Mechanical Energy Balance CHE315 Mechanical Energy Balance

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

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

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

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

Mechanical Energy Balance So: CHE315 Mechanical Energy Balance

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

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

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

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

Mechanical Energy Balance CHE315 Mechanical Energy Balance

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