Heat Transfer in a Slab
|
|
- Curtis Knight
- 5 years ago
- Views:
Transcription
1 Heat Transfer in a Slab Consider a large planar solid whose thickness (y-direction) is L. What is the temperature history of the slab if it is suddenly brought into contact with a fluid at temperature T? The transient conduction equation is T t = α 2 T y 2 at t =,T = T at y =, T = T 1 for t > at y = L, T = T 1 Let s make the problem dimensionless. θ = T T 1 T T 1 ; η = y L ; τ = αt L 2 The temperature can be expressed as so that the problem reposed is t = α 2 θ y 2 θ = 1 at τ = θ = at η = θ = as η = 1 ChE 333 1
2 How do we solve the equation? τ 2 θ η 2 = YG τ 2 YG η 2 = G dy dτ Yd2 G dη 2 = Suppose z has the form Θ = Y(τ)G(η) 1 dy Y dτ = 1 d 2 G = λ 2 G dη 2 dy dτ = λ2 Y ; d2 G dη 2 The equation is separable in the form = λ 2 G Integrating each of the equations we obtain Y(τ) = Ke λ2τ and G(η) = Asin(λη) +Bcos(λη) The solution for y(θ,η) has the form θ(η, τ) = Asin(λη) +Bcos(λη) e λ2 τ We can construct the exact solution using the boundary conditions θ(, τ) = Asin() +Bcos() e λ2τ = It follows that B must be if the condition is true for all θ > ChE 333 2
3 Now the other boundary condition θ(1, τ) = Asin(λ) e λ2τ = Now this is true for all τ > if and only if sin(λ) = but sin(λ) = only where λ = nπ where n =, 1, 2,... This means there are a countable infinity of solutions so that θ(η, τ) = Σ n= 1 e λ2τ A n sin(nπη) To obtain the coefficients A n, we need to use the initial condition. θ = 1 at τ < Σ n= 1 z(η, ) = A n sin(nπη) = 1 To determine the coefficients, we can use the orthogonality properties of the sine and cosine functions. (See Appendix) 1 1 sin(nπξ)sin(mπξ)dξ = for m n π for m = n ChE 333 3
4 We integrate 1 θ(η, )sin(mπη)dξ 1 = sin(mπη)dη You might remember that the first sine integral is non-zero if and only if n = m. Now the equation for A n is Σ n= 1 A n 1 sin(nπη)sin(mπη)dη 1 = sin(mπη)dη A n = 1 sin(nπη)dη = 1 sin 2 (nπη)dη nπ nπ sin(x)dx sin 2 (x)dx The result for the definite integrals follow from what I gave above. It follows that A n = 4 π 1 n n We saw earlier that the solution can be described as: ChE 333 4
5 θ(η, τ) = 4 π Σ n = 1 2n +1 e 2n π 2 τ sin( 2n + 1 πη) We have already examined how the sum converges. For τ >.2, only one term suffices to describe the solution. We can look at many different classes of problems. The general problem for transient heat transfer in a slab is one posed as T t = α 2 T y 2 at t =, T = T at y =, k T y = h T T 1 for t > at y = L 2, T y = The dimensionless form is: The solution is of the form = α t 2 θ y 2 θ = 1 at τ= = Bi θ at η = η η = as η = 1 θ(η, τ) = C n e ζ n 2 τ sin(ζ 2 n η) Σn = C n = 4 sin ζ n 2ζ n + sin 2 ζ n and ζ n tan ζ n = Bi ChE 333 5
6 Again the approximate solution is the one-term solution θ(η, τ) C 1 e ζ 1 2 τ sin(ζ 2 1 η) This argument is the same for any transient 1-dimensional heat transfer problems involving cylinders, planes or spheres. Examples Infinite Cylinder = 1 τ η η η η θ = 1 at τ= in η, 1 η = Bi θ at η = 1 η = at η = The solution is θ(η, τ) = C n e ζ n 2 τ J (ζ n η) Σn = C n = 2 ζ n J 1 ζ n 2 J 2 ζ n + J 1 2 ζ n and ζ n J 1 ζ n J ζ n = Bi An approximate one-term solution is θ(η, τ) = C 1 e ζ 1 2 τ J (ζ 1 η) Note that along the center line θ (, τ) = C 1 e ζ 1 2 τ ChE 333 6
7 So that the simpler representation is θ(η, τ) = θ J (ζ 1 η) ChE 333 7
8 Sphere τ = η 1 2 η η 2 η θ = 1 at τ= in η, 1 η = Bi θ at η = 1 η = at η = The solution is θ(η, τ) = Σn = C n e ζ n τsin(ζ 2 n η) ζ n η C n = 4 sin ζ n ζ n cos ζ n 2 ζ n + sin 2ζ n and 1 ζ n cot ζ n = Bi The Approximate Solution The center temperature is θ(η, τ) = C 1 e ζ 1 2 τsin(ζ 1 η) ζ 1 η θ (, τ) = C 1 e ζ 1 2 τ So that the temperature can be expressed as sin(ζ θ(η, τ) = θ 1 η) ζ 1 η ChE 333 8
9 Short Time Solutions Consider a large planar solid whose extent (y-direction) is very large. What is the temperature history of the slab if it is suddenly brought into contact with a fluid at temperature T a? The transient conduction equation is T t = α 2 T y 2 at t =, T = T at y =, k T y = h T T a T a as y, T = T Let s make the problem dimensionless. The temperature can be expressed as θ = T T a T T a so that the problem reposed is t = α 2 θ y 2 θ = 1 at t = y = Bi θ at y = θ = 1 as y ChE 333 9
10 Solutions We noted earlier that the equation can be solved by a combination of variables supposing that Τ = Τ(η,t) and we saw that the the appropriate choice for η is η = y 4αt The solutions for a number of different cases are as follows: Case 1 Constant Surface Temperature (T = T s ) T T s T i T s = erf y αt q " s t = k T s T i παt Case 2 Constant Surface Heat Flux (q s = q ) T T s = 2 q" k αt π e y 2 4αt q" y k erfc y 4αt Case 3 Surface Convection θ = erf y 4t + exp y + t erfc y 4t + t ChE 333 1
11 where y = hy k s and t = h k s 2 αt ChE
12 Surface temperature of a Cooling Sheet Polyethylene is extruded and coated onto an insulated substrate, moving at 2 cm/sec. The molten polymer is coated at a uniform temperature T of 4 F. Cooling is achieved by blowing air at a temperature T a of 8 F. Earlier heat transfer studies determined that the heat transfer coefficient, h, is.8 cal/cm-sec- F. The coating thickness B is.1 cm. At what point downstream does the surface temperature, T() fall to 144 F? Data T = 4 F h = 3.35 kw/m 2 - K B =.1 cm. T a = 8 F k s =.33 W/m- K α = m 2 /sec The Biot number can be estimated as: Bi = hb k s = = 1.15 The dimensionless surface temperature ratio is θ s = T T a T T a = =.2 The Gurney-Lurie Chart 11.4c yields for Bi 1, the ratio of the surface temperature to the mid-plane temperature However, since θ θ =.15 we can calculate the mid-plane temperature 1, from the relation for θ s which is θ s = θ θ 1 =.2 θ 1 This gives a midplane-temperature of θ 1 =.2/.15 > 1...Nonsense What s wrong??? We did a lot of things wrong. ChE
13 First of all the solution we used involved only 1 term of an infinite series... θ 1 = A 1 e β 1 2 x Fo sin β 1 ξ = θ 1 sin β 1 ξ We also get into trouble if we use such an equation for a short time solution. Therefore avoid the charts for small x Fo and large Biot numbers. The short time solution we presented in the last lecture had the form. θ = erf y 4t + exp y + t erfc y 4t + t where y = hb k s y B and t = hb k s 2 αt B 2 Now for this case, y = and we can use figure We can determine that the value of t at which Θ =.2. We observe that t 1/2 = 2.65 and consequently t = 7.65 Recall that ρc p = k/α = 2.5 MJ/m 2 - K. This leads to t = hb k s 2 αt B 2 = 7.65 = h k s 2 αt We calculate that the time passed is t =.52 seconds and since d = Vt, the distance is d = (2 cm/s).52 sec = 1.4 cm. ChE
14 An alternative method We can use the complete Fourier expansion, not just one term. θ = Σn =1 4 sin λ n cos λ 2λ n + sin 2λ n ξ e λ 2 nx Fo n λ n tan λ n = Bi The first set of eigenvalues are n λ n If we calculate the first three terms of the Fourier expansion, we obtain θ(1) =.178e 2.4x Fo +.155e 18.5x Fo For Θ =.2, by trial and error, we obtain x Fo =.68. If we calculate the time, we get.52 sec. The same as the short time solution. This allows us a measure of short time. as for a slab 4 αt B 2 1 or x Fo 1 16 ChE
Unsteady State Heat Conduction in a Bounded Solid How Does a Solid Sphere Cool?
Unstead State Heat Conduction in a Bounded Solid How Does a Solid Sphere Cool? We examined the cooling a sphere of radius R. Initiall the sphere is at a uniform temperature T 0. It is cooled b convection
More informationUnsteady State Heat Conduction in a Bounded Solid
Unsteady State Heat Conduction in a Bounded Solid Consider a sphere of radius R. Initially the sphere is at a uniform temperature T. It is cooled by convection to an air stream at temperature T a. What
More informationMECH 375, Heat Transfer Handout #5: Unsteady Conduction
1 MECH 375, Heat Transfer Handout #5: Unsteady Conduction Amir Maleki, Fall 2018 2 T H I S PA P E R P R O P O S E D A C A N C E R T R E AT M E N T T H AT U S E S N A N O PA R T I - C L E S W I T H T U
More information3.3 Unsteady State Heat Conduction
3.3 Unsteady State Heat Conduction For many applications, it is necessary to consider the variation of temperature with time. In this case, the energy equation for classical heat conduction, eq. (3.8),
More informationConsider a volume Ω enclosing a mass M and bounded by a surface δω. d dt. q n ds. The Work done by the body on the surroundings is.
The Energy Balance Consider a volume Ω enclosing a mass M and bounded by a surface δω. δω At a point x, the density is ρ, the local velocity is v, and the local Energy density is U. U v The rate of change
More informationChapter 5 Time-Dependent Conduction
Chapter 5 Time-Dependent Conduction 5.1 The Lumped Capacitance Method This method assumes spatially uniform solid temperature at any instant during the transient process. It is valid if the temperature
More informationChapter 4: Transient Heat Conduction. Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University
Chapter 4: Transient Heat Conduction Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University Objectives When you finish studying this chapter, you should be able to: Assess when the spatial
More informationParallel Plate Heat Exchanger
Parallel Plate Heat Exchanger Parallel Plate Heat Exchangers are use in a number of thermal processing applications. The characteristics are that the fluids flow in the narrow gap, between two parallel
More informationQUESTION ANSWER. . e. Fourier number:
QUESTION 1. (0 pts) The Lumped Capacitance Method (a) List and describe the implications of the two major assumptions of the lumped capacitance method. (6 pts) (b) Define the Biot number by equations and
More informationChapter 3: Transient Heat Conduction
3-1 Lumped System Analysis 3- Nondimensional Heat Conduction Equation 3-3 Transient Heat Conduction in Semi-Infinite Solids 3-4 Periodic Heating Y.C. Shih Spring 009 3-1 Lumped System Analysis (1) In heat
More informationIntroduction to Heat and Mass Transfer. Week 8
Introduction to Heat and Mass Transfer Week 8 Next Topic Transient Conduction» Analytical Method Plane Wall Radial Systems Semi-infinite Solid Multidimensional Effects Analytical Method Lumped system analysis
More informationTRANSIENT HEAT CONDUCTION
TRANSIENT HEAT CONDUCTION Many heat conduction problems encountered in engineering applications involve time as in independent variable. This is transient or Unsteady State Heat Conduction. The goal of
More informationSession 5 Heat Conduction in Cylindrical and Spherical Coordinates I
Session 5 Heat Conduction in Cylindrical and Spherical Coordinates I 1 Introduction The method of separation of variables is also useful in the determination of solutions to heat conduction problems in
More informationHeat Conduction in semi-infinite Slab
1 Module : Diffusive heat and mass transfer Lecture 11: Heat Conduction in semi-infinite Slab with Constant wall Temperature Semi-infinite Solid Semi-infinite solids can be visualized as ver thick walls
More informationNumerical Heat and Mass Transfer
Master degree in Mechanical Engineering Numerical Heat and Mass Transfer 02-Transient Conduction Fausto Arpino f.arpino@unicas.it Outline Introduction Conduction ü Heat conduction equation ü Boundary conditions
More informationReview: Conduction. Breaking News
CH EN 3453 Heat Transfer Review: Conduction Breaking News No more homework (yay!) Final project reports due today by 8:00 PM Email PDF version to report@chen3453.com Review grading rubric on Project page
More informationConduction Heat Transfer. Fourier Law of Heat Conduction. x=l Q x+ Dx. insulated x+ Dx. x x. x=0 Q x A
Conduction Heat Transfer Reading Problems 10-1 10-6 10-20, 10-48, 10-59, 10-70, 10-75, 10-92 10-117, 10-123, 10-151, 10-156, 10-162 11-1 11-2 11-14, 11-20, 11-36, 11-41, 11-46, 11-53, 11-104 Fourier Law
More informationElementary Non-Steady Phenomena
Elementary Non-Steady (Transient) Phenomena (T) Elementary Non-Steady Phenomena Because Transport deals with rates it is often the case that we must consider non-steady (or transient) operation (when the
More informationIntroduction to Heat and Mass Transfer. Week 7
Introduction to Heat and Mass Transfer Week 7 Example Solution Technique Using either finite difference method or finite volume method, we end up with a set of simultaneous algebraic equations in terms
More informationTime-Dependent Conduction :
Time-Dependent Conduction : The Lumped Capacitance Method Chapter Five Sections 5.1 thru 5.3 Transient Conduction A heat transfer process for which the temperature varies with time, as well as location
More informationIntroduction to Heat and Mass Transfer. Week 8
Introduction to Heat and Mass Transfer Week 8 Next Topic Transient Conduction» Analytical Method Plane Wall Radial Systems Semi-infinite Solid Multidimensional Effects Analytical Method Lumped system analysis
More informationConduction Heat Transfer. Fourier Law of Heat Conduction. Thermal Resistance Networks. Resistances in Series. x=l Q x+ Dx. insulated x+ Dx.
Conduction Heat Transfer Reading Problems 17-1 17-6 17-35, 17-57, 17-68, 17-81, 17-88, 17-110 18-1 18-2 18-14, 18-20, 18-34, 18-52, 18-80, 18-104 Fourier Law of Heat Conduction insulated x+ Dx x=l Q x+
More informationChapter 4 TRANSIENT HEAT CONDUCTION
Heat and Mass Transfer: Fundamentals & Applications Fourth Edition Yunus A. Cengel, Afshin J. Ghajar McGraw-Hill, 2011 Chapter 4 TRANSIENT HEAT CONDUCTION LUMPED SYSTEM ANALYSIS Interior temperature of
More information1 Solutions in cylindrical coordinates: Bessel functions
1 Solutions in cylindrical coordinates: Bessel functions 1.1 Bessel functions Bessel functions arise as solutions of potential problems in cylindrical coordinates. Laplace s equation in cylindrical coordinates
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationFIND: (a) Sketch temperature distribution, T(x,t), (b) Sketch the heat flux at the outer surface, q L,t as a function of time.
PROBLEM 5.1 NOWN: Electrical heater attached to backside of plate while front surface is exposed to convection process (T,h); initially plate is at a uniform temperature of the ambient air and suddenly
More informationFind c. Show that. is an equation of a sphere, and find its center and radius. This n That. 3D Space is like, far out
D Space is like, far out Introspective Intersections Inverses Gone wild Transcendental Computations This n That 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 c Find
More information1 R-value = 1 h ft2 F. = m2 K btu. W 1 kw = tons of refrigeration. solar = 1370 W/m2 solar temperature
Quick Reference for Heat Transfer Analysis compiled by Jason Valentine and Greg Walker Please contact greg.alker@vanderbilt.edu ith corrections and suggestions copyleft 28: You may copy, distribute, and
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
More informationConsider a volume Ω enclosing a mass M and bounded by a surface δω. d dt. q n ds. The Work done by the body on the surroundings is
The Energy Balance Consider a volume enclosing a mass M and bounded by a surface δ. δ At a point x, the density is ρ, the local velocity is v, and the local Energy density is U. U v The rate of change
More information[Yadav*, 4.(5): May, 2015] ISSN: (I2OR), Publication Impact Factor: (ISRA), Journal Impact Factor: 2.114
IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY IMPROVEMENT OF CLASSICAL LUMPED MODEL FOR TRANSIENT HEAT CONDUCTION IN SLAB USING HERMITE APROXIMATION Rakesh Krishna Yadav*,
More informationPlot of temperature u versus x and t for the heat conduction problem of. ln(80/π) = 820 sec. τ = 2500 π 2. + xu t. = 0 3. u xx. + u xt 4.
10.5 Separation of Variables; Heat Conduction in a Rod 579 u 20 15 10 5 10 50 20 100 30 150 40 200 50 300 x t FIGURE 10.5.5 Example 1. Plot of temperature u versus x and t for the heat conduction problem
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationASSUMPTIONS: (1) One-dimensional, radial conduction, (2) Constant properties.
PROBLEM 5.5 KNOWN: Diameter and radial temperature of AISI 00 carbon steel shaft. Convection coefficient and temperature of furnace gases. FIND: me required for shaft centerline to reach a prescribed temperature.
More informationChapter 10: Steady Heat Conduction
Chapter 0: Steady Heat Conduction In thermodynamics, we considered the amount of heat transfer as a system undergoes a process from one equilibrium state to another hermodynamics gives no indication of
More informationThe temperature of a body, in general, varies with time as well
cen58933_ch04.qd 9/10/2002 9:12 AM Page 209 TRANSIENT HEAT CONDUCTION CHAPTER 4 The temperature of a body, in general, varies with time as well as position. In rectangular coordinates, this variation is
More informationPhysics 250 Green s functions for ordinary differential equations
Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which
More informationQuantum Physics Lecture 8
Quantum Physics ecture 8 Steady state Schroedinger Equation (SSSE): eigenvalue & eigenfunction particle in a box re-visited Wavefunctions and energy states normalisation probability density Expectation
More informationSolving the Heat Equation (Sect. 10.5).
Solving the Heat Equation Sect. 1.5. Review: The Stationary Heat Equation. The Heat Equation. The Initial-Boundary Value Problem. The separation of variables method. An example of separation of variables.
More informationENSC 388. Assignment #8
ENSC 388 Assignment #8 Assignment date: Wednesday Nov. 11, 2009 Due date: Wednesday Nov. 18, 2009 Problem 1 A 3-mm-thick panel of aluminum alloy (k = 177 W/m K, c = 875 J/kg K, and ρ = 2770 kg/m³) is finished
More informationIntroduction to Heat and Mass Transfer. Week 9
Introduction to Heat and Mass Transfer Week 9 補充! Multidimensional Effects Transient problems with heat transfer in two or three dimensions can be considered using the solutions obtained for one dimensional
More informationUNIVERSITY OF SOUTH CAROLINA
UNIVERSITY OF SOUTH CAROLINA ECHE 460 CHEMICAL ENGINEERING LABORATORY I Heat Transfer Analysis in Solids Prepared by: M. L. Price, and Professors M. A. Matthews and T. Papathansiou Department of Chemical
More informationMath 201 Assignment #11
Math 21 Assignment #11 Problem 1 (1.5 2) Find a formal solution to the given initial-boundary value problem. = 2 u x, < x < π, t > 2 u(, t) = u(π, t) =, t > u(x, ) = x 2, < x < π Problem 2 (1.5 5) Find
More informationSpecific heat capacity. Convective heat transfer coefficient. Thermal diffusivity. Lc ft, m Characteristic length (r for cylinder or sphere; for slab)
Important Heat Transfer Parameters CBE 150A Midterm #3 Review Sheet General Parameters: q or or Heat transfer rate Heat flux (per unit area) Cp Specific heat capacity k Thermal conductivity h Convective
More informationsin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationEntropy Generation Analysis of Transient Heat Conduction in a Solid Slab with Fixed Temperature Boundary Conditions
WSEAS RASACIOS on HEA and MASS RASFER Entropy Generation Analysis of ransient Heat Conduction in a Solid Slab with Fixed emperature Boundary Conditions SOMPOP JARUGHAMMACHOE Mechanical Engineering Department
More informationINSTRUCTOR: PM DR. MAZLAN ABDUL WAHID TEXT: Heat Transfer A Practical Approach by Yunus A. Cengel Mc Graw Hill
M 792: IUO: M D. MZL BDUL WID http://www.fkm.utm.my/~mazlan X: eat ransfer ractical pproach by Yunus. engel Mc Graw ill hapter ransient eat onduction ssoc rof Dr. Mazlan bdul Wahid aculty of Mechanical
More informationBuoyancy-induced Flow:
Buoyancy-induced Flow: Natural Convection in a Unconfined Space If we examine the flow induced by heat transfer from a single vertical flat plat, we observe that the flow resembles that of a boundary layer.
More informationTotal energy in volume
General Heat Transfer Equations (Set #3) ChE 1B Fundamental Energy Postulate time rate of change of internal +kinetic energy = rate of heat transfer + surface work transfer (viscous & other deformations)
More information6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities
Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 3 Notes
Jim Lambers ENERGY 8 Spring Quarter 7-8 Lecture 3 Notes These notes are based on Rosalind Archer s PE8 lecture notes, with some revisions by Jim Lambers. Introduction The Fourier transform is an integral
More information(Refer Slide Time: 01:17)
Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. 7 Heat Conduction 4 Today we are going to look at some one dimensional
More informationNAME DATE PERIOD. Trigonometric Identities. Review Vocabulary Complete each identity. (Lesson 4-1) 1 csc θ = 1. 1 tan θ = cos θ sin θ = 1
5-1 Trigonometric Identities What You ll Learn Scan the text under the Now heading. List two things that you will learn in the lesson. 1. 2. Lesson 5-1 Active Vocabulary Review Vocabulary Complete each
More informationTHE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m.
THE UNIVERSITY OF WESTERN ONTARIO London Ontario Applied Mathematics 375a Instructor: Matt Davison Final Examination December 4, 22 9: 2: a.m. 3 HOURS Name: Stu. #: Notes: ) There are 8 question worth
More informationChapter 06: Analytic Trigonometry
Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is oneto-one. Are any of our trigonometric
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More informationOverview. Review Multidimensional PDEs. Review Diffusion Solutions. Review Separation of Variables. More diffusion solutions February 2, 2009
More diffusion solutions February, 9 More Diffusion Equation Solutions arry Caretto Mechanical Engineering 5B Seinar in Engineering Analysis February, 9 Overview Review last class Separation of variables
More informationOverview of Fourier Series (Sect. 6.2). Origins of the Fourier Series.
Overview of Fourier Series (Sect. 6.2. Origins of the Fourier Series. Periodic functions. Orthogonality of Sines and Cosines. Main result on Fourier Series. Origins of the Fourier Series. Summary: Daniel
More information4sec 2xtan 2x 1ii C3 Differentiation trig
A Assignment beta Cover Sheet Name: Question Done Backpack Topic Comment Drill Consolidation i C3 Differentiation trig 4sec xtan x ii C3 Differentiation trig 6cot 3xcosec 3x iii C3 Differentiation trig
More informationASSUMPTIONS: (1) Homogeneous medium with constant properties, (2) Negligible radiation effects.
PROBEM 5.88 KNOWN: Initial temperature of fire clay bric which is cooled by convection. FIND: Center and corner temperatures after 50 minutes of cooling. ASSUMPTIONS: () Homogeneous medium with constant
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationAlgebraically Explicit Analytical Solution of Three- Dimensional Hyperbolic Heat Conduction. Equation
Adv. Theor. Appl. Mech., Vol. 3, 010, no. 8, 369-383 Algebraically Explicit Analytical Solution of Three- Dimensional Hyperbolic Heat Conduction Equation Seyfolah Saedodin Department of Mechanical Engineering,
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationSection 7.2 Addition and Subtraction Identities. In this section, we begin expanding our repertoire of trigonometric identities.
Section 7. Addition and Subtraction Identities 47 Section 7. Addition and Subtraction Identities In this section, we begin expanding our repertoire of trigonometric identities. Identities The sum and difference
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationPeriodic functions: simple harmonic oscillator
Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω2 0y = 0 Solution can be written in various ways: y(t) = Ae iω 0t y(t) = A cos
More informationPhysics 6303 Lecture 11 September 24, LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation
Physics 6303 Lecture September 24, 208 LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation, l l l l l l. Consider problems that are no axisymmetric; i.e., the potential depends
More informationMathematical methods and its applications Dr. S. K. Gupta Department of Mathematics Indian Institute of Technology, Roorkee
Mathematical methods and its applications Dr. S. K. Gupta Department of Mathematics Indian Institute of Technology, Roorkee Lecture - 56 Fourier sine and cosine transforms Welcome to lecture series on
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More information4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
More informationMore with Angles Reference Angles
More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o
More informationd 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.
4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal
More informationLecture 11: Fourier Cosine Series
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without eplicit written permission from the copyright owner ecture : Fourier Cosine Series
More informationLecture 16: Bessel s Inequality, Parseval s Theorem, Energy convergence
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. ot to be copied, used, or revised without explicit written permission from the copyright owner. ecture 6: Bessel s Inequality,
More informationPrinciples of Food and Bioprocess Engineering (FS 231) Exam 2 Part A -- Closed Book (50 points)
Principles of Food and Bioprocess Engineering (FS 231) Exam 2 Part A -- Closed Book (50 points) 1. Are the following statements true or false? (20 points) a. Thermal conductivity of a substance is a measure
More information1 Current Flow Problems
Physics 704 Notes Sp 08 Current Flow Problems The current density satisfies the charge conservation equation (notes eqn 7) thusinasteadystate, is solenoidal: + =0 () =0 () In a conducting medium, we may
More informationSummary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer
1. Nusselt number Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer Average Nusselt number: convective heat transfer Nu L = conductive heat transfer = hl where L is the characteristic
More informationq x = k T 1 T 2 Q = k T 1 T / 12
Conductive oss through a Window Pane q T T 1 Examine the simple one-dimensional conduction problem as heat flow through a windowpane. The window glass thickness,, is 1/8 in. If this is the only window
More informationExercise 11. Isao Sasano
Exercise Isao Sasano Exercise Calculate the value of the following series by using the Parseval s equality for the Fourier series of f(x) x on the range [, π] following the steps ()-(5). () Calculate the
More information4. Analysis of heat conduction
4. Analysis of heat conduction John Richard Thome 11 mars 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Conduction 11 mars 2008 1 / 47 4.1 The well-posed problem Before we go further with
More informationLecture 4. Section 2.5 The Pinching Theorem Section 2.6 Two Basic Properties of Continuity. Jiwen He. Department of Mathematics, University of Houston
Review Pinching Theorem Two Basic Properties Lecture 4 Section 2.5 The Pinching Theorem Section 2.6 Two Basic Properties of Continuity Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu
More informationTransport Phenomena II
Transport Phenomena II Andrew Rosen April 25, 2014 Contents 1 Temperature Distributions with More Than One Independent Variable 3 1.1 The Microscopic Energy Balance.................................. 3
More informationHomework 7 Solutions
Homework 7 Solutions # (Section.4: The following functions are defined on an interval of length. Sketch the even and odd etensions of each function over the interval [, ]. (a f( =, f ( Even etension of
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationMath 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses
Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses Instructor: Sal Barone School of Mathematics Georgia Tech May 22, 2015 (updated May 22, 2015) Covered sections: 3.3 & 3.5 Exam 1 (Ch.1 - Ch.3) Thursday,
More informationSolving Nonhomogeneous PDEs (Eigenfunction Expansions)
Chapter 12 Solving Nonhomogeneous PDEs (Eigenfunction Expansions) 12.1 Goal We know how to solve diffusion problems for which both the PDE and the s are homogeneous using the separation of variables method.
More information2 General Relativity. 2.1 Curved 2D and 3D space
22 2 General Relativity The general theory of relativity (Einstein 1915) is the theory of gravity. General relativity ( Einstein s theory ) replaced the previous theory of gravity, Newton s theory. The
More informationA proof for the full Fourier series on [ π, π] is given here.
niform convergence of Fourier series A smooth function on an interval [a, b] may be represented by a full, sine, or cosine Fourier series, and pointwise convergence can be achieved, except possibly at
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationRing-shaped crack propagation in a cylinder under nonsteady cooling
High Performance Structures and Materials III 5 Ring-shaped crack propagation in a cylinder under nonsteady cooling V. A. Zhornik, Yu. A. Prokopenko, A. A. Rybinskaya & P. A. Savochka Department of Theoretical
More informationUniversity of Rome Tor Vergata
University of Rome Tor Vergata Faculty of Engineering Department of Industrial Engineering THERMODYNAMIC AND HEAT TRANSFER HEAT TRANSFER dr. G. Bovesecchi gianluigi.bovesecchi@gmail.com 06-7259-727 (7249)
More information6.1: Reciprocal, Quotient & Pythagorean Identities
Math Pre-Calculus 6.: Reciprocal, Quotient & Pythagorean Identities A trigonometric identity is an equation that is valid for all values of the variable(s) for which the equation is defined. In this chapter
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationLast Update: April 7, 201 0
M ath E S W inter Last Update: April 7, Introduction to Partial Differential Equations Disclaimer: his lecture note tries to provide an alternative approach to the material in Sections.. 5 in the textbook.
More information1. Partial differential equations. Chapter 12: Partial Differential Equations. Examples. 2. The one-dimensional wave equation
1. Partial differential equations Definitions Examples A partial differential equation PDE is an equation giving a relation between a function of two or more variables u and its partial derivatives. The
More information