Transport II Lecture Notes (2014) T. K. Nguyen Chemical and Materials Engineering Cal Poly Pomona Contents Chapter 1: Introduction to Heat Transfer 1.1 Introduction 1-1 1.2 Importance of Thermodynamic 1-2 1.3 Dilemma in the Study of Heat Transfer 1-2 Example 1.3-1: Final temperature in a oil bath 1-3 Example 1.3-2: Time to melt a slab of ice 1-6 1.4 Reason for Studying Heat Transfer 1-7 1.5 Modes of Heat Transfer 1-7 1.6 Nanoscale Origin of Heat Conduction 1-9 1.7 Thermal Conductivity 1-10 Chapter 2: Constitutive Relation Between q and T 2.1 Conduction 2-1 2.2 Convection 2-4 2.3 Thermal Radiation 2-7 2.4 Energy Balance 2-9 Example 2.4-1 Cylinder and piston system 2-10 First Law as a Rate Equation 2-11 Example 2.4-2 Uniform heat generation in an electrical wire 2-12 Example 2.4-3 One-dimensional steady state conduction 2-14 Example 2.4-4 Radiation heat transfer with a plate 2-15 2.5 Temperature of a Focal Plane Array 2-17 2.6 Examples in Conduction, Convection, and Radiations 2-23 Example 2.6-1 Baseplate heater 2-23 Example 2.6-2 Heating of silicon wafer 2-24 Example 2.6-3 Reacting spherical canister 2-25 Example 2.6-4 Heat dissipiatation in electronic devices 2-26 Example 2.6-5 Heat dissipiatation in computers 2-27 Chapter 3: Conduction Differential Equation 3.1 Derivation 3-1 Example 3.1-1 Equation for one-dimensional conduction 3-6 Example 3.1-2 Heat transfer across a wall 3-9 Example 3.1-3 Heat transfer in a quarter cylinder 3-12 Example 3.1-4 Differential equation for a quarter cylinder 3-13 3.2 Boundary and Initial Conditions 3-15 Example 3.2-1 Heat transfer through a slab 3-18 Example 3.2-2 Temperature distribution along a plate 3-21 Example 3.2-3 Temperature gradient at a surface 3-22 i Chapter 4: Analysis of 1-Dimensional Conduction 4.1 Steady State, Constant Properties, No Heat Generation 4-1 4.2 Cartesian System, Isothermal Surfaces 4-1 4.3 Radial System, Isothermal Surfaces 4-2 4.4 Spherical System, Isothermal Surfaces 4-4 4.5 Electrical and Thermal Analogy 4-6 Example 4.5-1 Thermal circuit for multi-layered wall 4-8 Example 4.5-2 Thermal circuit for composite cylinders 4-9 Example 4.5-3 Thermal circuit for wall with radiation 4-10 4.6 Overall Heat Transfer Coefficient 4-11 Example 4.6-1 Thermal circuit for multi-layered wall 4-13 4.7 Systems with Heat Sources 4-15 Example 4.7-1 Rectangular plate with heat generation 4-15 Example 4.7-2 Temperature distribution in a solid cylinder 4-19 Example 4.7-3 Maximum temperature in a copper wire 4-23 Example 4.7-4 Heat transfer with a strip heater 4-25 Example 4.7-5 Heat transfer through a teacup 4-26 Example 4.7-6 Evaporation time in an IC engine 4-27 Chapter 5: Analysis of Fins and “Extended Surfaces” 5.1 Introduction 5-1 5.2 Heat Transfer for Fins of Uniform Cross-Sectional Area 5-4 Example 5.2-1 Heat transfer from a long cylindrical fin 5-4 Example 5.2-2 Heat transfer from a finite cylindrical fin 5-5 Example 5.2-3 Heat transfer from two sides of a fin 5-7 5.3 Fin Performance 5-9 Example 5.3-1 Efficiency of a rectangular fin 5-10 Example 5.3-2 Efficiency of a triangular fin 5-12 Example 5.3-3 Heat transfer from a stack of fins 5-14 Chapter 6: Two Dimensional, Steady-State Conduction 6.1 The Energy Balance Method 6-1 Example 6.1-1 Steady-state conduction in a column 6-1 Example 6.1-2 Steady-state conduction 6-5 6.2 Partial Differential Equation Solver Software 6-7 Example 6.2-1 Steady-state conduction using COMSOL 6-8 ii Chapter 7: Unsteady-State Conduction 7.1 Introduction 7-1 7.2 The Lumped Capacitance Method 7-2 Example 7.2-1 Heat leaving a sphere by radiation 7-3 Human Body Temperature Regulation 7-6 Example 7.2-2 Heat loss from a body 7-6 Review: First Order Linear Ordinary Differential Equations 7-7 Example 7.2-3 Superheated liquid droplets 7-8 7.3 Differential Energy Balance 7-11 Review: The Newton-Raphson Method 7-14 Example 7.3-1 Cell damage in living tissue 7-18 7.4 Approximate Solutions 7-21 Total Energy Transfer 7-21 Example 7.4-1 Heat transfer in a slab 7-26 Example 7.4-2 Heat transfer in a semi-infinite medium 7-28 Example 7.4-3 Heat transfer through one surface of a slab 7-29 Example 7.4-4 Heat loss to a copper block 7-30 Example 7.4-5 Heat transfer in a finite cylinder 7-31 Chapter 8: Convective Heat Transfer 8.1 Introduction 8-1 8.2 Boundary Layer Concept 8-1 8.3 Correlations for Heat Convection 8-7 External Forced Convection Flow 8-8 Internal Forced Convection Flow 8-10 External Free Convection Flow 8-11 8.4 Correlations for Boiling and Condensation 8-12 Film Boiling 8-12 Film Condensation on a Vertical Plate 8-13 Film Condensation on Radial Systems 8-14 8.5 Examples of Convection Systems 8-15 Example 8.5-1 Water evaporating from a lake 8-15 Example 8.5-2 Heat loss from a scuba diver 8-16 Example 8.5-3 Heat loss from a cylinder 8-18 Example 8.5-4 Free convection from a horizontal plate 8-20 iii Chapter 9: Heat Exchangers 9.1 Introduction 9-1 9.2 Heat Exchanger types 9-4 9.3 Analysis of Heat Exchangers 9-9 Example 9.3-1: Different exchanger configurations 9-17 Example 9.3-2: Length of tube in heat exchanger 9-19 Example 9.3-3: Heat transfer rate for a counter flow exchanger 9-20 9.4 The Effectiveness-NTU Method 9-22 Example 9.4-1: Effectiveness-NTU relation for exchanger 9-26 Example 9.4-2: Counter-flow heat exchange in whales 9-29 Example 9.4-3: LMTD and Effectiveness-NTU method 9-31 Chapter 10: Radiation Heat Transfer 10.1 Introduction 10-1 10.2 Blackbody Radiation 10-2 10.3 Radiation Intensity 10-5 Emissive Power 10-5 Irradiation and Radiosity 10-6 Example 10.3-1 Irradiation and radiant energy on a surface 10-7 Example 10.3-2 Irradiation received by a detector 10-8 Example 10.3-3 Total irradiation at the earth’s surface 10-9 Example 10.3-4 Solid angle for irradiation 10-10 Example 10.3-5 Solar irradiation at the earth’s surface 10-11 Example 10.3-6 Estimation of the earth’s surface temperature 10-12 Appendix A. Previous Exams (2008) B. Previous Exams (2009) C. Previous Exams (2010) D. Previous Exams (2011) E. Previous Exams (2012) iv Chapter 1 Introduction to Heat Transfer 1. 1 Definition of heat transfer Heat transfer is the science which seeks to predict the rate of energy transfer between material bodies as a result of a temperature difference. Definition of 'heat' Heat is energy in transit solely as a result of a temperature difference. Definition of 'temperature' Temperature is a measure of the mean kinetic energy of molecules. Absolute zero (0oK) is a state of complete motionless of molecules. 'Rate' 'Rate' implies an element of speed, how fast an event happens, and time. 'System' A system is any designated region of a continuum of fixed mass. The boundaries of a system may be deformable but they always enclose the same mass. In thermodynamics, the universe can be divided into two parts. One part is the system, the other part is the rest of the universe called the surroundings. Surroundings Boundary System Figure 1.1 Schematic diagram of the "universe", showing a system and the surroundings. 'Control volume' A 'control volume' is also any designated region of a continuum except that it may permit matter to cross its boundaries. If the boundaries of a control volume are such that matter may not enter or leave the control volume, the control volume is identical to a system. In these respects, a 'system' is a subset of a 'control volume'. 'Equilibrium' 'Equilibrium' means that there are no spatial differences in the variables that describe the condition of the system, also called the 'state' of a system, such as its pressure, temperature, volume, and mass (P, T, V, m), and that any changes which occur do so infinitesimally slowly. 1-1 1.2 Importance of Thermodynamics in the Study of Heat Transfer Thermodynamics is the science which seeks to predict the amount of energy needed to bring about a change of state of a system from one equilibrium state to another. While thermodynamics tells us nothing about the mechanisms of energy transfer, rates of change, and time associated with a system changing from one equilibrium state to another, it is still the lynch- pin that allow us to answer these questions. Heat transfer analysis is based on the laws of thermodynamics. Because heat is transferred only when a temperature difference exists, and thus when a system or control volume is not in equilibrium, we have to acknowledge that there is an inherent dilemma in the application of the laws of thermodynamics to systems or control volumes that are not in equilibrium.