Engineering Sciences 22 — Systems
Summer, 2003
Reacting Systems
1. Thinking about reacting systems.
Analysis of reacting systems is usually taught primarily based on industrial reactors (e.g. in the
petrochemical industry). However, it a substantial diversity of reacting systems, including
biological and environmental as well as industrial systems can be analyzed and understood using
the same methods. Regardless of the type of reacting system, important considerations include:
the reaction kinetics, the feeding schedule, and the mixing regime. Reaction kinetics are modeled
using rate laws, which are the component equations for reacting systems. Feeding schedule and
mixing regime are important for determining the material balance (accounting equations).
Analysis of different types of physical systems in a common framework is a central feature of this
course. Notwithstanding this, each type of system has some features that are particular to it. In
the case of chemical systems, concentrations are often the only state variables of interest, and rate
laws are very often the only kind of system element considered. Although in many systems,
system equations can be written is state-variable form or other forms, for reacting systems statevariable form is almost universally used.
2. State Variables
Concentration.
As we shall see, reaction rates are a function of the concentration. For a given system or system
element, flow in and flow out are usually proportional to the concentration in the entering and
leaving streams, respectively. Thus concentration is a particularly important state variable in
reacting systems. In the gas phase, definitions of concentration may make it a function of
temperature, pressure, and volume. Things are simpler for liquid-phase reactions, where
concentration is usually independent of temperature, pressure, and volume (or nearly so). In
this course, we will almost exclusively consider liquid-phase reactions.
Our calculations using rate equations will assume concentrations in terms of moles per volume
(e.g., moles/L or moles/m3). Other common units of concentration in the liquid phase such as
mass per volume (e.g. g/L, or g/m3), can be used for material balance accounting equations,
but must be converted to concentration in moles for use with the rate laws discussed Section 4.
In a practical reacting system, pressure, volume, and temperature affect the reaction and may
change as the reaction proceeds. This, the system is actually a mixed-mode system, including
thermal and fluid sub-systems, as well as a chemical sub-system. We will consider mixed-mode
systems towards the end of the course, but for now we will focus on concentrations as the only
state variable of interest. You may assume that concentration can be considered to be
independent of T, P, and V in all examples in this course unless explicitly told otherwise. You
may also assume that volume is constant unless explicitly told otherwise.
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3. Engineering accounting: the material balance
3.1 Basic equation
Final amount – Initial amount = Input – Output + Generation – Consumption
In rate form,
Rate of accumulation = Input rate – Output rate + Net formation rate
The amount of material in a system is specified as concentration × volume, CV.
the basic material balance equation is
Hence,
d
(CV ) = Cin Qin – CoutQ out + Vr f
dt
f
where Q = volumetric flow rate (volume/time), r = net rate of formation
(material/volume-time). For reactions involving multiple species, multiple entry/exit
paths, and multiple reactions, there is a material balance for each species, a flow term for
each path, and a rate for each reaction:
d
(C A V ) =
dt
∑i Cin ,iQin,i
–
∑j Cout , jQout , j
+ V ∑ rA ,k
f
k
By definition, the rates rf are positive for products (which increase over the course a
reaction) and negative for reactants (which are depleted). A particular species could be a
product of one reaction and a reactant in another, and would consequently have two rate
terms in the material balance, one positive and one negative.
4.1 Common special cases
• Constant system volume,
d
dC
(CV ) = V
dt
dt
• Equal inflow and outflow rates, Qin = Qout = Q
dC
Q
=
(Cin – Cout ) + r f
dt
V
• Batch reactor: Q = 0. (Just load up the tank and let it run, with no inflow or outflow)
dC
f
= r
dt
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4. Rate laws
Rate laws express the rates rf in terms of the concentrations C of the reacting species. They are
the equivalent of the component laws in other kinds of lumped systems.
Material balances
Rate Laws
}
=>
System model (simultaneous ODEs)
4.1 A simple rate law
rA = kC A CB2
rA = rate of formation of A per unit volume ( moles/(time • volume) )
CA, CB = concentrations of A and B (moles/volume)
k = rate constant (units as required to make the rate law dimensionally correct) — may
depend on other parameters, such as temperature or light intensity.
RA is positive, indicating that A is formed (not consumed) at a rate proportional to product
of A’s concentration, and proportional to the second power of B’s concentration.
4.2 Example: A reaction described by the mechanism
k
A + 2B → C
The complete set of rate laws for the depletion of A and B, and the formation of C, are
2
rA = – kC A CB
2
rB = – 2kCA CB
rC = + kC A CB2
•
The rates for A and B are negative, because A and B are consumed in the reaction.
The rate for C is positive, because it is being produced by the reaction.
•
Because two molecules of B are consumed for each one of A, its rate is twice A’s.
Hence, the factor of 2 in B’s rate equation. We now need to write the concentrations in
terms of moles for this to work right.
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•
Summer, 2003
Why the exponents?
The reaction rate is determined by the probability that the reaction occurs. In order
for the reaction to go forward, one molecule of A and two molecules of B must be in
the same place at the same time. The whereabouts of any molecule of A is assumed to
be independent of that of any molecule of B, and according to the laws of probability,
the probability of two independent events simultaneously occurring is the product of
their individual probabilities, i.e.,
P(reaction occurs) = P(A is there) × P(2 Bs are there)
Furthermore, the molecules of B are assumed independent of each other, so
P(2 Bs are there) = P(one B is there) × P(another B is there) = (P(B is there))
2
Therefore,
P(reaction occurs) = P(A is there) × (P(B is there))
2
Now, the probability that a molecule of A is in the right place is proportional to the
concentration of A, and similarly for B. So finally,
2
r = P(reaction occurs) = k CA × CB
which is the rate law.
4.3 Reversible reactions
If the reaction is reversible, so that the mechanism is
aA + bB ↔ cC + dD
with kf governing the forward reaction and kr the reverse, the rate law has two terms,
r = ± k f CaA CBb m krCCcCDd
i.e., for the formation of C, we have
rC = + ck f CaA CbB – ckrCCcCDd
The reverse reaction retards the net formation of C, so it contributes a negative term to the
rate. Furthermore, because c molecules are formed/consumed in the reaction, there is a
factor of c in front of each term.
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5. Idealized types of reacting systems. Idealized types of reacting systems can be defined based
on the feeding schedule and mixing regime as follows:
Mixing Regime
Complete Mixing
No Axial Mixing
None
“Batch”
Feeding Schedule
Continuous
“Continuous Stirred-Tank
Reactor” (CSTR)
“Plug Flow”
These idealized types of reacting systems are convenient conceptual reference points. Behavior
corresponding to that predicted for these idealized system types can be achieved in industrial
reactors, although it often is not achieved, and is less commonly achieved in reacting systems
occurring in nature. Even when such idealized behavior is not observed, modeling in terms of an
idealized system type is usually a departure point for analyzing and understanding the system.
As we shall see, different terms in the material balance can be cancelled for different reactor types.
Batch systems. The idealized batch system is assumed to be well mixed, implying that there is no
variation with spatial position – or, stated mathematically, that all spatial derivatives are equal to
zero. Thus a batch system at a given time is characterized by one and only one concentration for
each chemical species present and one and only one reaction rate (evaluated at the concentrations
and other conditions in the reactor at that time) for each reaction. Thus, it is a lumped system,
and can be analyzed by the methods addressed in ENGS 22.
In a batch reactor, no material enters or leaves the system boundary for the duration of the
reaction. Hence the general form of the material balance (equation [11]) for a batch system is
d(VCA)
= VirA,i
[15]
dt
For a constant volume system this may be simplified further to give
dCA
[16]
dt = rA,i
And for a constant volume batch system with one reaction we have
dCA
dt = rA
[17]
Continuous stirred-tank systems. The term “continuous stirred-tank reactor” (often abbreviated
“CSTR”) is obviously drawn from an industrial context, but is commonly used to refer to any
continuously-fed, well mixed reacting system whether or not it is a tank. Both CSTRs and plug
flow systems may be classified as “continuous flow reactors”. The term “stirred-tank” thus
differentiates the CSTR, which is well mixed in all spatial directions, from a plug flow system,
which has no mixing in the axial direction. Based on this mixing, the CSTR is, like the batch
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reactor, considered a lumped system characterized by one and only one concentration for each
chemical species present.
The flow removed from a CSTR, Qout, has the same concentration as the fluid in the reactor, or
CA,out = CA,reactor
[19]
Material balance equations of the form of [11] through [14] are applicable to CSTRs, depending on
which simplifying assumptions are applicable.
We are often interested in the steady-state behavior of CSTRs. At steady-state, by definition, all
time derivatives are equal to zero. Thus for example, the material balance for a CSTR with
constant volume, equal input and output flows, and one reaction (from equation [14]) is:
0 = Q(CA,in - CA,out) + VrA
[20]
The quantity V/Q has units of time and corresponds to the average time that a fluid element
spends in the reactor, or mean residence time. Since this quantity is of both intuitive and practical
value, we give it a symbol, θ :
θ = V/Q
[21]
Combining [20] and [21], we may write, for a steady-state, constant volume CSTR with one
reaction, and equal input and output flows
CA, in - CA,out
θ = −r C
A ( A,out )
[22]
Plug flow systems. Plug flow reactors may be thought of as a pipe with no mixing along the axis of
the pipe. The name “plug flow” refers to the fact that a fluid element moves through a plug flow
reactor as a “plug” of constant volume and no interaction with the upstream and downstream
fluid elements.
In a plug flow reactor concentration varies can vary in both space and time. Thus, in general,
analyzing them might require partial differential equations, and thus would fall outside the
domain of lumped systems analysis. However, because there is no interaction between the
different “plugs,” it is like we have a series of little batch reactors, each going down the pipe one
after another. The very same equations used to describe a batch reactor can therefore be used to
describe a plug-flow reactor.
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Let’s look a little more carefully and formally at the analysis. With no axial mixing, a plug flow
reactor analyzed in a reference frame that moves at the velocity of the fluid varies only with the
time the fluid has spent in the reactor, tPF, given by
tPF = l/v
[23]
where
l = length traveled after the fluid has been in the reactor for a period of time tPF (< L, the total
length of the system)
v = fluid velocity in the axial direction
Considered in this reference frame, the material balance for a plug flow system is the same as for a
batch system, as given by equations [15], [16], and [17], except that the variable t in the batch
material balance (defined in a stationary reference frame) is replaced by tPF.
6. Summary. Formulation of problems involving reacting systems involves writing the material
balance in a form appropriate to the problem taking into consideration factors including:
• What reactor type (e.g. batch, CSTR, plug flow) is being modeled?
• Are there single or multiple input flows?
• Is the volume constant?
• Is there one reaction or more than one reactions?
• Are the input and output flows (if any) equal?
For a problem to be adequately specified (i.e. solvable), the number of material balance equations
must equal to the number of concentrations in the rate laws used.
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7. Sizing Reactors
Although not as fundamental as rate laws or material balances, the subject of sizing reactors is
easily understood and is one demonstration, among many, of the utility of lumped analysis of
reacting systems. The volume, V, required to achieve a desired rate of production, R
(material/time) is calculated by:
V=
desiredrateofproduction R
= P
volumetricproductivity
[25]
where
P is the volumetric productivity (material/(volume•time)).
The volumetric productivity for a product A of interest, PA, is calculated by
PA =
CA, final - C A,initial
τ
[26]
where τ = treaction for a batch system, θ for a CSTR, and tPF(L) for a plug flow system.
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