INSTRUCTOR: Here we're gonna look at a basic
Rankine cycle where we're gonna generate steam in a boiler.
It's at a high pressure and high temperature,
and we're gonna feed that to a turbine to generate work,
and then we're gonna condense the steam, re-pressurize it,
and pump it back to the boiler.
And the question is for this cycle,
for the conditions we're given here,
what's the thermal efficiency if
the turbine is adiabatic and reversible?
So, the first thing we wanna do is,
let's draw a diagram, label what we know.
And I'm gonna pause and draw that diagram.
So, in the diagram I've labeled the information
that we're given, such as the pressure and temperature
leaving the boiler, the pressure leaving the turbine,
and the fact that,
leaving the condenser, we have a saturated liquid
and we're assuming constant pressure through the condenser.
So, now, what we wanna do to calculate efficiency
is we want to know the work, the net work generated,
and we wanna know the heat added.
So, net work is two places.
We generate, of course, work from the turbine
but we use up a small amount in the pump.
And heat generated's in the boiler.
So, the equation that we're interested in
is determining the efficiency.
We use that symbol.
And so, this is the net shaft work.
Put absolute value sign because the efficiency is positive,
but net work, when we generate "work" in notation,
the sign notation "work" is gonna be negative.
So, a couple other things we can do is
we can go to the steam tables and can write down
the information that we know from the steam tables
and then we can use that information in our calculation.
So, for example, enthalpy of saturated liquid,
and I looked that up in the steam tables,
226 kilojoules per kilogram,
and we're also gonna need the volume of the liquid
because we're gonna have to calculate the work
using the liquid volume.
And the volume, cubic centimeters per kilogram.
The other information that we can find
from the steam tables is the enthalpy at three.
And this value, again, 3399 kilojoules per kilogram.
Now, because we have a turbine
that's adiabatic and reversible.
And what this means is that S3,
the entropy entering, is equal to S4.
And so, S3 we can read, again, from the steam tables,
for the superheated steam
6.73 kilojoules per kilogram degree Kelvin.
And so, that means S4 is the same value.
So, first thing, let's calculate
the work we need for the pump.
And so, the work for the pump is going to be
the volume times the pressure change because it's a liquid.
And the volume we just looked up in the steam tables.
The pressure change is going from
15 kilopascals to eight megapascals.
So, let me pause and put those numbers in.
So, I have converted the 15 kilo pascals
to .015 megapascals.
Now, when I do this calculation,
what I'm going to have to do is make some conversions.
So, let's multiply this by one joule
is a cubic centimeter megapascal.
And then, since we're working in kilojoules,
10 to the third joules is one kilojoule.
So, if I multiply that out, I get the work for the pump.
This is 8.1 kilojoules per kilogram.
It's a small number.
Doesn't take much energy to increase
the pressure for a liquid.
So, let's now create a little more room to work.
If I wanna calculate enthalpy at one,
which is the value entering the boiler,
H1 is going to be H5.
That leaving, or that entering the pump,
plus the work of the pump.
So, H5 we looked up in the steam tables,
work of the pump we just calculated.
So, this is in, again, kilojoules per kilogram,
so 234 kilojoules per kilogram is H1.
Now, if I do a balance around the boiler,
H3 is the value leaving and we know what that is, 3399,
so, an energy balance around the boiler
is going to be delta H equals Q.
There's no work in the boiler.
And delta H will be the final H3 minus H1.
That's QH, and I'll substitute the numbers in.
So, we have one of the numbers we need
for the calculation of efficiency.
We now need the work generated by the turbine and so,
in order to do that, we have the energy, enthalpy,
entering the turbine, that's 3399.
We need to calculate leaving and the way we do that
is we know the pressure and we know the entropy,
and so, we can go to the steam tables
at this pressure and calculate entropy.
If we look at the steam tables,
we see this entropy is between that of a liquid
and that of a vapor, so we're gonna have some quality.
So, let's do that calculation.
So, the entropy, 6.73 kilojoules per kilogram Kelvin,
that's what's leaving the turbine 'cause that's what entered
and so, that at 15 kilopascals
is somewhere between liquid and vapor,
so we have one minus X, we'll call the fraction liquid,
times entropy of the liquid at those conditions
and X, which is quality, is the fraction of vapor.
Now, I'm just gonna substitute the numbers in
from the steam tables
at 15 kilopascals saturation conditions.
So, here's the value for the quality and once I have that,
I can calculate the enthalpy
leaving the turbine by the same process.
In other words, enthalpy four is gonna be one minus X
times enthalpy at the liquid
plus X times enthalpy of the vapor.
So, let's substitute those numbers in,
again, using the steam tables, saturation conditions,
at 15 kilopascals to get enthalpies of the liquid and vapor.
So, now we have the enthalpy leaving the turbine,
we can calculate the work
because the energy balance for the turbine
is the work is equal to the change in enthalpy.
Because it's adiabatic and exit is H4,
the entrance is H3, so I'll substitute those numbers in.
So, here's the work from the turbine.
Notice it's negative and so, because we're removing energy
in the form of work from the turbine.
So, now, our efficiency, which is the net work,
absolute value over QH.
Net work is gonna be work we get out from the turbine,
plus the work that we have to put in
to pressurize the water.
So, energy for the pump.
Absolute value of that divided by 3165 for QH.
These are all kilojoules per kilogram, 0.382.
So, that's the efficiency that we wanted to calculate.
Only 38% of the energy that we put in the boiler
in the form of heat, we get out net
in the form of work that we can then use.