10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 11: Non-isothermal Reactors, equilibrium limitations, and stability
This lecture covers: Derivation of energy balances for ideal reactors; equilibrium
conversion, adiabatic and non-adiabatic reactor operation.
Non-isothermal Reactors
N rxns
dN i N streams
= ∑ Fi ,m + Vcv ∑ υi ,l rl
dt
m =1
l =1
stoichiomet
stoichiometric
ric
coefficient
rl - depends on concentration
-T
- catalyst
N streams
dU cvtotal
dV
+ P cv = ∑ H mconc (Tm ) Fmtotal + Q +Ws + (other
dx
dt
m=1
energy terms)
flow work
intensive
extensive
work
expansion
work
heat
shaft
work
do work Æ
Ws negative
If small control volume, pressure constant.
P1
P2
In Vcv1 has a
P1 ≠ P2
other control
volumes
fixed P
Figure 1. Schematic of a PFR with small control volumes, each with a fixed P.
PFR has many small control volumes, each with its own constant P.
For isothermal –
–
–
–
Q adjusted to keep T constant
Practical – have big cooling bath
or just operate at a particular temperature found after reactor
built
⇒ not a good strategy, for design we want to know ahead of
time
before assumed uniform T, actually have hot spots
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Where is T? In U cv
total
and
rl (T ) .
dU cvtotal dU cvtotal dT N streams ⎛ dU cvtotal ⎞ dN i
=
+ ∑ ⎜
⎟
dt
dT dt
i=1 ⎝ dN i ⎠ dt
heat capacity
of system
intensive
contribution of
each species
substitute for
dN i
dt
dY
= F (Y )
dt
Want
Assume ideal mixtures
U cvtotal ≈ ∑ N iU i (Tcv )
extensive
intensive
dU cvtotal
= U i (Tcv )
dN i
If P=Constant (Isobaric)
dH total d (U + PV ) dU dP
dV
=
=
+
V +P
dt
dt
dt N
dt
dt
0
total
cv
dU
dt
+P
dVcv
dt
↓
total
dH
dt
Assume isobaric, all ideal mixtures, neglecting K.E., P.E., other energies
⎛ N species
⎞ dTcv N streams N species
N
C
= ∑ ∑ Fi ,m ( H i (Tm ) − H i (Tcv ) )
⎜ ∑ i p ,i ⎟
m
i
⎝ i
⎠ dt
−
N streams N rxns
∑ ∑ H (T
i
i
∑υ
i
i ,l
H i (Tcv ) ≡ Δ H rxn (Tcv )
cv
)Vcvυi ,l rl (Tcv ) + Q +Ws
l
stoichiometric coefficient
l
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 11
Page 2 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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−
N streams N rxns
∑ ∑ H (T
i
i
l
cv
)Vcvυi ,l rl (Tcv ) = −∑ Vcv rl (Tcv )Δ H rxn (Tcv )
l
l
Assume
Q ≅ UA(Ta − Tcv )
heat
transfer
coefficient
(conduction)
coolant
reactor
area of
contact
W s ≈ 0
(As a stirrer, heat negligible)
If designing engines W s ≠ 0 .
Now just put into MATLAB and solve
Chapter 8 in Fogler
– lots of special case equations
– be careful of assumptions
Special case: Start up CSTR to a steady state
want to know ultimate T
N streams N species
dTcv
= 0 ≅ ∑ ∑ Fi ,m ( H i (Tm ) − H i (Tcv ) ) − ∑ Vcv rl Δ H rxn +UA(Ta − Tcv )
dt
l
m
i
All depend on TCV
When we reach steady state, no more accumulation
FA, in − FA, out + rAV = 0 at steady-state
See Fogler: 8.2.3
If just one reaction, one input stream, one output stream, and the system is at
steady-state:
XA =
UA(T − Ta ) + ∑ Fi , input C p , i (T − Tin )
FAo (−ΔH rxn )
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 11
Page 3 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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In this special case, conversion and T linear
1 reaction making heat as product is made.
When ΔH rxn = (-) Exothermic, reactor is hotter than cooling reactor (heat transfer
important)
(+) Endothermic, reactor must be heated so that reaction will run
G (T ) ≡ (−ΔH rxn )(−rA V FAo )
Generation
⎛
⎛ Fi , in
⎞⎜
UA
R(T ) = ⎜ ∑
C p, i ⎟ ⎜1 +
FAo
∑ Fi, inC p, i
⎝
⎠⎜ ⎜
K
⎝
⎞
⎟
⎟ (T − Tc )
⎟
⎟
⎠
Heat removal
K =0
K = Big
Tc =
Adiabatic
Cooling
KTa + Tin
1+ K
R(T ) linear with T
G (T ) → constant at high T
- not linear with T
Three steady-state solutions
R(T )
cool
a lot
G (T )
•
−ΔH rxn
•
•
T
Figure 2. Graph of G(T) versus T. Three steady-state points are shown where R(T)
intersects with the heat of reaction.
With multiple steady states must consider stability.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 11
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].