Incompressible Flow inPipes and Channels By Farhan Ahmad [email protected] Department of Chemical Engineering, University of Engineering & Technology Lahore Significance Industrial processes - flow of fluids through pipes, conduits, and processing equipment. Circular cross-section Non-circular cross-section Flow of fluids in 2 Totally or partially filled pipes, Layers down vertically inclined surfaces, Through beds of solids, and Agitated vessels. 3 5 6 Flow of Incompressible Fluids in Pipe – Shear-Stress Distribution Consider the steady flow of a viscous fluid at constant density in fully developed flow through a horizontal tube. Visualize a disk-shaped element of fluid, concentric with the axis of the tube. 7 Flow of Incompressible Fluids in Pipe – Shear-Stress Distribution 8 At wall After subtraction Relation between τ and r At r =0 , τ = 0 9 Relation between Skin Friction and Wall Shear Pressure Drop Apply the balance 10 Friction Factor ratio of the wall shear stress to the product of the density and the velocity head. 11 Relations between Skin Friction Parameter 12 Flow in Pipe Laminar Turbulent Fluid may be • Newtonian • Non-Newtonian 13 Laminar Flow of Newtonian Fluids Velocity Distribution Average velocity Momentum and Kinetic energy correction factors 14 Laminar Flow of Newtonian Fluids – Velocity Distribution Circular cross-section Local velocity u depends on radius r Consider a thin ring of radius r and width dr According to Newton's law of viscosity 15 Using both equations Integrate with boundary condition u = 0, at r = rw 16 Maximum velocity Maximum velocity is at the center of pipe i.e., at r = 0 Relation of local to maximum velocity 17 Graphical representation 18 Average Velocity 𝑺= 𝝅𝒓𝒘 𝟐 19 Kinetic energy correction factor For Laminar Flow α = 2 20 Momentum correction factor For Laminar Flow β = 4/3 21 Hagen-Poiseuille Equation 22 Laminar Flow for Non-Newtonian Liquids Velocity variations with radius for power law fluids The pressure difference for power law fluids 23 Laminar Flow for Non-Newtonian Liquids 24 Laminar Flow for Non-Newtonian Liquids Bingham-plastic fluids: The general shape of the curve of u versus r in case of Bingham-plastic fluids is; In the central portion - no velocity variation with the radius the velocity gradient is confined to an annular space between the central portion and tube wall. The center portion is moving in plug flow. 25 The velocity distribution is; The shear diagram is; 26 Laminar Flow for Non-Newtonian Liquids Bingham-plastic fluids: For the velocity variation in the annular space between the tube wall and the plug, the following equation applies; and 27 Turbulent Flow in Pipes and Closed Channels Viscous Sublayer Buffer layer Turbulent core 28 Velocity Distribution for Turbulent Flow Newtonian fluid Turbulent flow at Reynolds No 10000 Smooth pipe Velocity gradient is zero at centerline Turbulent core – eddies – large but of low intensity Transition zone – eddies – small but intense Kinetic energy At centerline - isentropic turbulence – anisotropic in turbulence core 29 Velocity Distribution for Turbulent Flow It is customary to express the velocity distribution in turbulent flow not as velocity vs. distance but in terms of dimensionless parameters defined by the following eqns; 30 Velocity Distribution for Turbulent Flow For the velocity distribution in the laminar sublayer; An empirical equation for the so-called buffer layer is; An equation proposed by Prandtl for the turbulent core is; 31 Velocity Distribution for Turbulent Flow 32 Flow Quantities for Turbulent Flow Average Velocity: 33 Flow Quantities for Turbulent Flow Reynolds Number – Friction Factor Law for Smooth Pipe: Von Karman equation 34 Flow Quantities for Turbulent Flow Kinetic Energy and Momentum Correction Factors: For turbulent flow f is of the order of 0.004, and for this value 35 and flow. both are assumed to be unity in case of turbulent Flow Quantities for Turbulent Flow Relation between Maximum velocity and Average Velocity: 36 Flow Quantities for Turbulent Flow Effect of Roughness: In turbulent flow, a rough pipe leads to a larger friction factor for a given Reynolds number than a smooth pipe does. If a rough pipe is smoothed, the friction factor is reduced. When further smoothing brings about no further reduction in friction factor for a given Reynolds number, the tube is said to be hydraulically smooth. 37 Flow Quantities for Turbulent Flow Effect of Roughness: Roughness parameter k f is a function of both NRe and the relative roughness k/D, where D is the diameter of the pipe. 38 Flow Quantities for Turbulent Flow Effect of Roughness: All clean, new commercial pipes seem to have the same type of roughness. Each material of construction has its own characteristic roughness parameter. Old, foul and corroded pipe can be very rough, and the character of the roughness differs from that of clean pipe. Roughness has no appreciable effect on the friction factor for laminar flow unless k is so large that the measurement of the 39 diameter becomes uncertain. Friction Factor Chart 40 Friction Factor Chart For Laminar flow – straight line with slope -1 For turbulent flow the lowest line represents the friction factor for smooth tubes. A much more convenient empirical equation for this line is the relation; Over a range of Reynolds number from about 50,000 to 1 × 106 Over a range of Reynolds number from about 3000 to 3 × 106 41 Friction Factor Chart For Power Law Fluids Comparing the above two equations 42 Friction Factor Chart 43 Drag Reduction in Turbulent Flow 44 Effect of Heat transfer / Non-isothermal flow When the fluid is either heated or cooled by a conduit wall hotter or colder than the fluid, the velocity gradient is changed. The effect on the velocity gradients is especially pronounced with liquids where viscosity is a strong function of temperature. 45 1. The Reynolds number is calculated on the assumption that the fluid temperature equals the mean bulk temperature, which is defined as the arithmetic average of the inlet and outlet temperatures. 2. The friction factor corresponding to the mean bulk temperature is divided by a factor Effect of Heat transfer / Non-isothermal flow < 46 Flow through Channels of Non-Circular cross-sections Equivalent Diameter: It is four times the hydraulic radius. Hydraulic Radius: It is the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel. 47 Friction Factor in Flow through Channels of NonCircular Cross-Sections 48 Friction Factor in Flow through Channels of NonCircular Cross-Sections For circular cross-section = 1.0 For Parallel planes = 1.5 49 Friction from changes in velocity and direction Change in velocity direction or magnitude Additional resistance to skin friction Boundary layer separation Sudden expansion Sudden contraction Fittings and valves 50 Friction loss from sudden expansion (1) 51 Friction loss from sudden expansion (2) 52 Friction loss from sudden contraction Vena contracta Kc is contraction loss coefficient For laminar flow, this coefficient < 0.1 For turbulent flow 53 Effect of fitting and valves 54 Form friction losses in Bernoulli’s equation 55 Separation from Velocity Decrease 56 Minimizing Contraction Losses 57 Minimizing Expansion Losses 58 Couette Flow 59 Layer Flow with Free Surface 60 Layer Flow with Free Surface 61 Layer Flow with Free Surface 62 Layer Flow with Free Surface 63 Reynolds Number 64