354033470X
Solving Direct and
Inverse Heat Conduction
Problems
Copyright Page
Preface
Table of contents
Nomenclature
Part I Heat Conduction Fundamentals
1 Fourier Law
Literature
2 Mass and Energy Balance Equations
2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity
2.2 Inner Energy Balance Equation
2.2.1 Energy Balance Equations in Three Basic Coordinate Systems
2.3 Hyperbolic Heat Conduction Equation
2.4 Initial and Boundary Conditions
2.4.1 First Kind Boundary Conditions (Dirichlet Conditions)
2.4.2 Second Kind Boundary Conditions von Neumann Conditions)
2.4.3 Third Kind Boundary Conditions
2.4.4 Fourth Kind Boundary Conditions
2.4.5 Non-Linear Boundary Conditions
2.4.6 Boundary Conditions on the Phase Boundaries
Literature
3 The Reduction of Transient Heat Conduction Equations and Boundary Conditions
3.1 Linearization of a Heat Conduction Equation
3.2 Spatial Averaging of Temperature
3.2.1 A Body Model with a Lumped Thermal Capacity
3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness
3.2.3 Heat Conduction Equation for a Circular Fin with Uniform Thickness
3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity
Literature
4 Substituting Heat Conduction Equation by Two-Equations System
4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient
4.2 One-Dimensional Inverse Transient Heat Conduction Problem
Literature
5 Variable Change
Literature
Part II Exercises. Solving Heat Conduction Problems
6 Heat Transfer Fundamentals
Exercise 6.1 Fourier Law in a Cylindrical Coordinate System
Solution
Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation
Solution
Exercise 6.3 Heat Transfer Through a Flat Single-Layeredand Double-Layered Wall
Solution
Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall
Solution
Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe
Solution
Exercise 6.6 Radiant TubeTemperature
Solution
Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall
Solution
Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity
Solution
Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor
Solution
Exercise 6.10 Determining Heat FluxBy Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity
Solution
Exercise 6.11 One-Dimensional Steady-State Plate Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources
Solution
Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources
Solution
Exercise 6.13 Inverse Steady-State Heat Conduction Problem in a Pipe
Solution
Exercise 6.14 General Equation of Heat Conduction in Fins
Solution
Exercise 6.15 Temperature Distribution and Efficiency of a Straight Fin with Constant Thickness
Solution
Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device
Solution
Exercise 6.17 Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness
Solution
Exercise 6.18 Approximated Calculation of a Circular Fin Efficiency
Solution
Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins
Solution
Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method
Solution
Exercise 6.21 Calculating Rectangular Fin Efficiency
Solution
Exercise 6.22 HeatTransfer Coefficient in Exchangers with Extended Surfaces
Solution
Exercise 6.23 Calculating Overall HeatTransfer Coefficient in a Fin Plate Exchanger
Solution
Exercise 6.24 Overall HeatTransfer Coefficient for a Longitudinally Finned Pipewith a Scale Layer on an Inner Surface
Solution
Exercise 6.25 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe
Solution
Exercise 6.26 Determining One-Dimensional Temperature Distribution in a Flat Wall by Means of Finite Volume Method
Solution
Exercise 6.27 Determining One-Dimensional Temperature Distribution in a Cylindrical Wall By Means of Finite Volume Method
Solution
Exercise 6.28 Inverse Steady-State Heat Conduction Problem for a Pipe Solved by Space-Marching Method
Solution
Exercise 6.29 Temperature Distribution and Efficiency of a Circular Fin with Temperature-Dependent Thermal Conductivity
Solution
Literature
7 Two-Dimensional Steady-State Heat Conduction. Analytical Solutions
Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness
Solution
Exercise 7.2 Temperature Distribution in a Straight Fin with Constant Thickness and Insulated Tip
Solution
Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip
Solution
Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler
Solution
Literature
8 Analytical Approximation Methods. Integral Heat Balance Method
Exercise 8.1 Temperature Distribution within a Rectangular Cross-Section of a Bar
Solution
Exercise 8.2 Temperature Distribution in an Infinitely Long Fin of Constant Thickness
Solution
Exercise 8.3 Determining Temperature Distribution in a Boiler's Water-Wall Tube by Means of Functional Correction Method
Solution
Literature
9 Two-Dimensional Steady-State Heat Conduction. Graphical Method
Exercise 9.1 Temperature Gradient and Surface-Transmitted Heat Flow
Solution
Exercise 9.2 Orthogonality of Constant Temperature Line and Constant Heat Flux
Solution
Exercise 9.3 Determining Heat Flow between Isothermal Surfaces
Solution
Exercise 9.4 Determining Heat Loss Through a Chimney Wall; Combustion Channel (Chimney) with Square Cross-Section
Solution
Exercise 9.5 Determining Heat Loss Through Chimney Wall with a Circular Cross-Section
Solution
Literature
10 Two-Dimensional Steady-State Problems. The Shape Coefficient
Exercise 10.1 Buried Pipe-to-Ground Surface Heat Flow
Solution
Exercise 10.2 Floor Heating
Solution
Exercise 10.3 Temperature of a Radioactive Waste Container Buried Underground
Solution
Literature
11 Solving Steady-State Heat Conduction Problems by Means of Numerical Methods
Exercise 11.1 Description of the Control Volume Method
Solution
a) Heat balance equation- Cartesian coordinates
b) Heat balance equation-cylindrical coordinates
Exercise 11.2 Determining Temperature Distribution in a Square Cross-Section of a Long Rod by Means of the Finite Volume Method
Solution
Exercise 11.3A Two-Dimensional Inverse Steady-State Heat Conduction Problem
Solution
Exercise 11.4 Gauss-Seidel Method and Over-Relaxation Method
Solution
Exercise 11.5 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Uniform Thickness by Means of the Finite Volume Method
Solution
Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney
Solution
Exercise 11.7 Pseudo-Transient Determination of Steady StateTemperatureDistribution in a Square Cross-Section of a Chimney; Heat Transfer by Convection and Radiation on an Outer Surface of a Chimney
Solution
Exercise 11.8 Finite Element Method
Historical Development of FEM
Solution
Exercise 11.9 Linear Functions That Interpolate Temperature Distribution (Shape Functions) Inside Triangular and Rectangular Elements
Solution
Exercise 11.10 Description of FEM Based on Galerkin Method
Solution
Exercise 11.11 Determining Conductivity Matrix for a Rectangular and Triangular Element
Solution
a) Conductivity matrix [Kec] for a finite rectangular element
b) Conductivity matrix [Kec] for a finite triangular element
Solution
a) Rectangular finite element
b) Triangular finite element
Exercise 11.13 Determining Vector {fqe} with Respect to Volumetric and Point Heat Sources in a Rectangular and Triangular Element
Solution
a) Rectangular element
b) Triangular element
Solution
a) Finite rectangular element
b) Finite triangular element
Exercise 11.15 Methods for Building Global Equation System in FEM
Solution
Exercise 11.16 Determining Temperature Distribution in a Square Cross-Section of an Infinitely Long Rod by Means of FEM, in which the Global Equation System is Constructed using Method I (from Ex. 11.15)
Solution
Exercise 11.17 Determining Temperature Distributionin an In finitely Long Rod with Square Cross-Sectionby Means of FEM, in which the Global Equation System is Constructed using Method II (from Ex. 11.15)
Solution
Exercise 11.18 Determining Temperature Distribution by Means of FEM in an Infinitely Long Rod with Square Cross-Section, in which Volumetric Heat Sources Operate
Solution
Exercise 11.19 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Constant Thickness by Means of FEM
Solution
Exercise 11.20 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Straight Fin with Constant Thickness (ANSYS Program)
Solution
Exercise 11.21 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Hexagonal Fin with Constant Thickness (ANSYS Program)
Solution
Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS Program)
Solution
Literature
12 Finite Element Balance Method and Boundary Element Method
Exercise 12.1 Finite Element Balance Method
Solution
Exercise 12.2 Boundary Element Method
Solution
Exercise 12.3 Determining Temperature Distribution in Square Region by Means of FEM Balance Method
Solution
Exercise 12.4 Determining Temperature Distribution in a Square Region using Boundary Element Method
Solution
Literature
13 Transient Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings
Exercise 13.1 Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings
Solution
Exercise 13.2 Heat Exchange between a Body with Lumped Thermal Capacity and Surroundings with Time-Dependent Temperature
Solution
Exercise 13.3 Determining Temperature Distribution of a Body with Lumped Thermal Capacity, when the Temperature of a Medium Changes Periodically
Solution
Exercise 13.4 Inverse Problem: Determining Temperature of a Medium on the Basis of Temporal Thermometer Indicated Temperature History
Solution
Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple
Solution
Exercise 13.6 Determining the Time It Takes to Cool Body Down to a Given Temperature
Solution
Exercise 13.7 Temperature Measurement Error of a Medium whoseTemperature Changes at Constant Rate
Solution
Exercise 13.8 Temperature Measurement Error of a Medium whose Temperature Changes Periodically
Solution
Exercise 13.9 Inverse Problem: Calculating Temperature of a Medium whose Temperature Changes Periodically, on the Basis of Temporal Temperature History Indicated by a Thermometer
Solution
Exercise 13.10 Measuring Heat Flux
Solution
Literature
14 Transient Heat Conduction in Half-Space
Exercise 14.1 Laplace Transform
Solution
Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature
Solution
Exercise 14.3 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increasein Heat Flux
Solution
Exercise 14.4 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Temperature of a Medium
Solution
Exercise 14.5 Formula Derivation for Temperature Distribution in a Half-Space when Surface Temperature isTime-Dependent
Solution
Exercise 14.6 Formula Derivation for a Quasi-Steady State Temperature Field in a Half-Space when Surface Temperature Changes Periodically
Solution
Exercise 14.7 Formula Derivation for Temperature of Two Contacting Semi-Infinite Bodies
Solution
Exercise 14.8 Depth of Heat Penetration
Solution
Exercise 14.9 Calculating Plate Surface Temperature Under the Assumption that the Plate is a Semi-Infinite Body
Solution
Exercise 14.10 Calculating Ground Temperature at a Specific Depth
Solution
Exercise 14.11 Calculating the Depth of Heat Penetration in the Wall of a Combustion Engine
Solution
Exercise 14.12 Calculating auasi-Steady-State Ground Temperature at a Specific Depth when Surface Temperature Changes Periodically
Solution
Exercise 14.13 Calculating Surface Temperature at the Contact Point of Two Objects
Solution
Literature
15 Transient Heat Conduction in Simple-Shape Elements
Exercise 15.1 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 3rd Kind
Solution
Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind
Solution
Exercise 15.3 Calculating Plate Surface Temperature and Average Temperature Across the Plate Thickness by Means of the Provided Graphs
Solution
Exercise 15.4 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind
Solution
Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind
Solution
Exercise 15.6 Calculating Temperature in an Infinitely Long Cylinder using the Annexed Diagrams
Solution
Exercise 15.7 Formula Derivation for a Temperature Distribution in a Sphere with Boundary Conditionsof 3rd Kind
Solution
Exercise 15.8 A Program for Calculating Temperature Distribution and Its Change Rate in a Sphere with Boundary Conditions of 3rd Kind
Solution
Exercise 15.9 Calculating Temperature of a Sphere using the Diagrams Provided
Solution
Exercise 15.10 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind
Solution
Exercise 15.11 A Program and Calculation Results for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind
Solution
Exercise 15.12 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind
Solution
Exercise 15.13 Program and Calculation Results for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind
Solution
Exercise 15.14 Formula Derivation for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind
Solution
Exercise 15.15 Program and Calculation Results for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind
Solution
Exercise 15.16 Heating Rate Calculations for a Thick-Walled Plate
Solution
Exercise 15.17 Calculating the Heating Rate of a Steel Shaft
Solution
Exercise 15.18 Determining Transients of Thermal Stresses in a Cylinder and a Sphere
Solution
Exercise 15.19 Calculating Temperature and Temperature Change Rate in a Sphere
Solution
Exercise 15.20 Calculating Sensor Thickness for Heat Flux Measuring
Solution
Literature
16 Superposition Method in One-Dimensional Transient Heat Conduction Problems
Exercise 16.1 Derivation of Duhamel Integral
Solution
Exercise 16.2 Derivation of an Analytical Formula for a Half-Space Surface Temperature when Medium's Temperature Undergoes a Linear Change in the Function of Time
Solution
Exercise 16.3 Derivation of an Approximate Formula for a Half-Space Surface Temperature with an Arbitrary Change in Medium's Temperature in the Function of Time
Solution
Exercise 16.4 Definition of an Approximate Formulafor a Half-Space Surface Temperature when Temperature of a Medium Undergoes a Linear Change in the Function of Time
Solution
Exercise 16.5 Application of the Superposition Method when Initial Body Temperature is Non-Uniform
Solution
Exercise 16.6 Description of the Superposition Method Applied to Heat Transfer Problems with Time-Dependent Boundary Conditions
Solution
Example 1
Example 2
Example 3
Example 4
Exercise 16.7 Formula Derivation for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse
Solution
Exercise 16.8 Formula Derivation for a Half-Space Surface Temperature with a Mixed Step-Variable Boundary Condition in Time
Solution
Exercise 16.9 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Triangular Pulse and the Calculationof This Temperature
Solution
Exercise 16.10 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Rectangular Pulse; Temperature Calculation
Solution
Exercise 16.11 A Program and Calculation Results for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse
Solution
Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time
Solution
Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided
Solution
Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier
Solution
Literature
17 Transient Heat Conduction in a Semi-Infinite Body. The Inverse Problem
Exercise 17.1 Measuring Heat Transfer Coefficient. The Transient Method
Solution
Exercise 17.2 Deriving a Formula for Heat Fluxon the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function
Solution
Exercise 17.3 Deriving Heat Flux Formula on the Basis of a Measured and Polynomial-Approximated Half-Space Surface Temperature Transient
Solution
Exercise 17.4 Formula Derivation for a Heat Flux Periodically Changing in Time on the Basis of a Measured Temperature Transient at a Point Located under the Semi-Space Surface
Solution
Exercise 17.5 Deriving a Heat Flux Formula on the Basis of Measured Half-Space Surface Temperature Transient, Approximated by a Linear and Square Function
Solution
Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method
Solution
Exercise 17.7 Determining Heat Fluxon the Basis of a Measured Time Transient of the Half-Space Temperature, Approximated by a Piecewise Linear Function
Solution
Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature
Solution
Literature
18 Inverse Transient Heat Conduction Problems
Exercise 18.1 Derivation of Formulas for Temperature Distribution and Heat Flux in a Simple-Shape Bodies on the Basis of a Measured Temperature Transient in a Single Point
Solution
Exercise 18.2 Formula Derivation for a Temperature of a Medium when Linear Time Change in Plate Surface Temperature is Assigned
Solution
Exercise 18.3 Determining Temperature Transient of a Medium for Which Plate Temperature at a Point with a Given Coordinate Changes According to the Prescribed Function
Solution
Exercise 18.4 Formula Derivation for a Temperature of a Medium, which is Warming an Infinite Plate; Plate Temperature at a Point with a Given Coordinate Changes at Constant Rate
Solution
Exercise 18.5 Determining Temperature and Heat Flux on the Plate Front Face on the Basis of a measured Temperature Transient on an Insulated BackSurface; Heat Flowon the Plate Surface is in the Form of a Triangular Pulse
Solution
Exercise 18.6 Determining Temperature and Heat Flux on the Surface of a Plate Front Faceon the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flowon the Plate Surface is in the Form of a Rectangular Pulse
Solution
Exercise 18.7 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes in a Linear Way
Solution
Exercise 18.8 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes According to the Square Function Assigned
Solution
Literature
19 Multidimensional Problems. The Superposition Method
Exercise 19.1 The Application of the Superposition Method to Multidimensional Problems
Solution
Boundary Conditions of 1st and 3rd Kind
Boundary Conditions of 2nd Kind
Exercise 19.2 Formula Derivation for Temperature Distribution in a Rectangular Region with a Boundary Condition of 3rd Kind
Solution
Exercise 19.3 Formula Derivation for Temperature Distribution in a Rectangular Region with Boundary Conditions of 2nd Kind
Solution
Exercise 19.4 Calculating Temperature in a Steel Cylinder of a Finite Height
Solution
Exercise 19.5 Calculating Steel Block Temperature
Solution
20 Approximate Analytical Methods for Solving Transient Heat Conduction Problems
Exercise 20.1 Description of an Integral Heat Balance Method by Means of a One-Dimensional Transient Heat Conduction Example
Solution
Exercise 20.2 Determining Transient Temperature Distribution in a Flat Wall with Assigned Conditions of 1st, 2nd and 3rd Kind
Solution
Example 1
Example 2
Example 3
Example 4
Exercise 20.3 Determining Thermal Stresses in a Flat Wall
Solution
Literature
21 Finite Difference Method
Exercise 21.1 Methods of Heat Flux Approximation on the Plate Surface
Solution
Exercise 21.2 Explicit Finite Difference Methodwith Boundary Conditions of 1st, 2nd and 3rd Kind
Solution
Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method
Solution
Exercise 21.4 Solving Two-Dimensional Problems by Means of the Implicit Difference Method
Solution
Exercise 21.5 Algorithm and a Program for Solving a Tridiagonal Equation System by Thomas Method
Solution
Exercise 21.6 Stability Analysis of the Explicit Finite Difference Method by Means of the von Neumann Method
Solution
Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program
Solution
Exercise 21.8 Calculating One-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program
Solution
Exercise 21.9 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program; Algebraic Equation System is Solved by Gaussian Elimination Method
Solution
Exercise 21.10 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program; Algebraic Equation System Solved by Over-Relaxation Method
Solution
Literature
22 Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM)
Exercise 22.1 Description of FEM Based on GalerkinMethod Used for Solving Two-Dimensional Transient Heat Conduction Problems
Solution
Exercise 22.2 Concentrated (Lumped) Thermal Finite Element Capacity in FEM
Solution
Exercise 22.3 Methods for Integrating Ordinary Differential Equations with Respect to Time Used in FEM
Solution
Exercise 22.4 Comparison of FEM Based on Galerkin Method and Heat Balance Method with Finite Volume Method
Solution
Exercise 22.5 Natural Coordinate System for One-Dimensional, Two-Dimensional Triangular and Two-Dimensional Rectangular Elements
Solution
a. One-dimensional elements
b. Two-dimensional tetragonal elements
c. Two-dimensional triangular elements
Exercise 22.6 Coordinate System Transformations and Integral Calculations by Means of the Gauss-Legendre Quadratures
Solution
a. One-dimensional elements
b. Tetragonal elements
c. Triangular elements
Exercise 22.7 Calculating Temperature in a Complex ShapeFin by Means of the ANSVS Program
Solution
Literature
23 Numerical-Analytical Methods
Explicit Method
Implicit Method
Crank-Nicolson Method
Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method
Solution
Exercise 23.2 Numerical-Analytical Method for Integrating a Linear Ordinary Differential Equation System
Solution
a. Approximating u(t) with a step function
b. Approximating u(t) with a piecewise linear function
Exercise 23.3 Determining Steel Plate Temperature by Means of the Method of Lines, while the Plate is Cooled by Air and Boiling Water
Solution
Exercise 23.4 Using the Exact Analytical Method and the Method of Lines to Determine Temperature of a Cylindrical Chamber
Solution
Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines
Solution
Exercise 23.6 Determining Temperature Distribution in a Cylindrical Chamber with Constant and Temperature Dependent Thermo-Physical Properties by Means of the Method of Lines
Solution
Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines
Solution
Literature
24 Solving Inverse Heat Conduction Problems by Means of Numerical Methods
Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems
Solution
a. Division of an inverse region into two control volumes (Fig. 24.2a)
b. Division of an inverse region into three control volumes (Fig. 24.2b)
c. Division of an inverse region into four control volumes (Fig. 24.2c)
Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems
Solution
Exercise 24.3 Weber Method Step-Marching Methods in Space
Solution
Exercise 24.4 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Measured Temperature on a Thermally Insulated Back Plate Surface; Heat Flux is in the Shapeof a Rectangular Pulse
Solution
Exercise 24.5 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Temperature Measurement on an Insulated Back Plate Surface; Heat Flux is in the Shape of a Triangular Pulse
Solution
Literature
25 Heat Sources
Exercise 25.1 Determining Formula for Transient Temperature Distribution Around an Instantaneous (Impulse) Point Heat Source Active in an Infinite Space
Solution
Exercise 25.2 Determining Formula for Transient Temperature Distribution in an Infinite Body Produced by an Impulse Surface Heat Source
Solution
Exercise 25.3 Determining Formula for Transient Temperature Distribution Around Instantaneous Linear Impulse Heat Source Active in an Infinite Space
Solution
Exercise 25.4 Determining Formula for Transient Temperature Distribution Around a Point Heat Source, which Lies in an Infinite Space and is Continuously Active
Solution
Exercise 25.5 Determining Formula for a Transient Temperature Distribution Triggered by a Surface Heat Source Continuously Active in an Infinite Space
Solution
Exercise 25.6 Determining Formula for a Transient Temperature Distribution Around a Continuously Active Linear Heat Source with Assigned Power q1 Per Unit of Length
Solution
Exercise 25.7 Determining Formula for Quasi-Steady StateTemperature Distribution Caused by a Point Heat Source with a Power Q0 that Moves at Constant Velocity v in Infinite Space or on the Half Space Surface
Solution
Exercise 25.8 Determining Formula for Transient Temperature Distribution Produced by a Point Heat Source with Power Qo that Moves at Constant Velocity v in Infinite Spaceor on the Half Space Surface
Solution
Exercise 25.9 Calculating Temperature Distribution along a Straight Line Traversed by a Laser Beam
Solution
Exercise 25.10 Quasi-Steady State Temperature Distribution in a Plate During the Welding Process; A Comparison between the Analytical Solution and FEM
Solution
Literature
26 Melting and Solidification (Freezing)
Exercise 26.1 Determination of a Formula which Describes the Solidification (Freezing) and Melting of a Semi-Infinite Body (the Stefan Problem)
Solution
Exercise 26.2 Derivation of a Formula that Describes the Solidification (Freezing) of a Semi-Infinite Body Under the Assumption that the Temperature of a Liquid is Non-Uniform
Solution
Exercise 26.3 Derivation of a Formula that Describe Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer
Solution
Exercise 26.4 Derivation of Formulas that Describe Solidification (Freezing) of Simple-Shape Bodies: Plate, Cylinder and Sphere
Solution
a. Plate
b. Cylinder
Exercise 26.5 Ablation of a Semi-Infinite Body
Solution
Exercise 26.6 Solidification of a Falling Droplet of Lead
Solution
Exercise 26.7 Calculating the Thickness of an Ice Layer After the Assigned Time
Solution
Exercise 26.8 Calculating Accumulated Energy in a Melted Wax
Solution
Exercise 26.9 Calculating Fish Freezing Time
Solution
Literature
Appendix A Basic Mathematical Functions
A.1. Gauss Error Function
A.2. Hyperbolic Functions
A.3. Bessel Functions
Literature
Appendix B Thermo-Physical Properties of Solids
B.1. Tables of Thermo-Physical Properties of Solids
B.2. Diagrams
B.3. Approximated Dependencies for Calculating Thermo Physical Properties of a Steel [8]
Density p at temperature 20ยฐC
Specific heat capacity c in a temperature function
Longitudinal elasticity module (Young's modulus) E in function of temperature
Poisson ratio v in function of temperature
Literature
Appendix C Fin Efficiency Diagrams (for Chap. 6, part II)
Literature
Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, Part II)
Appendix E Subprogram for Solving Linear Algebraic Equations System using Gauss Elimination Method (for Chap. 6, Part II)
Subprogram for solving linearalgebraic equations system using Gauss method
Appendix F Subprogram for Solving a Linear Algebraic Equations System by Means of Over Relaxation Method
Subprogram SOR section appendix f subprogram,for solving a linear algebraic equations system by means of over-relaxation method
Appendix G Subprogram for Solving an Ordinary Differential Equations System of 1st Order using Runge-Kutta Method of 4th Order (for Chap. 11, Part II)
Subprogram for solving an ordinary differential equations system of 1st order using Runge-Kutta method of 4th order
Appendix H Determining inverse Laplace Transform (for Chap. 15, part II)
Literature