shell momentum balances
DESCRIPTION
Shell Momentum Balances. Outline. Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube. Convective Momentum Transport. Recall: MOLECULAR MOMENTUM TRANSPORT. - PowerPoint PPT PresentationTRANSCRIPT
Shell Momentum Balances
Outline
1.Convective Momentum Transport
2.Shell Momentum Balance
3.Boundary Conditions
4.Flow of a Falling Film
5.Flow Through a Circular Tube
Convective Momentum Transport
Recall: MOLECULAR MOMENTUM TRANSPORT
Convective Momentum Transport: transport of momentum by bulk flow of a fluid.
Outline
1.Convective Momentum Transport
2.Shell Momentum Balance
3.Boundary Conditions
4.Flow of a Falling Film
5.Flow Through a Circular Tube
Shell Momentum Balance
rate of momentum rate of momentum
in by convective out by convective
transport transport
1. Steady and fully-developed flow is assumed.2. Net convective flux in the direction of the flow
is zero.
rate of momentum rate of momentumforce of gravity
in by molecular out by molecular 0acting on system
transport transport
Outline
1.Convective Momentum Transport
2.Shell Momentum Balance
3.Boundary Conditions
4.Flow of a Falling Film
5.Flow Through a Circular Tube
Boundary Conditions
Recall: No-Slip Condition (for fluid-solid interfaces)
Additional Boundary Conditions:For liquid-gas interfaces:“The momentum fluxes at the free liquid surface is zero.”
For liquid-liquid interfaces:“The momentum fluxes and velocities at the interface are continuous.”
Flow of a Falling Film
Liquid is flowing down an inclined plane of length L and width W.
δ – film thickness
Vz will depend on x-direction onlyWhy?
zx
y
Assumptions:1. Steady-state flow2. Incompressible fluid3. Only Vz component is significant4. At the gas-liquid interface, shear rates are negligible 5. At the solid-liquid interface, no-slip condition6. Significant gravity effects
Flow of a Falling Film
zx
y
δ
W
L
τxz ǀ x
τxz ǀ x + δ
zx
y
τij flux of j-momentum in the positive i-direction
Flow of a Falling Film
zx
y
δ
W
Lzx
y
τij flux of j-momentum in the positive i-direction
τyz ǀ y=0
τyz ǀ y=W
Flow of a Falling Film
zx
y
δ
W
Lzx
y
τij flux of j-momentum in the positive i-direction
τzz ǀ z=0
τzz ǀ z=L ρg cos α
Flow of a Falling Film
P(W∙δ)|z=0 – P(W∙δ)|z=L +(τxzǀ x )(W*L) – (τxz ǀ x +Δx )(W∙L) + (τyzǀ y=0 )(L*δ) – (τyz ǀ y=W )(L∙δ) + (τzz ǀ z=0)(W* δ) – (τzz ǀ z=L)(W∙δ) + (W L∙ ∙δ)(ρgcos α) = 0
Dividing all the terms by W∙L∙δ and noting that the direction of flow is along z:
𝜏𝑥𝑧∨¿𝑥+𝛿−
𝜏𝑥𝑧∨¿𝑥
𝛿=𝜌𝑔cos𝛼 ¿
¿
rate of momentum rate of momentumforce of gravity
in by molecular out by molecular 0acting on system
transport transport
Flow of a Falling Film
𝜏𝑥𝑧∨¿𝑥+𝛿−
𝜏𝑥𝑧∨¿𝑥
∆ 𝑥=𝜌𝑔cos𝛼 ¿
¿
𝑑 (𝜏 ¿¿ 𝑥𝑧)𝑑𝑥
=𝜌𝑔 cos𝛼 ¿
Boundary conditions:@ x = 0 x = x
𝜏𝑥𝑧=𝜌 𝑥𝑔cos𝛼
If we let Δx 0,
Integrating and using the boundary conditions to evaluate,
Flow of a Falling Film
𝜏𝑥𝑧=𝜌 𝑥𝑔cos𝛼
For a Newtonian fluid, Newton’s law of viscosity is
𝜏𝑥𝑧=−𝜇𝑑𝑣 𝑧
𝑑𝑥
Substitution and rearranging the equation gives
𝑑𝑣𝑧
𝑑𝑥=−(𝜌 𝑔cos𝛼𝜇 )𝑥
Flow of a Falling Film
𝑑𝑣𝑧
𝑑𝑥=−(𝜌 𝑔cos𝛼𝜇 )𝑥
Solving for the velocity,
𝑣 𝑧=−(𝜌 𝑔cos𝛼2𝜇 )𝑥2+𝐶2Boundary conditions:@ x = δ, vz = 0
Flow of a Falling Film
How does this profile look like?
Compute for the following:
Average Velocity:
0 0,
0 0
W
zz
z z ave W
v dxdyv dAv v
dA dxdy
Flow of a Falling Film
Compute for the following:
Mass Flowrate:
z zm vdA W v
Flow Between Inclined Plates
θ
Lδ
z
x
Derive the velocity profile of the fluid inside the two stationary plates. The plate is initially horizontal and the fluid is stationary. It is suddenly raised to the position shown above. The plate has width W.
Outline
1.Convective Momentum Transport
2.Shell Momentum Balance
3.Boundary Conditions
4.Flow of a Falling Film
5.Flow Through a Circular Tube
Flow Through a Circular Tube
Liquid is flowing across a pipe of length L and radius R.
Assumptions:1. Steady-state flow2. Incompressible fluid3. Only Vx component is significant4. At the solid-liquid interface, no-slip condition
Recall: Cylindrical Coordinates
Flow Through a Circular Tube
rate of momentum rate of momentumforce of gravity
in by molecular out by molecular 0acting on system
transport transport
0
1 2
:
: z z L
rz rzr r r
pressure PA PA
net momentumflux A A
0
Adding all terms together:
2 2 2 2 0rz rzz z L r r rP r r P r r rL rL
Flow Through a Circular Tube
0
2 2 2 2 0rz rzz z L r r rP r r P r r rL rL
0
0
Dividing by 2 :
0
Let 0:
0
rz rzz z L r r r
Lrz
L r
P P r rr
L r
x
P P dr r
L dr
Flow Through a Circular Tube
0 0Lrz
P P dr r
L dr
0
201
0 1
Solving:
2
2
Lrz
Lrz
Lrz
d P Pr r
dr L
P Pr r C
L
P P Cr
L r
BOUNDARY CONDITION!At the center of the pipe, the flux is zero (the velocity profile attains a maximum value at the center).
10 0Cr
C1 must be zero!
Flow Through a Circular Tube
0
2L
rz
P Pr
L
0
202
From the definition of flux:
Plugging in:
2
4
zrz
z L
Lz
dvdr
dv P Pr
dr L
P Pv r C
L
BOUNDARY CONDITION!At r = R, vz = 0.
202
202
04
4
L
L
P PR C
L
P PC R
L
2 20 0
4 4L L
z
P P P Pv r R
L L
Flow Through a Circular Tube
2 20
4L
z
P Pv R r
L
Compute for the following:
Average Velocity:2
0 0, 2
0 0
R
zz
z z ave R
v rdrdv dAv v
dA rdrd
Hagen-Poiseuille Equation
20
32L
ave
P Pv D
L
Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D
What if…?
The tube is oriented vertically.
What will be the velocity profile of a fluid whose direction of flow is in the +z-direction (downwards)?