Here, we're gonna describe what the energy balance
looks like for a stirred tank reactor.
And we're gonna do first a general balance of this
would be an unsteady-state tank,
and the idea is we have a feed coming in,
where F is the molar flow rate of component I,
multiple components flowing in,
and zero, that's the inlet value,
and T zero is the temperature coming
into the stirred-tank reactor.
The flow rate leaving should be a difference,
because we're carrying out a reaction,
so for each of the components,
the temperature likewise is different,
depending on how much heat transfer takes place,
is whether we run it adiabatically or isothermally,
and then the concentration's leaving,
and I put these down because inside the reactor,
we should have the same temperature,
and the same concentration
because we're assuming this is completely well-mixed,
there's no gradients within the reactor.
So as we add the feed,
then there goes the step-change in temperature.
V is the volume of the contents of the reactor,
and N sub I is the number of moles
of component I in the reactor.
Now, to make things simpler,
and so we can better understand the process here,
we're assuming constant heat capacity values,
heat capacity for each component,
we're also gonna ignore kinetic energy and potential energy,
we're gonna assume those are negligible.
Which is a pretty good assumption
for lots of chemical reactors.
Now, we're going to also assume
that we have no phase changes.
Well, this is an assumption that may or may not
be a good one, because there are certainly reactors
in which feed is a liquid
and what some product leaves is a vapor,
and then we'd have to take phase changes into account,
but to understand the energy balances first,
we'll do the simpler case with no phase changes.
The other notation we should be aware of,
I'm gonna use T sub R as a reference temperature.
And you'll see why we're doing that.
We're going to assume we have some jacket around
the tank for heat transfer,
and we're gonna assume there's a coolant temperature,
and the flow is high enough that that doesn't change,
otherwise we could also write an energy balance
for the coolant.
And now, if we look at the energy balance,
the first term is the change in energy within the tank,
so this is the number of moles in the tank,
of each component, right?
We're doing the summation for each component.
This is heat capacity,
and this is the derivative of the temperature in the tank
with time because we're looking at an unsteady-state system
so that temperature's gonna change with time,
and that's gonna be equal then to terms for flow-in,
and flow-out.
So let me first write the energy flow-in.
This is the molar flow rate of component I,
multiple components with flow-in in general,
of course, heat capacity again,
and then, let me correct this.
So the feed temperature minus our reference temperature,
our reference temperature is going to be arbitrary,
and we'll see how we can pick it to make
the equation perhaps easier to deal with.
We also have a flow-out term,
this again molar flow rate.
But this is the molar flow rate leaving.
Right, so this is moles per time, heat capacity,
this is now the temperature within the reactor,
which is identical to the temperature leaving.
The next term is the heat reaction term.
This is heat of reaction,
where Nth will be change of reaction,
per mole of reactant A.
This is the rate of formation of A.
If A is reacting, then this term is gonna be negative,
and this then is the volume.
So this rate reaction,
we'll see this as something like
minus rate constant times concentration to the first power.
So, this is the heat reaction at temperature T R,
that reference temperature,
since heat reaction in general can be
dependent on temperature.
Well, the other two terms that would show up would be,
heat transfer, in this case heat exchange between
the coolant that's flowing around the outside of the tank,
and then work being added to the system.
Well, work may be quite small, but we do have a stirrer,
that can input some work, in many cases that's pretty small
compared to heats of reaction,
but if we have a very viscous solution,
it's certainly possible that the work could be significant.
So this is the energy balance.
The only other term that we might want to define would be
the Q dot, would be a heat transfer co-efficient area
for heat transfer.
I'm gonna define it this way.
Temperature minus coolant temperature.
So, if the coolant temperature is lower
than a reactor temperature, we're removing heat.
This is the area for heat transfer,
and this is the heat transfer co-efficient.
This equation then simplifies,
for different types of reactors,
the general equation I've written would be
for the start-up of the continuous stirred-tank reactor.
If we are running at steady-state CSTR,
then this first term would be zero,
this term would be zero for steady-state CSTR.
If instead, we're running a batch reactor,
then the energy flowed terms in and out would be zero,
and we'd have the change energy in the tank
related to heat reaction Q and W,
and if we ran a semi-batch reactor,
then perhaps there's energy flow-in but not energy flow-out,
so we might end up crossing out one of these terms,
just depending on how the system's set up.
There could be no energy flow-in,
but there could be energy flow-out, for example.
So, if we start with this general energy balance,
then we can derive the energy balance for batch,
semi-batch and steady-state CSTRs.
What I'm gonna do is just write down
the simplified equations here.
So here are the equations, batch reactor.
We've taken out the flow terms.
Steady-state CSTR, we've taken out the derivative term.
Now, if I pick the reference temperature
equal to the feed temperature,
then it simplifies and we only have the flow-out term.
You could look at this energy balance,
is if we carry out the reaction at temperature T zero,
this would be the energy.
And then we took the products and heated them
from T zero to T.
You know if it was adiabatic, you know all that energy
would go into heating up the products
from the feed temperature to the exit temperature.
And the Semi-batch reactor is just the direct repeat
of the unsteady-state CSTR,
because it depends on which terms are zero
in the semi-batch as to which terms we cross out.