This question asks us what is the initial quality of steam within this box? It's a rigid
box. It contains water and it contains water vapor and the two are in equilibrium and it
tells you that the volume of the box is one and a half cubic meters the total mass within
the box the mass of the liquid water plus the mass of the water vapor is equal to 15 kg
and it tells you that the temperature of the two in equilibrium is 100
degrees C. So what is the initial quality of steam in the box? And to answer this you first
need to understand what quality means and it simply means what fraction of the mass
within this box consists of vapor so a quality of 100 percent would mean that the box was entirely
comprised of what we call steam or water vapor if the quality were zero it would mean that
the box was completely filled with liquid water, and if the quality was 50 percent it
means that half of the mass would be comprised of vapor and half the mass, 50 percent of
the mass would be comprised of water, so that does not mean that the box would be half full
of water and half full of vapor it would mean that half of the mass is comprised of water,
and because water is so much more dense than the vapor you might only see just a small
puddle of water on the bottom and that is still a quality of 50 percent, so to solve
this problem I try to figure out intensive variables that I can use to relate this to
the steam table and one thing I really like to do is to draw a couple of columns denoting
what I know about the problem and in state one I know I've got a specific volume of 1.5
cubic meters divided by 15 kg so specific volume of 0.1 cubic meters per
kg. I know I've got two intensive variables, temperature and specific volume and then I've
got water and water vapor in equilibrium, I can consult the saturated water temperature
table, this temperature table is organized so I can look up temperatures really quickly,
so the first column, temperature, and I see the temperature that I'm interested in at 100
degrees C and I look up the saturated pressure of that and this is right around one atmosphere
of pressure and it gives you some more information like the specific volume gives you the internal
energy, the enthalpy, the entropy of the system, but for right now what we're interested in
is the specific volume of the system. It gives you two columns and the first column is the
specific volume of the saturated liquid and the second column is the specific volume of
saturated vapor, so it's important to notice that the specific volume of vapors are always
greater than the specific volumes of liquid so if we look at 100 degrees C the specific
volume for the liquid is 0.001 m^3/kg, that's the inverse of the density,
conversely we look at the specific volume of the saturated vapor and we're looking at
something about 3 orders of magnitude greater in volume so what this is saying is that if
I took one kg of water it would have a volume of 0.001 m^3 or a volume
of one liter and I boiled all that water and I captured all that steam in a big bubble
that would mean that the volume would grow from 0.001 to about 1.7 m^3. I can
use that information now to solve my problem and the easiest way to do it is to use this
ratio, and it's a ratio of the difference in specific volumes, so this is the specific
volume of the system and it's the difference between this in the specific volume of the
liquid and its divided by the specific volume of pure vapor and the difference between that
and the specific volume of the liquid. One thing to note is that if I had a condition
in which my specific volume was equal to that of the vapor then the numerator and the denominator
would be equal and the quality then would be equal to one, and if I had a specific volume
that was equal to the specific volume of the liquid I would have a quality of zero, so
in this case I run the calculations and I've got a quality of about 0.059 or about six
percent so about six percent of the mass within this box is comprised of vapor, so I can also
do this calculation using EES and its just one line of code here. Let's define a variable
X1 and use the quality function and we'll work with water, type it out and remember
don't use H2O because it will treat it like an ideal gas, and this is a two phase equilibria
mixture and the ideal gas law does not apply in this case. Lets type out water, you could
also type out steam, but I'll type out water, and we've got a temperature equal to 100 degrees
C I always use units to make sure I'm doing things correctly and I've got a specific volume
V which is equal to 1.5 m^3 divided by 15 kg so push the calculator
button and indeed I get a quality of 0.05916 which jives with our hand calculations from
that table. So the second problem is asking what is the temperature if we add heat to
this system, what is the temperature when all of the water boils off, the first thing
I do is draw a P-V diagram, so I've got pressure on the Y axis, we got specific volume on the
X axis and then I draw this two-phase envelope so what you know now is that you've got subcooled
liquid here we've got a two-phase region up here and superheated vapor, and you know the
specific volume is about point 1 m^3/kg so somewhere on here we're located
on some vertical line and I'll call this 0.1 m^3/kg, and it's also telling
me that I've got a temperature of 100 degrees C so on these diagrams remember there's isotherms
that come down, they come horizontally across in the two-phase region and then they continue
down off to the right. So this is the isotherm of 100 degrees C, I've now located the thermodynamic
state of the system on this P-V diagram and that initial state is right at the intersection
of a specific volume of point 1 in a temperature of 100 degrees C it says state two occurs
when the last droplet of water vaporizes. So I've got one column of information let me
draw another column state two, the specific volume of state two has to be the same as
the specific volume of state one and that's because no mass has left the system and the
volume hasn't changed, I want to figure out where state two is located, so I know it still
has to be somewhere on this vertical blue line but I don't know where just yet and I
have to think about the problem for a second and the key word here is when all the water
boils off. So in other words when we're just at the water vapor equilibrium line so this
black line on the right hand side so we'll start out at this X and it moves upward as
you add heat to the system and eventually it will cross this intersection right here
and that is the thermodynamic state for state two. All I know is the specific volume and
I know that I've got saturated vapor. So we can look at this chart and I know the specific
volume of the vapor and it's 0.1 m^3/kg so I'm looking somewhere in this
table, can I find something as I come down, temperature is coming up, the specific volume
is going down and the smallest one that I see on this table is 0.127 m^3/kg
and that occurs of a temperature of 200 degrees C and when I look just qualitatively
at this it looks like I'm incrementing about 0.015 or something like that and it looks
like I'd move down a couple more rows just eyeballing it I bet I've got a temperature
of 210 degrees C. So with the table I've got my best guess is I've got a temperature of
210 degrees C. So let's do this using EES, you'll see that we'll get a more accurate
answer, let's give this a variable, call it T1 use the temperature function, we're going
to use water, I know I've got the same specific volume, I also know the quality and that quality
is equal to one, so at this point I've got enough information to define my system and
I can have it return a temperature for me, so if I check the units, I have established
units of Celsius for that so I'm ready to go so I'm going to calculate it and my guess
at this point based on what I've looked at from the table is I'll see a temperature.
Let's take a look at that, I'll push the calculator button to run it and indeed something really
close, we had a temperature of 212.2 degrees C so this is the temperature that occurs when
the last droplet of water evaporates from the system.