When performing a design of a binary distillation
column we used the graphical
McCabe-Thiele
method.
This involved drawing two
operating
lines
and the
$q$
-line.
In the tutorial, the
$q$
-line
had a
special value of
$q=1$, corresponding to a liquid feed.
Of course there are many features of the
VLE data,
operating lines,
and
$q$
-line
which have physical
interpretations.
We will now look at the $q$
value and the
$q$
-line
in more detail.
The
$q$
-line
results from a mass
balance around the feed tray of the column.
\begin{align*}
y &= x\frac{q}{q-1} - \frac{x_F}{q-1}
\end{align*}
Remember:
Constant molar overflow was assumed to
derive this.
When the feed stream enters the column it will
flash
to its equilibrium state at the
operating
pressure
of the column.
It is no coincidence then that the
$q$
-line
has the appearance of a flash drum
operating line.
But what exactly does the value $q$
represent?
Initially, I described $q$
as the fraction of feed
stream which remains liquid when entering the column.
Therefore $q F$
moles of feed flow down as liquid
into the stripping section of the column…
…
and
$(1-q)F$
moles of feed vapourise and rise
up into the enrichment section of the column (see right).
From this we can come up with a more general
definition of $q$
and
$(1-q)$
as
the change in the
liquid and vapour flow-rates in the column relative to the
feed stream flow-rate.
This definition is important as it is possible to
have $q$
-values greater than one and less than zero!
For example, let us consider the case when the feed
stream is a sub-cooled liquid at the column pressure.
The entering feed is below its boiling point and will
cause rising vapour to condense until the mixture reaches its
boiling point again.
This will mean that the liquid flowing back down the
column will consist of:
All of the liquid from the enrichment section.
All of the feed stream.
AND some fraction of the rising vapour.
The effective value of $q$
in this case must be
greater than 1!
The counter-example is if the feed stream is
superheated vapour at the column operating pressure.
This will cause the saturated liquid on the tray to
boil to vapour until the mixture reaches equilibrium.
This will mean that the vapour flowing up the column
will consist of:
All of the vapour from the stripping section.
All of the feed stream.
AND some fraction of the falling liquid.
The effective value of $q$
in this case must be
less than 0!
The appearance of the
$q$
-line
can be
categorised into:
Sub-cooled liquid feed ($q>1$).
Saturated liquid feed ($q=1$).
Two-phase feed ($0<q<1$).
Saturated vapour feed ($q=0$).
Super-heated vapour feed ($q<0$).
We can also see that there must be limits to the values
of $q$, and these correspond to complete boiling of the falling
liquid phase or complete condensation of the rising vapour
phase.
Without the counter-flowing vapour and liquid phases
the column cannot function and we cannot find a solution for
its design.
The appearance of the
$q$
-line
for
various values of
$q$.
In case you had some difficulty plotting $q=1$
from the
$q$
-line
equation…
\begin{align*}
y &= x\frac{q}{q-1} - \frac{x_F}{q-1}
\end{align*}
…
it is much more convenient to multiply the
$q$
-line
equation by
$(q-1)$
to give
\begin{align*}
(q-1)y &= x q - x_F
\end{align*}
Here it is easy to see that for $q=1$
we have
\begin{align*}
x=x_F
\end{align*}
regardless of the
$y$
value, so this is a vertical line!
For $q=0$
we have
\begin{align*}
y=x_F
\end{align*}
which is a horizontal line.
To calculate the value of $q$, we need to examine the
enthalpy of the feed stream
\begin{align*}
q = \frac{h_{F,sat. vapour}-h_F}{h_{fg}}
\end{align*}
where
$h_{F,vapour}$
is the specific enthalpy of a saturated
vapour feed,
$h_F$
is the actual specific enthalpy of the
feed, and
$h_{fg}$
is the latent heat of vapourisation in the
column (assumed constant on a stage (good old constant molar
overflow)).
We can see that $q=1$
corresponds to
$h_F=h_{F,sat. liquid}$
and
$q=0$
corresponds to
$h_F=h_{F,sat. vapour}$.
But we can even calculate $q$
for
sub-cooled/super-heated streams.
The hydraulic and mechanical design of a
plate
distillation column is a balancing act between minimising
pressure drop (operating cost), while keeping construction
costs low, and avoiding
weeping
and
entraining
/
flooding.
The construction costs come about from the use of
complex tray designs (see right), expensive materials, or
large columns, or many trays (inefficient designs).
Each tray type design is a trade-off between
pressure drop,
efficiency,
cost, and
liquid hold-up.
Some typical plate configurations.
Sieve plates
(top right) are simple,
cheap-to-build designs. They do not retain liquid well at low
vapour flow-rates/pressure drops, but facilitate a good
contact between the phases.
Valve cap
trays (middle) use small valves to improve
liquid hold-up at low pressure drops/vapour flow-rates
(
$1.5\times$
cost of seive plates).
Bubble cap
trays use more complex valves which
can also shape the bubbles to maximise the contact between the
phases, resulting in higher efficiencies (
$3\times$
cost of
seive plates).
Bubble cap trays can support very low vapour
flow-rates/pressure drops due to their excellent valves, and
so support high
turn-down
ratios.
Some typical plate configurations.
Chimney trays
(bottom-left) are not used for
mass-transfer, but are used to increase hold-up times at
section of the column (usually to perform a gas-liquid
separation).
The different tray designs offer different flow
characteristics in the column, but the primary concerns are
efficiency, pressure drop and liquid hold-up.
Lets take a closer look at the operation of a plate.
Some typical plate configurations.
In the “active area” of the plate, there is a layer
of liquid which is in contact with the vapour phase.
If the pressure drop of the vapour phase is not high
enough, the hydrodynamic head of this liquid layer will be
sufficient to cause liquid to
weep
through the plate,
possibly bypassing the rest of the stage.
In the downcomer stage, there is a head of liquid from
the upper stage ready to flow on to the current plate.
If the pressure drop is higher than the hydrodynamic
head in the downcomer, the liquid in the downcomer will be
blown up the column and the column will
flood.
Cross-flow plate configuration, taken from C&R
vol. 6, Fig.11.17.
It is possible to plot diagrams similar to the
multiphase flow maps introduced in EG3019 for column
operation.
In addition to
flooding
and
weeping, there
is
entrainment
where the liquid is sprayed out of the
column.
Coning
is where the vapour blows through the
liquid phase as a cone of vapour, significantly reducing the
contact time.
The area of satisfactory operation is strongly
affected by the plate design and fluid properties.
For example, if the fluid foams excessively
entrainment will become a major issue.
Sieve plate performance diagram, taken from C&R
vol. 6, Fig.11.17.
We cannot discuss detailed estimates of the flooding
or weeping limits of the column until a column design is
chosen.
But we cannot select a column design without a rough
estimate of its operational parameters.
To provide an initial estimate of
column height,
the spacing between trays is roughly estimated. Columns over
1 m in diameter have a spacing of 0.3–0.6 m (0.5 m/stage is a
good initial estimate).
The inner column diameter can be estimated by placing
a limit on the
superficial vapour flow rate
inside the
column.
\begin{align*}
D= \sqrt{\frac{4 \dot{V}}{\pi \rho_{vapour} \left\langle
v_{vapour}\right\rangle}}
\end{align*}
Sieve plate performance diagram, taken from C&R
vol. 6, Fig.11.17.
An initial estimate for this superficial vapour flow
rate can be obtained from empirical relations such as the one below:
\begin{align*}
\left\langle v_{vapour}\right\rangle^{max} = \left(-0.171 l_t^2+0.27 l_t-0.047\right)\sqrt{\frac{\rho_{liquid}-\rho_{vapour}}{\rho_{vapour}}}
\end{align*}
where
$l_t$
is the tray spacing (0.5–1.5 m/stage).
Obviously, this is a maximum vapour flow rate, so care
must be taken to stay below this in the section of the column with
the highest vapour flow rate.
With these rough design estimates, detailed design
calculations may begin.
For more detailed design notes for plate columns, see C&R Vol. 2
Sec. 11 and Vol. 6 Sec. 11.
