In this problem we're going to solve an unsteady state energy balance.
And the particular system is, we have a tank that's connected by a valve to a flowing gas
line.
So the tank is 5 liters and it starts out with nitrogen at 1 bar pressure and it's at
200 degrees C and then we have nitrogen flowing at 15 bar pressure continuously flowing through
this pipe.
This nitrogen is at 150 degrees C. So nitrogen is flowing this way.
So what we're going to do is open this valve, let the pressure reach 15 bar here, close
the valve.
This happens very quickly, essentially no heat transfer.
And our question is what is the temperature in the tank when we've equalized the pressure.
So the temperature will change if we are feeding 150 degrees centigrade nitrogen into a tank
containing nitrogen at 200 degrees centigrade.
So this is going to require us to do both a mass balance and an energy balance.
So let's look at the mass balance first.
Now we're assuming, and this is a reasonable pressure for nitrogen to assume this, that
we have an ideal gas.
And so we can calculate the number of moles in the tank.
So this is in the tank at the start, of course that's the initial pressure in the tank, the
volume in the tank is fixed, and then the initial temperature in the tank.
So let put these numbers in and calculate, first, the number of moles that start.
And so I've made sure the units cancel, we calculate the number of moles initially.
We can now write the number of moles finally, again using the ideal gas law.
So the number of moles finally is in terms of that final temperature, which we don't
know yet, that's what we are trying to calculate.
So now we need to write down an energy balance for an unsteady system.
And the way we're going to write that is initial energy, you have so many moles, and the internal
energy.
We use the internal energy for the system.
So the U is energy per mole times the number of moles.
And then we are going to add energy to the system.
Now this is important, we have so many moles that flow in, we have to determine this.
And because it's energy flowing into the system we write in terms of enthalpy.
It's an important part of the first law.
Amount of energy initially, amount of energy we added, then final energy is again internal
energy per mole times the number of moles, finally.
Now the number of moles that we've added, here, must equal the final number of moles
minus the initial number of moles.
So that we end with the unknown the the final temperature because we can write the final
number of moles in terms of the final temperate.
And now we want to substitute in values for internal energy and enthalpy and we are going
to have to pick a reference, or one way to approach this, let me say, is to pick a reference
temperature and then write enthalpies and internal energy in reference to that reference
temperature.
And there are no absolute values of enthalpy and internal energy, we always work with differences,
so we are looking for differences relative to our reference temperature, essentially
the same thing we do when we use the steam tables.
Again with the analogy of steam tables at my reference temperature, which I picked as
150 degrees because I realized that I will make something simplify.
I'm going to make my internal energy zero at 150 degrees C. I could have picked any
temperature as long as I am consistent and use it.
Now I want to calculate each of these values.
Let's calculate first the enthalpy at the reference temperature, of course that's the
internal energy at the reference temperature plus PV evaluated at the reference temperature.
Internal energy we said is zero, PV of course for an ideal gas, this is per mole so it's
RT at the reference temperature.
Keep in mind we are using absolute temperature, so let's substitute the values in so we can
calculate the reference enthalpy.
So I've substituted in the values and it's important I used absolute temperature, and
I have now the reference elthalpy.
The initial internal energy would be internal energy at the reference temperature plus the
constant volume heat capacity, initial temperature minus the reference temperature.
And, likewise, for the final, the same calculation but the final temperature.
Ideal gas change in internal energy is that Cv value, heat capacity at constant volume
times the temperature difference.
So let's substitute in the numbers here.
I can calculate a numerical value for that initial internal energy, the final internal
energy is in terms of the final temperature in kelvin.
Other term in my energy balance is enthalpy coming in which is the same as the reference
enthalpy because we picked 150 degrees.
So now I can substitute this value, this value, this value, my final number of moles and the
initial number of moles, which is here, back into the energy balance and I can solve for
the temperature.
Let me substitute those numbers in.
And so we can see the only unknown is the final temperature.
I only indicated units in some of the terms in the interest of space but we look at each
of these, the units are going to be joules.
So now it's just a matter of calculating the numerical values, solve for temperature.
So I've calculated the final temperature, 303 degrees centigrade.
So let's go back and look at the initial problem.
Tank starts at 200 degrees and we are adding nitrogen flow at 150 yet the final temperature
is 300 degrees centigrade, to 2 significant figures, let's call it 200 degrees centigrade.
So what's happening here, how is that possible.
So what we are seeing is that the nitrogen, as it flows into the tank, it's doing work
on the gas in the tank, compressing it.
That's adding energy, and that raises the temperature, and significantly raises the
temperature.
And so we feed 150 degrees C gas but we end up with a final tank at 300 degrees C assuming
no heat loss to the surroundings.