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CHEM-UA 652: Thermodynamics and Kinetics
Notes for Lecture 26
I.
EXAMPLE: ADIABATIC CSTRS
Problem: Consider the following first-order irreversible reaction taking place in an adiabatic CSTR:
k
A−
→B+C
(1)
Pure A is fed into the CSTR at a volumetric flow rate of 10 m3 /s and a temperature, T0 = 350 K. The volume of the reactor
is 100 m3 . What is the steady-state conversion, X, and temperature, T? Additional information: Cp,A = 3 J/mol K, Cp,B = 2
J/mol K, Cp,C = 2 J/mol K. ∆Hrxn (350K) = −1500 J/mol, Ea = 40, 000 J/mol, k = 1 × 10−3 1/s at T = 350 K.
Solution: The design equation for a first-order irreversible reaction is given by
X=
where
kτ
1 + kτ
(2)
1
1
k(T ) = k(350K)e−Ea ( RT − 350R )
(3)
Combining the above expression yields
X=
1
1
k(350K)e−Ea ( RT − 350R ) τ
1
1
1 + k(350K)e−Ea ( RT − 350R ) τ
(4)
The energy balance for the CSTR can be written as
0 = FA0 Cp,A (T0 − T ) − [∆Hrxn (350K) + ∆Cp (T − 350K)] FA0 X
(5)
where ∆Cp = Cp,C + Cp,B − Cp,A . Rearranging in terms of X yields
X=
Cp,A (T0 − T )
∆Hrxn (350K) + ∆Cp (T − 350K)
(6)
Equations 4 and 6 can be solved simultaneously for X and T. Figure 1 displays plots of X versus T from Equation 4 (solid line)
and Equation 6 (dashed line). As can be seen from the figure, there are three combinations of X and T that will satisfy the
two equations, indicating that there are multiple steady-states at which the reactor can operate.
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Conversion, X
0.8
0.6
0.4
0.2
0
400
500
600
700
Temperature (K)
800
900
FIG. 1. Conversion as a function of temperature calculated from the mass balance (solid line) and the energy balance (dashed
line).
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II.
PLUG FLOW REACTORS (PFRS)
Another type of reactor used in industrial processes is the plug flow reactor (PFR). Like the CSTRs, a constant flow of
reactants and products enter and exit the reactor. In PFRs, however, the reactor contents are not continuously stirred.
Instead, chemical species are flowed along a tube as a plug, as shown in Figure 2. As the plug of fluid flows through the PFR,
reactants are converted into products.
FIG. 2. Diagram of a PFR.
PFRs are characterized by their length, L, cross-sectional area, A, and linear flow rate of fluids through the reactor, u.
For the purpose of our analysis, we will assume PFRs to be cylindrical, that there are no radial variations in the velocity,
concentration, temperature or reaction rate along the reactor, and that there is no pressure drop or density variation along the
reactor. Based on these assumptions, we can define the linear flow rate of fluid, u, through the tube as
u=
v
A
(7)
where v is the volumetric flow rate and A is the cross-sectional area of the tube. The molar flow rate of species j along the
reactor length can be defined in terms of u as
Fj (x) = uA[j]
(8)
where [j] is a function of x.
A.
Mass balance for a PFR
As in the case of CSTRs, we can write a mass balance on species j in the PFR as
[accumulation of species j] = [flow of species j in] - [flow of species j out] + [generation of species j]
Because the concentration of species j varies along the length of the PFR, let us first consider a section of the PFR, ∆x. If
is sufficiently small, we can make the approximation that the reaction rate, r , is constant within ∆x. We can the write
mass balance as
dNj
= Fj (x) − Fj (x + ∆x) + A∆xrj
dt
Substituting Equation 8 into the above expression and setting dNj /dt = 0, we can write the steady-state mass balance for
PFR in terms of the concentration of species j:
Au ([j]x − [j]x+∆x ) + A∆xrj = 0
∆x
the
(9)
the
(10)
In the limit where ∆x → 0, we can arrive at the design equation for PFRs:
d[j]
1
= rj
dx
u
(11)
Examining Equation 11, we can see that the extent of conversion of reactants will depend on the length of the reactor, the
linear flow rate, u, and the reaction rate.
B.
Fractional conversion in PFRs
For a first-order irreversible reaction in which A → B, we can write Equation 11 as
d[A]
1
= − k[A]
dx
u
(12)
Rearranging the above equation,
Z
[A]
[A]0
d[A]
k
=−
[A]
u
x
Z
0
dx0
(13)
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and integrating
ln
[A]x
kx
=−
[A]0
u
(14)
we can write the dependence of the concentration, [A], along x as
[A]x = [A]0 e−
kx
u
[A]L = [A]0 e−
kL
u
(15)
and the final concentration, [A]L as
Recognizing that
L
u
(16)
is equal to the residence time, τ , for PFRs, we can also write the above equation as
[A]L = [A]0 e−kτ
(17)
[A]L = [A]0 (1 − X)
(18)
Plugging in
we can also write the euqation in terms of the fractional conversion
X = 1 − e−kτ
(19)
Figure 3 displays the concentration profile of species A along the length of the reactor. As can be seen from the figure, the
concentration profile of species A in a PFR is identical to that in a batch reactor, with the exception that the x-axis is the
length of the reactor instead of time. In fact, a PFR is a batch reactor in which we switch from a stationary coordinate system
as a function of time to a moving coordinate system as function of distance, such that dt → dx/u. Thus,
(20)
Concentration of A
d[A]
d[A]
→u
= −r([A])
dt
dx
Length along reactor, x
FIG. 3. Concentration profile of species A along the length of a PFR for a first-order irreversible reaction.
C.
Comparison of CSTRs and PFRs
For a first-order irreversible reaction, recall that the residence time, τ , for a CSTR is
τCST R =
[A]0 − [A]
VCST R
=
v
k[A]
(21)
while for a PFR, we can rearrange Equation 17 in terms of τ :
τP F R =
LP F R
VP F R
1 [A]0
=
= ln
u
v
k [A]
(22)
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For equal volumetric flowrates into and out of the reactors, the ratio of the residence times of CSTRs and PFRs is equal to the
ratio of the volumes of the reactors
[A]0 − [A]
τCST R
X
VCST R
=
=
=
1
τP F R
VP F R
0
[A]ln [A]
(1
−
X)ln
[A]
1−X
(23)
Figure 4 displays the plot of VCST R /VP F R as a function of the fractional conversion, X. As can be seen from the figure, the
ratio is always positive, indicating that to achieve the same fractional conversion, the volume of a CSTR must be larger than
the volume of a PFR. At high fractional conversion values, the volume required for a CSTR increases rapidly compared the the
volume of a PFR. If reactor volume is the only criterion for deciding the type of reactor to use, clearly PFRs are the optimal
choice. However, when one considers material costs and ease of operation, CSTRs may still be preferred for some applications.
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VCSTR/VPFR
20
15
10
5
0
0
0.2
0.4
0.6
Conversion, X
0.8
1
FIG. 4. Ratio of the volume of a CSTR to a the volume of a PFR as a function of fractional conversion for a first-order
irreversible reaction.