1. Since most thermodynamic properties are a function of two or more variables (e.g. the volume of an ideal gas is a function of P, N, and T), you will need to review your multi-variable calculus. Calculate (∂z/∂x)y, (∂z/∂y)x, (∂y/∂x)z, in terms of x, y, z, and any relevant constants for
a) xy = a/z, where a is a constant,
b) z = ax/y - b/y2, where a, and b are constants.
2) Do complete derivations of the changes in mechanical energy below.
A: A water balloon of volume 30 mL (30 x 10-6 m3) is lifted vertically a distance of 1 m in the earth's gravitational field. Calculate the change of potential energy of the balloon. You will need to calculate the mass of the balloon; assume the water is at a temperature of 20oC and a pressure of 1.01 bar (atmospheric pressure), and the mass of the rubber is negligible. Put your answer in SI units; two significant figures are all that is required.
B: The balloon is dropped now from this height. Calculate the kinetic energy of the balloon the moment before it hits a bucket on the ground by using conservation of energy. Calculate the velocity of the balloon at this same time.
C: The balloon hits the bucket and the kinetic energy of the balloon is transferred to kinetic energy of the water molecules within the (now broken) balloon. This enhanced motion of the molecules shows itself as an increase in temperature. Using a heat capacity for water of 4.2 kJ/kg K, calculate the temperature of the water in the bucket.
D: A gas initially at 1 bar of pressure is compressed by 1 ml (= 1 x 10-6 m3). Assume that the volume of gas is so large that the pressure does not change during this compression. How much energy was inputted into the gas during this compression?
E: What if the pressure changed during the compression? Write down the equation that you would use to calculate the amount of energy required to perform this compression.
F: Using the equation you wrote down in E, you can now perform a similar energy calculation as in part D, but for a finite volume of gas. Calculate the energy required to *isothermally* compress 100 ml of an ideal gas (a gas that conforms to PV=NRT) to a size of 99 ml. You will have to perform a simple integration.
3) Mass balance and unit conversions.
Barton Springs pool is a 3 acre pool filled by multiple springs with a cumulative flow of currently 15 cfs (cubic feet per second). The water exits at 70 F at atmospheric pressure, you can assume the density of water is 1000 kg/m3.The water level in the pool is maintained by a manmade dam at the outlet that can raised or lowered. The depth varies throughout the pool due to its natural bottom.
Since it is natural, there can be a buildup of algae on the bottom. Thus every Thursday morning the outlet is lowered by 3 ft and the shallow end is cleaned with a power washer. The pool needs to open again at 7PM Thursday evening. At what time does the outlet need to be raised to its original position (3 ft higher) on Thursday to make sure it is at its regular height for the opening time? You can assume when the pool is filling that no water is exiting the pool.
4) Although some of you are not taking PGE 310, Formation and Solution of Geosystems Engineering Problems (colloquially known as MATLAB), we will still have a small amount of MATLAB in this course. If you are in this group, don't fret, the demands are not large here and I will lead you through.
We will get started with this first assignment. All that is required is for you to run a MATLAB function on your particular name. This will get you used to working in the environment.
On Canvas, there is a folder in the Files section named "matlab routines". I will put matlab routines throughout the semester. For this assignment, there is routine labeled name2num.m . The .m suffix is for a matlab routine; the title is a brief description of what it does, in this case, it converts a name to two numbers. You can open the routine in a text editor, or even better the text editor of matlab to see its guts, it is not very difficult.
OK, finally your assignment. It is very simple. You are to run this function with the input of your first name, and then again run it again with the input of your second name. For me the syntax of running this from the command line is
[first,last] = name2num('David')
And then
[first,last] = name2num('DiCarlo')
MATLAB will output two numbers each time. They are numbers associated with the first and last letter of each name. Just print out the evidence and hand in with the rest of the assignment.