Shell Momentum Balances

1. Shell Momentum Balances

2. Outline Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube

3. Convective Momentum Transport Recall: MOLECULAR MOMENTUM TRANSPORT Convective Momentum Transport: transport of momentum by bulk flow of a fluid.

4. Outline Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube

5. Shell Momentum Balance Steady and fully-developed flow is assumed. Net convective flux in the direction of the flow is zero.

6. Outline Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube

7. Boundary Conditions Recall: No-Slip Condition (for fluid-solid interfaces) Additional Boundary Conditions: For liquid-gas interfaces: “The momentum fluxes at the free liquid surface is zero.” For liquid-liquid interfaces: “The momentum fluxes and velocities at the interface are continuous.”

8. Flow of a Falling Film Liquid is flowing down an inclined plane of length L and width W. δ – film thickness Vz will depend on x-direction only Why? y z x Assumptions: Steady-state flow Incompressible fluid Only Vz component is significant At the gas-liquid interface, shear rates are negligible At the solid-liquid interface, no-slip condition Significant gravity effects

9. Flow of a Falling Film τijflux of j-momentum in the positive i-direction y z x y z L x τxz ǀ x + δ τxz ǀ x δ W

10. Flow of a Falling Film τij flux of j-momentum in the positive i-direction y z x y z L τyz ǀ y=0 x δ τyz ǀ y=W W

11. Flow of a Falling Film τij flux of j-momentum in the positive i-direction y z x τzz ǀ z=0 y z L x δ W τzz ǀ z=L ρgcosα

12. Flow of a Falling Film P(W∙δ)|z=0 – P(W∙δ)|z=L + (τxzǀx )(W*L) – (τxzǀ x +Δx )(W∙L) + (τyzǀy=0 )(L*δ) – (τyzǀ y=W )(L∙δ) + (τzz ǀ z=0)(W*δ) – (τzz ǀz=L)(W∙δ) + (W∙L∙δ)(ρgcosα) = 0 Dividing all the terms by W∙L∙δand noting that the direction of flow is along z:

13. Flow of a Falling Film If we let Δx0, Integrating and using the boundary conditions to evaluate, Boundary conditions: @ x = 0 x = x

14. Flow of a Falling Film For a Newtonian fluid, Newton’s law of viscosity is Substitution and rearranging the equation gives

15. Flow of a Falling Film Solving for the velocity, Boundary conditions: @ x = δ, vz= 0

16. Flow of a Falling Film How does this profile look like? Compute for the following: Average Velocity:

17. Flow of a Falling Film Compute for the following: Mass Flowrate:

18. Flow Between Inclined Plates z x θ δ L Derive the velocity profile of the fluid inside the two stationary plates. The plate is initially horizontal and the fluid is stationary. It is suddenly raised to the position shown above. The plate has width W.

19. Outline Convective Momentum Transport Shell Momentum Balance Boundary Conditions Flow of a Falling Film Flow Through a Circular Tube

20. Flow Through a Circular Tube Liquid is flowing across a pipe of length L and radius R. Assumptions: Steady-state flow Incompressible fluid Only Vx component is significant At the solid-liquid interface, no-slip condition

21. Recall: Cylindrical Coordinates

22. Flow Through a Circular Tube

23. Flow Through a Circular Tube

24. Flow Through a Circular Tube BOUNDARY CONDITION! At the center of the pipe, the flux is zero (the velocity profile attains a maximum value at the center). C1must be zero! 

25. Flow Through a Circular Tube BOUNDARY CONDITION! At r = R, vz = 0.

26. Flow Through a Circular Tube Compute for the following: Average Velocity:

27. Hagen-Poiseuille Equation Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D

28. What if…? The tube is oriented vertically. What will be the velocity profile of a fluid whose direction of flow is in the +z-direction (downwards)?