Design a passive gate system for the problem of Example 2.20, using a different solution t...

Design a passive gate system for the problem of Example 2.20, using a different solution than the one shown.

EXAMPLE 2.20

The water level of a reservoir is to be regulated by a spillway that will open and allow flow when the water level is too high, but stays closed otherwise. The system must be a passive system. That is, no sensors or actuators that require an external electric power source may be used. The specifications are as shown in Figure 2.21, with a maximum allowable height of 4 m above the spillway, which is a square passageway of size by 2 m by 2 m. Design this spillway.

SOLUTION So what options are available to the engineer of this problem? Obviously the principles of hydrostatics must be employed, but beyond that no

FIGURE 2.21 Geometry of the problem for which a passive drain gate must be designed.

specific design has been specified. Many previous problems in this chapter dealt with calculating forces and moments on plates, so you may think of installing a metal plate in such a way that it will open under the desired conditions. Even with this decision made, there are still many ways to implement the design. The plate could be hinged at the top, the bottom, or somewhere in the middle. A counterweight connected through a rope-and-pulley system could be used to hold the gate shut, or a float could be used on the water side that is connected to the gate. Perhaps a stopper could be positioned on the wall of the spillway near the gate to ensure that it only moves in one direction or to keep it from opening too far. There are many possible solutions; one is sketched in Figure 2.22.

For this solution, a square gate has been placed in the open channel, with a stopper behind it so that the gate can only open clockwise. The gate is hinged exactly at its centroid, which is important since the line of action of a force always acts below the centroid, so that the moment due to hydrostatic force will tend to open the gate clockwise. To prevent the gate from opening prematurely, a counterweight has been attached to the gate 0.5 m above the hinge, which will pull the gate counterclockwise and tend to keep it shut. Now the size of the counterweight must be chosen, so the maximum magnitude of the allowed hydrostatic force on the gate should be calculated. The area of the gate is A = (2 m) (2 m) = 4 m2. When

FIGURE 2.22 Solution to the spillway design problem using a gate, stopper, and weight.

the depth above the gate is 4 m, the center of the gate will be 5 m below the water surface, so the pressure there is PCG = (5 m)(9800 N/m3) = 49,000 Pa. Using the centroid method, we find the force on the gate: F = PCG A = 196,000 N. Next, we must calculate the moment the hydrostatic force exerts on the gate so that the center of pressure can be located. From Example 2.14, we find

and the depth to the center of pressure is hCP = 993,100 N-m/196,000 N = 5.067 m, or 0.067 m below the hinge. Summing moments on the hinge to find the magnitude of the counterweight that will just keep the gate closed gives W(0.5 m) = 196,000 N(0.067 m), which results in W = 26,200 N. A slightly smaller value of W should probably be used, to account for friction in the pulley.