pressure at position (1) to 100kPa and using the same pipe characteristics and losses as before, what is the maximum height ∆h, a constant flow rate of Q = 1 L/s can achieve without the addition of a pump?

(Part 1)
For a person swimming in the water, assume that drag is entirely due to skin friction. Assume that the human body can be represented as a submerged, thin, flat rectangular plate of dimension 30 cm x 180 cm with drag occurring on both sides of the plate. Use a constant swimming speed of 2 m/s.

Complete the following:

a) Calculate the drag force of the person swimming.

b) Find the power of the person swimming.

(Part B)
For a person running (in the air), assume that drag is entirely due to form drag. Assume that the human body can be represented as a rectangular plate of dimension 30 cm x 180 cm. Use a constant running speed of 6 m/s.

Complete the following:

a) Calculate the drag force of the person running.

b) Find the power of the person running.