Falling Film Example from Birds's "Transport Phenomena"

2. Problem Example 2.2-2
Transport Phenomena

3. Presented By
Shahid Alam 201-3CH-97
Nabeel Ahmad 2013-CH-93
Fidda Hussain 2013-CH-91
M.Amir 2013-CH-41
M.Rizwan 2013-CH-71

4. Rework the Falling Film Problem for situation in which Viscosity
depends upon Position in Following Manner :
)(


x
a
oe

 Is the viscosity
at surface of
the film.
o
Is a constant which tells
how rapidly viscosity
changes as Changes
a
a

6. Direction
of Gravity
β
x
z
Diagram

7. xz
y
Flow
Area
Thickness of
shell=∆x
Length of shell=L
Width of shell=W

8. Solution 05
01
02
03
04
STA
RT

9. Momentum Balance on Shell
Rate of z-momentum In by viscous Transport at the surface
at x
Rate of z-momentum Out by flow at surface x + ∆x
b
a
Surface Area Shear (force/area) at the surface x
Surface Area
Shear (force/area) at the surface
x+∆x

10. Rate of z-momentum In by flow at the surface z=0
Rate of z-momentum Out by flow at surface z=L
Mass flow=A.v.ρ velocity
c
d
Mass flow=A.v.ρ velocity
Momentum Balance on Shell

11. Gravity Force acting on the fluid
Mass=Volume *Density
mg
β
x
z
e
Momentum Balance on Shell

12. Momentum Balance
For steady state the equation shell momentum Balance is
Rate of
momentum
Into the
shell
Rate of
momentum
Out of the
shell
Sum of all
the Forces
acting on
shell
Rate of
accumula-
tion
0
1
Putting all the values from equtions a,b,c,d,e into equation 1

13. Momentum Balance
 As the fluid is incompressible and no slip condition at wall,so
this implies that
and
ρ=Constant
 Therefore above equation becomes

14. Momentum Balance
 Divided both sides by LW∆x

15. Differential equation for
Momentum Flux Distribution
Taking limit as ∆x 0
According to the definition of Derivative
So above eqution becomes
2
1st order
differential
equation for
momentum flux
distribution

16. By using Newton’s Law of Viscosity


cosg
dx
d xz






1cos Cxgxz  
0x 0xz 01C

17.  cosgxxz   cosgx
dx
dvz









cosgx
dx
dv
e zx
a
o 










ax
o
z
e
gx
dx
dv







 cos
dxxe
g
dv
ax
o
z 

 

 cos ].[
cos
dxe
aa
e
x
g
v
ax
ax
o
z 

 






18. ]...[
cos
a
e
a
e
a
xg
v
axax
o
z


 



]..[
cos
22
2
ce
a
e
a
xg
v
axax
o
z 

  



19. 0zv
]..[ 2
2
2
2
2




 aa
e
a
e
a
c 
]
11
[. 2
2
2
aa
ec a
 
)]
11
(.[
cos
2
2
2
2
aa
ee
a
e
a
xg
v a
axax
o
z 

 


 
x

20. 



















 2
2
2
2 11
.
1
.
cos
aa
e
aa
x
e
g
v a
ax
o
z 



 


















 22
2
111cos
aa
x
e
aa
e
g
v
ax
a
o
z

 
........1 32
 xxxex
0a

21. 



















 )
1
...)(
!2
)(
1(
11
...
!3!2
10lim
cos
2
2
2
322
aa
x
ax
ax
aa
aa
aa
g
v
o
z






















 3
3
2
2
24
43
3
32
2
2
2
3
2
2
22
322
!.3
.
!.2
1
....
.!.3!.2!.3!.2
1
!.30!.2
1
0lim
cos

 xax
a
x
aa
xa
a
xa
a
ax
a
x
a
a
a
a
a
a
aa
aa
a
a
a
a
g
v
o
z





















 ...
62
1
6262
1
62
1
1
0lim
cos
3
3
2
2
24
42
3
3
2
2
2
3
2
2
22
322

 axx
a
x
a
xaaxx
a
x
a
a
a
a
a
a
aa
a
a
a
a
a
g
v
o
z

22. 





 2
22
2
1
2
1cos

 xg
v
o
z






 2
0
2
)(1
2
cos

 xg
vz


















 ...
3
.1
2
1
...
3
1
2
1
0lim
cos
3
3
2
22
a
xx
aa
g
v
o
z



23. 




















)
11
(
)
1
(
1
11cos
2
2
2
0
2
aa
e
aa
x
e
aa
e
g
v
a
ax
a
z




0x





















)
1
1(
1
.
)
1
(
1
.
1
11cos
2
0
2
aa
e
a
x
a
e
aa
e
g
v
a
ax
a
z





24. 





















)
1
1(
)
1
(
1
11cos
)1(
2
0
2
a
a
x
e
aa
e
g
v
x
a
a
z




0x






 2
0
2
max
11cos
aa
e
g
v a



25. 















)
1
1(
)
1
(
1
)1(
max
a
a
x
e
v
v
x
a
z 


26. Average Velocity


 w
w
z
z
dxdy
xdydv
v
0 0
0 0




 0
1
dxvv zz
])
1
1
1
.(1[[
1
0
)1(
max





 

dx
a
a
x
evv
x
a
z

27. ])
1
1
1
(.[
2
0
)1(
0
max
dx
a
a
x
exdx
v
v
x
a
z


 






]]..[
1
[
0
)1(
2
0
)1(
max
dx
a
x
e
x
e
a
av
v
x
a
x
a
z 











].
1
2.
1
.[
1
)1()1(
)1(
2
max
max 




 xdxe
a
xdxe
a
ex
a
av
vv
x
a
x
a
x
a
z






28. ].
1
.
2
.[
1
)1()1()1(
2
max
max 



 xdxe
a
xdxe
a
e
a
x
a
av
vv
x
a
x
a
x
a
z
 
].)
12
(.[
1
)1()1(
2
max
max 



 xdxe
aa
e
a
x
a
av
vv
x
a
x
a
z








 xdxe
aaa
av
e
a
x
a
av
vv
x
a
x
a
z .)
12
(
1
..
1
)1(
max
)1(
2
max
max



29. ].)[
12
(
1
..
1
)1(
)1(
max
)1(
2
max
max 







 dxe
aa
e
x
aaa
av
e
a
x
a
av
vv
x
a
x
ax
a
z


 


])[
12
(
1
..
1
)1(
2
2
)1(
max
)1(
2
max
max





 
 x
a
x
a
x
a
z e
a
e
a
x
aaa
av
e
a
x
a
av
vv
])[
1
2(
1
.
1 2
22
max2max
max
aaaa
av
a
av
vvz

 





30. ]
2
)
221
([
cos
332
2
aaaa
e
g
v a
o
avg 


]
11
[.
cos
]
2
)
221
([
cos
2
2
332
2
max
aa
e
g
aaaa
e
g
v
v
a
o
a
oavg








31. 2
332
max
11
2
)
221
aa
eaaaa
v
v a
avg




32.  
2
0 0
)(

 Wvdxdyvw zz
]
2
)
221
([
cos
. 332
2
aaaa
e
g
Ww a
o





33. ]
2
)
221
([
cos
332
32
aaaa
e
g
Ww a
o




.
m
q 

34. Force on the solid
a
e
gWL
F 

 cos
  x
z
dx
dv
A
F
|








 a
oe
g
WLF
cos

35. a
e
gWL
F 

 cos