1. Mechanical Energy Balance
CHE315
Mechanical Energy Balance

2. 2.9 Shell momentum balance inside a pipe
Objective:
To Derive a Velocity Profile (eqn.)for Newtonian Fluids flowing inside a pipe
CHE315
Mechanical Energy Balance

3. 2.9 Shell momentum balance inside a pipe
Let us consider the following simplifications:
Incompressible Newtonian fluid
One dimensional flow
Laminar flow
Fully developed flow (no entrance effect and velocity is independent from x)
CHE315
Mechanical Energy Balance

4. 2.9 Shell momentum balance inside a pipe
Fully developed flow (no entrance effect and velocity is independent from x)
CHE315
Mechanical Energy Balance

5. Net momentum efflux = rate of momentum out - rate of momentum in
The pressure forces =
Net momentum efflux = rate of momentum out - rate of momentum in
CHE315
Mechanical Energy Balance

6. Mechanical Energy Balance
So:
CHE315
Mechanical Energy Balance

7. Mechanical Energy Balance
If the momentum flux cannot be infinite at r = 0, Then C must be zero:
CHE315
Mechanical Energy Balance

8. Mechanical Energy Balance
The shear stress profile
Newtonian fluids:
Equating the two equations:
Integrating and using the boundary condition vx (r=R) = 0:
CHE315
Mechanical Energy Balance

9. This result means that the velocity distribution is parabolic
This result means that the momentum flux varies linearly with the radius
This result means that the velocity distribution is parabolic
CHE315
Mechanical Energy Balance

10. Mechanical Energy Balance
Using the definition of the average velocity:
We obtain in this case, the Hagen-Poiseuille equation:
CHE315
Mechanical Energy Balance

11. Mechanical Energy Balance
CHE315
Mechanical Energy Balance

12. Quiz #2: Sec. 2.7 Next Class (Monday) 26/11/1432H
CHE315
Mechanical Energy Balance