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)?