Second Law of Transport 2 Law (variation in space-time) nd
9
DT 2 D!u Navier-Stokes = k∇ T +µφu ! p ρ = µ∇ !u+ρ!g − ρC ∇p Dt Dt DT 2 Fourier’s ρC = k∇ T +µφ − r∆H DCA Dt 2 = D ∇ C +ν r DC A A A = D ∇ C +ν r 2 Fick Dt Dt Dϕ = σ∇ ϕ Dϕ Dt 2 = σ∇ ϕ Fundamental equations of the system Dt
2 2 2 ∂ S ∂ S ∂ S 2 ! ! ∇ S = ∇ · ∇S = + 2 + 2 2 ∂x ∂y ∂z
DS ∂S ∂S ∂S ∂S ∂S ! = + !u · ∇S = + ux + uy + uz Dt ∂t ∂t ∂x ∂y ∂z Transport Phenomena
10/31/08
Dt
= σ∇2 ϕ
Equations of Motion 12
Fundamental equations of the system D!u 2 ! ρ = µ∇ !u+ρ!g − ∇p Dt DT 2 ρCp = k∇ T +µφu − r∆Hrxn Dt DCA 2 = DA ∇ CA +νA r Dt ! · (ρ!u) = 0 ∇ Transport Phenomena
10/31/08
Example 13
Start-up in circular tube Cylindrical
coordinates Only vz(r) ≠ 0
r
z
Example: startup in circular tube only uz (r) != 0 ! " ! ∂uz ∂u 1 ∂ ∂uz z ! ρ + uz! = ∆p + µ r ∂t r ∂r ∂r ! ∂z In S.S. conditions (∂/∂t = Transport 0) Phenomena
θ
10/31/08
Example Example: startup in circular tube only u (r) != 0 z
14
! ∂uz ∂u z ! ρ + uz! = ∂t ! ∂z
1 ∂ ∆p + µ r ∂r
!
∂uz r ∂r
"
r
z
In S.S. conditions (∂/∂t = 0) ! " ∂ ∂uz ∆p r =− r ∂r ∂r µ ∂uz ∆p r"2 A" r" =− + " ∂r µ 2 "r ∆p r2 uz = − +B µ 4 $ ∆p # 2 2 uz = R −r 4µ
B.C. B.C.
θ
∂uz = f inite ∂r uz = 0
Transport Phenomena
r=0 r=R
10/31/08
θ
r
z
vz/vmax(r=0)
1.0
0.5
0.0 -1.0
0 0.0001 0.001 0.01 0.05 0.1 0.2 0.5 infinite
-0.8
-0.6
-0.4
-0.2
0.0
r/R
15
Transport Phenomena
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0.2
0.4
0.6
0.8
1.0
Numerical Solution 16
Finite elements approach Software packages
Fluent CFD Physics
Set up the correct problem and boundary conditions Many times is not necessary to know the exact T,C,v of every single point in your system
Transport Phenomena
10/31/08
Boundary Layer 17
δ
All the phenomena happen in the boundary layer Everything
outside the boundary layer is in equilibrium and constant
The boundary layer is very “thin” The
size of the boundary layer (δ) is smaller than the characteristic size of the system Order of magnitude approach Taylor
series Transport Phenomena
10/31/08
Reynold’s Number 18
Boundary layer approach Reinolds Number δ
D"u ρ = µ∇2"u Dt ρu
! ∆u !
≈µ
! ∆u !
x δ2 ! "2 δ µ ≈ = 1/Re x ρux Prandtl Number
ρux Re = µ ρux Re =
Transport Phenomena
10/31/08
µ
δ x
µ ≈ = 1/Re ρux
Prandtl’s Number
ρux Re = µ
Prandtl Number 19
uT = uδT /δ
DT ρCp = k∇2 T Dt 1/2 ! ! ∆T ∆T ! ! ρCp uT ≈k 2 x δT δT k ρCp u ≈ 2 δx δT ! "3 δT k ≈ δ Cp µ δT ≈ P r−1/3 δ
δ δT
Cp µ Pr = k Cp µ Pr = Transport Phenomena 10/31/08 k
Dt ! ! ∆T ∆T ! ! ρCp uT ≈k 2 x δT
Schmidt’s Number…
20
Cp µ Pr = k
δT k ρCp u ≈ 2 x δT ! "3 δT k ≈ Re−1/2 δ Cp µ δT ≈ P r−1/3 δ
δ
… As usual everything is the same
Schmidt Number δC ≈ Sc−1/3 δ
Cp µ Pr = k
δC
ρDA Sc = ρD Sc = µ µ A
Transport Phenomena
10/31/08
Nusselt’s Number 21
Considering the the fluxes: fluxes: Considering
δ δT
∂T ∂T = −k −k = h∆T h∆T qq = = ∂x ∂x ! ! ∆T ! ∆T ! ≈ k! ! ∆T ! hh! ∆T ≈k δ δTT hx x δδ hx x Nuu = = ≈ N ≈ δ δδTT kk δ !"#$ !"#$ !"#$ !"#$ 1/2 1/3 Re1/2 Prr1/3 Re P
In general general In
KA x x δ Sh = ≈ DA δ δC !"#$ !"#$ Re1/2 Sc1/3
Transport Phenomena
10/31/08
Sherwood’s Number 22
Considering the the fluxes: fluxes: Considering
δ δC
∂T ∂T = −k −k = h∆T h∆T qq = = ∂x ∂x ! ! ∆T ! ∆T ! ≈ k! ! ∆T ! hh! ∆T ≈k δ δTT hx x δδ hx x Nuu = = ≈ N ≈ δ δδTT kk δ !"#$ !"#$ !"#$ !"#$ 1/2 1/3 Re1/2 Prr1/3 Re P
In general general In
KA x x δ Sh = ≈ DA δ δC !"#$ !"#$ Re1/2 Sc1/3
Transport Phenomena
10/31/08
hx x δ Nu = ≈ k δ δT !"#$ !"#$
In General… Re1/2 P r1/3
23
In general
hx KA x 1/3 Nu = = f (Re) P r Sh = = f (Re) Sc1/3 k DA 1/2 Re Plate in laminar flow 2 + Re1/2 Sphere in laminar flow f (Re) = 0.8 0.005Re Packed bed in turbulent flow · · · · · · · · · · · · Extension to macroscopic balances ) Transport Phenomena 10/31/08 * * Q˙ = q dA = A U ∆T 1/U = 1/h + s /k
Extension to Macroscopic Balances 24
Transport Phenomena
10/31/08
hx KA x 1/3 Nu = = f (Re) P r Sh = = f (Re) Sc1/3 k DA 25 1/2 Re Plate in laminar flow 2 + Re1/2 Sphere in laminar flow f (Re) = 0.8 0.005Re Packed bed in turbulent flow · · · · · · · · · · · ·
Overall Transfer Coefficient
Extension to macroscopic balances ) * * Q˙ = q dA = A U ∆T 1/U = 1/hi + si /ki A
1/U = 1/hint + s/kmetal + 1/hext Transport Phenomena
10/31/08
Nu =
hx Nu = = f (Re) P r1/3 k 26 1/2 Re 2 + Re1/2 f (Re) = 0.8 0.005Re · · · · · · · · · · · ·
= f (Re) P r k KA x Sh = = f (Re) Sc1/3 D A 1/2 Re Plate 2 + Re1/2 Plate in laminar flow Sphere f (Re) = 0.8 Sphere in laminar flow 0.005Re Packed · · · ·flow Packed bed in turbulent ········
Log-Mean Temperature Difference
Extension to macroscopic balanc ) Extension to macroscopic balances ˙ Q= q dA = A U ∆T ) A * * Q˙ = q dA = A U ∆T 1/U = 1/hi + si /ki 1/U = 1/hint + s/kmetal + 1/hext A
1/U = 1/hint + s/kmetal + 1/hext
∆T1 − ∆T2 ∆Tln = ln (∆T1 /∆T2 )
Transport Phenomena
10/31/08
Conclusions 27
Momentum, Heat and Mass Transfer Highly
inter-correlated
Similar
physical principles Similar driving force Similar mathematical formulas 2nd
order PDE – numerical solutions Highly dependent from fluid-dynamic properties Simplifications Boundary
layer Dimensionless numbers Transport Phenomena
10/31/08
28
PART II Methanol Synthesis
Transport Phenomena
10/31/08
Methanol Production 29
Produced from SynGas Catalyst
Cu/ZnO-Al2O3 Equilibrium limited P
= 50 – 100 bar T = 200 – 300 ˚C
Main Commodity 20
Million ton/year
Typical plant size is 1 Million ton/year 12
m3/s Transport Phenomena
10/31/08
Catalyst specs 30
KATALCO 51-8 Methanol Synthesis Catalyst Long life as a result of the optimized formulation incorporating a patented use of a fourth component MgO
Product Benefits • •
Low operating temperatures minimize the catalyst sintering rate
•
High catalyst selectivity gives very low impurities in the crude product
•
Close approach to equilibrium is achieved and maintained
•
Easy to reduce and start-up
Product Uses
•
Synthesis of methanol from H2, CO and CO2 mixtures arising from steam reforming of hydrocarbons, coal gasification or POx
General Description
•
KATALCO 51-8 is a copper catalyst on a ZnO-Al2O3 support with a MgO promoter Physical Properties (Typical) KATALCO 51-8 Cylindrical pellet 5.4 mm 5.2 mm 1250 kg/m 3 80 kgf
Form Diameter Length Bulk Density Crush Strength (axial)
Shipping & Handling
Chemical Composition (Typical) CuO: Al2 O3 : ZnO: MgO:
64 wt% 10 wt% 24 wt% 2 wt%
•
Avoid contact with skin and clothing. Avoid breathing dust. Do not take internally. Please refer to the relevant Material Safety Data Sheet for further information.
•
KATALCO 51-8 catalyst is available in non-returnable polythene lined mild steel drums or bulk bags for easy loading.
Transport Phenomena
10/31/08
Reactor types 31
Transport Phenomena
10/31/08
Multi-Bed Reactor 32
6:96
)";=?@==BC )DEBFGHBG)@HGC)IAE>JKLDHGFM
6:27 6:25 6:24
N!C'$
6:29 6:26 6:67 6:65 6:64 6:69 6:66 456
476
866
896
846
#OPQ
856
Transport Phenomena
876
566
596
10/31/08
!
Reactor dimensioning 33
!"#$%&'()*+',"-#) 1 m/s ! wanted gas velocity
)
m2 section – approx 4 m diameter #JKG=BCB)5>=>D)A;=@5?=>DQH) )