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Course: ME 521, Fall 2007School: Penn State
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In Due: class, Friday October 26, 2007 Name ME 521 Fall Semester, 2007 Homework Set # 7 Professor J. M. Cimbala For instructor or TA use only: Problem Score Points 1 15 2 15 3 20 4 50 Total: 100 1. 2. 3. 4. (15 pts) Text problem 3.7. Do two parts, i.e. split the stream function into an inner and outer part, and match the two at r = R. For consistency, assume = 0 at the origin. (a) Calculate for r R. (b)...
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In Due: class, Friday October 26, 2007
Name
ME 521 Fall Semester, 2007
Homework Set # 7
Professor J. M. Cimbala
For instructor or TA use only: Problem Score Points 1 15 2 15 3 20 4 50 Total: 100
1.
2. 3.
4.
(15 pts) Text problem 3.7. Do two parts, i.e. split the stream function into an inner and outer part, and match the two at r = R. For consistency, assume = 0 at the origin. (a) Calculate for r R. (b) Calculate for r R. (15 pts) Text problem 4.7. Note: Stay in variables until the end. Please start with the control volume form of the linear momentum equation, as given in class. (20 pts) Text problem 5.4. Do two parts to this problem: (a) Find the vortex path, as asked for in the text. (b) Sketch the path of a vortex which passes through location (x = 1, y = 1), and has positive circulation . Indicate the direction of movement along the path. (50 pts) Consider a viscous, incompressible, Newtonian fluid, initially at rest. An infinitely long solid cylinder of radius R (infinitely long in the z-direction out of the page) starts rotating at constant angular velocity at time t = 0. After a very long time, the flow can be considered state. steady , u Ignore gravity in this problem, and assume that the flow is axisymmetric (no dependence). Also assume two-dimensionality (no z dependence). Note: For full credit, r, ur you must show all your work, including all algebra. R (a) Find ur. Hint: Appendix B1 is useful in this problem. (b) Find u. Hint: The reverse product rule is helpful here. (c) Find the rate of work per unit depth (into the page) added to the fluid by the spinning cylinder. Hint: All the added work is added right at the cylinder surface. The components of the deviatoric stress tensor are given in Appendix B1. (d) Find the rate of viscous dissipation of kinetic energy per unit volume. (e) Find the total rate of viscous dissipation of kinetic energy per unit depth in the entire flow field. (f) Discuss briefly the physical significance of your answer to Part (e). Hint: Compare to Part (c). (g) Find the pressure field, p(r) two different ways. (Your answer should be the same for the two ways, of course.) (h) Calculate the circulation using a circular path C of radius r centered around the origin at some radial location r in the fluid (r > R).
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Penn State - ME - 521
Due: In class, Friday November 30, 2007NameME 521 Fall Semester, 2007Homework Set # 10Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 15 2 30 3 25 4 30 Total: 1001.2.3.(15 pts) Text problem 6.3. The reason
Penn State - ME - 521
Due: In class, Friday September 14, 2007NameME 521 Fall Semester, 2007Homework Set # 2Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 30 2 15 3 15 4 15 5 15 6 10 Total: 1001.2.3.(30 pts) Mathematical manip
Penn State - ME - 521
t (min) 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240c data (mg/m3) 220 173 109 94 60 41 39 20 18 10 9 6 4 3 3 2 1
Cornell - MATH - 1920
Formula PageWednesday, January 26, 2005 2:46 AMA B = |A|*|B| cos () cos () = cos (-) sin () = - sin (-) Law of Cosines: |A| + |B| - 2|A|*|B| cos = |(B - A)|CIRCLE centered at 3, -5 with a radius of 7. (x - 3) + (y + 5) = 49Plane in space: Ve
Penn State - ME - 521
Vorticity Production at a Wall How Does Vorticity get Produced?Author: John M. Cimbala, Penn State University Latest revision: 11 October 2007Note: Consider steady, incompressible, Newtonian flow in which gravity is neglected and the wall is stat
Penn State - ME - 521
Sample simultaneous equation solver using Excel: J. M. Cimbala, August 2001 Here is how I set this up to solve three equations simultaneously: Tools-Add-Ins-Solver Add-in-OK Set up cells B27 through D27 with initial guesses (I used 1.0 for each varia
Penn State - ME - 521
Homework Set 3, Problem 5, Fall 2006 J. M. Cimbalatime 8:00 8:05 8:10 8:15 8:20 8:25 8:30 8:35 8:40 8:45 8:50 8:55 9:00 9:05 9:10 9:15 9:20 9:25 9:30 9:35 9:40 9:45 9:50 9:55 10:00 10:05 10:10 10:15 10:20 10:25 10:30 10:35 10:40 10:45 10:50 10:55 1
Penn State - ME - 521
M E 521: Foundations of Fluid Mechanics I, Fall Semester 2007Latest update: 29 August 2007Lectures: Text: Prerequisite: Web Pages:Instructor: TA/Grader:Mon., Wed., Fri.: 1:25-2:15 p.m., Room 220 Hammond [Note the time and room change.] Fluid M
Penn State - ME - 521
Particle number distribution for Homework 10, Problem 5, ME 405, Fall 2006 J. M. CimbalaGiven: j = class (bin number) 1 2 3 4 5 6 7 8 9 10 11 12 13 D p ,min,j (lower limit) 0.4 0.8 1.2 2 3 4 5 6 8 10 12 15 20 D p ,max,j n j = count (upper per class
Penn State - ME - 521
Control Volume Example: Flat Plate Boundary LayerAuthor: John M. Cimbala, Penn State University Latest revision: 20 September 2007Given:Consider flow over a two-dimensional flat plate. The freestream flow upstream of the plate is steady, uniform
Cornell - MATH - 1920
14.3. Path Independence, Potential fcns. and Conservative FieldsTuesday, April 12, 2005 12:07 PM
Penn State - ME - 521
Example of how to solve simultaneous equations using Excel J. M. Cimbala, August 2005 Consider the following set of 3 simultaneous equations for 3 unknowns, a, b, and c: 3a + b - 2c = 4 4a - 3b + 2c = 0 2a - 4c = 5 Although it is not necessary, it is
Penn State - ME - 521
Due: In class, Friday September 28, 2007NameME 521 Fall Semester, 2007Homework Set # 4Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 15 2 10 3 10 4 15 5 10 6 40 Total: 1001. 2.3. 4.5.(15 pts) Text problem
Penn State - ME - 521
ME 405 Class Handout - Particle Size Statistics for Grouped Data, J. M. Cimbala, Latest revision: November 2006(Based on data from William C. Hinds, Aerosol Technology, Wiley, New York, 1982.) Note: All sizes are diameters in micrometers (or microns
Penn State - ME - 521
Leibniz Theorem and the Reynolds Transport Theorem for Control VolumesAuthor: John M. Cimbala, Penn State University Latest revision: 20 September 20071-D Leibniz TheoremThe one-dimensional form of the Leibniz theorem allows us to differentiate a
Penn State - ME - 521
Various Forms of the Bernoulli EquationAuthor: John M. Cimbala, Penn State University Latest revision: 25 September 2007Note: The fluid is assumed to be Newtonian, and gravity is assumed to act in the negative z direction [positive z is "up"]. All
Cornell - MATH - 1920
13.6 Triple Integrals in Cylindrical and Spherical CoordinatesWednesday, April 06, 2005 5:19 PM
Penn State - ME - 521
Due: In class, Friday November 9, 2007NameME 521 Fall Semester, 2007Homework Set # 9Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 15 2 15 3 20 4 20 5 15 6 15 Total: 1001. 2.(15 pts) Text problem 3.8. Note: D
Cornell - MATH - 1920
14.2. Vector Fields, Work, Circulation and FlowThursday, May 12, 2005 1:40 PM
Penn State - ME - 521
Due: In class, Friday October 5, 2007NameME 521 Fall Semester, 2007Homework Set # 5Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 15 2 15 3 50 4 20 Total: 1001.2.(15 pts) Text problem 3.1. For consistency,
Penn State - ME - 521
Eigenvalues, Eigenvectors, and Tensor InvariantsAuthor: John M. Cimbala, Penn State University Latest revision: 20 September 2007Let unprimed coordinates x1, x2, and x3 denote the original axes. Let primed coordinates x1, x2, and x3 denote the rot
Cornell - MATH - 1920
13.3 - Double Integrals in Polar FormWednesday, April 06, 2005 5:51 PM
Penn State - ME - 521
Due: In class, Friday September 7, 2007NameME 521 Fall Semester, 2007Homework Set # 1Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 10 2 20 3 10 4 10 5 10 6 10 7 20 8 10 Total: 1001.2.3.4.5. 6. 7.(10
Cornell - MATH - 1920
13.2. Areas, Moments and Centers of MassWednesday, April 06, 2005 5:51 PM
Penn State - ME - 521
Two-D stagnation point similarity solution using the Runge-Kutta methodJ. M. Cimbala Set the origin to 1 instead of zero (for vectors and arrays): ORIGIN := 1Note also that the number of displayed digits is adjusted by Format-Number, change Displa
Penn State - ME - 521
Axes Transformation Rules and Tensor Notation Rules for Cartesian TensorsAuthor: John M. Cimbala, Penn State University Latest revision: 20 September 2007Axes Transformation RulesLet unprimed coordinates x1, x2, and x3 denote the original axes. L
Penn State - ME - 521
Due: In class, Friday December 7, 2007NameME 521 Fall Semester, 2007Homework Set # 11Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 10 2 15 3 10 4 30 5 35 Total: 1001.(10 pts) Consider creeping flow (Stokes f
Penn State - ME - 521
Blasius flat plate boundary layer similarity solution by the Runge-Kutta methodJ. M. Cimbala The equation to solve is f' + cff' = 0, where prime denotes d/d. Here, let c = 1/2. c := 0.5 ORIGIN := 1We will define a vector Y which contains three unk
Penn State - ME - 521
The Heat Flux-Vorticity AnalogyAuthor: John M. Cimbala, Penn State University Latest revision: 10 October 2007Note: For simplicity, consider two-dimensional incompressible, Newtonian flow in the x-y plane.1. Heat Fluxy Consider a wall that is
Penn State - ME - 521
Motion of Fluid ParticlesAuthor: John M. Cimbala, Penn State University Latest revision: 20 September 2007A fluid particle can undergo four different types of motion, which we express as rates: rate of translation, rate of dilatation, rate of shea
Penn State - ME - 521
Due: In class, Friday September 21, 2007NameME 521 Fall Semester, 2007Homework Set # 3Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 10 2 20 3 15 4 10 5 45 Total: 1001.2.3.4.(10 pts) Text problem 2.2. N
Penn State - ME - 521
Elementary Planar Irrotational Flows in Complex VariablesAuthor: John M. Cimbala, Penn State University Latest revision: 17 October 2007Note: Consider steady, incompressible, irrotational, Newtonian fluid flow in which gravity is neglected. The fl
Penn State - ME - 521
Simple Runge-Kutta solution of an ODE J. M. Cimbala, September 2001 The equation to be solved is dy/dt = B - Ay, where in general A and B can be functions of time. The data are plotted on a separate sheet called "plot of y vs t". tstart = deltat = y(
Penn State - ME - 521
ME 405 Fall 2006Professor John M. Cimbala Lecture 2911/13/2006Today, we will: Complete the hood/exhaust fan example problem from last time Go over Exam 2 Discuss Computational Fluid Dynamics in Chapter 10Results of Exam 2: Mean: 72.7 Lo
Penn State - ME - 521
Example Problem 5.4 - Smokers in a conference room J. M. Cimbala1000Constants: Q= V= S per cigarrette = c (0) = ca = Time increment for plot: t = 1 min3 6.3 m /min 3 33.3 m 1100 g/min/cigarrette 3 20 g/m 3 20 g/m900800700 c ( g/m3)600 500
Cornell - MATH - 1920
12.8 - Extreme Values and Saddle PointsSaturday, April 09, 2005 8:25 PM
Penn State - ME - 521
ME 405 Fall 2006Professor John M. Cimbala Lecture 3812/06/2006Today, we will: Discuss equations of particle motion in Section 8.4 Demonstrate some particle trajectory calculations using Mathcad Discuss terminal settling velocity in Sectio
Cornell - MATH - 1920
12.7- Directional Derivatives, Gradient vectors and Tangent PlanesFriday, February 25, 2005 10:48 PM
Penn State - ME - 521
EES windows for the EES file Sample_simple_set_of_simultaneous_equations.EES J. M. Cimbala, August 2005
Penn State - ME - 521
Grade efficiency of the cleaner is given: Additional columns for calculation of mass fraction and grade efficiency analysis:Mass distribution at the outlet of the air cleaner: g (D p ,j )out / (M j /m t )out = m j ,out / D(D p ,j ) m j ,out = (mas
Penn State - ME - 521
Section 3.5 Thermal Comfort and Heat StressTable 3.6 Metabolic rate as a function of physical activity for a 70 kg adult man (abstracted from ASHRAE, 1997). activity sleeping seated, quiet standing, relaxed walking about the office seated, heavy lim
Penn State - ME - 521
Equations for the Conservation of EnergyAuthor: John M. Cimbala, Penn State University Latest revision: 25 September 2007Mechanical Energy EquationNon-conservative form: D 1 ( 2 ui ui ) = ui gi + ui xij Dt jConservative form: where1 2 (
Penn State - ME - 521
Due: In class, Friday October 12, 2007NameME 521 Fall Semester, 2007Homework Set # 6Professor J. M. CimbalaFor instructor or TA use only: Problem Score Points 1 15 2 15 3 15 4 25 5 30 Total: 1001.2.(15 pts) Text problem 3.2. Do two part
Penn State - ME - 521
Nearly Incompressible Laminar Flow Equations and Solution TechniqueAuthor: John M. Cimbala, Penn State University Latest revision: 31 October 2007Assumptions and Approximations The fluid is Newtonian with constant properties (, , k, , , CP). Th
Cornell - MATH - 1920
11.3. Arc length and Unit tangent vectorTuesday, February 22, 2005 6:07 PM
Cornell - MATH - 1920
10.3 Dot Product:Friday, January 28, 2005 12:37 AMFriday, January 28, 2005 12:37 AM10.5 Lines and PlanesFriday, January 28, 2005 12:37 AM
Penn State - ME - 521
Example How to determine if there is a violation of the exposure limits or standards. Note: Here we use ACGIH TLV standards, which are not enforceable. However, the same kind of analysis can be performed with OSHA PEL values, which are enforceable.
Penn State - ME - 521
Example How to determine if there is a violation of the exposure limits or standards. Note: Here we use ACGIH TLV standards, which are not enforceable. However, the same kind of analysis can be performed with OSHA PEL values, which are enforceable.