flow of falling film, transport phenomenon, navier stokes equation derivation

1. Mehran university of engineering & technology, jamshoro
Department of Chemical engineering……!
Transport phenomenon
Presented by:
Abu Umeer (14ch42)
u.ahmed8@yahoo.com
Zeeshan Shaikh (14ch115)
Awais jamali (14ch06)
Ghulam yaseen (14ch24)
Shahzain tunio (14ch45)
Presented to:
Engr: Sikandar Mustafa Almani
Falling Film

2. 2
Flow of a Falling Film
The first example is the flow of a falling film . Consider the
Flow of a liquid at steady state along an inclined plate.
We are looking for (1) Volume flow rate
(2) Velocity profile perpendicular to the inclined plate

3. 3
Take a control
volume from
the system
Assumptions:
(1) Constant temperature  Constant density , viscosity .
(2) Laminar Flow
(3) Neglect entrance & exit effects .

4. 4
Start from the Navier-Stokes equation
For vz :
vx=0 vz is not function of z vz is not function of y
0(∵steady state) vy=0 Assuming vz is not function of z
uniform pressure
Driving force-
Gravity only
GDE (Governing Differential Equation)
z
zzzz
z
z
y
z
x
z
g
z
v
y
v
x
v
zz
v
v
y
v
v
x
v
v
t
v
























)( 2
2
2
2
2
2


02
2



 z
z
g
x
v

0cos2
2



  g
x
vz
gz β g
cos
g
gz

5. 5
2nd order ,  needs 2 Boundary Conditions.
B.C.
0,)2(
000,0)1(








z
zz
xz
vx
x
v
x
v
x



6. 6





cos
1
cos
2
2
2
2
g
x
v
g
x
v
z
z






From GDE
,vz is a function of x only
Integrate
dx
dv
x
v zz




1cos
1
cxg
dx
dvz
 

From B.C.(1)  c1=0
xg
dx
dvz
 

cos
1

7. 7
Integrate again
From B.C. (2)
Velocity Profile
2
2
cos
2
1
cxgvz  

])(1[cos
2
1
cos
2
1
cos
2
1
cos
2
1
cos
2
1
0
22
22
2
2
2
2











x
gv
gxgv
gc
cgv
z
z
z





8. 8
When x=0 , vz=vz,max
For average velocity vz
For volume flow rate
2
0
3
2
0
22
0
cos
3
1
]
3
[cos
2
1
|]
3
1
[cos
2
1
])(1[(cos
2
111




















gv
g
x
xg
dx
x
gdxvv
z
zz
2
max, cos
2
1


 gvz
3
cos
3
1
)( 

 

wgwvQ z

9. Success is a lousy teacher.
It seduces smart people into thinking they can’t lose.