10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 25: Course Review
Review
Fundamental Equations:
Acc = Flow In – Flow Out + Reaction
∂nm
= Fm ,o − Fm ,out + ∫∫∫ rm ( x, y, z , t ) dxdydz
∂t
infinitesimal volume
∂Cm
= ∇ ⋅ Fm + rm
∂t
K eq = e−ΔGrxn
RT
( K eq is unitless)
νn
N products
⎛ Pm
⎞
⎜ 1 bar ⎟
∏
⎝
⎠
K eq = m =1
−ν j
reactants
⎛ Pj
⎞
∏
⎜ 1 bar ⎟
⎠
j =1 ⎝
Kc =
k forward
kreverse
H 2O2 → H 2 + O2
n
P
=
V RT
⎛ PH 2
⎞⎛ PO2
⎞
⎜ 1 bar ⎟⎜ 1 bar ⎟
⎠⎝
⎠
K eq = ⎝
⎛ PH 2O2
⎞
⎜
1 bar ⎟⎠
⎝
Kc =
[ H 2 ][O2 ]
[ H 2O2 ]
units K c [ =]
mol
L
Convection dominated:
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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Constant P, constant T, constant reactor V
Fm ≈ ν Cm
∇ ⋅ Wm ≈ D∇ 2Cm + U ⋅∇Cm
Pressure drop in Packed Bed:
Ergun Equation:
⎤
G 1 − φ ⎡150 (1 − φ ) μ
∂P
=−
+ 1.75G ⎥
3 ⎢
pg c D p φ ⎣⎢
Dp
∂z
⎦⎥
∂U cvtot
∂V
+ P cv = ∑ Fj H j + Q + Ws
∂t
∂t
( U cv depends on T)
∂T ⎪⎧ν C (Tin − Tout ) + Q + ∑ rk ΔH rxn ⎪⎫
=⎨
⎬
∂t ⎪⎩
Ctotal
⎪⎭
where
tot
C is the heat capacity
Special Cases:
Perfectly Homogeneous (“well stirred”, “perfectly mixed”)
Æ no flows, “batch reactor”
∂nm
= rm ( C (t ) ) V
∂t
Æ CSTR, no t-dependence
0 = Fin − Fout + r ( Cout )
Homogeneous in x,y, not in z (no t-dependence)
PFR (typically gives higher productivity than CSTR)
∂Fm
= Arm ( C ( z ) )
∂z
“sort of” PFR
∂Fm
= Arm ( C avg ( z ) ) Ω( z )
∂z
•
bubble •
where Ω ( z ) is the effectiveness factor
cat
Figure 1. a) mass transfer from gas to liquid b) mass transfer into catalyst particle
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 25
Page 2 of 3
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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D
∂ 2Cm ⎛ 2 ⎞ ∂Cm
+⎜ ⎟
+ rm ( C ( r ) ) = 0
∂r 2 ⎝ r ⎠ ∂r
-
nondimensionalize
some solutions in book
(18.03)
guess solution, plug in to verify
matlab
Thiele modulus
φ2 ≡
rm ( Csurface )
Dsolid Cm , s R 2
- if small (<1): reaction limited, ignore effectiveness factor Ω (internal diffusion
fast)
- if big: transport matters!
Ffrom = Ak L (Cinterface − Cbulk )
bubble
kL A Æ correlations
(sphere-packed bed)
Finto
= kc A(Cbulk − Cs )
particle
kc ∼
D
δ
Converting the second-order differential equation into first-order ordinary differential
equations for MatLab solvers:
∂Cm
= qm
∂r
D
∂qm 2
+ qm + rm = 0
∂r r
2
−
qm + rm
∂qm
r
=
Æ
Æ MATLAB: ode15s
∂r
D
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 25
Page 3 of 3
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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