In this screencast we are going to set up
and solve an energy balance on the condenser,
and we will use sensible and latent heat to
help us with the solution. If we want to calculate
the cooling duty that is required to condense
and cool acetone from 100 C to 25 C at atmospheric
pressure, and we know the heat of vaporization
for acetone at its normal boiling point is
30.2 kJ/mol. So by definition the normal boiling
point is the boiling point at atmospheric pressure.
The process is illustrated here in the flow
chart below, and we can see that we have both
a change in temperature, as well as a change
in phase. Both of which elicit a change in specific
enthalpy, and we are trying to solve for Q.
So our open system energy balance in this
case reduces to Q is equal to delta H, which
assumes that there is no change in kinetic
or potential energy, and we can also write
that q is equal to n dot, which is the molar
flow rate time the change in specific enthalpy,
which is the enthalpy on a per mole basis.
Now if we had information about the specific
enthalpy of acetone at any given temperature
and pressure as well we do as water in the
case that we have the steam table. This would
be very easy we could look up the specific
enthalpy of acetone at the outlet. Look at
the specific enthalpy of acetone at the inlet
and take the difference and substitute into
our energy balance. Without that information
available to use we have to get more creative
and set up a hypothetical path that allows
for the calculation of the specific enthalpy
for this process. So the key property that
allows us to set up a hypothetical path is
that enthalpy is a state function, which means
with a change an enthalpy it only depends
on the initial and final condition. In this
case our initial we have acetone in the vapor
phase our temperature is 100 degC, and our
pressure is in atm, and we are going to the
liquid phase at 25 C, and we are keeping pressure
constant at 1 atm. So our overall change in
enthalpy for the process we can call delta
H, and we can set up any number of hypothetical
steps that is necessary to get from the same
initial condition to the final condition.
A key piece of information that is going to
help us in this processes is what temperature
does the phase change occur. We have the heat
of vaporization at it's normal boiling point.
So we need to know what temperature does that
occurs at. In this case the normal boiling
point for acetone is 56 degrees. We can look
up this information in a variety of different
text books or chemical engineering hand books.
So we need to get to the temperature at which
the phase change occurs. We can get there
in one step by changing the temperature and
keeping the phase constant. So we have an
intermediate step here were we have the vapor
at the boiling point, which is 56 degC, and
we are keeping the pressure constant at an
atm. At the boiling point we are going to
change phase. So we have a second step. We
are changing phase to the liquid phase at
the same temperature, and our third step we
are changing the temperature of the liquid
from 56 to 25 while keeping the phase constant.
So we have 3 hypothetical steps here. Each
of these have there own change in specific
enthalpy, which we will call delta H1, 2,
and 3. And by the definition of the state
function the overall enthalpy here is going
to be the sum of these 3 steps. So we can
coperate that into our energy balance, and
write that Q is equal to n dot times delta
H1, plus delta H2, plus delta H3. Two of our
three step involve a change in temperature,
and any time that we are adding heat. That
causes a change in temperature we are referring
to sensible heat. Sensible heat and the change
in specific enthalpy associated with sensible
heat is related to heat capacity. So we can
calculate a change in specific enthalpy over
a specific temperature interval. If we integrate
over our temperature of integral. So from
T1 to T2. We integrate the heat capacity,
which is a function of temperature with respect
to temperature, and the heat capacity and
the dependence of temperature is generally
expressed as a polynomial function. There
are several different polynomials function
that can be used. In table B2 in F and R the
expression for the heat capacity that is used
is; for acetone we can look up the values
for a through d from table B2. So from table
B2 we can see that the coefficient for acetone
vapor and liquid are different. For both the
vapor and the liquid the coefficients are
given in the tables below, which we will use
in our sensible heat calculations. In our
first step we are cooling acetone vapor. From
the inlet condition at 100 to the boiling
point. So we can calculate delta H1 by integrating
from 100 to the boiling point of 56. The heat
capacity were we have 4 coefficients. If we
evaluate this integral and put in our coefficients
and limits of integration we will find that
the sensible heat associated with this process
is -3.2 kJ/mol. Our second step involve the
change in phase, which I will hold off of
for now, and go to the other sensible heat
calculation, which is the cooling of the liquid.
We want to calculate delta H3. We are integrating
the heat capacity from the boiling point of
56 to the outlet condition at 25, and here
we have 2 coefficients. So we are integrating
a plus bT with respect to temperature, and
if we evaluate this integral and plug in the
coefficients we find that the sensible heat
for this step is negative 4.06 kJ/mol. The
last change in enthalpy that we need to calculate
which is delta H2. Which is associated with
the phase change. Anytime that we are adding
heat and we are changing phase we are referring
to the latent heat. We know some information
about the phase change, and that heat that
is absorbed when acetone is vaporized at the
boiling point, and at the beginning we were
given the heat of vaporization as 30.2 kJ/mol,
and that is specifically at the normal boiling
point at 56 deg and atm pressure. So in this
case we are condensing. We are not vaporizing,
but we can still use this information if we
consider that enthalpy is a state function.
If we consider the path of heat of vaporization
represents the transition from a liquid to
a vapor, and a state function by definition
only depend on the initial and final condition.
So if we reverse our final and initial condition,
and we consider our path in this case to be
the transition from a vapor to a liquid then
the change in enthalpy associated with this
path must be equal to the opposite of the
heat of vaporization. To calculate delta H2
it is going to be equal to the opposite of
the heat of vaporization at the same temperature,
which is 56, and so that is -30.2 kJ/mol.
So scrolling down I have redrawn our hypothetical
path, and I have shown the results from our
sensible and latent heat calculation. Which
we can use to calculate our Q. So we showed
that energy balance is Q is equal to n dot
times the sum of each specific enthalpy change
for our steps. So plugging in our values.
We know that the molar flow rate is 100 mol/sec,
and we have 3 change in enthalpies that we
calculated. So that gives us a final value
for our cooling duty of negative 3808 kJ/s.
Here the negative makes sense because we are
drawing energy from the system and we can
see from our hypothetical path that most of
the cooling duty that is required is associated
with a change in phase.
