Report

Shell Momentum Balances Outline 1.Convective Momentum Transport 2.Shell Momentum Balance 3.Boundary Conditions 4.Flow of a Falling Film 5.Flow Through a Circular Tube Convective Momentum Transport Recall: MOLECULAR MOMENTUM TRANSPORT Convective Momentum Transport: transport of momentum by bulk flow of a fluid. Outline 1.Convective Momentum Transport 2.Shell Momentum Balance 3.Boundary Conditions 4.Flow of a Falling Film 5.Flow Through a Circular Tube Shell Momentum Balance rate of momentum rate of momentum in by convective out by convective transport transport rate of momentum rate of momentum force of gravity in by molecular out by molecular acting on system 0 transport transport 1. Steady and fully-developed flow is assumed. 2. Net convective flux in the direction of the flow is zero. Outline 1.Convective Momentum Transport 2.Shell Momentum Balance 3.Boundary Conditions 4.Flow of a Falling Film 5.Flow Through a Circular Tube 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.” Flow of a Falling Film z y x Liquid is flowing down an inclined plane of length L and width W. δ – film thickness Vz will depend on x-direction only Why? Assumptions: 1. Steady-state flow 2. Incompressible fluid 3. Only Vz component is significant 4. At the gas-liquid interface, shear rates are negligible 5. At the solid-liquid interface, no-slip condition 6. Significant gravity effects 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 Flow of a Falling Film τij flux of j-momentum in the positive i-direction y z x y z L x τyz ǀ y=0 δ τyz ǀ y=W W 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 τzz ǀ z=L W ρg cos α Flow of a Falling Film rate of momentum rate of momentum force of gravity in by molecular out by molecular acting on system 0 transport transport P(W∙δ)|z=0 – P(W∙δ)|z=L + (τxzǀ x )(W*L) – (τxz ǀ x +Δx )(W∙L) + (τyzǀ y=0 )(L*δ) – (τyz ǀ y=W )(L∙δ) + (τzz ǀ z=0)(W* δ) – (τzz ǀ z=L)(W∙δ) + (W∙L∙δ)(ρgcos α) = 0 Dividing all the terms by W∙L∙δ and noting that the direction of flow is along z: |+ − | = cos Flow of a Falling Film |+ − | = cos ∆ If we let Δx 0, ( ) = cos Integrating and using the boundary conditions to evaluate, Boundary conditions: @ x = 0 = 0 x = x = = cos Flow of a Falling Film = cos For a Newtonian fluid, Newton’s law of viscosity is = − Substitution and rearranging the equation gives cos =− Flow of a Falling Film cos =− Solving for the velocity, Boundary conditions: @ x = δ, v z = 0 cos 2 = − + 2 2 2 cos 2 = (1 − ( ) ) 2 Flow of a Falling Film 2 cos 2 = (1 − ( ) ) 2 Compute for the following: Average Velocity: v z v z , ave W 0 0 v dA v dxdy dA dxdy z z W 0 0 How does this profile look like? Flow of a Falling Film 2 cos 2 = − (1 − ( ) ) 2 Compute for the following: Mass Flowrate: m v z dA W v z Flow Between Inclined Plates z x θ L δ 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. Outline 1.Convective Momentum Transport 2.Shell Momentum Balance 3.Boundary Conditions 4.Flow of a Falling Film 5.Flow Through a Circular Tube Flow Through a Circular Tube Liquid is flowing across a pipe of length L and radius R. Assumptions: 1. Steady-state flow 2. Incompressible fluid 3. Only Vx component is significant 4. At the solid-liquid interface, no-slip condition Recall: Cylindrical Coordinates Flow Through a Circular Tube rate of momentum rate of momentum force of gravity in by molecular out by molecular acting on system 0 transport transport pressure : PA z 0 PA z L net momentum flux : rz A1 r rz A2 r r Adding all terms together: P 2 r r z 0 P 2 r r z L rz 2 rL r rz 2 rL r r 0 Flow Through a Circular Tube P 2 r r z 0 P 2 r r z L rz 2 rL r rz 2 rL r r 0 Dividing by 2 Lr : P z 0 P z L rz r r rz r r r 0 r L r Let x 0 : d P0 PL r rz r 0 dr L Flow Through a Circular Tube d P0 PL r rz r 0 L dr Solving: d P0 PL rz r r dr L P P rz r 0 L r 2 C1 2L C1 P0 PL rz r r 2L BOUNDARY CONDITION! At the center of the pipe, the flux is zero (the velocity profile attains a maximum value at the center). C1 00 r C1 must be zero! Flow Through a Circular Tube P0 PL rz r 2L From the definition of flux: dv z rz dr Plugging in: P0 PL dv z r dr 2L P0 PL 2 vz r C2 4 L BOUNDARY CONDITION! At r = R, vz = 0. P P 0 0 L R 2 C2 4 L P P C2 0 L R 2 4 L P0 PL 2 P0 PL 2 vz r R 4L 4L Flow Through a Circular Tube P0 PL 2 2 vz R r 4L Compute for the following: Average Velocity: v z v z , ave 2 R 0 0 v dA v rdrd dA rdrd z z 2 R 0 0 Hagen-Poiseuille Equation vave P0 PL 2 D 32L Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D What if…? The tube is oriented vertically. What will be the velocity profile of a fluid whose direction of flow is in the +zdirection (downwards)?