Friction Losses Flow through Conduits Incompressible Flow CBE

Friction Losses Flow through Conduits Incompressible Flow CBE 150 A – Transport Spring Semester 2014

Goals • Calculate frictional losses for laminar and turbulent flow through circular and non-circular pipes • Define the friction factor in terms of flow properties • Calculate the friction factor for laminar and turbulent flow • Define and calculate the Reynolds number for different flow situations • Derive the Hagen-Poiseuille equation CBE 150 A – Transport Spring Semester 2014

Flow Through Circular Conduits Consider the steady flow of a fluid of constant density in fully developed flow through a horizontal pipe and visualize a disk of fluid of radius r and length d. L moving as a free body. Since the fluid posses a viscosity, a shear force opposing the flow will exist at the edge of the disk CBE 150 A – Transport Spring Semester 2014

Balances Mass Balance → CBE 150 A – Transport Spring Semester 2014

Balances Momentum Balance CBE 150 A – Transport Spring Semester 2014

Momentum Balance (contd) If we imagine that the fluid disk extends to the wall, Fw is just due to the shear stress τw acting over the length of the disk. Equating and solving for p over a length of pipe L. CBE 150 A – Transport Spring Semester 2014

Mechanical Energy Balance CBE 150 A – Transport Spring Semester 2014

Viscous Dissipation (Frictional Loss) Equation Combining the Momentum and MEB results: • Applies to laminar or turbulent flow • Good for Newtonian or Non-Newtonian fluids • Only good for friction losses as result of wall shear. Not proper for fittings, expansions, etc. CBE 150 A – Transport Spring Semester 2014

The Friction Factor tw is not conveniently determined so the dimensionless friction factor is introduced into the equations. CBE 150 A – Transport Spring Semester 2014

Fanning Friction Factor • • • Increases with length Decreases with diameter – Only need L, D, V and f to get friction loss Valid for both laminar and turbulent flow Valid for Newtonian and Non-Newtonian fluids CBE 150 A – Transport Spring Semester 2014

Calculation of f for Laminar Flow First we need the velocity profile for laminar flow in a pipe. We’ll rely on Chapter 8 for that result. Recall our earlier result: CBE 150 A – Transport Spring Semester 2014

Laminar Flow Find Bulk Velocity (measurable quantity). CBE 150 A – Transport Spring Semester 2014

Reynolds Number Osbourne Reynolds (1842 -1912) CBE 150 A – Transport Spring Semester 2014

Laminar Flow ←Newtonian Fluid CBE 150 A – Transport Spring Semester 2014

Hagen-Poiseuille (Laminar Flow) Recall again: Use: Measurement of viscosity by measuring p and q through a tube of known D and L for Laminar flow. CBE 150 A – Transport Spring Semester 2014

Turbulent Flow When flow is turbulent, the viscous dissipation effects cannot be derived explicitly as in laminar flow, but the following relation is still valid. The problem is that we can not write a closed form solution for the friction factor f. Must use correlations based on experimental data. CBE 150 A – Transport Spring Semester 2014

Friction Factor Turbulent Flow For turbulent flow f = f( Re , k/D ) where k is the roughness of the pipe wall. Material Roughness, k inches Cast Iron 0. 01 Galvanized Steel 0. 006 Commercial Steel Wrought Iron 0. 0018 Drawn Tubing 0. 00006 Note, roughness is not dimensionless. Here, the roughness is reported in inches. CBE 150 A – Transport Spring Semester 2014

How Does k/D Affect f (Text Figure 13. 1) CBE 150 A – Transport Spring Semester 2014

Friction Factor Turbulent Flow As and alternative to Moody Chart use Churchill’s correlation: CBE 150 A – Transport Spring Semester 2014

Friction Factor Turbulent Flow A less accurate but sometimes useful correlation for estimates is the Colebrook equation. It is independent of velocity or flow rate, instead depending on a combined dimensionless quantity CBE 150 A – Transport Spring Semester 2014

Flow Through Non-Circular Conduits Rather than resort to deriving new correlations for the friction factor, an approximation is developed for an ‘equivalent’ diameter Deq with which to calculate the Reynolds number and the friction factor. where: • RH = hydraulic radius • S = cross-sectional area • Lp = wetted perimeter Note: Do not use Deq to calculate cross-sectional area or for laminar flow situations. CBE 150 A – Transport Spring Semester 2014

Examples Circular Pipe Rectangular Ducts CBE 150 A – Transport Spring Semester 2014

Example 1 Water flows horizontally at a rate of 600 gal/min through 400 feet of 5 in. diameter Schedule 40 cast-iron pipe. Find the average (bulk) velocity and the pressure drop. 600 GPM 5 in. 400 ft. CBE 150 A – Transport Spring Semester 2014

Text Appendix M CBE 150 A – Transport Spring Semester 2014

10 Minute Problem My father is installing a sprinkler system at his lake house. The pump pulls water from the lake through a feed line and delivers 12 GPM to the sprinkler system distribution line at a point in the front yard. For the sprinkler system to operate properly, the pressure at the branch point must be 90 psig. What horsepower pump does he buy ? 40 ft. 25 ft. 10 ft. CBE 150 A – Transport Tubing lengths: Lake to pump suction – 50 ft. Pump to distribution line – 150 ft. Spring Semester 2014