Macroscopic Momentum Balances CBE 150 A Transport Spring

Macroscopic Momentum Balances CBE 150 A – Transport Spring Semester 2014

Momentum Accumulation = In – Out + {Generation – Consumption} Which terms can be eliminated? None: Momentum is not necessarily conserved But why? • It (momentum) can be generated by forces acting on system • It is a vector quantity. Exerted momentum is based on flow direction CBE 150 A – Transport Spring Semester 2014

Momentum So how is momentum transferred to a system? • Bulk Flow • Forces on System • Shear Stress on Walls Fluid flows easier in center of pipe • Friction • Gravity CBE 150 A – Transport Spring Semester 2014

Momentum is a vector quantity Must write balances in all three directions CBE 150 A – Transport Spring Semester 2014

Momentum At Steady State CBE 150 A – Transport Spring Semester 2014

Momentum CBE 150 A – Transport Spring Semester 2014

Bulk Flow Momentum Flow Rate Balance CBE 150 A – Transport Spring Semester 2014

• Why M? Newton’s Law CBE 150 A – Transport Spring Semester 2014

Forces Acting on Systems Surface Forces - Pressure Volume Forces ← Force pushing fluid into system (pressure is a scalar) Gravity ← Written for flow in upward direction Depends on ρ & V Surface Forces Shear Stress ← Written for net force on wall of channel CBE 150 A – Transport Spring Semester 2014

Momentum Balances Summation of Forces (Unidirectional Flow) Steady State Momentum Balance • Upward flow • • M = ρ u 2 S CBE 150 A – Transport Spring Semester 2014

Variable Flow at Cross Section Average velocity is not exactly correct CBE 150 A – Transport Spring Semester 2014

Variable Cross Sectional Velocity When is flow non-uniform? Laminar Flow Parabolic Flow β= When is flow approximately uniform? Turbulent Flow Plug Flow CBE 150 A – Transport β≈ Spring Semester 2014

Example Y X A B The police are using a fire hose to disperse a unruly gathering of Stanford students after a Cal victory. The fire hose delivers 0. 01 m 3/s of water at a velocity of 30 m/s. A member of the crowd has picked up a garbage can lid and is using it as a shield to deflect the flow. She is holding the lid by its edges with the dome facing the hose. In this configuration (A) the water splits with all the flow going at right angles to the incoming stream. Her Phi Beta Kappa boy friend tells here to flip the lid around and hold it by the handle (B) (this will produce a “x” velocity component of water leaving the lid of negative 15 m/s) She obeys her boy friend and quickly ends up on her rear end. What happened ? ? ? CBE 150 A – Transport Spring Semester 2014

Momentum Balance – 10 minute problem A 1 P 1 y v 1 x Gravity A 2 q v 2 P 2 Develop expressions for the forces acting on the fitting in the “x” and “y” directions CBE 150 A – Transport Spring Semester 2014