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

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

CBE 150A – Transport Spring Semester 2014 Balances Momentum Balance

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

CBE 150A – Transport Spring Semester 2014 Mechanical Energy Balance

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

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

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

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

CBE 150A – 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 150A – 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 150A – 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. Note, roughness is not dimensionless. Here, the roughness is reported in inches. Material Roughness, k inches Cast Iron0.01 Galvanized Steel0.006 Commercial Steel Wrought Iron Drawn Tubing

CBE 150A – Transport Spring Semester 2014 How Does k/D Affect f (Text Figure 13.1)

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

CBE 150A – 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 150A – 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 D eq with which to calculate the Reynolds number and the friction factor. where: R H = hydraulic radius S= cross-sectional area L p = wetted perimeter Note: Do not use Deq to calculate cross-sectional area or for laminar flow situations.

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

CBE 150A – 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. 400 ft. 5 in. 600 GPM

CBE 150A – Transport Spring Semester 2014 Text Appendix M

CBE 150A – Transport Spring Semester 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. Tubing lengths: Lake to pump suction – 50 ft. Pump to distribution line – 150 ft.