Lecture 12 F. Morrison CM3110 10/27/2019 CM3110 Transport I Part I: Fluid Mechanics More Complicated Flows II: External Flow (or applying fluid-mechanics problemsolving to a new category of flows) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 © Faith A. Morrison, Michigan Tech U. More complicated flows II: From Nice to Powerful r Nice: A z cross-section A: r z Learning to solve one particular problem (or a group of related problems) L vz(r) fluid R Powerful: Solving never-before-solved problems. 2 © Faith A. Morrison, Michigan Tech U. 1 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II A real flow problem (external). What is the speed of a sky diver? (Morrison, Example 8.1) 3 © Faith A. Morrison, Michigan Tech U. More complicated flows II A real flow problem (external). What is the speed of a sky diver? Can we use anything we learned from flow-throughconduits to tell us how to solve this new problem? (Morrison, Example 8.1) 4 © Faith A. Morrison, Michigan Tech U. 2 Lecture 12 F. Morrison CM3110 10/27/2019 Flow through Conduits More complicated flows II So far we have talked about internal flows •ideal flows (Poiseuille flow in a tube) •real flows (turbulent flow in a tube) Strategy for handling real flows: Dimensional analysis and data correlations, π Re How did we arrive at correlations? non-Dimensionalize ideal flow; use to guide expts on similar non-ideal flows; take data; develop empirical correlations from data What do we do with the correlations? use in MEB; calculate pressure-drop flow-rate relations 5 © Faith A. Morrison, Michigan Tech U. F. A. Morrison, An Introduction to Fluid Mechanics, Cambridge University Press 2013. 6 © Faith A. Morrison, Michigan Tech U. 3 Lecture 12 F. Morrison CM3110 10/27/2019 Solve a wide variety of practical problems with data correlations. Pumping loops, piping networks, pump characteristic curves, non-circular cross sections, flow through porous media, etc. F. A. Morrison, An Introduction to Fluid Mechanics, Cambridge University Press 2013. 7 © Faith A. Morrison, Michigan Tech U. More complicated flows II A real flow problem (external). What is the speed of a sky diver? We can apply the conduit process to this new problem. (Morrison, Example 8.1) 8 © Faith A. Morrison, Michigan Tech U. 4 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II A real flow problem (external). What is the speed of a sky diver? Drag (fluid force) Gravity Apply the physics: ππ π (Morrison, Example 8.1) 9 © Faith A. Morrison, Michigan Tech U. More complicated flows II A real flow problem (external). What is the speed of a sky diver? ππ π At terminal velocity π Drag 0 0 ππ πΉ (fluid force) πΉ Gravity ππ The fluids part of the problem is, what is πΉ ? (drag in flow around an obstacle) (Morrison, Example 8.1) 10 © Faith A. Morrison, Michigan Tech U. 5 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II A real flow problem (external). What is the speed of a sky diver? Can we address this new problem (new to us) . . . At terminal velocity π Drag 0 0 ππ πΉ (fluid force) πΉ Gravity ππ The fluids part of the problem is, what is πΉ ? (drag in flow around an obstacle) (Morrison, Example 8.1) More complicated flows II Flow through Conduits So far we have talked about internal flows •ideal flows (Poiseuille flow in a tube) •real flows (turbulent flow in a tube) Strategy for handling real flows: Dimensional analysis and data correlations, π Re How did we arrive at correlations? non-Dimensionalize ideal flow; use to guide expts on similar non-ideal flows; take data; develop empirical correlations from data By using the same approach as used for pipe flow? What do we do with the correlations? use in MEB; calculate pressure-drop flow-rate relations 11 © Faith A. Morrison, Michigan Tech U. Flow around Obstacles More complicated flows II Now, we will talk about external flows •ideal flows (flow around a sphere) •real flows (turbulent flow around a sphere, sky diver, other obstacles) Strategy for handling real flows: Dimensional analysis and data correlations How did we arrive at correlations? non-Dimensionalize ideal flow; use to guide expts on similar non-ideal flows; take data; develop empirical correlations from data What do we do with the correlations? calculate drag – free-stream velocity relations 12 © Faith A. Morrison, Michigan Tech U. 6 Lecture 12 F. Morrison CM3110 10/27/2019 Flow around Obstacles More complicated flows II Now, we will talk about external flows •ideal flows (flow around a sphere) •real flows (turbulent flow around a sphere, sky diver, other obstacles) Let’s try Strategy for handling real flows: Dimensional analysis and data correlations How did we arrive at correlations? non-Dimensionalize ideal flow; use to guide expts on similar non-ideal flows; take data; develop empirical correlations from data What do we do with the correlations? calculate drag – free-stream velocity relations 13 © Faith A. Morrison, Michigan Tech U. What is the steady state velocity field around a sphere dropping through an incompressible Newtonian fluid π 14 © Faith A. Morrison, Michigan Tech U. 7 Lecture 12 F. Morrison CM3110 10/27/2019 (equivalent to sphere falling through a liquid) Steady flow of an incompressible, Newtonian fluid around a sphere What is the steady state velocity field when an incompressible, Newtonian fluid flows around a stationary sphere? What is the drag on the sphere? The upstream velocity is π£ . π (Morrison, Example 8.2) flow π 15 © Faith A. Morrison, Michigan Tech U. (equivalent to sphere falling through a liquid) Steady flow of an incompressible, Newtonian fluid around a sphere π§ π, π, π ο±ο π¦ •spherical coordinates •symmetry in the π direction •calculate π£ and drag on sphere •upstream π£ π£ π π£ flow π π£ π£ π£ (Morrison, Example 8.2) 16 © Faith A. Morrison, Michigan Tech U. 8 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) π§ π, π, π ο±ο π¦ flow 17 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) Microscopic Balances (Microscopic balances on an arbitrary control volume) S dS Continuity Equation (mass) nΜ V οΆο² ο« v ο οο² ο½ ο ο² ο¨ο ο v ο© οΆt Equation of Motion (momentum) Newtonian οΆ ο¦ οΆv ο« v ο ο v ο· ο½ οοP ο« οο 2 v ο« ο² g fluid οΈ ο¨ οΆt ο²ο§ Navier-Stokes Equation 18 © Faith A. Morrison, Michigan Tech U. 9 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) Continuity (spherical coordinates) 19 www.chem.mtu.edu/~fmorriso/cm310/Navier.pdf © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) Navier-Stokes (spherical coordinates) 20 www.chem.mtu.edu/~fmorriso/cm310/Navier.pdf © Faith A. Morrison, Michigan Tech U. 10 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) Gravity π π§ π, π, π ο±ο πΜ sin ππΜ (See inside back cover; do some algebra) π¦ π π cos ππΜ ππΜ πcos ππΜ πsin ππΜ πcos π πsin π 0 (Morrison, Example 8.2) flow π 21 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) Because the flow is not unidirectional, we have to consider the left-hand-side of the Navier-Stokes Steady flow of an incompressible, Newtonian fluid around a sphere Equation of Motion: ο¦ οΆv οΆ ο« v ο οv ο· ο½ οοP ο« οο 2 v ο« ο² g ο¨ οΆt οΈ ο²ο§ (do we have to?) 22 © Faith A. Morrison, Michigan Tech U. 11 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) π£ π£ π£ π£ Eqn of Continuity: Eqn of Motion: πcos π πsin π 0 π π π π, π ο¨ ο© Steady flow of an incompressible, Newtonian fluid around a sphere ο¦ 1 οΆ r 2vr 1 οΆvο± sin ο± οΆ ο§ 2 ο·ο½0 ο§ r οΆr ο« r sin ο± οΆο± ο·οΈ ο¨ Creeping Flow ο¦ οΆv οΆ ο« v ο οv ο· ο½ οοP ο« οο 2 v ο« ο² g ο¨ οΆt οΈ ο²ο§ steady state SOLVE neglect inertia BC1: no slip at sphere surface BC2: velocity goes to π£ far from sphere 23 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) SOLUTION: Creeping Flow οΆ ο¦ ο© 3 R 1 ο¦ R οΆ3οΉ ο· ο§ v οͺ1 ο cos ο± ο« οΊ ο§ ο· ο· ο§ ο₯οͺ 2 r 2 ο¨ r οΈ οΊ ο» ο· ο§ ο« ο© 3 R 1 ο¦ R οΆ3οΉ ο· ο§ v ο½ ο§ ο vο₯οͺ1 ο ο ο§ ο· οΊ sin ο± ο· οͺο« 4 r 4 ο¨ r οΈ οΊο» ο· ο§ 0 ο· ο§ ο· ο§ 0 ο· ο§ οΈrο±ο¦ ο¨ 2 3 οvο₯ ο¦ R οΆ P ο½ P0 ο ο²gr cosο± ο ο§ ο· cosο± 2 R ο¨rοΈ Morrison, Example 8.2; complete solution steps in Denn, Process Fluid Mechanics (Prentice Hall, 1980) around a sphere (we neglected inertia, i.e. LHS of Navier-Stokes; this has consequences) all the stresses can be calculated from π£ πΜ π π»π£ Π π»π£ ππΌΜ³ πΜ (see the usual handout for stress components) 24 © Faith A. Morrison, Michigan Tech U. 12 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) SOLUTION: Velocity Field for Creeping Flow around a sphere • No slip at the walls of the sphere • Far from the sphere the flow does not feel the presence of the sphere (we neglected inertia, i.e. LHS of Navier-Stokes; this has consequences) 25 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) To get the force on the sphere (drag), we ask, What is the total z-direction force on the sphere? π§ π, π, π ο±ο π¦ πΉ π⋅Π ππ flow 26 © Faith A. Morrison, Michigan Tech U. 13 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) To get the force on the sphere, we ask, π§ π, π, π ο±ο Let’s try What is the total z-direction force on the sphere? π¦ πΉ π⋅Π ππ flow 27 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) πΉ π⋅Π ππ π§ π, π, π ο±ο π¦ flow 28 © Faith A. Morrison, Michigan Tech U. 14 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) πΉ π⋅Π ππ π§ π, π, π ο±ο π¦ π ? surface ππ Π flow ? ? ? π⋅Π ? 29 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) SOLUTION: Creeping Flow around a sphere What is the total z-direction force on the sphere? integrate over the entire sphere surface vector stress on a microscopic surface of unit normal πΜ total vector force on sphere 2ο° ο° Fο½ total z-direction force on the sphere ππ ⋅ π ο² ο² ο©ο«eˆ ο ο¨ο΄ Μ ο PI ο©οΉο» r 0 0 total stress at a point in the fluid r ο½R R 2 sinο± dο± dοͺ evaluate at the surface of the sphere 30 © Faith A. Morrison, Michigan Tech U. 15 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) πΉ π⋅Π ππ π§ π, π, π ο±ο π¦ flow 31 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) π§ π, π, π ο±ο π¦ πΉ π⋅Π ππ π π , π sin π flow πΉ π 3ππ£ sin π 2π ππππ 0 32 © Faith A. Morrison, Michigan Tech U. 16 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) π§ π, π, π ο±ο π¦ πΉ π⋅Π ππ π π , π sin π flow πΉ 3ππ£ sin π 2π π ππππ 0 Note: π , π , π are a function of π, π (see text page 614) 33 © Faith A. Morrison, Michigan Tech U. Slow flow around a sphere (Stokes Flow) Force on a sphere (creeping flow limit) comes from pressure comes from shear stresses form drag eˆz ο F ο½ Fz ο½ 4 ο° R 3 ο² g ο« 2ο°ο Rv ο₯ ο« 4ο°ο Rv ο₯ 3 buoyant force stationary terms 0 when π£ 0 (this is famous) See Morrison, p613-17 friction drag kinetic terms Stokes law: kinetic force ≡ πΉ 6πππ π£ 34 © Faith A. Morrison, Michigan Tech U. 17 Lecture 12 F. Morrison CM3110 10/27/2019 Slow flow around a sphere (Stokes Flow) Force on a sphere (creeping flow limit) comes from pressure comes from shear stresses form drag eˆz ο F ο½ Fz ο½ 4 ο° R 3 ο² g ο« 2ο°ο Rv ο₯ ο« 4ο°ο Rv ο₯ 3 friction drag buoyant force kinetic terms stationary terms 0 when π£ 0 Stokes law: kinetic force ≡ πΉ (this is famous) 6πππ π£ 35 See Morrison, p613-17 © Faith A. Morrison, Michigan Tech U. More complicated flows II A real flow problem (external). What is the speed of a sky diver? Back to our problem: Drag (fluid force) π§ π₯ Gravity ππ π (Morrison, Example 8.1) 36 © Faith A. Morrison, Michigan Tech U. 18 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II A real flow problem (external). What is the speed of a sky diver? ππ 0 π ππ πΉ Drag (fluid force) π§ πΉ We can get this from the creeping flow solution (Stokes flow) π₯ Gravity ππ Creeping flow drag: πΜ ⋅ πΉ 4 ππ ππ 3 πΉ 6πππ π£ 37 (Morrison, Example 8.1) © Faith A. Morrison, Michigan Tech U. More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? ππ 0 0 0 4ππ π 3 π π ππ 4ππ ππ 3 π£ π π₯ πΉ 0 0 6ππ ππ£ π π· π 18π 0 0 0 From the creeping flow solution (see inside front cover) (Morrison, Example 8.1) π§ Gravity 38 © Faith A. Morrison, Michigan Tech U. 19 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? Gravity π£ π π π· π 18π π§ π₯ π£ 14,000ππβ From the creeping flow solution (Stokes flow, no inertia) 39 (Morrison, Example 8.1) © Faith A. Morrison, Michigan Tech U. More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? Gravity π£ π π π· π 18π From the creeping flow solution (Stokes flow, no inertia) (Morrison, Example 8.1) π§ π₯ π£ 14,000ππβ (wrong) (oh well, nice try) 40 © Faith A. Morrison, Michigan Tech U. 20 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? Gravity π π£ π§ π₯ π π· π 18π π£ 14,000ππβ (wrong) From the creeping flow solution But, wait! . . . 41 (Morrison, Example 8.1) © Faith A. Morrison, Michigan Tech U. More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? Gravity ππ 0 ππ Now, we will talk about external flows •ideal flows (flow around a sphere) •real flows (turbulent flow around a sphere, other obstacles) Strategy for handling real flows: Dimensional analysis and data correlations How did we arrive at correlations? non-Dimensionalize ideal flow; use to guide expts on similar non-ideal flows; take data; develop empirical correlations from data What do we do with the correlations? calculate drag - superficial velocity relations (Morrison, Example 8.1) π₯ π Flow around Obstacles More complicated flows II π§ πΉ How about if we do dimensional analysis, measure data correlations for non-creeping flow, and then use the correlations to determine πΉ ? 42 © Faith A. Morrison, Michigan Tech U. 21 Lecture 12 F. Morrison CM3110 10/27/2019 Fast flow around a sphere (dimensional analysis) •Nondimensionalize eqns of change: ο¦ οΆv* * * * οΆ 1 *2 * 1 * ο§ * ο« v ο ο v ο· ο½ οο*P* ο« ο v ο« g ο§ οΆt ο· Re Fr ο¨ οΈ •Nondimensionalize eqn for πΉ , define dimensionless kinetic force f ο½ C D •conclude f=f(Re) or CD=CD(Re) Steady flow of an incompressible, Newtonian fluid around a sphere Turbulent Flow : ο½ drag coefficient Fz,kinetic ο°D2 ο¦ 1 2 οΆ ο§ ο²vο₯ ο· 4 ο¨2 οΈ •take data, plot, develop correlations 43 © Faith A. Morrison, Michigan Tech U. Fast flow around a sphere (dimensional analysis) Steady flow of an incompressible, Newtonian fluid around a sphere What does this look like in Creeping flow? (we have the solution) Creeping flow: π πΆ πΉ, Creeping Flow Stokes law 6πππ π£ ππ· 1 ππ£ 4 2 24 π π From the creeping flow solution 44 © Faith A. Morrison, Michigan Tech U. 22 Lecture 12 F. Morrison CM3110 10/27/2019 Fast flow around a sphere (dimensional analysis) Steady flow of an incompressible, Newtonian fluid around a sphere How do we apply to Turbulent flow? (we need to take data) Turbulent flow: Turbulent Flow Calculate πΆ from terminal velocity of a falling sphere (see Figure 8.13) At terminal speed the net weight is exactly balanced by the viscous retarding force. πΉ π π πΆ πΆ π π all measurable quantities π 4 π·π 3 π£ (see Example 8.4) 45 © Faith A. Morrison, Michigan Tech U. Fast flow around a sphere (dimensional analysis) •take data, plot, develop correlations Steady flow of an incompressible, Newtonian fluid around a sphere Creeping Flow Steady flow of an incompressible, Newtonian fluid around a sphere Turbulent Flow Data of πΆ Re ? 46 © Faith A. Morrison, Michigan Tech U. 23 Lecture 12 F. Morrison CM3110 10/27/2019 Fast flow around a sphere (dimensional analysis) Steady flow of an incompressible, Newtonian fluid around a sphere graphical correlation 24 Re 47 McCabe et al., Unit Ops of Chem Eng, 5th edition, p147 © Faith A. Morrison, Michigan Tech U. Fast flow around a sphere (dimensional analysis) Steady flow of an incompressible, Newtonian fluid around a sphere graphical correlation 24 Re We see the creeping-flow result for Re 2 McCabe et al., Unit Ops of Chem Eng, 5th edition, p147 48 © Faith A. Morrison, Michigan Tech U. 24 Lecture 12 F. Morrison CM3110 10/27/2019 Fast flow around a sphere (dimensional analysis) correlation equations creeping vortices wake flow Steady flow of an incompressible, Newtonian fluid around a sphere 24 π π 18.5π π π 0.44 π π π π . 2 500 0.10 π π π π 500 200,000 BSL, p194 •use correlations in engineering practice •particle settling (See Denn, Bird et al., Perry’s) •entrained droplets in distillation columns •particle separators •drop coalescence Dr. Morrison developed a single, combined correlation Denn, Process Fluid Mechanics, 1980 Bird, Stewart, Lightfoot, Transport Phenomena, 1960 and 2006 49 © Faith A. Morrison, Michigan Tech U. Fast flow around a sphere (dimensional analysis) Morrison Correlation F. A. Morrison An Introduction to Fluid Mechanics, Cambridge 2013, Figure 8.13, p625 50 © Faith A. Morrison, Michigan Tech U. 25 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II Drag (fluid force) A real flow problem (external). What is the speed of a sky diver? (Instead of creeping flow, use πͺπ« Re for noncreeping flow) 0 0 ππ ππ 0 π π ππ πππ π₯ πΉ 0 0 ππ£ π΄ πΆ 2 4 π π£ π§ Gravity π π·π 3ππΆ 0 0 0 Applicable in NON-creeping flow 51 (Morrison, Example 8.5) © Faith A. Morrison, Michigan Tech U. More complicated flows II Drag A real flow problem (external). What is the speed of a sky diver? (fluid force) Gravity π£ 4 π π§ π π·π π₯ 3ππΆ π£ 107ππβ Right! (or close, anyway) (Morrison, Example 8.5) 52 © Faith A. Morrison, Michigan Tech U. 26 Lecture 12 F. Morrison CM3110 10/27/2019 More complicated flows II Powerful: Solving never-before-solved problems. Left to explore: • What is non-creeping flow like? (boundary layers) • Viscosity dominates in creeping flow, what about the flow where inertia dominates? (potential flow) • What about mixed flows (viscous+inertial)? (boundary layers) • What about really complex flows (curly)? (vorticity; irrotational+circulation) 53 © Faith A. Morrison, Michigan Tech U. Fast flow around a sphere (dimensional analysis) F. A. Morrison An Introduction to Fluid Mechanics, Cambridge 2013, Figure 8.23, p650 54 © Faith A. Morrison, Michigan Tech U. 27 Lecture 12 F. Morrison CM3110 10/27/2019 Fast flow around a sphere (dimensional analysis) a) c) b) d) e) 55 © Faith A. Morrison, Michigan Tech U. We have learned somethings that are very powerful. More complicated flows II: From Nice to Powerful r Nice: A z cross-section A: r z Learning to solve one particular problem (or a group of related problems) L vz(r) 1. External flows⇒ use drag coefficient for real external flows fluid R Powerful: Solving never-before-solved problems. 2. In general, “simple” problems can lead to solutions to “complex” problems through dimensional analysis (and data correlation) End of Exam 3 topics; see Homeworks to apply 56 © Faith A. Morrison, Michigan Tech U. 28 Lecture 12 F. Morrison CM3110 10/27/2019 One more essential topic in Fluid Mechanics: Boundary Layers 57 © Faith A. Morrison, Michigan Tech U. 29
