Download DRIVING FORCES FOR THE TRANSPORT PHENOMENA What is

DRIVING FORCES FOR THE TRANSPORT PHENOMENA
What is the driving force for momentum transport?
Velocity Gradient!!!
What is the driving force for heat transfer?
Temperature Gradient!!!
What is the driving force for mass transport?
Concentration Gradient!!!
MOMENTUM BALANCE
• A small control volume of fluid is chosen.
• Choose a coordinate system.
• Momentum balance for this system is written to develop a
DIFFERENTIAL EQUATION.
• Use BOUNDARY CONDITIONS to obtain algebraic
relations from the solutions of the differential equations
• Solve the algebraic relations to determine engineering
characteristics of the system such as velocity distributions
Æshear stress at the fluid-solid interface.
THE MOMENTUM BALANCE FOR STEADY STATE FLOW
rate of momentum in - rate of momentum out + sum of forces
acting on the system = 0
Stress components acting on
a small element.
Equilibrium leads to symmetry
in off-diagonal stress
components.
FLOW OF A FALLING FILM
FLOW OF A FALLING FILM
Rate of momentum in across surface at x
(momentum transport due to viscosity)
(LW)(τxz)|x
Rate of momentum out across surface at x+∆x
(momentum transport due to viscosity)
(LW)(τxz)|x+∆x
Rate of momentum in across surface at z = 0
(due to fluid motion)
Rate of momentum out across surface at z = L
(due to fluid motion)
Gravity force acting on fluid
(W∆xvz)(ρvz)|z=0
(W∆xvz)(ρvz)|z=L
(LW∆x)(ρg cosβ)
FLOW OF A FALLING FILM
0
0
(LW)(τxz)|x - (LW)(τxz)|x+∆x + (W∆xvz)(ρvz)|z=0 - (W∆xvz)(ρvz)|z=L
+ (LW∆x)(ρg cosβ) = 0
If the equation is divided by LW∆x and if ∆X is infinitely smallÆ
lim
∆Χ → 0
τ xz
x + ∆x
- τ xz
∆X
x
=
d τ xz
dx
= ρ g cos β
The equation is integrated to yield
τxz = ρgx cosβ + C1
This equation gives the momentum flux or alternatively the
shear-stress distribution.
FLOW OF A FALLING FILM
The equation is integrated to yield
τxz = ρgx cosβ + C1
How to find the value of the integration constant, C1?
Use Boundary Condition(s) !!!
At x = 0, τxz = 0 Æ C1 = 0
The momentum flux is
τxz = ρgx cosβ
FLOW OF A FALLING FILM
If the fluid is Newtonian and laminar then
τxz = −η dvz / dx
Substituting this expression for τxz,
dvz / dx = - ρgx cosβ / η
vz = - (ρg cosβ) x2 / 2η + C2
Use Boundary Condition(s) to determine the integration
constant!!!
At x = δ, vz = 0 Æ C2 = (ρg cosβ) δ2 / 2η
∴the velocity distribution is
vz = [(ρgδ2cosβ) / 2η] [1−(x/δ)2] PARABOLIC
FLOW OF A FALLING FILM
The maximum velocity
vzmax = (ρgδ2cosβ) / 2η
The average velocity
2
2


x
g
1
ρ g δ cos β
ρ
δ
cos β
 
v z = ∫ v z dx =
1 −   dx =
∫
δ 0
2η
3η
0 
  δ  
δ
δ
The volume flow rate, Q,
ρ gW δ 3 cos β
Q = v z (W δ ) =
3η