q x = k T 1 T 2 Q = k T 1 T / 12

Size: px
Start display at page:

Download "q x = k T 1 T 2 Q = k T 1 T / 12"

Transcription

1 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 in a room 9x12x8 or 864 ft 3, the area of the window is 2 ft x 3 ft or 6 ft 2. Recall that q x is the heat flux and that k is the thermal conductivity T 2 q x = k dt dx x The energy at steady state yielded q x = k T 1 T 2 The room is well heated and the temperature is uniform, so the heat flow through the windowpane is Q = k T 1 T 2 A If the room temperature is 60 F and the exterior temperature is 20 F, and k is 0.41 Btu/hr-ft2- F then Q = / 12 6 = 9444 Btu hr Questions Is this a large rate? How can you tell whether it is large or not? ChE 333 1

2 Energy balance on the Room How long does it take for the room temperature to change from 60 F to 45 F? To make this estimate, we need to solve an energy balance on the room. A simple analysis yields d dt VρC p T 1 T ref = Q Recognizing that heat capacity density are essentially constant, the equation becomes dt 1 dt = ka VρC p T 1 T 2 Note that τ = VρC p ka Data T 2 = 20 F T 10 = 60 F T 1 = 45 F k = 0.41 BTU/ft-h- F A = 6ft^2 V = 864 ft^3 = in. ρ air = 0.07 lb/ft^3 C pair = 0.24 BTU/lb.- F and that τ has units of time. At the outset, T 1 = T 10 = 60 F The solution of the differential equation representing the energy balance is T 1 T 2 T 10 T 2 = e t τ To solve for the time required to get to 46 F, we need all the data in the table. t τ = ln T 1 T 2 T 10 T 2 = ln = 0.47 It follows that t = 0.47τ = 1.75 minutes. ChE 333 2

3 Heat Conduction in a Composite Solid q T T 0 T 2 Examine the simple one-dimensional conduction problem as heat flow through a thermally insulated windowpane. Each layer of window glass thickness,, is 1/16 in. The insulation layer of air between the two panes is also 1/16 in. Recall that q x is the heat flux and that k is the thermal conductivity T T3 The heat flow through the glass is given by x q x = k dt dx The energy at steady state yielded q x = k T 1 T 2 In layer 1 Q = k 1 T 0 T 1 δ 1 A In layer 2 Q = k 2 T 1 T 2 δ 2 A and in layer 3 by Q = k 3 T 2 T 3 δ 3 Then we can rewrite the equations in this form A Q δ 3 = T k 2 T 3 A ; Q δ 2 = T 3 k 1 T 2 A ; Q δ 1 = T 2 k 0 T 1 A 1 If we add the three equations, we obtain Q δ 1 + δ 2 + δ 3 = T A k 1 k 2 k 0 T 3 3 ChE 333 3

4 We can consider the thickness/conductivity as a resistance so that et Ri = δ i Ak i then δ 1 Ak 1 + δ 2 Ak 2 + δ 3 Ak 3 = R 1 + R 2 + R 3 = R t The heat flow is then of the following form : Q = q x = q x '' A = T 0 - T 3 Rt This is like a problem of current flow in a series circuit. In the single pane problem discussed in ecture 1, we noted that the resistance, δ/k, was 1/(192(0.41) = hr-ft 2 - F/Btu. Recall that for the problem of cooling the room, τ was 1.75 minutes. The thermal conductivity of air is Btu/hr-ft- F. so that δ 1 = δ 3 = k 1 k ; δ 2 k 2 = = as a consequence the reciprocal of the overall resistance is (0.0254) = Then we see that τ = (1.min) (0.746/0.0254) = min ChE 333 4

5 Tr T i Ta The Convective Boundary Condition Again consider a windowpane, but now there is a heat transfer limitation at one boundary described by a boundary condition. q x '' = - h T r -T i Conduction through the glass is described by The flux is constant at any cross-section so that we can write Solving for the temperatures we get q x '' = - k T i-t a δ T i - T a = δ k q x '' ; T r -T i = q x '' h T r - T a = δ k q x '' + q x '' Solving for q x, the relation becomes h = q x '' δ k + 1 h = q x A δ k + 1 h q x A = q x '' = T r- T a δ k + 1 h ChE 333 5

6 q x A = q x '' = h T r- T a hδ k + 1 Which modified shows a correction to the heat transfer coefficient modulated by the conduction problem Bi = hδ k The dimensionless number in the denominator is the Biot number, a ratio of the convective heat transfer coefficient to the equivalent heat transfer coefficient due to conduction.? What happens when we take into account the convection on both sides of the window? What do we do if we wish to transfer the heat transfer through the window sash (wooden frame around the window). ChE 333 6

7 Heat Transfer across a Composite Cylindrical Solid. In the case of heat transfer in a cylinder, there is radial symmetry do that heat conduction is important only in the radial direction. R 2 R 3 T 0 R 1 T 1 T2 T3 The heat flux in the radial direction is given by Fourier s law. q r '' = - k dt dr The total heat flow through any circular surface is constant Q = k 2πr dt = constant = C dr Rearranging we obtain a relation for the temperature gradient dt = C dr k 2πr which upon separation of variables is dt = C dr k 2π r An indefinite integration yields the temperature profile. T = The boundary conditions are C k 2π ln r + a 1 at r = R 1, T = T 1 ; at r = R 2, T = T 2 q" r2 = q" r2 at r = R 3, T = T 3 q" r3 = h(t 3 - T 0 ) ChE 333 7

8 so that T 1 T 2 = C k 1 2π ln R 1 ; T R 2 T 3 = C 2 k 2 2π ln R 2 R 1 It follows that T 1 T 0 = C 2π 1 k 1 ln R 2 R k 2 ln R 3 R h This can be expressed as Q = 2π 1 k 1 ln R 2 R k 2 ln R 3 R hr 3 T 1 T 0 Optimal Insulation on a Pipe Is there an optimal thickness for the exterior insulation? In the context of the problem just formulated, is there a best value for R 3? Note that Q = f(r 3 ). To find an extremum, dq = 0 and d2 Q 2 < 0 Some algebra yields: hr 3 = 1 dr 3 dr 3 k 2 It offers a critical radius for R 3 = k 2 /h beyond which the heat loss increases. ChE 333 8

Consider 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

Consider 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

University of Rome Tor Vergata

University 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 information

Conduction Heat Transfer. Fourier Law of Heat Conduction. Thermal Resistance Networks. Resistances in Series. x=l Q x+ Dx. insulated x+ Dx.

Conduction 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 information

Chapter 10: Steady Heat Conduction

Chapter 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 information

Review: Conduction. Breaking News

Review: 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 information

Time-Dependent Conduction :

Time-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 information

Numerical Heat and Mass Transfer

Numerical 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 information

Chapter 2 HEAT CONDUCTION EQUATION

Chapter 2 HEAT CONDUCTION EQUATION Heat and Mass Transfer: Fundamentals & Applications 5th Edition in SI Units Yunus A. Çengel, Afshin J. Ghajar McGraw-Hill, 2015 Chapter 2 HEAT CONDUCTION EQUATION Mehmet Kanoglu University of Gaziantep

More information

Heat processes. Heat exchange

Heat processes. Heat exchange Heat processes Heat exchange Heat energy transported across a surface from higher temperature side to lower temperature side; it is a macroscopic measure of transported energies of molecular motions Temperature

More information

ASSUMPTIONS: (1) Homogeneous medium with constant properties, (2) Negligible radiation effects.

ASSUMPTIONS: (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 information

Chapter 3: Steady Heat Conduction

Chapter 3: Steady Heat Conduction 3-1 Steady Heat Conduction in Plane Walls 3-2 Thermal Resistance 3-3 Steady Heat Conduction in Cylinders 3-4 Steady Heat Conduction in Spherical Shell 3-5 Steady Heat Conduction with Energy Generation

More information

Chapter 2: Heat Conduction. Dr Ali Jawarneh Department of Mechanical Engineering, Hashemite University

Chapter 2: Heat Conduction. Dr Ali Jawarneh Department of Mechanical Engineering, Hashemite University Chapter : Heat Conduction Equation Dr Ali Jawarneh Department of Mechanical Engineering, Hashemite University Objectives When you finish studying this chapter, you should be able to: Understand multidimensionality

More information

FIND: (a) Sketch temperature distribution, T(x,t), (b) Sketch the heat flux at the outer surface, q L,t as a function of time.

FIND: (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 information

Chapter 3: Steady Heat Conduction. Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University

Chapter 3: Steady Heat Conduction. Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University Chapter 3: Steady Heat Conduction Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University Objectives When you finish studying this chapter, you should be able to: Understand the concept

More information

Transport processes. 7. Semester Chemical Engineering Civil Engineering

Transport processes. 7. Semester Chemical Engineering Civil Engineering Transport processes 7. Semester Chemical Engineering Civil Engineering 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume Analysis 4. Differential Analysis of Fluid Flow 5. Viscous

More information

Radiant Heating Panel Thermal Analysis. Prepared by Tim Fleury Harvard Thermal, Inc. October 7, 2003

Radiant Heating Panel Thermal Analysis. Prepared by Tim Fleury Harvard Thermal, Inc. October 7, 2003 Radiant Heating Panel Thermal Analysis Prepared by Tim Fleury Harvard Thermal, Inc. October 7, 2003 Analysis Objective Perform a thermal test on a small sample of the concrete to determine the Thermal

More information

Introduction to Heat and Mass Transfer. Week 5

Introduction to Heat and Mass Transfer. Week 5 Introduction to Heat and Mass Transfer Week 5 Critical Resistance Thermal resistances due to conduction and convection in radial systems behave differently Depending on application, we want to either maximize

More information

Chapter 2 HEAT CONDUCTION EQUATION

Chapter 2 HEAT CONDUCTION EQUATION Heat and Mass Transfer: Fundamentals & Applications Fourth Edition Yunus A. Cengel, Afshin J. Ghajar McGraw-Hill, 2011 Chapter 2 HEAT CONDUCTION EQUATION Mehmet Kanoglu University of Gaziantep Copyright

More information

( ) PROBLEM C 10 C 1 L m 1 50 C m K W. , the inner surface temperature is. 30 W m K

( ) PROBLEM C 10 C 1 L m 1 50 C m K W. , the inner surface temperature is. 30 W m K PROBLEM 3. KNOWN: Temperatures and convection coefficients associated with air at the inner and outer surfaces of a rear window. FIND: (a) Inner and outer window surface temperatures, T s,i and T s,o,

More information

Chapter 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 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 information

QUESTION ANSWER. . e. Fourier number:

QUESTION 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 information

Chapter 5 Time-Dependent Conduction

Chapter 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 information

University of New Mexico Mechanical Engineering Spring 2012 PhD qualifying examination Heat Transfer

University of New Mexico Mechanical Engineering Spring 2012 PhD qualifying examination Heat Transfer University of New Mexico Mechanical Engineering Spring 2012 PhD qualifying examination Heat Transfer Closed book. Formula sheet and calculator are allowed, but not cell phones, computers or any other wireless

More information

Shell Balances Spherical Geometry. Remember, we can substitute Fourier's Laws anytime to get: dt

Shell Balances Spherical Geometry. Remember, we can substitute Fourier's Laws anytime to get: dt Shell Balances Spherical Geometry ChE B S.S. Heat conduction with source term: Try spherical geometry using a shell balance: Input Output Source Qr r Qr rδr Sr 4 π r Δ r r Remember, we can substitute Fourier's

More information

1. Nusselt number and Biot number are computed in a similar manner (=hd/k). What are the differences between them? When and why are each of them used?

1. Nusselt number and Biot number are computed in a similar manner (=hd/k). What are the differences between them? When and why are each of them used? 1. Nusselt number and Biot number are computed in a similar manner (=hd/k). What are the differences between them? When and why are each of them used?. During unsteady state heat transfer, can the temperature

More information

Chapter 2: Heat Conduction Equation

Chapter 2: Heat Conduction Equation -1 General Relation for Fourier s Law of Heat Conduction - Heat Conduction Equation -3 Boundary Conditions and Initial Conditions -1 General Relation for Fourier s Law of Heat Conduction (1) The rate of

More information

Chapter 2: Steady Heat Conduction

Chapter 2: Steady Heat Conduction 2-1 General Relation for Fourier s Law of Heat Conduction 2-2 Heat Conduction Equation 2-3 Boundary Conditions and Initial Conditions 2-4 Variable Thermal Conductivity 2-5 Steady Heat Conduction in Plane

More information

AR/IA 241 LN 231 Lecture 4: Fundamental of Energy

AR/IA 241 LN 231 Lecture 4: Fundamental of Energy Faculty of Architecture and Planning Thammasat University A/IA 24 LN 23 Lecture 4: Fundamental of Energy Author: Asst. Prof. Chalermwat Tantasavasdi. Heat For a specific substance, the heat given to the

More information

Conductors and Dielectrics

Conductors and Dielectrics 5.1 Current and Current Density Conductors and Dielectrics Electric charges in motion constitute a current. The unit of current is the ampere (A), defined as a rate of movement of charge passing a given

More information

Introduction to Heat and Mass Transfer. Week 9

Introduction 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 information

Conduction 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. 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 information

MECH 375, Heat Transfer Handout #5: Unsteady Conduction

MECH 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 information

TRANSIENT HEAT CONDUCTION

TRANSIENT 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 information

AC vs. DC Circuits. Constant voltage circuits. The voltage from an outlet is alternating voltage

AC vs. DC Circuits. Constant voltage circuits. The voltage from an outlet is alternating voltage Circuits AC vs. DC Circuits Constant voltage circuits Typically referred to as direct current or DC Computers, logic circuits, and battery operated devices are examples of DC circuits The voltage from

More information

1 Conduction Heat Transfer

1 Conduction Heat Transfer Eng690 - Formula Sheet 2 Conduction Heat Transfer. Cartesian Co-ordinates q x xa x A x dt dx R th A 2 T x 2 + 2 T y 2 + 2 T z 2 + q T T x) plane wall of thicness 2, x 0 at centerline, T s, at x, T s,2

More information

Department of Energy Fundamentals Handbook

Department of Energy Fundamentals Handbook Department of Energy Fundamentals Handbook THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW, Module 2 TABLE OF CONTENTS TABLE OF CONTENTS LIST OF FIGURES... iii LIST OF TABLES... REFERENCES... OBJECTIVES...

More information

Introduction to Heat and Mass Transfer. Week 7

Introduction 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 information

1 Conduction Heat Transfer

1 Conduction Heat Transfer Eng6901 - Formula Sheet 3 (December 1, 2015) 1 1 Conduction Heat Transfer 1.1 Cartesian Co-ordinates q x = q xa x = ka x dt dx R th = L ka 2 T x 2 + 2 T y 2 + 2 T z 2 + q k = 1 T α t T (x) plane wall of

More information

Arctice Engineering Module 3a Page 1 of 32

Arctice Engineering Module 3a Page 1 of 32 Welcome back to the second part of the second learning module for Fundamentals of Arctic Engineering online. We re going to review in this module the fundamental principles of heat transfer. Exchange of

More information

Liquid or gas flow through pipes or ducts is commonly used in heating and

Liquid or gas flow through pipes or ducts is commonly used in heating and cen58933_ch08.qxd 9/4/2002 11:29 AM Page 419 INTERNAL FORCED CONVECTION CHAPTER 8 Liquid or gas flow through pipes or ducts is commonly used in heating and cooling applications. The fluid in such applications

More information

STEADY HEAT CONDUCTION IN PLANE WALLS

STEADY HEAT CONDUCTION IN PLANE WALLS FIGUE 3 STEADY HEAT CONDUCTION IN PLANE WALLS The energy balance for the wall can be expressed as ate of ate of heat trans fer heat trans fer into the wall out of the wall ate of change of the energy of

More information

ESRL Module 8. Heat Transfer - Heat Recovery Steam Generator Numerical Analysis

ESRL Module 8. Heat Transfer - Heat Recovery Steam Generator Numerical Analysis ESRL Module 8. Heat Transfer - Heat Recovery Steam Generator Numerical Analysis Prepared by F. Carl Knopf, Chemical Engineering Department, Louisiana State University Documentation Module Use Expected

More information

HEAT EXCHANGER. Objectives

HEAT EXCHANGER. Objectives HEAT EXCHANGER Heat exchange is an important unit operation that contributes to efficiency and safety of many processes. In this project you will evaluate performance of three different types of heat exchangers

More information

Solutions to PS 2 Physics 201

Solutions to PS 2 Physics 201 Solutions to PS Physics 1 1. ke dq E = i (1) r = i = i k eλ = i k eλ = i k eλ k e λ xdx () (x x) (x x )dx (x x ) + x dx () (x x ) x ln + x x + x x (4) x + x ln + x (5) x + x To find the field for x, we

More information

Parallel Plate Heat Exchanger

Parallel 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 information

MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS. 1 2 x + 1. y = + 1 = x 1/ = 1. y = 1 2 x 3/2 = 1. into this equation would have then given. y 1.

MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS. 1 2 x + 1. y = + 1 = x 1/ = 1. y = 1 2 x 3/2 = 1. into this equation would have then given. y 1. MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS ) If x + y =, find y. IMPLICIT DIFFERENTIATION Solution. Taking the derivative (with respect to x) of both sides of the given equation, we find that 2 x + 2 y y =

More information

ENGI 3424 First Order ODEs Page 1-01

ENGI 3424 First Order ODEs Page 1-01 ENGI 344 First Order ODEs Page 1-01 1. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with respect

More information

Specific heat capacity. Convective heat transfer coefficient. Thermal diffusivity. Lc ft, m Characteristic length (r for cylinder or sphere; for slab)

Specific 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 information

Review. Spring Semester /21/14. Physics for Scientists & Engineers 2 1

Review. Spring Semester /21/14. Physics for Scientists & Engineers 2 1 Review Spring Semester 2014 Physics for Scientists & Engineers 2 1 Notes! Homework set 13 extended to Tuesday, 4/22! Remember to fill out SIRS form: https://sirsonline.msu.edu Physics for Scientists &

More information

Session 5 Heat Conduction in Cylindrical and Spherical Coordinates I

Session 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 information

The temperature of a body, in general, varies with time as well

The 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 information

Transient Heat Conduction in a Circular Cylinder

Transient Heat Conduction in a Circular Cylinder Transient Heat Conduction in a Circular Cylinder The purely radial 2-D heat equation will be solved in cylindrical coordinates using variation of parameters. Assuming radial symmetry the solution is represented

More information

PDHengineer.com Course M-3003

PDHengineer.com Course M-3003 PDHengineer.com Course M-3003 Heat Transfer Fundamentals To receive credit for this course This document is the course text. You may review this material at your leisure either before or after you purchase

More information

CBSE X Mathematics All India 2012 Solution (SET 1) Section D

CBSE X Mathematics All India 2012 Solution (SET 1) Section D CBSE X Mathematics All India 0 Solution (SET ) Section D Q9. The numerator of a fraction is less than its denominator. If is added to the denominator, the fraction is decreased by. Find the fraction. 5

More information

Thermal Unit Operation (ChEg3113)

Thermal Unit Operation (ChEg3113) Thermal Unit Operation (ChEg3113) Lecture 3- Examples on problems having different heat transfer modes Instructor: Mr. Tedla Yeshitila (M.Sc.) Today Review Examples Multimode heat transfer Heat exchanger

More information

1 CHAPTER 4 THERMAL CONDUCTION

1 CHAPTER 4 THERMAL CONDUCTION 1 CHAPTER 4 THERMAL CONDUCTION 4. The Error Function Before we start this chapter, let s just make sure that we are familiar with the error function erf a. We may need it during this chapter. 1 Here is

More information

Principles 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) 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 information

ELEC9712 High Voltage Systems. 1.2 Heat transfer from electrical equipment

ELEC9712 High Voltage Systems. 1.2 Heat transfer from electrical equipment ELEC9712 High Voltage Systems 1.2 Heat transfer from electrical equipment The basic equation governing heat transfer in an item of electrical equipment is the following incremental balance equation, with

More information

Figure 1.1. Relation between Celsius and Fahrenheit scales. From Figure 1.1. (1.1)

Figure 1.1. Relation between Celsius and Fahrenheit scales. From Figure 1.1. (1.1) CHAPTER I ELEMENTS OF APPLIED THERMODYNAMICS 1.1. INTRODUCTION. The Air Conditioning systems extract heat from some closed location and deliver it to other places. To better understanding the principles

More information

Conduction Heat Transfer HANNA ILYANI ZULHAIMI

Conduction Heat Transfer HANNA ILYANI ZULHAIMI + Conduction Heat Transfer HNN ILYNI ZULHIMI + OUTLINE u CONDUCTION: PLNE WLL u CONDUCTION: MULTI LYER PLNE WLL (SERIES) u CONDUCTION: MULTI LYER PLNE WLL (SERIES ND PRLLEL) u MULTIPLE LYERS WITH CONDUCTION

More information

Thermodynamics Heat Transfer

Thermodynamics Heat Transfer Thermodynamics Heat Transfer Lana Sheridan De Anza College April 30, 2018 Last time heat transfer conduction Newton s law of cooling Overview continue heat transfer mechanisms conduction over a distance

More information

ragsdale (zdr82) HW5 ditmire (58335) 1

ragsdale (zdr82) HW5 ditmire (58335) 1 ragsdale (zdr82) HW5 ditmire (58335) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 (part 1 of 2) 10.0

More information

Lecture D Steady State Heat Conduction in Cylindrical Geometry

Lecture D Steady State Heat Conduction in Cylindrical Geometry Conduction and Convection Heat Transfer Prof. S.K. Som Prof. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology Kharagpur Lecture - 08 1D Steady State Heat Conduction

More information

Handout 5: Current and resistance. Electric current and current density

Handout 5: Current and resistance. Electric current and current density 1 Handout 5: Current and resistance Electric current and current density Figure 1 shows a flow of positive charge. Electric current is caused by the flow of electric charge and is defined to be equal to

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 03 Finned Surfaces Fausto Arpino f.arpino@unicas.it Outline Introduction Straight fin with constant circular cross section Long

More information

Fluid Mechanics II Viscosity and shear stresses

Fluid Mechanics II Viscosity and shear stresses Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small

More information

Lumped parameter thermal modelling

Lumped parameter thermal modelling Chapter 3 umped parameter thermal modelling This chapter explains the derivation of a thermal model for a PMSM by combining a lumped parameter (P) model and an analytical distributed model. The whole machine

More information

Quiz 4 (Discussion Session) Phys 1302W.400 Spring 2018

Quiz 4 (Discussion Session) Phys 1302W.400 Spring 2018 Quiz 4 (Discussion ession) Phys 1302W.400 pring 2018 This group quiz consists of one problem that, together with the individual problems on Friday, will determine your grade for quiz 4. For the group problem,

More information

If there is convective heat transfer from outer surface to fluid maintained at T W.

If there is convective heat transfer from outer surface to fluid maintained at T W. Heat Transfer 1. What are the different modes of heat transfer? Explain with examples. 2. State Fourier s Law of heat conduction? Write some of their applications. 3. State the effect of variation of temperature

More information

Total energy in volume

Total 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 information

One-Dimensional, Steady-State. State Conduction without Thermal Energy Generation

One-Dimensional, Steady-State. State Conduction without Thermal Energy Generation One-Dimensional, Steady-State State Conduction without Thermal Energy Generation Methodology of a Conduction Analysis Specify appropriate form of the heat equation. Solve for the temperature distribution.

More information

ASSUMPTIONS: (1) One-dimensional, radial conduction, (2) Constant properties.

ASSUMPTIONS: (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 information

dq = λa T Direction of heat flow dq = λa T h T c

dq = λa T Direction of heat flow dq = λa T h T c THERMAL PHYSIS LABORATORY: INVESTIGATION OF NEWTON S LAW OF OOLING Along a well-insulated bar (i.e., heat loss from the sides can be neglected), it is experimentally observed that the rate of heat flow

More information

Heat Transfer. Solutions for Vol I _ Classroom Practice Questions. Chapter 1 Conduction

Heat Transfer. Solutions for Vol I _ Classroom Practice Questions. Chapter 1 Conduction Heat ransfer Solutions for Vol I _ lassroom Practice Questions hapter onduction r r r K K. ns: () ase (): Higher thermal conductive material is inside and lo thermal conductive material is outside K K

More information

Lecture 4: Classical Illustrations of Macroscopic Thermal Effects. Heat capacity of solids & liquids. Thermal conductivity

Lecture 4: Classical Illustrations of Macroscopic Thermal Effects. Heat capacity of solids & liquids. Thermal conductivity Lecture 4: Classical Illustrations of Macroscopic Thermal Effects Heat capacity of solids & liquids Thermal conductivity References for this Lecture: Elements Ch 3,4A-C Reference for Lecture 5: Elements

More information

Phys460.nb Back to our example. on the same quantum state. i.e., if we have initial condition (5.241) ψ(t = 0) = χ n (t = 0)

Phys460.nb Back to our example. on the same quantum state. i.e., if we have initial condition (5.241) ψ(t = 0) = χ n (t = 0) Phys46.nb 89 on the same quantum state. i.e., if we have initial condition ψ(t ) χ n (t ) (5.41) then at later time ψ(t) e i ϕ(t) χ n (t) (5.4) This phase ϕ contains two parts ϕ(t) - E n(t) t + ϕ B (t)

More information

(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2.

(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2. Math 180 Written Homework Assignment #8 Due Tuesday, November 11th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,

More information

Evaluating this approximately uniform field at the little loop s center which happens to lie on the big loop s axis we find

Evaluating this approximately uniform field at the little loop s center which happens to lie on the big loop s axis we find PHY 35 K. Solutions for problem set #1. Problem 7.: a) We assume the small loop is so much smaller than the big loop or the distance between the loops that the magnetic field of the big loop is approximately

More information

Lectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 6

Lectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 6 Lectures on Nuclear Power Safety Lecture No 6 Title: Introduction to Thermal-Hydraulic Analysis of Nuclear Reactor Cores Department of Energy Technology KTH Spring 2005 Slide No 1 Outline of the Lecture

More information

Table of Contents. Foreword... xiii. Preface... xv

Table of Contents. Foreword... xiii. Preface... xv Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations...

More information

Internal Flow: Heat Transfer in Pipes

Internal Flow: Heat Transfer in Pipes Internal Flow: Heat Transfer in Pipes V.Vuorinen Aalto University School of Engineering Heat and Mass Transfer Course, Autumn 2016 November 15 th 2016, Otaniemi ville.vuorinen@aalto.fi First about the

More information

One dimensional steady state diffusion, with and without source. Effective transfer coefficients

One dimensional steady state diffusion, with and without source. Effective transfer coefficients One dimensional steady state diffusion, with and without source. Effective transfer coefficients 2 mars 207 For steady state situations t = 0) and if convection is not present or negligible the transport

More information

Use of Phase-Change Materials to Enhance the Thermal Performance of Building Insulations

Use of Phase-Change Materials to Enhance the Thermal Performance of Building Insulations Introduction Use of Phase-Change Materials to Enhance the Thermal Performance of Building Insulations R. J. Alderman, Alderman Research Ltd., Wilmington, DE David W. Yarbrough, R&D Services, Inc., Cookeville,

More information

Thermal Systems Design

Thermal Systems Design Thermal Systems Design Fundamentals of heat transfer Radiative equilibrium Surface properties Non-ideal effects Internal power generation Environmental temperatures Conduction Thermal system components

More information

PHYS 212 Final Exam (Old Material) Solutions - Practice Test

PHYS 212 Final Exam (Old Material) Solutions - Practice Test PHYS 212 Final Exam (Old Material) Solutions - Practice Test 1E If the ball is attracted to the rod, it must be made of a conductive material, otherwise it would not have been influenced by the nearby

More information

4. Analysis of heat conduction

4. 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 information

ECE309 INTRODUCTION TO THERMODYNAMICS & HEAT TRANSFER. 10 August 2005

ECE309 INTRODUCTION TO THERMODYNAMICS & HEAT TRANSFER. 10 August 2005 ECE309 INTRODUCTION TO THERMODYNAMICS & HEAT TRANSFER 0 August 2005 Final Examination R. Culham & M. Bahrami This is a 2 - /2 hour, closed-book examination. You are permitted to use one 8.5 in. in. crib

More information

1 One-Dimensional, Steady-State Conduction

1 One-Dimensional, Steady-State Conduction 1 One-Dimensional, Steady-State Conduction 1.1 Conduction Heat Transfer 1.1.1 Introduction Thermodynamics defines heat as a transfer of energy across the boundary of a system as a result of a temperature

More information

Lecture 7 - Separable Equations

Lecture 7 - Separable Equations Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only

More information

Exam 2 Solutions. Note that there are several variations of some problems, indicated by choices in parentheses.

Exam 2 Solutions. Note that there are several variations of some problems, indicated by choices in parentheses. Exam 2 Solutions Note that there are several variations of some problems, indicated by choices in parentheses. Problem 1 Part of a long, straight insulated wire carrying current i is bent into a circular

More information

6 Chapter. Current and Resistance

6 Chapter. Current and Resistance 6 Chapter Current and Resistance 6.1 Electric Current... 6-2 6.1.1 Current Density... 6-2 6.2 Ohm s Law... 6-5 6.3 Summary... 6-8 6.4 Solved Problems... 6-9 6.4.1 Resistivity of a Cable... 6-9 6.4.2 Charge

More information

Number of pages in the question paper : 05 Number of questions in the question paper : 48 Modeling Transport Phenomena of Micro-particles Note: Follow the notations used in the lectures. Symbols have their

More information

Unsteady State Heat Conduction in a Bounded Solid

Unsteady 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 information

Dimensionless Numbers

Dimensionless Numbers 1 06.10.2017, 09:49 Dimensionless Numbers A. Salih Dept. of Aerospace Engineering IIST, Thiruvananthapuram The nondimensionalization of the governing equations of fluid flow is important for both theoretical

More information

ENGI 2422 First Order ODEs - Separable Page 3-01

ENGI 2422 First Order ODEs - Separable Page 3-01 ENGI 4 First Order ODEs - Separable Page 3-0 3. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with

More information

Yell if you have any questions

Yell if you have any questions Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 Before Starting All of your grades should now be posted

More information

3.3 Unsteady State Heat Conduction

3.3 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 information

The Electrodynamics of a Pair of PV Modules with Connected Building Resistance

The Electrodynamics of a Pair of PV Modules with Connected Building Resistance Proc. of the 3rd IASME/WSEAS Int. Conf. on Energy, Environment, Ecosystems and Sustainable Development, Agios Nikolaos, Greece, July 24-26, 2007 563 he Electrodynamics of a Pair of s with Connected Building

More information

THE WAVE EQUATION. d = 1: D Alembert s formula We begin with the initial value problem in 1 space dimension { u = utt u xx = 0, in (0, ) R, (2)

THE WAVE EQUATION. d = 1: D Alembert s formula We begin with the initial value problem in 1 space dimension { u = utt u xx = 0, in (0, ) R, (2) THE WAVE EQUATION () The free wave equation takes the form u := ( t x )u = 0, u : R t R d x R In the literature, the operator := t x is called the D Alembertian on R +d. Later we shall also consider the

More information

PROBLEM 3.8 ( ) 20 C 10 C m m m W m K W m K 1.4 W m K. 10 W m K 80 W m K

PROBLEM 3.8 ( ) 20 C 10 C m m m W m K W m K 1.4 W m K. 10 W m K 80 W m K PROBLEM 3.8 KNOWN: Dimensions of a thermopane window. Room and ambient air conditions. FIND: (a) Heat loss through window, (b) Effect of variation in outside convection coefficient for double and triple

More information