3.044 MATERIALS PROCESSING
LECTURE 16
Navier-Stokes Equation (1-D):
∂vx
∂ 2 vx 1 ∂P
Fx
=ν 2 −
+
∂t
∂y
ρ ∂x
ρ
2
∂vx
∂ vx
∂P
ρ
= μ 2 −
∂t
∂y
∂x
Inertial Force kg2m
s
1
m3
Viscous Force
Pressure Force
Simplest Case:
ρ
∂ 2 vx
∂vx
=μ 2
∂t
∂y
Non-Dimensional Length:
V
x
, where L =
L
A
y
Y =
L
X=
Non-Dimensional Time:
τ =t
Date: April 11th, 2012.
V0
L
1
+
Fx
Body Force
N
m3
2
LECTURE 16
Velocity:
vx
v0
v0 ∂vx
v0 ∂Vx ∂vx
v02 ∂Vx
=
=
=
L ∂τ
L ∂τ ∂Vx
L ∂τ
2
v0 ∂ V x
= 2
L ∂Y 2
v0 ∂ 2 V x
=μ 2
L ∂Y 2
Vx =
∂vx
∂vx ∂τ
=
∂t
∂τ ∂t
2
∂ vx
∂ 2 Vx
= v0 2
∂y
∂y 2
2
v ∂Vx
ρ 0
L ∂τ
∂Vx
=
∂τ
Reynold’s Number:
Re =
μ
ρLv0
∂ 2 Vx
∂Y 2
inertial force
ρ L v0
=
μ
viscous force
Flow in a Tube:
Geometry
Critical Re
channel
∼ 1000
tube
∼ 2100
1 free surface (e.g. falling film)
∼ 20
Below Recrit : definitely laminar
Above Recrit : might be turbulent, needs perturbation
3.044 MATERIALS PROCESSING
3
ΔP 2
R − r2
4Lμ
ΔP R
=
2 L
vx (r) =
τrx
Kinetic Force:
Fk =
τ dA
Fk = ΔP πR2
What if Flow is Turbulent?
“Plug Flow”:
⇒ Cannot solved explicitly for vx (r)
Generalized Drag Force:
Fk =
f
A
K
friction factor
char. area of momentum transfer char. kinetic energy of flow
16
2
Laminar: fL = Re
KE of flow = 21 ρvavg
A = 2πRL
4
LECTURE 16
How do you use Fk ?
(P1 − P2 ) πR2 = f AK
⇒ Provides a relationship between ΔP and vavg
Summary:
exactly
If Laminar: Fk can be calculated
vx (r) → τrx → Fx = τ dA
If Turbulent: Use empirical f from experiment or simulation
f = f (Re, Geometry)
Flow Past Objects:
Fk = Ffriction +
shear griping
sides
Fform
dynamic pressure
of impinging flow
Fk = (f AK)friction + (f AK)form
3.044 MATERIALS PROCESSING
Solve for: (f AK)friction
Laminar: Solve exactly with Stokes Law
24
Re
π
A = d2
4
1 2
K = ρv
2
Fk, fric = 3πμvd
ffric =
Turbulent: negligible
Solve for: (f AK)form
Laminar: Fform ≈ 0
Turbulent:
fform 0.44
π
A = d2
4
1 2
K = ρv
2
π
Fform = fform d2 ρv 2
8
5
6
LECTURE 16
Summary:
Fk = Ffric + Fform
laminar
term
turbulent
term
where ffric is a function of the Reynolds Number
and fform is a function of Geometry
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http://ocw.mit.edu
3.044 Materials Processing
Spring 2013
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