shell balances

CHEE 3363Spring 2014Handout 19

�Reading: Fox 8.1--8.4

Learning objectives for lecture

State the criterion for the transition from laminar to turbulent �

�

2

Reynolds number Re =

laminarturbulent

This transition is approximate

3

[

τyx +∂τyx

∂y

(

dy

2

)]

dx dz

[

τyx +∂τyx

∂y

(

−

dy

2

)]

dx dz

[

p +∂p

∂x

(

−

dx

2

)]

dy dz

[

p +∂p

∂x

(

dx

2

)]

dy dz

p, τyx

(“shell balance”) using the macroscopic equationsh

x

y

�

�

�

4

5

3

yof x

6

4

7

5L

8

6

9

y in

7

stationary top plate U = 0

8

L

9

y in

p2πr dr

(

p +∂p

∂xdx

)

2πr dr

τrx2πr dx

(

τrx +dτrx

drdr

)

2π(r + dr) dx

p, τrx

h xr

�

�

�

2R

In xexcept surface force term FSx vanish

annulus

2

3

function only of x(uniform pressure at each section)

function only of r(fully developed)

4

5

r = 0

6

r in

maximum at

Goal�

Need to account for losses due to friction�

system�

�

shear force�

�

�

�

20

(2) (3)

�

�

equations of motion��

boundary layer

Entrance length 2

(2) (3)