Shell Momentum Balance

Shell Momentum Balance - Shell Momentum Balance for Steady...

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1 Common boundary conditions a. At a solid-liquid interface: ‘no-slip’ condition is frequently assumed. b. At a liquid-liquid interface: ‘no-slip’ condition is frequently assumed and viscous stresses are continuous at the interface. c. At a gas-liquid interface: viscous stress may be negligible for small velocity gradient of gas liquid 1 liquid 2 interface ij ττ = liquid gas 0 = = Shell Momentum Balance for Steady Laminar Flow Applications of momentum principle Flow of falling film 1. Define domain boundary 2. Choose a coordinate system X Z y Shell Momentum Balance for Steady Laminar Flow

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2 3. Make assumptions a. Film thickness is uniform b. Fluid is incompressible and Newtonian b. Pressure is uniform in y and z direction, and may vary with x c. One dimensional flow with z as the main flow direction: d. Velocity is uniform in y and z direction: /0 / zz uy uz ∂= =∂ ∂ 0 x y uu = = z u Shell Momentum Balance for Steady Laminar Flow 4. List non-zero components ( , , ) z zg x g y g z u uF F F x g F Shell Momentum Balance for Steady Laminar Flow
3 Shell Momentum Balance for Steady Laminar Flow 5. Write a balance for z-momentum • For z-momentum transferred across z-plane at z = 0 (influxes) () ( )() ( ) 0 0 0 0 0 mol in z zz z z z z con in z z z u M Wx p pWx z MW x u u τµ ρ = = = = = ⎛⎞ =∆ + = ⎜⎟ ⎝⎠ and at z = L (outfluxes) 0 0 mol mol out in z Lz con con out in z MM == = = Net z-momentum transferred through the control volume in z-direction ( ) ( ) 00 0 total con con mol mol net in out in out z z z z L L M M M =− +− = Shell Momentum Balance for Steady Laminar Flow ( ) 0 mol con in in z zx z z x z x x x x x du L W L L u u dx = = and at x = x+ x (outfluxes) • For z-momentum transferred across x-plane at x = x (influxes) 0 mol con out out z z z xx L W L M +∆ =

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Shell Momentum Balance - Shell Momentum Balance for Steady...

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