Pressure Drop in Reactors - University of Utahmddeo/chen3553/handouts/prdrop.pdf · Pressure Drop...
3
Click here to load reader
Transcript of Pressure Drop in Reactors - University of Utahmddeo/chen3553/handouts/prdrop.pdf · Pressure Drop...
Pressure Drop in Reactors 0
0 0
0
0
0
0
0
( ) for isothermal reactors(1 )
For packed-bed reactors,
If we are considering the reaction, , which is first order(1 )
(1 )(1 )
(1 )
i A i ii
A A
AA A
A
F F X PCv v X P
dXF rdW
A BkC X Pr kC
X PkC XdX P
dW X P
νε
ε
ε
Θ += =
+
′= −
→−′− = =
+/ −
=+ 0 0 0 0
1
1 (1 )(1 )
( , )
In this equation, conversion is a function of pressure. Hence, we need an additional relationship between pressure and conversion.
A
k X PC v v X P
dX f X PdW
ε−
=/ +
=
The additional relationship is obtained through the Ergun Equation
3
1 150(1 ) 1.75p p
dP G Gdz D D
φ φ µρ φ
⎡ ⎤⎛ ⎞− −= − +⎢ ⎥⎜ ⎟
⎝ ⎠ ⎢ ⎥⎣ ⎦
Porosity
(1 ) Solid volume/total bed volume Gas density
Particle diameter
Viscosity of the fluid Length along the reactor Superficial velocity (volumetric flow/cross sectional area)
pD
zu
G u
φφ
ρ
µ
ρ
−− −−−
−−−
= ⋅ = 2
g Mass flux = units ( )cm s−
This equation can be reduced to 2 ( , )dP f X PdW
=
For any flow reactor, 0m m=
0 0
00
0 0
0 0 0 00
0
3
00
0 0
0 30
Ideal gas law
1 150(1 ) 1.75
Substitute for
Where,
1 150(1 ) 1.
T
T
T
T
p p
T
T
p p
v v
P FTv vP T F
v T FPv P T F
dP G Gdz D D
P FdP Tdz P T F
GD D
ρ ρ
ρρ ρ
φ φ µρ φ
ρ
β
φ φ µβρ φ
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠⎛ ⎞= = ⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞− −= − +⎢ ⎥⎜ ⎟
⎝ ⎠ ⎢ ⎥⎣ ⎦
⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
⎛ ⎞− −= +⎜ ⎟
⎝ ⎠
b
00
0 0
75
Let (1 )Bulk Density: (1 )
1(1 )
C c
c
T
C c T
G
W A z
P FdP TdW A P T F
φ ρρ φ ρ
βφ ρ
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
= − ×= −
⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠
0
0
0
0 0
0
2(1 )
2
This equation must be used for multiple reactions.
C c
T
T
A P
P FdP TdW T FP
P
βαφ ρ
α
≡−
⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠⎜ ⎟
⎝ ⎠
0
0
0 0
00 0 0
0
00
0
0
0
0
02
If,
2
(1 )
1
(1 )2
For isothermal reactors,
(1 ) ( , )2
This is the second ODE solved
T
T
AT T A T
T
AA
T
T
T
Py P
P FdP TdW T y F
FF F F X F XF
F yFF XF
PdP T XdW T y
PdP X f P XdW y
α
ε
ε
α ε
α ε
=
⎛ ⎞ ⎛ ⎞= − ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= + ∂ = + ∂
∂ = ∂ =
= +
⎛ ⎞= − +⎜ ⎟
⎝ ⎠
= − + =
i i i
i
for single reaction applications.
Analytical solutions are possible when 0 or when 1.Xε ε= When this is true,
( )
0
0
0
02
0
2
0
1/ 2 0
0 0
1/ 2
0
0 0
2( )
( )2
1
21 where, =(1 )
21
c c
PdPPdW P
Pd PPP dW
Pd PdW
P WP
P WP A P
zPP P
α
α
α
α
βα αϕ ρ
β
−=
= −
⎛ ⎞⎜ ⎟⎝ ⎠ = −
⎛ ⎞ = −⎜ ⎟⎝ ⎠
= −−
⎛ ⎞= −⎜ ⎟⎝ ⎠
Since, .(1 ) c cW A zϕ ρ= −
