Do this problem using EES and get properties using the substance $$water$.$

You want to build a potato gun using a $4\mathrm{.}0$ inch schedule $80$ pipe that you've found. Schedule $80$

pipe has an outer diameter of $4\mathrm{.}50$ inch and a wall thickness $0\mathrm{.}357$ inch; therefore, the inner

diameter is D$\_$in$\_$inch $=3\mathrm{.}79$ inch. The pipe has a length of L $=6$ ft and you have insulated its

external surface. The potato is jammed into the barrel so that it sits a distance a$1=10$ inch from

the bottom of the pipe. The potato is held in place by a pin which can be removed in order to

initiate the launch process. The potato has mass mp $=0\mathrm{.}75$ lbm$.$

You are evaluating various options for launching the potato. One option is to inject some lighter

fluid into the space between the potato and the bottom of the pipe and light it$,$ causing an

explosion that launches the potato. You could also launch the potato using a source of

compressed air. For this problem, let$$s evaluate the option of connecting the pipe to one of the

steam lines that run through the heating system in your building. $($Note that all of these all fall

into the category of don't try this at home.$)$

The bottom cap of the pipe is connected to the steam line in your building through a valve.

According to the building maintenance man, the steam in your building is at pressure Ps $=85$

psig $($yes$,$ the g means gage pressure, not absolute pressure$)$ and has a quality of $1.$

Process $1$: Charging the gun

The first part of the launch process involves filling the charge volume $($i$.$e$.,$ the volume between

the bottom of the potato and the bottom of the pipe$)$ with steam. Initially, the charge volume

contains some air, but for the purposes of this problem you can ignore this air and assume that

the charge volume begins the process completely empty $($i$.$e$.,$ there is no mass at all in the charge

volume at the start of process $1).$ You open the valve and allow the charge volume to fill with

steam from the source.

Process $1,$ charging the gun.

Process $1$ concludes when the steam reaches state $1,$ where the pressure in the charge volume is

equal to the source pressure, P$1=$ Ps ; at this point you close the valve. You may assume that

there is no heat transfer from the charge volume to the surroundings during this process. You are

using a big squishy potato, so you can assume that it forms a tight seal between the inner

diameter and the potato and there is no leakage of steam around the potato.

a$.)$ What is the temperature of the steam in the charge volume at the end of process $1,$

T$\_1\_$C $($C$)?$ Hint: it$$s not the same as the temperature of the steam in the line.

b$.)$ What is the entropy generated by the charging process, S$\_$gen $($J$/$K$)?$ This includes

the entropy generated by the flow through the valve as well as entropy generated by

mixing in the charge volume $$ choose your system so that you include both sources

of entropy generation.

Process $2$: Launching the potato

Once the gun is charged, the valve is closed and then, after a suitable countdown, the pin is

removed in order to launch the potato. The launch process is over once the potato reaches the

end of the gun $($i$.$e$.,$ a$2=$ L $=6$ ft$)$ at which point the steam is at state $2.$

Process $2,$ launching the potato.

filled with steam

at P$1=$ Ps $(1)$

potato launch velocity, v

a$2=$ L

$(2)$

Again, you may assume that none of the steam leaks out of the gun during the launch. Further,

we are going to assume that the steam is always internally in equilibrium during the launch

process so that there is no entropy generated within the steam as it expands from state $1$ to state

$2$; note that this provides a reasonable upper bound on the performance of your gun. Assume

that there is no heat transfer from the steam during the launch.

c$.)$ What is the temperature of the steam at state $2,$ T$\_2\_$C $($C$)?$

d$.)$ What is the work transfer $($J$)$ from the steam to the potato during process $2,$ W $($J$)?$

Note that the pressure in the gun chamber is not constant because the potato is being

accelerated throughout the process $($i$.$e$.,$ this is not the case of a free floating piston$)$;

however, you should be able to compute the work transfer using an energy balance.

e$.)$ What is the velocity that the potato has as it is launched from the gun, vel $($m$/$s$)?$

You may neglect friction between the potato and the bore of the gun. You will

probably want to do an energy balance on the potato $-$ don$$t forget that the potato

experiences a work transfer with the environment.

f$.)$ If the potato experiences no air resistance after it is launched, then what is the

maximum height that it will reach $($relative to the exit of the gun$),$ H$\_$ft $($ft$)?$