CM3110 Transport Processes and Unit Operations I Professor Faith Morrison Department of Chemical Engineering Michigan Technological University CM3110 - Momentum and Heat Transport CM310 Heat and Mass Transport www.chem.mtu.edu/~fmorriso/cm310/cm310.html 1 Why study fluid mechanics? 015 1
Why study fluid mechanics? It s required for my degree 3 Why study fluid mechanics? It s required for my degree (too literal) 4 015
Why study fluid mechanics? It s required for my degree (too literal) Fluids are involved in engineered systems Image from: money.cnn.com Image from: newegg.com 5 Why study fluid mechanics? It s required for my degree (too literal) Fluids are involved in engineering systems (many devices that employ fluids can be operated and maintained and sometimes designed without detailed mathematical analysis) 6 015 3
Why study fluid mechanics? Modern engineering systems are complex and often cannot be operated and maintained without analytical understanding Design of new systems will come from high-tech innovation, which can only come from detailed, analytical understanding of how physics/nature works Image: planetforward.ca Image: wikipedia.org 7 Membrane Oxygenator (Heart-Lung Machine) spent blood fresh blood heat exchanger O MO pump membrane oxygenator oxygen goes into blood; carbon dioxide comes out Secondary flow drives the O mass transfer 8 015 4
Artificial Heart Image: health.howstuffworks.com/medicine/modern/artificial-heart.htm 9 Microfluidics Lab on a Chip Sensors, diagnostics www.nature.com/nmeth/journal/v4/n8/full/nmeth0807-665.html 10 015 5
And more.... Helicopters Airplanes Quieter fans Flexible body armor Undersea oil drilling Surgery Food processing Plastics D and 3D printing Battery manufacture Celestial exploration Volcanos Biomedical devices (stents, artificial organs, prosthetics) Sensor development 11 Image from: en.wikipedia.org Where to start? We ve already started. 1 015 6
Where to start? There are problems that can be addressed with a macroscopic energy balance: The Mechanical Energy Balance Assumptions: Δp Δ v Wson, gδzf friction m p p v v W 1 1 son,,1 g(z z 1) F1 m 1. single-input, single output. Steady state 3. Constant density (incompressible fluid) 4. Temperature approximately constant 5. No phase change, no chemical rxn 6. Insignificant amounts of heat transferred 13 For example: Flow in Pipes and Fittings 1 75 ft 0 ft tank 8 ft pump ID=3.0 in ID=.0 in 50 ft 40 ft 1. Single-input, single output. Steady state 3. Constant density (incompressible fluid) 4. Temperature approximately constant 5. No phase change, no chemical rxn 6. Insignificant amounts of heat transferred Mechanical Energy Balance 14 015 7
For example: Centrifugal Pumps What flow rate does a centrifugal pump produce? system Answer: Depends on how much work it is asked to do. pump H ( ft) valve 50% open valve full open Calculate with the Mechanical Energy Balance (CM10, CM315) H ) (,1 V ( ) H d, s V V ( gal / min) 15 We can apply the MEB to many important engineering systems Image from: www.directindustry,com Image from: www.processindustryforum.com MEB Assumptions: 1. single-input, single output. Steady state 3. Constant density (incompressible fluid) 4. Temperature approximately constant 5. No phase change, no chemical rxn 6. Insignificant amounts of heat transferred Image from: www.directindustry.com Calculate: Work, pressures, flows Image from: www.epa.gov 16 015 8
The Mechanical Energy Balance (Review) Δp Δ v W gδzf m p p 1 1 son, v v W g(z z 1) F1 m friction son,,1 Where do we get this? This is the friction due to wall drag (straight pipes) and fittings and valves. 17 The Mechanical Energy Balance Friction Term (Review) The friction has been measured and published in this form: Straight pipes: Use literature plot of f L v as a function of Fstraight pipes 4f D Reynolds Number Fittings and Valves: F fittings, valves K f v Use literature tables of K f for laminar and turbulent flow 18 015 9
(Review) friction Friction term in Mechanical Energy Balance length of straight pipe L Ffriction 4f Kf n i i D i v (see McCabe et al., or Morrison Chapter 1, or Perry s Chem Eng Handbook) number of each type of fitting Note f is a function of velocity) (from literature; the Moody chart) friction-loss coefficients (from literature; see McCabe et al., Geankoplis, or Morrison Chapter 1) Note that friction overall is directly a function of velocity) If the velocity changes within the system (e.g. pipe diameter changes), then we need different friction terms for each velocity Data are organized in terms of two dimensionless parameters: (Review) Flow rate Pressure Drop Reynolds Number v D Re z Fanning Friction Factor f 1 4 L D P P 0 1 L v z density v z average velocity D pipe diameter viscosity P P pressure drop 0 L L pipe length 015 10
Moody Chart: Data Correlation for Friction in Straight Pipes (Review) 16 Re Moody Chart (image from: Geankoplis) Friction Loss from Fittings (source: Morrison, Chapter 1; originally from Perry s Handbook) 015 11
Example 1 What is the pressure change over 50 meters of 1/ inch inner-diameter straight pipe? The average velocity is 5. ft/s and the pipe is smooth. 3 Example 1 What is the pressure change over 50 meters of 1/ inch inner-diameter straight pipe? The average velocity is 5. ft/s and the pipe is smooth. ANSWER: 18 psi 4 015 1
Example What is the flow rate at the drain from a constant-head tank with a fluid level h? You may neglect frictional losses. 5 Example What is the flow rate at the drain from a constant-head tank with a fluid level h? You may neglect frictional losses. ANSWER: 6 015 13
(Review) For more examples: see HW1; Prerequisite readings Exam 1: Next Tues 6:30-8:00pm Last year s exam and solution is on the web. TA help session is Monday night Topics: vectors, linear algebra, integration, MEB, fluid statics 7 p p v v W g(z z 1) F son 1 m 1 1 The Mechanical Energy Balance (MEB) is a macroscopic analysis. It is limited in application:,,1 1. single-input, single output. Steady state 3. Constant density (incompressible fluid) 4. Temperature approximately constant 5. No phase change, no chemical rxn 6. Insignificant amounts of heat transferred friction It cannot determine flow patterns It does not model momentum exchanges It cannot be adapted to systems other than those for which it was designed (see list above) 8 015 14
Energy balances (the MEB) can only take us so far with fluids modeling (due to assumptions). To understand complex flows, we must use the MOMENTUM balance. Image from: whatsupwiththat.com Image from: wwwmath.mtu.edu Image from: www.13rf.com Naruto Whirlpools, Japan Image from: commons.wikipedia.org 9 Momentum Balance: Newton s nd Law of Motion PH 100: apply to individual bodies CM 3110: apply to a continuum Image from: www.texturex.com See also: http://youtu.be/6kknnjfpgto 30 015 15
Fluid Mechanics control volume,, Continuum (density, velocity, stress fields) Control volume Stress in a fluid at a point (stress tensor) Stress and deformation (Newtonian constitutive equation) Microscopic and macroscopic momentum balances Internal flows pipes, conduits External flows drag, boundary layers Advanced fluid mechanics complex shapes 31 Momentum... is a vector Microscopic momentum balance v v v P t v g Ch 6 Macroscopic momentum balance # streams # streams d P Acos v vˆ panˆ RMCV g dt i1 i1 Ai Ai Ch 9 So we need vector math. 3 3 015 16
Vectors v v v v x y z xyz v1 v v 3 13 vr v v z r z Note: v v v x 1 r (usually) Same vector, different coordinate systems, different components. v v vector magnitude v v vˆ unit vector We choose coordinate systems for convenience. 33 Fluid velocity is a vector field,, 0.01 0. 0.6 1..0 3.0 Vector plot of the velocity field in creeping flow around a sphere The flow is a steady upward flow; the length and direction of the vector indicates the velocity at that location. 34 creeping flow (sphere) 015 17
Vectors Cartesian coordinate system (three ways of writing the same thing, the Cartesian basis vectors) We do algebra with the basis vectors the same way as with other quantities The Cartesian basis vectors are constant (do not change with position) 35 Vectors Cylindrical coordinate system z z r P y x The cylindrical basis vectors are variable (depend on position) x rcos eˆ cos ˆ sin ˆ r ex e (see inside y back cover; y r sin eˆ sineˆ cos ˆ x e also, supplemental y handouts) z z eˆ eˆ z z 36 015 18
Vectors Spherical coordinate system Note: spherical coordinate system in use by the fluid mechanics community uses 0 as the angle from the =axis to the point. (see inside back cover; The spherical basis vectors are variable (with position) also, supplemental handouts) x r sin cos eˆ sin cos ˆ sinsin ˆ cos ˆ r ex ey ez y r sin sin eˆ coscoseˆ cossin ˆ ( sin ) ˆ x ey ez z rcos eˆ sin eˆ coseˆ x y 37 Fluid Velocity is a Vector Field Velocity magnitude and direction vary with position 0.01 0. 0.6 1..0 3.0,, 38 creeping flow (sphere) 015 19
Example 3: At positions (1,45 o,0) and (1,90 o,0) in the,, coordinate system, the velocity vector of a fluid is given by What is this vector in the usual coordinate system? 0 1 0 39 Example 3: At positions (1,45 o,0) and (1,90 o,0) in the,, coordinate system, the velocity vector of a fluid is given by What is this vector in the usual coordinate system? 0 1 0 ANSWERS: 0 1 0 0 40 015 0
We use Calculus in Fluid Mechanics to: 1. Calculate flow rate,. Calculate average velocity, 3. Express forces on surfaces due to fluids (vectors) 4. Express torques on surfaces due to fluids (vectors) 41 1. Calculate Flow rate: or V Q General: Q area ( v nˆ ) d( area) Tube flow: R Q v ( r) rdrd 0 0 z ˆ v n is the component of v in the direction normal to the area 4 015 1
Common surface shapes: rectangular : d( area) dxdy circular : d( area) r drd surface of cylinder : d( area) Rddz spherical : d( area) ( rd ) r sind r sindd (see inside back cover; also, supplemental handouts) 43 Example 4: Calculate the flow rate in flow down an incline plane of width W. Momentum balance calculation gives: g cos( ) v z ( x) H x (we will learn how to get this equation for ; here it is given) H 44 015
Example 4: Calculate the flow rate in flow down an incline plane of width W. Momentum balance calculation gives: g cos( ) v z ( x) H x (we will learn how to get this equation for ; here it is given) H ANSWER: 3 45. Calculate Average velocity: v General: v Q area Tube flow: v Q R area is the cross-sectional area normal to flow 46 015 3
Example 5: The shape of the velocity profile for a steady flow in a tube is found to be given by the function below. Over the range 0 <r<10 mm, (R=10mm), what is the average value of the velocity? 47 Example 5: The shape of the velocity profile for a steady flow in a tube is found to be given by the function below. Over the range 0 <r<10 mm, (R=10mm), what is the average value of the velocity? ANSWER: 48 015 4
3. Express forces on surfaces due to fluids Total fluid force on a surface Π Π Total stress tensor 49 (p81) Example 6: In a liquid of density, what is the net fluid force on a submerged sphere (a ball or a balloon)? What is the direction of the force and how does the magnitude of the fluid force vary with fluid density? air H 0 f z x 50 015 5
Solution: We will be able to do this in this course (Ch4, p57). From expression for force due to fluid, obtain: (in spherical coordinates) Total fluid force on a surface Π sin We can do the math from here. (Calc 3) 51 Solution: We will be able to do this in this course (Ch4, p57). From expression for force due to fluid, obtain: (in spherical coordinates) Total fluid force on a surface Π sin ANSWER: (see p83) 0 0 4 3 5 015 6
4. Express torques on surfaces due to fluids total fluid torque T ˆ R n ds onasurface R lever arm pi total stress tensor S at surface (Points from axis of rotation to position where torque is applied) We will learn to write the stress tensor for our systems; then we can calculate stresses, torques. 53 Example 7, Torque in Couette Flow: A cup-and-bob apparatus is widely used to measure viscosities for fluids. For the apparatus below, what is the torque needed to turn the inner cylinder (called the bob) at an angular speed of? 54 015 7
Torque in Couette Flow Solution: 1. Solve for velocity field (microscopic momentum bal). Calculate stress tensor 3. Formulate equation for torque (an integral) 4. Integrate 5. Apply boundary conditions 55 See problem 6. p487 Torque in Couette Flow Solution: Velocity solution: T v v pi 0 R r R v 1R r 56 0 rz total fluid torque T ˆ R n ds on a surface S at surface What is lever arm, R? etc 015 8
See problem 6. p487 Torque in Couette Flow Solution: total fluid torque T ˆ R n ds on a surface S at surface ANSWER: (see p308) 4 Ω 1 0 0 1 57 Summary of Quick Start A: Mechanical Energy Balance friction 1. SI-SO, steady, incompressible, no rxn, no Δ, no. Macroscopic 3. Choose points 1 and wisely 4. Solve for or, or, velocity, elevation B): Use Calculus in Fluid Mechanics to 1. Calculate flow rate. Calculate average velocity 3. Express forces on surfaces due to fluids 4. Express torques on surfaces due to fluids Δp Δ v Wson, gδzf m p p v v W 1 1 g(z z 1) F1 m son,,1 58 015 9
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