Fluid Mechanics: Flow Rate in Pipes

• Fluid Science III uses Fluid Mechanics: Fundamentals and Applications by Yunus A. Çengel and John M. Cimbala, 4th Edition, 2018 as its prescribed textbook.
• Mr. L Masheane can be contacted at [email protected] or 051 507 3683 and is located in BHP 152.

Objectives

• After completing this learning unit, students will be able to:
• Determine the flow rate in a pipe for Laminar and Turbulent Flow.
• Understand dimensionless Reynolds number and its significance.
• Have a deeper understanding of Entrance and Fully developed region.
• Recommended chapter for study is Book A: Study Chapter 8, specifically problems 8.32, 8.35, 8.36, 8.38 on page 425 of Source A.

Introduction

• Liquid or gas flow through pipes or ducts is commonly used in heating and cooling applications and fluid distribution networks.
• The fluid is usually forced to flow by a fan or pump through a flow section.
• Particular attention should be paid to friction, which is directly related to the pressure drop and head loss during flow through pipes and ducts.
• The pressure drop is then used to determine the pumping power requirement.
• Analyzing flow through various ducts and vessels in chemical processing plants depends on the conservation of mass.

Conservation of Mass

• The value of the average velocity Vavg at some streamwise cross-section is determined from the requirement that the conservation of mass principle be satisfied.
• An equation representing this is ṁ = ρVavg Ac = ∫Ac ρu(r) dAc.
• The average velocity for incompressible flow in a circular pipe of radius R can be calculated using Vavg = (∫Ac ρu(r)dAc) / (ρAc) = (∫0R ρu(r)2πr dr) / (ρπR2) = (2 / R^2) ∫0R u(r)r dr.
• For inviscid or ideal fluids, the velocity distribution is uniform.
• For ideal fluids, the volume under the velocity profile is cylindrical.
• In other fluids, the velocity profile for the cross section is a paraboloid.
• Volumetric flow is represented by the volume under the velocity profile.
• Average velocity Vavg is defined as the average speed through a cross section.
• For fully developed laminar pipe flow, Vavg is half of the maximum velocity.

Laminar and Turbulent Flows

• Fluid flow is streamlined at low velocities but turns chaotic as the velocity is increased above a critical value.
• In Laminar flow, fluid particles follow straight-line paths since fluid flows in thin layers.
• In Turbulent flow, fluid particles follow erratic paths which change direction in space and time.
• When a candle is extinguished, the smoke initially rises vertically from the wick with laminar flow before bending in the transition and curling in the turbulent phase.

Reynold's Experiment

• British Engineer Osborne Reynolds (1842-1912) devised an experiment to demonstrate laminar and turbulent flow differences.
• Reynolds's experiment apparatus consisted of:
• A constant head tank filled with water
• A small tank containing dye
• A horizontal glass tube provided with a bell mouthed entrance
• A regulating valve
• Red-dyed liquid flows vertically downward and then rightward into inlet A, then flows through a horizontal pipe to outlet B.
• In laminar flow, the red-dyed liquid appears as a solid band along the centerline of the pipe.
• In transitional flow, the red-dyed liquid forms bent and broken ribbons along the centerline.
• In turbulent flow, the red-dyed liquid initially forms a thin band along the centerline, before spreading radially and dissipating.

Reynolds Number

• The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid.
• The flow regime primarily depends on the ratio of inertial forces to viscous forces (Reynolds number).
• The Reynolds number can be calculated using Re = (Inertial forces) / (Viscous forces) = (Vavg D) / v = (ρVavg D) / μ.
• At large Reynolds numbers, the inertial forces are large relative to the viscous forces, resulting in random and rapid fluid fluctuations (turbulent).
• At small or moderate Reynolds numbers, the viscous forces are large enough to suppress these fluctuations and keep the fluid “in line” (laminar).
• The Reynolds number at which the flow becomes turbulent is called the Critical Reynolds number, Recr.
• The value of the critical Reynolds number is different for different geometries and flow conditions.
• The Reynolds number can be viewed as the ratio of inertial forces to viscous forces acting on a fluid element.
• For flow through noncircular pipes, the Reynolds number is based on the hydraulic diameter.
• Hydraulic diameter is calculated using Dh = (4Ac) / p.
• For circular pipes, Dh = (4Ac) / p = (4(πD²/4)) / (πD) = D.
• The hydraulic diameter Dh = 4Ac/p is defined such that it reduces to ordinary diameter for circular tubes.
• For flow in a circular pipe:
• Re < 2300 indicates laminar flow
• 2300 < Re < 4000 indicates transitional flow
• Re > 4000 indicates turbulent flow
• In the transitional flow region of 2300 ≤ Re ≤ 4000, the flow switches between laminar and turbulent seemingly randomly.

The Entrance Region

• Velocity boundary layer refers to the region of the flow in which the effects of the viscous shearing forces caused by fluid viscosity are felt.
• Boundary layer region is where the viscous effects and the velocity changes are significant.
• Irrotational (core) flow region denotes where the frictional effects are negligible, and the velocity remains essentially constant in the radial direction.
• Hydrodynamic entrance region refers to the region from the pipe inlet to the point at which the boundary layer merges at the centerline.
• The length of the hydrodynamic entrance region is the hydrodynamic entry length Lh.
• Hydrodynamically developing flow is flow in the entrance region, where the velocity profile develops.
• Hydrodynamically fully developed region is the region beyond the entrance region in which the velocity profile is fully developed and remains unchanged.
• Flow is fully developed when both the velocity profile and the normalized temperature profile remain unchanged.
• In the fully developed flow region of a pipe, the velocity profile does not change downstream, and thus the wall shear stress remains constant as well.
• The pressure drop is higher in the entrance regions of a pipe.
• The effect of the entrance region is always to increase the average friction factor for the entire pipe.
• The variation of wall shear stress in the flow direction for flow in a pipe from the entrance region into the fully developed region can be charted.

Entrance Lengths

• The hydrodynamic entry length is usually taken to be the distance from the pipe entrance to where the wall shear stress (and thus the friction factor) reaches within about 2 percent of the fully developed value.
• Hydrodynamic entry length for laminar flow: (Lh, laminar) / D ≈ 0.05Re
• Hydrodynamic entry length for turbulent flow: (Lh, turbulent) / D = 1.359Re^(1/4)
• Approximation using hydrodynamic entry length for turbulent flow: (Lh, turbulent) / D ≈ 10
• The pipes used in practice are usually several times the length of the entrance region.
• Flow through the pipes is often assumed to be fully developed for the entire length of the pipe.
• This simplistic approach gives reasonable results for long pipes.
• It may give poor results for short ones since it underpredicts the wall shear stress and thus the friction factor.