Problem
Water in a cylindrical container is hanging by a rod from the ceiling at point O shown in the figure. The container is initially full, and initially displaced 20 degree . There is a hole in the bottom where water pours out and the top of the container is open to the atmosphere. As the container swings it empties and the water level h decreases.
The cylindrical tank has a diameter D and length b of 4 cm and 0.5 m, respectively. The distance from the ceiling to the bottom of the tank L is 5 m. The hole diameter d in the bottom of the tank is 1 mm. The combined rod and empty container's center of gravity is identified by cg and its mass is 0.0063 kg.
You are to develop a model. Use a control volume analysis of mass, linear momentum, and angular momentum to predict the change in its period as a function of time, the number of swings it makes before it empties, and the time it takes to empty. You are to make the following
assumptions:
i. Water is incompressible and has a density of 998 kg/m3.
ii. The linear momentum in the container's axis direction is negligible such that the hydrostatic pressure in the tank and the centrifugal force are balanced by the fluid momentum exiting the container.
iii. No friction between the fluid and container walls.
iv. Water's free surface remains perpendicular to the container's axis.
v. No pressure drop through hole.
In addition to the results to be determined listed above, you should address the following issues:
a. Does the period increase or decrease as the water exits the container? Explain.
b. Does it take longer to drain if the container is swinging or stationary. Explain.
c. How does the e ffect of the initial angle ø have on the time to empty. Explain.
Format
The format for the project report is given below. It should include at least the information listed.
1. Title Page Include project title, student name, course name, date, and instructor.
2. Table of Contents
3. Problem Definition In this section you should include:
a. A problem statement with a picture describing the problem.
b. Discussion of the specific problems to be addressed that were stated in the original project handout.
c. Discussion and identification of the governing equation with initial conditions.
4. Solution Method
You should include:
a. Discussion of how the defined model will be solved.
b. Discuss in detail the Runge-Kutta method.
5. Results In this section you should include:
a. Graphical representation of the solution.
b. Discussion of every figure.
c. Results of hand calculations to estimate validity of numerical solution.
d. Also include the discussion of the answers to questions posed on the original problem statement.
6. Conclusions
7. Appendix In this section you should include:
a. Program listing.
b. Hand calculations
Note:
i. Everything must be typed.
ii. 1.4 Mb floppy disk with matlab program.
iii. Equations must be created using Equation editor and numbered.
iv. Figures and tables should be in text of paper.
v. Figures must be numbered and have caption following figure number.
vi. Tables must be numbered and have caption preceding table number.
vii. 1.5 line spacing
viii. Times Roman 12 pt font
ix. straight left and right margins
x. Figures and tables should be completely discussed in heart of paper.